We study central configurations in the four body problem, i.e., configurations in which the forces on all the bodies point to a fixed, single point in space. The newly formulated pair-space formalism yields a set of vectorial equations that fully characterize such configurations. We investigate a sub-class of solutions in which at least two pairs of inter-body distances are equal. The only such non-collinear configurations are the tetrahedron (the unique non-planar configuration), kites and the isosceles trapezium. The specific shapes (internal angles) are determined by the ratio of the masses of the bodies. Mathematical expression are given for all these relations.

In connection with the still unsolved multidimensional corona problem for algebras of bounded holomorphic functions on convex domains, we study the solvability of the Bézout equation for the algebra of bounded holomorphic functions on the polydisk with values in a complex Banach algebra. Assuming local solvability of the Bézout equation on a special open cover of the maximal ideal space of the algebra, we combine a dimension-induction scheme with a careful analysis of the topological structure of this space to glue local solutions into a global one. As a corollary, we obtain the solvability of the Bézout equation for a broader class of subalgebras containing the slice algebra of bounded holomorphic functions, the case of the latter having been previously proved by the first author

In real Lie theory, matrices that admit a real logarithm reside in the identity component $\mathrm{GL}_n(\mathbb{R})_+$ of the general linear group $\mathrm{GL}_n(\mathbb{R})$, with logarithms in the Lie algebra $\mathfrak{gl}_n(\mathbb{R})$. The exponential map \[ \exp : \mnr \to \mathrm{GL}_n(\mathbb{R}) \] provides a fundamental link between the Lie algebra and the Lie group, with the logarithm as its local inverse. In this paper, we characterize all bijective linear maps $\varphi : \mnr \to \mnr$ that preserve the class of matrices admitting a real logarithm (principal logarithm). We show that such maps are exactly those of the form \[ \varphi(A) = c\, P A P^{-1} \quad \text{or} \quad \varphi(A) = c\, P A^{T} P^{-1}, \] for some $P \in \mathrm{GL}_n(\mathbb{R})$ and $c > 0$. The proof proceeds in two stages. First, we analyze preservers within the class of standard linear transformations. Second, using Zariski denseness, we prove that any bijective linear map preserving matrices with real logarithms (principal logarithm) must preserve $\mathrm{GL}_n(\mathbb{R})$, which then implies the map is of the standard form.

During their investigation of power-sequence terraces, Anderson and Preece briefly mention a construction of a terrace for the cyclic group $\mathbb{Z}_n$ when $n$ is odd and $2n+1$ is prime; it is built using the discrete logarithm modulo $2n+1$. In this short note we see that this terrace gives rise to an orthogonal double cover (ODC) for the complete graph $K_n$ by Hamiltonian paths. This gives infinitely many new values for which such an ODC is known.

We develop efficient hierarchical preconditioners for optimal control problems governed by partial differential equations with uncertain coefficients. Adopting a discretize-then-optimize framework that integrates finite element discretization, stochastic Galerkin approximation, and advanced time-discretization schemes, the approach addresses the challenge of large-scale, ill-conditioned linear systems arising in uncertainty quantification. By exploiting the sparsity inherent in generalized polynomial chaos expansions, we derive hierarchical preconditioners based on truncated stochastic expansion that strike an effective balance between computational cost and preconditioning quality. Numerical experiments demonstrate that the proposed preconditioners significantly accelerate the convergence of iterative solvers compared to existing methods, providing robust and efficient solvers for both steady-state and time-dependent optimal control applications under uncertainty.

We derive a variational formulation for the scalar wave equation in the second-order formulation on bounded Lipschitz domains and homogeneous initial conditions. We investigate a variational framework in a bounded space-time cylinder $Q$ with a new solution space and the test space $L^2(Q)$ for source terms in $L^2(Q)$. Using existence and uniqueness results in $H^1(Q)$, we prove that this variational setting fits the inf-sup theory, including an isomorphism as solution operator. Moreover, we show that the new solution space is not a subspace of $H^2(Q)$. This new uniqueness and solvability result is not only crucial for discretizations using space-time methods, including least-squares approaches, but also important for regularity results and the analysis of related space-time boundary integral equations, which form the basis for space-time boundary element methods.

We investigate model reduction of parametric linear time-invariant (LTI) dynamical systems. When posed in the frequency domain, this problem can be formulated as seeking a low-order rational function approximation of a high-order rational function. We propose to use a standard reduced basis method (RBM) to construct this low-order rational function. Algorithmically, this procedure is an iterative greedy approach, where the greedy objective is evaluated through an error estimator that exploits the linearity of the frequency domain representation. The greedy framework is motivated through theoretical results of rational approximability of functions. This framework provides a principled approach to rational compression of high-order rational functions, and provides a computational pathway for model reduction of parametric LTI systems.

We utilize the structure of quasiautomorphic forms over an arbitrary Hecke triangle group to define a new vector analogue of an automorphic form. We supply a proof of the functional equations that hold for these functions modulo the group generators.

We classify gradings on the Hecke category that refine the standard integer grading. We also classify object-preserving autoequivalences of the Hecke category. We obtain a natural bigrading on the Hecke category which is related to the Frobenius automorphism. We also obtain an exotic grading in special characteristic that can be used to categorify many Hecke algebras with unequal parameters, including all Hecke algebras with unequal parameters for all finite and affine Weyl groups. This paper is a replacement for arXiv:2305.08278, which is now obsolete.

Given a graph $G$ and a target graph $H$, an $H$-coloring of $G$ is an adjacency-preserving vertex map from $G$ to $H$. The number of $H$-colorings of $G$, $\hom(G,H)$, has been studied for many classes of $G$ and $H$. In particular, extremal questions of maximizing and minimizing $\hom(G,H)$ have been considered when $H$ is a clique or $G$ is a tree. In this paper, we develop a new technique using automorphisms of $H$ to show that $\hom(T,H)$ is minimized by paths as $T$ varies over trees on a fixed number of vertices. We introduce the term Hoffman-London to refer to graphs that are minimal in this sense. In particular, we define an automorphic similarity matrix which is used to compute $\hom(T,H)$ and give matrix conditions under which $H$ is Hoffman-London. We then apply this technique to identify several families of graphs that are Hoffman-London, including loop threshold graphs and some with applications in statistical physics (e.g. the Widom-Rowlinson model). By combining our approach with a few other observations, we fully characterize the minimizing trees for all graphs $H$ on three or fewer vertices.

Inverse problems are important mathematical problems that seek to recover model parameters from noisy data. Since inverse problems are often ill-posed, they require regularization or incorporation of prior information about the underlying model or unknown variables. Proximal operators, ubiquitous in nonsmooth optimization, are central to this because they provide a flexible and convenient way to encode priors and build efficient iterative algorithms. They have also recently become key to modern machine learning methods, e.g., for plug-and-play methods for learned denoisers and deep neural architectures for learning priors of proximal operators. The latter was developed partly due to recent work characterizing proximal operators of nonconvex priors as subdifferential of convex potentials. In this work, we propose to leverage connections between proximal operators and Hamilton-Jacobi partial differential equations (HJ PDEs) to develop novel deep learning architectures for learning the prior. In contrast to other existing methods, we learn the prior directly without recourse to inverting the prior after training. We present several numerical results that demonstrate the efficiency of the proposed method in high dimensions.

In this paper we introduce a two-parameter family of Mehler kernels and connect them to a class of Baouendi-Grushin flows in fractal dimension. We also highlight a link with a geometric fully nonlinear equation and formulate two questions.

We show that if an endomorphism $f:\mathbb{T}^2 \to \mathbb{T}^2$ is absolutely partially hyperbolic, then it has a center foliation. Moreover, the center foliation is leaf conjugate to that of its linearization.

In this article, we disprove a conjecture of F. Alarcón and D. Anderson and give a complete classification of the prime ideals in the one variable polynomial semiring with coefficients in Boolean semifield. We group the prime ideals of $\mathbb{B}[x]$ into three classes, indexed by integers.

We study the Reflect-Reflect-Relax (RRR) algorithm in its small-step (flow-limit) regime. In the smooth transversal setting, we show that the transverse dynamics form a hyperbolic sink, yielding exponential decay of a natural gap measure. Under uniform geometric assumptions, we construct a tubular neighborhood of the feasible manifold on which the squared gap defines a strict Lyapunov function, excluding recurrent dynamics and chaotic behavior within this basin. In the discrete setting, the induced flow is piecewise constant on W-domains and supports Filippov sliding along convergent boundaries, leading to finite-time capture into a solution domain. We prove that small-step RRR is a forward-Euler discretization of this flow, so that solution times measured in rescaled units converge to a finite limit while iteration counts diverge, explaining the emergence of iteration-optimal relaxation parameters. Finally, we introduce a heuristic mesoscopic framework based on percolation and renormalization group to organize performance deterioration near the Douglas-Rachford limit.

Given a m-dimensional Gaussian process and polynomial m variables with real coefficients, we calculate the induced path odered exponenial in two different ways: one is purely algebraic in spirit and the other one is diagrammatic in spirit and uses multigraph labelings (and is inspired by the use of Feynman diagrams in quantum field theory).

We introduce the first example of algebraically constructed hierarchical quasi-cyclic codes. These codes are built from Reed-Solomon codes using a 1964 construction of superimposed codes by Kautz and Singleton. We show both the number of levels in the hierarchy and the index of these Reed-Solomon derived codes are determined by the field size. We show that this property also holds for certain additional classes of polynomial evaluation codes. We provide explicit code parameters and properties as well as some additional bounds on parameters such as rank and distance. In particular, starting with Reed-Solomon codes of dimension $k=2$ yields hierarchical quasi-cyclic codes with Tanner graphs of girth 6. We present a table of small code parameters and note that some of these codes meet the best known minimum distance for binary codes, with the additional hierarchical quasi-cyclic structure. We draw connections to similar constructions in the literature, but importantly, while existing literature on related codes is largely simulation-based, we present a novel algebraic approach to determining new bounds on parameters of these codes.

We study the existence of positive eigenvalues with associated nonnegative mild eigenfunctions for a class of abstract initial value problems in Banach spaces with functional, possibly nonlocal, initial conditions. The framework includes periodic, multipoint, and integral average conditions. Our approach relies on nonlinear analysis, topological methods, and the theory of strongly continuous semigroups, yielding results applicable to a wide range of models. As an illustration, we apply the abstract theory to a reaction-diffusion equation with a nonlocal initial condition arising from a heat flow problem.

We study the question of the least number of random edges that need to be added to a Pósa-Seymour graph, that is, a graph with minimum degree exceeding $\frac k{k+1}n$, to secure the existence of the $m$-th power of a Hamiltonian cycle, $m>k$. It turns out that, depending on $k$ and $m$, this quantity may be captured by two types of thresholds, with one of them, called over-threshold, becoming dominant for large $m$. Indeed, for each $k\ge2$ and $m>m_0(k)$, we establish asymptotically tight lower and upper bounds on the over-thresholds (provided they exist) and show that for infinitely many instances of $m$ the two bounds coincide. In addition, we also determine the thresholds for some small values of $k$ and $m$.

We prove that for every fixed integer $s$ and every planar graph $H$, the class of $H$-induced-minor-free and $K_{1,s}$-induced-subgraph-free graphs has polylogarithmic tree-independence number. This is a weakening of a conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht.

We present an algorithm to explore various properties of the numerical semigroups with a given maximum primitive. In particular, we count the number of such numerical semigroups and verify that there is no counterexample to Wilf's conjecture among the numerical semigroups with maximum primitive up to \(60\).

We investigate the moduli space of holomorphic $GL(1|1)$ Higgs bundles over a compact Riemann surface. The supergroup $GL(1|1)$, the simplest non-trivial example beyond abelian cases, provides an ideal setting for developing supergeometric analogues of classical results in Higgs bundle theory. We derive an explicit description of the moduli space and we study the analogue of the Narasimhan-Seshadri theorem as well as the nonabelian Hodge correspondence. Furthermore, we formulate and solve the corresponding Hitchin equations, demonstrating their compatibility with fermionic contributions. As a highlight, we discuss the related Hitchin system on $\mathbb{P}^1$ and its integrability.

We present a relation between stable rank one and real rank zero via the method of tracial oscillation. Let $A$ be a simple separable $C^*$-algebra of stable rank one. We show that $A$ has tracial approximate oscillation zero and, as a consequence, the tracial sequence algebra $l^\infty(A)/J_A$ has real rank zero, where $J_A$ is the trace-kernel ideal with respect to 2-quasitraces. We also show that for a $C^*$-algebra $B$ that has non-trivial 2-quasitraces, $B$ has tracial approximate oscillation zero is equivalent to $l^\infty(B)/J_B$ has real rank zero.

We express the branch points cross ratio of cyclic Mumford curves as quotients of $p$-adic theta functions evaluated at the p-adic period matrix

Let l be a prime number and let E and E' be l-isogenous elliptic curves defined over Q. In this paper we determine the proportion of primes p for which E(F_p) is isomorphic to E'(F_p). Our techniques are based on those developed in \cite{ck} and \cite{rnt}.

This is the first in a sequence of papers devoted to stable envelopes in critical cohomology and critical $K$-theory for symmetric GIT quotients with potentials and related geometries, and their applications to geometric representation theory and enumerative geometry. In this paper, we construct critical stable envelopes and establish their general properties, including compatibility with dimensional reductions, specializations, Hall products, and other geometric constructions. In particular, for tripled quivers with canonical cubic potentials, the critical stable envelopes reproduce those on Nakajima quiver varieties. These set up foundations for applications in subsequent papers.

A multiplicative subset S of a ring R is called strongly multiplicative if (\cap_{i \in \Delta} s_i R) \cap S is non-empty for each family (s_i)_{i \in \Delta} of S. In this paper, we study how these sets help stabilize localization and ideal operations. We show that localization and arbitrary intersections commute, meaning S^{-1}(\cap I_\alpha) = \cap S^{-1}I_\alpha$ for any family of ideals, if and only if S is strongly multiplicative. Furthermore, we characterize total quotient rings and strongly zero-dimensional rings in terms of strongly multiplicative sets. We also answer an open question by Hamed and Malek about whether this condition is needed for S-minimal primes to exist. In addition, we prove a Strong Krull's Separation Lemma, which guarantees a maximal ideal disjoint from S. Finally, we demonstrate that if S is a strongly multiplicative set and S is not contained in U(R), then S-minimal primes are not classical prime ideals, and we provide an algorithmic approach to constructing such ideals.

Yeliussizov has classified the positive specializations of symmetric Grothendieck functions, defined in several different ways, providing a K-theoretic lift of the classical Edrei-Thoma theorem. This note studies the analogous classification problem for Ikeda and Naruse's K-theoretic Schur P- and Q-functions, which are the shifted versions of symmetric Grothendieck functions. Our results extend a shifted variant of the Edrei-Thoma theorem due to Nazarov. We also discuss an application to the problem of determining the extreme harmonic functions on a filtered version of the shifted Young lattice.

In this paper, we investigate the instability of the trivial steady states to the incompressible viscous fluid with Navier-slip boundary conditions. For the linear instability, the existence of infinitely many normal mode solutions to the linearized equations is shown via the operator method of Lafitte and Nguyen (2022). Hence, we prove the nonlinear instability by adapting the framework of Desjardins and Grenier (2003) studying some classes of viscous boundary layers to obtain two separated solutions at escaping time. Our work performs a different approach from that of Ding, Li and Xin (2018).

A theorem of Hoischen states that given a positive continuous function $\varepsilon:\mathbb{R}\to\mathbb{R}$, an integer $n\geq 0$, and a closed discrete set $E\subseteq\mathbb{R}$, any $C^n$ function $f:\mathbb{R}\to\mathbb{R}$ can be approximated by an entire function $g$ so that for $k=0,\dots,n$, and $x\in\mathbb{R}$, $|D^{k}g(x)-D^{k}f(x)|<\varepsilon(x)$, and if $x\in E$ then $D^{k}g(x)=D^{k}f(x)$. The approximating function $g$ is entire and hence piecewise monotone. Building on earlier work, for $n\leq 3$, we determine conditions under which when $f$ is piecewise monotone we can choose $g$ to be comonotone with $f$ (increasing and decreasing on the same intervals), and under which the derivatives of $g$ can be taken to be comonotone with the corresponding derivatives of $f$ if the latter are piecewise monotone. The proof for $n\leq 3$ establishes the theorem for all $n$, assuming a conjecture (shown in previous work with Haris and Madhavendra to hold for $n\leq 3$) regarding the set of $2(n+1)$-tuples $(f(0),Df(0),\dots,D^nf(0),f(1),Df(1),\dots,D^nf(1))$ of the values at the endpoints of the derivatives of a $C^n$ function $f$ on $[0,1]$ for which $D^nf$ is increasing and not constant.

We study the sine-Gordon measure defined on each homotopy class. The energy space decomposes into infinitely many such classes indexed by the topological degree $Q \in \mathbb{Z}$. Even though the sine-Gordon action admits no minimizer in homotopy classes with $|Q| \ge 2$, we prove that the Gibbs measure on each class nevertheless concentrates and exhibits Ornstein-Uhlenbeck fluctuations near the multi-soliton manifold in the joint low-temperature and infinite-volume limit. Furthermore, we show that the joint distribution of the multi-soliton centers coincides with the ordered statistics of independent uniform random variables, so that each soliton's location follows a Beta distribution.

Generating samples from complex and high-dimensional distributions is ubiquitous in various scientific fields of statistical physics, Bayesian inference, scientific computing and machine learning. Very recently, Huang et al. (IEEE Trans. Inform. Theory, 2025) proposed new Schrödinger-Föllmer samplers (SFS), based on the Euler discretization of the Schrödinger-Föllmer diffusion evolving on the unit interval $[0, 1]$. There, a convergence rate of order $\mathcal{O}(\sqrt{h})$ in the $L^2$-Wasserstein distance was obtained for the Euler discretization with a uniform time step-size $h>0$. By incorporating a temperature parameter, different samplers are introduced in this paper, based on the Euler discretization of the Schrödinger-Föllmer process with temperatures. As revealed by numerical experiments, high temperatures are vital, particularly in sampling from multimodal distributions. Further, a novel approach of error analysis is developed for the time discretization and an enhanced convergence rate of order ${ \mathcal{O}(h)}$ is obtained in the $L^2$-Wasserstein distance, under certain smoothness conditions on the drift. This significantly improves the existing order-half convergence in the aforementioned paper. Unlike Langevin samplers, SFS is of gradient-free, works in a unit interval $[0, 1]$ and does not require any ergodicity. Numerical experiments confirm the convergence rate and show that, the SFS substantially outperforms vanilla Langevin samplers, particularly in sampling from multimodal distributions.

In this work we compute the conserved currents and charges associated to the action of an infinitesimal isometry (Killing field) in Einstein--Cartan--Palatini gravity. We offer a new approach to these quantities through the formalism of $L_\infty$-algebras and the work of Ćirić, Giotopoulos, Radovanović, and Szabo, and Costello and Gwilliam. We demonstrate our approach by computing the entropy of the Schwarzchild black hole. Along the way, we prove a purely algebraic result about the existence and utility of a higher (a full $\infty$) version of the adjoint action of an $L_\infty$-algebra.

Consider the task of generating samples from a tilted distribution of a random vector whose underlying distribution is unknown, but samples from it are available. This finds applications in fields such as finance and climate science, and in rare event simulation. In this article, we discuss the asymptotic efficiency of a self-normalized importance sampler of the tilted distribution. We provide a sharp characterization of its accuracy, given the number of samples and the degree of tilt. Our findings reveal a surprising dichotomy: while the number of samples needed to accurately tilt a bounded random vector increases polynomially in the tilt amount, it increases at a super polynomial rate for unbounded distributions.

We determine all premodular subcategories and modular tensor subcategories in the module categories of Virasoro vertex operator algebras $L(c_{p,q},0)$ and the module categories of the simple current extensions of $L(c_{p,p+1},0)$.

It was recently proved that every claw-free cubic graph admits a (1, 1, 2, 2)-packing coloring--that is, its vertex set can be partitioned into two 1-packings and two 2-packings. This result was established by Brešar, Kuenzel, and Rall [Discrete Mathematics 348 (8) (2025), 114477]. In this paper, we provide a simpler and shorter proof.

In this paper we resolve a conjecture of Zhi-Wei Sun concerning the integrality and arithmetic structure of certain trigonometric determinants. Our approach builds on techniques developed in our previous work, where trigonometric determinants were studied via special values of Dirichlet $L$-functions. The method is refined by establishing a connection between odd characters modulo $4n$ and even characters modulo $n$. The results highlight a close connection between trigonometric determinant matrices, Fourier-analytic structures, and arithmetic invariants.

This paper continues the study initiated in [ISZ25] on the moduli of surfaces admitting lc-trivial fibrations. Using the techniques developed in [ISZ25], we (1) provide a classification of the surfaces appearing on the boundary of the KSBA-moduli space of elliptic surfaces with a bisection (2) recover the results of a series of papers on the moduli stacks of elliptic surfaces with a section [AB22, Inc20, Bru15]. Notably, our proof of (2) avoids the use of explicit steps of an MMP, such as the "La Nave flip" from [LN02], which plays a central role in [AB22,Inc20].

We develop an optimal regularity theory for parabolic partial differential equations in weighted mixed norm Sobolev-Zygmund spaces. The results extend the classical Schauder estimates to coefficients that are merely measurable in time and to the critical case of integer-order regularity. In addition, nonzero initial data are treated in the optimal trace space via a sharp trace theorem.

We develop the notion of a "filtered cospan" as an algebraic object that stands in the same relation to interlevel persistence modules as filtered chain complexes stand with respect to sublevel persistence modules. This relation is expressed via a functor from a category of filtered cospans to a category of persistence modules that arise in Bauer-Botnan-Fluhr's study of relative interlevel set homology. We associate a filtered cospan to a Morse function $f:X\to [-\Lambda,\Lambda]$ such that $\partial X$ is the union of the regular level sets $f^{-1}(\{\pm\Lambda\})$; this allows us to capture the interlevel persistence of such a function in terms of data associated to Morse chain complexes. Similar filtered cospans are associated to simplicial and singular chain complexes, and isomorphism theorems are proven relating these to each other and to relative interlevel set homology. Filtered cospans can be decomposed, under modest hypotheses, into certain standard elementary summands, giving rise to a notion of persistence diagram for filtered cospans that is amenable to computation. An isometry theorem connects interleavings of filtered cospans to matchings between these persistence diagrams.

In the study of walks with small steps confined to multidimensional orthants, a certain group of transformations plays a central role. In particular, several techniques to potentially compute the generating function, including the orbit sum method, can only be applied when this group is finite. In this note, we present three new results concerning this group. First, in two dimensions, we provide a complete characterization of the weight parameters that yield a finite group. In higher dimensions, we show that whenever the group is finite, it must necessarily be isomorphic to a simpler reflection group. Finally, in dimension three, we give a full classification of the parameters leading to a finite group that also satisfies an additional Weyl property.

In this paper, we discover a secondary term in the asymptotic formula for the mean value of Hecke--Maass special $L$-values $ L (1/2+it_f, f) $ with the average over $f (z)$ in an orthonormal basis of (even or odd) Hecke--Maass cusp forms of Laplace eigenvalue $1/4 + t_f^2$ ($t_f > 0$). To be explicit, we prove $$ \sum_{t_f \leqslant T} \omega_f L (1/2+it_f, f) = \frac {T^2} {\pi^2} + \frac {8T^{3/2}} {3\pi^{3/2} } + O \big(T^{1+\varepsilon}\big), $$ for any $\varepsilon > 0$, where $\omega_f$ are the harmonic weights. This provides a new instance of (large) secondary terms in the moments of $L$-functions -- it was known previously only for the smoothed cubic moment of quadratic Dirichlet $L$-functions. The proof relies on an explicit formula for the smoothed mean value of $L (1/2+it_f, f)$.

We study various categories of Whittaker modules over the queer Lie superalgebras $\mathfrak q(n)$. We formulate standard Whittaker modules and reduce the problem of composition factors of these standard Whittaker modules to that of Verma modules in the BGG categories $\mathcal O$ of $\mathfrak q(n)$. We also obtain an analogue of Losev-Shu-Xiao decomposition for the finite $W$-superalgebras $U(\mathfrak q(n), E)$ of $\mathfrak q(n)$ associated to an odd nilpotent element $E\in \mathfrak q(n)_{\bar{1}}$. As an application, we establish several equivalences of categories of Whittaker $\mathfrak q(n)$-modules and analogues of BGG category of $U(\mathfrak q(n), E)$-modules. In particular, we reduce the multiplicity problem of Verma modules over $U(\mathfrak q(n), E)$ to that of the Verma modules in the BGG categories $\mathcal O$ of $\mathfrak q(n)$.

We show that it is impossible to classify topological conjugacy relation of minimal homeomorphisms on the torus by countable structures.

Let $RG$ be the group ring of an arbitrary group $G$ over an associative non-commutative ring $R$ with identity. In this paper, we have obtained the necessary and sufficient conditions under which $RG$ is Jordan nilpotent of index $4$.

We study the collection of measures obtained via push-forward along a map between smooth varieties over p-adic fields. We investigate when the stalks of this collection are finite-dimensional. We provide an algebro-geometric criterion ensuring this property. This criterion is formulated in terms of a canonical subvariety of the cotangent bundle of the source of the map.

This paper proposes a Chebyshev polynomial expansion framework for the recovery of a continuous angular power spectrum (APS) from channel covariance. By exploiting the orthogonality of Chebyshev polynomials in a transformed domain, we derive an exact series representation of the covariance and reformulate the inherently ill-posed APS inversion as a finite-dimensional linear regression problem via truncation. The associated approximation error is directly controlled by the tail of the APS's Chebyshev series and decays rapidly with increasing angular smoothness. Building on this representation, we derive an exact semidefinite characterization of nonnegative APS and introduce a derivative-based regularizer that promotes smoothly varying APS profiles while preserving transitions of clusters. Simulation results show that the proposed Chebyshev-based framework yields accurate APS reconstruction, and enables reliable downlink (DL) covariance prediction from uplink (UL) measurements in a frequency division duplex (FDD) setting. These findings indicate that jointly exploiting smoothness and nonnegativity in a Chebyshev domain provides an effective tool for covariance-domain processing in multi-antenna systems.

This paper establishes the Local Asymptotic Normality (LAN) property for the mixed fractional Brownian motion under high-frequency observations with Hurst index $H \in (0, 3/4)$. The simultaneous estimation of the volatility and the Hurst index encounters a degeneracy problem in the Fisher information matrix.

We propose a set of algebraic equations describing eigenvalues and eigenstates of a relativistic evolution operator for a two-dimensional $q$-oscillator Kagomé lattice. Evolution operator is constructed with the help of $q$-oscillator solution of the Tetrahedron Equation. We focus on the unitary regime of the evolution operator, so our results are related to 3d integrable systems of the quantum mechanics. Our conjecture is based on a two-dimensional lattice version of the coordinate Bethe-Ansatz.

In this short review paper the detailed analysis of six two-dimensional quantum {\it superintegrable} systems in flat space is presented. It includes the Smorodinsky-Winternitz potentials I-II (the Holt potential), the Fokas-Lagerstrom model, the 3-body Calogero and Wolfes (equivalently, $G_2$ rational, or $I_6$) models, and the Tremblay-Turbiner-Winternitz (TTW) system with integer index $k$. It is shown that all of them are exactly-solvable, thus, confirming the Montreal conjecture (2001); they admit algebraic forms for the Hamiltonian and both integrals (all three can be written as differential operators with polynomial coefficients without a constant term), they have polynomial eigenfunctions with the invariants of the discrete symmetry group of invariance taken as variables, they have hidden (Lie) algebraic structure $g^{(k)}$ with various $k$, and they possess a (finite order) polynomial algebras of integrals. Each model is characterized by infinitely-many finite-dimensional invariant subspaces, which form the infinite flag. Each subspace coincides with the finite-dimensional representation space of the algebra $g^{(k)}$ for a certain $k$. In all presented cases the algebra of integrals is a 4-generated $(H, I_1, I_2, I_{12}\equiv[I_1, I_2])$ infinite-dimensional algebra of ordered monomials of degrees 2,3,4,5, which is a subalgebra of the universal enveloping algebra of the hidden algebra.

We study the critical branching random walk on $\mathbb{Z}^d$ started from a distant point $x$ and conditioned to hit some compact set $K$ in $\mathbb{Z}^d$. We are interested in the occupation time in $K$ and present its asymptotic behaviors in different dimensions. It is shown in this work that the occupation time is of order $\|x\|^{4-d}$ in dimensions $d\leq 3$, of order $\log\|x\|$ in dimension $d=4$, and of order 1 in dimensions $d\geq 5$. The corresponding weak convergences are also established. These results answer a question raised by Le Gall and Merle (Elect. Comm. in Probab. 11 (2006), 252-265).

Let $k$ be a unital commutative ring. In this paper, we study polynomial functors from the category of finitely generated free nilpotent groups to the category of $k$-modules, focusing on comparisons across different nilpotency classes and polynomial degrees. As a consequence, we obtain refinements of parts of the results of Baues and Pirashvili on polynomial functors over free nilpotent groups of class at most 2, which also recover several folklore results for free groups and free abelian groups. Furthermore, we investigate a modular analogue, formulated using dimension subgroups over a field of positive characteristic instead of lower central series. To prove the main results, we establish general criteria that guarantee equivalences between the categories of polynomial functors of different degrees or with different base categories. They are described by using a two-sided ideal of a monad associated with the base category, which encodes polynomiality of a specific degree. Inspired by the main results, we also investigate an analogous ideal for analytic functors, and show that, in most cases, no such an ideal exists.

We establish $L^p$ error estimates for monotone numerical schemes approximating Hamilton-Jacobi equations on the $d$-dimensional torus. Using the adjoint method, we first prove a $L^1$ error bound of order one for finite-difference and semi-Lagrangian schemes under standard convexity assumptions on the Hamiltonian. By interpolation, we also obtain $L^p$ estimates for every finite $p>1$. Our analysis covers a broad class of schemes, improves several existing results, and provides a unified framework for discrete error estimates.

This paper studies the policy mirror descent (PMD) method, which is a general policy optimization framework in reinforcement learning and can cover a wide range of policy gradient methods by specifying difference mirror maps. Existing sample complexity analysis for policy mirror descent either focuses on the generative sampling model, or the Markovian sampling model but with the action values being explicitly approximated to certain pre-specified accuracy. In contrast, we consider the sample complexity of policy mirror descent with temporal difference (TD) learning under the Markovian sampling model. Two algorithms called Expected TD-PMD and Approximate TD-PMD have been presented, which are off-policy and mixed policy algorithms respectively. Under a small enough constant policy update step size, the $\tilde{O}(\varepsilon^{-2})$ (a logarithm factor about $\varepsilon$ is hidden in $\tilde{O}(\cdot)$) sample complexity can be established for them to achieve average-time $\varepsilon$-optimality. The sample complexity is further improved to $O(\varepsilon^{-2})$ (without the hidden logarithm factor) to achieve the last-iterate $\varepsilon$-optimality based on adaptive policy update step sizes.

We develop and analyze a single-loop algorithm for minimizing the sum of a Lipschitz differentiable function $f$, a prox-friendly proper closed function $g$ (with a closed domain on which $g$ is continuous) and the composition of another prox-friendly proper closed function $h$ (whose domain is closed on which $h$ is continuous) with a continuously differentiable mapping $c$ (that is Lipschitz continuous and Lipschitz differentiable on the convex closure of the domain of $g$). Such models arise naturally in many contemporary applications, where $f$ is the loss function for data misfit, and $g$ and $h$ are nonsmooth functions for inducing desirable structures in $x$ and $c(x)$. Existing single-loop algorithms mainly focus either on the case where $h$ is Lipschitz continuous or the case where $h$ is an indicator function of a closed convex set. In this paper, we develop a single-loop algorithm for more general possibly non-Lipschitz $h$. Our algorithm is a single-loop variant of the successive difference-of-convex approximation method (SDCAM) proposed in [22]. We show that when $h$ is Lipschitz, our algorithm exhibits an iteration complexity that matches the best known complexity result for obtaining an $(\epsilon_1,\epsilon_2,0)$-stationary point. Moreover, we show that, by assuming additionally that dom $g$ is compact, our algorithm exhibits an iteration complexity of $\tilde{O}(\epsilon^{-4})$ for obtaining an $(\epsilon,\epsilon,\epsilon)$-stationary point when $h$ is merely continuous and real-valued. Furthermore, we consider a scenario where $h$ does not have full domain and establish vanishing bounds on successive changes of iterates. Finally, in all three cases mentioned above, we show that one can construct a subsequence such that any accumulation point $x^*$ satisfies $c(x^*)\in$ dom $h$, and if a standard constraint qualification holds at $x^*$, then $x^*$ is a stationary point.

The determination of $ES(7)$ is the first open case of the planar Erdős--Szekeres problem, where the general conjecture predicts $ES(7)=33$. We present a SAT encoding for the 33-point case based on triple-orientation variables and a 4-set convexity criterion for excluding convex 7-gons, together with convex-layer anchoring constraints. The framework yields UNSAT certificates for a collection of anchored subfamilies. We also report pronounced runtime variability across configurations, including heavy-tailed behavior that currently dominates the computational effort and motivates further encoding refinements.

We show that the Kac particle system converges, as the number of particles tends to infinity, to the solution of the homogeneous Boltzmann equation, in the regime of moderately soft potentials, $\gamma \in (-2,0)$ with the common notation. This proves the propagation of chaos. We adapt the recent work of Imbert, Silvestre and Villani, to show that the Fisher information is nonincreasing in time along solutions to the Kac master equation. This estimate allows us to control the singularity of the interaction.

For large enough (but fixed) prime powers $q$, and trace functions to squarefree moduli in $\mathbb{F}_q[u]$ with slopes at most $1$ at infinity, and no Artin--Schreier factors in their geometric global monodromy, we come close to square-root cancellation in short sums. A special case is a function field version of Hooley's Hypothesis $R^*$ for short Kloosterman sums. As a result, we are able to make progress on several problems in analytic number theory over $\mathbb{F}_q[u]$ such as Mordell's problem on the least residue class not represented by a polynomial and the variance of short Kloosterman sums.

Let $M$ be a smooth manifold, let $TM$ be its tangent bundle and $T^{*}M$ its cotangent bundle. This paper investigates integrability conditions for generalized metrics, generalized almost para-complex structures, and generalized Hermitian structures on the generalized tangent bundle of $M$, $E=TM \oplus T^{*}M$. In particular, two notions of integrability are considered: integrability with respect to the Courant bracket and integrability with respect to the bracket induced by an affine connection. We give sufficient criteria that guarantee the integrability for the aforementioned generalized structures, formulated in terms of properties of the associated $2$-form and connection. Extensions to the pseudo-Riemannian setting and consequences for generalized Hermitian and Kähler structures are also discussed. We also describe relationship between generalized metrics and weak metric structures.

We use formal microlocalization to describe the Fourier--Laplace transformation between rank 3 and rank 2 D-module representations of Painleve systems. We conclude the existence of biregular morphism between the corresponding de Rham complex structures.

As wireless communication applications evolve from traditional multipath environments to high-mobility scenarios like unmanned aerial vehicles, multiplexing techniques have advanced accordingly. Traditional single-carrier frequency-domain equalization (SC-FDE) and orthogonal frequency-division multiplexing (OFDM) have given way to emerging orthogonal time-frequency space (OTFS) and affine frequency-division multiplexing (AFDM). These approaches exploit specific channel structures to diagonalize or sparsify the effective channel, thereby enabling low-complexity detection. However, their reliance on these structures significantly limits their robustness in dynamic, real-world environments. To address these challenges, this paper studies a random multiplexing technique that is decoupled from the physical channels, enabling its application to arbitrary norm-bounded and spectrally convergent channel matrices. Random multiplexing achieves statistical fading-channel ergodicity for transmitted signals by constructing an equivalent input-isotropic channel matrix in the random transform domain. It guarantees the asymptotic replica MAP bit-error rate (BER) optimality of AMP-type detectors for linear systems with arbitrary norm-bounded, spectrally convergent channel matrices and signaling configurations, under the unique fixed point assumption. A low-complexity cross-domain memory AMP (CD-MAMP) detector is considered, leveraging the sparsity of the time-domain channel and the randomness of the equivalent channel. Optimal power allocations are derived to minimize the replica MAP BER and maximize the replica constrained capacity of random multiplexing systems. The optimal coding principle and replica constrained-capacity optimality of CD-MAMP detector are investigated for random multiplexing systems. Additionally, the versatility of random multiplexing in diverse wireless applications is explored.

In this paper we study a family of one-dimensional stationary cubic nonlinear Schrödinger (NLS) equations with periodic potentials and linear part displaying Dirac points in the dispersion relation. By introducing a suitable periodic perturbation, one can open a spectral gap around the Dirac-point energy. This allows to construct standing waves of the NLS equation whose leading-order profile is a modulation of Bloch waves by means of the components of a spinor solving an appropriate cubic nonlinear Dirac (NLD) equation. We refer to these solutions as Dirac solitons. Our analysis thus provides a rigorous justification for the use of the NLD equation as an effective model for the original NLS equation.

The explosive growth of artificial intelligence (AI) workloads in modern data centers demands a radical transformation of interconnect architectures. Traditional copper and optical wiring face fundamental challenges in latency, power consumption, and rigidity, constraining the scalability of distributed AI clusters. This article introduces a vision for Terahertz (THz) Wireless Data Center (THz-WDC) that combines ultra-broadband capacity, one-hop low-latency communication, and energy efficiency in the short-to-medium range (1-100m). Performance and technical requirements are first articulated, including up to 1 Tbps per link, aggregate throughput up to 10 Tbps via spatial multiplexing, sub-50 ns single-hop latency, and sub-10 pJ/bit energy efficiency over 20m. To achieve these ambitious goals, key enabling technologies are explored, including digital-twin-based orchestration, low-complexity beam manipulation technologies, all-silicon THz transceivers, and low-complexity analog baseband architectures. Moreover, as future data centers shift toward quantum and chiplet-based modular architectures, THz wireless links provide a flexible mechanism for interconnecting, testing, and reconfiguring these modules. Finally, numerical analysis is presented on the latency and power regimes of THz versus optical and copper interconnects, identifying the specific distance and throughput domains where THz links can surpass conventional wired solutions. The article concludes with a roadmap toward wireless-defined, reconfigurable, and sustainable AI data centers.

Given a graph G equals (V,E), a subset S subset of V is a dominating set if every vertex in V minus S is adjacent to some vertex in S. The dominating set with the least cardinality, gamma, is called a gamma-set which is commonly known as a minimum dominating set. The dominion of a graph G, denoted by zeta(G), is the number of its gamma-sets. Some relations between these two seemingly distinct parameters are established. In particular, we present the dominions of paths, some cycles and the join of any two graphs.

The existence of the Herman ring of a function adds interest and complexity to the dynamics of the function. We present a detailed and understandable summary of the core discoveries and recent developments on the Herman ring of rational and transcendental meromorphic functions. It is demonstrated that the Herman ring is intriguing on its own and valuable in terms of overall dynamics. Finally, the results and potential future research problems are briefly discussed.

We propose two new alternative numerical schemes to solve the coupled Einstein-Euler equations in the Generalized Harmonic formulation. The first one is a finite difference (FD) Central Weighted Essentially Non-Oscillatory (CWENO) scheme on a traditional Cartesian mesh, while the second one is an ADER (Arbitrary high order Derivatives) discontinuous Galerkin (DG) scheme on 2D unstructured polygonal meshes. The latter, in particular, represents a preliminary step in view of a full 3D numerical relativity calculation on moving meshes. Both schemes are equipped with a well-balancing (WB) property, which allows to preserve the equilibrium of a priori known stationary solutions exactly at the discrete level. We validate our numerical approaches by successfully reproducing standard vacuum test cases, such as the robust stability, the linearized wave, and the gauge wave tests, as well as achieving long-term stable evolutions of stationary black holes, including Kerr black holes with extreme spin. Concerning the coupling with matter, modeled by the relativistic Euler equations, we perform a classical test of spherical accretion onto a Schwarzschild black hole, as well as an evolution of a perturbed non-rotating neutron star, demonstrating the capability of our schemes to operate also on the full Einstein-Euler system. Altogether, these results provide a solid foundation for addressing more complex and challenging simulations of astrophysical sources through DG schemes on unstructured 3D meshes.

This paper aims at developing exactly energy-conservative and structure-preserving finite volume schemes for the discretisation of first-order symmetric-hyperbolic and thermodynamically compatible (SHTC) systems of partial differential equations in continuum physics. Due to their thermodynamic compatibility the class of SHTC systems satisfies an additional conservation law for the total energy and many PDE in this class of equations also satisfy stationary differential constraints (involutions). First, we propose a simple semi-discrete cell-centered HTC finite volume scheme that employs collocated grids and that is compatible with the total energy conservation law, but which does not satisfy the involutions. Second, we develop a fully discrete semi-implicit finite volume scheme that conserves total energy and which can be proven to satisfy also the involution constraints exactly at the discrete level. This method is a vertex-based staggered semi-implicit scheme that preserves the basic vector calculus identities $\nabla \cdot \nabla \times A = 0$ and $\nabla \times \nabla \phi = 0$ for any vector and scalar field, respectively, exactly at the discrete level and which is also exactly totally energy conservative. The main key ingredient of the proposed implicit scheme is the fact that it uses a discrete version of the symmetric-hyperbolic Godunov-form of the governing PDE system. This leads naturally to sequences of symmetric and positive definite linear algebraic systems to be solved inside an iterative fixed-point method used in each time step. We apply our new schemes to three different SHTC systems. In particular, we consider the equations of nonlinear acoustics, the nonlinear Maxwell equations in the absence of charges and a nonlinear version of the Maxwell-GLM system. We also show some numerical results to provide evidence of the stated properties of the proposed schemes.

We introduce a new goodness-of-fit test for count data on $\mathbb{N}$ for the Zeta distribution with unknown parameter. The test is built on a Stein-type characterization that uses, as Stein operator, the infinitesimal generator of a birth-death process whose stationary distribution is Zeta. The resulting $L^2$-type statistic is shown to be omnibus consistent, and we establish the limit null behavior as well as the validity of the associated parametric bootstrap procedure. In a Monte Carlo simulation study, we compare the proposed test with the only existing Zeta-specific procedure of Meintanis (2009), as well as with more general competitors based on empirical distribution functions, kernel Stein discrepancies and other Stein-type characterizations.

In this mostly expository article, we provide a new account of our proof with Minsky and Sisto that mapping class groups and Teichmüller spaces admit bicombings. More generally, we explain how the hierarchical hull of a pair of points in any colorable hierarchically hyperbolic space is quasi-isometric to a finite CAT(0) cube complex of bounded dimension, with the added property that perturbing the pair of points results in a uniformly bounded change to the cubical structure. Our approach is simplified and new in many aspects.

We study large values of quadratic character sums with summation lengths exceeding the square root of the modulus. Assuming the Generalized Riemann Hypothesis, we obtain a new Omega result.

Let $ p \ge 5 $ be a prime and let $ b, c \in \mathbb{Z} $. Denote by $ T_k(b,c) $ the generalized central trinomial coefficient, i.e., the coefficient of $ x^k $ in $ (x^2 + bx + c)^k $. In this paper, we establish congruences modulo $ p^3 $ and $ p^4 $ for sums of the form $$ \sum_{k=0}^{p-1} (2k+1)^{2a+1}\,\varepsilon^{k}\,\frac{T_k(b,c)^4}{d^{2k}}, $$ where $ a \in \left\lbrace 0,1\right\rbrace $, $ \varepsilon \in \{1,-1\} $, and $ d = b^2 - 4c $ satisfies $ p \nmid d $. In particular, for the special case $ b = c = 1 $, we show that \begin{align*} \sum_{k=0}^{p-1}\left( 2k+1\right) ^{3} \frac{T_{k}^4}{9^k}\equiv -\frac{3p}{4}+\frac{3p^2}{4}\left( \frac{q_p(3)}{4}-1\right) \pmod{p^3}, \end{align*} where $T_k$ is the central trinomial coefficient and $q_p(a)$ is the Fermat quotient.

We prove that any $C^{1+\alpha}$ transitive conservative partially hyperbolic diffeomorphism of a closed 3-manifold with virtually solvable fundamental group is ergodic. Consequently, in light of \cite{FP-classify}, this establishes the equivalence between transitivity and ergodicity for $C^{1+\alpha}$ conservative partially hyperbolic diffeomorphisms in \emph{any} closed 3-manifold. Moreover, we provide a characterization of compact accessibility classes under transitivity, thereby giving a precise classification of all accessibility classes for transitive 3-dimensional partially hyperbolic diffeomorphisms.

Sampling based on score diffusions has led to striking empirical results, and has attracted considerable attention from various research communities. It depends on availability of (approximate) Stein score functions for various levels of additive noise. We describe and analyze a modular scheme that reduces score-based sampling to solving a short sequence of ``nice'' sampling problems, for which high-accuracy samplers are known. We show how to design forward trajectories such that both (a) the terminal distribution, and (b) each of the backward conditional distribution is defined by a strongly log concave (SLC) distribution. This modular reduction allows us to exploit \emph{any} SLC sampling algorithm in order to traverse the backwards path, and we establish novel guarantees with short proofs for both uni-modal and multi-modal densities. The use of high-accuracy routines yields $\varepsilon$-accurate answers, in either KL or Wasserstein distances, with polynomial dependence on $\log(1/\varepsilon)$ and $\sqrt{d}$ dependence on the dimension.

We determine the minimal polynomial of each element of the double cover $G$ of the symmetric or alternating group in every irreducible spin representation of $G$.

This short note includes three results: $(1)$ If a reflexive sheaf $\mathcal{E}$ on a log terminal Kähler-Einstein variety $(X,\omega)$ is slope stable with respect to a singular Kähler-Einstein metric $\omega$, then $\mathcal{E}$ admits an $\omega$-admissible Hermitian-Yang-Mills metric. $(2)$ If a K-stable log terminal projective variety with big anti-canonical divisor satisfies the equality of the Miyaoka-Yau inequality in the sense of \cite{IJZ25}, then its anti-canonical model admits a quasi-étale cover from $\mathbb{C}P^n$. $(3)$ There exists a holomorphic rank 3 vector bundle on a compact complex surface which is semistable for some nef and big line bundle, but it is not semistable for any ample line bundles.

Phase retrieval seeks to reconstruct a signal from phaseless intensity measurements and, in applications where measurements contain errors, demands stable reconstruction. We study local stability of phase retrieval in reproducing kernel Hilbert spaces. Motivated by Grohs-Rathmair's Cheeger-type estimate for Gabor phase retrieval, we introduce a kernel Cheeger constant that quantifies connectedness relative to kernel localization. This notion yields a clean stability certificate: we establish a unified lower bound over both real and complex fields, and in the real case also an upper bound, each in terms of the reciprocal kernel Cheeger constant. Our framework treats finite- and infinite-dimensional settings uniformly and covers discrete, semi-discrete, and continuous measurement domains. For generalized wavelet phase retrieval from (semi-)discrete frames, we bound the kernel Cheeger constant by the Cheeger constant of a data-dependent weighted graph. We further characterize phase retrievability for generalized wavelet transforms and derive a simple sufficient criterion for wavelet sign retrieval in arbitrary dimension for transforms associated with irreducibly admissible matrix groups.

In this Note, we present recent developments in the Renormalization Theory of quadratic polynomials and discuss their applications, with an emphasis on the MLC conjecture, the problem of local connectivity of the Mandelbrot set, and on its geometric counterparts.

The string hypothesis for Bethe roots represents a cornerstone in the study of quantum integrable systems, providing access to physical quantities such as the ground-state energy and the finite-temperature free energy. While the $t-W$ scheme and the inhomogeneous $T-Q$ relation have enabled significant methodological advances for systems with broken $U(1)$ symmetry, the underlying physics induced by symmetry breaking remains largely unexplored, due to the previously unknown distributions of the transfer-matrix roots. In this paper, we propose a new approach to determining the patterns of zero roots and Bethe roots for the $\Lambda-\theta$ and inhomogeneous Bethe ansatz equations using tensor-network algorithms. As an explicit example, we consider the isotropic Heisenberg spin chain with non-diagonal boundary conditions. The exact structures of both zero roots and Bethe roots are obtained in the ground state for large system sizes, up to ($N\simeq 60$ and $100$). We find that even in the absence of $U(1)$ symmetry, the Bethe and zero roots still exhibit a highly structured pattern. The zero roots organize into bulk strings, boundary strings, and additional roots, forming two dominant lines with boundary-string attachments. Correspondingly, the Bethe roots can be classified into four distinct types: regular roots, line roots, arc roots, and paired-line roots. These structures are associated with a real-axis line, a vertical line, characteristic arcs in the complex plane, and boundary-induced conjugate pairs. Comparative analysis reveals that the $t-W$ scheme generates significantly simpler root topologies than those obtained via off-diagonal Bethe Ansatz.

We introduce a notion of restricted pyramid configurations for computing the 1-leg Donaldson-Thomas $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex. We study a special type of restricted pyramid configurations with the prescribed 1-leg partitions, and find one unique class of them satisfying the symmetric interlacing property. This leads us to obtain an explicit formula for a class of 1-leg Donaldson-Thomas $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex through establishing its connection with 1-leg Donaldson-Thomas $\mathbb{Z}_4$-vertex using the vertex operator methods of Okounkov-Reshetikhin-Vafa and Bryan-Young.

We discuss a class of discrete Gaussian free fields on Hamming graphs, where interactions are determined solely by the Hamming distance between vertices. The purpose of examining this class is that it differs significantly from the commonly discussed spin system on the integer lattice with nearest-neighbour interactions. After introducing general results on the partition function and covariance for the class of Gaussian free fields, we present detailed properties of some specific models. Group-theoretic arguments and the Fourier transform give some explicit results.

We prove that the unreduced singular instanton homology $I^\sharp(Y,K;\mathbb{Z})$ has $2$-torsion for any null-homologous fibered knot $K$ of genus $g>0$ in a closed $3$-manifold $Y$ except for $\#^{2g}S^1\times S^2$. The main technical result is a formula of $I^\sharp(Y,K;\mathbb{C})$ via sutured instanton theory, by which we can compare the dimensions of $I^\sharp(Y,K;\mathbb{F}_2)$ and $I^\sharp(Y,K;\mathbb{C})$. As a byproduct, we show that $I^\sharp(S^3,K;\mathbb{C})$ for a knot $K\subset S^3$ admitting lens space surgeries is determined by the Alexander polynomial, while some special cases of torus knots have been previously studied by many people. Another byproduct is that the next-to-top Alexander grading summand of instanton knot homology $KHI(S^3,K,g(K)-1)$ is non-vanishing when $K$ has unknotting number one, which generalizes the Baldwin--Sivek's result in the fibered case. Finally, we discuss the relation to the Heegaard Floer theory.

