We present an adaptive geometry in which the yardstick co-deforms with space itself, formulated on cellular spaces where length is a count: distances are shortest cell-crossing counts. No cell shape, angles, or embedding are assumed; the framework is deliberately micro-agnostic. Curvature and deformation are inferred operationally by comparing a measured radius to a radius reconstructed from boundary/area/volume counts; the linear dimension of a cell serves as the single universal unit of length, yielding unified small-ball/small-sphere estimators in 2D/3D/4D. We prove that the count metric on locally finite complexes is geodesic, show flatness on uniform lattices, and establish stability of distances and curvature estimators under small local perturbations. As a bridge to the smooth setting, a line-density field induces a conformal metric $g = e^{2u} g_0$ that reproduces the same operational quantities. We outline a Ricci-like construction from directional slices and give a spherically symmetric illustrative example consistent with Schwarzschild-type spatial behavior. Overall, the model provides an intrinsic, micro-agnostic calculus linking discrete measurements to continuum notions with guarantees, including Gromov--Hausdorff control under mild regularity assumptions.

This paper is concerned with the axiomatic basis of structures within Hypercompositional Algebra. It is proven that the axioms employed in the definition of numerous hypercompositional structures lack independence. Accordingly, novel definitions are introduced in this work which minimize the established definitions by reducing the necessary set of axioms.

This paper is devoted to the investigation of multidimensional analogues of refined Bohr-type inequalities for bounded holomorphic mappings on the unit polydisc $\mathbb{P}\Delta(0;1_n)$. We provide a definitive resolution to the Bohr phenomenon in several complex variables by determining sharp radii for functional power series involving the class of Schwarz functions $\omega_{n,m}\in\mathcal{B}_{n,m}$ and the local modulus $|f(z)|$. By employing the directional derivative operator $\partial_uf(z) = \sum_{k=1}^{n} u_k \frac{\partial f(z)}{\partial z_k}$, where $u=(u_1,u_2,\ldots,u_n)\in\mathbb{C}^n$ such that $|u_1|+|u_2|+\ldots+|u_n|=1$, we obtain refined growth estimates for derivatives that generalize well-known univariate results to $\mathbb{C}^n$. The optimality of the obtained constants is rigorously verified, demonstrating that all established radii are sharp.

In this paper, the concept of a picture fuzzy subgroup of a group is studied, and the notion of the direct product of picture fuzzy subgroups is introduced. Several characterisations of the direct product of picture fuzzy subgroups are established using the $(r, s, t)$-cut sets of picture fuzzy sets.

Motivated by Thomassen's well-known line graph conjecture, many researchers have explored sufficient conditions for claw-free graphs to be Hamiltonian or Hamilton-connected. In 1994, Ageev proved that every $2$-connected claw-free graph with domination number at most $2$ is Hamiltonian. In this paper, we extend this line of research to $3$-connected graphs by establishing the best possible upper bound on the domination number that guarantees Hamiltonicity. Specifically, we show that, except for some well-defined exceptional graphs, every $3$-connected claw-free graph $G$ with domination number at most $5$ is Hamiltonian. Furthermore, we prove that, apart from a few exceptional cases, every $3$-connected claw-free graph $G$ with domination number at most $4$ is Hamilton-connected, thereby generalizing earlier results of Zheng, Broersma, Wang and Zhang and Vrána, Zhan and Zhang. We further investigate the Hamiltonian properties of line graphs of $3$-hypergraphs, and prove that every 3-connected line graph of a 3-hypergraph with domination number at most $4$ is Hamiltonian.

The Sombor index, a degree-based topological descriptor introduced by Gutman in 2021, lacks closed-form expressions for complex hierarchical trees with multi-level pendant structures and nonuniform degree distributions, despite extensive results for simpler families such as paths, stars, cycles, and basic caterpillars. For a simple graph $\mathcal{G}$, the Sombor index is defined as \[ \mathrm{SO}(\mathcal{G}) = \sum_{uv \in E(\mathcal{G})} \sqrt{d(v)^2 + d(u)^2}. \] In this work, we derive a general recursive formula for the Sombor index of multi-level pendant-augmented path trees. These trees are constructed from a spine path $\mathcal{P}_n$ ($n \ge 2$) in which each vertex has degree $2+k$ and are iteratively augmented over $m \ge 1$ hierarchical levels. Pendants attached to odd-indexed spine vertices branch with replication factor $k$ and terminal degree $\ell_i$, whereas those stemming from even-indexed vertices incorporate an initial offset $\ell_1>2$ that propagates through subsequent levels. These results significantly advance the theoretical and computational study of degree-based topological descriptors in iteratively constructed graphs.

A quick way to compute generating functions related to Pell-Padovan tetranacci numbers and classical sequences of recursions of order two is provided. Eight special instances can be computed at once.

We give new, short proofs of recent permutation polynomial results of Bousalmi, Bayad, and Derbal by reducing the verification to explicit computations on a three-element multiplicative subgroup via Zieve's fiber criterion. Building on this approach, we develop a general framework -- combining Zieve's theorem with the AGW criterion -- for constructing complete permutation polynomials over finite fields through a fiber decomposition over the cube roots of unity. A scalar specialization of the criterion yields families that are easy to produce and verify. We illustrate the construction with concrete examples and show through counterexamples that the underlying arithmetic conditions are sharp.

It is well known that there is no closed form analytic expression for the perimeter of an ellipse. In 1927, Srinivasa Ramanujan provides two approximations to the perimeter of an ellipse that are amazingly accurate. However, he does not provide an explanation of how he arrived at those expressions. In this paper, we will try to provide such an explanation that is likely how he derived those expressions. Using insights from our analysis, we improve on these approximations. To the best of our knowledge, ours is the first attempt to explain Ramanujan's ellipse perimeter formula and our approximation is uniformly better than his expression.

Qin established the geometric realization of entire quantum groups via perverse sheaves, which further give rise to dual canonical bases with integral and positive structure constants for quantum groups of type ADE. In this paper, we prove that the dual canonical bases of (Drinfeld double) quantum groups coincide with Berenstein--Greenstein's double canonical bases, by reinterpreting their intricate algebraic construction via the geometry of NKS quiver varieties. This result settles several conjectures therein, including those on positivity and invariance under braid group actions.

We investigate the singularities of two-ruled hypersurfaces in the Euclidean four-space. By considering the points that minimize the distance between adjacent rulings, we obtain a characterization the striction curve. We introduce the notion of pseudo-non-degenerate two-ruled hypersurfaces and examine their fundamental properties. We show that two-ruled hypersurfaces constructed from a curve equipped with a Frenet-type frame, via height functions, are pseudo-non-degenerate. Furthermore, we study properties of the original curve through the striction curves and the singularities of pseudo-non-degenerate two-ruled hypersurfaces constructed in this manner.

This paper provides the first algebraic characterization of an algebra of cohomological Hecke operators associated with modifications of coherent sheaves on a smooth surface $X$ along a fixed proper curve $Z \subset X$ (possibly singular and reducible), establishing a direct connection with Yangians. It is based on the theory of equivariant nilpotent cohomological Hall algebras $\mathbf{HA}^T_{X,Z}$, developed by the same authors. More precisely, let $X$ be a resolution of a Kleinian singularity (for example, $X = T^\ast\mathbb{P}^1$) and let $Z$ be the exceptional divisor. One of the main results of this paper is an explicit isomorphism $\mathbf{HA}^T_{X,Z} \simeq \mathbb{Y}^+_\infty$, where $\mathbb{Y}^+_\infty$ is a completed, nonstandard, positive half of the affine Yangian $\mathbb{Y}(\mathfrak{g})$ of the corresponding affine ADE Lie algebra $\mathfrak{g}$. Furthermore, the generators of $\mathbf{HA}^T_{X,Z}$--given by fundamental classes of substacks of zero-dimensional sheaves and of pushforwards of line bundles on $Z$--are expressed explicitly in terms of Yangian generators. Our main tools, which may be of independent interest, are: (i) a `continuity' theorem describing the behavior of cohomological Hall algebras of objects in the heart of $t$-structures $\tau_n$ when the sequence $(\tau_n)_n$ converges, in an appropriate sense, to a fixed $t$-structure $\tau_\infty$; (ii) the definition of a multi-parameter Yangian $\mathbb{Y}_Q$ for an arbitrary quiver $Q$, given by generators and relations; (iii) a theorem relating the algebraic action of the braid group $B_Q$ on the Yangian $\mathbb{Y}_Q$ to the action of $B_Q$ on the equivariant 2-dimensional cohomological Hall algebra $\mathbf{HA}^T_Q$ of $Q$, where the latter can be described in terms of derived reflection functors of the bounded derived category of modules over the preprojective algebra of $Q$.