This paper explores computational methods for solving the Longest Vector Problem (LVP) and Closest Vector Problem (CVP) in $p$-adic fields. Leveraging the non-Archimedean property of $p$-adic norms, we propose a polynomial time algorithm to compute orthogonal bases for $p$-adic lattices when the $p$-adic field is given by a minimal polynomial. The method utilizes the structure of maximal orders and $p$-radicals in extension fields of $\mathbb{Q}_{p}$ to efficiently construct uniformizers and residue field bases, enabling rapid solutions for LVP and CVP. In addition, we introduce the characterization of norms on vector spaces over $\mathbb{Q}_p$.

MDS codes play a central role in practice due to their broad applications. To date, most known MDS codes are generalized Reed-Solomon (GRS) codes, leaving codes that are not equivalent to GRS codes comparatively less understood. Studying this non-GRS regime is therefore of intrinsic theoretical interest, and is also practically relevant since the strong algebraic structure of GRS codes can be undesirable in cryptographic settings. Among the known non-GRS codes, twisted generalized Reed-Solomon (TGRS) codes and Roth-Lempel codes are two representative families of non-GRS codes that have attracted significant attention. Though substantial work has been devoted to the construction and structural analysis of TGRS and Roth-Lempel codes, comparatively little attention has been paid to their decoding, and many problems remain open. In this paper, we propose list and unique decoding algorithms for TGRS codes and Roth-Lempel codes based on the Guruswami-Sudan algorithm. Under suitable parameter conditions, our algorithms achieve near-linear running time in the code length, improving upon the previously best-known quadratic-time complexity. Our TGRS decoder supports fixed-rate TGRS codes with up to O(n^2) twists, substantially extending prior work that only handled the single-twist case. For Roth-Lempel codes, we provide what appears to be the first efficient decoder. Moreover, our list decoders surpass the classical unique-decoding radius for a broad range of parameters. Finally, we incorporate algebraic manipulation detection (AMD) codes into the list-decoding framework, enabling recovery of the correct message from the output list with high probability.

This paper examines the simplest case of total differential equations that appears in the theory of foliation structures, without imposing the smoothness assumptions. This leads to a peculiar asymmetry in the differentiability of solutions. To resolve this asymmetry, this paper focuses on the differentiability of the integral manifold. When the system is locally Lipschitz, a solution is ensured to be only locally Lipschitz, but the integral manifolds must be $C^1$. When the system is $C^k$, we can only ensure the existence of a $C^k$ solution, but the integral manifolds must be $C^{k+1}$. In addition, we see a counterexample in which the system is $C^1$, but there is no $C^2$ solution. Moreover, we characterize a minimizer of an optimization problem whose objective function is a quasi-convex solution to a total differential equation. In this connection, we examine two necessary and sufficient conditions for the system in which any solution is quasi-convex.

Node-Kayles is a well-known impartial combinatorial game played on graphs, where players alternately select a vertex and remove it along with its neighbors. By the Sprague-Grundy theorem, every position of an impartial game corresponds to a non-negative integer called its Grundy value. In this paper, we investigate the Grundy value sequences of $n$-regular trees as well as graphs formed by joining two $n$-regular trees with a path of length $k$. We derive explicit formulas and recursive relations for the associated Grundy value sequences. Furthermore, we prove that these sequences are eventually periodic and determine both their preperiod lengths and their periods.

We establish explicit unconditional results on the graphic properties of the prime gap sequence. Let \( p_n \) denote the \( n \)-th prime number (with $p_0=1$) and \( \mathrm{PD}_n = (p_\ell - p_{\ell-1})_{\ell=1}^n \) be the sequence of the first \( n \) prime gaps. Building upon the recent work by Erdős \emph{et al}, which proved the graphic nature of \( \mathrm{PD}_n \) for large \( n \) unconditionally, and for all \( n \) under RH, we provide the first explicit unconditional threshold such that:\\ (1) For all \( n \geq \exp\exp(30.5) \), \( \mathrm{PD}_n \) is graphic.\\ (2) For all \( n \geq \exp\exp(34.5) \), every realization \( G_n \) of \( \mathrm{PD}_n \) satisfies that \( (G_n, p_{n+1}-p_n) \) is DPG-graphic. Our proofs utilize a more refined criterion for when a sequence is graphic, and better estimates for the first moment of large prime gaps proven through an explicit zero-free region and explicit zero-density estimate for the Riemann zeta function.

We investigate random spatially coupled low-density parity-check (SC-LDPC) code ensembles over finite fields. Under different variable-node edge-spreading rules, the random Tanner graphs of several coupled ensembles are defined by multiple independent, uniformly random monomial maps. The two main coupled ensembles considered are referred to as the standard coupled ensemble and the improved coupled ensemble. We prove that both coupled ensembles exhibit asymptotically good minimum distance and minimum stopping set size. Theoretical and numerical results show that the improved coupled ensemble can achieve better distance performance than the standard coupled ensemble. We introduce the essential preliminaries and analytical tools needed to analyze the iterative decoding threshold of coupled ensembles over any finite field. We consider a class of memoryless channels with special symmetry, termed q-ary input memoryless symmetric channels (QMSCs), and show that, for these channels, the distribution of channel messages (in form of probability vectors) likewise exhibits this symmetry. Consequently, we define symmetric probability measures and their reference measures on a finite-dimensional probability simplex, analyze their foundational properties and those of their linear functionals, endow their respective spaces with metric topologies, and conduct an in-depth study of their degradation theory. Based on our analytical framework, we establish a universal threshold saturation result for both of the coupled ensembles over a q-ary finite field on QMSCs. Specifically, as the coupling parameters increase, the belief-propagation threshold of a coupled system saturates to a well-defined threshold that depends only on the underlying ensemble and the channel family.

We investigate the existence of nonnegative bump solutions to the sublinear elliptic equation \[ \begin{cases} -\Delta v - K(x)v + |v|^{q-2}v = 0 & \text{in } \mathbb{R}^N, \\ v(x) \to 0 & \text{as } |x| \to \infty, \end{cases} \] where $q \in (1,2)$, $ N \geq 2$, and the potential $K \in L^p_{\mathrm{loc}}(\mathbb{R}^N)$ with $p > N/2$ is a function without any symmetry assumptions. Under the condition that $\|K - 1\|_{L^p_{\mathrm{loc}}}$ is sufficiently small, we construct infinitely many solutions with arbitrarily many bumps. The construction is challenged by the sensitive interaction between bumps, whose limiting profiles have compact support. The key to ensuring their effective separation lies in obtaining sharp estimates of the support sets. Our method, based on a truncated functional space, provides precisely such control. We derive qualitative local stability estimates in region-wise maximum norms that govern the size of each bump's essential support, confining its core to a designated region and minimizing overlap. Crucially, these estimates are uniform in the number of bumps, which is the pivotal step in establishing the existence of solutions with infinitely many bumps.

We investigate the Hitchin hyperkähler metric on the moduli space of strongly parabolic $\mathfrak{sl}(2,\C)$-Higgs bundles on the $n$-punctured Riemann sphere and its degeneration obtained by scaling the parabolic weights $t\alpha$ as $t\to0$. Using the parabolic Deligne--Hitchin moduli space, we show that twistor lines of hyperpolygon spaces arise as limiting initial data for twistor lines at small weights, and we construct the corresponding real-analytic families of $\lambda$-connections. On suitably shrinking regions of the moduli space, the rescaled Hitchin metric converges, in the semiclassical limit, to the hyperkähler metric on the hyperpolygon space $\mathcal X_\alpha$, which thus serves as the natural finite-dimensional model for the degeneration of the infinite-dimensional hyperkähler reduction. Moreover, higher-order corrections of the Hitchin metric in this semiclassical regime can be expressed explicitly in terms of iterated integrals of logarithmic differentials on the punctured sphere.

Using geometric Eisenstein series, foundational work of Arinkin and Gaitsgory constructs cuspidal-Eisenstein decompositions for ind-coherent nilpotent sheaves on the de Rham moduli of local systems. This article extends these constructions to coherent (not ind-coherent) nilpotent sheaves on the Dolbeault, Hodge and twistor moduli from non-abelian Hodge theory. We thus account for Higgs bundles, Hodge filtrations and hyperkähler rotations of local systems. In particular, our constructions are shown to decompose a hyperholomorphic sheaf theory of so-called BBB-branes into cuspidal and Eisenstein components. Our work is motivated, on the one hand, by the `classical limit' or `Dolbeault geometric Langlands conjecture' of Donagi and Pantev, and on the other, by attempts to interpret Kapustin and Witten's physical duality between BBB-branes and BAA-branes in 4D supersymmetric Yang--Mills theories as a mathematical statement within the geometric Langlands program.

We study minimum vertex-degree conditions in 3-uniform hypergraphs for (tight) spanning components and (combinatorial) surfaces. Our main results show that a 3-uniform hypergraph $G$ on $n$ vertices contains a spanning component if $\delta_1(G) \gtrsim \tfrac{1}{2} \binom{n}{2}$ and a spanning copy of any surface if $\delta_1(G) \gtrsim \tfrac{5}{9} \binom{n}{2}$, which in both cases is asymptotically optimal. This extends the work of Georgakopoulos, Haslegrave, Montgomery, and Narayanan who determined the corresponding minimum codegree conditions in this setting.

We present a new Schwarz Lemma for bounded domains with Bergman metrics. The key ingredient of our proof is the Cauchy-Schwarz inequality from probability theory.

A sign pattern is a matrix that has entries from the set $\{+,-,0\}$. An $n\times n$ sign pattern $\mathcal{P}$ is called consistent if every real matrix in its qualitative class has exactly $k$ real eigenvalues and $n-k$ nonreal eigenvalues for some integer $k$, with $1\leq k\leq n$. In the article \cite{1}, the authors established a necessary condition for irreducible, tridiagonal patterns with a $0$-diagonal to be consistent. Subsequently, they proposed that this condition is also sufficient for such patterns to be consistent. In this article, we first demonstrate that this proposition does not hold. We characterize all irreducible, tridiagonal sign patterns with a $0$-diagonal of order at most five that are consistent. Moreover, we establish useful, necessary conditions for irreducible, combinatorially symmetric sign patterns to be consistent. Finally, we introduce the class $\Delta$ of all $2$-consistent sign patterns and provide several necessary conditions for sign patterns to belong to this class.

We introduce and study a loopless multigraph associated with a split graph $S$: the factor graph of $S$, denoted by $\Phi(S)$, which encodes the combinatorial information about 2-switch transformations over $S$. This construction provides a cleaner, compact and non-redundant alternative to the graph $A_4(S)$ by Barrus and West, for the particular case of split graphs. If $\Phi(S)$ is simple and connected, we obtain a precise description of the underlying structure of $S$, particularly when $\Phi(S)$ is complete, highlighting the usefulness of the factor graph for understanding 2-switch dynamics in balanced and indecomposable split graphs, as well as its 2-switch-degree classification.

We study affine control systems on smooth manifolds and their complete lifts to the tangent bundle, providing an explicit geometric description of the solutions of the lifted system. We show that, although controllability of the complete lift implies controllability of the original system, the lifted system is never controllable due to intrinsic geometric constraints. By introducing chain controllability, we prove that controllability of the original system guarantees chain controllability of its complete lift.

A sign pattern is a matrix whose entries are from the set $\{+,-, 0\}$. A square sign pattern $A$ is called sign $k$-potent if $k$ is the smallest positive integer for which $A^{k+1}=A$, and for $k=1$, $A$ is called sign idempotent. In 1993, Eschenbach \cite{01} gave an algorithm to construct sign idempotent sign patterns. However, Huang \cite{02} constructed an example to show that matrices obtained by Eschenbach's algorithm were not necessarily sign idempotent. In \cite{03}, Park and Pyo modified Eschenbach's algorithm to construct all reducible sign idempotent sign patterns. In this paper, we give an example to establish that the modified algorithm by Park and Pyo does not always terminate in a single iteration; the number of iterations, depending on the order of the sign pattern, could be large. In this paper, we give a new algorithm that terminates in a single iteration to construct all possible sign idempotent sign patterns. We also provide an algorithm for constructing sign $k$-potent sign patterns. Further, we give some necessary and sufficient conditions for a sign $k$-potent sign pattern to allow $k$-potence.

In this note we establish that the word problem is algorithmically decidable for finitely generated just infinite groups given by a recursively enumerable set of relations. The proof does not use the Wilson--Grigorchuk theorem on the classification of just infinite groups, and the argument proceeds directly from the definition, using ideas from classical results on decidability of the word problem: Kuznetsov's theorem on simple groups and the Dyson--Mostowski theorem on residually finite groups.

We apply the formalism of Alper-Halpern-Leistner-Heinloth to construct proper good moduli spaces for moduli stacks of Bridgeland semistable orthosymplectic complexes on a complex smooth projective variety, which we propose as a candidate for compactifying moduli spaces of principal bundles for the orthogonal and symplectic groups. We also prove some results on good moduli spaces of fixed point stacks and mapping stacks from finite groupoids.

This paper proposes the use of combinatorial techniques from tropical geometry to build the 120 tritangent planes to a given smooth algebraic space sextic. Although the tropical count is infinite, tropical tritangents come in 15 equivalence classes, each containing the tropicalization of exactly eight classical tritangents. Under mild genericity conditions on the tropical side, we show that liftings of tropical tritangents are defined over quadratic extensions of the ground field over which the input sextic curve is defined. When the input curve is real, we prove that every complex liftable member of a given tropical tritangent class either completely lifts to the reals or none of its liftings are defined over the reals. As our main application we use these methods to build examples of real space sextics with 64 and 120 totally real tritangents, respectively. The paper concludes with a discussion of our results in the arithmetic setting.

Let $n\geq 2$ be an integer, $p$ be a prime number and $K$ be a finite extension of $\mathbb{Q}_p$. Motivated by Schraen's thesis and Gehrmann's definition of automorphic simple $\mathscr{L}$-invariants, we study the first non-vanishing extension groups between a pair of locally $K$-analytic generalized Steinberg representations of $\mathrm{GL}_n(K)$. We study subspaces of these extension groups defined by using either relative conditions with respect to Lie subalgebras of $\mathfrak{s}\mathfrak{l}_{n}$ (isomorphic to $\mathfrak{s}\mathfrak{l}_{m}$ for some $2\leq m<n$) or maps between locally $K$-analytic generalized Steinberg representations of $\mathrm{GL}_n(K)$ with different highest weights. The applications of these computations are two-fold. On one hand, we prove that a certain universal successive extension of filtered $(\varphi,N)$-modules can be realized as the space of homomorphisms from a suitable shift of the dual of locally $K$-analytic Steinberg representation into the de Rham complex of the Drinfeld upper-half space, generalizing one main result of Schraen's thesis from $\mathrm{GL}_{3}(\mathbb{Q}_p)$ to $\mathrm{GL}_{n}(K)$. On the other hand, we give a definition of higher $\mathscr{L}$-invariants for $\mathrm{GL}_n(K)$ (which we call Breuil-Schraen $\mathscr{L}$-invariants) and discuss its possible explicit relation to Fontaine-Mazur $\mathscr{L}$-invariants, using ideas from Breuil-Ding's higher $\mathscr{L}$-invariants for $\mathrm{GL}_{3}(\mathbb{Q}_p)$.

The aim of this note is to present the simple observation that a slight refinement of the Contraction Mapping Principle allows one to recover the precise convergence rate in the Picard-Lindelöf Theorem.

Recently, the weak Drazin inverse and its characterization have been crucial studies for matrices of index k. In this article, we have revisited W-weighted DMP and MPD inverses and constructed a general class of unique solutions to certain matrix equations. Moreover, we have generalized the W-weighted Drazin inverse of Meng, 2017 using the minimal rank Wweighted weak Drazin inverse. In addition to that, we have derived several equivalent properties of W-weighted DMP and MPD inverses for minimal rank W-weighted weak Drazin inverse of rectangular matrices. Furthermore, some projection-based results are discussed for the characterization of minimal rank W-weighted Drazin inverse, along with some new expressions that are derived for MPD and DMP inverses. Thereby, we have elaborated certain expressions of the perturbation formula for W-weighted weak MPD and DMP inverses. As an application, we establish the reverse and forward order laws using the W-weighted weak Drazin inverse and the minimal rank W-weighted weak Drazin inverse, and apply these results to solve certain matrix equation.

Existing methods for nonconvex bilevel optimization (NBO) require prior knowledge of first- and second-order problem-specific parameters (e.g., Lipschitz constants and the Polyak-Łojasiewicz (PŁ) parameters) to set step sizes, a requirement that poses practical limitations when such parameters are unknown or computationally expensive. We introduce the Adaptive Fully First-order Bilevel Approximation (AF${}^2$BA) algorithm and its accelerated variant, A${}^2$F${}^2$BA, for solving NBO problems under the PŁ conditions. To our knowledge, these are the first methods to employ fully adaptive step size strategies, eliminating the need for any problem-specific parameters in NBO. We prove that both algorithms achieve $\mathcal{O}(1/\epsilon^2)$ iteration complexity for finding an $\epsilon$-stationary point, matching the iteration complexity of existing well-tuned methods. Furthermore, we show that A${}^2$F${}^2$BA enjoys a near-optimal first-order oracle complexity of $\tilde{\mathcal{O}}(1/\epsilon^2)$, matching the oracle complexity of existing well-tuned methods, and aligning with the complexity of gradient descent for smooth nonconvex single-level optimization when ignoring the logarithmic factors.

We investigate when a maximum distance separable ($MDS$) code over $F_q$ is also completely regular ($CR$). For lengths $n=q+1$ and $n=q+2$ we provide a complete classification of the $MDS$ codes that are $CR$ or at least uniformly packed in the wide sense ($UPWS$). For the more restricted case $n\leq q$ with $q\leq 5$ we obtain a full classification (up to equivalence) of all nontrivial $MDS$ codes: there are none for $q=2$; only the ternary Hamming code for $q=3$; four nontrivial families for $q=4$; and exactly six linear $MDS$ codes for $q=5$ (three of which are $CR$ and one admits a self-dual version). Additionally, we close two gaps left open in a previous classification of self-dual $CR$ codes with covering radius $\rho\leq 3$: we precisely determine over which finite fields the $MDS$ self-dual completely regular codes with parameters $[2,1,2]_q$ and $[4,2,3]_q$ exist.

For a simple graph G = (V, E), a coloring of vertices of G using two colors, say red and blue, is called a quasi neighborhood balanced coloring if, for every vertex of the graph, the number of red neighbors and the number of blue neighbors differ by at most one. In addition, there must be at least one vertex in G for which this difference is exactly one. If a graph G admits such a colouring, then G is said to be a quasi-neighbourhood balanced colored graph. We also define variants of such a coloring, like uniform quasi neighborhood balanced coloring, positive quasi neighborhood balanced coloring and negative quasi neighborhood balanced coloring based on the color of the extra neighbor of every vertex of odd degree of the graph G. We present several examples of graph classes that admit the various variants of quasi neighborhood balanced coloring. We also discuss various graph operations involving such graphs. Furthermore, we prove that there is no forbidden subgraph characterization for the class of quasi neighborhood balanced coloring and show that the problem of determining whether a given graph has such a coloring is NP-complete.

When optimizing a nonlinear objective, one can employ a neural network as a surrogate for the nonlinear function. However, the resulting optimization model can be time-consuming to solve globally with exact methods. As a result, local search that exploits the neural-network structure has been employed to find good solutions within a reasonable time limit. For such methods, a lower per-iteration cost is advantageous when solving larger models. The contribution of this paper is two-fold. First, we propose a gradient-based algorithm with lower per-iteration cost than existing methods. Second, we further adapt this algorithm to exploit the piecewise-linear structure of neural networks that use Rectified Linear Units (ReLUs). In line with prior research, our methods become competitive with -- and then dominant over -- other local search methods as the optimization models become larger.

A subspace $U$ of $\mathbb{F}_q^n$ is called \textit{cyclically covering} if the whole space $\mathbb{F}_q^n$ is the union of the cyclic shifts of $U$. The case $\mathbb{F}_q^n$ itself is the only covering subspace, is of particular interest. Recently, Huang solved this problem completely under the condition $\gcd(n, q)=1$ using primitive idempotents and trace functions, and explicitly posed the non-coprime case as an open question. This paper provides a complete answer to Huang's question. We prove that if $n = p^k m$ where $p = \operatorname{char}(\mathbb{F}_q)$ and $\gcd(m, p)=1$, then $h_q(p^k m) = 0$ if and only if $h_q(m) = 0$. This result fully reduces the non-coprime case to the coprime case settled by Huang. Our proof employs the structure theory of cyclic group algebras in modular characteristic.

The classic algorithm [Papadimitriou, this http URL '81] for IPs has a running time $n^{O(m)}(m\cdot\max\{\Delta,\|\textbf{b}\|_{\infty}\})^{O(m^2)}$, where $m$ is the number of constraints, $n$ is the number of variables, and $\Delta$ and $\|\textbf{b}\|_{\infty}$ are, respectively, the largest absolute values among the entries in the constraint matrix and the right-hand side vector of the constraint. The running time is exponential in $m$, and becomes pseudo-polynomial if $m$ is a constant. In recent years, there has been extensive research on FPT (fixed parameter tractable) algorithms for the so-called $n$-fold IPs, which may possess a large number of constraints, but the constraint matrix satisfies a specific block structure. It is remarkable that these FPT algorithms take as parameters $\Delta$ and the number of rows and columns of some small submatrices. If $\Delta$ is not treated as a parameter, then the running time becomes pseudo-polynomial even if all the other parameters are taken as constants. This paper explores the trade-off between time and accuracy in solving an IP. We show that, for arbitrary small $\varepsilon>0$, there exists an algorithm for IPs with $m$ constraints that runs in ${f(m,\varepsilon)}\cdot\textnormal{poly}(|I|)$ time, and returns a near-feasible solution that violates the constraints by at most $\varepsilon\Delta$. Furthermore, for $n$-fold IPs, we establish a similar result -- our algorithm runs in time that depends on the number of rows and columns of small submatrices together with $1/\varepsilon$, and returns a solution that slightly violates the constraints. Meanwhile, both solutions guarantee that their objective values are no worse than the corresponding optimal objective values satisfying the constraints. As applications, our results can be used to obtain additive approximation schemes for multidimensional knapsack as well as scheduling.

In the present paper, we consider a class of Markov processes on the discrete circle which has been introduced by König, O'Connell and Roch. These processes describe movements of exchangeable interacting particles and are discrete analogues of the unitary Dyson Brownian motion: a random number of particles jump together either to the left or to the right, with trajectories conditioned to never intersect. We provide asymptotic mixing times for stochastic processes in this class as the number of particles goes to infinity, under a sub-Gaussian assumption on the random number of particles moving at each step. As a consequence, we prove that a cutoff phenomenon holds independently of the transition probabilities, subject only to the sub-Gaussian assumption and a minimal aperiodicity hypothesis. Finally, an application to dimer models on the hexagonal lattice is provided.

We study the contact Lefschetz condition on compact contact solvmanifolds, as introduced by B.\ Cappelletti-Montano, A.\ De Nicola and I.\ Yudin. We seek to fill the gap in the literature concerning Benson-Gordon type results, characterizing $1$-Lefschetz contact solvmanifolds. We prove that the $1$-Lefschetz condition on Lie algebras is preserved via $1$-dimensional central extensions by a symplectic cocycle, thereby establishing that a unimodular symplectic Lie algebra $(\mathfrak{h}, \omega)$ is $1$-Lefschetz if and only if its contactization $(\mathfrak{g}, \eta)$ is $1$-Lefschetz. We achieve this by showing an explicit relation for the relevant cohomology degrees of $\mathfrak{h}$ and $\mathfrak{g}$. Using this, we show how the commutators $[\mathfrak{h},\mathfrak{h}]$ and $[\mathfrak{g},\mathfrak{g}]$ are related, especially when the $1$-Lefschetz condition holds. By specializing to the nilpotent setting, we prove that $1$-Lefschetz contact nilmanifolds equipped with an invariant contact form are quotients of a Heisenberg group, and deduce that there are many examples of compact $K$-contact solvmanifolds not admitting compatible Sasakian structures. We also construct examples of completely solvable $1$-Lefschetz solvmanifolds, some having the $2$-Lefschetz property and some failing it.

We introduce a general probabilistic framework for discrete-time, infinite-horizon discounted Mean Field Type Games (MFTGs) with both global common noise and team-specific common noises. In our model, agents are allowed to use randomized actions, both at the individual level and at the team level. We formalize the concept of Mean Field Markov Games (MFMGs) and establish a connection between closed-loop policies in MFTGs and Markov policies in MFMGs through different layers of randomization. By leveraging recent results on infinite-horizon discounted games with infinite compact state-action spaces, we prove the existence of an optimal closed-loop policy for the original MFTG when the state spaces are at most countable and the action spaces are general Polish spaces. We also present an example satisfying our assumptions, called Mean Field Drift of Intentions, where the dynamics are strongly randomized, and we establish the existence of a Nash equilibrium using our theoretical results.

This paper investigates finiteness conditions for gentle and skew-gentle algebras. First, we prove that a skew-gentle algebra is $\tau$-tilting finite if and only if it is representation-finite, which extends the result for gentle algebras by Plamondon (2019). Second, using surface models, we characterize silting-discreteness for the perfect derived categories of graded gentle and skew-gentle algebras. Specifically, for a graded gentle algebra, silting-discreteness is equivalent to its associated surface being of genus zero with non-zero winding numbers for all simple closed curves. We further extend this geometric characterization to graded skew-gentle algebras via orbifold surface models.

Given a group $G$ and $V$ a representation of $G$, denote the number of indecomposable summands of $V^{\otimes k}$ by $b_k^{G, V}$. Given a tilting representation $T$ of $\text{SL}_2(K)$ where $K=\overline{K}$ and of characteristic $p>2$, we show that $Ck^{-\alpha_p}(\text{dim} T)^k<b_k^{T, \text{SL}_2(K)}<Dk^{-\alpha_p}(\text{dim} T)^k$ for some $C, D>0$ where $\alpha_p=1-(1/2)\log_p(\frac{p+1}{2}).$

Kastermans proved that consistently $\bigoplus_{\aleph_1} \mathbb{Z}_2$ has a cofinitary representation. We present a short proof that $\bigoplus_{\mathfrak{c}} \mathbb{Z}_2$ always has an arithmetic cofinitary representation. Further, for every finite group $F$ we construct an arithmetic maximal cofinitary group of isomorphism type $(\ast_{\mathfrak{c}} \mathbb{Z}) \times F$. This answers an implicit question by Schrittesser and Mejak whether one may construct definable maximal cofinitary groups not decomposing into free products.

We study two long-standing conjectures concerning lower bounds for the Betti numbers of a graded module over a polynomial ring. We prove new cases of these conjectures in codimensions five and six by reframing the conjectures as arithmetic problems in the Boij-Söderberg cone. In this setting, potential counterexamples correspond to explicit Diophantine obstructions arising from the numerics of pure resolutions. Using number-theoretic methods, we completely classify these obstructions in the codimension three case revealing some delicate connections between Betti tables, commutative algebra and classical Diophantine equations. The new results in codimensions five and six concern Gorenstein algebras where a study of the variety determined by these Diophantine equations is sufficient to resolve the conjecture in this case.

In this work we study quadratic Lie algebras that contain the Heisenberg Lie algebra $\h_m$ as an ideal. We give a procedure for constructing these kind of quadratic Lie algebras and prove that any quadratic Lie algebra $\g$ that contains the Heisenberg Lie algebra as an ideal is constructed by using this procedure. We state necessary and sufficiency conditions to determine whether an indecomposable quadratic Lie algebra is the Heisenberg Lie algebra extended by a derivation. In addition, we state necessary and sufficiency conditions to determine whether the quotient $\g/\h_m$ admits an invariant metric and we also study the case when the nilradical of the Lie algebra $\g$ is equal to $\h_m$.

We establish two exact correspondences between reverse-mode automatic differentiation (backpropagation evaluated at a fixed forward pass) and compositions of projection maps in Kullback--Leibler (KL) geometry. In both settings, message passing defined by alternating KL projections enforces agreement and factorization constraints. In the first setting, backpropagation arises as the differential of a KL projection map on a delta-lifted factorization. In the second setting, on complete and decomposable sum--product networks, backpropagation coincides with exact probabilistic inference, and the backward values admit an interpretation as Lagrange multipliers of a KL I-projection problem.

The charge functions for n-dimensional partitions are known for n=2,3,4 in the literature. We give the expression for arbitrary odd dimension in a recent work, and now further conjecture a formula for all even dimensional cases. This conjecture is proved rigorously for 6D, and numerically verified for 8D.

Let $\mathcal{P}_k(n)$ denote the set of partitions of $n$ whose largest part is bounded by $k,$ which are in well-known bijection with $(k+1)$-cores $\mathcal{C}_k$. We study a growth process on $\mathcal{C}_k$, whose stationary distribution is the $k$-Plancherel measure, which is a natural extension of the Plancherel measure in the context of $k$-Schur functions. When $k\to\infty$ it converges to the Plancherel measure for partitions, a limit studied first by Vershik-Kerov. However, when $k$ is fixed and $n\to \infty$, we conjecture that it converges to a shape close to the limit shape from the uniform growth of partitions, as studied by Rost. We show that the limiting behavior, for fixed $k$, is governed by a finite Markov chain with $k!$ states over a subset of the $k$-bounded partitions or equivalently as a TASEP over cyclic permutations of length $k+1$. This paper initiates the study of these processes, state some theorems and several intriguing conjectures found by computations of the finite Markov chain.

We study the period map from infinitesimal deformations of a scheme $X$ over a perfect field $k$ to those of the associated $k$-linear $\infty$-category $\mathrm{QC}(X)$. For quasicompact, smooth, and separated $X$, we identify the corresponding map on tangent fibres with the dual HKR map $\mathrm{R}\Gamma(X, \mathrm{T}_X)[1] \to \mathrm{HH}^{\bullet}(X/k)[2]$, and give conditions for injectivity on homotopy groups. As applications, we prove liftability along square-zero extensions to be a derived invariant (at least when $\mathrm{char}(k) \ne 2$), and exhibit cases where the entire (classical) deformation functor of $X$ is a derived invariant; this partially answers a question of Lieblich.

We explicitly construct a heat kernel as a Neumann series for certain function spaces, such as $L^{1}$, $L^{2}$, and Hilbert spaces, associated to a locally compact Hausdorff space $\mathfrak{X}$ with Borel $\sigma$-algebra $\mathcal{B}$, and endowed with additional measure-theoretic data. Our approach is an adaptation of classical work due to Minakshishundaram and Pleijel, and it requires as input a parametrix or small time approximation to the heat kernel. The methodology developed in this article applies to yield new instances of heat kernel constructions, including normalized Laplacians on finite and infinite graphs as well as Hilbert spaces with reproducing kernels.

In three articles published in CNJ in 2012 and 2016 , we discussed some links between mathematical sciences, coin minting and numismatics. This article is a continuation of this cycle. It tells the story of selected important developments in the history of geometry using modern commemorative coins as a background and illustration.

We establish sharp large deviation asymptotics for the maximum order statistic of independent and identically distributed heavy-tailed random variables, valid for all Borel subsets of the right tail. This result yields exact decay rates for exceedance probabilities at thresholds that grow faster than the natural extreme-value scaling. As an application, we derive the polynomial rate of decay of ruin probabilities in insurance portfolios where insolvency is driven by a single extreme claim.

This article develops several functional models for a given $\Gamma_n$-contraction. The first model is motivated by the Douglas functional model for a contraction. We then establish factorization results that clarify the relationship between a minimal isometric dilation and an arbitrary isometric dilation of a contraction. Using these factorization results, we obtain a Sz.-Nagy-Foias type functional model for a completely non-unitary $\Gamma_n$-contraction, as well as Schäffer type functional model for $\Gamma_n$-contraction.

Let $A$ be a finite-dimensional associative $k$-algebra with identity. The primary aim of this paper is to study the rationality properties of the group of all $k$-algebra automorphisms of $A$, as an affine algebraic group over an arbitrary field $k$. We investigate mainly the $R$-equivalence property of the identity component of $\mathrm{Aut}_{k}(A)$ over a perfect field $k$.

Bumpless pipe dreams (BPDs) are combinatorial objects used in the study of Schubert and Grothendieck polynomials. Weigandt recently introduced a co-BPD object associated to each BPD and used them to give an analogue to the change of bases formulas of Lenart and Lascoux between these polynomials. She posed the problem of characterizing the set of permutations whose BPDs have only reduced co-BPDs. We give a pattern-avoidance characterization for these permutations using a set of seven patterns.

Let $A$ be a finite-dimensional commutative associative algebra with unity over an algebraically closed field $\mathbb{K}$. The purpose of the paper is to study the solvability of $G_A$, where $G_A$ is the identity component of $\text{Aut}_\mathbb{K}(A)$. Inspired by Pollack's work, Saorín and Asensio have started this study for a commutative associative algebra $A$ when $\text{dim}(R/R^2)=2$, where $R$ is the Jacobson radical of $A$. In this paper, we give new sufficient conditions on $A$ so that $G_A$ is solvable without any restriction on $\text{dim}(R/R^2)$.

In this paper, we prove that the Euclidean distance between two independent random vectors uniformly distributed on $l_p^n$-balls $(1 \leq p \leq \infty)$ or on its boundary satisfies a central limit theorem as $n$ tends to $\infty$. Also, we give a compact proof of the case of the sphere, which was proved by Hammersley. Furthermore, we complement our central limit theorem by providing large deviation principles for the cases $p \geq 2$.

Let $F$ be a non-archimedean local field or a finite field. In this article, we obtain an explicit and complete set of double coset representatives for $S\backslash GL_{2n}(F)/Q$ where $S$ is the Shalika subgroup and $Q$ a maximal parabolic subgroup of the group $GL_{2n}(F)$ of invertible $2n\times 2n$ matrices. We compute the cardinality of $S\backslash GL_{2n}(F)/Q$ and also give an alternate perspective on the double cosets arising intrinsically from certain subgroups which are relevant for applications in representation theory. Finally, if $Q$ is a maximal parabolic subgroup of the type $(r,2n-r),$ we prove that $S\backslash GL_{2n}(F)/Q$ is in one to one correspondence with $\Delta S_n\backslash S_{2n}/S_{r}\times S_{2n-r}$ leading to a Bruhat decomposition. The results and proofs discussed in this article are valid over any arbitrary field $F$ even though our motivation is from representation theory of $p$-adic and finite linear groups.

We study the problem of estimating the score function using both implicit score matching and denoising score matching. Assuming that the data distribution exhibiting a low-dimensional structure, we prove that implicit score matching is able not only to adapt to the intrinsic dimension, but also to achieve the same rates of convergence as denoising score matching in terms of the sample size. Furthermore, we demonstrate that both methods allow us to estimate log-density Hessians without the curse of dimensionality by simple differentiation. This justifies convergence of ODE-based samplers for generative diffusion models. Our approach is based on Gagliardo-Nirenberg-type inequalities relating weighted $L^2$-norms of smooth functions and their derivatives.

Formal verification of deep neural networks is increasingly required in safety-critical domains, yet exact reasoning over piecewise-linear (PWL) activations such as ReLU suffers from a combinatorial explosion of activation patterns. This paper develops a solver-grade methodology centered on \emph{incremental certificate learning}: we maximize the work performed in a sound linear relaxation (LP propagation, convex-hull constraints, stabilization), and invoke exact PWL reasoning only through a selective \emph{exactness gate} when relaxations become inconclusive. Our architecture maintains a node-based search state together with a reusable global lemma store and a proof log. Learning occurs in two layers: (i) \emph{linear lemmas} (cuts) whose validity is justified by checkable certificates, and (ii) \emph{Boolean conflict clauses} extracted from infeasible guarded cores, enabling DPLL(T)-style pruning across nodes. We present an end-to-end algorithm (ICL-Verifier) and a companion hybrid pipeline (HSRV) combining relaxation pruning, exact checks, and branch-and-bound splitting. We prove soundness, and we state a conditional completeness result under exhaustive splitting for compact domains and PWL operators. Finally, we outline an experimental protocol against standardized benchmarks (VNN-LIB / VNN-COMP) to evaluate pruning effectiveness, learned-lemma reuse, and exact-gate efficiency.

We develop an equivariant Lagrangian Floer theory for Liouville sectors that have symmetry of a Lie group $G$. Moreover, for Liouville manifolds with $G$-symmetry, we develop a correspondence theory to relate the equivariant Lagrangian Floer cohomology upstairs and Lagrangian Floer cohomology of its quotient. Furthermore, we study the symplectic quotient in the presence of nodal type singularities and prove that the equivariant correspondence gives an isomorphism on cohomologies which was conjectured by Lekili-Segal.

We study the mean-field limit for a class of agent-based models describing flocking with nonlinear velocity alignment. Each agent interacts through a communication protocol $\phi$ and a non-linear coupling of velocities given by the power law $A(\bv) = |\bv|^{p-2}\bv$, $p > 2$. The mean-field limit is proved in two settings -- deterministic and stochastic. We then provide quantitative estimates on propagation of chaos for deterministic case in the case of the classical fat-tailed kernels, showing an improved convergence rate of the $k$-particle marginals to a solution of the corresponding Vlasov equation. The stochastic version is addressed with multiplicative noise depending on the local interaction intensity, which leads to the associated Fokker-Planck-Alignment equation. Our results extend the classical Cucker-Smale theory to the nonlinear framework which has received considerable attention in the literature recently.

The reflection coefficient is an important thermal property of materials, especially at the nanoscale, and determining this property requires solving an inverse problem based on macroscopic temperature measurements. In this manuscript, we investigate the stability of this inverse problem to infer the reflection coefficient in the phonon transport equation. We show that the problem becomes ill-posed as the system transitions from the ballistic to the diffusive regime, characterized by the Knudsen number converging to zero. Such a stability estimate clarifies the discrepancy observed in previous studies on the well-posedness of this inverse problem. Furthermore, we quantify the rate at which the stability deteriorates with respect to the Knudsen number and confirm the theoretical result with numerical evidence.

A valuable feature of the tubal tensor framework is that many familiar constructions from matrix algebra carry over to tensors, including SVD and notions of rank. Most importantly, it has been shown that for a specific family of tubal products, an Eckart-Young type theorem holds, i.e., the best low-rank approximation of a tensor under the Frobenius norm is obtained by truncating its tubal SVD. In this paper, we provide a complete characterization of the family of tubal products that yield an Eckart-Young type result. We demonstrate the practical implications of our theoretical findings by conducting experiments with video data and data-driven dynamical systems.

We construct a smooth nontrivial mixed partially hyperbolic system and explicitly identify its skeleton. This example shares characteristics with the classical examples. Moreover, the support of each physical measure contains three fixed points with mutually distinct unstable indices. By appropriately perturbing the skeleton, we provide an example where the number of physical measures varies upper semi-continuously. The general framework of mixed partially hyperbolic systems has been studied in theorem.

In Fritz & Rischel, Infinite products and zero-one laws in categorical probability, the problem was posed of finding an interesting Markov category which is causal and has all (small) Kolmogorov products (there Problem 6.7). Here we give an example where the deterministic subcategory is the category of Stone spaces (i.e. the dual of the category of Boolean algebras) and the kernels correspond to a restricted class of Kleisli arrows for the Radon monad. We look at this from two perspectives. First via pro-completions and Stone spaces directly. Second via duality with Boolean and algebras and effect algebras.

Relation between the real eigenvalues of the non-backtracking matrix and those of the non-backtracking Laplacian is considered with respect to node clustering. For this purpose we use the real eigenvalues of the transition probability matrix (when the random walk goes through the oriented edges with the rule of ``not going back in the next step'') which have a linear relation to those of the non-backtracking Laplacian of Jost,Mulas. ``Inflation--deflation'' techniques are also developed for clustering the nodes of the non-backtracking graph. With further processing, it leads to the clustering of the nodes of the original graph, which usually comes from a sparse stochastic block model of Bordenave,Decelle.

We construct a four-dimensional lattice-gas model with finite-range interactions that has non-periodic, ``quasicrystalline'' Gibbs states at low temperatures. Such Gibbs states are probability measures which are small perturbations of non-periodic ground-state configurations corresponding to tilings of the plane with Ammann's aperiodic tiles. Our construction is based on the correspondence between probabilistic cellular automata and Gibbs measures on their space-time trajectories, and a classical result on noise-resilient computing with cellular automata. The cellular automaton is constructed on the basis of Ammann's tiles, which are deterministic in one direction, and has non-periodic space-time trajectories corresponding to each valid tiling. Repetitions along two extra dimensions, together with an error-correction mechanism, ensure stability of the trajectories subjected to noise.

Simple sufficient conditions are given that ensure the uniform continuity in distribution for Borel transformations of random fields.

In honor of Zhiren Wang on the occasion of being awarded the Brin Prize, we report on his exciting and deep work on rigidity of higher rank abelian groups and lattices in higher rank semisimple groups.

This thesis studies skein relations in cluster algebras arising from punctured surfaces. We introduce skein-type identities expressing cluster variables associated with incompatible curves on a surface in terms of cluster variables corresponding to compatible arcs. Incompatibility arises from phenomena such as intersections, self-intersections, and opposite taggings at punctures. To establish these identities, we develop a combinatorial-algebraic framework that relates loop graphs to certain representations. These skein relations can then be applied to investigate structural properties of cluster algebras from punctured surfaces. In particular, they can be used to prove the existence of bases satisfying natural positivity and compatibility conditions. This extends existing work on surface cluster algebras by incorporating punctures in the interior of the surface, thereby enlarging the class of cluster algebras for which such skein relations and bases can be constructed.

In this paper, we extend the classical quadrilateral based hierarchical Poincaré-Steklov (HPS) framework to triangulated geometries. Traditionally, the HPS method takes as input an unstructured, high-order quadrilateral mesh and relies on tensor-product spectral discretizations on each element. To overcome this restriction, we introduce two complementary high-order strategies for triangular elements: a reduced quadrilateralization approach which is straightforward to implement, and triangle based spectral element method based on Dubiner polynomials. We show numerically that these extensions preserve the spectral accuracy, efficiency, and fast direct-solver structure of the HPS framework. The method is further extended to time dependent and evolving surfaces, and its performance is demonstrated through numerical experiments on reaction-diffusion systems, and geometry driven surface evolution.

This survey presents recent developments concerning the Shafarevich conjecture, non-abelian Hodge theories, hyperbolicity, and the topology of complex algebraic varieties, as well as the interplay among these areas. More precisely, we present the main ideas and techniques involved in the linear versions of the following conjectures: the Shafarevich conjecture, the Chern-Hopf-Thurston conjecture, Kollár's conjecture on the holomorphic Euler characteristic, the de Oliveira-Katzarkov-Ramachandran conjecture, and Campana's nilpotency conjecture. In addition, we discuss characterizations of the hyperbolicity of complex quasi-projective varieties via representations of their fundamental groups, together with the generalized Green-Griffiths-Lang conjecture in the presence of a big local system.

We study deterministic constructions of graphs for which the unique completion of low rank matrices is generically possible regardless of the values of the entries. We relate the completability to the presence of some patterns (particular unions of self-avoiding walks) in the subgraph of the lattice graph generated from the support of the bi-adjacency matrix. The construction makes it possible to design infinite families of graphs on which exact and stable completion is possible for every fixed rank matrix through the sum-of-squares hierarchy.

In the first part of the article we establish the existence in the sense of sequences of solutions in $H^{2}(R)$ for some nonhomogeneous linear differential equation in which one of the terms has the argument translated by a constant. It is shown that under the reasonable technical conditions the convergence in $L^{2}(R)$ of the source terms implies the existence and the convergence in $H^{2}(R)$ of the solutions. The second part of the work deals with the solvability in the sense of sequences in $H^{2}(R)$ of the integro-differential equation in which one of the terms has the argument shifted by a constant. It is demonstrated that under the appropriate auxiliary assumptions the convergence in $L^{1}(R)$ of the integral kernels yields the existence and the convergence in $H^{2}(R)$ of the solutions. Both equations considered involve the second order differential operator with or without the Fredholm property depending on the value of the constant by which the argument gets translated.

Decentralized optimization over directed graphs is essential for applications such as robotic swarms, sensor networks, and distributed learning. In many practical scenarios, the underlying network is a Time-Varying Broadcast Network (TVBN), where only row-stochastic mixing matrices can be constructed due to inaccessible out-degree information. Achieving exact convergence over TVBNs has remained a long-standing open question, as the limiting distribution of time-varying row-stochastic mixing matrices depends on unpredictable future graph realizations, rendering standard bias-correction techniques infeasible. This paper resolves this open question by developing the first algorithm that achieves exact convergence using only time-varying row-stochastic matrices. We propose PULM (Pull-with-Memory), a gossip protocol that attains average consensus with exponential convergence by alternating between row-stochastic mixing and local adjustment. Building on PULM, we develop PULM-DGD, which converges to a stationary solution at $\mathcal{O}(\ln(T)/T)$ for smooth nonconvex objectives. Our results significantly extend decentralized optimization to highly dynamic communication environments.

We consider a reflected process in the positive orthant driven by an exogenous jump process. For a given input process, we show that there exists a unique minimal strong solution to the given particle system up until a certain maximal stopping time, which is stated explicitly in terms of the dual formulation of a linear programming problem associated with the state of the system. We apply this model to study the ruin time of interconnected insurance firms, where the stopping time can be interpreted as the failure time of a reinsurance agreement between the firms. Our work extends the analysis of the particle system in Baker, Hambly, and Jettkant (2025) to the case of jump driving processes, and the existence result of Reiman (1984) beyond the case of sub-stochastic reflection matrices.

The year 2025 marked the passing of two towering figures of twentieth-century mathematical physics, Rodney Baxter and Chen-Ning Yang. Yang reshaped modern physics through the introduction of non-abelian gauge theory and, independently, through the consistency conditions underlying what is now called the Yang-Baxter equation. Baxter transformed those conditions into a systematic theory of exact solvability in statistical mechanics and quantum integrable systems. This article is written in memory of Baxter and Yang, whose work revealed how local consistency principles generate global mathematical structure. We review the Yang-Mills formulation of gauge theory, its mass obstruction and resolution via symmetry breaking, and the geometric framework it engendered, including instantons, Donaldson-Floer theory, magnetic monopoles, and Hitchin systems. In parallel, we trace the emergence of the Yang-Baxter equation from factorised scattering to solvable lattice models, quantum groups, and Chern-Simons theory. Rather than separate narratives, gauge theory and integrability are presented as complementary manifestations of a shared coherence principle, an ongoing journey from gauge symmetry toward mathematical unity.

We investigate the steady self-propelled motion of a rigid body immersed in a three-dimensional incompressible viscous fluid governed by the Navier-Stokes equations. The analysis is performed in a body-fixed reference frame, so that the fluid occupies an exterior domain and the propulsion mechanism is modeled through nonhomogeneous Navier-slip boundary conditions at the fluid-body interface. Such conditions provide a realistic description of propulsion in microfluidic and rough-surface regimes, where partial slip effects are significant. Under suitable smallness assumptions on the boundary flux and on the normal component of the prescribed surface velocity, we establish the existence of weak steady solutions to the coupled fluid-structure system. A key analytical ingredient is the derivation of a Korn-type inequality adapted to exterior domains with rigid-body motion and Navier-slip interfaces, which yields uniform control of both the fluid velocity and the translational and rotational velocities of the body. Beyond existence, we provide a necessary and sufficient condition under which a prescribed slip velocity on the body surface induces nontrivial translational or rotational motion of the rigid body. This is achieved through the introduction of a finite-dimensional thrust space, defined via auxiliary exterior Stokes problems with Navier boundary conditions, which captures the effective contribution of boundary-driven flows to the rigid-body motion. Our results clarify how boundary effects generate propulsion and extend the classical Dirichlet-based theory to the Navier-slip setting.

A new algorithm for exactly sampling from the set of proper colorings of a graph is presented. This is the first such algorithm that has an expected running time that is guaranteed to be linear in the size of a graph with maximum degree \( \Delta \) when the number of colors is greater than \( 3.637 \Delta + 1\).

We investigate deep composite polynomial approximations of continuous but non-differentiable functions with algebraic cusp singularities. The functions in focus consist of finitely many cusp terms of the form $|x-a_j|^{\alpha_j}$ with rational exponents $\alpha_j\in(0,1)$ on a real-analytic background. We propose a constructive approximation scheme that combines a division-free polynomial iteration for fractional powers with an outer layer for the analytic polynomial fitting. Our main result shows that this composite structure achieves exponential convergence in the the number of scalar coefficients in the inner and outer polynomial layers. Specifically, the $L^p([-1,1])$ approximation error, decays exponentially with respect to the parameter budget, in contrast to the algebraic rates obtained by classical single-layer polynomial approximation for cusp-type functions. Numerical experiments for both single and multiple cusp configurations confirm the theoretical rates and demonstrate the parameter efficiency of deep composite polynomial constructions.

We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Specifically, we show that if the tangent flow at a singular point is a multiplicity-one cylinder, then in a neighborhood of that point the flow is mean-convex, its time-slices arise as level sets of a continuous function, and all nearby tangent flows are cylindrical. Moreover, we establish a canonical neighborhood theorem near such points, which characterizes the flow via local models. We also obtain a more uniform version of the Mean Convex Neighborhood Conjecture, which only requires closeness to a cylinder at some initial time and yields a quantitative version of this structural description. Our proof relies on a complete classification of ancient, asymptotically cylindrical flows. We prove that any such flow is non-collapsed, convex, rotationally symmetric, and belongs to one of three canonical families: ancient ovals, the bowl soliton, or the flying wing translating solitons. Central to our method is a refined asymptotic analysis and a novel \emph{leading mode condition,} together with a new ``induction over thresholds'' argument. In addition, our approach provides a full parameterization of the space of asymptotically cylindrical flows and gives a new proof of the existence of flying wing solitons. Our method is independent of prior work and, together with our prequel paper, this work is largely self-contained.

This study proposes a unified stochastic framework for approximating and computing the gradient of every smooth function evaluated at non-independent variables, using $\ell_p$-spherical distributions on $\R^d$ with $d, p\geq 1$. The upper-bounds of the bias of the gradient surrogates do not suffer from the curse of dimensionality for any $p\geq 1$. Also, the mean squared errors (MSEs) of the gradient estimators are bounded by $K_0 N^{-1} d$ for any $p \in [1, 2]$, and by $K_1 N^{-1} d^{2/p}$ when $2 \leq p \ll d$ with $N$ the sample size and $K_0, K_1$ some constants. Taking $\max\left\{2, \log(d) \right\} < p \ll d$ allows for achieving dimension-free upper-bounds of MSEs. In the case where $d\ll p< +\infty$, the upper-bound $K_2 N^{-1} d^{2-2/p}/ (d+2)^2$ is reached with $K_2$ a constant. Such results lead to dimension-free MSEs of the proposed estimators, which boil down to estimators of the traditional gradient when the variables are independent. Numerical comparisons show the efficiency of the proposed approach.

The Kadar--Yu algebras are a physically motivated sequence of towers of algebras interpolating between the Brauer algebras and Temperley--Lieb algebras. The complex representation theory of the Brauer and Temperley--Lieb algebras is now fairly well understood, with each connecting in a different way to Kazhdan--Lusztig theory. The semisimple representation theory of the KY algebras is also understood, and thus interpolates, for example, between the double-factorial and Catalan combinatorial realms. However the non-semisimple representation theory has remained largely open, being harder overall than the (already challenging) Brauer case. In this paper we determine generalised Chebyshev-like forms for the determinants of gram matrices of contravariant forms for standard modules. This generalises the root-of-unity paradigm for Temperley--Lieb algebras (and many related algebras); interpolating in various ways between this and the `integral paradigm' for Brauer algebras. The standard module gram determinants give a huge amount of information about morphisms between standard modules, making thorough use of the powerful homological machinery of towers of recollement (ToR), with appropriate gram determinants providing the ToR `bootstrap'. As for the Brauer and TL cases the representation theory has a strongly alcove-geometric flavour, but the KY cases guide an intriguing generalisation of the overall geometric framework.

The square of a graph $G$, denoted $G^2$, has the same vertex set as $G$ and has an edge between two vertices if the distance between them in $G$ is at most $2$. Thomassen [12] showed that $\chi(G^2) \leq 7$ if $G$ is a subcubic planar graph. A natural question is whether $\chi_{\ell}(G^2) \leq 7$ or not if $G$ is a subcubic planar graph. Recently Kim and Lian [11] showed that $\chi_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 6. And Jin, Kang, and Kim [10] showed that $\chi_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph without 4-cycles and 5-cycles. In this paper, we show that the square of a subcubic planar graph without 5-cycles is 7-choosable, which improves the results of [10] and [11].

We construct a diagrammatic categorification of the spherical module over the Hecke algebra. We establish a basis for the morphism spaces of this category, and prove that it is equivalent to an existing algebraic spherical category.

For a finite group $G$, let $a_n(G)$ be the number of subgroups of order $n$ and define $\zeta_G(s)=\sum_{n\ge 1} a_n(G)n^{-s}$. Examples are known of non-isomorphic finite groups with the same group zeta function. However, no general criterion is known for when two finite groups have the same group zeta function. Fix integers $m,n\ge 1$ and a prime $p$, and consider the metacyclic $p$-groups of split type $G(p,m,n,k)$ defined by $ G(p,m,n,k)=\langle a,b \mid a^{p^{m}}=b^{p^{n}}=\mathrm{id}, b^{-1}ab=a^{k}\rangle$. For fixed $m$ and $n$, we characterize the pairs of parameters $k_1,k_2$ for which $\zeta_{G(p,m,n,k_1)}(s)=\zeta_{G(p,m,n,k_2)}(s)$.

We find a partial extension of Poincaré duality theorem of Jell-Rau-Shaw to hypersurfaces obtained by non-primitive Viro's combinatorial patchworking. We define a classification of the triangulations of a lattice polytope by a level of primitivity and we find a partial Poincaré duality for patchworkings depending on the level of primitivity of the triangulation. Our notion of primitivity is defined modulo a certain integral domain, it is weaker than the classical definition of primitivity. We obtain also a generalization of the complete Poincaré duality over a certain integral domain to hypersurfaces obtained by patchworkings which are primitive modulo this integral domain. In particular, our corollary is that any tropical hypersurface obtained by patchworking from a triangulation of a simple lattice polytope satisfies complete Poincaré duality over the field of rationals, which is a converse of a theorem of Aksnes.

Timesteppers constitute a powerful tool in modern computational science and engineering. Although they are typically used to advance the system forward in time, they can also be viewed as nonlinear mappings that implicitly encode steady states and stability information. In this work, we present an extension of the matrix-free framework for calculating, via timesteppers, steady states of deterministic systems to stochastic particle simulations, where intrinsic randomness prevents direct steady state extraction. By formulating stochastic timesteppers in the language of optimal transport, we reinterpret them as operators acting on probability measures rather than on individual particle trajectories. This perspective enables the construction of smooth cumulative- and inverse-cumulative-distribution-function ((I)CDF) timesteppers that evolve distributions rather than particles. Combined with matrix-free Newton-Krylov solvers, these smooth timesteppers allow efficient computation of steady-state distributions even under high stochastic noise. We perform an error analysis quantifying how noise affects finite-difference Jacobian action approximations, and demonstrate that convergence can be obtained even in high noise regimes. Finally, we introduce higher-dimensional generalizations based on smooth CDF-related representations of particles and validate their performance on a non-trivial two-dimensional distribution. Together, these developments establish a unified variational framework for computing meaningful steady states of both deterministic and stochastic timesteppers.

We construct solutions to the three-dimensional Euler equations exhibiting anomalous dissipation in finite time through a vanishing viscosity limit. Inspired by \cite{BDL23} and \cite{cheskidov2023dissipation}, we extend the \(2\frac{1}{2}\)-dimensional constructions and establish an Onsager-critical energy criterion adapted to such flows, showing its sharpness. Moreover, we provide a fully three-dimensional dissipative Euler example, sharp in Onsager's sense, driven by a slightly rough external force, following the framework of \cite{CL21}.

Lattices induced by coverings arise naturally in matroid theory and combinatorial optimization, providing a structured framework for analyzing relationships between independent sets and closures. In this paper, we explore the structural properties of such lattices, with a particular focus on their rank structure, covering relations, and enumeration of elements per level. Leveraging these structural insights, we investigate necessary and sufficient conditions under which the lattice induced by a covering is isomorphic to classical geometric lattices, including the lattice of partitions, the lattice of subspaces of a vector space over a finite field, and the Dowling lattice. Our results provide a unified framework for comparing these combinatorial structures and contribute to the broader study of lattice theory, matroids, and their applications in combinatorics.

Convolution admits a natural formulation as a functional operation on matrices. Motivated by the functional and entrywise calculi, this leads to a framework in which convolution defines a matrix transform that preserves positivity. Within this setting, we establish results parallel to the classical theories of Pólya--Szegő, Schoenberg, Rudin, Loewner, and Horn in the context of entrywise calculus. The structure of our transform is governed by a Cayley--Hamilton-type theory valid in commutative rings of characteristic zero, together with a novel polynomial-matrix identity specific to convolution. Beyond these analytic aspects, we uncover an intrinsic connection between convolution and the Bruhat order on the symmetric group, illuminating the combinatorial aspect of this functional operation. This work extends the classical theory of entrywise positivity preservers and operator monotone functions into the convolutional setting.

We completely characterize the boundedness of the log-concave-tailed canonical processes. The corresponding new majorizing measure theorem for log-concave-tailed canonical processes is proved using the new tree structure. Moreover, we introduce the new growth condition. Combining this condition with the dual majorizing measure theorem proven in the paper, we have developed a polynomial-time algorithm for computing expected supremum of the log-concave canonical processes.

Let the root of the word $w$ be the smallest prefix $v$ of $w$ such that $w$ is a prefix of $vvv...$. $per(w)$ is the length of the root of $w$. For any $n\ge5$, an $n$-ary threshold word is a word $w$ such that for any factor (subword) $v$ of $w$ the condition $\frac{|v|}{per(v)}\le\frac{n}{n-1}$ holds. Dejean conjecture (completely proven in 2009) states for $n\ge5$ that exists infinitely many of $n$-ary TWs. This manuscript is based on the author's student works (diplomas of 2011 (bachelor's thesis) and 2013 (master's thesis) years) and presents an edited version (in Russian) of these works with some improvements. In a 2011 work proposed new methods of proving of the Dejean conjecture for some odd cases $n\ge5$, using computer verification in polynomial time (depending on $n$). Moreover, the constructed threshold words (TWs) are ciclic/ring TWs (any cyclic shift is a TW). In the 2013 work, the proof method (of 2011) was improved by reducing the verification conditions. A solution for some even cases $n\ge6$ is also proposed. A 2013 work also proposed a method to construct stronger TWs, using a TW tree with regular exponential growth. Namely, the TWs, where all long factors have an exponent close to 1.

We discuss propagation of space-time singularities for the quantum harmonic oscillator with time-dependent metric and potential perturbations. Reformulating the quasi-homogeneous wave front set according to Lascar (1977) in a semiclassical manner, we obtain a characterization of its appearance in comparison with the unperturbed system. The idea of our proof is based on the argument of Nakamura (2009), which was originally devised for the analysis of spatial singularities of the Schrödinger equation, however, the application is non-trivial since the time is no more a parameter, but takes a part in the base variables.

We show that Blokhuis' quadratic upper bound for two-distance sets is sharp over finite fields in almost all dimensions. Our construction complements Lisoněk's higher-dimensional maximal constructions that were carried out in Lorentz spaces.

John E. Littlewood posted the question {\em ``Is it possible in 3-space for seven infinite circular cylinders of unit radius each to touch all the others? Seven is the number suggested by counting constants.''} Bozóki, Lee, and Rónyai constructed a configuration of 7 mutually touching unit cylinders. The best-known upper bounds show that at most 10 unit cylinders in $\mathbb{R}^3$ can mutually touch. We consider this problem in higher dimensions, and obtain exponential (in $d$) upper bounds on the number of mutually touching cylinders in $\mathbb{R}^d$. Our method is fairly flexible, and it makes use of the fact that cylinder touching can be expressed as a combination of polynomial equalities and non-equalities.

A difference set with parameters $(v, k, \lambda)$ is a subset $D$ of cardinality $k$ in a finite group $G$ of order $v$, such that the number $\lambda$ of occurrences of $g \in G$ as the ratio $d^{-1}d'$ in distinct pairs $(d, d')\in D\times D$ is independent of $g$. We prove the nonexistence of $(120, 35, 10)$-difference sets, which has been an open problem for 70 years since Bruck introduced the notion of nonabelian difference sets. Our main tools are 1. a generalization of the category of finite groups to that of association schemes (actually, to that of relation partitions), 2. a generalization of difference sets to equi-distributed functions and its preservation by pushouts along quotients, 3. reduction to a linear programming in the nonnegative integer lattice with quadratic constraints.

We study a variational problem for the Landau--Lifshitz energy with Dzyaloshinskii--Moriya interactions arising in 2D micromagnetics, focusing on the Bogomol'nyi regime. We first determine the minimal energy for arbitrary topological degree, thereby revealing two types of phase transitions consistent with physical observations. In addition, we prove the uniqueness of the energy minimizer in degrees $0$ and $-1$, and nonexistence of minimizers for all other degrees. Finally, we show that the homogeneous state remains stable even beyond the threshold at which the skyrmion loses stability, and we uncover a new stability transition driven by the Zeeman energy.

We study the long-time mixing behavior of the stochastic nonlinear Schrödinger equation in $\mathbb{R}^d$, $d\le 3$. It is well known that, under a sufficiently strong damping force, the system admits unique ergodicity, although the rate of convergence toward equilibrium has remained unknown. In this work, we address the mixing property in the regime of large damping and establish that solutions are attracted toward the unique invariant probability measure at polynomial rates of arbitrary order. Our approach is based on a coupling strategy with pathwise Strichartz estimates.

Motivated by the notion of intermediate dimensions introduced by Falconer et al., we introduce a continuum of topological entropies that are intermediate between the (Bowen) topological entropy and the lower and upper capacity topological entropies. This is achieved by restricting the families of allowable covers in the definition of topological entropy by requiring that the lengths of all strings used in a particular cover satisfy \( N \le n < N/\theta + 1 \), where \( \theta \in [0,1] \) is a parameter. When \( \theta = 1 \), only covers using strings of the same length are allowed, and we recover the lower and upper capacity topological entropies; when \( \theta = 0 \), there are no restrictions, and the definition coincides with the topological entropy. We first establish a quantitative inequality for the upper and lower intermediate topological entropies, which mirrors the corresponding result for intermediate dimensions. As a consequence, the intermediate topological entropies are continuous on $(0,1]$, though discontinuity may arise at $0$; an illustrative example is provided to demonstrate this phenomenon. We then investigate several fundamental properties of the intermediate topological entropies for nonautonomous dynamical systems, including the power rule, monotonicity and product formulas. Finally, we derive an inequality relating intermediate entropies with respect to factor maps.

In this paper, we construct certain rational or integral elements in the motivic cohomology of superelliptic curves which are quotient curves of abelian coverings of $\mathbb{P}^1$ minus $n+2$ points, and prove that these elements are non-trivial by expressing their regulators in terms of Appell-Lauricella hypergeometric functions. We also check that such elements are integral under a mild assumption. We also give various numerical examples for the Beilinson conjecture on special values of $L$-functions of the superelliptic curves by using hypergeometric expressions.

We present the notion of injective hom-complexity, leading to a connection between the covering number of a group and the sectional number of a group homomorphism, and provide estimates for computing this invariant.

This paper studies minimizing solutions to a two dimensional Allen-Cahn system on the upper half plane, subject to Dirichlet boundary conditions, \begin{equation*} \Delta u-\nabla_u W(u)=0, \quad u: \mathbb{R}_+^2\to \mathbb{R}^2,\ u=u_0 \text{ on } \partial \mathbb{R}_+^2, \end{equation*} where $W: \mathbb{R}^2\to [0,\infty)$ is a multi-well potential. We give a complete classification of such half-space minimizing solutions in terms of their blow-down limits at infinity. In addition, we characterize the asymptotic behavior of solutions near the associated sharp interfaces.

We study the unitary almost Mathieu operator (UAMO), a one-dimensional quasi-periodic unitary operator arising from a two-dimensional discrete-time quantum walk on $\mathbb Z^2$ in a homogeneous magnetic field. In the positive Lyapunov exponent regime $0\le \lambda_1<\lambda_2\le 1$, we establish an arithmetic localization statement governed by the frequency exponent $\beta(\omega)$. More precisely, for every irrational $\omega$ with $\beta(\omega)<L$, where $L>0$ denotes the Lyapunov exponent, and every non-resonant phase $\theta$, we prove Anderson localization, i.e. pure point spectrum with exponentially decaying eigenfunctions. This extends our previous arithmetic localization result for Diophantine frequencies (for which $\beta(\omega)=0$) to a sharp threshold in frequency.

This technical report presents a comprehensive study of SDPT3, a widely used open-source MATLAB solver for semidefinite-quadratic-linear programming, which is based on the interior-point method. It includes a self-contained and consistent description of the algorithm, with mathematical notation carefully aligned with the implementation. The aim is to offer a clear and structured reference for researchers and developers seeking to understand or build upon the implementation of SDPT3.

In Part I, we established the construction of the Prequantum Groupoid for simply connected spaces. This second part extends the theory to arbitrary connected parasymplectic diffeological spaces $(\mathrm{X}, \omega)$. We identify the obstruction to the existence of the Prequantum Groupoid as the non-additivity of the integration of the prequantum form $\mathbf{K}\omega$ on the space of loops. By defining a Total Group of Periods $\mathrm{P}_\omega$ directly on the space of paths, which absorbs the periods arising from the algebraic relations of the fundamental group, we construct a Prequantum Groupoid $\mathbf{T}_\omega$ with connected isotropy isomorphic to the torus of periods $\mathrm{T}_\omega = \mathbf{R}/\mathrm{P}_\omega$. Furthermore, we propose that this groupoid $\mathbf{T}_\omega$ constitutes the Quantum System itself. The classical space $\mathrm{X}$ is embedded as the Skeleton of units, surrounded by a Quantum Fog of non-identity morphisms. We prove that the group of automorphisms of the Quantum System is isomorphic to the group of symmetries of the Dynamical System, $\mathrm{Diff}(\mathrm{X}, \omega)$.

Let $\varepsilon>0$ and, for an odd prime $p$, set $$ S_\ell(p):=\sum_{n\le \ell}\left(\frac{n}{p}\right). $$ Define the first-passage time $$ f_\varepsilon(p):=\min\{\ell\ge 1:\ S_\ell(p)<\varepsilon\ell\}. $$ We prove that there exists a constant $c_\varepsilon>0$ such that, as $x\to\infty$, $$ \sum_{p\le x} f_\varepsilon(p)\sim c_\varepsilon \frac{x}{\log x}. $$

In this paper, we investigate the existence and characterization of the value for a two-player zero-sum differential game with symmetric incomplete information on a continuum of initial positions and with signal revelation. Before the game starts, the initial position is chosen randomly according to a probability measure with compact support, and neither player is informed of the chosen initial position. However, they observe a public signal revealing the current state as soon as the trajectory of the dynamics hits a target set. We prove that, under a suitable notion of signal-dependent strategies, the value of the game exists, and the extended value function of the game is the unique viscosity solution of an associated Hamilton-Jacobi-Isaacs equation that satisfies a boundary condition.

This paper develops a discrete data-driven approach for solving the inverse source problem of the wave equation with final time measurements. Focusing on the $L^2$-Tikhonov regularization method, we analyze its convergence under two different noise models, using noisy discrete spatial observations. By exploiting the spectral decomposition of the forward operator and introducing a noise separation technique into the variational framework, we establish error bounds for the reconstructed solution $u$ and the source term $f$ without requiring classical source conditions. Moreover, an expected convergence rate for the source error is derived in a weaker topology. We also extend the analysis to the fully discrete case with finite element discretization, showing that the overall error depends only on the noise level, regularization parameter, time step size, and spatial mesh size. These estimates provide a basis for selecting the optimal regularization parameter in a data-driven manner, without a priori information. Numerical experiments validate the theoretical results and demonstrate the efficiency of the proposed algorithm.

We establish periodic quasiconformal extension theorems for periodic orientation-preserving quasisymmetric self homeomorphisms of quasicircles or quasi-round carpets. As applications, we prove that, if $f$ is a periodic orientation-preserving quasisymmetric self homeomorphism of a quasi-round carpet $S$ of measure zero in $\mathbb{C}$, which has a fixed point in the outer peripheral circle of $S$, then $f$ is the identity on $S$. Moreover, we prove that, if $f$ is a quasisymmetric self homeomorphism of a square carpet $S$ of measure zero in a rectangle ring, which fixes each of the four vertices of the outer peripheral circle of $S$, then $f$ is the identity on $S$. An analogous rigidity problem for the $\mathbb{C}^*$-square carpets is discussed.

We propose a unified four-dimensional (4D) spatiotemporal formulation for time-dependent convection-diffusion problems that preserves underlying physical structures. By treating time as an additional space-like coordinate, the evolution problem is reformulated as a stationary convection-diffusion equation on a 4D space-time domain. Using exterior calculus, we extend this framework to the full family of convection-diffusion problems posed on $H(\textbf{grad})$, $H(\textbf{curl})$, and $H(\text{div})$. The resulting formulation is based on a 4D Hodge-Laplacian operator with a spatiotemporal diffusion tensor and convection field, augmented by a small temporal perturbation to ensure nondegeneracy. This formulation naturally incorporates fundamental physical constraints, including divergence-free and curl-free conditions. We further introduce an exponentially-fitted 4D spatiotemporal flux operator that symmetrizes the convection-diffusion operator and enables a well-posed variational formulation. Finally, we prove that the temporally-perturbed formulation converges to the original time-dependent convection-diffusion model as the perturbation parameter tends to zero.

In this article, we generalize the following problem, which is called the rational angle bisection problem, to the $n$-dimensional space $k^n$ over a subfield $k$ of $\mathbb R$: on the coordinate plane, for which rational numbers $a$ and $b$ are the slopes of the angle bisectors between two lines with slopes $a$ and $b$ rational? First, we give a few characterizations of when the angle bisectors between two lines with direction vectors in $k^n$ have direction vectors in $k^n.$ To find solutions to the problem in the case when $k = \mathbb Q,$ we also give a formula for the integral solutions of $x_1{}^2+\dots +x_n{}^2 = dx_{n+1}{}^2,$ which is a generalization of the negative Pell's equation $x^2-dy^2 = -1,$ where $d$ is a square-free positive integer. Second, by applying the above characterizations, we give a necessary and sufficient condition for the incenter of a given $n$-simplex with $k$-rational vertices to be $k$-rational. On the coordinate plane, we prove that every triangle with $k$-rational vertices and incenter can be obtained by scaling a triangle with $k$-rational side lengths and area, which is a generalization of a Heronian triangle. We also state certain fundamental properties of a few centers of a given triangle with $k$-rational vertices.

In this paper, we study the momentum distribution of an electron gas in a $3$-dimensional torus. The goal is to compute the occupation number of Fourier modes for some trial state obtained through random phase approximation. We obtain the mean-field analogue of momentum distribution formulas for electron gas in [Daniel and Voskov, Phys. Rev. 120, (1960)] in high density limit and [Lam, Phys. Rev. \textbf{3}, (1971)] at metallic density. Our findings are related to recent results obtained independently by Benedikter, Lill and Naidu, and our analysis applies to a general class of singular potentials rather than just the Coulomb case.

Contextual bandits are a central framework for sequential decision-making, with applications ranging from recommendation systems to clinical trials. While nonparametric methods can flexibly model complex reward structures, they suffer from the curse of dimensionality. We address this challenge using a single-index model, which projects high-dimensional covariates onto a one-dimensional subspace while preserving nonparametric flexibility. We first develop a nonasymptotic theory for offline single-index regression for each arm, combining maximum rank correlation for index estimation with local polynomial regression. Building on this foundation, we propose a single-index bandit algorithm and establish its convergence rate. We further derive a matching lower bound, showing that the algorithm achieves minimax-optimal regret independent of the ambient dimension $d$, thereby overcoming the curse of dimensionality. We also establish an impossibility result for adaptation: without additional assumptions, no policy can adapt to unknown smoothness levels. Under a standard self-similarity condition, however, we construct a policy that remains minimax-optimal while automatically adapting to the unknown smoothness. Finally, as the dimension $d$ increases, our algorithm continues to achieve minimax-optimal regret, revealing a phase transition that characterizes the fundamental limits of single-index bandit learning.

Motivated by the work of Baronti et al. [J. Math. Anal. Appl. 252(2000) 124-146], where they defined the supremum of an arithmetic mean of the side lengths of a triangle, summing antipodal points on the unit sphere, we introduce a new geometric constant for Banach spaces, utilizing the Heinz means that interpolate between the geometric and arithmetic means associated with Birkhoff orthogonality. We discuss the bounds in Banach spaces and find the values of constant in Hilbert spaces. We obtain the characterization of uniformly non-square spaces. We investigate the correlation between our notion of the Heinz mean constant and other well-known terms, viz., the modulus of convexity, modulus of smoothness, and rectangular constant. Furthermore, we also give a characterization of the Radon plane with an affine regular hexagonal unit sphere.

We study finite-sum nonlinear programs whose decision variables interact locally according to a graph or hypergraph. We propose MP-Jacobi (Message Passing-Jacobi), a graph-compliant decentralized framework that couples min-sum message passing with Jacobi block updates. The (hyper)graph is partitioned into tree clusters. At each iteration, agents update in parallel by solving a cluster subproblem whose objective decomposes into (i) an intra-cluster term evaluated by a single min-sum sweep on the cluster tree (cost-to-go messages) and (ii) inter-cluster couplings handled via a Jacobi correction using neighbors' latest iterates. This design uses only single-hop communication and yields a convergent message-passing method on loopy graphs. For strongly convex objectives we establish global linear convergence and explicit rates that quantify how curvature, coupling strength, and the chosen partition affect scalability and provide guidance for clustering. To mitigate the computation and communication cost of exact message updates, we develop graph-compliant surrogates that preserve convergence while reducing per-iteration complexity. We further extend MP-Jacobi to hypergraphs; in heavily overlapping regimes, a surrogate-based hyperedge-splitting scheme restores finite-time intra-cluster message updates and maintains convergence. Experiments validate the theory and show consistent improvements over decentralized gradient baselines.

A $t$-fold blocking set of the finite Desarguesian plane $\mathrm{PG}(2,p^n)$, $p$ prime, is a set of points meeting each line of the plane in at least $t$ points. The minimum size of such sets is of interest for numerous reasons; however, even the minimum size of nontrivial blocking sets (i.e. $1$-fold blocking sets not containing a line) in \(\mathrm{PG}(2,p^n)\) is an open question when $n\geq 5$ is odd. For $n>1$ the conjectured lower bound for this size is $(p^n+p^{n(s-1)/s}+1)$, where $p^{n/s}$ is the size of the largest proper subfield of $\mathbb{F}_{p^n}$. Since the union of $t$ pairwise disjoint nontrivial blocking sets is a $t$-fold blocking set, it is conjectured that when $p^{n/s}$ is large enough w.r.t. $t$, then the minimum size of a $t$-fold blocking set in $\mathrm{PG}(2,p^n)$ is $t(p^n+p^{n(s-1)/s}+1)$. If $n$ is even, then the decomposition of the plane into disjoint Baer subplanes gives a $t$-fold blocking set of this size. However, for odd $n$, the existence of such sets is an unsolved problem in most cases. In this paper, we construct $3$-fold blocking sets of conjectured size. These blocking sets are obtained as the disjoint union of three linear blocking sets of Rédei type, and they lie on the same orbit of the projectivity $(x:y:z)\mapsto (z:x:y)$.

We count the maximal lattices over $p$-adic fields and the rational number field. For this, we use the theory of Hecke series for a reductive group over nonarchimedean local fields, which was developed by Andrianov and Hina-Sugano. By treating the Euler factors of the counting Dirichlet series for lattices, we obtain zeta functions of classical groups, which were earlier studied with $p$-adic cone integrals. When our counting series equals the existing zeta functions of groups, we recover the known results in a simple way. Further we obtain some new zeta functions for non-split even orthogonal and odd orthogonal groups.

We show that any holomorphic germ $f \colon (X,x_0) \to (Y,y_0)$ of topological degree $1$ between normal surface singularities can be written as $f=\pi \circ \sigma$, where $\pi \colon Y' \to (Y,y_0)$ is a modification and $\sigma \colon (X,x_0) \to (Y',y_1)$ is a local isomorphism sending $x_0$ to a point $y_1 \in \pi^{-1}(y_0)$. A result by Fantini, Favre and myself guarantees that when $f$ is a selfmap, then $(X,x_0)$ is a sandwiched singularity. We give here an alternative proof based on the construction of the associated Kato surfaces, and valuative dynamics.

Fisher's fiducial probability has recently received renewed attention under the name confidence. In this paper, we reformulate it within an extended-likelihood framework, a representation that helps to resolve many long-standing controversies. The proposed formulation accommodates multi-dimensional parameters and shows how higher-order approximations can be used to refine standard asymptotic confidence statements.

We consider nonlocal operators of the form \begin{equation*} L_t u(x) = \int_{\mathbb{R}^d} \left( u(x+y)-u(x)-\nabla u(x)\cdot y^{(\sigma)} \right) \nu_t(dy), \end{equation*} where $\nu_t$ is a general Lévy measure of order $\sigma \in(0,2)$. We allow this class of Lévy measures to be very singular and impose no regularity assumptions in the time variable. Continuity of the operators and the unique strong solvability of the corresponding nonlocal parabolic equations in $L_p$ spaces are established. We also demonstrate that, depending on the ranges of $\sigma$ and $d$, the operator can or cannot be treated in weighted mixed-norm spaces.

In this paper, we study combinatorial aspects of reduced plane curves known as $\mathscr{M}$-curves. This notation is a natural generalization of maximizing plane curves which are well-known in the theory of algebraic surfaces. We focus here on $\mathscr{M}$-arrangements of conics and lines with ordinary singularities of multiplicity less than five and we provide various numerical constraints on their existence, particularly in terms of their weak combinatorics. Moreover, we study in detail the scenario when our $\mathscr{M}$-arrangements consist of lines and just one conic.

In this paper, for $p> 1 $ and $r \ge 1$ we provide a complete characterization of the positive Borel measures $\mu$ on the unit ball $\B_n$ of $\mathbb {C}^n$ for which the induced Toeplitz operator $T_\mu$ is $r$-summing on the Bergman space $A^{p}$. We prove that the $r$-summing norm of $T_\mu: A^p\to A^p$ is equivalent to $\|\widetilde{\mu}\|_{L^{\kappa}(d\lambda)}$, where $\kappa$ is a positive number determined by $p$ and $r$. As some preliminary, we describe when a Carleson embedding $J_\mu: A^p \to L^q(\mu) (1\le p, q\le 2)$ is $r$-summing, which extends the main result in [B. He, et al, Absolutely summing Carleson embeddings on Bergman spaces, Adv. Math., 439, 109495 (2024)].

We first review the convolution fast-Fourier-transform (CFFT) approach for the numerical solution of backward stochastic differential equations (BSDEs) introduced in (Hyndman and Oyono Ngou, 2017). We then propose a method for improving the boundary errors obtained when valuing options using this approach. We modify the damping and shifting schemes used in the original formulation, which transforms the target function into a bounded periodic function so that Fourier transforms can be applied successfully. Time-dependent shifting reduces boundary error significantly. We present numerical results for our implementation and provide a detailed error analysis showing the improved accuracy and convergence of the modified convolution method.

In this paper, we consider a class of constrained multiobjective optimization problems, where each objective function can be expressed by adding a possibly nonsmooth nonconvex function and a differentiable function with Lipschitz continuous gradient, then subtracting a weakly convex function. This encompasses multiobjective optimization problems involving difference-of-convex (DC) functions, which are prevalent in various applications due to their ability to model nonconvex problems. We first establish necessary and sufficient optimality conditions for these problems, providing a theoretical foundation for algorithm development. Building on these conditions, we propose a proximal subgradient algorithm tailored to the structure of the objectives. Under mild assumptions, the sequence generated by the proposed algorithm is bounded and each of its cluster points is a stationary solution.

We consider products of $n$ random Hermitian matrices which generalize the one-matrix model and show its relation to Hurwitz numbers which count ramified coverings of certain type. Namely, these Hurwitz numbers count $2k$-fold ramified coverings of the Riemann sphere with arbitrary ramification type over $0$ and $\infty$ and ramifications related to the partition $(2^k)$ (``brickworks'' - involution without fixed points) elsewhere. Products of normal random matrices are also considered.

In this paper, we introduce Steklov and Neumann isocapacitary constants for the $p$-Laplacian on compact manifolds. These constants yield two-sided bounds for the $(p,\alpha)$-Sobolev constants, which degenerate to upper and lower bounds for the first nontrivial Steklov and Neumann eigenvalues of the $p$-Laplacian when $\alpha= 1$.

The classical quasi-shuffle algebra for multiple zeta values have a well-known Hopf algebra structure. Recently, the shuffle algebra for multiple zeta values are also equipped with a Hopf algebra structure. This paper shows that these two Hopf algebras are isomorphic.

We establish an invariance principle connecting boundary random walks on $\mathbb N$ with Feller's Brownian motions on $[0,\infty)$. A Feller's Brownian motion is a Feller process on $[0,\infty)$ whose excursions away from the boundary $0$ coincide with those of a killed Brownian motion, while its behavior at the boundary is characterized by a quadruple $(p_1,p_2,p_3,p_4)$. This class encompasses many classical models, including absorbed, reflected, elastic, and sticky Brownian motions, and further allows boundary jumps from $0$ governed by the measure $p_4$. For any Feller's Brownian motion that is not purely driven by jumps at the boundary, we construct a sequence of boundary random walks whose appropriately rescaled processes converge weakly to the given Feller's Brownian motion.

Risk management often plays an important role in decision making under uncertainty. In quantitative risk management, assessing and optimizing risk metrics requires efficient computing techniques and reliable theoretical guarantees. In this paper, we introduce several topics on quantitative risk management and review some of the recent studies and advancements on the topics. We consider several risk metrics and study decision models that involve the metrics, with a main focus on the related computing techniques and theoretical properties. We show that stochastic optimization, as a powerful tool, can be leveraged to effectively address these problems.

Let $F$ be a non-archimedean local field or a finite field. Let $\pi$ be a principal series representation of $GL_{2n}(F)$ induced from any of its maximal standard parabolic subgroups. Let $N$ be the unipotent radical of the maximal parabolic subgroup $P$ of $GL_{2n}(F)$ corresponding to the partition $(n,n).$ In this article, we describe the structure of the twisted Jacquet module $\pi_{N,\psi}$ of $\pi$ with respect to $N$ and a non-degenerate character $\psi$ of $N.$ We also provide a necessary and sufficient condition for $\pi_{N,\psi}$ to be non-zero and show that the twisted Jacquet module is non-zero under certain assumptions on the inducing data. As an application of our results, we obtain the structure of twisted Jacquet modules of certain non-generic irreducible representations of $GL_{2n}(F)$ and establish the existence of their Shalika models. We conclude our article with a conjecture by Dipendra Prasad classifying the smooth irreducible representations of $GL_{2n}(F)$ with a non-zero twisted Jacquet module.

We study locally countable acyclic measure-class-preserving (mcp) Borel graphs by analyzing their "topography" -- the interaction between the geometry and the associated Radon--Nikodym cocycle. We identify three notions of topographic significance for ends in such graphs and show that the number of nonvanishing ends governs both amenability and smoothness. More precisely, we extend the Adams dichotomy from the pmp to the mcp setting, replacing the number of ends with the number of nonvanishing ends: an acyclic mcp graph is amenable if and only if a.e. component has at most two nonvanishing ends, while it is nowhere amenable exactly when a.e. component has a nonempty perfect (closed) set of nonvanishing ends. We also characterize smoothness: an acyclic mcp graph is essentially smooth if and only if a.e. component has no nonvanishing ends. Furthermore, we show that the notion of nonvanishing ends depends only on the measure class and not on the specific measure. At the heart of our analysis lies the study of acyclic countable-to-one Borel functions. Our critical result is that, outside of the essentially two-ended setting, all back ends in a.e. orbit are vanishing and admit cocycle-finite geodesics. We also show that the number of barytropic ends controls the essential number of ends for such functions. This leads to a surprising topographic characterization of when such functions are essentially one-ended. Our proofs utilize mass transport, end selection, and the notion of the Radon--Nikodym core for acyclic mcp graphs, a new concept that serves as a guiding framework for our topographic analysis.

We investigate the structure of the commutative Banach algebra formed as the direct sum of integrable radial functions on the disc and the radial operators on the Bergman space, endowed with the convolution from quantum harmonic analysis as the product. In particular, we study the Gelfand theory of this algebra and discuss certain properties of the appropriate Fourier transform of operators which naturally arises from the Gelfand transform.

In this article, we establish maximal inequalities and deduce ergodic theorems for state-preserving actions of amenable, locally compact, second-countable groups on tracial non-commutative $L^1$-spaces. As a further consequence, in combination with the Neveu decomposition, we obtain a stochastic ergodic theorem for amenable group actions.

Reconfigurable intelligent surfaces (RISs) mounted on unmanned aerial vehicles (UAVs) can reshape wireless propagation on-demand. However, their performance is sensitive to UAV jitter and cascaded channel uncertainty. This paper investigates a downlink multiple-input single-output UAV-mounted RIS system in which a ground multiple-antenna base station (BS) serves multiple single-antenna users under practical impairments. Our goal is to maximize the expected throughput under stochastic three-dimensional UAV jitter and imperfect cascaded channel state information (CSI) based only on the available channel estimates. This leads to a stochastic nonconvex optimization problem subject to a BS transmit power constraint and strict unit-modulus constraints on all RIS elements. To address this problem, we design a model-free deep reinforcement learning (DRL) framework with a contextual bandit formulation. A differentiable feasibility layer is utilized to map continuous actions to feasible solutions, while the reward is a Monte Carlo estimate of the expected throughput. We instantiate this framework with constrained variants of deep deterministic policy gradient (DDPG) and twin delayed deep deterministic policy gradient (TD3) that do not use target networks. Simulations show that the proposed algorithms yield higher throughput than conventional alternating optimization-based weighted minimum mean-square error (AO-WMMSE) baselines under severe jitter and low CSI quality. Across different scenarios, the proposed methods achieve performance that is either comparable to or slightly below the AO-WMMSE benchmark, based on sample average approximation (SAA) with a relative gap ranging from 0-12%. Moreover, the proposed DRL controllers achieve online inference times of 0.6 ms per decision versus roughly 370-550 ms for AO-WMMSE solvers.

The phase reduction technique is essential for studying rhythmic phenomena across various scientific fields. It allows the complex dynamics of high-dimensional oscillatory systems to be expressed by a single phase variable. This paper provides a detailed review and synthesis of phase reduction with two main goals. First, we develop a solid geometric framework for the theory by creating isochrons, which are the level sets of the asymptotic phase, using the Graph Transform theorem. We show that isochrons form an invariant, continuous structure of the basin of attraction of a stable limit cycle, helping to clarify the concept of the asymptotic phase. Second, we systematically explain how to derive the first-order phase reduction for weakly perturbed and coupled systems. In the end, we discuss the limitations of the first-order approach, particularly its restriction to very small perturbations and the issue of vanishing coupling terms in certain networks. We finish by outlining the framework and importance of higher-order phase reductions. This establishes a clear link from classical theory to modern developments and sets the stage for a more in-depth discussion in a future publication.

Muller's ratchet, in its prototype version, models a haploid, asexual population whose size~$N$ is constant over the generations. Slightly deleterious mutations are acquired along the lineages at a constant rate, and individuals carrying less mutations have a selective advantage. In the classical variant, an individual's selective advantage is proportional to the difference between the population average and the individual's mutation load, whereas in the ratchet with {\em tournament selection} only the signs of the differences of the individual mutation loads matter. In a parameter regime which leads to slow clicking (i.e. to a loss of the currently fittest class at a rate $\ll 1/N$) we prove that the rescaled process of click times of the tournament ratchet converges as $N\to \infty$ to a Poisson process. Central ingredients in the proof are a thorough analysis of the metastable behaviour of a two-type Moran model with selection and deleterious mutation (which describes the size of the fittest class up to its extinction time) and a lower estimate on the size of the new fittest class at a clicktime.

Let $k$ be a field with characteristic different from $2$. In this paper, we describe the $k$-rational orbit spaces in some irreducible prehomogeneous vector spaces $(G,V)$ over $k$, where $G$ is a connected reductive algebraic group defined over $k$ and $V$ is an irreducible rational representation of $G$ with a Zariski dense open orbit. We parametrize all composition algebras over the field $k$ in terms of the orbits in some of these representations. This leads to a parametric description of the reduced Freudenthal algebras of dimensions $6$ and $9$ over $k$ (if $\text{char}(k)\neq 2,3$). We also get a parametrization for the involutions of the second kind defined on a central division $K$-algebra $B$ with center $K$, a quadratic extension of the underlying field $k$.

Extended admissible groups belong to a particular class of graphs of groups that admit a decomposition generalizing those of non-geometric 3-manifold groups and Croke-Kleiner admissible groups. In this paper, we study several coarse-geometric aspects of extended admissible groups. We show that changing the gluing edge isomorphisms does not affect the quasi-isometry type of these groups. We also prove that, under mild conditions on the vertex groups, extended admissible groups exhibit large-scale nonpositive curvature, thereby answering a question posed by Nguyen-Yang. As an application, our results enlarge the class of extended admissible groups known to admit well-defined quasi-redirecting boundaries, a notion recently introduced by Qing-Rafi. In addition, we compute the divergence of extended admissible groups, generalizing a result of Gersten from non-geometric 3-manifold groups to this broader setting. Finally, we study several aspects of subgroup structure in extended admissible groups.

We study constant-factor approximation algorithms for the Bin Packing Problem with Setups (BPPS). First, we show that adaptations of classical BPP heuristics can have arbitrarily poor worst-case performance on BPPS instances. Then, we propose a two-phase heuristic for the BPPS that applies an {\alpha}-approximation algorithm for the BPP to the items of each class and then performs a merging phase on the open bins. We prove that this heuristic is a 2 {\alpha}-approximation algorithm for the BPPS.

Let $k$ be a field with $\text{char}(k)\neq 2$. We prove that all maximal flags of composition algebras over $k$, appear as the $k$-rational $Sp_{6}$-orbits in a Zariski-dense $Sp_{6}$-invariant subset $V^{ss}\subset V=\wedge^{3}V_{6}$, where $V_{6}$ is the standard $6$-dimensional irreducible representation of $Sp_{6}$. This gives an arithmetic interpretation for the orbit spaces of the semi-stable sets in the prehomogeneous vector spaces $(Sp_{6}\times GL_{1}^{2},V)$ and $(GSp_{6}\times GL_{1}^{2},V)$. We also get all reduced Freudenthal algebras of dimensions $6$ and $9$, represented by the same orbit spaces.

Let $ G $ be a connected Lie group with real Lie algebra $ \mathfrak{g}$. Suppose $G$ is also a complex manifold. We obtain explicit holomorphic sectional and bisectional curvature formulas of left-invariant strongly pseudoconvex complex Finsler metrics $F$ on $G$ in terms of the complex Lie algebra $\mathfrak{g}^{1,0}$; we also obtain a necessary and sufficient condition for $F$ to be a Kähler-Finsler metric and a weakly Kähler-Finsler metric, respectively. As an application, we obtain the rigidity result: if $F$ is a left-invariant strongly pseudoconvex complex Finsler metric on a complex Lie group $G$, then $F$ must be a complex Berwald metric with vanishing holomorphic bisectional curvature; moreover, $F$ is a Kähler-Berwald metric iff $G$ is an Abelian complex Lie group.

This PhD Thesis consists of two parts. The first part focuses on novel algebraic and geometric approaches to the classification problem of coboundary Lie bialgebras up to Lie algebra automorphisms. More specifically, Grassmann, graded algebra and algebraic invariant techniques are discussed. Using these algebraic methods, equivalence classes of r-matrices for three-dimensional coboundary Lie bialgebras are studied. Moreover, particular higher-dimensional cases, e.g. $\mathfrak{so}(2,2)$ and $\mathfrak{so}(3,2)$, are partially analysed. From the geometric perspective, the main role is played by the newly introduced notion: the Darboux family. This powerful tool allows an efficient and thorough study of equivalence classes of r-matrices for four-dimensional indecomposable coboundary Lie bialgebras. In order to showcase its ability to tackle decomposable examples, $\mathfrak{gl}_2$ is additionally studied. The second part of the Thesis sketches interesting directions for applications of r-matrices. Firstly, it is illustrated how r-matrices might be useful to describe foliated Lie-Hamilton systems. Secondly, the role of r-matrices in deformations of certain cases of Lie systems is discussed. In particular, based on the general procedure for deformations of Lie-Hamilton systems, its extension to Jacobi-Lie systems is suggested and supported by the detailed computation of the deformed Schwarz equation.

In 1871, Saint-Venant introduced the shallow water equations. Since then, the global classical solutions for arbitrary large initial data of the multi-dimensional viscous Saint-Venant system have remained a well-known open problem. It was only recently that [Huang-Meng-Zhang, http:arXiv:2512.15029, 2025], under the assumption of radial symmetry, first proved the existence of global classical solutions for arbitrary large initial data to the initial-boundary value problem of the two-dimensional viscous shallow water equations. At the same time, [Chen-Zhang-Zhu, http:arXiv:2512.18545, 2025] also independently proved the existence of global large solutions to the Cauchy problem of this system. Notably, in the work of Huang-Meng-Zhang, they also established the existence of global classical solutions for arbitrary large initial data to the isentropic compressible Navier-Stokes equations satisfying the BD entropy equality in both two and three dimensions, and the viscous shallow water equations are precisely a specific class of isentropic compressible fluids subject to the BD entropy equality. In this paper, we prove a new BD entropy inequality for a class of non-isentropic compressible fluids, which can be regarded as a generalization of the shallow water equations with transported entropy. Employing new estimates on the lower bound of density different from that of Huang-Meng-Zhang's work, we show the "viscous shallow water system with transport entropy" will admit global classical solutions for arbitrary large initial data to the spherically symmetric initial-boundary value problem in both two and three dimensions. Our results also relax the restrictions on the dimension and adiabatic index imposed in Huang-Meng-Zhang's work on the shallow water equations, extending the range from $N=2,\ \gamma \ge \frac{3}{2}$ to $N=2,\ \gamma > 1$ and $N=3,\ 1<\gamma<3$.

In this article, we define the concept of an $S$-$k$-irreducible ideal and $S$-$k$-maximal ideal in a commutative semiring. We also establish several results concerning $S$-$k$-primary ideals and prove the existence theorem and the $S$-version of the uniqueness theorem using localization, for $S$-$k$-primary decompositions. Also we show that the $S$-radical of every $S$-primary ideal is a prime ideal of $R$. Moreover, we investigate the structure of $S$-primary ideals in principal ideal semidomain and prove that each such ideal can be expressed of the form, $I = (vp^n)$, $n\in \mathbf{N}$ and for some $p \in \mathbf P -\mathbf P_S$ and $v\in R$ such that $(v)\cap S\neq \varnothing $, where $\mathbf P$ is the set of all irreducible (prime) elements of R and for a multiplicative subset $S\subsetneq R$, the set $\mathbf P_S$ defined by $\mathbf P_S=\{p\in \mathbf P : (p) \cap S \neq \varnothing \}$.

In this paper, we study purely discontinuous symmetric Markov processes on closed subsets of ${\mathbb R}^d$, $d\ge 1$, with jump kernels of the form $J(x,y)=|x-y|^{-d-\alpha}{\mathcal B}(x,y)$, $\alpha\in (0,2)$, where the function ${\mathcal B}(x,y)$ may blow up at the boundary of the state space. This extends the framework developed recently for conservative self-similar Markov processes on the upper half-space to a broader geometric setting. Examples of Markov processes that fall into our general framework include traces of isotropic $\alpha$-stable processes in $C^{1,\rm Dini}$ sets, processes in Lipschitz sets arising in connection with the nonlocal Neumann problem, and a large class of resurrected self-similar processes in the closed upper half-space. We establish sharp two-sided heat kernel estimates for these Markov processes. A fundamental difficulty in accomplishing this task is that, in contrast to the existing literature on heat kernels for jump processes, the tails of the associated jump measures in our setting are not uniformly bounded. Thus, standard techniques in the existing literature used to study heat kernels are not applicable. To overcome this obstacle, we employ recently developed weighted functional inequalities specifically designed for jump kernels blowing up at the boundary.

The thin-film equation $\partial_t u = -\nabla \cdot (u^n \nabla \Delta u)$ describes the evolution of the height $u=u(x,t)\geq 0$ of a viscous thin liquid film spreading on a flat solid surface. We prove Hölder continuity of energy-dissipating weak solutions to the thin-film equation in the physically most relevant case of two spatial dimensions $d=2$. While an extensive existence theory of weak solutions to the thin-film equation was established more than two decades ago, even boundedness of weak solutions in $d=2$ has remained a major unsolved problem in the theory of the thin-film equation. Due the fourth-order structure of the thin-film equation, De Giorgi-Nash-Moser theory is not applicable. Our proof is based on the hole-filling technique, the challenge being posed by the degenerate parabolicity of the fourth-order PDE.

We consider a system of four one-dimensional inelastic hard spheres evolving on the real line $\mathbb{R}$, and colliding according to a scattering law characterized by a fixed restitution coefficient $r$. We study the possible orders of collisions when the inelastic collapse occurs, relying on the so-called $\mathfrak{b}$-to-$\mathfrak{b}$ mapping, a two-dimensional dynamical system associated to the original particle system which encodes all the possible collision orders. We prove that the $\mathfrak{b}$-to-$\mathfrak{b}$ mapping is a piecewise projective transformation, which allows one to perform efficient numerical simulations of its orbits. We recover previously known results concerning the one-dimensional four-particle inelastic hard sphere system and we support the conjectures stated in the literature concerning particular periodic orbits. We discover three new families of periodic orbits that coexist depending on the restitution coefficient, we prove rigorously that there exist stable periodic orbits for the $\mathfrak{b}$-to-$\mathfrak{b}$ mapping for restitution coefficients larger than the upper bounds previously known, and we prove the existence of quasi-periodic orbits for this mapping.

In this article, we study how to compute the number of $K$-rational points with a given $j$-invariant on an arbitrary modular curve. As an application, for each positive integer $n$, we determine the list of possible numbers of cyclic $n$-isogenies an elliptic curve over some number field can admit. Similarly, for an odd prime power $p^k$, we calculate the possible values for the number of points above some $j$-invariant on Cartan modular curves $X_{\mathrm s}(p^k)$, $X_{\mathrm{ns}}(p^k)$ and their normalizers. Combining known results about images of Galois representations of CM elliptic curves with our work, we also devise a simple algorithm to determine the number of rational CM points on any modular curve.

We introduce a variant of the Kurepa family. We then use one such family to construct a ccc indestructible property associated with a complete coherent Suslin tree $S$. Moreover, in every ccc forcing extension that preserves Suslin of $S$, forcing with $S$ induces a strong negative partition relation.

In this article, we study higher-order Bol's inequality for radial normal solutions to a singular Liouville equation. By applying these inequalities along with compactness arguments, we derive necessary and sufficient conditions for the existence of radial normal solutions to a singular $Q$-curvature problem. Moreover, under suitable assumptions on the $Q$-curvature, we obtain uniform bounds on the total $Q$-curvature.

We consider the problem of finding the maximum number $e_d(n)$ of pairs of touching circles in a packing of $n$ congruent circles of diameter $d$ in the hyperbolic plane of curvature $-1$. In the Euclidean plane, the maximum comes from a spiral construction of the tiling of the plane with equilateral triangles (Harborth 1974), with a similar result in the hyperbolic plane for the values of $d$ corresponding to the order-$k$ triangular tilings (Bowen 2000). We present various upper and lower bounds for $e_d(n)$ for all values of $d > 0$. In particular, we prove that if $d > 0.66114\dots$ except for $d=0.76217\dots$, then the number of touching pairs is less than the one coming from a spiral construction in the order-$7$ triangular tiling, which we conjecture to be extremal. We also give a lower bound $e_d(n) > (2+\varepsilon_d)n$ where $\varepsilon_d > 1$ for all $d > 0$.