We introduce a concept of asymptotic mean of digits (symbols) in the $Q_s$--representation of a real number, that is a generalization of the $s$--adic representation and have a self-similar geometry. We discuss its relationship with the frequencies of digits and formulate problems related to the concept. We study the topological, metric, and fractal properties of the set of real numbers that have no asymptotic mean of $Q_s$--symbols. Also we study topological, metric and fractal properties of the sets of real numbers that have asymptotic mean of $Q_3$--symbols which is equal to value of digit frequency of number.

We study topological, metric and fractal properties of set of numbers $[0;1]$ with given asymptotic mean of digits in their ternary representation. We investigate connection of these numbers and numbers with a given frequency of digits.

In the paper we describe some properties of function $$ y=r(x)=\lim_{n\to\infty}\frac{1}{n}\sum^{\infty}_{k=1}\alpha_k(x), \text{ where } x=\sum^{\infty}_{k=1}\alpha_k(x)4^{-k} $$ of $4-$adic digits asymptotic mean of fractional part of real number $x$, particularly properties of it's level sets $ S_{\theta}=\left\{x: r(x)=\theta,\: \theta=const, \: 0\leqslant\theta\leqslant 3\right\}, $ if all $4-$adic digits frequencies exist, i.e. $$ \nu_i(x)=\lim_{n\to\infty}n^{-1}\#\{k: \alpha_k(x)=i, i\leqslant n\}, \:\: i=0,1,2,3. $$ We provided an algorithm of constructing point from the set $S_{\theta}$, and proved continuality and every where density of the set. We found conditions of zero and full Lebesgue measure and estimates of Hausdorff--Besicovitch fractal dimension.

These notes are an expanded version of the lectures held in Tromso, in May 2025 at the "Lie-Stormer Summer School : Invariant Theory from classics to modern developments", in the framework of TiME events. We emphasize the analogy between binary quartics and ternary cubics (and subsequently modular forms) based on their harmonic and equianharmonic invariants. Triangle groups are presented in both the elliptic and the hyperbolic setting with their associated tilings. The topics include the discussion of a short Hilbert paper on polynomials which are powers, that was proposed to the participants. The appendix contains some exercises, with sketches of solutions, and a section devoted to Pfaffians edited by Vincenzo Galgano.

The cosmohedron was recently proposed as a polytope underlying the cosmological wavefunction for $\text{Tr}(\Phi^3)$ this http URL faces were conjectured to be in bijection with Matryoshkas, which are obtained from a subdivision of a polygon by sequentially wrapping groups of polygons into larger polygons. In this paper we prove the correctness of this construction, and elucidate its combinatorial structure. Cosmohedra generalize to a wider class of $\mathcal{X}$ in $Y$ polytopes, where we chisel a polytope from the family $\mathcal{X}$ at each vertex of a polytope $Y$. We sketch a new application of these chiseled polytopes to the physics of ultraviolet divergences in loop-integrated Feynman amplitudes.

We study three natural properties that measure the robustness of asymptotic bases of order 2: having divergent representation function, being decomposable as a union of two bases, and containing a minimal basis. Erdős and Nathanson showed that sufficiently rapid growth of the representation function (specifically, $r_A(n) \ge C \log n$ for appropriate $C$) implies both decomposability and the existence of a minimal basis. We prove that for weaker growth rates, these three properties are independent. The construction proceeds via an inductive scheme on exponentially growing intervals.

Partially ordered patterns (POPs) play an important role in the study of permutation patterns, providing a convenient framework for describing large families of classical patterns. The problem of enumerating permutations that avoid POPs has therefore attracted considerable attention in the literature. In particular, Gao and Kitaev resolved many counting problems for POP-avoiding permutations of lengths 4 and 5, linking the enumeration to a wide range of other combinatorial objects. Motivated by their work, we initiate the study of permutations that simultaneously avoid two POPs belonging to the class of flat POPs. We establish a connection between permutations avoiding such POPs and the $k$-Fibonacci numbers. Moreover, we provide a bijection between permutations avoiding these POPs and certain restricted permutations, which allows us to use the method developed by Baltić to derive the generating function for permutations avoiding these POPs. Finally, we obtain enumerative results for separable permutations avoiding these two POPs, of lengths up to 5, with respect to six statistics, thereby extending the results of Gao et al. on the avoidance of a single flat POP in separable permutations. Notably, when both patterns are of length 5, the respective generating function is a rational function, with the sum in the numerator (resp., denominator) containing 293 (resp., 17) monomials.

In the paper, we prove the existence of radial solutions to \begin{equation}\notag%\label{main-eq-abstarct} %\begin{aligned} -\Delta_p u+({\rm sgn}(p-s)+V(x))|u|^{p-2}u+\lambda |u|^{s-2}u=|u|^{q-2}u\qquad\text{in}\,\R^N \\ %\int_{\R^N}|u|^sdx&=\rho^s %\end{aligned} \end{equation} with prescribed $L^s(\R^N)$-norm, where $N\ge 3,\,p\in[2,N),\,s\in(1,p],\,q\in(p\frac{N+s}{N},\frac{Np}{N-p})$ and $V:\R^N\to\R$ is a suitable radial potential. We stress that $V$ is required to be radial but not necessarily bounded, and there are no assumptions about its sign. The case $V\equiv 0$ is also included. The proof is variational and relies on a min-max argument. A key-tool is the Pohozaev identity, which is shown to be true for any solution under quite weak assumptions about the potential $V$. This identity is proved with the aid of a new global boundedness result for subsolutions to a suitable $p$-Laplace equation.

A hybrid framework is developed that highlights and unifies the most important aspects of the Noether correspondence between symmetries and conserved integrals in Lagrangian and Hamiltonian mechanics. Several main results are shown: (1) a modern form of Noether's theorem is presented that uses only the equations of motion, with no knowledge required of an explicit Lagrangian; (2) the Poisson bracket is formulated with Lagrangian variables and used to express the action of symmetries on conserved integrals; (3) features of point symmetries versus dynamical symmetries are clarified and explained; (4) both autonomous and non-autonomous systems are treated on an equal footing. These results are applied to dynamical systems that are locally Liouville integrable. In particular, they allow finding the complete Noether symmetry group of such systems.

We present a structure-aware parallel presolve framework specialized to arrowhead linear programs (AHLPs) and designed for high-performance computing (HPC) environments, integrated into the parallel interior point solver PIPS-IPM++. Large-scale LPs arising from automated model generation frequently contain redundancies and numerical pathologies that necessitate effective presolve, yet existing presolve techniques are primarily serial or structure-agnostic and can become time-consuming in parallel solution workflows. Within PIPS-IPM++, AHLPs are stored in distributed memory, and our presolve builds on this to apply a highly parallel, distributed presolve across compute nodes while keeping communication overhead low and preserving the underlying arrowhead structure. We demonstrate the scalability and effectiveness of our approach on a diverse set of AHLPs and compare it against state-of-the-art presolve implementations, including PaPILO and the presolve implemented within Gurobi. Even on a single machine, our presolve significantly outperforms PaPILO by a factor of 18 and Gurobi's presolve by a factor of 6 in terms of shifted geometric mean runtime, while reducing the problems by a similar amount to PaPILO. Using a distributed compute environment, we outperform Gurobi's presolve by a factor of 13.

We consider nonconvex real valued functions whose truncations are either quasiconvex or even convex starting with a certain level. Among them, the $C^2$-smooth functions whose level sets are all completely contained in the positive definite region of their Hessian matrices, starting with a certain level, are good examples of such functions. For such a function we show the injectivity of its restricted gradient to a large subset of the positive definite region of its Hessian matrices.

Let $T_P$ be the theory of beautiful pairs of algebraically closed fields of fixed characteristic. It is known that for real tuples in models of $T_P$, SU-rank coincides with Morley rank and can be computed effectively. Building on Pillay's geometric description (2007) of imaginaries in $T_P$, we define an additive rank on imaginaries of $T_P$, called the geometric rank. It takes values in $\omega*\mathbb N + \mathbb Z$ and coincides with SU-rank on real tuples. It refines SU-rank and characterizes forking in $T_P^{\mathrm{eq}}$, from which we derive an explicit criterion for determining forking independence.

We undertake a detailed analysis of a reaction-advection-diffusion (RAD) equation from the viewpoint of pulse-response studies, with particular attention to effects due to the advection velocity. Our boundary-value problem is a mathematical model for a system consisting of a narrow reactor tube into which a short pulse of reactant gas is injected at one end and a mixture of reaction product and unreacted gas flows out at the opposite end. Exit flow properties such as moments and peak characteristics are obtained analytically as functions of the Péclet number. The description of a standard transport curve\ -- -including diffusion and advection but no reaction\ -- -can serve as the baseline for further characterization of chemical activity. This characterization is done here for a first order irreversible reaction. Among our main observations is that chemical activity is easily obtained from the ratio of the exit flow curve in the presence of reaction over the standard transport curve.