We obtain novel criteria for the existence of local bifurcation for periodic solutions of Hamiltonian systems by a comparison principle of the spectral flow. Our method allows to find the appearance of new solutions by a simple inspection of the coefficients of the system.

In this paper, we develop boundedness estimates for Fourier integral operators on Fourier Lebesgue spaces when the associated canonical relation is parametrised by a complex phase function. Our result constitutes the complex analogue of those obtained for real canonical relations by Rodino, Nicola, and Cordero. We prove that, under the spatial factorization condition of rank $\varkappa$, the corresponding Fourier integral operator is bounded on the Fourier Lebesgue space $\mathcal{F}L^p,$ provided that the order $m$ of the operator satisfies that $ m \leq -\varkappa\left|\frac{1}{p}-\frac{1}{2}\right|, 1 \leq p \leq \infty. $ This condition on the order $m$ is sharp.

This paper is the second in a two-part solution to Almgren's conjecture on the existence of area-minimizing submanifolds with fractal singular sets. In part one, we construct area-minimizing submanifolds with fractal singular sets on certain special manifolds. Here we continue our work and show that area-minimizing submanifolds with fractal singular sets exist on almost any smooth manifold.

We propose a decision-theoretic framework for computational complexity, complementary to classical theory: moving from syntactic exactness (Turing / Shannon) to semantic simulability (Le Cam). While classical theory classifies problems by the cost of exact solution, modern computation often seeks only decision-valid approximations. We introduce a framework where "computation" is viewed as the efficient simulation of a target statistical experiment within a bounded risk distortion (Le Cam deficiency). We formally define computational deficiency ($\delta_{\text{poly}}$) and use it to construct the complexity class LeCam-P (Decision-Robust Polynomial Time), characterizing problems that may be syntactically hard but semantically easy to approximate. We show that classical Karp reductions can be viewed as zero-deficiency simulations, and that approximate reductions correspond to bounded deficiency. Furthermore, we establish the No-Free-Transfer Inequality, showing that strictly invariant representations inevitably destroy decision-relevant information. This framework offers a statistical perspective on approximation theory, bridging the gap between algorithmic complexity and decision theory.

A graph $G$ factors into graphs $H$ and $K$ via a matrix product if $A = BC$, where $A$, $B$, and $C$ are the adjacency matrices of $G$, $H$, and $K$, respectively. The graph $G$ is prime if, in every such factorization, one of the factors is a perfect matching that is, it corresponds to a permutation matrix. We characterize all prime graphs, then using this result we classify all factorable forests, answering a question of Akbari et al. [\emph{Linear Algebra and its Applications} (2025)]. We prove that every torus is factorable, and we characterize all possible factorizations of grids, addressing two questions posed by Maghsoudi et al. [\emph{Journal of Algebraic Combinatorics} (2025)].

To every finite-dimensional $\mathbb C$-algebra $\Lambda$ of finite representation type we associate an affine variety. These varieties are a large generalization of the varieties defined by "$u$ variables" satisfying "$u$-equations", first introduced in the context of open string theory and moduli space of ordered points on the real projective line by Koba and Nielsen, rediscovered by Brown as "dihedral co-ordinates", and recently generalized to any finite type hereditary algebras. We show that each such variety is irreducible and admits a rational parametrization. The assignment is functorial: algebra quotients correspond to monomial maps among the varieties. The non-negative real part of each variety has boundary strata that are controlled by Jasso reduction. These non-negative parts naturally define a generalization of open string integrals in physics, exhibiting factorization and splitting properties that do not come from a worldsheet picture. We further establish a family of Rogers dilogarithm identities extending results of Chapoton beyond the Dynkin case.

In this paper, we propose an efficient numerical approach for solving a specific type of quartic inhomogeneous polynomial optimization problem inspired by practical applications. The primary contribution of this work lies in establishing an inherent equivalence between the quartic inhomogeneous polynomial optimization problem and a multilinear optimization problem (MOP). This result extends the equivalence between fourth-order homogeneous polynomial optimization and multilinear optimization in the existing literature to the equivalence between fourth-order inhomogeneous polynomial optimization and multilinear optimization. By leveraging the multi-block structure embedded within the MOP, a tensor-based proximal alternating minimization algorithm is proposed to approximate the optimal value of the quartic problem. Under mild assumptions, the convergence of the algorithm is rigorously proven. Finally, the effectiveness of the proposed algorithm is demonstrated through preliminary computational results obtained using synthetic datasets.

We provide a complete classification of the homogeneous $3$-local representations of the twin group $T_n$, the virtual twin group $VT_n$, and the welded twin group $WT_n$, for all $n\geq 4$. Beyond this classification, we examine the main characteristics of these representations, particularly their irreducibility and faithfulness. More deeply, we show that all such representations are reducible, and most of them are unfaithful. Also, we find necessary and sufficient conditions of the first two types of the classified representations of $T_n$ to be irreducible in the case $n=4$. The obtained results provide insights into the algebraic structure of these three groups.

We propose a structure-preserving parametric approximation for geometric flows with general anisotropic effects. By introducing a hyperparameter $\alpha$, we construct a unified surface energy matrix $\hat{\boldsymbol{G}}_k^\alpha(\theta)$ that encompasses all existing formulations of surface energy matrices, and apply it to anisotropic curvature flow. We prove that $\alpha=-1$ is the unique choice achieving optimal energy stability under the necessary and sufficient condition $3\hat{\gamma}(\theta)\geq\hat{\gamma}(\theta-\pi)$, while all other $\alpha\neq-1$ require strictly stronger conditions. The framework extends naturally to general anisotropic geometric flows through a unified velocity discretization that ensures energy stability. Numerical experiments validate the theoretical optimality of $\alpha=-1$ and demonstrate the effectiveness and robustness.

We propose a finite volume stochastic collocation method for the random Euler system. We rigorously prove the convergence of random finite volume solutions under the assumption that the discrete differential quotients remain bounded in probability. Convergence analysis combines results on the convergence of a deterministic FV method with stochastic compactness arguments due to Skorokhod and Gyöngy-Krylov.

We describe totally compatible structures on the Jacobson radical of the incidence algebra of a finite poset over a field. We show that such structures are in general non-proper.

We consider commutative Frobenius pseudomonoids in the bicategory of spans, and we show that they are in correspondence with 2-Segal cosymmetric sets. Such a structure can be interpreted as a coherent 2-dimensional topological quantum field theory taking values in the bicategory of spans. We also describe a construction that produces a 2-Segal cosymmetric set from any partial monoid equipped with a distinguished element.

Passive and bistatic radar systems are often limited by strong clutter and direct-path interference that mask weak moving targets. Conventional cancellation methods such as the extensive cancellation algorithm require careful tuning and can distort the delay-Doppler response. This paper introduces a convex optimization framework that adaptively synthesizes per-cell delay-Doppler filters to suppress clutter while preserving the canonical cross-ambiguity function (CAF). The approach formulates a quadratic program that minimizes distortion of the CAF surface subject to linear clutter-suppression constraints, eliminating the need for a separate cancellation stage. Monte Carlo simulations using common communication waveforms demonstrate strong clutter suppression, accurate CFAR calibration, and major detection-rate gains over the classical CAF. The results highlight a scalable, CAF-faithful method for adaptive clutter mitigation in passive radar.

This paper studies the following chemotaxis-fluid system in a two-dimensional bounded domain $\Omega$: \begin{equation*} \begin{cases} n_t + u \cdot \nabla n &= \Delta n - \chi \nabla \cdot \left (n \frac{\nabla c}{c^k} \right ) + r n - \frac{\mu n^2}{\log^\eta(n+e)}, c_t + u \cdot \nabla c &= \Delta c - \alpha c + \beta n, u_t + u \cdot \nabla u &= \Delta u - \nabla P + n \nabla \phi + f, \nabla \cdot u &= 0, \end{cases} \end{equation*} where $r, \mu, \alpha, \beta, \chi$ are positive parameters, $k, \eta \in (0,1)$, $\phi \in W^{2,\infty}(\Omega)$, and $f \in C^1\left(\bar{\Omega}\times [0, \infty)\right) \cap L^\infty\left(\Omega \times (0, \infty)\right)$. We show that, under suitable conditions on the initial data and with no-flux/no-flux/Dirichlet boundary conditions, this system admits a globally bounded classical solution. Furthermore, the system possesses an absorbing set in the topology of $C^0(\bar{\Omega}) \times W^{1, \infty}(\Omega) \times C^0(\bar{\Omega}; \mathbb{R}^2)$.

Optimal control provides a principled framework for transforming dynamical system models into intelligent decision-making, yet classical computational approaches are often too expensive for real-time deployment in dynamic or uncertain environments. In this work, we propose a method based on self-supervised neural operators for open-loop optimal control problems. It offers a new paradigm by directly approximating the mapping from system conditions to optimal control strategies, enabling instantaneous inference across diverse scenarios once trained. We further extend this framework to more complex settings, including dynamic or partially observed environments, by integrating the learned solution operator with Model Predictive Control (MPC). This yields a solution-operator learning method for closed-loop control, in which the learned operator supplies rapid predictions that replace the potentially time-consuming optimization step in conventional MPC. This acceleration comes with a quantifiable price to pay. Theoretically, we derive scaling laws that relate generalization error and sample/model complexity to the intrinsic dimension of the problem and the regularity of the optimal control function. Numerically, case studies show efficient, accurate real-time performance in low-intrinsic-dimension regimes, while accuracy degrades as problem complexity increases. Together, these results provide a balanced perspective: neural operators are a powerful novel tool for high-performance control when hidden low-dimensional structure can be exploited, yet they remain fundamentally constrained by the intrinsic dimensional complexity in more challenging settings.

In this paper, we construct cw-expansive homeomorphisms on compact surfaces of genus greater than or equal to zero with a fixed point whose local stable set is connected but not locally connected. This provides an affirmative answer to question posed by Artigue [3]. To achieve this, we generalize the construction from the example of Artigue, Pacifico and Vieitez [6], obtaining examples of homeomorphisms on compact surfaces of genus greater than or equal to two that are 2-expansive but not expansive. On the sphere and the torus, we construct new examples of cw2-expansive homeomorphisms that are not N -expansive for all N greater than or equal to one.

We obtain some $d\ge2$ such that every graph $G$ with no induced copy of the five-vertex path $P_5$ has chromatic number at most $\omega(G)^d$, thereby resolving an open problem of Gyárfás from 1985. The proof consists of three main ingredients: $\bullet$ an analogue of Rödl's theorem for the chromatic number of $P_5$-free graphs, proved via the ``Gyárfás path'' argument; $\bullet$ a decomposition argument for $P_5$-free graphs that allows one to grow high-chromatic anticomplete pairs indefinitely or to capture a polynomially chromatic-dense induced subgraph; and $\bullet$ a ``chromatic density increment'' argument that uses the Erdős-Hajnal result for $P_5$ as a black box.

We investigate a correspondence between solutions $\lambda(x,y)$ of the Liouville equation \[ \Delta \lambda = -\varepsilon e^{-4\lambda}, \] and the Weierstrass representations of spacelike ($\varepsilon = 1$) and timelike ($\varepsilon = -1$) minimal surfaces with diagonalizable Weingarten map in the three-dimensional Lorentz--Minkowski space $\mathbb{L}^3$. Using complex and paracomplex analysis, we provide a unified treatment of both causal types. We study the action of pseudo-isometries of $\mathbb{L}^3$ on minimal surfaces via Möbius-type transformations, establishing a correspondence between these transformations and rotations in the special orthochronous Lorentz group. Furthermore, we show how local solutions of the Liouville equation determine the Gauss map and the associated Weierstrass data. Finally, we present explicit examples of spacelike and timelike minimal surfaces in $\mathbb{L}^3$ arising from solutions of the Liouville equation.

Let $\nu_1,\nu_2,\dots$ be a sequence of probabilities on the nonnegative integers, and $X=(X_1,X_2, \dots)$ be a sequence of independent random variables $X_i$ with law $\nu_i$. For $\lambda>0$ denote $Z^\lambda_i:= \sum_x \lambda^x\nu_i(x)$ and $\lambda^{\max}:= \sup\{\lambda>0: Z^\lambda_i<\infty \text{ for all }i\}$, and assume $\lambda^{\max}>1$. For $\lambda<\lambda^{\max}$, define the tilted probability $\nu_i^{\lambda}(x):= \lambda^x\nu_i(x)/Z^{\lambda}_i$, and let $X^\lambda$ be a sequence of independent variables $X^\lambda_i$ with law $\nu^{\lambda}_i$, and denote $S^\lambda_n:= X^{\lambda}_1+\dots+X^{\lambda}_n$, with $S_n=S^1_n$. Choose $\lambda^*\in(1,\lambda^{\max})$ and denote $R^*_n:= E (S^{\lambda^*}_n)$. The Gibbs Conditioning Principle (GCP) holds if $P(X\in\cdot|S_n>R^*_n)$ converges weakly to the law of $X^{\lambda^*}$, as $n\to\infty$. We prove the GCP for log-concave $\nu_i$'s, meaning $\nu_i(x+1)\,\nu_i(x-1) \le ( \nu_i(x))^2$, subject to a technical condition that prevents condensation. The canonical measures are the distributions of the first $n$ variables, conditioned on their sum being $k$. Efron's theorem states that for log-concave $\nu_i$'s, the canonical measures are stochastically ordered with respect to $k$. This, in turn, leads to the ordering of the conditioned tilted measures $P(X^\lambda\in\cdot|S^\lambda_n>R^*_n)$ in terms of $\lambda$. This ordering is a fundamental component of our proof.

In this paper, we prove that for every $C^1$ vector field preserving an ergodic hyperbolic invariant measure which is not supported on singularities, if the Oseledec splitting of the ergodic hyperbolic invariant measure is a dominated splitting, then the ergodic hyperbolic invariant measure can be approximated by periodic measures, and the Lyapunov exponents of the ergodic hyperbolic invariant measure can also be approximated by the Lyapunov exponents of those periodic measures.

This work characterizes the general form of a bijective linear map $\Psi:\mathscr{M}_n(\mathbb{C}) \to \mathscr{M}_n(\mathbb{C})$ such that $[\Psi(A_1),~\Psi(A_2)]=D_2$ whenever $[A_1,~A_2]=D_1$ where $D_1~\text{and}~D_2$ are fixed matrices. Additionally, let $\mathscr{H}_1$ and $\mathscr{H}_2$ be the infinite-dimensional complex Hilbert spaces. We characterize the bijective linear map $\Psi: \mathscr{B}(\mathscr{H}_1) \to \mathscr{B}(\mathscr{H}_2)$ where $\Psi(A_1) \circ ~\Psi(A_2)=D_2$ whenever $A_1\circ ~A_2=D_1$ and $D_1~\text{and}~D_2$ are fixed operators.

We describe the spectrum of ordinary Diophantine exponents for $d$-dimensional lattices. The result reduces the problem to two-dimensional case and uses argument of metric theory.

The classical Melan equation modeling suspension bridges is considered. We first study the explicit expression and the uniform positivity of the analytical solution for the simplified ``less stiff'' model, based on which we develop a monotone iterative technique of lower and upper solutions to investigate the existence, uniqueness and approximability of the solution for the original classical Melan this http URL applicability and the efficiency of the monotone iterative technique for engineering design calculations are discussed by verifying some examples of actual bridges. Some open problems are suggested.

We study a continuous time stochastic optimal control problem under partial observations that are available only at discrete time instants. This hybrid setting, with continuous dynamics and intermittent noisy measurements, arises in applications ranging from robotic exploration and target tracking to epidemic control. We formulate the problem on the space of beliefs (information states), treating the controller's posterior distribution of the state as the state variable for decision making. On this belief space we derive a Pontryagin maximum principle that provides necessary conditions for optimality. The analysis carefully tracks both the continuous evolution of the state between observation times and the Bayesian jump updates of the belief at observation instants. A key insight is a relationship between the adjoint process in our maximum principle and the gradient of the value functional on the belief space, which links the optimality conditions to the dynamic programming approach on the space of probability measures. The resulting optimality system has a prediction and update structure that is closely related to the unnormalised Zakai equation and the normalised Kushner-Stratonovich equation in nonlinear filtering. Building on this analysis, we design a particle based numerical scheme to approximate the coupled forward (filter) and backward (adjoint) system. The scheme uses particle filtering to represent the evolving belief and regression techniques to approximate the adjoint, which yields a practical algorithm for computing near optimal controls under partial information. The effectiveness of the approach is illustrated on both linear and nonlinear examples and highlights in particular the benefits of actively controlling the observation process.

We use a recent result of Bader and Sauer on coboundary expansion to prove residually finite three-dimensional Poincaré duality groups never have property (T). This implies such groups are never Kähler. The argument applies to fundamental groups of (possibly non-aspherical) compact 3-manifolds, giving a new proof of a theorem of Fujiwara that states if the fundamental group of a compact 3-manifold has property (T), then that group is finite. The only consequence of geometrization needed in the proof is that 3-manifold groups are residually finite.

We study the Chern-Weil theory for the primitive cohomology of a symplectic manifold. First, given a symplectic manifold, we review the superbundle-valued forms on this manifold and prove a primitive version of the Bianchi identity. Second, as the main result, we prove a transgression formula associated with the boundary map of the primitive cohomology. Third, as an application of the main result, we introduce the concept of primitive characteristic classes and point out a further direction.

Motivated by a problem originating in the study of defect structures in nematic liquid crystals, we describe and study a numerical algorithm for the resolution of a Plateau-like problem. The energy contains the area of a two-dimensional surface $T$ and the length of its boundary $\partial T$ reduced by a prescribed curve to make our problem non-trivial. We additionally include an obstacle $E$ for $T$ and pose a surface energy on $E$. We present an algorithm based on the Alternating Direction Method of Multipliers that minimizes a discretized version of the energy using finite elements, generalizing existing TV-minimization methods. We study different inclusion shapes demonstrating the rich structure of minimizing configurations and provide physical interpretation of our findings for colloidal particles in nematic liquid crystal.

We define a filtration on the variational bicomplex according to jet order. The filtration is preserved by the interior Euler operator, which is not a module homomorphism with respect to the ring of smooth functions on the jet space. However, the induced maps on the graded components of this filtration are. Furthermore, the space of functional forms in the image of the interior Euler operator inherits a filtration. Though the filtered subspaces are not submodules either, the graded components are isomorphic to linear spaces which do have module structures. This works for any fixed degree of the functional forms. In this way, the condition that a functional form vanishes can be stated concisely with a module basis. We work out explicitly two examples: one for functional forms of degree two in relation to the Helmholtz conditions and the other of arbitrary degree but with jet order one.

Let $X$ be a compact complex manifold of dimension $n$ and let $m$ be a positive integer with $m\leq n$. Assume that $X$ admits a Kähler metric $\omega$ and a weakly positive, $\partial\bar\partial$-closed, smooth $(n-m,\,n-m)$-form $\Omega$. We introduce the notions of $(\omega,\,\Omega)$-Hermite-Einstein holomorphic vector bundles and $(\omega,\,\Omega)$(-semi)-stable coherent sheaves on $X$ by generalising the classical definitions depending only on $\omega$. We then prove that the $(\omega,\,\Omega)$-Hermite-Einstein condition implies the $(\omega,\,\Omega)$-semi-stability of a holomorphic vector bundle and its splitting into $(\omega,\,\Omega)$-stable subbundles. This extends a classical result by Kobayashi and Lübke to our generalised setting.

Motivated by the question of defining a $p$-adic string worldsheet action in genus one, we define a Laplacian operator on the Tate curve, and study its Green's function. We show that the Green's function exists. We provide an explicit formula for the Green's function, which turns out to be a non-Archimedean counterpart of the Archimedean Green's function on a flat torus.

We present a data-driven method for spectral analysis of the Koopman operator based on direct construction of the pseudo-resolvent from time-series data. Finite-dimensional approximation of the Koopman operator, such as those obtained from Extended Dynamic Mode Decomposition, are known to suffer from spectral pollution. To address this issue, we construct the pseudo-resolvent operator using the Sherman-Morrison-Woodbury identity whose norm serves as a spectral indicator, and pseudoeigenfunctions are extracted as directions of maximal amplification. We establish convergence of the approximate spectrum to the true spectrum in the Hausdorff metric for isolated eigenvalues, with preservation of algebraic multiplicities, and derive error bounds for eigenvalue approximation. Numerical experiments on pendulum, Lorenz, and coupled oscillator systems demonstrate that the method effectively suppresses spectral pollution and resolves closely spaced spectral components.

We use numerics to construct solitary waves in Dirac--Klein--Gordon (in one and three spatial dimensions) and study the dependence of energy and charge on $\omega$. For the construction, we use the iterative procedure, starting from solitary waves of nonlinear Dirac equation, computing the corresponding scalar field, and adjusting the coupling constant. We also consider the case of massless scalar field, when the iteration procedure could be compared with the shooting method. We use the virial identities to control the error of simulations. We also discuss possible implications from the obtained results for the spectral stability of solitary waves.

Similarity notions between vertices in a graph, such as structural and regular equivalence, are one of the main ingredients in clustering tools in complex network science. We generalise structural and regular equivalences for undirected hypergraphs and provide a characterisation of structural and regular equivalences of undirected graphs and hypergraphs through neighbourhood graphs and Ollivier-Ricci curvature. Our characterisation sheds new light on these similarity notions opening a new avenue for their exploration. These characterisations also enable the construction of a possibly wide family of regular partitions, thereby offering a new route to a task that has so far been computationally challenging.

Let $G$ be a finite group. Let $K/k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C \subseteq \mathrm{Gal}(K/k) \simeq G$ be a conjugacy invariant subset. It is well known that there exists an unramified prime ideal $\mathfrak{p}$ of $k$ with Frobenius element lying in $C$ and norm satisfying $\mathrm{N}\mathfrak{p} \ll |\mathrm{Disc}(K)|^{\alpha}$ for some constant $\alpha = \alpha(G,C)$. There is a rich literature establishing unconditional admissible values for $\alpha$, with most approaches proceeding by studying the zeros of $L$-functions. We give an alternative approach, not relying on zeros, that often substantially improves this exponent $\alpha$ for any fixed finite group $G$, provided $C$ is a union of rational equivalence classes. As a particularly striking example, we prove that there exist absolute constants $c_1,c_2 > 0$ such that for any $n\geq 2$ and any conjugacy class $C \subset S_n$, one may take $\alpha(S_n,C) = c_1 \exp(-c_2n)$. Our approach reduces the core problem to a question in character theory.

We consider the problem of discretizing evolution operators of linear delay equations with the aim of approximating their spectra, which is useful in investigating the stability properties of (nonlinear) equations via the principle of linearized stability. We develop a general convergence analysis based on a reformulation of the operators by means of a fixed-point equation, providing a list of hypotheses related to the regularization properties of the equation and the convergence of the chosen approximation techniques on suitable subspaces. This framework unifies the proofs for some methods based on pseudospectral discretization, which we present here in this new form. To exemplify the generality of the framework, we also apply it to a method of weighted residuals found in the literature, which was previously lacking a formal convergence analysis.

We prove a Cartier duality for gerbes of algebraic and analytic vector bundles as an anti-equivalence of Hopf algebras in the category of kernels of analytic stacks. As an application, we prove that the category of solid quasi-coherent sheaves on the Hodge-Tate stack of a smooth rigid variety over an algebraically closed field $C$ of mixed characteristic $(0,p)$ is equivalent to the category of weight $1$ sheaves on Bhatt-Zhang's Simpson gerbe.

Motivated by the work of Cheng--Fang--Wang--Yu on the hypersingular Bergman projection, we develop a real-variable and dyadic framework for hypersingular operators in regimes where strong-type estimates fail at the critical line. The main new input is a hypersingular sparse domination principle combined with Bourgain's interpolation method, which provides a flexible mechanism to establish critical-line (and endpoint) estimates. In the unit disc setting with $1<t<3/2$, we obtain a full characterization of the $(p,q)$ mapping theory for the dyadic hypersingular maximal operator $\mathcal M_t^{\mathcal D}$, in particular including estimates on the critical line $1/q-1/p=2t-2$ and a weighted endpoint criterion in the radial setting. We also prove endpoint estimates for the hypersingular Bergman projection \[ K_{2t}f(z)=\int_{\mathbb D}\frac{f(w)}{(1-z\overline w)^{2t}}\,dA(w), \] including a restricted weak-type bound at $(p,q)=\bigl(\tfrac{1}{3-2t},1\bigr)$. Finally, we introduce a class of hypersingular cousin of sparse operators in $\mathbb R^n$ associated with \emph{graded} sparse families, quantified by the sparseness $\eta$ and a new structural parameter (the \emph{degree}) $K_{\mathcal S}$, and we characterize the corresponding strong/weak/restricted weak-type regimes in terms of $(n,t,\eta,K_{\mathcal S})$. Our real-variable perspective addresses to an inquiry raised by Cheng--Fang--Wang--Yu on developing effective real-analytic tools in the hypersingular regime for $K_{2t}$, and it also provides a new route toward the critical-line analysis of Forelli--Rudin type operators and related hypersingular operators in both real and complex settings.

Axial algebras of Monster type are a class of non-associative algebras which generalise the Griess algebra, whose automorphism group is the largest sporadic simple group, the Monster. The $2$-generated algebras, which are the building blocks from which all algebras in this class can be constructed, have recently been classified by Yabe; Franchi and Mainardis; and Franchi, Mainardis and McInroy. There are twelve infinite families of examples as well as the exceptional Highwater algebra and its cover, however their properties are not well understood. In this paper, we detail the properties of each of these families, describing their ideals and quotients, subalgebras and idempotents in all characteristics. We also describe all exceptional isomorphisms between them. We give new bases for several of the algebras which better exhibit their axial features and provide code for others to work with them.

We give a proof of Fourier extension conjecture on the paraboloid in all dimensions bigger than 2 that begins with a decomposition suggested in Sawyer [Saw8] of writing a smooth Alpert projection as a sum of pieces whose Fourier extensions are localized. This is then used in the case d=3 to establish the trilinear equivalence of the Fourier extension conjecture given in C. Rios and E. Sawyer [RiSa1] and [RiSa3]. A key aspect of the proof is that the trilinear equivalence only requires an averaging over grids, which converts a difficult exponential sum into an oscillatory integral with periodic amplitude, that is then used to prove the localization on the Fourier side. Finally, we extend this argument to all dimensions bigger than 2 using bilinear analogues of the smooth Alpert trilinear inequalities, which generalize those in Tao, Vargas and Vega [TaVaVe].

We consider classification problems for manifolds and discrete subgroups of Lie groups from a descriptive set-theoretic point of view. This work is largely foundational in conception and character, recording both a framework for general study and Borel complexity computations for some of the most fundamental classes of manifolds. We show, for example, that for all $n\geq 0$, the homeomorphism problem for compact topological $n$-manifolds is Borel equivalent to the relation $=_{\mathbb{N}}$ of equality on the natural numbers, while the homeomorphism problem for noncompact topological $2$-manifolds is of maximal complexity among equivalence relations classifiable by countable structures. A nontrivial step in the latter consists of proving Borel measurable formulations of the Jordan--Schoenflies and surface triangulation theorems. Turning our attention to groups and geometric structures, we show, strengthening results of Stuck--Zimmer and Andretta--Camerlo--Hjorth, that the conjugacy relation on discrete subgroups of any noncompact semisimple Lie group is essentially countable universal. So too, as a corollary, is the isometry relation for complete hyperbolic $n$-manifolds for any $n\geq 2$, generalizing a result of Hjorth--Kechris. We then show that the isometry relation for complete hyperbolic $n$-manifolds with finitely generated fundamental group is, in contrast, Borel equivalent to the equality relation $=_{\mathbb{R}}$ on the real numbers when $n=2$, but that it is not concretely classifiable when $n=3$; thus there exists no Borel assignment of numerical complete invariants to finitely generated Kleinian groups up to conjugacy. We close with a survey of the most immediate open questions.

We introduce \textit{basic inequalities} for first-order iterative optimization algorithms, forming a simple and versatile framework that connects implicit and explicit regularization. While related inequalities appear in the literature, we isolate and highlight a specific form and develop it as a well-rounded tool for statistical analysis. Let $f$ denote the objective function to be optimized. Given a first-order iterative algorithm initialized at $\theta_0$ with current iterate $\theta_T$, the basic inequality upper bounds $f(\theta_T)-f(z)$ for any reference point $z$ in terms of the accumulated step sizes and the distances between $\theta_0$, $\theta_T$, and $z$. The bound translates the number of iterations into an effective regularization coefficient in the loss function. We demonstrate this framework through analyses of training dynamics and prediction risk bounds. In addition to revisiting and refining known results on gradient descent, we provide new results for mirror descent with Bregman divergence projection, for generalized linear models trained by gradient descent and exponentiated gradient descent, and for randomized predictors. We illustrate and supplement these theoretical findings with experiments on generalized linear models.

We prove that the local limit of the weighted spanning trees on any simple connected high degree almost regular sequence of electric networks is the Poisson(1) branching process conditioned to survive forever, by generalizing [NP22] and closing a gap in their proof. We also study the local statistics of the WST's on high degree almost balanced sequences, which is interesting even for the uniform spanning trees. Our motivation comes from studying an interpolation $\{\mathsf{WST}^{\beta}(G)\}_{\beta\in [0, \infty)}$ between UST(G) and MST(G) by WST's on a one-parameter family of random environments. This model has recently been introduced in [MSS24, Kús24], and the phases of several properties have been determined on the complete graphs. We show a phase transition of $\mathsf{WST}^{\beta_n}(G_n)$ regarding the local limit and expected edge overlaps for high degree almost balanced graph sequences $G_n$, without any structural assumptions on the graphs; while the expected total length is sensitive to the global structure of the graphs. Our general framework results in a better understanding even in the case of complete graphs, where it narrows the window of the phase transition of [Mak24].

We present a versatile framework to study strong existence and uniqueness for stochastic differential equations (SDEs) in Hilbert spaces with irregular drift. We consider an SDE in a separable Hilbert space $H$ \begin{equation*} dX_t= (A X_t + b(X_t))dt +(-A)^{-\gamma/2}dW_t,\quad X_0=x_0 \in H, \end{equation*} where $A$ is a self-adjoint negative definite operator with purely atomic spectrum, $W$ is a cylindrical Wiener process, $b$ is $\alpha$-Hölder continuous function $H\to H$, and a nonnegative parameter $\gamma$ such that the stochastic convolution takes values in $H$. We show that this equation has a unique strong solution provided that $\alpha > 2\gamma/(1+\gamma)$. This substantially extends the seminal work of Da Prato and Flandoli (2010) as no structural assumption on $b$ is imposed. To obtain this result, we do not use infinite-dimensional Kolmogorov equations but instead develop a new technique combining Lê's theory of stochastic sewing in Hilbert spaces, Gaussian analysis, and a method of Lasry and Lions for approximation in Hilbert spaces.

We study fixed point biased involutions that avoid a pattern. For every pattern of length three we obtain limit theorems for the asymptotic distribution of the (appropriately centered and scaled) number of fixed points of a random fixed point biased involution avoiding that pattern. When the pattern being avoided is either $321$, $132$, or $213$, we find a phase transition depending on the strength of the bias. We also obtain a limit theorem for distribution of fixed points when the pattern is $123\cdots k(k+1)$ for any $k$ and partial results when the pattern is $(k+1)k\cdots 321$.

The splitting field of an elliptic surface $\mathcal{E}/\mathbb{Q}(t)$ is the smallest finite extension $\mathcal{K} \subset \mathbb{C}$ such that all $\mathbb{C}(t)$-rational points are defined over $\mathcal{K}(t)$. In this paper, we provide a symbolic algorithmic approach to determine the splitting field and a set of $68$ linearly independent generators for the Mordell--Weil lattice of Shioda's elliptic surface $Y^2=X^3 +t^{360} +1$. This surface is noted for having the largest known rank 68 for an elliptic curve over $\mathbb{C}(t)$. Our methodology utilizes the known decomposition of the Mordell-Weil Lattice of this surface into Lattices of ten rational elliptic surfaces and one $K3$ surface. We explicitly compute the defining polynomials of the splitting field, which reach degrees of 1728 and 5760, and verify the results via height pairing matrices and specialized symbolic software packages.

Fix a finite field $\mathbb{F}$. Let $\mathrm{VI}$ be a skeleton of the category of finite dimensional $\mathbb{F}$-vector spaces and injective $\mathbb{F}$-linear maps. We study $\mathrm{VI}^m$-modules over a noetherian commutative ring in the nondescribing characteristic case. We prove that if a finitely generated $\mathrm{VI}^m$-module is generated in degree $\leqslant d$ and related in degree $\leqslant r$, then its regularity is bounded above by a function of $m$, $d$, and $r$. A key ingredient of the proof is a shift theorem for finitely generated $\mathrm{VI}^m$-modules.

How do the geometric properties of a domain impact the spectrum of an operator defined on it? How do we compute accurate and reliable approximations of these spectra? The former question is studied in spectral geometry, and the latter is a central concern in numerical analysis. In this short expository survey we revisit the process of eigenvalue approximation, from the perspective of computational spectral geometry. Over the years a multitude of methods -- for discretizing the operator and for the resultant discrete system -- have been developed and analyzed in the field of numerical analysis. High-accuracy and provably convergent discretization approaches can be used to examine the interplay between the spectrum of an operator and the geometric properties of the spatial domain or manifold it is defined on. While computations have been used to guide conjectures in spectral geometry, in recent years approximation-theoretic tools and validated computations are also being used as part of proof strategies in spectral geometry. Given a particular spectral feature of interest, should we discretize the original problem, or seek a reformulation? Of the many possible approximation strategies, which should we choose? These choices are inextricably linked to the objective: on the one hand, rapid, specialized methods are often ideal for conjecture formulation (prioritizing efficiency and accuracy), whereas schemes with guaranteed, computable error bounds are needed when computation is incorporated into a proof strategy. We also review instances where the demanding requirements of spectral geometry -- the need for rigorous error control or the robust calculation of higher eigenvalues -- motivate new developments in numerical analysis.

In this note, we present a characterization of semistable unitary operators on $L^2(\mathbb{R})$, under the assumption that the operator is (i) translation-invariant, (ii) symmetric, and (iii) locally uniformly continuous (LUC) under dilation. As a consequence, we characterize one-parameter groups formed by such operators, which are of the form $e^{i\beta t|{d}/{dx}|^\alpha}$, with $\alpha,\beta\in\mathbb R$.

The aim of this article is to provide a firm mathematical foundation for the application of deep gradient flow methods (DGFMs) for the solution of (high-dimensional) partial differential equations (PDEs). We decompose the generalization error of DGFMs into an approximation and a training error. We first show that the solution of PDEs that satisfy reasonable and verifiable assumptions can be approximated by neural networks, thus the approximation error tends to zero as the number of neurons tends to infinity. Then, we derive the gradient flow that the training process follows in the ``wide network limit'' and analyze the limit of this flow as the training time tends to infinity. These results combined show that the generalization error of DGFMs tends to zero as the number of neurons and the training time tend to infinity.

Multicommodity capacitated network design (MCND) models can be used to optimize the consolidation of shipments within e-commerce fulfillment networks. In practice, fulfillment networks require that shipments with the same origin and destination follow the same transfer path. This unsplittable flow requirement complicates the MCND problem, requiring integer programming (IP) formulations in which binary variables replace continuous flow variables. To enhance the solvability of this variant of the MCND problem for large-scale logistics networks, this work focuses on strengthening dual bounds. We investigate the polyhedra of arc-set relaxations, and we introduce two new classes of valid inequalities that can be implemented within solution approaches. We develop one approach that dynamically adds valid inequalities to the root node of a reformulation of the MCND IP with additional valid metric inequalities. We show the effectiveness of our ideas with a comprehensive computational study using path-based fulfillment instances, constructed from data provided by a large U.S.-based e-commerce company, and the well-known arc-based Canad instances. Experiments show that our best solution approach for a practical path-based model reduces the IP gap by an average of 26.5% and 22.5% for the two largest instance groups, compared to solving the reformulation alone, demonstrating its effectiveness in improving the dual bound. In addition, experiments using only the arc-based relaxation highlight the strength of our new valid inequalities relative to the linear programming relaxation (LPR), yielding an IP-gap reduction of more than 85%.

This paper develops a discrete theory of real Riemann surfaces based on quadrilateral cellular decompositions (quad-graphs) and a linear discretization of the Cauchy-Riemann equations. We construct a discrete analogue of an antiholomorphic involution and classify the topological types of discrete real Riemann surfaces, recovering the classical results on the number of real ovals and the separation of the surface. Central to our approach is the construction of a symplectic homology basis adapted to the discrete involution. Using this basis, we prove that the discrete period matrix admits the same canonical decomposition $\Pi = \frac{1}{2} H + i T$ as in the smooth setting, where $H$ encodes the topological type and $T$ is purely imaginary. This structural result bridges the gap between combinatorial models and the classical theory of real algebraic curves.

In this article, we introduce a new class of smooth partially proper rigid analytic varieties over a $p$-adic field that satisfy Poincaré duality for étale cohomology with mod $p$-coefficients : the varieties satisfying "primitive comparison with compact support". We show that almost proper varieties, as well as p-adic (weakly admissible) period domains in the sense of Rappoport-Zink belong to this class. In particular, we recover Poincaré duality for almost proper varieties as first established by Li-Reinecke-Zavyalov, and we compute the étale cohomology with $\mathbb{F}_p$-coefficients of p-adic period domains, generalizing a computation of Colmez-Dospinescu-Niziol for Drinfeld's symmetric spaces. The arguments used in this paper rely crucially on Mann's six functors formalism for solid $\mathcal{O}^{+,a}/\pi$ coefficients.

In this paper, we study a multivariate gamma subordinator whose components are independent gamma processes subject to a random time governed by an independent negative binomial process. We derive the explicit expressions for its joint Laplace-Stieltjes transform, its probability density function and the associated governing differential equations. Also, we study a time-changed variant of the multivariate generalized counting process where the time is changed by an independent multivariate gamma subordinator. For this time-changed process, we obtain the corresponding Lévy measure and probability mass function. Later, we discuss an application of the time-changed multivariate generalized counting process to a shock model.

We identify universal polar dual pairs of spherical codes $C$ and $D$ such that for a large class of potential functions $h$ the minima of the discrete $h$-potential of $C$ on the sphere occur at the points of $D$ and vice versa. Moreover, the minimal values of their normalized potentials are equal. These codes arise from the known sharp codes embedded in the even unimodular extremal lattices $E_8$ and $\Lambda_{24}$ (Leech lattice). This embedding allows us to use the lattices' properties to find new universal polar dual pairs. In the process we extensively utilize the interplay between the binary Golay codes and the Leech lattice. As a byproduct of our analysis, we identify a new universally optimal (in the sense of energy) code in the projective space $\mathbb{RP}^{21}$ with $1408$ points (lines). Furthermore, we extend the Delsarte-Goethals-Seidel definition of derived codes from their seminal $1977$ paper and generalize their Theorem 8.2 to show that if a $\tau$-design is enclosed in $k\leq \tau$ parallel hyperplanes, then each of the hyperplane's sub-code is a $(\tau+1-k)$-design in the ambient subspace.

We prove an orbifold type decomposition theorem for the Hochschild homology of the symmetric powers of a small DG category $\mathcal{A}$. In noncommutative geometry, these can be viewed as the noncommutative symmetric quotient stacks of $\mathcal{A}$. We use this decomposition to show that the total Hochschild homology of the symmetric powers of $\mathcal{A}$ is isomorphic to the symmetric algebra $S^*(\mathrm{HH}_\bullet(\mathcal{A}) \otimes t \mathbb{k}[t])$. Our methods are explicit - we construct mutually inverse homotopy equivalences of the standard Hochschild complexes involved. These explicit maps are then used to induce from the symmetric algebra onto the total Hochschild homology the structures of the Fock space for the Heisenberg algebra of $\mathcal{A}$, of a Hopf algebra, and of a free $\lambda$-ring generated by $\mathrm{HH}_\bullet(\mathcal{A})$.

In this paper, we study the observability of compactly perturbed infinite dimensional systems. Assuming that a given infinite-dimensional system with self-adjoint generator is exactly observable we derive sufficient conditions on a compact self adjoint perturbation to guarantee that the perturbed system stays exactly observable. The analysis is based on a careful asymptotic estimation of the spectral elements of the perturbed unbounded operator in terms of the compact perturbation. These intermediate results are of importance themselves.

We introduce a new approach for analyzing ancient solutions and singularities of mean curvature flow that are locally modeled on a cylinder. Its key ingredient is a general mechanism, called the \emph{PDE--ODI principle}, which converts a broad class of parabolic differential equations into systems of ordinary differential inequalities. This principle bypasses many delicate analytic estimates used in previous work, and yields asymptotic expansions to arbitrarily high order. As an application, we establish the uniqueness of the bowl soliton times a Euclidean factor among ancient, cylindrical flows with dominant linear mode. This extends previous results on this problem to the most general setting and is made possible by the stronger asymptotic control provided by our analysis. In the other case, when the quadratic mode dominates, we obtain a complete asymptotic expansion to arbitrary polynomial order, which will form the basis for a subsequent paper. Our framework also recovers and unifies several classical results. In particular, we give new proofs of the uniqueness of tangent flows (due to Colding-Minicozzi) and the rigidity of cylinders among shrinkers (due to Colding-Ilmanen-Minicozzi) by reducing both problems to a single ordinary differential inequality, without using the Łojasiewicz-Simon inequality. Our approach is independent of prior work and the paper is largely self-contained.

We derive bilinear tau forms of the canonically quantized Painlevé equations, thereby relating them to those previously obtained from the $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations for the $\mathcal{N}=2$ supersymmetric gauge theory partition functions on a general $\Omega$-background. We fully fix the refined Painlevé/gauge theory dictionary by formulating the proper equations for the quantum nonautonomous Painlevé Hamiltonians. We also describe the symmetry structure of the quantum Painlevé tau functions and, as a byproduct of this analysis, obtain the $\mathbb{C}^2/\mathbb{Z}_2$ blowup relations in the nontrivial holonomy sector of the gauge theory.

An orthogonal n-frame is an ordered set of n pairwise orthogonal vectors. The set of all orthogonal n-frames in a d-dimensional quadratic vector space is an algebraic variety V(d,n). In this paper, we investigate the variety V(d,n) as well as the quadratic ideal I(d,n) generated by the orthogonality relations, which cuts out V(d,n). We classify the irreducible components of V(d,n), give criteria for the ideal I(d,n) to be prime or a complete intersection, and for the variety V(d,n) to be normal. We also give near-equivalent conditions for V(d,n) to be factorial. Applications are given to the theory of Lovász-Saks-Schrijver ideals.

For complex simulation problems, inferring parameters of scientific interest often precludes the use of classical likelihood-based techniques due to intractable likelihood functions. Simulation-based inference (SBI) methods forego the need for explicit likelihoods by directly utilizing samples from the simulator to learn posterior distributions over parameters $\mathbf{\theta}$ given observed data $\mathbf{x}_{\text{o}}$. Recent work has brought attention to diffusion models -- a type of generative model rooted in score matching and reverse-time stochastic dynamics -- as a flexible framework SBI tasks. This article reviews diffusion-based SBI from first principles to applications in practice. We first recall the mathematical foundations of diffusion modeling (forward noising, reverse-time SDE/ODE, probability flow, and denoising score matching) and explain how conditional scores enable likelihood-free posterior sampling. We then examine where diffusion models address pain points of normalizing flows in neural posterior/likelihood estimation and where they introduce new trade-offs (e.g., iterative sampling costs). The key theme of this review is robustness of diffusion-based SBI in non-ideal conditions common to scientific data: misspecification (mismatch between simulated training data and reality), unstructured or infinite-dimensional observations, and missingness. We synthesize methods spanning foundations drawing from Schrodinger-bridge formulations, conditional and sequential posterior samplers, amortized architectures for unstructured data, and inference-time prior adaptation. Throughout, we adopt consistent notation and emphasize conditions and caveats required for accurate posteriors. The review closes with a discussion of open problems with an eye toward applications of uncertainty quantification for probabilistic geophysical models that may benefit from diffusion-based SBI.

We propose a data-driven approach to identifying the functionally independent invariants that can be constructed from a tensor with a given symmetry structure. Our algorithm proceeds by first enumerating graphs, or tensor networks, that represent inequivalent contractions of a product of tensors, computing instances of these scalars using randomly generated data, and then seeking linear relations between invariants using numerical linear algebra. Such relations yield syzygies, or functional dependencies relating different invariants. We apply this approach in an extended case study of the independent invariants that can be constructed from an antisymmetric $3$-form $H_{\mu \nu \rho}$ in six dimensions, finding five independent invariants. This result confirms that the most general Lagrangian for such a $3$-form, which depends on $H_{\mu \nu \rho}$ but not its derivatives, is an arbitrary function of five variables, and we give explicit formulas relating other invariants to the five independent scalars in this generating set.

In this paper, we model the locations of five major banks in mainland France, two lucrative and three cooperative institutions based on socio-economic considerations. Locations of banks are collected using web scrapping and constitute a bivariate spatial point process for which we estimate nonparametrically summary functions (intensity, Ripley and cross-Ripley's K functions). This shows that the pattern is highly inhomogenenous and exhibits a clustering effect especially at small scales, and thus a significant departure to the bivariate (inhomogeneous) Poisson point process is pointed out. We also collect socio-economic datasets (at the living area level) from INSEE and propose a parametric modelling of the intensity function using these covariates. We propose a group-penalized bivariate composite likelihood method to estimate the model parameters, and we establish its asymptotic properties. The application of the methodology to the banking dataset provides new insights into the specificity of the cooperative model within the sector, particularly in relation to the theories of institutional isomorphism.

The best way to represent generic conformal blocks is provided by the free-field formalism, where they acquire a form of multiple Dotsenko-Fateev-like integrals of the screening operators. Degenerate conformal blocks can be described by the same integrals with special choice of parameters. Integrals satisfy various recurrent relations, which for the special choice of parameters reduce to closed equations. This setting is widely used in explaining the AGT relation, because similar integral representations exist also for Nekrasov functions. We extend this approach to the case of q-Virasoro conformal blocks on elliptic surface -- generic and degenerate. For the generic case, we check equivalence with instanton partition function of a 5d circular quiver gauge theory. For the degenerate case, we check equivalence with partition function of a defect in the same theory, also known as the Shiraishi function. We find agreement in both cases. This opens a way to re-derive the sophisticated equation for the Shiraishi function as the equation for the corresponding integral, what seems straightforward, but remains technically involved and is left for the future.

In this work, families of kinks are analytically identified in multifield theories with either polynomial or deformed sine-Gordon-type potentials. The underlying procedure not only allows us to obtain analytical solutions for these models, but also provides a framework for constructing more general families of field theories that inherit certain analytical information about their solutions. Specifically, this method combines two known field theories into a new composite field theory whose target space is the product of the original target spaces. By suitably coupling the fields through a superpotential defined on the product space, the dynamics in the subspaces become entangled while preserving original kinks as boundary kinks. Different composite field theories are studied, including extensions of well-known models to wider target spaces.

We study the Coulomb-branch sector of 3D $\mathcal{N}=4$ gauge theories with half-hypermultiplets in general pseudoreal representations $\mathbf{R}$ ("noncotangent" theories). This yields (short) quantization of the Coulomb branch and correlators of the Coulomb branch operators captured by the 1d topological sector. This is done by extending the hemisphere partition function approach to noncotangent matter. In this setting one must first cancel the parity anomaly, and overcome an obstacle that $(2,2)$ boundary conditions for half-hypers are generically incompatible with gauge symmetry. Using the Dirichlet boundary conditions for the gauge fields and a careful treatment of half-hypermultiplet boundary data, we describe the resulting shift/difference operators implementing monopole insertions (including bubbling effects) on $HS^3$, and use the $HS^3$ partition function as a natural module on which the Coulomb-branch operator algebra $\mathcal{A}_C$ is represented. As applications we derive generators and relations of $\mathcal{A}_C$ for $SU(2)$ theories with general matter (including half-integer spin representations), analyze theories with Coulomb branch $y^2=z(x^2-1)$, compute the Coulomb branch of an $A_n$ quiver with spin-$\frac32$ half-hypers, and check consistency of a general monopole-antimonopole two-point function.

Generative artificial intelligence (GenAI) is shifting from conversational assistants toward agentic systems -- autonomous decision-making systems that sense, decide, and act within operational workflows. This shift creates an autonomy paradox: as GenAI systems are granted greater operational autonomy, they should, by design, embody more formal structure, more explicit constraints, and stronger tail-risk discipline. We argue stochastic generative models can be fragile in operational domains unless paired with mechanisms that provide verifiable feasibility, robustness to distribution shift, and stress testing under high-consequence scenarios. To address this challenge, we develop a conceptual framework for assured autonomy grounded in operations research (OR), built on two complementary approaches. First, flow-based generative models frame generation as deterministic transport characterized by an ordinary differential equation, enabling auditability, constraint-aware generation, and connections to optimal transport, robust optimization, and sequential decision control. Second, operational safety is formulated through an adversarial robustness lens: decision rules are evaluated against worst-case perturbations within uncertainty or ambiguity sets, making unmodeled risks part of the design. This framework clarifies how increasing autonomy shifts OR's role from solver to guardrail to system architect, with responsibility for control logic, incentive protocols, monitoring regimes, and safety boundaries. These elements define a research agenda for assured autonomy in safety-critical, reliability-sensitive operational domains.

Non-parametric correlation coefficients have been widely used for analysing arbitrary random variables upon common populations, when requiring an explicit error distribution to be known is an unacceptable assumption. We examine an \(\ell_{2}\) representation of a correlation coefficient (Emond and Mason, 2002) from the perspective of a statistical estimator upon random variables, and verify a number of interesting and highly desirable mathematical properties, mathematically similar to the Whitney embedding of a Hilbert space into the \(\ell_{2}\)-norm space. In particular, we show here that, in comparison to the traditional Spearman (1904) \(\rho\), the proposed Kemeny \(\rho_{\kappa}\) correlation coefficient satisfies Gauss-Markov conditions in the presence or absence of ties, thereby allowing both discrete and continuous marginal random variables. We also prove under standard regularity conditions a number of desirable scenarios, including the construction of a null hypothesis distribution which is Student-t distributed, parallel to standard practice with Pearson's r, but without requiring either continuous random variables nor particular Gaussian errors. Simulations in particular focus upon highly kurtotic data, with highly nominal empirical coverage consistent with theoretical expectation.

This paper introduces a novel quasi-likelihood extension of the generalised Kendall \(\tau_{a}\) estimator, together with an extension of the Kemeny metric and its associated covariance and correlation forms. The central contribution is to show that the U-statistic structure of the proposed coefficient \(\tau_{\kappa}\) naturally induces a quasi-maximum likelihood estimation (QMLE) framework, yielding consistent Wald and likelihood ratio test statistics. The development builds on the uncentred correlation inner-product (Hilbert space) formulation of Emond and Mason (2002) and resolves the associated sub-Gaussian likelihood optimisation problem under the \(\ell_{2}\)-norm via an Edgeworth expansion of higher-order moments. The Kemeny covariance coefficient \(\tau_{\kappa}\) is derived within a novel likelihood framework for pairwise comparison-continuous random variables, enabling direct inference on population-level correlation between ranked or weakly ordered datasets. Unlike existing approaches that focus on marginal or pairwise summaries, the proposed framework supports sample-observed weak orderings and accommodates ties without information loss. Drawing parallels with Thurstone's Case V latent ordering model, we derive a quasi-likelihood-based tie model with analytic standard errors, generalising classical U-statistics. The framework applies to general continuous and discrete random variables and establishes formal equivalence to Bradley-Terry and Thurstone models, yielding a uniquely identified linear representation with both analytic and likelihood-based estimators.

We consider the design of mixing matrices to minimize the operation cost for decentralized federated learning (DFL) in wireless networks, with focus on minimizing the maximum per-node energy consumption. As a critical hyperparameter for DFL, the mixing matrix controls both the convergence rate and the needs of agent-to-agent communications, and has thus been studied extensively. However, existing designs mostly focused on minimizing the communication time, leaving open the minimization of per-node energy consumption that is critical for energy-constrained devices. This work addresses this gap through a theoretically-justified solution for mixing matrix design that aims at minimizing the maximum per-node energy consumption until convergence, while taking into account the broadcast nature of wireless communications. Based on a novel convergence theorem that allows arbitrarily time-varying mixing matrices, we propose a multi-phase design framework that activates time-varying communication topologies under optimized budgets to trade off the per-iteration energy consumption and the convergence rate while balancing the energy consumption across nodes. Our evaluations based on real data have validated the efficacy of the proposed solution in combining the low energy consumption of sparse mixing matrices and the fast convergence of dense mixing matrices.

In this paper, we construct a class of stochastic interpolation neural network operators (SINNOs) with random coefficients activated by sigmoidal functions. We establish their boundedness, interpolation accuracy, and approximation capabilities in the mean square sense, in probability, as well as path-wise within the space of second-order stochastic (random) processes \( L^2(\Omega, \mathcal{F},\mathbb{P}) \). Additionally, we provide quantitative error estimates using the modulus of continuity of the processes. These results highlight the effectiveness of SINNOs for approximating stochastic processes with potential applications in COVID-19 case prediction.

Quantum brushes are computational arts software introduced by Ferreira et al (2025) that leverage quantum behavior to generate novel artistic effects. In this outreach paper, we introduce the mathematical framework and describe the implementation of two quantum brushes based on variational quantum algorithms, Steerable and Chemical. While Steerable uses quantum geometric control theory to merge two works of art, Chemical mimics variational eigensolvers for estimating molecular ground energies to evolve colors on an underlying canvas. The implementation of both brushes is available open-source at this https URL and is fully compatible with the original quantum brushes.

The Fleet Size and Mix Vehicle Routing Problem (FSMVRP) is a prominent variant of the Vehicle Routing Problem (VRP), extensively studied in operations research and computational science. FSMVRP requires simultaneous decisions on fleet composition and routing, making it highly applicable to real-world scenarios such as short-term vehicle rental and on-demand logistics. However, these requirements also increase the complexity of FSMVRP, posing significant challenges, particularly in large-scale and time-constrained environments. In this paper, we propose a deep reinforcement learning (DRL)-based approach for solving FSMVRP, capable of generating near-optimal solutions within a few seconds. Specifically, we formulate the problem as a Markov Decision Process (MDP) and develop a novel policy network, termed FRIPN, that seamlessly integrates fleet composition and routing decisions. Our method incorporates specialized input embeddings designed for distinctdecision objectives, including a remaining graph embedding to facilitate effective vehicle employment decisions. Comprehensive experiments are conducted on both randomly generated instances and benchmark datasets. The experimental results demonstrate that our method exhibits notable advantages in terms of computational efficiency and scalability, particularly in large-scale and time-constrained scenarios. These strengths highlight the potential of our approach for practical applications and provide valuable inspiration for extending DRL-based techniques to other variants of VRP.

This paper presents a novel safe control framework that integrates Sliding Mode Control (SMC), High-Order Control Barrier Functions (HOCBFs) with state-dependent adaptiveness and a lightweight projection for collision-free navigation of an over-actuated 3-DOF marine surface vessel subjected to strong environmental disturbances (wind, waves, and current). SMC provides robustness to matched disturbances common in marine operations, while HOCBFs enforce forward invariance of obstacle-avoidance constraints. A fast half-space projection method adjusts the SMC control only when needed, preserving robustness and minimizing chattering. The approach is evaluated on a nonlinear marine platform model that includes added mass, hydrodynamic damping, and full thruster allocation. Simulation results show robust navigation, guaranteed obstacle avoidance, and computational efficiency suitable for real-time embedded use. For small marine robots and surface vessels with limited onboard computational resources-where execution speed and computational efficiency are critical-the SMC-HOCBF framework constitutes a strong candidate for safety-critical control.

For many cellular automata, it is possible to express the state of a given cell after $n$ iterations as an explicit function of the initial configuration. We say that for such rules the solution of the initial value problem can be obtained. In some cases, one can construct the solution formula for the initial value problem by analyzing the spatiotemporal pattern generated by the rule and decomposing it into simpler segments which one can then describe algebraically. We show an example of a rule when such approach is successful, namely elementary rule 156. Solution of the initial value problem for this rule is constructed and then used to compute the density of ones after $n$ iterations, starting from a random initial condition. We also show how to obtain probabilities of occurrence of longer blocks of symbols.

ReLU networks are piecewise-linear (PWL), enabling exact symbolic verification via \SMT(\LRA) or \MILP. However, safety claims in certification pipelines require not only correctness but also \emph{checkable evidence}. We develop a proof-carrying verification core for PWL neural constraints: (i) we formalize ReLU networks as unions of polyhedra indexed by activation patterns; (ii) we present exact \SMT/\MILP encodings and the canonical convex-hull relaxation for each bounded ReLU; and (iii) we introduce a certificate calculus in which bound tightening, stabilization, strengthening, and pruning steps emit explicit algebraic witnesses (LP dual multipliers and Farkas infeasibility certificates). Crucially, these witnesses are \emph{symbolic objects} that admit independent verification in exact arithmetic over $\Q$. We provide a symbolic certificate checker, normalization rules that preserve validity, and a compositional view of region-wise certificates as a global proof artifact for universal safety.

An effective model for describing the relativistic quantum dynamics of a radiating electron is developed via a relativistic generalization of the Lindblad master equation. By incorporating both radiation reaction and vacuum fluctuations into the Dirac equation within an open quantum system framework, our approach captures the Zitterbewegung of the electron, ensuing noncommutativity of its effective spatial coordinates, and provides the quantum analogue of the Landau-Lifshitz (LL) classical equation of motion with radiation reaction. We develop the corresponding phase-space representation via the relativistic Wigner function and derive the semiclassical limit through a Foldy-Wouthuysen transformation. The latter elucidates the signature of quantum vacuum fluctuations in the LL equation, and shows its relationship with the corrected Sokolov equation. Our results offer a robust framework for investigating quantum radiation reaction effects in ultrastrong laser fields.

Tensor networks provide a powerful new framework for classifying and simulating correlated and topological phases of quantum matter. Their central premise is that strongly correlated matter can only be understood by studying the underlying entanglement structure and its associated (generalised) symmetries. In essence, tensor networks provide a compressed, holographic description of the complicated vacuum fluctuations in strongly correlated systems, and as such they break down the infamous many-body exponential wall. These lecture notes provide a concise overview of the most important conceptual, computational and mathematical aspects of this theory.

Many advanced energy harvesting technologies rely on advanced control of thermal emission. Recently, it has been shown that the emissivity and absorptivity of thermal emitters can be controlled independently in nonreciprocal emitters. While significant progress has been made in engineering these nonreciprocal thermal emitters, realizing a highly efficient, pattern-free emitter capable of supporting dual-polarization nonreciprocal emission remains a challenging task. Existing solutions are largely based on metamaterials and exhibit polarization-dependent behavior. This work proposes pattern-free multilayer heterostructures combining magneto-optical and magnetic Weyl semimetal materials and systematically evaluates their nonreciprocal emission performance for p- and s-polarized waves. The findings show that omnidirectional polarization-independent nonreciprocity can be achieved utilizing multilayer structures with different magnetization directions that do not follow simple vector summation. To further enhance the performance, Pareto optimization is employed to tune the key design parameters, enabling the maximization of nonreciprocal thermal emission in a given wavelength range. This approach offers a versatile strategy for designing high-performance thermal emitters tailored for multi-objective optical functionalities.

High-resolution voxel-based micro-finite element ($\mu$FE) models derived from $\mu$CT imaging enable detailed investigation of bone mechanics but remain computationally challenging at anatomically relevant scales. This study presents a comprehensive $\mu$FE framework for large-scale biomechanical analysis of an intact New Zealand White (NZW) rabbit femur, integrating advanced segmentation, scalable finite element solvers, and experimental validation using predominantly open-source libraries. Bone geometries were segmented from $\mu$CT data using the MIA clustering algorithm and converted into voxel-based $\mu$FE meshes, which were solved using the open-source MFEM library with algorithms designed for large-scale linear elasticity systems. The numerical solutions were verified by comparing with a commercial finite element solver, and by evaluating the performance of full assembly and element-by-element formulations within MFEM. Models containing over $8\times10^{8}$ DOFs were solved using moderate HPC resources, demonstrating the feasibility of anatomically realistic $\mu$FE simulations at this scale. Resolution effects were investigated by comparing models with voxel sizes of 20, 40, and 80 $\mu$m, revealing that 40 $\mu$m preserves boundary displacement and principal strain distributions with minimal bias while significantly reducing computational cost. Sensitivity analyses further showed that segmentation parameters influence the global mechanical response. Finally, $\mu$FE predictions were coupled with Digital Image Correlation measurements on an NZW rabbit femur under compression to calibrate effective bone material properties at the micron scale. The results demonstrate that large-scale, experimentally informed $\mu$FE modeling can be achieved using open-source tools, providing a robust foundation for preclinical assessment of bone mechanics and treatment-related risks.

Inverse reinforcement learning (IRL) and dynamic discrete choice (DDC) models explain sequential decision-making by recovering reward functions that rationalize observed behavior. Flexible IRL methods typically rely on machine learning but provide no guarantees for valid inference, while classical DDC approaches impose restrictive parametric specifications and often require repeated dynamic programming. We develop a semiparametric framework for debiased inverse reinforcement learning that yields statistically efficient inference for a broad class of reward-dependent functionals in maximum entropy IRL and Gumbel-shock DDC models. We show that the log-behavior policy acts as a pseudo-reward that point-identifies policy value differences and, under a simple normalization, the reward itself. We then formalize these targets, including policy values under known and counterfactual softmax policies and functionals of the normalized reward, as smooth functionals of the behavior policy and transition kernel, establish pathwise differentiability, and derive their efficient influence functions. Building on this characterization, we construct automatic debiased machine-learning estimators that allow flexible nonparametric estimation of nuisance components while achieving $\sqrt{n}$-consistency, asymptotic normality, and semiparametric efficiency. Our framework extends classical inference for DDC models to nonparametric rewards and modern machine-learning tools, providing a unified and computationally tractable approach to statistical inference in IRL.

Random probability measures, together with their constructions, representations, and associated algorithms, play a central role in modern Bayesian inference. A key class is that of proper species sampling processes, which offer a relatively simple yet versatile framework that extends naturally to non-exchangeable settings. We revisit this class from a computational perspective and show that they admit exact finite mixture representations. In particular, we prove that any proper species sampling process can be written, at the prior level, as a finite mixture with a latent truncation variable and reweighted atoms, while preserving its distributional features exactly. These finite formulations can be used as drop-in replacements in Bayesian mixture models, recasting posterior computation in terms of familiar finite-mixture machinery. This yields straightforward MCMC implementations and tractable expressions, while avoiding ad hoc truncations and model-specific constructions. The resulting representation preserves the full generality of the original infinite-dimensional priors while enabling practical gains in algorithm design and implementation.

We study the interplay between holomorphic conformal field theory and dualities of 3D topological quantum field theories generalizing the paradigm of level-rank duality. A holomorphic conformal field theory with a Kac-Moody subalgebra implies a topological interface between Chern-Simons gauge theories. Upon condensing a suitable set of anyons, such an interface yields a duality between topological field theories. We illustrate this idea using the $c=24$ holomorphic theories classified by Schellekens, which leads to a list of novel sporadic dualities. Some of these dualities necessarily involve condensation of anyons with non-abelian statistics, i.e. gauging non-invertible one-form global symmetries. Several of the examples we discover generalize from $c=24$ to an infinite series. This includes the fact that Spin$(n^{2})_{2}$ is dual to a twisted dihedral group gauge theory. Finally, if $-1$ is a quadratic residue modulo $k$, we deduce the existence of a sequence of holomorphic CFTs at central charge $c=2(k-1)$ with fusion category symmetry given by $\mathrm{Spin}(k)_{2}$ or equivalently, the $\mathbb{Z}_{2}$-equivariantization of a Tambara-Yamagami fusion category.

Semantic communications conveys task-relevant meaning rather than focusing solely on message reconstruction, improving bandwidth efficiency and robustness for next-generation wireless systems. However, learned semantic representations can still leak sensitive information to unintended receivers (eavesdroppers). This paper presents a deep learning-based semantic communication framework that jointly supports multiple receiver tasks while explicitly limiting semantic leakage to an eavesdropper. The legitimate link employs a learned encoder at the transmitter, while the receiver trains decoders for semantic inference and data reconstruction. The security problem is formulated via an iterative min-max optimization in which an eavesdropper is trained to improve its semantic inference, while the legitimate transmitter-receiver pair is trained to preserve task performance while reducing the eavesdropper's success. We also introduce an auxiliary layer that superimposes a cooperative, adversarially crafted perturbation on the transmitted waveform to degrade semantic leakage to an eavesdropper. Performance is evaluated over Rayleigh fading channels with additive white Gaussian noise using MNIST and CIFAR-10 datasets. Semantic accuracy and reconstruction quality improve with increasing latent dimension, while the min-max mechanism reduces the eavesdropper's inference performance significantly without degrading the legitimate receiver. The perturbation layer is successful in reducing semantic leakage even when the legitimate link is trained only for its own task. This comprehensive framework motivates semantic communication designs with tunable, end-to-end privacy against adaptive adversaries in realistic wireless settings.

Dynamic multipliers can be used to guarantee the stability of Lurye systems with slope-restricted nonlinearities, but give no guarantee that the closed-loop system has finite incremental gain. We show that multipliers guarantee the closed-loop power gain to be bounded and quantifiable. Power may be measured about an appropriate steady state bias term, provided the multiplier does not require the nonlinearity to be odd. Hence dynamic multipliers can be used to guarantee such Lurye systems have low sensitivity to noise, provided other exogenous signals have constant steady state. For periodic excitation, the closed-loop response can apparently have a subharmonic or chaotic response. We revisit a class of multipliers that can guarantee a unique, attractive and period-preserving solution. We show the multipliers can be derived using classical tools and reconsider assumptions required for their application. Their phase limitations are inherited from those of discrete-time multipliers. The multipliers cannot be used at all frequencies unless the circle criterion can also be applied; this is consistent with known results about dynamic multipliers and incremental stability.

In this work, we develop a mathematical theory for the photonic Hall effect and prove the existence of guided electromagnetic waves at the interface of two honeycomb photonic crystals. The guided wave resembles the edge states in electronic systems: it is induced by the topological Hall effect, and the wave propagates along the interface but not in the bulk media. Starting from a symmetric honeycomb photonic crystal that attains Dirac points at the high-symmetry points of the Brillouin zone, $K$ and $K'$, we introduce two classes of perturbations for the periodic medium. The perturbations lift the Dirac degeneracy, forming a spectral band valley at the points $K$ and $K'$ with well-defined topological phase that depends on the sign of the perturbation parameters. By employing the layer potential techniques and spectral analysis, we investigate the existence of guided wave along an interface when two honeycomb photonic crystals are glued together. In particular, we elucidate the relationship between the existence of the interface mode and the nature of perturbations imposed on the two periodic media separated by the interface.

This paper presents novel interpretations to the field of classical signal processing of the Wigner-Ville transform as an information measurement tool. The transform's utility in detecting and localizing information-laden signals amidst noisy and cluttered backgrounds, and further providing measure of their information volumes, are detailed herein using Tsallis' entropy and information and related functionals. Example use cases in radio frequency communications are given, where Wigner-Ville-based detection measures can be seen to provide significant sensitivity advantage, for some shown contexts greater than 15~dB advantage, over energy-based measures and without extensive training routines. Such an advantage is particularly significant for applications which have limitations on observation resources including time/space integration pressures and transient and/or feeble signals, where Wigner-Ville-based methods would improve sensing effectiveness by multiple orders of magnitude. The potential for advancement of several such applications is discussed.

This study presents an evaluation of derivative-free optimization algorithms for the direct minimization of Hartree-Fock-Roothaan energy functionals involving nonlinear orbital parameters and quantum numbers with noninteger order. The analysis focuses on atomic calculations employing noninteger Slater-type orbitals. Analytic derivatives of the energy functional are not readily available for these orbitals. Four methods are investigated under identical numerical conditions: Powell's conjugate-direction method, the Nelder-Mead simplex algorithm, coordinate-based pattern search, and a model-based algorithm utilizing radial basis functions for surrogate-model construction. Performance benchmarking is first performed using the Powell singular function, a well-established test case exhibiting challenging properties including Hessian singularity at the global minimum. The algorithms are then applied to Hartree-Fock-Roothaan self-consistent-field energy functionals, which define a highly non-convex optimization landscape due to the nonlinear coupling of orbital parameters. Illustrative examples are provided for closed$-$shell atomic configurations, specifically the He, Be isoelectronic series, with calculations performed for energy functionals involving up to eight nonlinear parameters.

This paper proposes an alternative approach to formally establishing the correctness of the RSA public key cryptosystem. The methodology presented herein deviates slightly from conventional proofs found in existing literature. Specifically, this study explores the conditions under which the choice of the positive integer N, a fundamental component of RSA, can be extended beyond the standard selection criteria. We derive explicit conditions that determine when certain values of N are valid for the encryption scheme and explain why others may fail to satisfy the correctness requirements. The scope of this paper is limited to the mathematical proof of correctness for RSA-like schemes, deliberately omitting issues related to the cryptographic security of RSA.

This paper proposes a novel second-order optimization algorithm. It aims to address large-scale optimization problems in deep learning because it incorporates the OCP method and appropriately approximating the diagonal elements of the Hessian matrix. Extensive experiments on multiple standard visual localization benchmarks demonstrate the significant superiority of the proposed method. Compared with conventional optimiza tion algorithms, our framework achieves competitive localization accuracy while exhibiting faster convergence, enhanced training stability, and improved robustness to noise interference.

The orthogonal time frequency space (OTFS) signal is considered a promising solution for high-mobility wireless environments. It manages Doppler effects by utilizing delay-Doppler (DD) domain processing. However, the relatively long OTFS frame duration could introduce considerable sensing or communication latency when radar and communication are performed separately. By operating in a dual-functional radar and communication (DFRC) mode, the OTFS system performs sensing and data transmission simultaneously, thereby reducing the resulting latency. Nevertheless, the optimal OTFS DFRC signal strategy remains insufficiently explored. This paper investigates the optimal signal design for OTFS DFRC systems, focusing on pilot symbol design and data symbol power allocation. Specifically, we derive a channel capacity lower bound metric for communication that considers channel estimation errors in OTFS. For sensing, we derive an integrated sidelobe level (ISL), accounting for the randomness of the data symbols alongside the deterministic pilot symbols. Leveraging the above metrics, we formulate an optimization problem that balances radar and communication performance, and then solve it using an alternating optimization framework. We validate the proposed signal through numerical analysis and Monte Carlo simulations. Our analysis shows that OTFS DFRC enforces a deterministic pilot signal that is characterized by a concentrated peak in the DD domain, which furnishes a common structure in the DD domain facilitating sensing and channel estimation, with data multiplexed in other DD grids, thereby unifying sensing and communication within a single OTFS signal. Compared with conventional OTFS signals, the proposed OTFS DFRC signal expands the achievable sensing-communication performance region, delivering at least a 9.45 dB ISL suppression for sensing and a 4.82 dB SINR ratio gain for communication.

We present a comparative study of classical numerical solvers, such as Petviashvili's method or finite difference with Newton iterations, and neural network-based methods for computing ground states or profiles of solitary-wave solutions to the one-dimensional dispersive PDEs that include the nonlinear Schrödinger, the nonlinear Klein-Gordon and the generalized KdV equations. We confirm that classical approaches retain high-order accuracy and strong computational efficiency for single-instance problems in the one-dimensional setting. Physics-informed neural networks (PINNs) are also able to reproduce qualitative solutions but are generally less accurate and less efficient in low dimensions than classical solvers due to expensive training and slow convergence. We also investigate the operator-learning methods, which, although computationally intensive during training, can be reused across many parameter instances, providing rapid inference after pretraining, making them attractive for applications involving repeated simulations or real-time predictions. For single-instance computations, however, the accuracy of operator-learning methods remains lower than that of classical methods or PINNs, in general.

We define two new indecomposable (not fully reducible) ${\cal N}=8$, $d=1$ off-shell multiplets and consider the corresponding models of ${\cal N}=8$ supersymmetric mechanics with spin variables. Each multiplet is described off shell by a scalar superfield which is a nonlinear deformation of the standard scalar superfield $X$ carrying the $d=1$ multiplet ${\bf (1,8,7)}$. Deformed systems involve, as invariant subsets, two different off-shell versions of the irreducible multiplet ${\bf (8,8,0)}$. For both systems we present the manifestly ${\cal N}=8$ supersymmetric superfield constraints, as well as the component off- and on-shell invariant actions, which for one version exactly match those given in arXiv:2402.00539 [hep-th]. The two models differ off shell, but prove to be equivalent to each other on shell, with the spin variables sitting in the adjoint representation of the maximal $R$-symmetry group ${\rm SO}(8)$.

We apply methods of equivariant homotopy theory, which may not previously have found due attention in condensed matter physics, to classify first the fragile/unstable topological phases of 2D crystalline Chern insulator materials, and second the possible topological order of their fractional cousins. We highlight that the phases are given by the equivariant 2-Cohomotopy of the Brillouin torus of crystal momenta (with respect to wallpaper point group actions) -- which, despite the attention devoted to crystalline Chern insulators, seems not to have been considered before. Arguing then that any topological order must be reflected in the adiabatic monodromy of gapped quantum ground states over the covariantized space of these band topologies, we compute the latter in examples where this group is non-abelian, showing that any potential FQAH anyons must be localized in momentum space. We close with an outlook on the relevance for the search for topological quantum computing hardware. Mathematical details are spelled out in a supplement.

We study the situation when the T-dual of a toric Kähler geometry is a generalized Kähler geometry involving semi-chiral fields. We explain that this situation is generic for polycylinders, tori and related geometries. Gauging multiple isometries in this case requires the introduction of semi-chiral gauge fields on top of the standard ones. We then apply this technology to the generalized Kähler geometry of the $\eta$-deformed $\mathbb{CP}^{n-1}$ model, relating it to the Kähler geometry of its T-dual.

Large language models (LLMs) have made rapid progress in formal theorem proving, yet current benchmarks under-measure the kind of abstraction and library-mediated reasoning that organizes modern mathematics. In parallel with FATE's emphasis on frontier algebra, we introduce LeanCat, a Lean benchmark for category-theoretic formalization -- a unifying language for mathematical structure and a core layer of modern proof engineering -- serving as a stress test of structural, interface-level reasoning. Part I: 1-Categories contains 100 fully formalized statement-level tasks, curated into topic families and three difficulty tiers via an LLM-assisted + human grading process. The best model solves 8.25% of tasks at pass@1 (32.50%/4.17%/0.00% by Easy/Medium/High) and 12.00% at pass@4 (50.00%/4.76%/0.00%). We also evaluate LeanBridge which use LeanExplore to search Mathlib, and observe consistent gains over single-model baselines. LeanCat is intended as a compact, reusable checkpoint for tracking both AI and human progress toward reliable, research-level formalization in Lean.

We consider eigenfunctions of many-body system Hamiltonians associated with generalized (a-twisted) Cherednik operators used in construction of other Hamiltonians: those arising from commutative subalgebras of the Ding-Iohara-Miki (DIM) algebra. The simplest example of these eigenfunctions is provided by non-symmetric Macdonald polynomials, while generally they are constructed basing on the ground state eigenfunction coinciding with the twisted Baker-Akhiezer function being a peculiar (symmetric) eigenfunction of the DIM Hamiltonians. Moreover, the eigenfunctions admit an expansion with universal coefficients so that the dependence on the twist $a$ is hidden only in these ground state eigenfunctions, and we suggest a general formula that allows one to construct these eigenfunctions from non-symmetric Macdonald polynomials. This gives a new twist in theory of integrable systems, which usually puts an accent on symmetric polynomials, and provides a new dimension to the {\it triad} made from the symmetric Macdonald polynomials, untwisted Baker-Akhiezer functions and Noumi-Shiraishi series.

These lecture notes are a contribution to the proceedings of the school "Geometric, Algebraic and Topological Methods for Quantum Field Theory", held in Villa de Leyva, Colombia, from 31st of July to 9th of August 2023. Its intention is to put together several basic tools of classical integrability and contrast them with those available in the formulation of asymptotic symmetries and the definition of canonical charges in gauge theories. We consider as a working example the Chern-Simons theory in 3D dimensions, motivated by its various applications in condensed matter physics, gravity, and black hole physics. We review basic aspects of the canonical formulation, symplectic geometry, Liouville integrability, and Lax Pairs. We define the Hamiltonian formulation of the Chern-Simons action and the canonical generators of the gauge symmetries, which are surface integrals that subject to non-trivial boundary conditions, realize transformations that do change the physical state, namely large (or improper gauge transformations). We propose asymptotic conditions that realize an infinite set of abelian conserved charges associated with integral models. We review two different cases: the Korteweg-de Vries equation for its connection with the Virasoro algebra and fluid dynamics, and the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, as it embeds an infinite class of non-linear notable integrable evolution equations. We propose a concrete example for gravity in 3D with $\Lambda<0$, where we find a near-horizon asymptotic dynamics. We finalize offering some insights on the initial value problem, its connection with integrable systems and flat connections. We study some properties of the Monodromy matrix and recover the infinite KdV charges from the trace invariants extracted from the Monodromy evolution equation that can be written in a Lax form.

Erdős and Lov'asz asked whether there exists a "3-critical" 3-uniform hypergraph in which every vertex has degree at least 7. The original formulation does not specify what 3-critical means, and two non-equivalent notions have appeared in the literature and in later discussions of the problem. In this paper we resolve the question under both interpretations. For the transversal interpretation (criticality with respect to the transversal number), we prove that a 3-uniform hypergraph $H$ with $\tau(H)=3$ and $\tau(H-e)=2$ for every edge $e$ has at most 10 edges; in particular, $\delta(H)\le 6$, and this bound is sharp, witnessed by the complete 3-graph $K^{(3)}_5$. For the chromatic interpretation (criticality with respect to weak vertex-colourings), we give an explicit 3-uniform hypergraph on 9 vertices with $\chi(H)=3$ and minimum degree $\delta(H)=7$ such that deleting any single edge or any single vertex makes it 2-colourable. The criticality of the example is certified by explicit witness 2-colourings listed in the appendices, together with a short verification script.

Multi-agent target tracking in the presence of nonlinear dynamics and agent heterogeneity, where state-space dimensions may differ, is a challenging problem that traditional graph Laplacian methods cannot easily address. This work leverages the framework of cellular sheaves, a mathematical generalization of graph theory, to natively model such heterogeneous systems. While existing coordination sheaf frameworks focus on cooperative problems like consensus, this work extends them to the non-cooperative target-tracking problem. The tracking of multiple, unknown targets is formulated as a harmonic extension problem on a cellular sheaf, accommodating nonlinear dynamics and external disturbances for all agents. A decentralized control law is developed using the sheaf Laplacian, and a corresponding Lyapunov-based stability analysis is provided to guarantee tracking error convergence, with results validated by simulation.

Drifts of asset returns are notoriously difficult to model accurately and, yet, trading strategies obtained from portfolio optimization are very sensitive to them. To mitigate this well-known phenomenon we study robust growth-optimization in a high-dimensional incomplete market under drift uncertainty of the asset price process $X$, under an additional ergodicity assumption, which constrains but does not fully specify the drift in general. The class of admissible models allows $X$ to depend on a multivariate stochastic factor $Y$ and fixes (a) their joint volatility structure, (b) their long-term joint ergodic density and (c) the dynamics of the stochastic factor process $Y$. A principal motivation of this framework comes from pairs trading, where $X$ is the spread process and models with the above characteristics are commonplace. Our main results determine the robust optimal growth rate, construct a worst-case admissible model and characterize the robust growth-optimal strategy via a solution to a certain partial differential equation (PDE). We demonstrate that utilizing the stochastic factor leads to improvement in robust growth complementing the conclusions of the previous study by Itkin et. al. (arXiv:2211.15628 [q-fin.MF], forthcoming in $\textit{Finance and Stochastics}$), which additionally robustified the dynamics of the stochastic factor leading to $Y$-independent optimal strategies. Our analysis leads to new financial insights, quantifying the improvement in growth the investor can achieve by optimally incorporating stochastic factors into their trading decisions. We illustrate our theoretical results on several numerical examples including an application to pairs trading.

Persistent homology (PH) has recently emerged as a powerful tool for extracting topological features. Integrating PH into machine learning and deep learning models enhances topology awareness and interpretability. However, most PH methods on graphs rely on a limited set of filtrations, such as degree-based or weight-based filtrations, which overlook richer features like recurring information across the dataset and thus restrict expressive power. In this work, we propose a novel graph filtration called Frequent Subgraph Filtration (FSF), which is derived from frequent subgraphs and produces stable and information-rich frequency-based persistent homology (FPH) features. We study the theoretical properties of FSF and provide both proofs and experimental validation. Beyond persistent homology itself, we introduce two approaches for graph classification: an FPH-based machine learning model (FPH-ML) and a hybrid framework that integrates FPH with graph neural networks (FPH-GNNs) to enhance topology-aware graph representation learning. Our frameworks bridge frequent subgraph mining and topological data analysis, offering a new perspective on topology-aware feature extraction. Experimental results show that FPH-ML achieves competitive or superior accuracy compared with kernel-based and degree-based filtration methods. When integrated into graph neural networks, FPH yields relative performance gains ranging from 0.4 to 21 percent, with improvements of up to 8.2 percentage points over GCN and GIN backbones across benchmarks.

Higher-order ODE solvers have become a standard tool for accelerating diffusion probabilistic model (DPM) sampling, motivating the widespread view that first-order methods are inherently slower and that increasing discretization order is the primary path to faster generation. This paper challenges this belief and revisits acceleration from a complementary angle: beyond solver order, the placement of DPM evaluations along the reverse-time dynamics can substantially affect sampling accuracy in the low-neural function evaluation (NFE) regime. We propose a novel training-free, first-order sampler whose leading discretization error has the opposite sign to that of DDIM. Algorithmically, the method approximates the forward-value evaluation via a cheap one-step lookahead predictor. We provide theoretical guarantees showing that the resulting sampler provably approximates the ideal forward-value trajectory while retaining first-order convergence. Empirically, across standard image generation benchmarks (CIFAR-10, ImageNet, FFHQ, and LSUN), the proposed sampler consistently improves sample quality under the same NFE budget and can be competitive with, and sometimes outperform, state-of-the-art higher-order samplers. Overall, the results suggest that the placement of DPM evaluations provides an additional and largely independent design angle for accelerating diffusion sampling.

Imagine you walk in a plane. You move by making a step of a certain length per time interval in a chosen direction. Repeating this process by randomly sampling step length and turning angle defines a two-dimensional random walk in what we call comoving frame coordinates. This is precisely how Ross and Pearson proposed to model the movements of organisms more than a century ago. Decades later their concept was generalised by including persistence leading to a correlated random walk, which became a popular model in Movement Ecology. In contrast, Langevin equations describing cell migration and used in active matter theory are typically formulated by position and velocity in a fixed Cartesian frame. In this article, we explore the transformation of stochastic Langevin dynamics from the Cartesian into the comoving frame. We show that the Ornstein-Uhlenbeck process for the Cartesian velocity of a walker can be transformed exactly into a stochastic process that is defined self-consistently in the comoving frame, thereby profoundly generalising correlated random walk models. This approach yields a general conceptual framework how to transform stochastic processes from the Cartesian into the comoving frame. Our theory paves the way to derive, invent and explore novel stochastic processes in the comoving frame for modelling the movements of organisms. It can also be applied to design novel stochastic dynamics for autonomously moving robots and drones.

Biological systems encode function not primarily in steady states, but in the structure of transient responses elicited by time-varying stimuli. Overshoots, biphasic dynamics, adaptation kinetics, fold-change detection, entrainment, and cumulative exposure effects often determine phenotypic outcomes, yet are poorly captured by classical steady-state or dose-response analyses. This paper develops an input-output perspective on such "dynamic phenotypes," emphasizing how qualitative features of transient behavior constrain underlying network architectures independently of detailed parameter values. A central theme is the role of sign structure and interconnection logic, particularly the contrast between monotone systems and architectures containing antagonistic pathways. We show how incoherent feedforward (IFF) motifs provide a simple and recurrent mechanism for generating non-monotonic and adaptive responses across multiple levels of biological organization, from molecular signaling to immune regulation and population dynamics. Conversely, monotonicity imposes sharp impossibility results that can be used to falsify entire classes of models from transient data alone. Beyond step inputs, we highlight how periodic forcing, ramps, and integral-type readouts such as cumulative dose responses offer powerful experimental probes that reveal otherwise hidden structure, separate competing motifs, and expose invariances such as fold-change detection. Throughout, we illustrate how control-theoretic concepts, including monotonicity, equivariance, and input-output analysis, can be used not as engineering metaphors, but as precise mathematical tools for biological model discrimination. Thus we argue for a shift in emphasis from asymptotic behavior to transient and input-driven dynamics as a primary lens for understanding, testing, and reverse-engineering biological networks.

We investigate BPS states in 4d N=4 supersymmetric Yang-Mills theory and the corresponding (p, q) string networks in Type IIB string theory. We propose a new interpretation of the algebra of line operators in this theory as a tensor product of vector representations of a quantum toroidal algebra, which determines protected spin characters of all framed BPS states. We identify the SL(2,Z)-noninvariant choice of the coproduct in the quantum toroidal algebra with the choice of supersymmetry subalgebra preserved by the BPS states and interpret wall crossing operators as Drinfeld twists of the coproduct. Kontsevich-Soibelman spectrum generator is then identified with Khoroshkin-Tolstoy universal R-matrix.

On-shell diagrams are gauge invariant quantities which play an important role in the description of scattering amplitudes. Based on the principles of generalized unitarity, they are given by products of elementary three-point amplitudes where the kinematics of internal on-shell legs are determined by cut conditions. In the ${\cal N}=4$ Super Yang-Mills (SYM) theory, the dual formulation for on-shell diagrams produces the same quantities as canonical forms on the Grassmannian $G(k,n)$. Most of the work in this direction has been devoted to the planar diagrams, which dominate in the large $N$ limit of gauge theories. On the mathematical side, planar on-shell diagrams correspond to cells of the positive Grassmannian $G_+(k,n)$ which have been very extensively studied in the literature in the past 20 years. In this paper, we focus on the non-planar on-shell diagrams which are relevant at finite $N$. In particular, we use the triplet formulation of Maximal-Helicity-Violating (MHV) on-shell diagrams to obtain certain regions in the Grassmannian $G(2,n)$. These regions are unions of positive Grassmannians with different orderings (referred to as oriented regions). We explore the features of these unions, and show that they are pseudo-positive geometries, in contrast to positive geometry of a single oriented region. For all non-planar diagrams which are \emph{internally planar} there always exists a strongly connected geometry, and for those that are \emph{irreducible}, there exists a geometry with no spurious facets. We also prove that the already known identity moves, square and sphere moves, form the complete set of identity moves for all MHV on-shell diagrams.

A Detector Response Matrix (DRM) is a discrete representation of an instrument's Detector Response Function (DRF), which quantifies how many discrete energy depositions occur in a detector volume for a given distribution of particles incident on the detector. For simple radiation detectors that can count such energy depositions (such as scintillators, Proportional Counter Tubes (PCTs), etc), we consider the ideal counting DRF, $\mathbf{G}_\varphi (E_\mathrm{in}, E_\mathrm{dep})$, which relates the detector's counting histogram (number of energy depositions within a given channel) to an incident particles characterization, $\varphi$ (e.g. incident flux, fluence, intensity). From the counting DRF we can derive the counting DRM, the effective area, and the flash effective area (which measures the total energy deposited in the detector from a large, instantaneous fluence).

We introduce a Modewise Additive Factor Model (MAFM) for matrix-valued time series that captures row-specific and column-specific latent effects through an additive structure, offering greater flexibility than multiplicative frameworks such as Tucker and CP factor models. In MAFM, each observation decomposes into a row-factor component, a column-factor component, and noise, allowing distinct sources of variation along different modes to be modeled separately. We develop a computationally efficient two-stage estimation procedure: Modewise Inner-product Eigendecomposition (MINE) for initialization, followed by Complement-Projected Alternating Subspace Estimation (COMPAS) for iterative refinement. The key methodological innovation is that orthogonal complement projections completely eliminate cross-modal interference when estimating each loading space. We establish convergence rates for the estimated factor loading matrices under proper conditions. We further derive asymptotic distributions for the loading matrix estimators and develop consistent covariance estimators, yielding a data-driven inference framework that enables confidence interval construction and hypothesis testing. As a technical contribution of independent interest, we establish matrix Bernstein inequalities for quadratic forms of dependent matrix time series. Numerical experiments on synthetic and real data demonstrate the advantages of the proposed method over existing approaches.

We consider the extent to which we can learn from a completely randomized experiment whether everyone has treatment effects that are weakly of the same sign, a condition we call monotonicity. From a classical sampling perspective, it is well-known that monotonicity is untestable. By contrast, we show from the design-based perspective -- in which the units in the population are fixed and only treatment assignment is stochastic -- that the distribution of treatment effects in the finite population (and hence whether monotonicity holds) is formally identified. We argue, however, that the usual definition of identification is unnatural in the design-based setting because it imagines knowing the distribution of outcomes over different treatment assignments for the same units. We thus evaluate the informativeness of the data by the extent to which it enables frequentist testing and Bayesian updating. We show that frequentist tests can have nontrivial power against some alternatives, but power is generically limited. Likewise, we show that there exist (non-degenerate) Bayesian priors that never update about whether monotonicity holds. We conclude that, despite the formal identification result, the ability to learn about monotonicity from data in practice is severely limited.

Many applications involve estimating the mean of multiple binomial outcomes as a common problem -- assessing intergenerational mobility of census tracts, estimating prevalence of infectious diseases across countries, and measuring click-through rates for different demographic groups. The most standard approach is to report the plain average of each outcome. Despite simplicity, the estimates are noisy when the sample sizes or mean parameters are small. In contrast, the Empirical Bayes (EB) methods are able to boost the average accuracy by borrowing information across tasks. Nevertheless, the EB methods require a Bayesian model where the parameters are sampled from a prior distribution which, unlike the commonly-studied Gaussian case, is unidentified due to discreteness of binomial measurements. Even if the prior distribution is known, the computation is difficult when the sample sizes are heterogeneous as there is no simple joint conjugate prior for the sample size and mean parameter. In this paper, we consider the compound decision framework which treats the sample size and mean parameters as fixed quantities. We develop an approximate Stein's Unbiased Risk Estimator (SURE) for the average mean squared error given any class of estimators. For a class of machine learning-assisted linear shrinkage estimators, we establish asymptotic optimality, regret bounds, and valid inference. Unlike existing work, we work with the binomials directly without resorting to Gaussian approximations. This allows us to work with small sample sizes and/or mean parameters in both one-sample and two-sample settings. We demonstrate our approach using three datasets on firm discrimination, education outcomes, and innovation rates.

We formulate the covariant hydrodynamics equations describing the fluid dynamics as the problem of intersection theory on the infinite dimensional symplectic manifold associated with spacetime. This point of view separates the structures related to the equation of state, the geometry of spacetime, and structures related to the (differential) topology of spacetime. We point out a five-dimensional origin of the formalism of Lichnerowicz and Carter. Our formalism also incorporates the chiral anomaly and Onsager quantization. We clarify the relation between the canonical velocity and Landau $4$-velocity, the meaning of Kelvin's theorem. Finally, we discuss some connections to topological strings, Poisson sigma models, and topological field theories in various dimensions.

We formalise the self referential definition of physical laws using monotone operators on a lattice of theories, resolving the pathologies of naive set theoretic formulations. By invoking Tarski fixed point theorem, we identify physical theories as least fixed points of admissibility constraints derived from Galois connections. We demonstrate that QED and General Relativity can be represented in such a logical structure with respect to their symmetry and locality principles.

This paper explores the partition properties of roller coaster permutations, a class of permutations characterized by maximizing the number of alternating runs in all subsequences. We establish a connection between the structure of these permutations and their partition numbers, defined as the minimum number of monotonic subsequences required to cover the permutation. Our main result provides a theoretical upper bound for the partition number of a roller coaster permutation of length $n$, given by $P_{max}(n) \le \lfloor\frac{\lceil\frac{n-2}{2}\rceil}{2}\rfloor + 2$. We further present experimental data for $n < 15$ that suggests this bound is nearly sharp.

The existence of large-data weak entropy solutions to a nonisothermal immiscible compressible two-phase unsaturated flow model in porous media is proved. The model is thermodynamically consistent and includes temperature gradients and cross-diffusion effects. Due to the fact that some terms from the total energy balance are non-integrable in the classical weak sense, we consider so-called variational entropy solutions. A priori estimates are derived from the entropy balance and the total energy balance. The compactness is achieved by using the Div-Curl lemma.

In the present paper, we obtain a general lower bound for the $2$-adic valuation of the algebraic part of the central value of the complex $L$-series for the quadratic twists of any elliptic curve over $\mathbb{Q}$, showing that when the $2$-part of the product of Tamagawa factors grows, the $2$-part of the algebraic central $L$-value grows as well, in accordance with the Birch--Swinnerton-Dyer exact formula. This generalises a result of Coates--Kim--Liang--Zhao to all elliptic curves defined over $\mathbb{Q}$. We also prove the existence of an explicit infinite family of quadratic twists with analytic rank $0$ for a large family of elliptic curves.

The Pólya-Ostrowski group of a Galois number field $K$, is the subgroup $Po(K)$ of the ideal class group $Cl(K)$ of $K$ generated by the classes of all the strongly ambiguous ideals of $K$. The number field $K$ is called a Pólya field, whenever $Po(K)$ is trivial. In this paper, using some results of Bennett Setzer \cite{Bennett} and Zantema \cite{Zantema}, we give an explicit relation between the order of Pólya groups and the Hasse unit indices in real biquadratic fields. As an application, we refine Zantema's upper bound on the number of ramified primes in Pólya real biquadratic fields.

Let $G$ be an affine or hyperbolic rank 2 Kac--Moody group over a finite field $\mathbb F_q$. Let $X=X_{q+1}$ be the Tits building of $G$, the $(q+1)$--homogeneous tree, and let $\Gamma$ be a non-uniform lattice in $G$. When $\Gamma$ is a standard parabolic subgroup for the negative $BN$--pair, we define Eisenstein series on $\Gamma \backslash X$ and prove its convergence in a half space using Iwasawa decomposition of the Haar measure on $G$. A crucial tool is a description of the vertices of $X$ in terms of Iwasawa cells. We also prove meromorphic continuation of the Eisenstein series. This requires us to construct an integral operator on the Tits building $X$ and a truncation operator for the Eisenstein series. We also develop the functional analytic framework necessary for proving meromorphic continuation in our setting, by refining and extending Bernstein's Continuation Principle.

We propose a method for computing generating functions of genus-zero invariants of a gauged linear sigma model $(V, G, \theta, w)$. We show that certain derivatives of $I$-functions of quasimap invariants of $[V //_\theta G]$ produce $I$-functions (appropriately defined) of the GLSM. When $G$ is an algebraic torus we obtain an explicit formula for an $I$-function, and check that it agrees with previously computed $I$-functions in known special cases. Our approach is based on a new construction of GLSM invariants which applies whenever the evaluation maps from the moduli space are proper, and includes insertions from light marked points.

We compute the cohomology ring $H^*(U,\mathbb{Z}/n\mathbb{Z})$ for $U=X\setminus S$ where $X$ is the spectrum of the ring of integers of a number field $K$ and $S$ is a finite set of finite primes. As a consequence, we obtain an efficient way to compute presentations of $Q_2(G_S)$, where $G_S$ is Galois group of the maximal extension of $K$ unramified outside of a finite set of primes $S$, for varying $K$. This includes the following cases (for $p$ any prime dividing $n$): $\mu_p(\overline{K}) \not\subseteq K$; $S$ does not contain the primes above $p$; and $p=2$ with $K$ admitting real archimedean places. We also show how to recover the classical reciprocity law of the Legendre symbol from the graded commutativity of the cup product.

We address a system of equations modeling a compressible fluid interacting with an elastic body in dimension three. We prove the local existence and uniqueness of a strong solution when the initial velocity belongs to the space $H^{2+\epsilon}$ and the initial structure velocity is in $H^{1.5+\epsilon}$ , where $\epsilon \in (0, 1/2)$.

In this paper, we consider axisymmetric, swirl-free solutions of the Euler equation in four and higher dimensions. We show that in dimension $d\geq 4$, axisymmetric, swirl-free solutions of the Euler equation have properties which could allow finite-time singularity formation of a form that is excluded when $d=3$, and we prove a conditional blowup result for axisymmetric, swirl-free solutions of the Euler equation in dimension $d\geq 4$. The condition which must be imposed on the solution in order to imply blowup becomes weaker as $d\to +\infty$, suggesting the dynamics are becoming much more singular as the dimension increases.

We construct small resolutions of the moduli space $\overline{Q}_n$ of stable scaled $n$-marked lines of Ziltener and Ma'u--Woodward and of the moduli space $\overline{P}_n$ of stable $n$-marked ${\mathbb G}_a$-rational trees introduced in earlier work. The resolution of $\overline{P}_n$ is the augmented wonderful variety corresponding to the graphic matroid of the complete graph. The resolution of $\overline{Q}_n$ is a further blowup, also a wonderful model of an arrangement in ${\mathbb P}^{n-1}$.

The fundamental geometry of self-similar sets becomes significantly more complex when the generating contractive maps include non-trivial rotational components. A well-known family exemplifying this complexity is that of the dragon curves in the plane. In this paper, we prove that every dragon curve has a polygonal convex hull. Moreover, we completely characterize their convex hulls.

Closed-loop optimal control design for high-dimensional nonlinear systems has been a long-standing challenge. Traditional methods, such as solving the associated Hamilton-Jacobi-Bellman equation, suffer from the curse of dimensionality. Recent literature proposed a new promising approach based on supervised learning, by leveraging powerful open-loop optimal control solvers to generate training data and neural networks as efficient high-dimensional function approximators to fit the closed-loop optimal control. This approach successfully handles certain high-dimensional optimal control problems but still performs poorly on more challenging problems. One of the crucial reasons for the failure is the so-called distribution mismatch phenomenon brought by the controlled dynamics. In this paper, we investigate this phenomenon and propose the Progressive Optimal Path Sampling (POPS) method to mitigate this problem. We theoretically prove that this enhanced sampling strategy outperforms both the vanilla approach and the widely used Dataset Aggregation (DAgger) method on the classical linear-quadratic regulator by a factor proportional to the total time duration. We further numerically demonstrate that the proposed sampling strategy significantly improves the performance on tested control problems, including the optimal landing problem of a quadrotor and the optimal reaching problem of a 7 DoF manipulator.

This is a companion paper to earlier work of the authors, which proved an integral surgery formula for framed instanton homology. First, we present an enhancement of the large surgery formula, a rational surgery formula for null-homologous knots in any 3-manifold, and a formula encoding a large portion of $I^\sharp(S^3_0(K))$. Second, we use the integral surgery formula to study the framed instanton homology of many 3-manifolds: Seifert fibered spaces with nonzero orbifold degrees, especially nontrivial circle bundles over any orientable surface, surgeries on a family of alternating knots and all twisted Whitehead doubles, and splicings with twist knots. Finally, we use the previous techniques and computations to study almost L-space knots, ${\it i.e.}$, the knots $K\subset S^3$ with $\dim I^\sharp(S_n^3(K))=n+2$ for some $n\in\mathbb{N}_+$. We show that an almost L-space knot of genus at least $2$ is fibered and strongly quasi-positive, and a genus-one almost L-space knot must be either the figure eight or the mirror of the $5_2$ knot in Rolfsen's knot table.

In this paper, we prove a complex version of the incidence estimate of Guth, Solomon and Wang for tubes obeying certain strong spacing conditions, and we use one of our new estimates to resolve a discretized variant of Falconer's distance set problem in $\mathbb{C}^2$.

We introduce an abstract framework of Cartesian squares beyond the context of fiber products, and use it to extend the notion of pullback from classical to compact quantum principal bundles. Based only on our abstract notion of a Cartesian square, we extend key concepts of Equivariant Topology, such as the pullback of a family of group actions, orbit spaces, slices and global sections, change of base and structure group, free actions, and the groupoid of compact principal bundles. Finally, we embed the thus extended Equivariant Topology inside the 2-category of Grothendieck categories in such a way that our notion of a Cartesian square becomes the appropriate Beck-Chevalley condition.