The classical model of a star is the Lane-Emden star with dynamics governed by the Euler-Poisson equations. We consider the case of a liquid star with a "stiffened gas" equation of state $p=\rho^\gamma-1$. We derive the full 3D linearised Euler-Poisson system around liquid Lane-Emden stars with no symmetry assumptions on the perturbations and show that the associated linear operator $\mathbf L$ is non-negative whenever the radial mode is non-negative. We show that $\mathbf L$ has an infinite-dimensional kernel each element of which corresponds to a linearly growing solution to the linearised system. When restricted to irrotational perturbations and modding out the three kernel elements corresponding to momentum conservation, however, we prove that $\mathbf L$ is strictly positive with coercivity bound $\langle\mathbf L\boldsymbol\theta,\boldsymbol\theta\rangle_{\bar\rho}\gtrsim\|\boldsymbol\theta\|_{L^2(B_R)}^2$. Hence we demonstrate that the liquid Lane-Emden stars are stable against non-radial irrotational perturbations whenever the purely radial mode is stable, improving upon previous results that dealt only with purely radial perturbations. However, the stability might not be as strong as one might hope, as we prove that $\|\nabla\boldsymbol\theta\|_{L^2(B_R)}^2$ cannot be controlled even in this case.

We study the problem of extending any order-preserving Lipschitz function that maps a subset of a partially ordered Hilbert space X into a Hadamard poset Y without increasing its Lipschitz constant and preserving its monotonicity. This sort of an extension is always possible when X is one-dimensional. However, when dim X is at least 2 and Y satisfies some fairly weak conditions, it holds (universally) if and only if the order of X is trivial. The conditions on Y are satisfied by any Hilbert poset. Therefore, as a special case of our main result, we find that there is no order-theoretic generalization of Kirszbraun's theorem.

We consider the local kinematics at fluid interfaces in two-phase flows within the sharp interface framework. In the considered case with phase change and slip at the interface, the governing velocity field is discontinuous at the phase boundary with possible jumps of the normal and the tangential components. This causes the associated initial value problems for the kinematic differential equation, governing the motion of fluid elements, to be ill-posed in general. Motivated by a corresponding example, where the velocity field is physically consistent regarding the balance of mass and momentum as well as the entropy inequality, we employ concepts from differential inclusions, to rigorously define co-moving sets within this framework. Based on this general two-phase flow setting, we proof a natural extension of the Reynolds transport theorem to this case.

We prove a ''dimension expansion'' version of the Elekes-Rónyai theorem for trivariate real analytic functions: If $f$ is a trivariate real analytic function, then $f$ is either locally of the form $g(h(x)+k(y)+l(z))$, or the following is true: whenever a Borel set $A\subset\mathbb{R}$ has Hausdorff dimension $\alpha\in \left(\frac{1}{2},1\right)$, $f(A\times A\times A)$ has dimension significantly larger than that of $A$, i.e. \begin{align*} \dim_Hf(A\times A\times A)\geq \alpha+\varepsilon(\alpha),\quad \text{for some } \varepsilon(\alpha)>0, \end{align*} Moreover, if $\alpha>\frac{2}{3}$, $f(A\times A\times A)$ has positive Lebesgue measure. This is a considerable extension of the result established by Koh, T. Pham, and Shen (J. Funct. Anal. 286 (2024)). We also obtain an alternative proof and an improvement for the Elekes-Rónyai type theorem for bivariate real analytic functions established by Raz and Zahl (Geom. Funct. Anal. 34 (2024)). We derive these from more general results, showing that various $k$-point configuration sets of thin sets have positive Lebesgue measure by exploiting the optimal $L^2$-based Sobolev estimates for the associated family of Fourier integral operators. Extending the framework developed by Greenleaf, Iosevich, and Taylor (Mathematika 68 (2022), Math. Z. 306 (2024)) to prove Mattila-Sjölin type theorems, we obtain Falconer-type results for many configuration sets on which the method would be vacuous if demanding nonempty interior. In particular, when $k=2$, we generalize the Falconer-type result for metric functions in $\mathbb{R}^d$ satisfying strong non-vanishing curvature conditions established by Eswarathasan, Iosevich, and Taylor (Adv. Math. 228 (2011)) and the asymmetric Mattila-Sjölin type results of Greenleaf, Iosevich, and Taylor (J. Geom. Anal. 31 (2021)) to a broader class of smooth functions of asymmetric form.

We introduce birational strong complete regularity and strong complete regularity, two numerical invariants for pairs of (relative) Fano type. They are defined using variants of qdlt Fano type models and the dimension of the dual complex of the reduced boundary, and can be viewed as Fano type refinements of Shokurov's complete regularity. We establish basic properties of these invariants and clarify its relation to models of qdlt Fano type appearing in K-stability. In particular, we prove that any pair with maximal birational strong complete regularity is $1$-complementary, and the thresholds where birational strong complete regularity or strong complete regularity jumps satisfy the ascending chain condition.

Centered weighted composition operators on $L^2$-spaces are characterized. The characterization is obtained without the assumption that the operator is a product of a multiplication and a composition operator. The concept of spectrally half-centered operators is introduced, and it is shown that unbounded weighted composition operators are spectrally half-centered provided their powers are closed and densely defined. A criteria for centered weighted shifts on directed trees of types I--IV are provided. Various examples are presented.

We consider isomorphism of controllable graphs and cospectrality of distance-regularized graphs (which are known to be distance-regular or distance-biregular) in relation to logical definability. While most characterizations of these equivalence relations for such graph classes are of algebraic and spectral flavor, here we inject tools from first-order logic, extending and unifying several existing results.

This is a technical note which extends the results of Kosygina, Mountford and Peterson (Ann. Probab., 51(5):1684-1728, 2023, Section 4) about generalized Pólya's urns from a specific weight function $w(n) = (n+1)^{-\alpha}$ to a general family of weight functions satisfying $(w(n))^{-1}=n^{\alpha}\left(1+2Bn^{-1}+O\left(n^{-2}\right)\right)$ as $n \to \infty$. The latter was considered by Tóth (Ann. Probab., 24(3):1324-1367, 1996) as a part of his study of polynomially self-repelling walks. This extension will be used in forthcoming developments concerning scaling limits of these walks and related processes.

This paper develops negative curvature methods for continuous nonlinear unconstrained optimization in stochastic settings, in which function, gradient, and Hessian information is available only through probabilistic oracles, i.e., oracles that return approximations of a certain accuracy and reliability. We introduce conditions on these oracles and design a two-step framework that systematically combines gradient and negative curvature steps. The framework employs an early-stopping mechanism to guarantee sufficient progress and uses an adaptive mechanism based on an Armijo-type criterion to select the step sizes for both steps. We establish high-probability iteration-complexity guarantees for attaining second-order stationary points, deriving explicit tail bounds that quantify the convergence neighborhood and its dependence on oracle noise. Importantly, these bounds match deterministic rates up to noise-dependent terms, and the framework recovers the deterministic results as a special case. Finally, numerical experiments demonstrate the practical benefits of exploiting negative curvature directions even in the presence of noise.

We consider deterministic particle dynamics with time evolving weights and their associated Kolmogorov equation and mean-field equation. We prove existence and unique- ness for the limit PDE alongside estimates on the growth of the logarithmic gradient as well as existence of weak solutions for the Kolmogorov equation satisfying an appropriate entropy inequality. We then apply these estimates and the relative entropy method as developed in [17], in order to derive the associated equation as a mean field limit. Our results cover both interactions and influence kernels with mild regularity assumptions.

Analogously to the quantum case considered in Cruz-de-la-Rosa and Guerrero-Poblete (Open Syst. Inf. Dyn. 32, 2550005, 2025), this work proposes a graph-theoretic approach to studying non-equilibrium properties in Markov chains. We prove that the kernel of the incidence matrix associated with the interaction graph of the chain, which consists of cycles, is isomorphic to the space of anti-symmetric matrices with rows sum to zero. The main contribution of this work is the introduction of the called cycle matrices, which constitute a basis for the space of matrices that describe the non-equilibrium.

We compute some particular examples of cohomological Chow groups for varieties with isolated singularities. For higher-dimensional varieties, we compute the cohomological Chow groups of codimension one, provided that the dual complex associated to the normal crossing divisor is contractible. For 3-dimensional varieties, we consider a weaker condition on the dual complex, namely $H^{2}(\Gamma(E))=0$.

In the context of differential fields of characteristic zero with several commuting derivations, we discuss the notion of $\#$-differential equations on parameterized D-torsors and their associated Galois extensions. Using model-theoretic methods, we observe that any generalized strongly normal extension (in the sense of Pillay [14] and, more generally, León Sánchez [9]) is the Galois extension of a parameterized D-torsor. Furthermore, we prove a parameterized version of a theorem of Kolchin on differential cohomology, itself of independent interest, and use it to provide a necessary and sufficient cohomological condition for when a generalized strongly normal extension is the Galois extension for a log-differential equation on its Galois group (as a parameterized D-group). We also present general model-theoretic versions of some of the main results.