Inspired by work of the first and second author, this paper studies the Gromov width of the disk cotangent bundle of spheroids and Zoll spheres of revolution. This is achieved with the use of techniques from integrable systems and embedded contact homology capacities.

The paper develops a second-order time-domain moment matching framework for the structure-preserving model reduction of second-order dynamical systems of high dimension, avoiding the first-order double-sized equivalent system. The moments of a second-order system are defined based on the solutions of second-order Sylvester equations, leading to families of parameterized second-order reduced models that match the moments of an original second-order system at selected interpolation points. Furthermore, a two-sided moment matching problem is addressed, providing a unique second-order reduced system that matches two distinct sets of interpolation points. We also construct the reduced second-order systems that match the moments of both the zero and first-order derivatives of the transfer function of the original second-order system. Finally, the Loewner framework is extended to second-order systems, where two parameterized families of models are presented that retain the second-order structure and interpolate sets of tangential data. The theory of the second-order time-domain moment matching is illustrated on vibrating systems.

The stochastic Lanczos quadrature method has garnered significant attention recently. Upon examination of the error analyses given by Ubaru, Chen and Saad and Cortinovis and Kressner, certain notable inconsistencies arise. It turns out that the former's results are valid for cases with symmetric quadrature nodes and may not be adequate for many practical cases such as estimating log determinant of matrices. This paper analyzes probabilistic error bound of the stochastic Lanczos quadrature method for cases with asymmetric quadrature nodes. Besides, an optimized error allocation technique is employed to minimize the overall number of matrix vector multiplications required by the stochastic Lanczos quadrature method.

We compute the mod $(p,v_1)$ and mod $(2,\eta,v_1)$ $\mathrm{THH}$ of many variants of the image-of-$J$ spectrum. In particular, we do this for $j_{\zeta}$, whose $\mathrm{TC}$ is closely related to the $K$-theory of the $K(1)$-local sphere. We find in particular that the failure for $\mathrm{THH}$ to satisfy $\mathbb{Z}_p$-Galois descent for the extension $j_{\zeta} \to \ell_p$ corresponds to the failure of the $p$-adic circle to be its own free loop space. For $p>2$, we also prove the Segal conjecture for $j_{\zeta}$, and we compute the $K$-theory of the $K(1)$-local sphere in degrees $\leq 4p-6$.

Let $F$ be a finite field, let $f$ be a function from $F$ to $F$, and let $a$ be a nonzero element of $F$. The discrete derivative of $f$ in direction $a$ is $\Delta_a f \colon F \to F$ with $(\Delta_a f)(x)=f(x+a)-f(x)$. The differential spectrum of $f$ is the multiset of cardinalities of all the fibers of all the derivatives $\Delta_a f$ as $a$ runs through $F^*$. An almost perfect nonlinear (APN) function is one for which the largest cardinality in its differential spectrum is $2$. Almost perfect nonlinear functions are of interest as cryptographic primitives. If $d$ is a positive integer, then the power function over $F$ with exponent $d$ is the function $f \colon F \to F$ with $f(x)=x^d$ for every $x \in F$. There is a small number of known infinite families of APN power functions. In this paper, we re-express the exponents for one such family in a more convenient form. This enables us not only to obtain the differential spectrum of each power function $f$ with an exponent in our family, but also to determine the elements that lie in an arbitrary fiber of the discrete derivative of $f$. This differential analysis, which is far more detailed than previous results, is achieved by composing the discrete derivative of $f$ with some permutations and a double covering of its domain to obtain a function whose fibers can more readily be analyzed.

We prove a priori bounds for Feigenbaum quadratic polynomials, i.e., infinitely renormalizable polynomials $f_c: z\mapsto z^2+c$ of bounded type. It implies local connectivity of the corresponding Julia sets $J(f_c)$ and MLC (local connectivity of the Mandelbrot set) at the corresponding parameters $c$. It also yields the scaling Universality, dynamical and parameter, for the corresponding combinatorics. The MLC Conjecture was open for the most classical period-doubling Feigenbaum parameter as well as for the complex tripling renormalizations. Universality for the latter was conjectured by Goldberg-Khanin-Sinai in the early 1980s.

We introduce the GEneralized Newton Inexact Operator Splitting solver (GeNIOS) for large-scale convex optimization. GeNIOS speeds up ADMM by approximately solving approximate subproblems: it uses a second-order approximation to the most challenging ADMM subproblem and solves it inexactly with a fast randomized solver. Despite these approximations, GeNIOS retains the convergence rate of classic ADMM and can detect primal and dual infeasibility from the algorithm iterates. At each iteration, the algorithm solves a positive-definite linear system that arises from a second-order approximation of the first subproblem and computes an approximate proximal operator. GeNIOS solves the linear system using an indirect solver with a randomized preconditioner, making it particularly useful for large-scale problems with dense data. Our high-performance open-source implementation in Julia allows users to specify convex optimization problems directly (with or without conic reformulation) and allows extensive customization. We illustrate GeNIOS's performance on a variety of problem types. Notably, GeNIOS is up to ten times faster than existing solvers on large-scale, dense problems.

The L'vov-Kaplansky conjecture states that the image of a multilinear noncommutative polynomial $f$ in the matrix algebra $M_n(K)$ is a vector space for every $n \in {\mathbb N}$. We prove this conjecture for the case where $f$ has degree $3$ and $K$ is an algebraically closed field of characteristic $0$.

This paper establishes Paley-Wiener perturbation theorems for probabilistic frames. The classical Paley-Wiener perturbation theorem shows that if a sequence is close to a basis in a Banach space, then this sequence is also a basis. Similar perturbation results have been established for frames in Hilbert spaces. In this work, we show that if a probability measure is sufficiently close to a probabilistic frame in an appropriate sense, then this probability measure is also a probabilistic frame. Moreover, we obtain explicit frame bounds for such probability measures that are close to a given probabilistic frame in the $2$-Wasserstein metric. This yields an alternative proof of the fact that the set of probabilistic frames is open in $\mathcal{P}_2(\mathbb{R}^n)$ under the $2$-Wasserstein topology.

It is known that each knot has a semimeander diagram (i. e. a diagram composed of two smooth simple arcs), however the number of crossings in such a diagram can only be roughly estimated. In the present paper we provide a new estimate of the complexity of the semimeander diagrams. We prove that for each knot $K$ with more than 10 crossings, there exists a semimeander diagram with no more than $0.31 \cdot 1.558^{\operatorname{cr}(K)}$ crossings, where $\operatorname{cr}(K)$ is the crossing number of $K$. As a corollary, we provide new estimates of the complexity of other meander-like types of knot diagrams, such as meander diagrams and potholders. We also describe an efficient algorithm for constructing a semimeander diagram from a given one.

This paper axiomatizes, in a two-stage setup, a new theory for decision under risk and ambiguity. The axiomatized preference relation $\succeq$ on the space $\tilde{V}$ of random variables induces an ambiguity index $c$ on the space $\Delta$ of probabilities, a probability weighting function $\psi$, generating the measure $\nu_{\psi}$ by transforming an objective probability measure, and a utility function $\phi$, such that, for all $\tilde{v},\tilde{u}\in\tilde{V}$, \begin{align*} \tilde{v}\succeq\tilde{u} \Leftrightarrow \min_{Q \in \Delta} \left\{\mathbb{E}_Q\left[\int\phi\left(\tilde{v}^{\centerdot}\right)\,\mathrm{d}\nu_{\psi}\right]+c(Q)\right\} \geq \min_{Q \in \Delta} \left\{\mathbb{E}_Q\left[\int\phi\left(\tilde{u}^{\centerdot}\right)\,\mathrm{d}\nu_{\psi}\right]+c(Q)\right\}. \end{align*} Our theory extends the rank-dependent utility model of Quiggin (1982) for decision under risk to risk and ambiguity, reduces to the variational preferences model when $\psi$ is the identity, and is dual to variational preferences when $\phi$ is affine in the same way as the theory of Yaari (1987) is dual to expected utility. As a special case, we obtain a preference axiomatization of a decision theory that is a rank-dependent generalization of the popular maxmin expected utility theory. We characterize ambiguity aversion in our theory.

For $P$ a poset, the dimension of $P$ is defined to be the least cardinal $\kappa$ such that $P$ is embeddable in a direct product of $\kappa$ totally ordered sets. We study the behavior of this function on finite-dimensional (not necessarily finite) posets. In general, the dimension dim($P$ x $Q$) of a product of two posets can be smaller than dim($P$) + dim($Q$), though no cases are known where the discrepancy is greater than 2. We obtain a result that gives upper bounds on the dimensions of certain products of posets, including cases where the discrepancy 2 is achieved. But the paper is mainly devoted to stating questions, old and new, about dimensions of product posets, noting implications among their possible answers, and introducing some related concepts that might be helpful in tackling these questions.

For two real symmetric matrices, their eigenvalue configuration is therelative arrangement of their eigenvalues on the real line. We consider the following problem: given two parametric real symmetric matrices and an eigenvalue configuration, find a simple condition on the parameters such that the two matrices have the given eigenvalue configuration. In this paper, we develop theory and give an algorithm for this problem. The output of the algorithm is a condition written in terms of the signatures of certain related symmetric matrices.

We study an example of a projective threefold with a non-isolated singularity and its derived category. The singular locus can be locally described as a line of surface nodes compounded with a threefold node at the origin. We construct a semiorthogonal decomposition where one component absorbs the singularity in the sense of Kuznetsov--Shinder, and the other components are equivalent to the derived categories of smooth projective varieties. The absorbing category is seen to be closely related to the absorbing category constructed for nodal varieties by Kuznetsov--Shinder, reflecting the geometry of the singularity. We further show that the semiorthogonal decomposition is induced by one on a geometric resolution, and briefly consider the properties of the absorbing category under smoothing.

Let $\mathfrak{g}$ be a symmetrizable Kac--Moody algebra. Given a root $\alpha$ and a real root $\beta$ of $\mathfrak{g}$, it is known that the $\beta$-string through $\alpha$, denoted $R_\alpha(\beta)$, is finite. Given an imaginary root $\beta$, we show that $R_\alpha(\beta)=\{\beta\}$ or $R_\alpha(\beta)$ is infinite. If $(\beta,\beta)<0$, we also show that the multiplicity of the root ${\alpha+n\beta}$ grows at least exponentially as $n\to\infty$. If $(\beta,\beta)=(\alpha, \beta) = 0$, we show that $R_\alpha(\beta)$ is bi-infinite and the multiplicities of $\alpha+n\beta$ are bounded. If $(\beta,\beta)=0$ and $(\alpha, \beta) \neq 0$, we show that $R_\alpha(\beta)$ is semi-infinite and the muliplicity of $\alpha+n\beta$ or $\alpha-n\beta$ grows faster than every polynomial as $n\to\infty$. We also prove that $\dim \mathfrak{g}_{\alpha+\beta} \geq \dim \mathfrak{g}_\alpha + \dim \mathfrak{g}_\beta -1$ whenever $\alpha \neq \beta$ with $(\alpha, \beta)<0$.

We prove that equality within the Minkowski inequality for asymptotically flat static manifolds is achieved only by slices of Schwarzschild space.

A factorisation problem in the symmetric group is central if conjugate permutations always have the same number of factorisations. We give the first fully combinatorial proof of the centrality of transitive star factorisations that is valid in all genera, which answers a natural question of Goulden and Jackson from 2009. We begin by showing that the set of star factorisations is equinumerous with a certain set of monotone factorisations, a new result. We give more than one proof of this, and, crucially, one of our proofs is bijective. As a corollary we obtain new formulae for some monotone double Hurwitz factorisations, and a new relation between Hurwitz and monotone Hurwitz factorisations. We also generalise a theorem of Goulden and Jackson from 2009 that states that the transitive power of Jucys-Murphy elements are central. Our theorem states that the transitive image of any symmetric function evaluated at Jucys-Murphy elements is central, which gives a transitive version of Jucys' original result from 1974.

In this paper, we derive decay rates for solutions to the incompressible Navier-Stokes equations and Hall-magnetohydrodynamic equations. We first improve the decay rate of weak solutions to these equations by refining the Fourier splitting method with initial data in the space of pseudo-measures. Additionally, we investigate these equations with initial data in the Lei-Lin spaces and establish decay rates for those solutions.

We consider the constrained-degree percolation (CDP) model on the hypercubic lattice. This is a continuous-time percolation model defined by a sequence $(U_e)_{e\in\mathcal{E}^d}$ of i.i.d. uniform random variables and a positive integer $k$, referred to as the constraint. The model evolves as follows: each edge $e$ attempts to open at a random time $U_e$, independently of all other edges. It succeeds if, at time $U_e$, both of its end-vertices have degrees strictly smaller than $k$. It is known \cite{hartarsky2022weakly} that this model undergoes a phase transition when $d\geq3$ for most nontrivial values of $k$. In this work, we prove that, for any fixed constraint, the number of infinite clusters at any time $t\in[0,1)$ is almost surely either 0 or 1. This uniqueness result implies the continuity of the percolation function in the supercritical regime, $t\in(t_c,1)$, where $t_c$ denotes the percolation critical threshold. The proof relies on a key time-regularity property of the model: the law of the process is continuous with respect to time for local events. In fact, we establish differentiability in time, thereby extending the result of \cite{SSS} to the CDP setting.

We study the approximation error $\varepsilon(x)=\operatorname{li}_{*}(x)-\operatorname{li}(x)$ arising from the classical Stieltjes asymptotic expansion for the logarithmic integral. Our analysis is based on the discrete values $\varepsilon_k=\varepsilon(e^{k})$ and their increments $\Delta_k=\varepsilon_{k+1}-\varepsilon_k,$ for which we derive new unconditional analytic bounds. Using precise integral representations for each increment $\Delta_k$, together with sharp upper and lower estimates for the associated kernel integrals, we obtain computable and uniform bounds for $\varepsilon_k$ for all $k\ge 1$, and hence for $\varepsilon(x)$ for all $x\ge e$. We prove the following unconditional bounds: $$\begin{array}{l} \displaystyle \frac{1}{3}\sqrt{\frac{2\pi}{\ln(x)}} + o\left(\frac{1}{\sqrt{\ln(x)}}\right) \le \varepsilon(x) \le \frac{1}{3}\sqrt{\frac{2\pi}{\ln(x)}} + o\left(\frac{1}{\sqrt{\ln(x)}}\right) \text{for all } e \le x \le e^{1000}, \end{array} $$ $$\begin{array}{l} \displaystyle \frac{1}{3}\sqrt{\frac{2\pi}{\ln(x)}} + o\left(\frac{1}{\sqrt{\ln(x)}}\right) - C_{l} \le \varepsilon(x) \le \frac{1}{3}\sqrt{\frac{2\pi}{\ln(x)}} + o\left(\frac{1}{\sqrt{\ln(x)}}\right) + C_{r} \text{for all } x>e^{1000} \text{ with } C_{l} = 0.0000035462\text{ and } C_{r}=0.0000021511. \end{array}$$ These results establish the first fully explicit global bounds for the Stieltjes approximation error. Finally, our findings strongly support the conjectural behaviour: $$ \varepsilon(x) = \frac{1}{3}\sqrt{\frac{2\pi}{\ln(x)}} + o\!\left(\frac{1}{\sqrt{\ln(x)}}\right), \qquad x\ge e. $$

In this study, we analyze a novel dynamical system inspired by Montgomery's pair correlation conjecture, modeling the spacings between nontrivial zeros of the Riemann zeta function via the GUE kernel $g(u) = 1 - \left( \frac{\sin(\pi u)}{\pi u} \right)^2 + \delta(u)$. The recurrence $x_{n+1} = 1 - \left( \frac{\sin(\pi/x_n)}{\pi/x_n} \right)^2 + \frac{1}{x_n}$ emulates eigenvalue repulsion as a quantum operator analogue realizing the Pólya-Hilbert conjecture. Bifurcation analysis and Lyapunov exponents reveal quantum-like chaos: near $x=0$, linearized dynamics $f(x) = 1 - \pi^2 x^2$ yield Gaussian Lyapunov function $V(x) = C_1 e^{-\pi^2 x^3/3}$ with LaSalle invariance bounding zeros in $[0,1]$; large $x$ exhibit exponential growth $\lambda_n \to \ln(\pi^2/6)$. Entropy analysis confirms GUE level repulsion with zero entropy for small initial conditions. Comparative validation against actual $\gamma_n$ achieves errors $<10^{-100}$, while spectral density $\rho(E) \sim \frac{\log E}{2\pi}$ matches zeta zero statistics. This bridges Montgomery pair correlation to quantum chaos, providing computational evidence for Riemann zero spacing distributions and supporting the quantum operator hypothesis for $\zeta(1/2+it)$.

This paper investigates the approximate controllability of linear fractional impulsive evolution equations in Hilbert spaces. The system under consideration involves the Caputo fractional derivative of order $0<\alpha\leq 1$, a closed linear operator generating a strongly continuous semigroup, and instantaneous state jumps governed by bounded linear impulse operators. We first derive an explicit representation of the mild solution by combining fractional solution operators with impulsive operators. Using this representation, we characterize the approximate controllability of the system through a necessary and sufficient condition expressed in terms of the convergence of an associated family of impulsive resolvent operators. This resolvent condition extends the classical criterion for approximate controllability to the fractional impulsive setting. To illustrate the applicability of our theoretical results, a concrete example is provided. The analysis presented here bridges the gap between the well-established theory for integer-order impulsive systems and the more complex fractional case, highlighting the distinct challenges and solutions arising from the interplay of fractional dynamics and impulsive effects.

A generalized particle system interacting with a massless Bose field is investigated. We assume regularity conditions for the commutation relations of the interaction and annihilation operators. It is proven that if the ground state exists, its multiplicity is finite.

A bottleneck in the theory of large-amplitude and multi-d viscous and relaxation shock stability is the development of nonlinear damping estimates controlling higher by lower derivatives. These have traditionally proceeded from time-evolution bounds based on Friedrichs symmetric and Kawashima or Goodman type energy estimates. Here, we propose an alternative program based on frequency-dependent pseudodifferential time-space damping estimates in the spirit of Kreiss. These are seen to be equivalent in the linear case to high-frequency spectral stability, and, just as for the constant-coefficient analysis of Kreiss, sharp in a pointwise, fixed-frequency, sense. This point of view leads to a number of simplifications and extensions using already-existing analysis. We point to the new issue of turning points, analogous to glancing points in the constant-coefficient case as an important direction for further development.

The Gromov-Wasserstein (GW) distance is a powerful tool for comparing metric measure spaces which has found broad applications in data science and machine learning. Driven by the need to analyze datasets whose objects have increasingly complex structure (such as node and edge-attributed graphs), several variants of GW distance have been introduced in the recent literature. With a view toward establishing a general framework for the theory of GW-like distances, this paper considers a vast generalization of the notion of a metric measure space: for an arbitrary metric space $Z$, we define a $Z$-network to be a measure space endowed with a kernel valued in $Z$. We introduce a method for comparing $Z$-networks by defining a generalization of GW distance, which we refer to as $Z$-Gromov-Wasserstein ($Z$-GW) distance. This construction subsumes many previously known metrics and offers a unified approach to understanding their shared properties. This paper demonstrates that the $Z$-GW distance defines a metric on the space of $Z$-networks which retains desirable properties of $Z$, such as separability, completeness, and geodesicity. Many of these properties were unknown for existing variants of GW distance that fall under our framework. Our focus is on foundational theory, but our results also include computable lower bounds and approximations of the distance which will be useful for practical applications.

The main purpose of this paper is to present a new and more uniform model-theoretic/combinatorial proof of the theorem ([5]): The randomization $T^{R}$ of a complete first-order theory $T$ with $NIP$ is a (complete) first-order continuous theory with $NIP$. The proof method is based on the significant use of a particular type of models of $T^{R}$, namely simple models, certain indiscernible arrays, and Rademacher mean width. Using simple models of $T^R$ gives the advantage of re-proving this theorem in a simpler and quantitative manner. We finally turn our attention to $NSOP$ in randomization. We show that based on the definition of $NSOP$ given [13], $T^R$ is stable if and only if it is $NIP$ and $NSOP$.

We provide a necessary and sufficient condition for the restricted wreath product $A\wr B$ to be $\mathcal{C}$-hereditarily conjugacy separable where $\mathcal{C}$ is an extension-closed pseudovariety of finite groups. Moreover, we prove that the Grigorchuk group is 2-hereditarily conjugacy separable. As an application, we demonstrate that the lamplighter groups and $\mathbb{Z} \wr \mathbb{Z}$ are hereditarily conjugacy separable (but not $p$-conjugacy separable for any prime $p$) which provides infinitely many new examples of solvable, non-polycyclic hereditarily conjugacy separable groups. Furthermore, we study wreath products of cyclic subgroup separable groups and the derived length of iterated wreath products of solvable groups with an abelian base group and, as an application, we give an explicit construction of non-polycyclic hereditarily conjugacy separable groups of arbitrary derived length as an iterated wreath products of abelian groups.

If $\mathscr A$ is a set of natural numbers of exponential density $\delta$, then the exponential density of all numbers of the form $x^3+a$ with $x\in\mathbb N$ and $a\in\mathscr A$ is at least $\min(1, \frac 13+\frac 56 \delta)$. This is a considerable improvement on the previous best lower bounds for this problem, obtained by Davenport more than 80 years ago. The result is the best possible for $\delta\ge \frac 45$.

We propose an extreme dimension reduction method extending the Extreme-PLS approach to the case where the covariate lies in a possibly infinite-dimensional Hilbert space. The ideas are partly borrowed from both Partial Least-Squares and Sliced Inverse Regression techniques. As such, the method relies on the projection of the covariate onto a subspace and maximizes the covariance between its projection and the response conditionally to an extreme event driven by a random threshold to capture the tail-information. The covariate and the heavy-tailed response are supposed to be linked through a non-linear inverse single-index model and our goal is to infer the index in this regression framework. We propose a new family of estimators and show its asymptotic consistency with convergence rates under the model. Assuming mild conditions on the noise, most of the assumptions are stated in terms of regular variation unlike the standard literature on SIR and single-index regression. Finally, our results are illustrated on a finite-sample study with synthetic functional data as well as on real data from the financial realm, highlighting the effectiveness of the dimension reduction for estimating extreme risk measures.

Fejér monotonicity is a well-established property often observed in sequences generated by optimization algorithms. In this paper, we study an extension of this property, called Fejér* monotonicity, which was initially proposed in [SIAM J. Optim., 34(3), 2535-2556 (2024)]. We discuss and explore its behavior within Hilbert spaces as a tool for optimization algorithms. Additionally, we investigate weak and strong convergence properties of this novel concept. Through illustrative examples and insightful results, we contrast Fejér* with weaker notions of quasi-Fejér-type monotonicity.

We prove vanishing results for Witten genera of string generalized complete intersections in homogeneous $\text{Spin}^c$-manifolds and in other $\text{Spin}^c$-manifolds with Lie group actions. By applying these results to Fano manifolds with second Betti number equal to one we get new evidence for a conjecture of Stolz.

The Lusternik-Schnirelmann category of a space was introduced to obtain a lower bound on the number of critical points of a $C^1$-function on a given manifold. Related to Lusternik-Schnirelmann category and motivated by topological robotics, the topological complexity (TC) of a space is a numerical homotopy invariant whose topological properties are an active field of research. The notions of sequential and parametrized topological complexity extend the ideas of topological complexity. While the definition of TC is closely related to Lusternik-Schnirelmann category, the connections of sequential and parametrized TC to critical point theory have not been fully explored yet. In this article we apply methods from Lusternik-Schnirelmann theory to establish various lower bounds on numbers of critical points of functions in terms of sequential and parametrized TCs. We carry out several consequences and applications of these bounds, among them a computation of the parametrized TC of the unit tangent bundles of $(4m-1)$-spheres.

This survey focuses on the geometric problem of log-surfaces, which are pairs consisting of a smooth projective surface and a reduced non-empty boundary divisor. In the first part, we focus on the geography problem for complex log-surfaces associated with pairs of the form $(\mathbb{P}^{2}, C)$, where $C$ is an arrangement of smooth plane curves admitting ordinary singularities. Specifically, we focus on the case in which $C$ is an arrangement consisting of smooth rational curves as its irreducible components. In the second part, containing original new results, we study log-surfaces constructed as pairs consisting of a complex projective $K3$ surface and a rational curve arrangement. In particular, we provide some combinatorial conditions for such pairs to have the log-Chern slope equal to $3$. Our survey is illustrated with many explicit examples of log-surfaces.

Suppose $G$ is a graph and $L$ is a list assignment for $G$. A request of $L$ is a function $r$ with nonempty domain $D\subseteq V(G)$ such that $r(v) \in L(v)$ for each $v \in D$. The triple $(G,L,r)$ is $\epsilon$-satisfiable if there exists a proper $L$-coloring $f$ of $G$ such that $f(v) = r(v)$ for at least $\epsilon|D|$ vertices in $D$. We say $G$ is $(k, \epsilon)$-flexible if $(G,L',r')$ is $\epsilon$-satisfiable whenever $L'$ is a $k$-assignment for $G$ and $r'$ is a request of $L'$. It is known that a graph $G$ is not $(k, \epsilon)$-flexible for any $k$ if and only if $\epsilon > 1/ \rho(G)$ where $\rho(G)$ is the Hall ratio of $G$. The list flexibility number of a graph $G$, denoted $\chi_{\ell flex}(G)$, is the smallest $k$ such that $G$ is $(k,1/ \rho(G))$-flexible. A fundamental open question on list flexibility numbers asks: Is there a graph with list flexibility number greater than its coloring number? In this paper, we show that the list flexibility number of any complete multipartite graph $G$ is at most the coloring number of $G$. We also initiate the study of list epsilon flexibility functions of complete bipartite graphs which was first suggested by Kaul, Mathew, Mudrock, and Pelsmajer in 2024. Specifically, we completely determine the list epsilon flexibility function of $K_{m,n}$ when $m \in \{1,2\}$ and establish some additional bounds for small $m$. Our proofs reveal a connection to list coloring complete bipartite graphs with asymmetric list sizes which is a topic that was explored by Alon, Cambie, and Kang in 2021.

Classical $(r,\delta)$-locally recoverable codes are designed for avoiding loss of information in large scale distributed and cloud storage systems. We introduce the quantum counterpart of those codes by defining quantum $(r,\delta)$-locally recoverable codes which are quantum error-correcting codes capable of correcting $\delta -1$ qudit erasures from sets of at most $r+ \delta -1$ qudits. We give a necessary and sufficient condition for a quantum stabilizer code $Q(C)$ to be $(r,\delta)$-locally recoverable. Our condition depends only on the puncturing and shortening at suitable sets of both the symplectic self-orthogonal code $C$ used for constructing $Q(C)$ and its symplectic dual $C^{\perp_s}$. When $Q(C)$ comes from a Hermitian or Euclidean dual-containing code, and under an extra condition, we show that there is an equivalence between the classical and quantum concepts of $(r,\delta)$-local recoverability. A Singleton-like bound is stated in this case and examples attaining the bound are given.

The Gas-Kinetic Scheme (GKS), widely used in computational fluid dynamics for simulating hypersonic and other complicated flow phenomena, is extended in this work to electromagnetic problems by solving Maxwell's equations. In contrast to the classical GKS formulation, the proposed scheme employs a discrete rather than a continuous velocity space. By evaluating a time-accurate numerical flux at cell interfaces, the proposed scheme attains second-order accuracy within a single step. Its kinetic formulation provides an inherently multidimensional framework, while the finite-volume formulation ensures straightforward extension to unstructured meshes. Through the incorporation of a collision process, the scheme exhibits lower numerical dissipation than classical flux-vector splitting (FVS) methods. Furthermore, the kinetic decomposition enables direct implementation of non-reflecting boundary conditions. The proposed scheme is validated against several benchmark problems and compared with established methods, including the Finite-Difference Time-Domain (FDTD) method and FVS. A lattice Boltzmann method (LBM) implementation is also included for comparative analysis. Finally, the technique is applied to simulate electromagnetic wave propagation in a realistic aircraft configuration, demonstrating its ability to model complex geometries.

Variational inference is a fast and scalable alternative to Markov chain Monte Carlo and has been widely applied to posterior inference tasks in statistics and machine learning. A traditional approach for implementing mean-field variational inference (MFVI) is coordinate ascent variational inference (CAVI), which relies crucially on parametric assumptions on complete conditionals. We introduce a novel particle-based algorithm for MFVI, named PArticle VI (PAVI), for nonparametric mean-field approximation. We obtain non-asymptotic error bounds for our algorithm. To our knowledge, this is the first end-to-end guarantee for particle-based MFVI.

This manuscript introduces Diophantine labeling, a new way of labeling of the vertices for finite simple undirected graphs with some divisibility condition on the edges. Maximal graphs admitting Diophantine labeling are investigated and their number of edges are computed. Some number-theoretic techniques are used to characterize vertices of maximum degree and nonadjacent vertices. Some necessary and sufficient conditions for vertices of equal degrees are found.

In this paper, we analyze the mean field backward stochastic differential equations (MFBSDEs) with double mean reflections, whose generator and constraints both depend on the distribution of the solution. When the generator is Lipschitz continuous, based on the backward Skorokhod problem with nonlinear constraints, we investigate the solvability of the doubly mean reflected MFBSDEs by constructing a contraction mapping. Furthermore, if the constraints are linear, the solution can also be constructed by a penalization method. For the case of quadratic growth, we obtain the existence and uniqueness results by using a fixed-point argument, the BMO martingale theory and the {\theta}-method.

In a sequence $S=\{(X,f_n)\}_{n\in\mathbb{N}}$ of dynamical systems sharing a common ambient space, the point $f^k_n(x)$ visited by a certain $x\in X$ depends on the iteration order $k$ and on the index $n$ specifying the system in $S$. If the sequence of maps $\{f_n\}_{n\in\mathbb{N}}$ is eventually constant at every point, the $f_n$-orbits show an emergent poset structure. A maximal initial segment of this poset is isomorphic to a certain countable ordinal $\ge\omega$. We study this transfinite emergent structure from the point of view of topological dynamics, investigating orbits, recurrence, limit sets, attractors and conjugacy.

This paper proposes a unified framework for designing robustness in optimization under uncertainty using gauge sets, convex sets that generalize distance and capture how distributions may deviate from a nominal reference. Representing robustness through a gauge set reweighting formulation brings many classical robustness paradigms under a single convex-analytic perspective. The corresponding dual problem, the upper approximator regularization model, reveals a direct connection between distributional perturbations and objective regularization via polar gauge sets. This framework decouples the design of the nominal distribution, distance metric, and reformulation method, components often entangled in classical approaches, thus enabling modular and composable robustness modeling. We further provide a gauge set algebra toolkit that supports intersection, summation, convex combination, and composition, enabling complex ambiguity structures to be assembled from simpler components. For computational tractability under continuously supported uncertainty, we introduce two general finite-dimensional reformulation methods. The functional parameterization approach guarantees any prescribed gauge-based robustness through flexible selection of function bases, while the envelope representation approach yields exact reformulations under empirical nominal distributions and is asymptotically exact for arbitrary nominal choices. A detailed case study demonstrates how the framework accommodates diverse robustness requirements while admitting multiple tractable reformulations.

Multimarginal optimal transport (MOT) has emerged as a useful framework for many applied problems. However, compared to the well-studied classical two-marginal optimal transport theory, analysis of MOT is far more challenging and remains much less developed. In this paper, we study the statistical estimation and inference problems for the entropic MOT (EMOT), whose optimal solution is characterized by the multimarginal Schrödinger system. Assuming only boundedness of the cost function, we derive sharp sample complexity for estimating several key quantities pertaining to EMOT (cost functional and Schrödinger coupling) from point clouds that are randomly sampled from the input marginal distributions. Moreover, with substantially weaker smoothness assumption on the cost function than the existing literature, we derive distributional limits and bootstrap validity of various key EMOT objects. As an application, we propose the multimarginal Schrödinger barycenter as a new and natural way to regularize the exact Wasserstein barycenter and demonstrate its statistical optimality.

We present a formalization of Borel determinacy in the Lean 4 theorem prover. The formalization includes a definition of Gale-Stewart games and a proof of Martin's theorem stating that Borel games are determined. The proof closely follows Martin's "A purely inductive proof of Borel determinacy".

Let $(x_1,y_1),\ldots,(x_n,y_n)$ be distinct non-constant and non-degenerate solutions of the classical Lotka-Volterra system \begin{equation}\notag \begin{split} x'&= axy + bx\\ y'&= cxy + dy, \end{split} \end{equation} where $a,b,c,d\in\mathbb{C}\setminus\{0\}$. We show that if $d$ and $b$ are linearly independent over $\mathbb{Q}$, then the solutions are algebraically independent over $\mathbb{C}$, that is $tr.deg_{\mathbb{C}}\mathbb{C}(x_1,y_1,\ldots,x_n,y_n)=2n$. As a main part of the proof, we show that the set defined by the system in universal differential fields, with $d$ and $b$ linearly independent over $\mathbb{Q}$, is strongly minimal and geometrically trivial. Our techniques also allows us to obtain partial results for some of the more general $2d$-Lotka-Volterra system.

Hamilton-Jacobi partial differential equations (HJ PDEs) play a central role in many applications such as economics, physics, and engineering. These equations describe the evolution of a value function which encodes valuable information about the system, such as action, cost, or level sets of a dynamic process. Their importance lies in their ability to model diverse phenomena, ranging from the propagation of fronts in computational physics to optimal decision-making in control systems. This paper provides a review of some recent advances in numerical methods to address challenges such as high-dimensionality, nonlinearity, and computational efficiency. By examining these developments, this paper sheds light on important techniques and emerging directions in the numerical solution of HJ PDEs.

The phenomenon where cells with elongated protrusions, such as neurons, communicate by contacting other cells and arrange themselves appropriately is termed cell sorting through haptotaxis. This phenomenon is described by partial differential equations involving nonlocal advection. In contrast, cell phenomena where cells communicate with other cells via chemical substances and arrange themselves appropriately are termed cell sorting through chemotaxis, typically modeled by chemotactic systems such as the Keller--Segel model. Although there are clear differences between haptotaxis and chemotaxis, similar behaviors are often observed. In this study, we investigate the relationship between haptotaxis and chemotaxis in cell sorting phenomena. Specifically, we analyze the connections between a nonlocal aggregation model for haptotaxis and a Keller--Segel type chemotaxis system. By demonstrating convergence under specific kernel approximations, we show how these distinct mechanisms can lead to comparable dynamic behaviors. In particular, we establish that the gradient of a given kernel can be approximated by linear combinations of gradients of fundamental solutions, which also provides a mathematical contribution of independent interest. This study provides a mathematical framework for understanding the interplay between haptotaxis and chemotaxis in cell sorting phenomena.

In this paper, we study the full asymptotic expansion of the partition functions of determinantal point processes defined on a polarized Kähler manifold. We show that the coefficients of the expansion are given by geometric functionals on Kähler metrics satisfying the cocycle identity, whose first variations can be expressed through the TYZ expansion coefficients of the Bergman kernel. In particular, these functionals naturally generalize the Mabuchi functional in Kähler geometry and the Liouville functional on Riemann surfaces. We further show that Futaki-type holomorphic invariants obstruct the existence of critical points of these geometric functionals, extending Lu's formula. We also verify that certain formulas remain valid up to the third coefficient without assuming polarization. Finally, we discuss the relation of our results to the quantum Hall effect (QHE), where the determinantal point process provides a microscopic model. In particular, we recover the higher-dimensional effective Chern-Simons actions derived in the physics literature and confirm a conjecture of Klevtsov on the form of the partition function asymptotics.

Green hydrogen has multiple use cases and is produced from renewable energy, such as solar or wind energy. It can be stored in large quantities, decoupling renewable energy generation from its use, and is therefore considered essential for achieving a climate-neutral economy. The intermittency of renewable energy generation and the stochastic nature of demand are, however, challenging factors for the dynamic planning of hydrogen storage and transportation. This holds particularly in the early-adoption phase when hydrogen distribution occurs through vehicle-based networks. We therefore address the Dynamic Inventory Routing Problem (DIRP) under stochastic supply and demand with direct deliveries for the vehicle-based distribution of hydrogen. To solve this problem, we propose a Constrained Reinforcement Learning (CRL) framework that integrates constraints into the learning process and incorporates parameterized post-decision state value predictions. Additionally, we introduce Lookahead-based CRL (LCRL), which improves decision-making over a multi-period horizon to enhance short-term planning while maintaining the value predictions. Our computational experiments demonstrate the efficacy of CRL and LCRL across diverse instances. Our learning methods provide near-optimal solutions on small scale instances that are solved via value iteration. Furthermore, both methods outperform typical deep learning approaches such as Proximal Policy Optimization, as well as classical inventory heuristics, such as (s,S)-policy-based and Power-of-Two-based heuristics. Furthermore, LCRL achieves a 10% improvement over CRL on average, albeit with higher computational requirements. Analyses of optimal replenishment policies reveal that accounting for stochastic supply and demand influences these policies, showing the importance of our addition to the DIRP.

This paper concerns the problem of reachability of a given state for a multiagent control system in $\mathbb{R}^d$. In such a system, at every time each agent can choose his/her velocity which depends both on his/her position and on the position of the whole crowd of agents (modeled by a probability measure on $ \mathbb{R}^d$). The main contribution of the paper is to study the above reachability problem with a given rate of attainability through a Lyapunov method adapted to the Wasserstein space of probability measures. As a byproduct we obtain a new comparison result for viscosity solutions of Hamilton Jacobi equations in the Wasserstein space.

The objective of this paper is to investigate the connection between penalty functions from stochastic optimal control, convex semigroups from analysis and convex expectations from probability theory. Our main result provides a one-to-one relation between these objects. As an application, we use the representation via penality functions and duality arguments to show that convex expectations are determined by their finite dimensional distributions. To illustrate this structural result, we show that Hu and Peng's axiomatic description of $G$-Lévy processes in terms of finite dimensional distributions extends uniquely to the control approach introduced by Neufeld and Nutz. Finally, we show that convex expectations with a Markovian structure are fully determined by their one-dimensional distributions, which give rise to a classical semigroup on the state space. As an application of this result, we establish a Laplace principle for entropic risk measures associated to controlled diffusions.

We construct symmetric square type $L$-series for vector-valued modular forms transforming under the Weil representation associated to a discriminant form. We study Hecke operators and integral representations to investigate their properties, deriving functional equations and infinite product expansions.

We show that every sufficiently large oriented graph $G$ with minimum indegree and outdegree both at least $(3|V(G)|-1)/8$ contains every orientation of a Hamilton cycle. This result improves the approximate bound established by Kelly and resolves a long-standing problem posed by Häggkvist and Thomason in 1995. The degree condition is tight and it can be improved to $(3|V(G)|-4)/8$ for Hamilton cycles that are nearly directed, generalizing a classic result by Keevash, Kühn and Osthus. Additionally, we derive a pancyclicity result for arbitrary orientations. More precisely, the above degree condition suffices to guarantee the existence of cycles of every possible orientation and every possible length unless $G$ is isomorphic to one of the exceptional oriented graphs.

We show that two coniveau filtrations on the mod 2 cohomology group of a smooth projective complex variety differ.

Connections between structural graph theory and finite model theory recently gained a lot of attention. In this setting, many interesting questions remain on the properties of dependent (NIP) hereditary classes of graphs, in particular related to first-order transductions. In this paper, we study modelizations (which are strong forms of transduction pairings) of classes of graphs by classes of structures. In particular, we consider models obtained by coupling a partial order and a colored graph (thus forming a partially ordered colored graph). Motivated by Simon's decomposition theorem of dependent types into a stable part and a distal (order-like) part, we conjecture that every dependent hereditary class of graphs admits a modelization in a monadically dependent coupling of a class of posets with bounded treewidth cover graphs and a monadically stable class of colored graphs. In this paper, we consider the first non-trivial case (classes with bounded linear cliquewidth) and prove that the conjecture holds in a strong form, the model class being a monadically dependent coupling of a class of disjoint unions of chains and a class of colored graphs with bounded pathwidth. We extend our study to classes that admit bounded-size bounded linear cliquewidth decompositions and prove that they have a modelization in a monadically dependent coupling of a class of disjoint unions of chains and a class of colored graphs with bounded expansion, the model class also admitting bounded-size bounded linear cliquewidth decompositions.

In constraint learning, we use a neural network as a surrogate for part of the constraints or of the objective function of an optimization model. However, the tractability of the resulting model is heavily influenced by the size of the neural network used as a surrogate. One way to obtain a more tractable surrogate is by pruning the neural network first. In this work, we consider how to approach the setting in which the neural network is actually a given: how can we solve an optimization model embedding a large and predetermined neural network? We propose surrogating the neural network itself by pruning it, which leads to a sparse and more tractable optimization model, for which we hope to still obtain good solutions with respect to the original neural network. For network verification and function maximization models, that indeed leads to better solutions within a time limit, especially -- and surprisingly -- if we skip the standard retraining step known as finetuning. Hence, a pruned network with worse inference for lack of finetuning can be a better surrogate.

We consider a cell-free massive multiple-input multiple-output (MIMO) system with multiple antennas on the users and access points (APs). In previous works, the downlink spectral efficiency (SE) has been evaluated using the hardening bound that requires no downlink pilots. This approach works well for single-antenna users. In this paper, we show that much higher SEs can be achieved if downlink pilots are sent when having multi-antenna users. The reason is that the effective channel matrix does not harden. We propose a pilot-based downlink estimation scheme, derive a new SE expression, and show numerically that it yields substantially higher performance when having correlated Rayleigh fading channels. In cases with multi-antenna users, the APs can either transmit the same or different data streams. The latter reduces the fronthaul signaling but comes with a SE loss. We propose precoding and combining schemes for these cases and consider whether channel knowledge is shared between the APs. Finally, we show numerically how the number of users, APs, and the number of antennas on users and APs affect the SE.

We study the distortion of intermediate dimension under supercritical Sobolev mappings and also under quasiconformal or quasisymmetric homeomorphisms. In particular, we extend to the setting of intermediate dimensions both the Gehring--Väisälä theorem on dilatation-dependent quasiconformal distortion of dimension and Kovalev's theorem on the nonexistence of metric spaces with conformal dimension strictly between zero and one. Applications include new contributions to the quasiconformal classification of Euclidean sets and a new sufficient condition for the vanishing of conformal box-counting dimension. We illustrate our conclusions with specific consequences for Bedford--McMullen carpets, samples of Mandelbrot percolation, and product sets containing a polynomially convergent sequence factor.

We study neural field equations, which are prototypical models of large-scale cortical activity, subject to random data. We view this spatially-extended, nonlocal evolution equation as a Cauchy problem on abstract Banach spaces, with randomness in the synaptic kernel, firing rate function, external stimuli, and initial conditions. We determine conditions on the random data that guarantee existence, uniqueness, and measurability of the solution in an appropriate Banach space, and examine the regularity of the solution in relation to the regularity of the inputs. We present results for linear and nonlinear neural fields, and for the two most common functional setups in the numerical analysis of this problem. In addition to the continuous problem, we analyse in abstract form neural fields that have been spatially discretised, setting the foundations for analysing uncertainty quantification (UQ) schemes.

Many datasets are naturally modeled as graphs, where vertices denote entities and edges encode pairwise interactions. However, some problems exhibit higher-order structure that lies beyond this framework. Among the simplest examples is line clustering, in which points in a Euclidean space are grouped into clusters well approximated by line segments. As any two points trivially determine a line, the relevant structure emerges only when considering higher-order tuples. To capture this, we construct a 3-uniform hypergraph by treating sets of three points as hyperedges whenever they are approximately collinear. This geometric hypergraph encodes information about the underlying line segments, which can be extracted using community recovery algorithms. We characterize the fundamental limits of line clustering and establish the near-optimality of hypergraph-based methods. In particular, we derive information-theoretic thresholds for exact and almost exact recovery for noisy observations from intersecting lines in the plane. Finally, we introduce a polynomial-time spectral algorithm that succeeds up to polylogarithmic factors of the information-theoretic bounds.

We solve two open problems of Valeriy Bardakov about Cayley graphs of racks and graph-theoretic realizations of right quasigroups. We also extend Didier Caucal's classification of labeled Cayley digraphs to right quasigroups and related algebraic structures like quandles. First, we characterize markings of graphs that realize racks. As an application, we construct rack-theoretic (di)graph invariants from permutation representations of graph automorphism groups. We describe how to compute these invariants with general results for path graphs and cycle graphs. Second, we show that all right quasigroups are realizable by edgeless graphs and complete (di)graphs. Using Schreier (di)graphs, we also characterize Cayley (di)graphs of right quasigroups Q that realize Q. In particular, all racks are realizable by their full Cayley (di)graphs. Finally, we give a graph-theoretic characterization of labeled Cayley digraphs of right-cancellative magmas, right-divisible magmas, right quasigroups, racks, quandles, involutory racks, and kei.

In this brief note, we investigate some constructions of Lyapunov functions for stochastic discrete-time stabilizable dynamical systems, in other words, controlled Markov chains. The main question here is whether a Lyapunov function in some statistical sense exists if the respective controlled Markov chain admits a stabilizing policy. We demonstrate some constructions extending on the classical results for deterministic systems. Some limitations of the constructed Lyapunov functions for stabilization are discussed, particularly for stabilization in mean. Although results for deterministic systems are well known, the stochastic case was addressed in less detail, which the current paper remarks on. A distinguishable feature of this work is the study of stabilizers that possess computationally tractable convergence certificates.

This paper introduces a geometric multigrid preconditioner for the Shifted Boundary Method (SBM) designed to solve PDEs on complex geometries. While SBM simplifies mesh generation by using a non-conforming background grid, it often results in non-symmetric and potentially ill-conditioned linear systems that are challenging to solve efficiently. Standard multigrid methods with pointwise smoothers prove ineffective for such systems due to the localized perturbations introduced by the shifted boundary conditions. To address this challenge, we introduce a Discontinuous Galerkin (DG) formulation for SBM that enables the design of a cell-wise multiplicative smoother within an $hp$-multigrid framework. The element-local nature of DG methods naturally facilitates cell-wise correction, which can effectively handle the local complexities arising from the boundary treatment. Numerical results for the Poisson equation demonstrate favorable performance with mesh refinement for linear ($p=1$) and quadratic ($p=2$) elements in both 2D and 3D, with iteration counts showing mild growth. However, challenges emerge for cubic ($p=3$) elements, particularly in 3D, where the current smoother shows reduced effectiveness.

Let $(M,g)$ be a closed Riemannian manifold of dimension at least $3$. Let $S$ be the union of the focal submanifolds of an isoparametric function on $(M,g)$. In this article we address the existence of solutions of the Hardy-Sobolev type equation $\Delta_g u+K(x)u=\frac{u^{q-1}}{\left(d_{S}(x)\right)^s}$, where $d_{S}(x)$ is the distance from $x$ to $S$ and $q>2$. In particular, we will prove the existence of infinite sign-changing solutions to the equation.

We investigate pre-local tabularity in normal extensions of the logic $\mathrm{S4}\times \mathrm{S4}$. We show that there are exactly four pre-locally tabular logics in normal extensions of products of finite height, and that every non-locally tabular logic in this family is contained in one of them. We also give an axiomatic criterion of local tabularity above the logic of products with Noetherian skeletons. Finally, we discuss examples of pre-locally tabular extensions of $\mathrm{S4}\times \mathrm{S4}$ outside this class, including logics with the converse and universal modalities.

In this paper we consider nonzero harmonic functions vanishing on some subsets of $\mathbb R^n$. We give a positive solution to Problem 151 from the Scottish Book posed by R. Wavre in 1936. In more detail, we construct a nonzero harmonic polynomial that vanishes on the edges of the unit cube. Moreover, using harmonic morphisms we build new nontrivial families of harmonic polynomials that vanish at the same set in the unit ball in $\mathbb R^n$ for all $n \geq 4$. This extends certain results by Logunov and Malinnikova. We also present new results on harmonic functions in the space whose zero sets are unions of affine codimension two subspaces.

We prove that the Dehn function of every finitely presented Bestvina-Brady group grows as a linear, quadratic, cubic, or quartic polynomial. In fact, we provide explicit criteria on the defining graph to determine the degree of this polynomial. As a consequence, we identify an obstruction that prevents certain Bestvina-Brady groups from admitting a CAT(0) structure.

Next-generation wireless technologies such as 6G aim to meet demanding requirements such as ultra-high data rates, low latency, and enhanced connectivity. Extremely Large-Scale MIMO (XL-MIMO) and Reconfigurable Intelligent Surface (RIS) are key enablers, with XL-MIMO boosting spectral and energy efficiency through numerous antennas, and RIS offering dynamic control over the wireless environment via passive reflective elements. However, realizing their full potential depends on accurate Channel State Information (CSI). Recent advances in deep learning have facilitated efficient cascaded channel estimation. However, the scalability and practical deployment of existing estimation models in XL-MIMO systems remain limited. The growing number of antennas and RIS elements introduces a significant barrier to real-time and efficient channel estimation, drastically increasing data volume, escalating computational complexity, requiring advanced hardware, and resulting in substantial energy consumption. To address these challenges, we propose a lightweight deep learning framework for efficient cascaded channel estimation in XL-MIMO systems, designed to minimize computational complexity and make it suitable for deployment on resource-constrained edge devices. Using spatial correlations in the channel, we introduce a patch-based training mechanism that reduces the dimensionality of input to patch-level representations while preserving essential information, allowing scalable training for large-scale systems. Simulation results under diverse conditions demonstrate that our framework significantly improves estimation accuracy and reduces computational complexity, regardless of the increasing number of antennas and RIS elements in XL-MIMO systems.

In this paper various Steiner systems $S(2,k,v)$ for $k = 6$ are collected and enumerated for specific constructions. In particular, two earlier unknown types of $1$-rotational designs are found for the groups $SL(2,5)$ and $((\mathbb Z_3 \times \mathbb Z_3) \rtimes \mathbb Z_3) \times \mathbb Z_5$. Also new Steiner systems $S(2,6,96), S(2,6,106), S(2,6,111)$ are listed.