This paper examines multi-armed bandits in which actions are taken at random discrete times. The model consists of $J$ independent arms. When an arm is operated, it must remain active for a random duration, modeled by the inter-arrival time of a (possibly arm-dependent) renewal process. For arms evolving as a Lévy process, we provide an explicit characterization of the Gittins index, which is known to yield an optimal strategy. Furthermore, when the inter-arrival times are exponential and the arms evolve as either a spectrally negative Lévy process, a reflected spectrally negative Lévy process, or a diffusion process, the Gittins index is explicitly characterized in terms of the scale function or diffusion characteristics, respectively. Numerical experiments are performed to support the theoretical results.

We prove that the unconditional uniqueness of mild solutions to the Navier-Stokes equations fails in all the Besov spaces with negative regularity index, by constructing non-trivial stationary singular solutions via convex integration. We also establish uniqueness of stationary weak solutions in an endpoint critical space. Similar results are proved for the fractional Navier-Stokes equations with arbitrarily large power of the Laplacian in both Lebesgue and Besov spaces.

We investigate the existence of minimal hypersurfaces in $\mathbb{S}^{n+1}$ that are generated by the isoparametric foliation of a subsphere $\mathbb{S}^n$. By considering a generalized rotational ansatz formed by the union of homothetic copies of isoparametric leaves, we reduce the minimal surface equation to an ordinary differential equation. We prove that this construction yields a closed embedded minimal hypersurface for any choice of isoparametric hypersurface $M \subset \mathbb{S}^n$. The resulting hypersurfaces have the topological type $S^1 \times M$, extending the known examples of minimal hypertori ($S^1\times S^k\times S^k$ and $S^1\times S^k\times S^l$) to a broader class of topologies determined by isoparametric structures.

Recent developments show that AI can prove research-level theorems in mathematics, both formally and informally. This essay urges mathematicians to stay up-to-date with the technology, to consider the ways it will disrupt mathematical practice, and to respond appropriately to the challenges and opportunities we now face.

Renewable power-to-hydrogen (ReP2H) enables large-scale renewable energy utilization and supports the decarbonization of hard-to-abate sectors, such as chemicals and maritime transport, via hydrogen-based renewable ammonia and methanol fuels. As a result, utility-scale ReP2H projects are expanding worldwide. However, off-grid ReP2H systems exhibit low inertia due to their converter-dominated nature, making frequency security a critical concern. Although recent studies show that electrolyzers can contribute to frequency regulation (FR), their support capability depends on operating states and loading levels, creating a trade-off between hydrogen output and frequency security. To address this challenge, this work develops a unified co-optimization framework for frequency security-aware production scheduling of utility-scale off-grid ReP2H systems coordinating heterogeneous electrolyzers. A system-level frequency response model is established to capture multi-stage FR from alkaline water electrolyzers (AWEs), proton exchange membrane electrolyzers (PEMELs), and other resources, including ammonia-fueled generators retrofitted in co-located chemical plants, battery energy storage, and wind turbines (WTs). Stage-wise transient frequency security constraints are derived, reformulated into tractable forms, and embedded into production scheduling, enabling coordinated on/off switching and load allocation across electrolyzers to maximize hydrogen output under uncertain renewable power input while enforcing frequency security constraints. Case studies based on real-world systems demonstrate that the proposed approach allows HPs to replace 55.52% and 96.85% of FR reserves from WTs and AFGs, respectively, while maintaining comparable hydrogen output. Year-long simulations show an average 28.96% increase in annual net profit resulting from reduced reliance on conventional reserves.

Measures on Fraïssé classes are a key input in the Harman--Snowden (2022) construction of tensor categories. Treelike Fraïssé classes provide a particularly tractable source of examples. In this paper, we complete the classification of measures on Cameron's elementary treelike classes. In particular, for the class $\partial \mathfrak{T}_3(n)$ of node-colored rooted binary tree structures with $n$ colors, we classify measures by an explicit bijection with directed rooted trees edge-labeled by $\{1, \dots, n\}$ with a distinguished vertex, yielding $(2n+2)^n$ distinct $\ZZ\left[\frac 12\right]$-valued measures. For each $n \geq 1$, we use a family of measures $\mu_n^I$ and their supports $\partial \mathfrak{T}_3(n)^{\mathrm{ord}}_I$ (where $I \subseteq \{1, \dots n\}$) to construct the Karoubi envelopes $\mathbf{Rep}(\partial \mathfrak{T}_3(n)^{\mathrm{ord}}_I;\mu^I_n)$, producing infinite families of semisimple tensor categories with superexponential growth that cannot be obtained via Deligne's interpolation of representation categories. We also prove the nonexistence of measures on the $n$-colored tree class $C_n\mathfrak{T}$ for $n \geq 2$ and the labeled tree class $L \mathfrak{T}$, extending Snowden's results for uncolored trees.

In this paper, we introduce the little $\mu$-function, which is obtained as a degenerate limit of the generalized $\mu$-function. We derive the little $\mu$-function as the image of the $q$-Borel summation of a divergent solution to the Ramanujan equation which is the most degenerate second order linear $q$-difference equations of Laplace type excluding those of constant coefficients. Moreover, we present several formulas, such as symmetries and connection formulas for the little $\mu$-function, similar to those for the generalized $\mu$-function. Furthermore, we establish contiguous relations related to the $q,t$-Fibonacci sequences and Wronskian relations involving the Rogers-Ramanujan continued fraction.

We introduce a semistability notion of the intrinsic reductions of a non-archimedean rational function at each non-classical point in the Berkovich projective line, and compute the intrinsic semistability loci for the iterations of a quadratic rational function using a reduction theoretic slope formula for the hyperbolic resultant function associated to those iterations. In particular, we establish a precise stationarity of those loci for iterated quadratic rational functions similar to that in the case of non-archimedean polynomial dynamics.

Focusing on the assessment of benthic algae blooms that decay subexponentially, we propose a tractable (solvable in a closed form) and well-defined (that does not diverge) environmental index for the impact assessment of long-memory processes under model uncertainties. Our target system generates long memory through an infinite superposition of multiscale processes. The sensitivity of the environmental index can be controlled by the degree of model uncertainty in terms of the relative entropy and nonexponential discount; hence, we apply a long-memory discount to evaluate long-memory processes. In our framework, the evaluation of the environmental index is reduced to finding a proper solution to an infinite-dimensional extended Hamilton-Jacobi-Bellman system. We can solve this system under sufficient conditions for the unique existence of sufficiently regular solutions, and numerically handle them by using a quantization technique. Finally, we present a demonstrative application of the proposed framework to benthic algae population dynamics in river environments based on a laboratorial experiment. This paper offers a tractable framework towards the assessment of persistent environmental phenomena.

We give an upper bound for the number of compact essential orientable non-isotopic surfaces, with Euler characteristic at least some constant $\chi$, properly embedded in a finite-volume hyperbolic 3-manifold $M$, closed or cusped. This bound is a polynomial function of the volume of $M$, with degree that depends linearly on $|\chi|$.

Locally decodable codes (LDCs) are error correction codes that allow recovery of any single message symbol by probing only a small number of positions from the (possibly corrupted) codeword. Relaxed locally decodable codes (RLDCs) further allow the decoder to output a special failure symbol $\bot$ on a corrupted codeword. While known constructions of RLDCs achieve much better parameters than standard LDCs, it is intriguing to understand the relationship between LDCs and RLDCs. Separation results (i.e., the existence of $q$-query RLDCs that are not $q$-query LDCs) are known for $q=3$ (Gur, Minzer, Weissenberg, and Zheng, arXiv:2512.12960, 2025) and $q \geq 15$ (Grigorescu, Kumar, Manohar, and Mon, arXiv:2511.02633, 2025), while any $2$-query RLDC also gives a $2$-query LDC (Block, Blocki, Cheng, Grigorescu, Li, Zheng, and Zhu, CCC 2023). In this work, we generalize and strengthen the main result in Grigorescu, Kumar, Manohar, and Mon (arXiv:2511.02633, 2025), by removing the requirement of linear codes. Specifically, we show that any $q$-query RLDC with soundness error below some threshold $s(q)$ also yields a $q$-query LDC with comparable parameters. This holds even if the RLDC has imperfect completeness but with a non-adaptive decoder. Our results also extend to the setting of locally correctable codes (LCCs) and relaxed locally correctable codes (RLCCs). Using our results, we further derive improved lower bounds for arbitrary RLDCs and RLCCs, as well as probabilistically checkable proofs of proximity (PCPPs).

In this article, the small ball probability is obtained for the collision local time of two independent symmetric $\alpha-$stable processes with parameters $\alpha_1,\alpha_2\in(0,2]$ satisfying $\max\{\alpha_1,\alpha_2\}>1$. The proof is based on obtaining the asymptotic behavior of moment generating function by contour integration.

We study the set of isomorphism classes of polarized superspecial abelian varieties $(A,\lambda)$ of a fixed dimension over $\mathbb{F}_p$ with Frobenius endomorphism $\pi_A=\sqrt{-p}$ and $\ker \lambda =\ker \pi_A$. This set plays an important role in the geometry of the supersingular locus, and the generalizations of Deuring's $2T-H$ Theorem by Ibukiyama and Katsura. We determine when this set is nonempty and classify its genera. Our method reduces the problems of superspecial abelian varieties to those of certain hermitian lattices by the lattice description established by Jordan et. al and Ibukiyama--Karemaker--Yu, and we treat these problems on the lattices concerned by arithmetic methods.

Analog error correction codes have been proposed for analog in-memory computing on resistive crossbars, which can accelerate vector-matrix multiplication for machine learning. Unlike traditional communication or storage channels, this setting involves a mixed noise model with small perturbations and outlier errors. A number of analog codes have been proposed for handling a single outlier, and several constructions have also been developed to address multiple outliers. However, the set of available code families remains limited, covering only a narrow range of code lengths and dimensions. In this paper, we study a recently proposed family of geometric codes capable of handling multiple outliers, and develop a geometric analysis that characterizes their m-height profiles.

We characterize matrix polynomials $P,Q$ such that the inequality $$ \left\Vert Q(D)u\right\Vert _{L^{2}}\leq C\left\Vert P(D)u\right\Vert _{L^{2}}\quad\text{for all }u\in C_c^\infty(\Omega), $$ holds on bounded open sets $\Omega$. We also characterize the operators $P,Q$ for which the linear continuous embedding above is compact, i.e., if $u_n\in C_c^\infty(\Omega)$ are such that $(P(D)u_n)_{n\geq 1}$ is bounded in $L^2$, then $(Q(D)u_n)_{n\geq 1}$ is strongly compact in $L^2$.

Masur and Minsky showed that the curve graph is quasi-isometric to the Teichmüller space electrified along its thin part, and hence the Teichmüller space is weakly relatively hyperbolic with respect to the thin part. In this paper, we extend this result to the $k$-multicurve graph by electrifying the Teichmüller space along the thin part where the extremal length of $k$ curves is sufficiently small. A key ingredient is a bound on the $k$-multicurve graph distance in terms of the intersection number, which is obtained by adapting the upper bound for the pants graph due to Lackenby and Yazdi.

Let $X/S$ be a smooth family of smooth projective varieties, where $S$ is a smooth affine curve over a field $k$ of characteristic $0.$ We relate the differential fundamental groupoid scheme of $X/k$ with the differential fundamental groupoid scheme of $S/k$ and the relative differential fundamental group of $X/S$ in a short exact sequence. This yields natural maps from the group cohomology of the geometric relative fundamental group to the Gauss-Manin connections. For families of curves of genus at least $1,$ we prove that these maps are isomorphisms thus give an interpretation of the Gauss-Manin connection in terms of cohomology of the differential fundamental group. As a consequence we show that, as a surface over $k$, $X$ after a little shrinking becomes de Rham $K(\pi,1).$

The kernel smoothing with large bandwidth values causes oversmoothing or underfitting in general. However, when irrelevant variables are included, the corresponding large bandwidth values are known to have an effect of shrinking them. This study investigates asymptotic properties of the kernel conditional density estimator and the regression estimator with large bandwidth matrix elements for cases of multi-index model. It is clarified that the optimal convergence rate of the estimators depends on not the number of the variables but the effective dimension without eliminating the irrelevant variables. Thus, the kernel conditional density estimator and regression estimator are demonstrated to equip the reduction of the curse of dimensionality by nature. Finite sample performances are investigated by a numerical study, and the bandwidth selection is discussed. Finally a case study on the Boston housing data is provided.

Let $Y_{1|1}$ be the Yangian associated to the general linear Lie superalgebra $\mathfrak{gl}_{1|1}$, defined over an algebraically closed field $\mathbbm{k}$ of characteristic $p>2$. In this paper, we classify the finite dimensional irreducible representations of the restricted super Yangian $Y_{1|1}^{[p]}$ and the restricted truncated shifted super Yangian $Y_{1|1,\ell}^{[p]}(\sigma)$.

In this paper, we introduced some notions on the n-Normed Spaces. Those are bounded k-linear (or multilinear) functionals and k-continuous (or multicontinuous) functions with k \in \mathbb{N}. We defined k-linear functionals under several types of boundedness, and constructed the corresponding dual spaces based on each type of boundedness. We then proved that these types of boundedness are actually equivalent. This means the boundedness of a multilinear functional can be verified using any of the equivalent notions of boundedness that we defined earlier. The equivalent also implies that all of the resulting dual spaces are identical as a set. We also defined two norms on the dual spaces and showed that both norms are equivalent. Moreover, we gave some examples of bounded k-linear functionals on an n-normed space and calculated their norms with respect to the types of boundedness. We also defined a new notion of k-continuous function in n-normed spaces. Then we gave a relation between the bounded k-linear functional and k-continuous function in n-normed spaces.

Exploiting the special features of four-dimensional Riemannian Geometry, we derive topological and rigidity results for hypersurfaces immersed in space forms of dimension 5: we characterize isoparametric hypersurfaces by means of the Weyl tensor, we prove sharp topological bounds on the Weyl functional and, inspired by a famous conjecture by Chern, we find estimates for the norm of the second fundamental form in terms of the Euler characteristic in the minimal, constant scalar curvature case, under some volume constraints. Finally, we prove some rigidity results by means of integral inequalities on the derivatives of the second fundamental form. We also extend some of the results to the case of a locally conformally flat 5-dimensional ambient space.

In this paper, we consider a dynamical system on the Riemann sphere that evolves through a set-valued map, namely a holomorphic correspondence. Analogous to the investigation of the dynamics effected by a continuous map defined on a compact metric space, wherein the concept of measure-theoretic entropy of the map and its utility in defining the pressure of a function are well-studied, we define the measure-theoretic entropy of a holomorphic correspondence and use the same to define the pressure of continuous functions. These ideas naturally lead to the formulation of a variational principle in the context of the dynamics of a holomorphic correspondence.

We consider a class of two-stage nonconvex nonsmooth stochastic conic program, where the objective functions in both stages can contain nonsmooth terms that are functions with easily computed proximal mappings, further composed with affine mappings. This kind of problem is capable of modeling various applications. Solving these problems, however, can be challenging due to the two-stage structure with possibly large number of scenarios, the nonconvex, nonsmooth and even non-Lipschitz discontinuous terms, as well as the conic constraints. In this paper, we define a KKT point of the problem, show that it is a necessary optimality condition under mild conditions, and transform it to an equivalent nonmonotone nonsmooth two-stage stochastic variational inequality (SVI). We then propose a successive difference-of-convex (SDC) method by making use of Moreau envelope to solve it, the subproblems of which are approximately solved by the progressive hedging method for solving maximal monotone two-stage SVI. We show the rigorous convergence of our method under suitable assumptions. An extension of Markowitz's mean-variance model is provided as an application and numerical results on it demonstrate the effectiveness of the model and the SDC method.

We construct linear codes over the finite field Fq from arbitrary simplicial complexes, establishing a connection between topological properties and fundamental coding parameters. First, we study the behaviour of the weights of codewords from a geometric point of view, interpreting them in terms of the combinatorial structure of the associated simplicial complex. This approach allows us to describe the minimum distance of the codes in terms of certain geometric features of the complex. Subsequently, we analyse how various topological operations on simplicial complexes affect the classical parameters of the codes. This study leads to the formulation of geometric criteria that make it possible to explicitly control and manipulate these parameters. Finally, as an application of the obtained results, we construct several families of optimal linear codes over F2 using these geometric methods. Thanks to the previously established geometric properties, we can precisely determine the parameters of these families.

We study the Equi-Baire one property families of Möbius transformations on the Riemann sphere. For a loxodromic map $f$, we show its iterates $\{f^n\}$ form an orbitally Equi-Baire one family on the attracting basin. For a one-parameter subgroup $\{f_t \}$, we prove it is Equi-Baire one on all compact sets of $\widehat{\mathbb{C}}$ if and only if the subgroup is relatively compact in $\mathrm{SL}(2,\mathbb{C})$. This provides a dynamical characterization of the Equi-Baire one condition for Möbius families.

Design of antenna structures for Internet of Things (IoT) applications is a challenging problem. Contemporary radiators are often subject to a number of electric and/or radiation-related requirements, but also constraints imposed by specifics of IoT systems and/or intended operational environments. Conventional approaches to antenna design typically involve manual development of topology intertwined with its tuning. Although proved useful, the approach is prone to errors and engineering bias. Alternatively, geometries can be generated and optimized without supervision of the designer. The process can be controlled by suitable algorithms to determine and then adjust the antenna geometry according to the specifications. Unfortunately, automatic design of IoT radiators is associated with challenges such as determination of desirable geometries or high optimization cost. In this work, a variable-fidelity framework for performance-oriented development of free-form antennas represented using the generic simulation models is proposed. The method employs a surrogate-assisted classifier capable of identifying a suitable radiator topology from a set of automatically generated (and stored for potential re-use) candidate designs. The obtained geometry is then subject to a bi-stage tuning performed using a gradient-based optimization engine. The presented framework is demonstrated based on six numerical experiments concerning unsupervised development of bandwidth-enhanced patch antennas dedicated to work within 5 GHz to 6 GHz and 6 GHz to 7 GHz bands, respectively. Extensive benchmarks of the method, as well as the generated topologies are also performed.