Let $A$ be a condensable algebra in a modular tensor category $\mathcal{C}$. We define an action of the fusion category $\mathcal{C}_A$ of $A$-modules in $\mathcal{C}$ on the morphism space $\mbox{Hom}_{\mathcal{C}}(x,A)$ for any $x$ in $\mathcal{C}$, whose characters are generalized Frobenius-Schur indicators. This fusion action can be considered on $A$, and we prove a categorical generalization of the Schur-Weyl duality for this action. For any fusion subcategory $\mathcal{B}$ of $\mathcal{C}_A$ containing all the local $A$-modules, we prove the invariant subobject $B=A^\mathcal{B}$ is a condensable subalgebra of $A$. The assignment of $\mathcal{B}$ to $A^\mathcal{B}$ defines a Galois correspondence between this kind of fusion subcategories of $\mathcal{C}_A$ and the condensable subalgebras of $A$. In the context of VOAs, we prove for any nice VOAs $U \subset A$, $U=A^{\mathcal{C}_A}$ where $\mathcal{C}=\mathcal{M}_U$ is the category of $U$-modules. In particular, if $U = A^G$ for some finite automorphism group $G$ of $A,$ the fusion action of $\mathcal{C}_A$ on $A$ is equivalent to the $G$-action on $A.$

The original Goodstein process is based on writing numbers in hereditary $b$-exponential normal form: that is, each number $n$ is written in some base $b\geq 2$ as $n=b^ea+r$, with $e$ and $r$ iteratively being written in hereditary $b$-exponential normal form. We define a new process which generalises the original by writing expressions in terms of a hierarchy of bases $B$, instead of a single base $b$. In particular, the `digit' $a$ may itself be written with respect to a smaller base $b'$. We show that this new process always terminates, but termination is independent of Kripke-Platek set theory, or other theories of Bachmann-Howard strength.

We study when Poincaré series for congruence subgroups do not vanish identically. We show that almost all Poincaré series with suitable parameters do not vanish when either the weight $k$ or the index $m$ varies in a dyadic interval. Crucially, analyzing the problem `on average' over these weights or indices allows us to prove non-vanishing in ranges where the index $m$ is significantly larger than $k^2$ - a range in which proving non-vanishing for individual Poincaré series remains out of reach of current methods.

Graphlets are subgraphs rooted at a fixed vertex. The number of occurrences of graphlets aligned to a particular vertex, called graphlet degree sequence (gds), gives a topological description of the surrounding of the analyzed vertex. Graphlet degree distribution (gdd) of a graph is a matrix containing graphlet degree sequence for all vertices in the given graph. A long standing open problem called reconstruction conjecture (RC) asks whether the structure of a graph is uniquely determined by the multiset of its vertex-deleted subgraphs. Graphlet degree distribution up to size (n - 1), (<= n - 1)-gdd, gives more information to reconstruct the graph and we use it to reconstruct any graph having a unique almost-asymmetric vertex-deleted subgraph, where almost-asymmetric means that at most one automorphism orbit has size larger than one. Moreover, we prove that any graph containing a vertex-cut of size 1 or any graph of order n having a vertex with degree at most 2 or at least n-2 is reconstructible from its (<= n - 1)-gdd, which expands results shown in the standard RC. We also discuss the relation between gdd and graph connectivity and the conditions on (<= 3)-gdd, whose breaking means that no graph with such gdd exists.

Preliminary version of a course in univariate real analysis, with 14 chapters and 1 appendix (Chapters 1-8 complete at present). 1. Infinite sums. Real numbers; 2. Limits of sequences and subsequences; 3. Arithmetic of limits. AK series; 4. Infinite series. Elementary functions; 5. Limits of functions. Asymptotic notation; 6. Continuous functions; 7. Derivatives; 8. Applications of mean value theorems; 9. Taylor polynomials and series. Real analytic functions; 10. Primitives of uniformly continuous functions; 11. Newton integral. Primitives of rational functions; 12. Riemann integral. Transcendence of the number e; 13. Riemann integral. Henstock--Kurzweil integral; 14. More applications of Riemann integral and A. Solutions to exercises.

For any extension of commutative rings $A\subseteq B$ we first naturally define a group $\Cl(A,B)$, that we call the ideal class group of this extension. Then we study the basic properties of this group. Next, using ideas from algebraic geometry, we prove that every commutative ring has a (faithfully flat) ring extension whose Picard group is trivial. These results have several interesting applications. In particular, ...

We establish sharp bilinear eigenfunction estimates for the Laplace-Beltrami operator on the standard three-sphere $\mathbb{S}^3$, eliminating the logarithmic loss that has persisted in the literature since the pioneering work of Burq, Gérard, and Tzvetkov over twenty years ago. This completes the theory of multilinear eigenfunction estimates on the standard spheres. Our approach relies on viewing $\mathbb{S}^3$ as the compact Lie group $\mathrm{SU}(2)$ and exploiting its representation theory. Motivated by applications to the energy-critical nonlinear Schrödinger equation (NLS) on $\mathbb{R} \times \mathbb{S}^3$, we also prove a refined anisotropic Strichartz estimate on the cylindrical space $\mathbb{R}_{x_1} \times \mathbb{T}_{x_2}$ of $L^\infty_{x_2}L^4_{t,x_1}$-type, adapted to certain spectrally localized functions. The argument relies on multiple sharp measure estimates and a robust kernel decomposition method. Combining these two key ingredients, we derive a refined bilinear Strichartz estimate on $\mathbb{R} \times \mathbb{S}^3$, which in turn yields small-data global well-posedness for the above mentioned NLS in the energy space.

In this paper, we establish a distributed functional optimization (DFO) theory over time-varying networks. The vast majority of existing distributed optimization theories are developed based on Euclidean decision variables. However, for many scenarios in machine learning and statistical learning, such as reproducing kernel spaces or probability measure spaces that use functions or probability measures as fundamental variables, the development of existing distributed optimization theories exhibit obvious theoretical and technical deficiencies. This paper addresses these issues by developing a novel general DFO theory on Banach spaces, allowing functional learning problems in the aforementioned scenarios to be incorporated into our framework for resolution. We study both convex and nonconvex DFO problems and rigorously establish a comprehensive convergence theory of distributed functional mirror descent and distributed functional gradient descent algorithm to solve them. Satisfactory convergence rates are fully derived. The work has provided generic analyzing frameworks for DFO. The established theory is shown to have crucial application value in the kernel-based distributed learning theory over networks.

To meet the demands of instantaneous control of instabilities over long time horizons in plasma fusion, we design a dynamic feedback control strategy for the Vlasov-Poisson system by constructing an operator that maps state perturbations to an external control field. In the first part of the paper, we propose learning such an operator using a neural network. Inspired by optimal control theory for linearized dynamics, we introduce a low-rank neural operator architecture and train it via adjoint state method. The resulting controller is effective at suppressing instabilities well beyond the training time horizon. To generalize control across varying initial data, we further introduce a novel cancellation-based control strategy that removes the destabilizing component of the electric field. This approach naturally defines an operator without requiring any training, ensures perturbation decay over infinite time, and demonstrates strong robustness under noisy feedback. Numerical experiments confirm the effectiveness of the method in both one- and multidimensional settings.

In this study, inextensible flows of curves in four-dimensional pseudo-Galilean space are expressed, and the necessary and sufficient conditions of these curve flows are given as partial differential equations. Also, the directional derivatives are defined in accordance with the Serret-Frenet frame in G41, the extended Serret-Frenet relations are expressed by using Frenet formulas in G41. Furthermore, the bending elastic energy functions are expressed for the same particle according to curve a(s,t).

The four types of homogeneity -- additive, multiplicative, exponential, and logarithmic -- are generalized as transformations describing how a function $f$ changes under scaling or shifting of its arguments. These generalized homogeneity functions capture different scaling behaviors and establish fundamental properties. Such properties include how homogeneity is preserved under function operations and how it determines the transformation behavior of related constructions like quotient functions. This framework extends the classical concept of homogeneity to a wider class of functional symmetries, providing a unified approach to analyzing scaling properties in various mathematical contexts.

A fundamental theorem of linear programming states that a feasible linear program is solvable if and only if its objective function is copositive with respect to the recession cone of its feasible set. This paper demonstrates that this crucial guarantee does not extend to Second-Order Cone Programs (SOCPs), a workhorse model in robust and convex optimization. We construct and analyze a rigorous counterexample derived from a robust linear optimization problem with ellipsoidal uncertainty. The resulting SOCP possesses a non-empty feasible set, a bounded objective, and an objective function that is copositive on its recession cone. Despite satisfying these classical conditions for solvability, the problem admits no optimal solution; its infimum is finite but unattainable. We trace this pathology directly to the non-polyhedral geometry of the second-order cone, which causes the image of the feasible set under the linear objective to be non-closed. We interpret the example explicitly within the context of robust optimization, discuss its significant practical implications for modeling and computation, and propose effective mitigation strategies via polyhedral approximation or regularization.

In this paper, we study the quasisymmetric Hausdorff minimality of homogeneous Moran sets. First, we obtain the Hausdorff dimension formula of two classes of homogeneous Moran sets which satisfy some conditions. Second, we show two special classes of homogeneous Moran sets with Hausdorff dimension 1 are quasisymmetrically Hausdorff minimal.

We analyze the cubic fragment $\mathcal D_3$ over $\mathbb N$ by isolating the uniform closure principle any total correct cubic solver would have to realize. In $\mathsf{HA}$ we give a fully constructive, additive and degree-controlled encoding of bounded verification: for each externally fixed bound, we effectively produce a finite system of degree-3 Diophantine equations whose solutions represent the existence of the corresponding finite proof or computation trace. The encoding is purely syntactic, using "gadgets" and "Carryless Pairing". In a classical metatheory (e.g. $\mathsf{PA}$) we show that the global solver hypothesis implies a uniform operator eliminating the bound inside $\mathcal D_3$, which is incompatible with standard non-uniformity/realizability constraints. Hence no uniform cubic can exist clasically.

Algebraic Multigrid (AMG) methods are state-of-the-art algebraic solvers for partial differential equations. Still, their efficiency depends heavily on the choice of suitable parameters and/or ingredients. Paradigmatic examples include the so-called strong threshold parameter $\theta$, which controls the algebraic coarse-grid hierarchy, as well as the smoother, i.e., the relaxation methods used on the fine grid to damp out high-frequency errors. In AMG, since the coarse grids are constructed algebraically (without geometric intuition), the smoother's performance is even more critical. For the linear systems stemming from polytopal discretizations, such as Polytopal Discontinuous Galerkin (PolyDG) and Virtual Element Methods (VEM), AMG sensitivity to such choices is even more critical due to the significant variability of the underlying meshes, which results in algebraic systems with different sparsity patterns. We propose a novel deep learning approach that automatically tunes the strong threshold parameter, as well as the smoother choice in AMG solvers, for linear systems of equations arising from polytopal discretizations, thereby maximizing AMG performance. We interpret the sparse matrix resulting from polytopal discretization as a grayscale image, and by applying pooling, our neural network extracts compact features that preserve the necessary information at a low computational cost. We test various differential problems in both two- and three-dimensional settings, with heterogeneous coefficients and polygonal/polyhedral meshes, and demonstrate that the proposed approach generalizes well. In practice, we demonstrate that we can reduce AMG solver time by up to $27\%$ with minimal changes to existing PolyDG and VEM codes.

We consider statistical inference for a class of continuous semimartingale regression models based on high-frequency observations subject to contamination by finite-activity jumps and spike noise. By employing density-power weighting and Hölder-inequality-based normalization, we propose easy-to-implement, robustified versions of the conventional Gaussian quasi-maximum-likelihood estimator that require only a single tuning parameter. We prove their asymptotic mixed normality at the standard rate of $\sqrt{n}$. It is theoretically shown that these estimators are simultaneously robust against contamination in both the covariate and response processes. Additionally, under suitable conditions on the selection of the tuning parameter, the proposed estimators achieve the same asymptotic distribution as the conventional estimator in the contamination-free case. Illustrative simulation results highlight the estimators' insensitivity to the choice of the tuning parameter.

Semantic communication shifts the focus from bit-level accuracy to task-relevant semantic delivery, enabling efficient and intelligent communication for next-generation networks. However, existing multi-modal solutions often process all available data modalities indiscriminately, ignoring that their contributions to downstream tasks are often unequal. This not only leads to severe resource inefficiency but also degrades task inference performance due to irrelevant or redundant information. To tackle this issue, we propose a novel task-aware distributed information bottleneck (TADIB) framework, which quantifies the contribution of any set of modalities to given tasks. Based on this theoretical framework, we design a practical coding scheme that intelligently selects and compresses only the most task-relevant modalities at the transmitter. To find the optimal selection and the codecs in the network, we adopt the probabilistic relaxation of discrete selection, enabling distributed encoders to make coordinated decisions with score function estimation and common randomness. Extensive experiments on public datasets demonstrate that our solution matches or surpasses the inference quality of full-modal baselines while significantly reducing communication and computational costs.

This work develops user-friendly a posteriori error estimates of finite element methods, based on smoothers of linear iterative solvers. The proposed method employs simple smoothers, such as Jacobi or Gauss--Seidel iteration, on an auxiliary finer mesh to process the finite element residual for a posteriori error control. The implementation has linear complexity and requires only a coarse-to-fine prolongation operator. For symmetric problems, we prove the reliability and efficiency of smoother-type error estimators under a saturation assumption. Numerical experiments for various PDEs demonstrate that the proposed smoother-type error estimators outperform residual-type estimators in accuracy and exhibit robustness with respect to parameters and polynomial degrees.

In this note we generalize the results of the recent work by Tom Bohman and Jacob Hofstad on the independence number in G(n, p) to the case of the random k-uniform hypergraph. Concentration in two values occurs in the regime $p>n^{-(k-1)k/(k+1)+\varepsilon}$.

We construct a quenching solution to the parabolic MEMS model \[ u_t = \Delta u - \frac{1}{u^2} \quad \text{in } \mathcal{B} \times (0,T), \quad u|_{\partial \mathcal{B}} = 1, \] where $\mathcal{B}$ is the unit disc in $\mathbb{R}^2$, and $T > 0$ denotes the quenching time. The constructed solution quenches only at the origin and admits the final profile \[ u(x,T) \sim \left(x_1^2 x_2^2 + \theta(x_1^6 + x_2^6)\right)^{\frac{1}{3}} \quad \text{as } |x| \to 0, \] where $\theta \in (0, \theta^*)$ for some $\theta^* > 0$. To our knowledge, this is the first example of a quenching solution with a genuinely non-radial profile. The proof relies on the construction of a good approximate solution, using a perturbative expansion in self-similar variables. We then justify the true solution that remains close to this approximation through a spectral analysis combined with a robust energy method.

We study bifurcations of twisted solutions in a continuum limit (CL) for the Kuramoto model (KM) of identical oscillators defined on nearest neighbor graphs, which may be deterministic dense, random dense or random sparse, when it may have phase-lag. We use the center manifold reduction, which is a standard technique in dynamical systems theory, and prove that the CL suffers bifurcations at which the one-parameter family of twisted solutions becomes unstable and a stable or unstable two-parameter family of modulated twisted solutions that oscillate or not depending on whether the phase-lag exists or not is born. We demonstrate the theoretical results by numerical simulations for the KM on deterministic dense, random dense and random sparse graphs.

We study interior $\varepsilon$-regularity and Type I blowup criteria for suitable weak solutions to the three-dimensional incompressible MHD equations. Our starting point is a direct iteration scheme for the classical Caffarelli--Kohn--Nirenberg scaled energy quantities $A,E,C$ and $D$, which yields $\varepsilon$-regularity criteria under smallness assumptions on the velocity field $u$ and boundedness assumptions on the magnetic field $b$, with the underlying scaling-invariant quantities chosen independently. As an intermediate step, we prove that finiteness of one such scaling-invariant quantity for each of $u$ and $b$ allows only Type I blowup, in the sense that $A(u,b;r)+E(u,b;r)+C(u,b;r)+D(p;r)<\infty$ for small $r$. This extends Seregin's Type I criteria for the Navier--Stokes equations to the MHD setting and provides a natural point of departure for the analysis of Type II blowup. By interpolation and embedding, we further obtain $\varepsilon$-regularity criteria and Type I characterisations in terms of general scaled mixed Lebesgue norms for $u$ and $b$, with independent exponent choices. While we do not aim to sharpen existing mixed-norm $\varepsilon$-regularity criteria, the present formulation offers a unified and comparatively direct route that is naturally compatible with the Type I framework; in particular, the mixed-norm Type I description does not follow from earlier mixed-norm $\varepsilon$-regularity proofs by a formal replacement of the smallness parameter.

This work investigates the global exponential stabilization of a degenerate Euler-Bernoulli beam subjected to a non uniform axial force and a delayed feedback control. First, we establish the well-posedness of the system by constructing an appropriate energy space in weighted Sobolev settings. Using Lümer-Phillips theorem, we prove that the linear operator associated with the problem generates a $\mathcal{C}_0$-semigroup of contractions. Second, we establish the uniform exponential stability of the system. By constructing a novel Lyapunov functional incorporating weighted integral terms, we demonstrate that the energy's system exponentially decays to zero and derive a precise decay rate estimate. This work provides a significant extension to the stability theory for complex distributed parameter systems.

We introduce a general diploid population model with self-fertilization and possible overlapping generations, and study the genealogy of a sample of $n$ genes as the population size $N$ tends to infinity. Unlike traditional approach in coalescent theory which considers the unconditional (annealed) law of the gene genealogies averaged over the population pedigree, here we study the conditional (quenched) law of gene genealogies given the pedigree. We focus on the case of high selfing probability and obtain that this conditional law converges to a random probability measure, given by the random law of a system of coalescing random walks on an exchangeable fragmentation-coalescence process of \cite{berestycki04}. This system contains the system of coalescing random walks on the ancestral recombination graph as a special case, and it sheds new light on the site-frequency spectrum (SFS) of genetic data by specifying how SFS depends on the pedigree. The convergence result is proved by means of a general characterization of weak convergence for random measures on the Skorokhod space with paths taking values in a locally compact Polish space.

Fejér's theorem guarantees norm convergence of Cesàro means of Taylor partial sums in the Hardy space, whereas such convergence generally fails in weighted Dirichlet-type spaces, especially in the higher-order setting. In this paper, we investigate summability problems in higher-order weighted Dirichlet spaces $\widehat{\mathcal{H}}_{\mu,m}$ and show that Taylor partial sums are not uniformly bounded in these spaces and may therefore diverge in norm. To restore convergence, we introduce a family of modified polynomials whose coefficients are adjusted by a suitable weight array. Under mild boundedness and variation assumptions on the weights, we establish norm convergence of the modified sums via a coefficient correspondence principle and a Local Douglas formula. As an application, when the weight measure $\mu$ is a finite sum of Dirac point masses, explicit formulas for the modified coefficients are obtained, yielding a Fejér-type summability theorem for higher-order weighted Dirichlet spaces.

In this study, we consider a multi-pursuer single-evader quantitative pursuit-evasion game with payoff function that includes only the terminal cost. The terminal cost is a function related only to the terminal position of the evader. This problem has been extensively studied in target defense games. Here, we prove that a candidate for the value function generated by geometric method is the viscosity solution of the corresponding Hamilton-Jacobi-Isaacs partial differential equation (HJI PDE) Dirichlet problem. Therefore, the value function of the game at each point can be computed by a mathematical program. In our work, the convexity of the terminal cost or the target is not required. The terminal cost only needs to be locally Lipschitz continuous. The cases in which the terminal costs or the targets are not convex are covered. Therefore, our result is more universal than those of previous studies, and the complexity of the proof is improved. We also discuss the optimal strategies in this game and present an intuitive explanation of this value function.

The question of whether there exists a finite group of order at least three in which every element except one is a commutator has remained unresolved in group theory. In this article, we address this open problem by developing an algorithmic approach that leverages several group theoretic properties of such groups. Specifically, we utilize a result of Frobenius and various necessary properties of such groups, combined with Plesken and Holt's extensive enumeration of finite perfect groups, to systematically examine all finite groups up to a certain order for the desired property. The computational core of our work is implemented using the computer system GAP (Groups, Algorithms, and Programming). We discover two nonisomorphic groups of order 368,640 that exhibit the desired property. Our investigation also establishes that this order is the minimum order for such a group to exist. As a result, this study provides a positive answer to Problem 17.76 in the Kourovka Notebook. In addition to the algorithmic framework, this paper provides a structural description of one of the two groups found.

We consider the central role of improving directions in solution methods for mixed integer bilevel linear optimization problems (MIBLPs). Current state-of-the-art methods for solving MIBLPs employ the branch-and-cut framework originally developed for solving mixed integer linear optimization problems. This approach relies on oracles for two kinds of subproblems: those for checking whether a candidate pair of leader's and follower's decisions is bilevel feasible, and those required for generating valid inequalities. Typically, these two types of oracles are managed separately, but in this work, we explore their close connection and propose a solution framework based on solving a single type of subproblem: determining whether there exists a so-called improving feasible direction for the follower's problem. Solution of this subproblem yields information that can be used both to check feasibility and to generate strong valid inequalities. Building on prior works, we expose the foundational role of improving directions in enforcing the follower's optimality condition and extend a previously known hierarchy of optimality-based relaxations to the mixed-integer setting, showing that the associated relaxed feasible regions coincide exactly with the closure associated with intersection cuts derived from improving directions. Numerical results with an implementation using a modified version of the open source solver MibS show that this approach can yield practical improvements.

Private information retrieval (PIR) is a mechanism for efficiently downloading messages while keeping the index of the desired message secret from the servers. PIR schemes have been extended to various scenarios with adversarial servers: PIR schemes where some servers are unresponsive or return noisy responses are called robust PIR and Byzantine PIR, respectively; PIR schemes where some servers collude to reveal the index are called colluding PIR. The information-theoretic upper bound on the download efficiency of these PIR schemes has been proved in previous studies. However, systematic ways to construct PIR schemes that achieve the upper bound are not known. In order to construct a capacity-achieving PIR schemes systematically, it is necessary to clarify the conditions that the queries should satisfy. This paper proves the necessary and sufficient conditions for capacity-achieving PIR schemes.

Raimi's theorem guarantees the existence of a partition of $\mathbb{N}$ into two parts with an unavoidable intersection property: for any finite coloring of $\mathbb{N}$, some color class intersects both parts infinitely many times, after an appropriate shift (translation). We establish a polynomial extension of this result, proving that such intersections persist under polynomial shifts in any dimension. Let $P^{(1)},\dots,P^{(f)}\in\mathbb{Z}[x]$ be non-constant polynomials with positive leading coefficients and $P^{(j)}(0)=0$ for every $j$. We construct a partition of $\mathbb{N}^k$ into an arbitrarily fixed finite number of pieces such that for any coloring of $\mathbb{N}^k$ with finitely many colors, there exist $x_0\in \mathbb{N}$ and a single color class that meets all partition pieces after shifts by $x_0+P^{(j)}(h)$ in each of the $k$ coordinate directions, for every $j$ and infinitely many values $h\in \mathbb{N}$. Our proof exploits Weyl's equidistribution theory, Pontryagin duality, and the structure of polynomial relation lattices. We also prove some finite analogues of the above results for abelian groups and $SL_2(\mathbb{F}_q)$.

An antidirected cycle in a digraph $G$ is a subdigraph whose underlying graph is a cycle, and in which no two consecutive edges form a directed path in $G$. Let $\sigma_{+-}(G)$ be the minimum value of $d^+(x)+d^-(y)$ over all pairs of vertices $x, y$ such that there is no edge from $x$ to $y$, that is, $$\sigma_{+-}(G)=\min\{d^+(x)+d^-(y): \{x,y\}\subseteq V(G), xy\notin E(G)\}.$$ In 1972, Woodall extended Ore's theorem to digraphs by showing that every digraph $G$ on $n$ vertices with $\sigma_{+-}(G)\geqslant n$ contains a directed Hamilton cycle. Very recently, this result was generalized to oriented graphs under the condition $\sigma_{+-}(G)\geqslant(3n-3)/4$. In this paper, we give the exact Ore-type degree threshold for the existence of antidirected Hamilton cycles in oriented graphs. More precisely, we prove that for sufficiently large even integer $n$, every oriented graph $G$ on $n$ vertices with $\sigma_{+-}(G)\geqslant(3n+2)/4$ contains an antidirected Hamilton cycle. Moreover, we show that this degree condition is best possible.

Let $C$ be a smooth projective absolutely irreducible curve of genus at least 2, defined over the rationals. For a number field $L$, we define the set of $L$-new points on $C$ to be $C(L)_{new} = \{P \in C(L) : \mathbb{Q}(P)=L\}$; this is the set of points on $C$ defined over $L$ but not any strictly smaller field. Let $n$ be at least 2. We conjecture that $C(L)_{new}$ is empty for 100 percent of degree $n$ number fields $L$ when ordered by absolute discriminant. For degrees $n=2$, $3$, we give sufficient criteria for our conjecture to hold in terms of an explicit model for $C$. For general $n$ we prove a theorem that harmonises with the conjecture. In particular, we verify our conjecture for $n=2$ and $C=X_0(N)$ for the $18$ values $N \ne 37$ such that $X_0(N)$ is hyperelliptic, and also for $n=3$ and $C=X_0(23)$, $X_0(29)$, $X_0(31)$, $X_0(64)$. Moreover, we prove the analogue of our conjecture for the unit equation, again with $n=3$.

We survey classical localization problems arising from quantum network models in symmetry class C and their mappings to history-dependent random walks on directed lattices. We describe how localization versus delocalization of trajectories can be analysed using percolation methods and combinatorial enumeration of path intersection patterns. In particular, we review results establishing almost sure finiteness of trajectories for parameters near criticality and polynomial bounds on the confinement length in cylindrical geometries.

We study the local statistical behavior of the super-Coulombic Riesz gas of particles in Euclidean space of arbitrary dimension, with inverse power distance repulsion integrable near $0$, and with a general confinement potential, in a certain regime of inverse temperature. Using a bootstrap procedure, we prove local laws on the next order energy and control on fluctuations of linear statistics that are valid down to the microscopic lengthscale, and provide controls for instance, on the number of particles in a (mesoscopic or microscopic) box, and the existence of a limit point process up to subsequences. As a consequence of the local laws, we derive an almost additivity of the free energy that allows us to exhibit for the first time a CLT for Riesz gases corresponding to small enough inverse powers, at small mesoscopic length scales, which can be interpreted as the convergence of the associated potential to a fractional Gaussian field. Compared to the Coulomb interaction case, the main new issues arise from the nonlocal aspect of the Riesz kernel. This manifests in (i) a novel technical difficulty in generalizing the transport approach of Leblé and the second author to the Riesz gas which now requires analyzing a degenerate and singular elliptic PDE, (ii) the fact that the transport map is not localized, which makes it more delicate to localize the estimates, (iii) the need for coupling the local laws and the fluctuations control inside the same bootstrap procedure.

The rapid scaling of artificial intelligence models has revealed a fundamental tension between model capacity (storage) and inference efficiency (computation). While classical information theory focuses on transmission and storage limits, it lacks a unified physical framework to quantify the thermodynamic costs of generating information from compressed laws versus retrieving it from memory. In this paper, we propose a theoretical framework that treats information processing as an enabling mapping from ontological states to carrier states. We introduce a novel metric, Derivation Entropy, which quantifies the effective work required to compute a target state from a given logical depth. By analyzing the interplay between Shannon entropy (storage) and computational complexity (time/energy), we demonstrate the existence of a critical phase transition point. Below this threshold, memory retrieval is thermodynamically favorable; above it, generative computation becomes the optimal strategy. This "Energy-Time-Space" conservation law provides a physical explanation for the efficiency of generative models and offers a rigorous mathematical bound for designing next-generation, energy-efficient AI architectures. Our findings suggest that the minimization of Derivation Entropy is a governing principle for the evolution of both biological and artificial intelligence.

We introduce the \emph{Generalized Latin Square Graph} $\Gamma(S)$ of a finite semigroup $S$. Since we record global factorization multiplicities and local alternative counts, we define three counting invariants $N_S,N_R,N_C$. This gives that we have a simple degree formula \[ \text{deg}(v)=2n-3+Q(v),\qquad Q(v)=N_S(s_k)-2N_R(v)-2N_C(v). \] We show that $\Gamma(S)$ is regular exactly when $Q$ is constant. We apply the framework to cancellative semigroups, bands, Brandt semigroups and null semigroups. For null semigroups, since we identify $\Gamma(S)\cong K_n\times K_n$, we compute the spectrum and energy. A concise computational appendix lists the \texttt{GAP} driver and representative outputs.

It is an old idea to use gradient flows or time-discretized variants thereof as methods for solving minimization problems. In some applications, for example in machine learning contexts, it is important to know that for generic initial data, gradient flow trajectories do not get stuck at saddle points. There are classical results concerned with the non-degenerate situation. But if the Hessian of the objective function has a non-trivial kernel at the critical point, then these results are inconclusive in general. In this paper, we show how relevant information can be extracted by ``blowing up'' the objective function around the non-strict saddle point, i.e., by a suitable non-linear rescaling that makes the higher order geometry visible.

We consider Eulerian cycles without transversal selfintersections in $4$-valent planar graphs. We prove that any cycle of this type in the graph of an ideal right-angled hyperbolic $3$-polytope corresponds to a hyperbolic link such that its complement consists of $4$-copies of this polytope glued according to its checkerboard coloring. Moreover, this link consists of trivially embedded circles bijectively corresponding to the vertices of the polytope. We prove that there is always at least one Eulerian cycle of this type and introduce a transformation of a cycle along conjugated vertices allowing to build new cycles from a given one. The $2$-fold branched covering space corresponding to our link is a small cover over some simple $3$-polytope. It corresponds to a Hamiltonian cycle on this polytope via the A.D. Mednykh's construction. We show that any Hamiltonian cycle on a compact right-angled hyperbolic $3$-polytope arises in this way, while in the finite volume case the necessary and sufficient condition is that at each ideal vertex it does not go straight. The link corresponding to a Hamiltonian cycle on a simple $3$-polytope always contains the Hopf link consisting of two circles. We consider links corresponding to Hamiltonian theta-graphs and Hamiltonian $K_4$-graphs on simple $3$-polytopes introduced by A.D. Mednykh and this http URL. Vesnin. We give a criterion when such a link consists of mutually unlinked circles and when it is trivial. We give a necessary condition for such a link to be hyperbolic. The simplest example is the Borromean rings corresponding to the Hamiltonian theta-graph on the cube. We introduce the notions of a nonselfcrossing Eulerian theta-graph and $K_4$-graph on a right-angled hyperbolic $3$-polytope of finite volume with $2$ or $4$ finite vertices and construct the corresponding hyperbolic link.

This work establishes a novel, unified theoretical framework for a class of high order embedded boundary methods, revealing that the Reconstruction for Off-site Data (ROD) treatment shares a fundamental structure with the recently developed shifted boundary polynomial correction [Ciallella, M., et al. (2023)]. By proving that the ROD minimization problem admits an equivalent direct polynomial correction formulation, we unlock two major advances. First, we derive a significant algorithmic simplification, replacing the solution of the minimization problem with a straightforward polynomial evaluation, thereby enhancing computational efficiency. Second, and most critically, this reformulation enables the first stability result for the ROD method when applied to the linear advection equation with discontinuous Galerkin discretization. Our analysis, supported by a comprehensive eigenspectrum study for polynomial degrees up to six, characterizes the stability region of the new ROD formulation. The theoretical findings, which demonstrate the stability constraints, are validated through targeted numerical experiments.

This paper surveys randomized algorithms in numerical linear algebra for low-rank decompositions of matrices and tensors. The survey begins with a review of classical matrix algorithms that can be accelerated by randomized dimensionality reduction, such as the singular value decomposition (SVD) or interpolative (ID) and CUR decompositions. Recent advances in randomized dimensionality reduction are discussed, including new methods of fast matrix sketching and sampling techniques, which are incorporated into classical matrix algorithms for fast low-rank matrix approximations. The extension of randomized matrix algorithms to tensors is then explored for several low-rank tensor decompositions in the CP and Tucker formats, including the higher-order SVD, ID, and CUR decomposition.

This paper develops a strong computational approach to simulate a three-dimensional nonlinear radiation-conduction model in optically thick media, subject to suitable initial and boundary conditions. The space derivatives are approximated by the mixed finite element method ($\mathcal{P}_{p}/\mathcal{P}_{p-1}$), while the interpolation technique is employed in two stages to approximate the time derivative. The proposed strategy is so-called, a strong two-stage explicit/implicit computational technique combined with mixed finite element method. Specifically, the new algorithm should be observed as a predictor-corrector numerical scheme. Additionally, it efficiently treats the time derivative term and provides a necessary requirement on time step for stability. Under this time step limitation, the stability is deeply analyzed whereas the convergence order is numerically computed in the $L^{2}$-norm. The theoretical results suggest that the developed approach is stable and temporal second-order accurate. Some numerical experiments are performed to confirm the theory, to establish that the constructed method is spatial fourth-order convergent and to demonstrate the practical applicability and computational efficiency of the numerical scheme.

In this work, we study the simplest example of the landscape of conformal field theories: one-dimensional CFTs with finite-dimensional state space. Following the definition of quantum field theory given by G. Segal, we formulate the condition under which a one-dimensional QFT (quantum mechanics) possesses conformal symmetry, and we give a complete classification of conformal Hamiltonians with finite rank. It turns out that correlation functions in such theories are polynomial functions of the underlying geometric data. Moreover, the one-dimensional conformal Ward identities determine their scaling behavior, so that the correlators of the conformal observables are, in fact, homogeneous polynomials.

This paper studies short brooms in edge-chromatic critical graphs. We prove that for any short broom in a $\Delta$-critical graph, at most one color is missing at more than one vertex. Moreover, this color (if exists) is missing at exactly two vertices. Applying this result, we verify the Vertex-splitting Conjecture for graphs with $\Delta \geq 2(n-1)/3$ and the Overfull Conjecture for $\Delta$-critical graphs satisfying $\Delta \geq (2n+5\delta-12)/3$.

To construct a BPS algebra with representations furnished by n-dimensional partitions, the first step is to construct the eigenvalue of the Cartan operators acting on them. The generating function of the eigenvalues is called the charge function. It has an important property that for each partition, the poles of the function correspond to the projection of the boxes which can be added to or removed from the partition legally. The charge functions of lower dimensional partitions, i.e., Young diagrams for 2D, plane partitions for 3D and solid partitions for 4D, are already given in the literature. In this paper, we propose an expression of the charge function for arbitrary odd dimensional partitions and have it proved for 5D case. Some explicit numerical tests for 7D and 9D case are also conducted to confirm our formula.

We show that, in contrast to the case of Laplace eigenfunctions, the nodal set of high energy eigenfunctions of the clamped plate problem is not necessarily dense, and can in fact exhibit macroscopic "nodal voids". Specifically, we show that there are small deformations of the unit disk admitting a clamped plate eigenfunction of arbitrarily high frequency that does not vanish in a disk of radius 0.44.

We prove a sharp upper bound on negative moments of sums of independent Steinhaus random variables (that is uniform on circles in the plane). Together with the series of earlier works: König-Kwapień (2001), Baernstein II-Culverhouse (2002), and König (2014), this closes the investigation of sharp $L_p-L_2$ Khinchin-type inequalities for the Steinhaus sums. Incidentally, we fix a mistake in an earlier paper, as well as provide an application to sharp bounds on Rényi entropy.

In this paper we investigate the complex exponential integral means spectrums of univalent functions in the unit disk. We show that all integral means spectrum (IMS) functionals for complex exponents on the universal Teichmüller space, the closure of the universal Teichmüller curve, and the universal asymptotic Teichmüller space are continuous. We also show that the complex exponential integral means spectrum of any univalent function admitting a quasiconformal extension to the extended complex plane is strictly less than the universal integral means spectrum. These extend some related results in our recent work \cite{Jin}. Here we employ a different and more direct approach to prove the continuity of IMS functional on the universal asymptotic Teichmüller space. Additionally, we completely determine the integral means spectrums of all univalent rational functions in the unit disk. As a consequence, we show that the Brennan conjecture is true for this class of univalent functions. Finally, we present some remarks and raise some problems and conjectures regarding IMS functionals on Teichmüller spaces, univalent rational functions, and a multiplier operator whose norm is closely related to the Brennan conjecture.

A. Einstein considered a manifold with a non-symmetric (0,2)-tensor $G=g+F$, where $g$ is a Riemannian metric and $F\ne0$, and a connection $\nabla$ with torsion $T$ such that $(\nabla_X G)(Y,Z)=-G(T(X,Y),Z)$. Guided by the almost Lie algebroid construction on a vector bundle, we define the basic concepts of Bochner's technique for Einstein's non-symmetric geometry, give a clear example of the Einstein's connection $\nabla$, prove Weitzenböck type decomposition formula and obtain vanishing results about the null space of the Bochner and Hodge type Laplacians.

This article gives a self-contained proof of Mostow Rigidity, at least modulo undergrad real analysis. The proof should be accessible to grad students interested in geometry and topology. It has no new research, but I think that this is an unusually clean and analytically light proof of this famous result. I am posting this because I think it will be useful to geometry/topology students.

A hybrid computational approach that integrates the finite element method (FEM) with least squares support vector regression (LSSVR) is introduced to solve partial differential equations. The method combines FEM's ability to provide the nodal solutions and LSSVR with higher-order Legendre polynomial kernels to deliver a closed-form analytical solution for interpolation between the nodes. The hybrid approach implements element-wise enhancement (super-resolution) of a given numerical solution, resulting in high resolution accuracy, while maintaining consistency with FEM nodal values at element boundaries. It can adapt any low-order FEM code to obtain high-order resolution by leveraging localized kernel refinement and parallel computation without additional implementation overhead. Therefore, effective inference/post-processing of the obtained super-resolved solution is possible. Evaluation results show that the hybrid FEM-LSSVR approach can achieve significantly higher accuracy compared to the base FEM solution. Comparable accuracy is a achieved when comparing the hybrid solution with a standalone FEM result with the same polynomial basis function order. The convergence studies were conducted for four elliptic boundary value problems to demonstrate the method's ability, accuracy, and reliability. Finally, the algorithm can be directly used as a plug-and-play method for super-resolving low-order numerical solvers and for super-resolution of expensive/under-resolved experimental data.

We study the deviation from proportionality of rows and columns in products of positive matrices. We prove a sharp, dimension-free bound showing that worst-case misalignment is already captured in dimension two and follows an explicit Mobius law, refining the classical Birkhoff-Bushell contraction theory.

A classical result of Tutte shows that every 4-connected planar graph is Hamiltonian. In recent years, there has been growing interest in extending classical Hamiltonian results from planar graphs to sparse graphs with drawings allowing crossings, such as $k$-planar graphs, where each edge is crossed at most $k$ times. For example, using different approaches, Hudák, Tomáš and Suzuki, as well as Noguchi and Suzuki, independently proved that every optimal 1-planar graph is Hamiltonian. Here, an optimal 1-planar graph refers to one that attains the maximum possible number of edges. In this paper, we establish results on the Hamiltonicity of optimal 2-planar graphs, that is, 2-planar graphs with the maximum number of edges. More precisely, we show that every 4-connected optimal 2-planar graph is Hamiltonian-connected. With vertex-connectivity 3, there exist infinitely many optimal 2-planar graphs that are non-Hamiltonian.

The \textit{Teleportation=Translation} conjecture posits that the recovery of information from a black hole is dual to a geometric translation in the emergent spacetime. In this paper, we establish this equivalence by constructing a continuous family of unitaries that bridges the discrete algebraic teleportation protocol and modular flow. We resolve the failure of dynamic idempotency, inherent in Type III von Neumann algebras, by employing the Haagerup-Kosaki crosse-product construction. This lift to the semifinite envelope yields a canonical, dynamically consistent path whose unique self-adjoint generator $\tilde{G}$ is proven to be twice the modular Hamiltonian difference, $\tilde{G}=2(K_{\tilde{\mathcal{M}}}-K_{\tilde{\mathcal{N}}})$. We establish this identity as a closed operator equivalence using Nelson's analytic vector theorem and quantify its structural robustness via Kosaki's non-commutative $L^p$ theory. Our results provide a concrete analytic mechanism for probing emergent geometry from quantum information, offering a kinematic framework naturally extendable to include gravitational back-reaction.

We provide a complete list of the connected components of the spaces of non-discriminant functions within standard versal deformations of function singularities of classes $X_9$, $J_{10}$ and $P_8^1$ (as well as a partial list for the remaining class, $P_8^2$). Thus, we prove (and improve in one particular case) the corresponding conjectures from the previous work \cite{para} with the same title. As an application, we enumerate all local Petrovskii lacunas near arbitrary parabolic singularities of wavefronts of hyperbolic PDEs. In particular, we discover a new local lacuna at the $P_8^2$ singularities. We also show that the complements of the discriminant varieties of $X_9^+$ and $P_8^1$ singularities have nontrivial one-dimensional homology groups, unlike all simple singularities.

This paper examines the classification of hyperbolic equations. We study a class of equations of the form $$\frac{\partial^2 u}{\partial x\partial y}=F\left(\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},u\right),$$ where $u(x,y)$ is the unknown function and $x,y$ are independent variables. The classification is based on the requirement for the existence of higher fifth-order symmetries. As a result, a list of four equations with the required conditions was obtained.

We consider sets of fixed CP, multilinear, and TT rank tensors, and derive conditions for when (the smooth parts of) these sets are smooth homogeneous manifolds. For CP and TT ranks, the conditions are essentially that the rank is sufficiently low. These homogeneous structures are then used to derive Riemannian metrics whose geodesics are both complete and efficient to compute.

This paper examines aspirational requirements for software addressing mixed-integer optimization problems constrained by the nonlinear Shallow Water partial differential equations (PDEs), motivated by applications such as river-flow management in hydropower cascades. Realistic deployment of such software would require the simultaneous treatment of nonlinear and potentially non-smooth PDE dynamics, limited theoretical guarantees on the existence and regularity of control-to-state mappings under varying boundary conditions, and computational performance compatible with operational decision-making. In addition, practical settings motivate consideration of uncertainty arising from forecasts of demand, inflows, and environmental conditions. At present, the theoretical foundations, numerical optimization methods, and large-scale scientific computing tools required to address these challenges in a unified and tractable manner remain the subject of ongoing research across the associated research communities. Rather than proposing a complete solution, this work uses the problem as a case study to identify and organize the mathematical, algorithmic, and computational components that would be necessary for its realization. The resulting framework highlights open challenges and intermediate research directions, and may inform both more circumscribed related problems and the design of future large-scale collaborative efforts aimed at addressing such objectives.

The notion of \emph{contact triad connection} on contact triads $(Q,\lambda,J_\xi)$ was introduced by Wang and the present author in early 2010's from scratch as the contact analog to the canonical connection of an almost Kähler manifolds. The connection facilitates the study of analysis of the contact instanton equation which ranges from local elliptic priori estimates, for both the interior [OW2,OW3] and the boundary [OY], to the generic perturbation theory of the asymptotic operators [KO], which also encompasses the case of pseudoholomorphic curves on noncompact symplectic manifolds with cylindrical ends [OK]. The main purpose of the present paper is to give a simpler and more canonical construction of the contact triad connection by first giving its characterization in terms of the associated \emph{almost contact structure} and then providing its construction using the almost contact moving frame in the framework of contact geometry.

This paper develops a general mathematical framework for pseudo-parabolic gradient systems with state-dependent dynamics. The state dependence is induced by variable coefficient fields in the governing energy functional. Such coefficients arise naturally in scientific and technological models, including state-dependent mobilities in KWC-type grain boundary motion and variable orientation-adaptation operators in anisotropic image denoising. We establish two main results: the existence of energy-dissipating solutions, and the uniqueness and continuous dependence on initial data. The proposed framework yields a general well-posedness theory for a broad class of nonlinear evolutionary systems driven by state-dependent operators. As illustrative applications, we present an anisotropic image-denoising model and a new pseudo-parabolic KWC-type model for anisotropic grain boundary motion, and prove that both fit naturally within the abstract structure of $(\mathrm{S})_\nu$.

We solve the capillary $L_p$-Christoffel--Minkowski problem in the half-space for $1<p<k+1$ in the class of even hypersurfaces. A crucial ingredient is a non-collapsing estimate that yields lower bounds for both the height and the capillary support function. Our result extends the capillary Christoffel--Minkowski existence result of \cite{HIS25}.

This work proposes a unified control architecture that couples a Reinforcement Learning (RL)-driven controller with a disturbance-rejection Extended State Observer (ESO), complemented by an Event-Triggered Mechanism (ETM) to limit unnecessary computations. The ESO is utilized to estimate the system states and the lumped disturbance in real time, forming the foundation for effective disturbance compensation. To obtain near-optimal behavior without an accurate system description, a value-iteration-based Adaptive Dynamic Programming (ADP) method is adopted for policy approximation. The inclusion of the ETM ensures that parameter updates of the learning module are executed only when the state deviation surpasses a predefined bound, thereby preventing excessive learning activity and substantially reducing computational load. A Lyapunov-oriented analysis is used to characterize the stability properties of the resulting closed-loop system. Numerical experiments further confirm that the developed approach maintains strong control performance and disturbance tolerance, while achieving a significant reduction in sampling and processing effort compared with standard time-triggered ADP schemes.

We study the \emph{Bipartite Degree Realization} (BDR) problem: given a graphic degree sequence $D$, decide whether it admits a realization as a bipartite graph. While bipartite realizability for a fixed vertex partition can be decided in polynomial time via the Gale--Ryser theorem, the computational complexity of BDR without a prescribed partition remains unresolved. We address this question through a parameterized analysis. For constants $0 \le c_1 \le c_2 \le 1$, we define $\mathrm{BDR}_{c_1,c_2}$ as the restriction of BDR to degree sequences of length $n$ whose degrees lie in the interval $[c_1 n, c_2 n]$. Our main result shows that $\mathrm{BDR}_{c_1,c_2}$ is solvable in polynomial time whenever $0 \le c_1 \le c_2 \le \frac{\sqrt{c_1(c_1+4)}-c_1}{2}$, as well as for all $c_1 > \tfrac12$. The proof relies on a reduction to extremal \emph{least balanced degree sequences} and a detailed verification of the critical Gale--Ryser inequalities, combined with a bounded subset-sum formulation. We further show that, assuming the NP-completeness of unrestricted BDR, the problem $\mathrm{BDR}_{c_1,c_2}$ remains NP-complete for all $0 < c_2 < \frac{1}{2}$ and $c_1 < 1 - c_2 - \sqrt{1-2c_2}$. % This establishes a sharp conditional boundary between tractable and intractable parameter regimes. Our results clarify the algorithmic landscape of bipartite degree realization and contribute to the broader study of potentially bipartite graphic degree sequences.

This paper reviews recent results and open problems on the conductor of finite group characters, highlighting their connections to one another and to broader topics in the representation theory of finite groups.

Piecewise quasipolynomial growth of Presburger counting functions combines with tame persistent homology module theory to conclude piecewise quasipolynomial behavior of constructible families of finely graded modules over constructible commutative semigroup rings. Functorial preservation of constructibility for families under local cohomology, $\operatorname{Tor}$, and $\operatorname{Ext}$ yield piecewise quasipolynomial, quasilinear, or quasiconstant growth statements for length of local cohomology, $a$-invariants, regularity, depth; length of $\operatorname{Tor}$ and Betti numbers; length of $\operatorname{Ext}$ and Bass numbers; associated primes via $v$-invariants; and extended degrees, including the usual degree, Hilbert-Samuel multiplicity, arithmetic degree, and homological degree.

We introduce and systematically study \emph{hierarchical filtrations} of vector bundles on smooth projective varieties. These are filtrations by saturated subsheaves of equal rank whose successive quotients are torsion sheaves supported in codimension one. The associated numerical invariant, called \emph{hierarchical depth}, measures the maximal length of such filtrations. We establish general bounds for hierarchical depth in terms of the determinant class and provide exact formulas for smooth curves and varieties of Picard rank one. A key technical result concerns the commutativity of elementary transforms along disjoint divisors and their role in constructing filtrations. For surfaces, we analyze the behavior of hierarchical depth under birational morphisms and prove that it transforms additively along the steps of the minimal model program. In particular, we obtain an explicit formula relating the depth on a surface to that on its minimal model via exceptional divisor contributions. As an application, we connect hierarchical depth to degeneracies in algebraic--geometric codes and show that birational simplification via the MMP leads to effective improvements of code parameters. This establishes hierarchical depth as a new bridge between birational geometry, vector bundle theory, and coding theory.

We study \emph{unimodular fake} $\mu's$, i.e. multiplicative functions $\mathfrak f: \N \to \mathbb{S}^1 \cup \{0\} $ determined by a fixed sequence $\{\varepsilon_k\}_{k\ge 0} \subset \mathbb{S}^1 \, \cup \, \{0\}$ via the rule $\mathfrak f(p^k)=\varepsilon_k$ for every prime $p$ and $k \ge 0$. The Dirichlet series of a given $\mathfrak f$ admits the Euler product \[ F_{\mathfrak f}(s)=\sum_{n\ge1}\frac{\mathfrak f(n)}{n^s} =\prod_p g_{\mathfrak f}(p^{-s}),\qquad g_{\mathfrak f}(u)=\sum_{k\ge0}\varepsilon_k u^k, \] and the canonical zeta-factorization \[ F_{\mathfrak f}(s)=\zeta(s)^{\,z}\,\zeta(2s)^{\,w}\,G_{\mathfrak f}(s), \qquad z=\varepsilon_1,\ \ w=\varepsilon_2-\frac{\varepsilon_1(\varepsilon_1+1)}{2}, \] where $G_{\mathfrak f}(s)$ is a holomorphic Euler product on $\Re s>1/3$. Assuming the Riemann hypothesis and Simple Zeros Conjecture, we derive an explicit formula for $A_{\mathfrak f}^{\exp}(x)= \sum_{n \ge 1} \mathfrak f(n) \, e^{-n/x} $ of the form \[ A_{\mathfrak f}^{\exp}(x) -\Delta_1(x;z,w) = \Delta_{1/2}(x;z,w)\;+\;\sum_{\rho}\Delta_\rho(x;z,w,\mathfrak f)\;+\;\mathcal E(x). \] To our knowledge, our expansion is the first extension of the Selberg-Delange method for Dirichlet series of the form $\zeta(s)^{\,z}\,\zeta(2s)^{\,w}\,G(s)$ that, beyond the main term from $s=1$, also extracts lower-order contributions from the singularities on the critical line $\Re(s)=1/2$. On the other hand, we introduce a notion of \emph{bias} at the natural scale $x^{1/2}(\Log x)^{w-1}$ and obtain an explicit criterion distinguishing \emph{persistent}, \emph{apparent}, and \emph{unbiased} behavior in this regime.