Development of modern antennas is a cognitive process that intertwines experience-driven determination of topology and tuning of its parameters to fulfill the performance specifications. Alternatively, the task can be formulated as an optimization problem so as to reduce reliance of geometry selection on engineering insight. In this work, a bi-stage framework for automatic generation of antennas is considered. The method determines free-form topology through optimization of interconnections between components (so-called pixels) that constitute the radiator. Here, the process involves global optimization of connections between pixels followed by fine-tuning of the resulting topology using a surrogate-assisted local-search algorithm to fulfill the design re-quirements. The approach has been demonstrated based on two case studies concerning development of broadband and dual-band monopole antennas.

We introduce linearly decomposable (LD) generalized pairs, which serve as a workable substitute for rational decompositions in the non-NQC setting. Using LD generalized pairs, together with a refinement of special termination and Kollár-type gluing theory, we prove the existence of flips for log canonical generalized pairs without assuming the klt condition, the NQC condition, or $\mathbb Q$-factoriality. Together with the cone and contraction theorems, this yields the existence of the minimal model program for arbitrary log canonical generalized pairs.

We introduce a class of algebras over a field $\mathbb{F}$ related to directed graphs in which all edges are labeled by nonzero elements of the field $\mathbb{F}$. If all labels are different from $1$, these algebras are axial algebras. We determine their fusion laws, prove them to be simple in almost all cases, and determine their automorphism group under some conditions on the degrees and girth of the graph. A construction of a class of these graphs with prescribed automorphism group enables us to construct for each group $G$ infinitely many simple (axial) algebras (with a fixed fusion law) such that the automorphism group of the algebra is isomorphic to $G$.

We calculate the genus zero cobordism-valued Gromov-Witten invariants of a point by refining the string equation on $\overline{\mathcal{M}}_{0,n}$ from the Chow ring to algebraic cobordism. This gives inductive formulas for cobordism-valued psi-class intersections on $\overline{\mathcal{M}}_{0,n}$, and in particular the cobordism classes $[\overline{\mathcal{M}}_{0,n}]$, and for their images in $K$-theory. Explicit formulas are given up to $n = 8$.

The stability of non-isolated equilibria to quasilinear parabolic problems of the form $u' = A(u)u + f(u)$ is established in interpolation spaces (and thus extending previous results relying on maximal regularity). The approach allows full flexibility in choosing the interpolation methods and requires only low regularity assumptions on the semilinear part $f$. Applications to concrete problems are presented, including the capillarity-driven Hele--Shaw problem and the fractional mean curvature flow.

Error-correcting codes are a method for representing data, so that one can recover the original information even if some parts of it were corrupted. The basic idea, which dates back to the revolutionary work of Shannon and Hamming about a century ago, is to encode the data into a redundant form, so that the original information can be decoded from the redundant encoding even in the presence of some noise or corruption. One prominent family of error-correcting codes are Reed-Solomon Codes which encode the data using evaluations of low-degree polynomials. Nearly six decades after they were introduced, Reed-Solomon Codes, as well as some related families of polynomial-based codes, continue to be widely studied, both from a theoretical perspective and from the point of view of applications. Besides their obvious use in communication, error-correcting codes such as Reed-Solomon Codes are also useful for various applications in theoretical computer science. These applications often require the ability to cope with many errors, much more than what is possible information-theoretically. List-decodable codes are a special class of error-correcting codes that enable correction from more errors than is traditionally possible by allowing a small list of candidate decodings. These codes have turned out to be extremely useful in various applications across theoretical computer science and coding theory. In recent years, there have been significant advances in list decoding of Reed-Solomon Codes and related families of polynomial-based codes. This includes efficient list decoding of such codes up to the information-theoretic capacity, with optimal list-size, and using fast nearly-linear time, and even sublinear-time, algorithms. In this book, we survey these developments.

We present a computational framework to investigate steady state distributions and perform stability analysis for random ordinary differential equations driven by parameter uncertainty. Using the nonlinear Rosenzweig McArthur predator prey model as a case study, we characterize the non-trivial equilibrium steady state of the system and investigate its complex distribution when the parameter probability densities are multi-modal mixture models with partially overlapping or separated components. In consequence, this application includes both, uncertainties and superpositions, of the system parameters. In addition, we present the stability analysis of steady states based on the eigenvalue distribution of the system's Jacobian matrix in this stochastic regime. The steady state posterior density and stability metrics are computed with a recently published Monte Carlo based numerical scheme specifically designed for random equation systems (Hoegele, 2026). Particularly, the simplicity of this stochastic extension of dynamic systems combined with a broadly applicable computational approach is demonstrated. Numerical experiments show the emergence of multi-modal steady state distributions of the predator prey model and we calculate their stability regions, illustrating the method's applicability to uncertainty quantification in dynamical systems.

The $hp$ local discontinuous Galerkin (LDG) method proposed by Castillo et al. [Math. Comp.,~71 (238): 455-478, 2002] has been shown to be an efficient approach for solving convection-diffusion equations. However, theoretical analysis indicates that, for solutions with limited spatial regularity, the method exhibits suboptimal convergence in $p$, suffering a loss of one order, comparing to numerical experiments. The purpose of this paper is to close the gap between theoretical estimates and numerical evidence. This is accomplished by establishing new approximation results for the associated Gauss-Radau projections of functions in suitable function spaces that can optimally characterize the regularity of singular solutions. We show that such a framework arises aturally and enables the study of various types of singular solutions, with full consistency between theoretical analysis and numerical results. This investigation sheds light on the resolution of the suboptimality in $p$ observed in the literature for several other types of DG schemes in different settings.

We conduct a topological-numerical analysis of global dynamics in a discrete-time two-gene Andrecut-Kauffman model. This model describes gene expression regulation through nonlinear interactions. We use rigorous numerical methods to construct Morse decomposition of the system across a wide range of parameters. We obtain qualitative results by effectively computing the Conley indices of the constructed isolating neighborhoods that form the Morse decomposition. We introduce new symbols to convey the information provided by the Conley index in an easy to understand schematic way. We additionally conduct numerical simulations aimed at confirming the presence of complex dynamical phenomena, including multistability and the existence of chaotic attractors. The results demonstrate the usefulness of topological methods in understanding the global structure of dynamics in a gene regulatory model and highlight the richness of dynamics that can be observed in such a system when parameter values change.

We prove that modules over an Artinian Gorenstein local ring $R$ have rational Poincaré series sharing a common denominator if $R/\soc(R)$ is a Golod ring. If $R$ is a Gorenstein local ring with square of the maximal ideal being generated by at most two elements, we show that modules over $R$ have rational Poincaré series sharing a common denominator. By a result of \c Sega, it follows that $R$ satisfies the Auslander-Reiten conjecture. We provide a different proof of a result of Rossi and \c Sega concerning rationality of Poincaré series of modules over compressed Gorenstein local rings. We also give a new proof of the fact that modules over Gorenstein local rings of codepth at most three have rational Poincaré series sharing a common denominator, which is originally due to Avramov, Kustin and Miller.

In accordance with the Cameron-Goethals-Seidel-Shult Classification Theorem, we extend the characterization of Hoffman colorability of line graphs from (Abiad, Bosma, Van Veluw, 2025) to all connected graphs with smallest eigenvalue at least $-2$; we give a characterization of Hoffman colorability of generalized line graphs, and we completely classify the Hoffman colorable exceptional graphs. The 245 Hoffman colorable exceptional graphs from this classification admit a natural partial ordering, and we determine the 29 graphs that are maximal in this respect, in a way similar to the classification of maximal ($E_8$-representable) exceptional graphs as described in (Cvetković, Rowlinson, Simić, 2004). Lastly, as a byproduct and also similarly as in (loc. cit.), we determine all 39 graphs that are maximal with respect to being representable in the $E_7$ root system.

We provide asymptotics for the number of faces of a certain family of Hanner polytopes. As a corollary, we come close to saturating the FLM inequality for a certain family of parameters.

For a cardinal $\kappa$, denote by $\mathbf{H}^\kappa$ the algebraic real hyperbolic space of dimension $\kappa$. For a topological group $\Gamma$, we study the set of continuous representations $\Gamma \to \operatorname{Isom}(\mathbf{H}^\kappa)$ up to continuous self-representations $\operatorname{Isom}(\mathbf{H}^\kappa)\to \operatorname{Isom}(\mathbf{H}^\kappa)$. The novelty of this work relies in considering simultaneously all cardinals, finite or infinite. We will endow this set of classes of representations with a natural topology, and show that this character variety is compact. This will also enable us to recover all previous compactifications of actions on $\mathbf{H}^n$ by certain actions on real trees for the equivariant Gromov-Hausdorff topology. A class of representations recovers in particular the homothety class of its marked length spectrum. We will define the notion of algebraic cross-ratio and prove a GNS-embedding result, enabling us to generalize some rigidity properties of the marked length spectrum. We will also introduce a notion of abstract cross-ratio, and use it to show that a wide class of groups $\Gamma$ (characterized by the existence of what we call a $3$-full action on a $\operatorname{CAT}(-1)$-space) admit at most one class of irreducible representations into $\operatorname{Isom}(\mathbf{H}^\kappa)$ whose boundedness properties are controlled by those of $(X,d)$. We will apply this to topological groups $\Gamma$ such as the isometry group $\operatorname{Isom}(\mathbf{H}^\kappa)$ itself, the automorphism group $\operatorname{Aut}(T_\omega)$ of the simplicial tree with countably infinite valency, and the automorphism group $\operatorname{PGL}_2(\mathbb{K}, \lvert\cdot \rvert)$ of the projective line over a non-Archimedean field.

We study the Klein-Gordon equation in one spatial and one temporal dimension. Physically, this equation describes the wave function of a relativistic spinless boson with positive rest mass. Mathematically, this is the most elementary hyperbolic partial differential equation, after the wave equation itself. Relative to the origin, the spacetime splits according to the light cones, and we find four quarter-planes, two of which are timelike while the remaining two are spacelike. Not unexpectedly, the solutions behave quite differently in the two types of quarter-planes. It turns out that the spacelike quarter-planes exhibit a Liouville phenomenon, where insufficient growth forces the solutions to display a certain kind of symmetry, where the values on the two linear edges are in a one-to-one relation. This phenomenon shares features with the classical Liouville theorem as well as the Phragmen-Lindelof principle for harmonic functions.

It follows from recent results of V. Bakhtin, R. Oleinik, and the second named author that, given a metric space $\mathcal{X}$, a continuous map $\gamma\colon [a,b] \to \mathcal{X}$ is a map of bounded variation if and only if $f \circ \gamma$ is a function of bounded variation for every Lipschitz function $f\colon\mathcal{X} \to \mathbb{R}$. In this note, we show that the continuity assumption is of crucial importance: for many interesting examples of metric spaces there are no analogs of that characterization without the continuity assumption on $\gamma$. The interesting examples are: $\ell_2$, infinite metric trees, and Laakso-type spaces. However, for ultrametric spaces the said characterization holds without any continuity assumptions.

To a height-$1$ formal group defined over the ring of integers of a finite extension $K$ of $\mathbb{Q}_p$ is attached its $p$-adic Tate module, which is a crystalline character of $\mathrm{Gal}(\overline K/K)$ of Hodge-Tate weight $1$. This association is an equivalence. We prove, over extensions whose ramification index is relatively prime to $p^2-p$, that the set of consistent sequences attached to a height-$1$ commuting pair $(f, u)$ of noninvertible and invertible formal power series is a crystalline character of weight $1$, for which $f$ is an endomorphism. As a result, we deduce a proof of a conjecture of Lubin in new cases.

Magnetic resonance imaging (MRI) is central to the diagnosis of multiple sclerosis, where the identification of biomarkers such as the central vein sign benefits from high-resolution images. However, most clinical brain MRI scans are performed using 1.5 T scanners, which provide lower sensitivity compared to higher-field systems. We propose a blind super-resolution framework to enhance real 1.5 T MRI images acquired in clinical settings, where only post-processed data are available and the degradation model is not fully known. The problem is formulated as a non-convex blind inverse problem involving the joint estimation of the high-resolution image and the blur kernel. Image regularization is handled through a Plug-and-Play strategy based on a pretrained denoiser, while suitable constraints are imposed on the blur kernel. To solve the resulting model, we design a heterogeneous alternating block-coordinate method in which the two variables are updated using different types of algorithms. Convergence properties are rigorously established. Experiments on FLAIR and SWI sequences acquired at 1.5 T show improved structural definition and enhanced visibility of clinically relevant features, with visual comparison against 3 T images.

We resolve a question posed by Benedetti and Sagan by constructing a signreversing involution on Takeuchi's expansion that yields the antipode for the ring of symmetric functions in terms of the Schur basis.

Let \( \ell_n(x) \) denote the maximal run-length among the first \( n \) digits of the Lüroth expansion of \( x\in(0,1] \). While \( \ell_n(x) \) grows logarithmically, we investigate the finer multifractal properties of the exceptional set where $\ell_n(x)$ exhibits linear growth. Specifically, we establish the Hausdorff dimension of the set \[ \left\{ x \in (0,1] : \liminf_{n \to \infty} \frac{\ell_n(x)}{n} = \alpha, \; \limsup_{n \to \infty} \frac{\ell_n(x)}{n} = \beta \right\}, \] for all \( 0 \le \alpha \le \beta \le 1 \).

We consider the non-homogeneous first-order differential equation with hysteresis described by the Krasnoselskii-Pokrovskii rate-independent hysteresis operator. Existence and uniqueness of solutions as well as the boundedness of solution in response to a bounded input are proved. The global stability of the equation is also investigated. Periodic solutions and their stability are studied in addition. The differential equation under analysis constitutes the so-called inversion-free feedforward control, which was proposed for mitigating arbitrary rate-independent hysteresis effects in the actuated systems. The experimentally identified non-smooth and non-strictly monotonic hysteresis of a magnetic shape memory alloy (MSMA) actuator serves as the case study. The performed analysis is settled in a series of theorems which are illustrated by numerical examples.

We compute the canonical form of the cosmological polytope for any graph in terms of the dual of the shifted cosmological polytope in two different ways. On the way, we provide an explicit coordinate description of the dual of the cosmological polytope. Moreover, we construct two triangulations of the dual cosmological polytope in terms of maximal and almost maximal tubings of the underlying graph. Though the existence of the first triangulation was already suggested by Arkani-Hamed, Benincasa and Postnikov, the second is completely new and, in particular, gives rise to a new expression of the canonical form of the cosmological polytope.

We study the translational motions of homonuclear diatomic molecules prepared in their ${}^3\Sigma$ electronic states, deeply bound vibrational states, and rotational states of well-defined parity. The trapping potential arises due to the interaction of the total spin of electrons and orbital angular momentum of nuclei with the trap's quadrupole magnetic field. The translational motion of a molecule is treated classically. We examine the Hamilton equations that govern the center of mass dynamics numerically and analytically. Using data of a hydrogen molecule at the ground vibrational state, we present global dynamics using the Poincaré section method and various types of trajectories: periodic, quasi-periodic and chaotic. We prove that the Hamiltonian system governing this motion is non-integrable. The particle's orbits are confined to a bound region of space that grows with energy, but for small energies (< 1.8 K), the motion is restricted to a processing chamber (a few centimetres). Solutions of equations of motion occurring on the symmetry axis and the horizontal plane are expressed in terms of Jacobi elliptic functions.

The uncontrolled proliferation of cancer cells and their interaction with healthy tissue poses a major challenge in oncology. This manuscript develops and analyzes mathematical models that describe tumor response to radiotherapy by incorporating the Linear Quadratic model for cell survival. To improve therapeutic efficiency, the theory of optimal control is introduced on a system of coupled differential equations, allowing for the comparison of constant versus optimized radiation strategies. The analytical study of these models provides insights into the expected dynamics under different treatment scenarios, while numerical simulations validate the theoretical results and highlight the benefits of optimal control in reducing tumor burden with minimized collateral damage.

We establish the asymptotic sharpness of a Nikolskii type inequality proved by A. Baranov and R. Zarouf for rational functions $f$ in the Wiener algebra of absolutely convergent Fourier series, with at most $n$ poles, all lying outside the dilated disc $\frac{1}{\lambda}\mathbb{D}$, where $\mathbb{D}$ denotes the open unit disc and $\lambda\in[0,1)$ is fixed. More precisely, this inequality tells that the Wiener norm of such functions is bounded by their $H^{2}$-norm -- i.e., their norm in the Hardy space of the disc -- times a factor of order $\sqrt{\frac{n}{1-\lambda}}$. In this paper, we construct explicit test functions showing that this bound cannot be improved in general: the inequality is asymptotically sharp as $n\to\infty$, up to a universal constant, for every fixed $\lambda\in[0,1)$.