Most intelligent reflecting surface (IRS)-aided indoor visible light communication (VLC) studies ignore the time delays introduced by reflected paths, even though these delays are inherent in practical wideband systems. In this work, we adopt a realistic assumption of IRS-induced time delay for physical layer security (PLS) enhancement. We consider an indoor VLC system where an IRS is used to shape the channel so that the reflected signals add constructively at the legitimate user and create intersymbol interference at eavesdroppers located inside the coverage area. The resulting secrecy capacity maximisation over the IRS element allocation is formulated as a complex combinatorial optimisation problem and is solved using deep reinforcement learning with proximal policy optimisation (PPO). The approach is evaluated for both colluding eavesdroppers, which combine their received signals, and non-colluding eavesdroppers, which act independently. Simulation results are shown for various simulation setups, which demonstrate significant secrecy capacity gains. In a worst-case scenario, where the eavesdroppers have stronger channels than the legitimate user, the proposed PPO-based IRS allocation improves secrecy capacity by 107\% and 235\% in the colluding and non-colluding cases, respectively, compared with allocating all IRS elements to the legitimate user. These results demonstrate that time-delay-based IRS control can provide a strong secrecy advantage in practical indoor VLC scenarios.

We study the limiting spectral distribution of the normalized Laplacian $\mathcal L$ of an Erdős-Rényi graph $G(n,p)$. To account for the presence of isolated vertices in the sparse regime, we define $\mathcal L$ using the Moore-Penrose pseudoinverse of the degree matrix. Under this convention, we show that the empirical spectral distribution of a suitably normalized $\mathcal L$ converges weakly in probability to the semicircle law whenever $np\to\infty$, thereby providing a rigorous justification of a prediction made in (Akara-pipattana and Evnin, 2023). Moreover, if $np>\log n+\omega(1)$, so that $G(n,p)$ has no isolated vertices with high probability, the same conclusion holds for the standard definition of $\mathcal L$. We further strengthen this result to almost sure convergence when $np=\Omega(\log n)$. Finally, we extend our approach to the Chung-Lu random graph model, where we establish a semicircle law for $\mathcal L$ itself, improving upon (Chung, Lu, and Vu 2003), which obtained the semicircle law only for a proxy matrix.

Cooperative collision avoidance between robots, or `agents,' in swarm operations remains an open challenge. Assuming a decentralized architecture, each agent is responsible for making its own decisions and choosing its control actions. Most existing approaches rely on a (wireless) communication network between (some of) the agents. In reality, however, communication is brittle. It may be affected by latency, further delays and packet losses, and transmission faults. Moreover, it is subject to adversarial attacks, such as jamming or spoofing. This paper proposes Contingency Model-based Control (CMC), a decentralized cooperative approach that does not rely on communication. Instead, the control algorithm is based on consensual rules that are designed for all agents offline, similar to traffic rules. For CMC, this includes the definition of a contingency trajectory for each robot, and perpendicular bisecting planes as collision avoidance constraints. The setup permits a full guarantee of recursive feasibility and collision avoidance between all swarm members in closed-loop operation. CMC naturally satisfies the plug & play paradigm, i.e., new robots may enter the swarm dynamically. The effectiveness of the CMC regime is demonstrated in two numerical examples, showing that the collision avoidance guarantee is intact and the robot swarm operates smoothly in a constrained environment.

We study the Euclidean Distance degree of algebraic neural network models from the perspective of algebraic geometry. Focusing on shallow networks with two neurons, quadratic activation, and scalar output, we identify the associated neurovariety with the second secant variety of a quadratic Veronese embedding. We introduce and analyze the virtual Euclidean Distance degree, a projective invariant defined as the sum of the polar degrees of the variety, which coincides with the usual Euclidean Distance degree for a generic choice of scalar product. Using intersection theory, Chern-Mather classes, and the Nash blow-up provided by Kempf's resolution, we reduce the computation of the virtual Euclidean Distance degree to explicit intersection numbers on a Grassmannian. Applying equivariant localization, we prove that this invariant depends stably polynomially on the input dimension. Numerical experiments based on homotopy continuation illustrate the dependence of the Euclidean Distance degree on the chosen metric and highlight the distinction between the generic and nongeneric cases, such as the Bombieri-Weyl metric.

In this article, we introduce a three-precision formulation of the General Alternating-Direction Implicit method (GADI) designed to accelerate the solution of large-scale sparse linear systems $Ax=b$. GADI is a framework that can represent many existing Alternating-Direction Implicit (ADI) methods. These methods are a class of linear solvers based on a splitting of $A$ such that the solution of the original linear system can be decomposed into the successive computation of easy-to-solve structured subsystems. Our proposed mixed precision scheme for GADI solves these subsystems in low precision to reduce the overall execution time while computing the residual and solution update in high precision to enable the solution to converge to high accuracy. We develop a rounding error analysis of mixed precision GADI that establishes the rates of convergence of the forward and backward errors to certain limiting accuracies. Our analysis also highlights the conditions on the splitting matrices under which mixed precision GADI is guaranteed to converge for a given set of precisions. We then discuss a systematic and robust strategy for selecting the GADI regularization parameter $\alpha$, whose adjustment is critical for performance. Specifically, our proposed strategy makes use of a Gaussian Process Regression (GPR) model trained on a dataset of low-dimensional problems to initialize $\alpha$. Finally, we proceed to a performance analysis of mixed precision GADI on an NVIDIA A100 GPU to validate our approach. Using low precision (Bfloat16 or FP32) to solve the subsystems, we obtain speedups of $2.6\times$, $1.7\times$, and $3.1\times$ over a full double precision GADI implementation on large-scale 2D, 3D convection-diffusion and complex reaction-diffusion problems (up to $1.3\times 10^{8}$ unknowns), respectively.

The entropically regularized optimal transport problem between probability measures on compact Euclidean subsets can be represented as an information projection with moment inequality constraints. This allows its Fenchel dual to be approximated by a sequence of convex, finite-dimensional problems using sieve methods, enabling tractable estimation of the primal value and dual optimizers from samples. Assuming only continuity of the cost function, I establish almost sure consistency of these estimators. I derive a finite-sample convergence rate for the primal value estimator, showing logarithmic dependence on sieve complexity, and quantify uncertainty for the dual optimal value estimator via matching stochastic bounds involving suprema of centered Gaussian processes. These results provide the first statistical guarantees for sieve-based estimators of entropic optimal transport, extending beyond the empirical Sinkhorn approach.

Motivated by a question of Erdös and inquiries by Beeson and Laczkovich, we explore the possible $N$ for which a triangle $T$ can tile into $N$ congruent copies of a triangle $R$. The \emph{reptile} cases (where $T$ is similar to $R$) and the \emph{commensurable-angles} cases (where all angles of $R$ are rational multiples of $\pi$) are well-understood. We tackle the most interesting remaining case, which is when $R$ contains an angle of $2\pi/3$ and when $T$ is one of $6$ ``sporadic'' specific triangles, of which only $2$ were known to have constructions. For each of these, we create a family of constructions and conjecture that they are the only possible $N$ that occur for these triangles.

By using our previous results on Lê modules and an upper-bound on the betti numbers which we proved with Lê, we investigate the cohomology of Milnor fibers and the internal local systems given by the vanishing cycles of hypersurfaces with one-dimensional singular sets.

This article improves the estimate of $|S_1(t_2)-S_1(t_1)|$, which is the definite integral of the argument of the Riemann zeta-function between $t_1$ and $t_2$. Estimates of this quantity are needed to apply Turing's method to compute the exact number of zeta zeros up to a given height.

Motivated by the constructions of binary sequences by utilizing the cyclic elliptic function fields over the finite field $\mathbb{F}_{2^{n}}$ by Jin \textit{et al.} in [IEEE Trans. Inf. Theory 71(8), 2025], we extend the construction to the cyclic elliptic function fields with odd characteristic by using the quadratic residue map $\eta$ instead of the trace map used therein. For any cyclic elliptic function field with $q+1+t$ rational points and any positive integer $d$ with $\gcd(d, q+1+t)=1$, we construct a new family of binary sequences of length $q+1+t$, size $q^{d-1}-1$, balance upper bounded by $(d+1)\cdot\lfloor2\sqrt{q}\rfloor+|t|+d,$ the correlation upper bounded by $(2d+1)\cdot\lfloor2\sqrt{q}\rfloor+|t|+2d$ and the linear complexity lower bounded by $\frac{q+1+2t-d-(d+1)\cdot\lfloor2\sqrt{q}\rfloor}{d+d\cdot\lfloor2\sqrt{q}\rfloor}$ where $\lfloor x\rfloor$ stands for the integer part of $x\in\mathbb{R}$.

Let $A$ be a subset of an additive abelian semigroup and let $hA$ be the $h$-fold sumset of $A$. The following question is considered: Let $(A_q)_{q=1}^{\infty}$ be a strictly decreasing sequence of sets in the semigroup and let $A = \bigcap_{q=1}^{\infty} A_q$. When does one have \[ hA = \bigcap_{q=1}^{\infty} hA_q \] for some or all $h \geq 2$?

A well-posedness and maximal regularity result for the time-periodic Cahn-Hilliard-Gurtin system in the half space is proved. For this purpose, we introduce a novel class of complementing boundary conditions, extending the classical Lopatinski\uı-Shapiro conditions from elliptic and parabolic theory to time-periodic mixed-order systems with general boundary conditions. Moreover, we show that the classical Lopatinski\uı-Shapiro conditions are in general insufficient for well-posedness of mixed-order systems.

Many real objects are modeled as discrete sets of points, such as corners or other salient features. For our main applications in chemistry, points represent atomic centers in a molecule or a solid material. We study the problem of classifying discrete (finite and periodic) sets of unordered points under isometry, which is any transformation preserving distances in a metric space. Experimental noise motivates the new practical requirement to make such invariants Lipschitz continuous so that perturbing every point in its epsilon-neighborhood changes the invariant up to a constant multiple of epsilon in a suitable distance satisfying all metric axioms. Since the given points are unordered, the key challenge is to compute all invariants and metrics in a near-linear time of the input size. We define the Pointwise Distance Distribution (PDD) for any discrete set and prove, in addition to the properties above, the completeness of PDD for all periodic sets in general position. The PDD can compare nearly 2 million crystals from the world's five largest databases within 2 hours on a modest desktop computer. The impact is upholding data integrity in crystallography because the PDD will not allow anyone to claim a `new' material as a noisy disguise of a known crystal.

We show how measurement and nonlocality can be explained consistently with macroscopic realism and no-signaling, and causal relations for macroscopic quantities. Considering measurement of a field amplitude $\hat{x}$, we derive theorems that lead to an equivalence between a quantum phase-space probability distribution Q(x,p,t) and stochastic trajectories for real amplitudes x and p propagating backwards and forwards in time, respectively. We present forward-backward stochastic simulations that motivate a Q-based model of reality. Amplification plays a key role in measurement. With amplification, contributions due to interference become unobservable, leading to branches that correspond to distinct eigenvalues. This elucidates how the system evolves from a superposition to an eigenstate, from which Born's rule follows. We deduce a hybrid causal structure involving causal deterministic relations for amplified variables, along with microscopic noise inputs and hidden loops for unobservable quantities. Causal consistency is confirmed. The simulations allow evaluation of a state inferred for the system, conditioned on a particular branch, from which we deduce a model for projection and collapse of the wave function. The theory is extended to Einstein-Podolsky-Rosen and Bell nonlocality. We demonstrate consistency with three weak local realistic premises: the existence of real properties (defined after operations that fix measurement settings); a partial locality implying no-signaling; elements of reality that apply to the predictions of a system by a meter, once meter-settings are fixed. A mechanism for non-locality is identified. Our work shows how forward-backward stochastic simulations lead to a hybrid causal structure, involving both deterministic causal relations and hidden stochastic loops, explaining measurement and entanglement, with paradoxes associated with retrocausality avoided.

We propose $\textsf{ScaledGD($\lambda$)}$, a preconditioned gradient descent method to tackle the low-rank matrix sensing problem when the true rank is unknown, and when the matrix is possibly ill-conditioned. Using overparametrized factor representations, $\textsf{ScaledGD($\lambda$)}$ starts from a small random initialization, and proceeds by gradient descent with a specific form of damped preconditioning to combat bad curvatures induced by overparameterization and ill-conditioning. At the expense of light computational overhead incurred by preconditioners, $\textsf{ScaledGD($\lambda$)}$ is remarkably robust to ill-conditioning compared to vanilla gradient descent ($\textsf{GD}$) even with overprameterization. Specifically, we show that, under the Gaussian design, $\textsf{ScaledGD($\lambda$)}$ converges to the true low-rank matrix at a constant linear rate after a small number of iterations that scales only logarithmically with respect to the condition number and the problem dimension. This significantly improves over the convergence rate of vanilla $\textsf{GD}$ which suffers from a polynomial dependency on the condition number. Our work provides evidence on the power of preconditioning in accelerating the convergence without hurting generalization in overparameterized learning.

Dallard, Milanič, and Štorgel [arXiv '22] ask if for every class excluding a fixed planar graph $H$ as an induced minor, Maximum Independent Set can be solved in polynomial time, and show that this is indeed the case when $H$ is any planar complete bipartite graph, or the 5-vertex clique minus one edge, or minus two disjoint edges. A positive answer would constitute a far-reaching generalization of the state-of-the-art, when we currently do not know if a polynomial-time algorithm exists when $H$ is the 7-vertex path. Relaxing tractability to the existence of a quasipolynomial-time algorithm, we know substantially more. Indeed, quasipolynomial-time algorithms were recently obtained for the $t$-vertex cycle, $C_t$ [Gartland et al., STOC '21] and the disjoint union of $t$ triangles, $tC_3$ [Bonamy et al., SODA '23]. We give, for every integer $t$, a polynomial-time algorithm running in $n^{O(t^5)}$ when $H$ is the friendship graph $K_1 + tK_2$ ($t$ disjoint edges plus a vertex fully adjacent to them), and a quasipolynomial-time algorithm running in $n^{O(t^2 \log n)+f(t)}$, with $f$ a single-exponential function, when $H$ is $tC_3 \uplus C_4$ (the disjoint union of $t$ triangles and a 4-vertex cycle). The former extends a classical result on graphs excluding $tK_2$ as an induced subgraph [Alekseev, DAM '07], while the latter extends Bonamy et al.'s result.

$\mathcal{N}=1$ superconformal minimal models are the first series of unitary conformal field theories (CFTs) extending beyond Virasoro algebra. Using coset constructions, we characterize CFTs in $\mathcal{N}=1$ superconformal minimal models using combinations of a parafermion theory, an Ising theory and a free boson theory. Supercurrent operators in the original theory also becomes sums of operators from each constituent theory. If we take our $\mathcal{N}=1$ superconformal theories as the neutral part of the edge theory of a fractional quantum Hall state, we present a systematic way of calculating its ground state wavefunction using free field methods. Each ground state wavefunction is known previously as a sum of polynomials with distinct clustering behaviours. Based on our decomposition, we find explicit expressions for each summand polynomial. A brief generalization to $S_3$ minimal models using coset construction is also included.

Kemeny (1959) introduced a topologically complete metric space to study ordinal random variables, particularly in the context of Condorcet's paradox and the measurability of ties. Building on this, Emond & Mason (2002) reformulated Kemeny's framework into a rank correlation coefficient by embedding the metric space into a Hilbert structure. This transformation enables the analysis of data under weak order-preserving transformations (monotonically non-decreasing) within a linear probabilistic framework. However, the statistical properties of this rank correlation estimator, such as bias, estimation variance, and Type I error rates, have not been thoroughly evaluated. In this paper, we derive and prove a complete U-statistic estimator in the presence of ties for Kemeny's \(\tau_{\kappa}\), addressing the positive bias introduced by tied ranks. We also introduce a consistent population standard error estimator. The null distribution of the test statistic is shown to follow a \(t_{(N-2)}\)-distribution. Simulation results demonstrate that the proposed method outperforms Kendall's \(\tau_{b}\), offering a more accurate and robust measure of ordinal association which is topologically complete upon standard linear models.

The Population Stability Index (PSI) is a widely used measure in credit risk modeling and monitoring within the banking industry. Its purpose is to monitor for changes in the population underlying a model, such as a scorecard, to ensure that the current population closely resembles the one used during model development. If substantial differences between populations are detected, model reconstruction may be necessary. Despite its widespread use, the origins and properties of the PSI are not well documented. Previous literature has suggested using arbitrary constants as a rule-of-thumb to assess resemblance (or "stability"), regardless of sample size. However, this approach too often calls for model reconstruction in small sample sizes while not detecting the need often enough in large sample sizes. This paper introduces an alternative discrepancy measure, the Population Resemblance statistic (PRS), based on the Pearson chi-square statistic. Properties of the PRS follow from the non-central chi-square distribution. Specifically, the PRS allows for critical values that are configured according to sample size and the number of risk categories. Implementation relies on the specification of a set of parameters, enabling practitioners to calibrate the procedure with their risk tolerance and sensitivity to population shifts. The PRS is demonstrated to be universally competent in a simulation study and with real-world examples.

It is well understood that, when numerically simulating SDEs with general noise, achieving a strong convergence rate better than $O(\sqrt{h})$ (where h is the step size) requires the use of certain iterated integrals of Brownian motion, commonly referred to as its "Lévy areas". However, these stochastic integrals are difficult to simulate due to their non-Gaussian nature and for a $d$-dimensional Brownian motion with $d > 2$, no fast almost-exact sampling algorithm is known. In this paper, we propose LévyGAN, a deep-learning-based model for generating approximate samples of Lévy area conditional on a Brownian increment. Due to our "Bridge-flipping" operation, the output samples match all joint and conditional odd moments exactly. Our generator employs a tailored GNN-inspired architecture, which enforces the correct dependency structure between the output distribution and the conditioning variable. Furthermore, we incorporate a mathematically principled characteristic-function based discriminator. Lastly, we introduce a novel training mechanism termed "Chen-training", which circumvents the need for expensive-to-generate training data-sets. This new training procedure is underpinned by our two main theoretical results. For 4-dimensional Brownian motion, we show that LévyGAN exhibits state-of-the-art performance across several metrics which measure both the joint and marginal distributions. We conclude with a numerical experiment on the log-Heston model, a popular SDE in mathematical finance, demonstrating that high-quality synthetic Lévy area can lead to high order weak convergence and variance reduction when using multilevel Monte Carlo (MLMC).

Resided at the intersection of multi-fidelity optimization (MFO) and Bayesian optimization (BO), MF BO has found a niche in solving expensive engineering design optimization problems, thanks to its advantages in incorporating physical and mathematical understandings of the problems, saving resources, addressing exploitation-exploration trade-off, considering uncertainty, and processing parallel computing. The increasing number of works dedicated to MF BO suggests the need for a comprehensive review of this advanced optimization technique. In this paper, we survey recent developments of two essential ingredients of MF BO: Gaussian process (GP) based MF surrogates and acquisition functions. We first categorize the existing MF modeling methods and MFO strategies to locate MF BO in a large family of surrogate-based optimization and MFO algorithms. We then exploit the common properties shared between the methods from each ingredient of MF BO to describe important GP-based MF surrogate models and review various acquisition functions. By doing so, we expect to provide a structured understanding of MF BO. Finally, we attempt to reveal important aspects that require further research for applications of MF BO in solving intricate yet important design optimization problems, including constrained optimization, high-dimensional optimization, optimization under uncertainty, and multi-objective optimization.

Ensemble methods combine the predictions of several base models. We study whether or not including more models always improves their average performance. This question depends on the kind of ensemble considered, as well as the predictive metric chosen. We focus on situations where all members of the ensemble are a priori expected to perform equally well, which is the case of several popular methods such as random forests or deep ensembles. In this setting, we show that ensembles are getting better all the time if, and only if, the considered loss function is convex. More precisely, in that case, the loss of the ensemble is a decreasing function of the number of models. When the loss function is nonconvex, we show a series of results that can be summarised as: ensembles of good models keep getting better, and ensembles of bad models keep getting worse. To this end, we prove a new result on the monotonicity of tail probabilities that may be of independent interest. We illustrate our results on a medical problem (diagnosing melanomas using neural nets) and a "wisdom of crowds" experiment (guessing the ratings of upcoming movies).

Expander graphs are fundamental in both computer science and mathematics, with a wide array of applications. With quantum technology reshaping our world, quantum expanders have emerged, finding numerous uses in quantum information theory, quantum complexity, and noncommutative pseudorandomness. The classical expander mixing lemma plays a central role in graph theory, offering essential insights into edge distribution within graphs and aiding in the analysis of diverse network properties and algorithms. This paper establishes the quantum analogue of the classical expander mixing lemma and its structural converse for quantum expanders.

In this paper, the method of gaps, a technique for deriving closed-form expressions in terms of information measures for the generalization error of supervised machine learning algorithms is introduced. The method relies on the notion of \emph{gaps}, which characterize the variation of the expected empirical risk (when either the model or dataset is kept fixed) with respect to changes in the probability measure on the varying parameter (either the dataset or the model, respectively). This distinction results in two classes of gaps: Algorithm-driven gaps (fixed dataset) and data-driven gaps (fixed model). In general, the method relies on two central observations: $(i)$~The generalization error is the expectation of an algorithm-driven gap or a data-driven gap. In the first case, the expectation is with respect to a measure on the datasets; and in the second case, with respect to a measure on the models. $(ii)$~Both, algorithm-driven gaps and data-driven gaps exhibit closed-form expressions in terms of relative entropies. In particular, algorithm-driven gaps involve a Gibbs probability measure on the set of models, which represents a supervised Gibbs algorithm. Alternatively, data-driven gaps involve a worst-case data-generating (WCDG) probability measure on the set of data points, which is also a Gibbs probability measure. Interestingly, such Gibbs measures, which are exogenous to the analysis of generalization, place both the supervised Gibbs algorithm and the WCDG probability measure as natural references for the analysis of supervised learning algorithms. All existing exact expressions for the generalization error of supervised machine learning algorithms can be obtained with the proposed method. Also, this method allows obtaining numerous new exact expressions, which allows establishing connections with other areas in statistics.

Extracting governing physical laws from computational or experimental data is crucial across various fields such as fluid dynamics and plasma physics. Many of those physical laws are dissipative due to fluid viscosity or plasma collisions. For such a dissipative physical system, we propose a framework to learn the corresponding laws of the systems based on their energy-dissipation laws, assuming either continuous data (probability density) or discrete data (particles) are available. Our methods offer several key advantages, including their robustness to corrupted/noisy observations, their easy extension to more complex physical systems, and the potential to address higher-dimensional systems. We validate our approaches through representative numerical examples and carefully investigate the impacts of data quantity and data property on model discovery.

We study the problem of solving matrix games of the form $\max_{\mathbf{w}\in\mathcal{W}}\min_{\mathbf{p}\in\Delta}\mathbf{p}^{\top}A\mathbf{w}$, where $A$ is some matrix and $\Delta$ is the probability simplex. This problem encapsulates canonical tasks such as finding a linear separator and computing Nash equilibria in zero-sum games. However, perhaps surprisingly, its inherent complexity (as formalized in the standard framework of oracle complexity [Nemirovski and Yudin, 1983]) is not well-understood. In this work, we first identify different oracle models which are implicitly used by prior algorithms, amounting to multiplying the matrix $A$ by a vector from either one or both sides. We then prove complexity lower bounds for algorithms under both access models, which in particular imply a separation between them. Specifically, we start by showing that algorithms for linear separability based on one-sided multiplications must require $\Omega(\gamma_A^{-2})$ iterations, where $\gamma_A$ is the margin, as matched by the Perceptron algorithm. We then prove that accelerated algorithms for this task, which utilize multiplications from both sides, must require $\tilde{\Omega}(\gamma_{A}^{-2/3})$ iterations, establishing the first oracle complexity barrier for such algorithms. Finally, by adapting our lower bound to $\ell_1$ geometry, we prove that computing an $\epsilon$-approximate Nash equilibrium requires $\tilde{\Omega}(\epsilon^{-2/3})$ iterations, which is an exponential improvement over the previously best-known lower bound due to Hadiji et al. [2024].

We study the Lagrangian Index Policy (LIP) for restless multi-armed bandits with long-run average reward. In particular, we compare the performance of LIP with the performance of the Whittle Index Policy (WIP), both heuristic policies known to be asymptotically optimal under certain natural conditions. Even though in most cases their performances are very similar, in the cases when WIP shows bad performance, LIP continues to perform very well. We then propose reinforcement learning algorithms, both tabular and NN-based, to obtain online learning schemes for LIP in the model-free setting. The proposed reinforcement learning schemes for LIP require significantly less memory than the analogous schemes for WIP. We calculate analytically the Lagrangian index for the restart model, which applies to the optimal web crawling and the minimization of the weighted age of information. We also give a new proof of asymptotic optimality in case of homogeneous arms as the number of arms goes to infinity, based on exchangeability and de Finetti's theorem.

We consider a multitype Galton-Watson process that allows for the mutation and reversion of individual types in discrete and continuous time. In this setting, we explicitly compute the time evolution of quantities such as the mean and distributions of different types. This allows us in particular to estimate the proportions of different types in the long run, as well as the distribution of the first time of occurrence of a given type as the tree size or time increases. Our approach relies on the recursive computation of the joint distribution of types conditionally to the value of the total progeny. In comparison with the literature on related multitype models, we do not rely on approximations.

In this work, we develop a novel mathematical framework for universal digital quantum computation using algebraic probability theory. We rigorously define quantum circuits as finite sequences of elementary quantum gates and establish their role in implementing unitary transformations. A key result demonstrates that every unitary matrix in \(\mathrm{U}(N)\) can be expressed as a product of elementary quantum gates, leading to the concept of a universal dictionary for quantum computation. We apply this framework to the construction of quantum circuits that encode probability distributions, focusing on the Grover-Rudolph algorithm. By leveraging controlled quantum gates and rotation matrices, we design a quantum circuit that approximates a given probability density function. Numerical simulations, conducted using Qiskit, confirm the theoretical predictions and validate the effectiveness of our approach. These results provide a rigorous foundation for quantum circuit synthesis within an algebraic probability framework and offer new insights into the encoding of probability distributions in quantum algorithms. Potential applications include quantum machine learning, circuit optimization, and experimental implementations on real quantum hardware.

We revisit the Markov Entropy Decomposition, a classical convex relaxation algorithm introduced by Poulin and Hastings to approximate the free energy in quantum spin lattices. We identify a sufficient condition for its convergence, namely the decay of the effective interaction. The effective interaction, also known as Hamiltonians of mean force, is a widely established correlation measure, and we show our decay condition in 1D at any temperature as well as in the high-temperature regime under a certain commutativity condition on the Hamiltonian building on existing results. This yields polynomial and quasi-polynomial time approximation algorithms in these settings, respectively. Furthermore, the decay of the effective interaction implies the decay of the conditional mutual information for the Gibbs state of the system. We then use this fact to devise a rounding scheme that maps the solution of the convex relaxation to a global state and show that the scheme can be efficiently implemented on a quantum computer, thus proving efficiency of quantum Gibbs sampling under our assumption of decay of the effective interaction.

We introduce new affine invariant ensemble Markov chain Monte Carlo (MCMC) samplers that are easy to construct and improve upon existing methods, especially for high-dimensional problems. We first propose a simple derivative-free side move sampler that improves upon popular samplers in the \texttt{emcee} package by generating more effective proposal directions. We then develop a class of derivative-based affine invariant ensemble Hamiltonian Monte Carlo (HMC) samplers based on antisymmetric preconditioning using complementary ensembles, which outperform standard, non-affine-invariant HMC when sampling highly anisotropic distributions. We provide asymptotic scaling analysis for high-dimensional Gaussian targets to further elucidate the properties of these affine invariant ensemble samplers. In particular, with derivative information, the affine invariant ensemble HMC can scale much better with dimension compared to derivative-free ensemble samplers.

For three decades, carrier-phase observations have been used to obtain the most accurate location estimates using global navigation satellite systems (GNSS). These estimates are computed by minimizing a nonlinear mixed-integer least-squares problem. Existing algorithms linearize the problem, orthogonally project it to eliminate real variables, and then solve the integer least-square problem. There is now considerable interest in developing similar localization techniques for terrestrial and indoor settings. We show that algorithms that linearize first fail in these settings and we propose several algorithms for computing the estimates. Some of our algorithms are elimination algorithms that start by eliminating the non-linear terms in the constraints; others construct a geometric arrangement that allows us to efficiently enumerate integer solutions (in polynomial time). We focus on simplified localization problems in which the measurements are range (distance) measurements and carrier phase range measurements, with no nuisance parameters. The simplified problem allows us to focus on the core question of untangling the nonlinearity and the integer nature of some parameters. We show using simulations that the new algorithms are effective at close ranges at which the linearize-first approach fails.

We introduce a new framework for quantum channel discrimination in an adversarial setting, where the tester plays against an adversary. We show that in asymmetric hypothesis testing, the optimal type-II error exponent is precisely characterized by a new notion of quantum channel divergence (termed the minimum output channel divergence). This serves as a direct analog of the quantum Stein's lemma in this new framework, and complements previous studies on ``best-case'' channel discrimination, thereby providing a complete understanding of the ultimate limits of quantum channel discrimination. Notably, the optimal error exponent can be achieved by simple non-adaptive adversarial strategies, and despite the need for regularization, it remains efficiently computable and satisfies the strong converse property in general. Furthermore, we show that entropy accumulation, a powerful tool in quantum cryptography, can be reframed as an adversarial channel discrimination problem, establishing a new connection between quantum information theory and quantum cryptography.

We present a new formulation of the lack-of-fit reduction in non-equilibrium thermodynamics using the path-integral formalism. The formulation is based on the Onsager-Machlup variational principle, and it allows us to find reduced dynamical equations by minimizing information discrepancy with respect to the detailed evolution. The reduced evolution consists of a Hamiltonian vector field and a gradient flow. The reduction method is illustrated on the Kac-Zwanzig model, where we show how irreversibility emerges from purely Hamiltonian evolution by ignoring some degrees of freedom. We also show how to generalize the Fisher information matrix and Kullback-Leibler divergence between two probability distributions to the case when the two distributions are related by the principle of maximum entropy, even in the case when the entropy is not of Boltzmann-Gibbs type (for instance Tsallis-Havrda-Charvat entropy).

The covariant characterization of the existence of gravitational radiation traversing infinity $\mathscr{J}$ in the presence of a negative cosmological constant is presented. It is coherent and consistent with the previous characterizations put forward for the cases of non-negative cosmological constant, relying on the properties of the asymptotic super-Poynting vector; or in more transparent terms, based on the intrinsic properties of the flux of tidal energy at infinity. The proposed characterization is fully satisfactory, it can be covariantly typified in terms of boundary data at infinity, and it can also be categorized according to the geometric properties of the rescaled Weyl tensor at $\mathscr{J}$. The cases with no incoming radiation entering from (or no outgoing radiation escaping at) $\mathscr{J}$ can similarly be determined in terms of the boundary data or geometric properties of the rescaled Weyl tensor. In particular, we identify the most general boundary conditions that, in an initial-boundary value problem, ensure absence of gravitational radiation traversing $\mathscr{J}$, namely (functional) proportionality between the Cotton-York tensor field and the holographic stress tensor field at $\mathscr{J}$. We also present novel conditions ensuring the absence of just incoming (outgoing) radiation at $\mathscr{J}$. These are given in a covariant way and also in terms of standard rescaled Weyl tensor scalars. The results are compatible with any matter content of the physical spacetime.

Machine learning has emerged as a transformative tool for solving differential equations (DEs), yet prevailing methodologies remain constrained by dual limitations: data-driven methods demand costly labeled datasets while model-driven techniques face efficiency-accuracy trade-offs. We present the Mathematical Artificial Data (MAD) framework, a new paradigm that integrates physical laws with data-driven learning to facilitate large-scale operator discovery. By exploiting DEs' intrinsic mathematical structure to generate physics-embedded analytical solutions and associated synthetic data, MAD fundamentally eliminates dependence on experimental or simulated training data. This enables computationally efficient operator learning across multi-parameter systems while maintaining mathematical rigor. Through numerical demonstrations spanning 2D parametric problems where both the boundary values and source term are functions, we showcase MAD's generalizability and superior efficiency/accuracy across various DE scenarios. This physics-embedded-data-driven framework and its capacity to handle complex parameter spaces gives it the potential to become a universal paradigm for physics-informed machine intelligence in scientific computing.

Heegaard splittings provide a natural representation of closed 3-manifolds by gluing two handlebodies along a common surface. These splittings can be equivalently given by two finite sets of meridians lying on the surface, which define a Heegaard diagram. We present a data structure to effectively represent Heegaard diagrams as normal curves with respect to triangulations of a surface, where the complexity is measured by the space required to express the normal coordinates' vectors in binary. This structure can be significantly more compact than triangulations of 3-manifolds, yielding exponential gains for certain families. Even with this succinct definition of complexity, we establish polynomial-time algorithms for comparing and manipulating diagrams, performing stabilizations, detecting trivial stabilizations and reductions, and computing topological invariants of the underlying manifolds, such as their fundamental and homology groups. We also contrast early implementations of our techniques with standard software programs for 3-manifolds, achieving faster algorithms for the average cases and exponential gains in speed for some particular presentations of the inputs.

High-fidelity simulations and physical experiments are essential for engineering analysis and design, yet their high cost often makes two critical tasks--global sensitivity analysis (GSA) and optimization--prohibitively expensive. This limitation motivates the common use of Gaussian processes (GPs) as proxy regression models that provide uncertainty-aware predictions from a limited number of high-quality observations. GPs naturally enable efficient sampling strategies that support informed decision-making under uncertainty by extracting information from a subset of possible functions for the model of interest. However, direct sampling from GPs is inefficient due to their infinite-dimensional nature and the high cost associated with large covariance matrix operations. Despite their popularity in machine learning and statistics communities, sampling from GPs has received little attention in the community of engineering optimization. In this paper, we present the formulation and detailed implementation of two notable sampling methods--random Fourier features and pathwise conditioning--for generating posterior samples from GPs at reduced computational cost. Alternative approaches are briefly described. Importantly, we detail how the generated samples can be applied in GSA, single-objective optimization, and multi-objective optimization. We show successful applications of these sampling methods through a series of numerical examples.

Spectral degeneracies, including diabolic (DP) and exceptional (EP) points, exhibit unique sensitivity to external perturbations, enabling powerful control and engineering of wave phenomena. We present a residue-based perturbation theory that quantifies complex resonance splitting of DP and EP type spectral degeneracies using generalized Wigner-Smith operators. We validate our theory using both analytic Hamiltonian models and numerical electromagnetic simulations, demonstrating excellent agreement across a range of cases. Our approach accurately predicts degenerate resonance splitting using only scattering data, offering a powerful framework for precision tuning, inverse design, and practical exploitation of non-Hermitian phenomena.

In Online Convex Optimization (OCO), when the stochastic gradient has a finite variance, many algorithms provably work and guarantee a sublinear regret. However, limited results are known if the gradient estimate has a heavy tail, i.e., the stochastic gradient only admits a finite $\mathsf{p}$-th central moment for some $\mathsf{p}\in\left(1,2\right]$. Motivated by it, this work examines different old algorithms for OCO (e.g., Online Gradient Descent) in the more challenging heavy-tailed setting. Under the standard bounded domain assumption, we establish new regrets for these classical methods without any algorithmic modification. Remarkably, these regret bounds are fully optimal in all parameters (can be achieved even without knowing $\mathsf{p}$), suggesting that OCO with heavy tails can be solved effectively without any extra operation (e.g., gradient clipping). Our new results have several applications. A particularly interesting one is the first provable and optimal convergence result for nonsmooth nonconvex optimization under heavy-tailed noise without gradient clipping. Furthermore, we explore broader settings (e.g., smooth OCO) and extend our ideas to optimistic algorithms to handle different cases simultaneously.

We propose a mechanism to control the motion of magnetic Skyrmions through the generation of a Skyrmion number current. This current is induced and tuned by an explicitly time-dependent Hamiltonian that includes a Zeeman term arising from the interaction between the spin system and circularly polarized light. To capture the effect, we apply a first-order perturbation method to the Landau-Lifshitz-Gilbert equation, using a breathing Skyrmion ansatz based on the Belavin-Polyakov profile. This approach reveals that the time-dependent deformation of the Skyrmion boundary produces an anisotropic breathing mode, which in turn generates a nonzero Skyrmion number current. The resulting dynamics in momentum space form a limit cycle, whose characteristics depend on the external magnetic field amplitude, the Heisenberg exchange coupling, and the Gilbert damping constant. Our formulation not only clarifies the topological origin of optically driven Skyrmion motion but also points to Skyrmion number currents as a low-dissipation alternative to electric currents for efficient Skyrmion control.

Modern multi-class image classification relies on high-dimensional CNN feature vectors, which are computationally expensive and obscure the underlying data geometry. Conventional graph-based classifiers degrade on natural multi-class images because typical graphs fail to preserve separability on feature manifolds with complex topology. We address this with a physics-inspired pipeline frozen MobileNetV2 embeddings are treated as Ising spins on a sparse Multi-Edge Type QC-LDPC graph forming a Random Bond Ising Model. The system is tuned to its Nishimori temperature identified where the smallest Bethe-Hessian eigenvalue vanishes. Our method rests on two innovations: we prove a spectral-topological correspondence linking graph trapping sets to invariants via the Ihara-Bass zeta function removing these structures boosts top-1 accuracy over four-fold in multi-class settings; we develop a quadratic-Newton estimator for the Nishimori temperature converging in around 9 Arnoldi iterations for a 6-times speedup enabling spectral embedding on scales like ImageNet-100. The resulting graphs compress 1280-dimensional MobileNetV2 features to 32 dimensions for ImageNet10 and 64 for ImageNet-100 We achieve 98.7% top-1 accuracy on ImageNet-10 and 84.92% on ImageNet-100 with a three-graph soft ensemble Versus MobileNetV2 our hard ensemble increases top-1 by 0.1% while cutting FLOPs by 2.67-times compared to ResNet50 the soft ensemble drops top1 by only 1.09% yet reduces FLOPs by 29-times. Novelty lies in (a) rigorously linking trapping sets to topological defects, (b) an efficient Nishimori temperature estimator and (c) demonstrating that topology-guided LDPC embedding produces highly compressed accurate classifiers for resource-constrained deployment

We study the design of interpolation schedules in the stochastic interpolants framework for flow and diffusion-based generative models. We show that while all scalar interpolation schedules achieve identical statistical efficiency under Kullback-Leibler divergence in path space after optimal diffusion coefficient tuning, their numerical efficiency can differ substantially. This motivates focusing on numerical properties of the resulting drift fields rather than purely statistical criteria for schedule design. We propose averaged squared Lipschitzness minimization as a principled criterion for numerical optimization, providing an alternative to kinetic energy minimization used in optimal transport approaches. A transfer formula is derived that enables conversion between different schedules at inference time without retraining neural networks. For Gaussian distributions, the optimized schedules achieve exponential improvements in Lipschitz constants over standard linear schedules, while for Gaussian mixtures, they reduce mode collapse in few-step sampling. We also validate our approach on high-dimensional invariant distributions from stochastic Allen-Cahn equations and Navier-Stokes equations, demonstrating robust performance improvements across resolutions.

We study asymptotics of the $d=4$, $\mathcal{N}=1$ superconformal index for toric quiver gauge theories. Using graph-theoretic and algebraic factorization techniques, we obtain a cycle expansion for the large-$N$ index in terms of the $R$-charge-weighted adjacency matrix. Applying saddle-point techniques at the on-shell $R$-charges, we determine the asymptotic degeneracy in the univariate specialization for $\hat{A}_{m}$, and along the main diagonal for the bivariate index for $\mathcal{N}=4$ and $\hat{A}_{3}$. In these cases we find $\ln |c_{n}| \sim \gamma n^{\frac{1}{2}}+ \beta \ln n + \alpha$ (Hardy-Ramanujan type). We also identify polynomial growth for $dP3$, $Y^{3,3}$ and $Y^{p,0}$, and give numerical evidence for $\gamma$ in further $Y^{p,p}$ examples. Finally, we generalize Murthy's giant graviton expansion via the Hubbard-Stratonovich transformation and Borodin-Okounkov formula to multi-matrix models relevant for quivers.

In classical mechanics, driven systems with dissipation often exhibit complex, fractal dynamics known as strange attractors. This paper addresses the fundamental question of how such structures manifest in the quantum realm. We investigate the quantum Duffing oscillator, a paradigmatic chaotic system, using the Caldirola-Kanai (CK) framework, where dissipation is integrated directly into a time-dependent Hamiltonian. By employing the Husimi distribution to represent the quantum state in phase space, we present the first visualization of a quantum strange attractor within this model. Our simulations demonstrate how an initially simple Gaussian wave packet is stretched, folded, and sculpted by the interplay of chaotic dynamics and energy loss, causing it to localize onto a structure that beautifully mirrors the classical attractor. This quantum "photograph" is inherently smoothed, blurring the infinitely fine fractal details of its classical counterpart as a direct consequence of the uncertainty principle. We supplement this analysis by examining the out-of-time-ordered correlator (OTOC), which shows that stronger dissipation clarifies the exponential growth associated with the classical Lyapunov exponent, thereby confirming the model's semiclassical behavior. This work offers a compelling geometric perspective on open chaotic quantum systems and sheds new light on the quantum-classical transition.

We consider the asymmetric version of the Popkov-Schütz two-lane lattice gas with general jump rates, subject to the stationary measure being of product form. This still leaves five free parameters. At density 1/2 the eigenvalues of the flux Jacobian are degenerate. We compute the second order expansion of the average fluxes at density 1/2 and thereby identify the universality classes.

We provide a short introduction to ``Lorentzian metric spaces" i.e., spacetimes defined solely in terms of the two-point Lorentzian distance. As noted in previous work, this structure is essentially unique if minimal conditions are imposed, such as the continuity of the Lorentzian distance and the relative compactness of chronological diamonds. The latter condition is natural for interpreting these spaces as low-regularity versions of globally hyperbolic spacetimes. Confirming this interpretation, we prove that every Lorentzian metric space admits a Cauchy time function. The proof is constructive for this general setting and it provides a novel argument that is interesting already for smooth spacetimes.

Artificial Intelligence (AI) and large language models (LLMs) are increasingly used in social and psychological research. Among potential applications, LLMs can be used to generate, customise, or adapt measurement instruments. This study presents a preliminary investigation of AI-generated questionnaires by comparing two ChatGPT-based adaptations of the Body Awareness Questionnaire (BAQ) with the validated human-developed version. The AI instruments were designed with different levels of explicitness in content and instructions on construct facets, and their psychometric properties were assessed using a Bayesian Graded Response Model. Results show that although surface wording between AI and original items was similar, differences emerged in dimensionality and in the distribution of item and test information across latent traits. These findings illustrate the importance of applying statistical measures of accuracy to ensure the validity and interpretability of AI-driven tools.

Over the past decade, a combinatorial framework for discrete, finite, and irreversibly aggregating systems has emerged. This work reviews its progress, practical applications, and limitations. We outline the approach's assumptions and foundations, based on direct enumeration of system states, contrasting with classical Smoluchowski and Marcus-Lushnikov methods. Using the constant kernel as an example, we derive combinatorial expressions for the average number of clusters of a given size and their standard deviation, and present the complete probability distribution for cluster counts. The method is then extended to several kernels (additive, product, linear-chain, condensation) by explicitly enumerating ways to form clusters of a given size. For general kernels, approximate solutions are obtained via recursive expressions, enabling predictions without explicit solutions. Applications to aerosol growth and planetesimal formation are demonstrated, with comparisons to numerical results. We summarize issues of validity and precision and propose open problems. The appendix includes partial Bell polynomials, generating functions, Lagrange inversion, potential applications, and links between combinatorial and scaling solutions of the Smoluchowski equation.

Recent studies have revealed synchronized multivalued behavior in thermodynamic, dynamical, and geometric quantities during the black hole first-order phase transition, which enables a diagnosis from different perspectives, yet its fundamental origin has remained poorly understood. By constructing a unified geometric framework integrating real analysis and covering space theory, we reveal the universal mathematical mechanism behind this phenomenon. We prove that this multivaluedness originates from two non-degenerate critical points in the temperature function $T(r_+)$, where $r_+$ is the horizon radius, which fold the parameter space into a three-sheeted covering structure. As a direct application, we propose that a black hole undergoes a first-order phase transition if and only if its $T(r_+)$ curve has two extrema. Accordingly, we establish a classification scheme, denoted $A1$, $A2$, and $B$ for black holes. This scheme offers a complementary perspective to classifications based on global topological invariants. Our work provides a theoretical foundation for diagnosing phase transitions via multivaluedness and establishes a unified geometric perspective on black hole thermodynamics, chaotic dynamics, and spacetime structure during first-order phase transitions.

We study the implicit bias of flow matching (FM) samplers via the lens of empirical flow matching. Although population FM may produce gradient-field velocities resembling optimal transport (OT), we show that the empirical FM minimizer is generally not a gradient field, even when each conditional flow is. Consequently, empirical FM is intrinsically not OT-optimal in the Benamou-Brenier sense. In view of this, we analyze the kinetic energy of generated samples. With Gaussian sources, both instantaneous and integrated kinetic energies exhibit exponential concentration, while heavy-tailed sources lead to polynomial tails. These behaviors are governed primarily by the choice of source distribution rather than the data. Overall, these notes provide a concise mathematical account of the structural and energetic biases arising in empirical FM.

As systems engineering (SE) objectives evolve from design and operation of monolithic systems to complex System of Systems (SoS), the discipline of Mission Engineering (ME) has emerged which is increasingly being accepted as a new line of thinking for the SE community. Moreover, mission environments are uncertain, dynamic, and mission outcomes are a direct function of how the mission assets will interact with this environment. This proves static architectures brittle and calls for analytically rigorous approaches for ME. To that end, this paper proposes an intelligent mission coordination methodology that integrates digital mission models with Reinforcement Learning (RL), that specifically addresses the need for adaptive task allocation and reconfiguration. More specifically, we are leveraging a Digital Engineering (DE) based infrastructure that is composed of a high-fidelity digital mission model and agent-based simulation; and then we formulate the mission tactics management problem as a Markov Decision Process (MDP), and employ an RL agent trained via Proximal Policy Optimization. By leveraging the simulation as a sandbox, we map the system states to actions, refining the policy based on realized mission outcomes. The utility of the RL-based intelligent mission coordinator is demonstrated through an aerial firefighting case study. Our findings indicate that the RL-based intelligent mission coordinator not only surpasses baseline performance but also significantly reduces the variability in mission performance. Thus, this study serves as a proof of concept demonstrating that DE-enabled mission simulations combined with advanced analytical tools offer a mission-agnostic framework for improving ME practice; which can be extended to more complicated fleet design and selection problems in the future from a mission-first perspective.

Quantum LDPC codes have attracted intense interest due to their advantageous properties for realizing efficient fault-tolerant quantum computing. In particular, sheaf codes represent a novel framework that encompasses all well-known good qLDPC codes with profound underlying mathematics. In this work, we generalize Poincaré duality from manifolds to both classical and quantum codes defined via sheaf theory on $t$-dimensional cell complexes. Viewing important code properties including the encoding rate, code distance, local testability soundness, and efficient decoders as parameters of the underlying (co)chain complexes, we rigorously prove a duality relationship between the $i$-th chain and the $(t-i)$-th cochain of sheaf codes. We further build multiplicative structures such as cup and cap products on sheaved chain complexes, inspired by the standard notions of multiplicative structures and Poincaré duality on manifolds. This immediately leads to an explicit isomorphism between (co)homology groups of sheaf codes via a cap product. As an application, we obtain transversal disjoint logical $\mathrm{C}Z$ gates with $k_{\mathrm{C}Z}=\Theta(n)$ on families of good qLDPC and almost-good quantum locally testable codes. Moreover, we provide multiple new methods to construct transversal circuits composed of $\mathrm{C}\mathrm{C}Z$ gates as well as for higher order controlled-$Z$ that are provably logical operations on the code space. We conjecture that they generate nontrivial logical actions, pointing towards fault-tolerant non-Clifford gates on nearly optimal qLDPC sheaf codes. Mathematically, our results are built on establishing the equivalence between sheaf cohomology in the derived-functor sense, Čech cohomology, and the cohomology of sheaf codes, thereby introducing new mathematical tools into quantum coding theory.

We derive a closed-form expression for the adjoint polynomials of torus knots and investigate their special properties. The results are presented in the very explicit double sum form and provide a deeper insight into the structure of adjoint invariants essential for the Vogel's universality of Chern-Simons theory.

The Lambda Value-at-Risk (Lambda$-VaR) is a generalization of the Value-at-Risk (VaR), which has been actively studied in quantitative finance. Over the past two decades, the Expected Shortfall (ES) has become one of the most important risk measures alongside VaR because of its various desirable properties in the practice of optimization, risk management, and financial regulation. Analogously to the intimate relation between ES and VaR, we introduce the Lambda Expected Shortfall (Lambda-ES), as a generalization of ES and a counterpart to Lambda-VaR. Our definition of Lambda-ES has an explicit formula and many convenient properties, and we show that it is the smallest quasi-convex and law-invariant risk measure dominating Lambda-VaR under mild assumptions. We examine further properties of Lambda-ES, its dual representation, and related optimization problems.

We present a space-time extension of a conservative Cartesian cut-cell finite-volume method for two-phase diffusion problems with prescribed interface motion. The formulation follows a two-fluid approach: one scalar field is solved in each phase with discontinuous material properties, coupled by sharp interface conditions enforcing flux continuity and jump laws. To handle moving boundaries on a fixed Cartesian grid, the discrete balance is written over phase-restricted space-time control volumes, whose geometric moments (swept volumes and apertures) are used as weights in the finite-volume operators. This construction naturally accounts for the creation and destruction of cut cells (fresh/dead-cell events) and yields strict discrete conservation. The resulting scheme retains the algebraic structure of the static cut-cell formulation while incorporating motion through local geometric weights and interface coupling operators. A series of verification and validation tests in two and three dimensions demonstrate super-linear accuracy in space, robust behavior under repeated topology changes and conservation across strong coefficient jumps and moving interfaces. The proposed space-time cut-cell framework provides a conservative building block for multiphase transport in evolving geometries and a foundation for future free-boundary extensions such as Stefan-type phase change.