We investigate the Maximal Entropy Simple Symmetric Exclusion Process (MESSEP) on a discrete ring with L sites and N indistinguishable particles. Its eigenfunctions are Schur polynomials evaluated at the L-th roots of unity, yielding an explicit spectral decomposition. The analysis relies on this eigenstructure and on the link between Schur polynomials and irreducible characters of the symmetric group, which forms the core algebraic tool for the scaling limits. In the low-density regime, where N is fixed and L tends to infinity, the rescaled dynamics converge to the Unitary Dyson Brownian Motion (UDBM). The electrostatic repulsion then emerges as an entropic force, providing a canonical microscopic derivation of the UDBM. In the hydrodynamic regime, where N is equivalent to $\alpha$L with $\alpha$ P p0, 1q, the empirical measure converges to a density solving a nonlinear, nonlocal transport equation. Its moment generating function satisfies a complex Burgers-type equation. As $\alpha$ tends to 0, this equation coincides with that governing the spectral distribution of the Free Unitary Brownian Motion (FUBM), thereby bridging discrete entropic exclusion dynamics and free unitary hydrodynamics. Overall, the MESSEP provides a unified canonical discrete framework connecting unitary Dyson motion and free unitary Brownian motion through nonlinear hydrodynamic limits, with Schur and character theory as the central algebraic structure.

For integers $n \ge 3$ and $r \ge 1$, let $\Gamma_{n,r}$ be the alternating-oriented digraph obtained by gluing $r$ directed $n$-cycles along a single edge in a staircase pattern, and let $A_{n,r}$ be its adjacency matrix. A canonical $n$-layer partition puts $A_{n,r}$ into an $n$-cyclic block form and isolates a cyclic product core $K_{n,r}$, so the nonzero spectrum of $A_{n,r}$ is obtained from that of $K_{n,r}$ by taking $n$th roots. We show that $K_{n,r}$ is totally nonnegative and irreducible, and hence its nonzero eigenvalues are real, positive, and simple. It follows that all nonzero eigenvalues of $A_{n,r}$ are simple and occur in $\exp(2\pi i/n)$-orbits, forming unions of regular $n$-gons in the complex plane. A one-step Schur complement yields a three-term recursion in $r$ for the characteristic polynomials $\Phi_{n,r} \in \mathbb{Z}[x]$. This determines both the multiplicity of the eigenvalue $0$ and the number of nonzero eigenvalues, and leads to a generating function with cubic denominator. Applying a Tran-type confinement theorem gives the uniform bound $\rho(A_{n,r}) \le (27/4)^{1/n}$ and the sharp limit $\displaystyle\lim_{r \to \infty} \rho(A_{n,r}) = (27/4)^{1/n}$ for each fixed $n$. Finally, specializing at $x=1$ relates $\Phi_{n,r}(1)$ to Padovan spiral numbers and yields a complete classification of rational nonzero eigenvalues.

Let $X$ be a conical symplectic variety admitting a crepant resolution $Y$. Based on the theory of universal Poisson deformations, we construct a complex manifold called the principal twistor model associated with $Y$. We prove a universality theorem for this model: if the regular locus of $X$ admits a hyperkähler cone metric, then the twistor space of any algebraic hyperkähler metric on $Y$ asymptotic to this cone metric is uniquely recovered by slicing the principal twistor model. As an application, we use this universality to study the moduli space of hyperkähler structures with asymptotic behavior, and show that it admits an inclusion into a finite-dimensional real vector space.

We propose an implicit-explicit trust region method for computing second-order stationary points of a class of Landau-type free energy functionals, which correspond to physically (meta-)stable phases. The proposed method is demonstrated through the Landau-Brazovskii (LB) model in this work, while broader applicability to more Landau models of the similar type is straightforwardly extended. The LB energy functional is discretized via the Fourier pseudospectral method, which yields a finite-dimensional nonconvex optimization problem. By exploiting the Hessian structure, specifically, that the interaction potential is diagonal in reciprocal space whereas the bulk energy is diagonal in physical space, we design an adaptive implicit-explicit solver for the trust region subproblem. This solver utilizes the fast Fourier transform to perform efficient matrix-vector products, significantly reducing computational complexity while ensuring provable convergence to the global minimizer of the subproblem. In contrast to existing algorithms that target first-order stationary points, our proposed method can converge to a second-order stationary state, corresponding to a local minimum with theoretical convergence guarantees. Numerical experiments on the LB model demonstrate that the proposed approach efficiently escapes saddle points and significantly outperforms existing first-order schemes. Furthermore, we successfully identify the stable region of the FDDD phase, a structure previously unreported in the LB phase diagram.

Data approximation is essential in fields such as geometric design, numerical PDEs, and curve modeling. Moving Least Squares (MLS) is a widely used method for data fitting; however, its accuracy degrades in the presence of discontinuities, often resulting in spurious oscillations similar to those associated with the Gibbs phenomenon. This work extends the integration of MLS with the Weighted Essentially Non-Oscillatory (WENO) method and with an innovative partition of unity approach to higher dimensions. We propose a data-dependent operator using the novel Non-Linear Partition of Unity based on Moving Least Squares method in $\mathbb{R}^n$, which improves accuracy near discontinuities and maintains high-order accuracy in smooth regions. We demonstrate some theoretical properties of the method and perform numerical experiments to validate its effectiveness.

Beamforming is a fundamental technology that not only enhances communication efficiency but also lays the foundation for massive multiple-input multiple-output~(MIMO) systems. However, its reliance on accurate channel state information (CSI) estimation introduces significant training overhead and feedback costs, especially in large-scale antenna systems. In this paper, we investigate positioning-assisted beamforming as a competitive alternative to the CSI-based methods, which circumvents the complicated CSI estimation. In particular, we analyze the outage probability of positioning-assisted systems with joint Gaussian beams and derive its closed-form expressions for both two-dimensional~(2D) and three-dimensional~(3D) scenarios. Based on these results, we also derive closed-form expressions for the optimal joint Gaussian beam pattern. The optimal solution is independent of the positioning error distribution in 2D scenarios but depends on it in 3D cases. Subsequently, the asymptotic performance of the approximation error is analyzed. Numerical results verify the derived outage probability expressions, and show the effectiveness of the beam pattern optimization.

We propose a method to identify nonlinear acyclic networks in continuous time when the dynamics are located on the edges and all the nodes are excited. We show that it is necessary and sufficient to measure all the sinks to identify any tree in continuous time when the functions associated with the dynamics are analytic and satisfy $f(0)=0$, which is analogous to the discrete-time case. For general directed acyclic graphs (DAGs), we show that it is necessary and sufficient to measure all sinks, assuming that the dynamics are not linear (a condition that can be relaxed for trees). Then, based on the measurement of higher order derivatives and nonzero initial conditions, we introduce a method for the identification of trees, which allows us to recover the nonlinear functions located in the edges of the network under the assumption of dictionary functions. Finally, we propose a method to identify multiple parallel paths of the same length between two nodes, which allow us to identify any DAG when combined with the algorithm for the identification of trees. Several examples are added to illustrate the results.

In cell-free massive MIMO, centralized precoding is {theoretically known} to {remarkably} outperform its distributed counterparts, albeit {with} high implementation complexity. However, this letter highlights a practical limitation {often overlooked:} {widely used closed-form} centralized {precoders} are typically derived under a sum-power constraint, which often demands unrealistic power allocation that exceeds hardware capabilities. {When two simple heuristics (global power scaling and local normalization) are applied to enforce the per-AP instantaneous power constraint}, the centralized performance superiority disappears, making distributed precoding {a robust option}.

Motivated by ill-posed PDEs such as $\mathrm{div} (v) = F$ we study locally convex topologies $\mathcal{T}_{\mathcal{C}}$ on real vector spaces $X$ that are a ``localized'' version of a locally convex topology $\mathcal{T}$ to members of a family $\mathcal{C}$ of convex subsets of $X$. The distributions $F$ arising as $\mathrm{div} (v)$ are expected to be the members of the dual of well-chosen $X$ with respect to an appropriate localized topology $\mathcal{T}_{\mathcal{C}}$. In this work, the emphasis is on studying the functional analytic properties of $\mathcal{T}_{\mathcal{C}}$, according to those of $\mathcal{T}$ and $\mathcal{C}$. For instance, we show that in all foreseen applications, $\mathcal{T}_{\mathcal{C}}$ is sequential but none of Fréchet-Urysohn, barrelled, and bornological. These awkward phenomena are illustrated explicitly on a specific example corresponding to the distributional divergence of continuous vector fields in $\mathbb{R}^m$. We also show that, essentially, $\mathcal{T}_{\mathcal{C}}$ is semireflexive if and only if members of $\mathcal{C}$ are $\mathcal{T}$-compact. This leads to an abstract existence theorem, thereby establishing a general scheme for characterizing those $F$ such that $\mathrm{div} (v) = F$ for various classes of regularity of $v$, various classes of domains, and various boundary conditions.

We obtain new lower and upper bounds for the numerical radius of a bounded linear operator $A$ on a complex Hilbert space, which refine the existing ones. In particular, if $w(A)$ and $\|A\|$ denote the numerical radius and operator norm of $A$, respectively, then we show that \begin{eqnarray*} \nu(A) + \frac{1}{4} \left\||A|^2+|A^*|^2\right\| \leq w^2(A) \leq \frac12 w\left(\frac{|A|+|A^*|}{2}A \right)+ \frac14 \left\| |A|^2+ \left( \frac{|A|+|A^*|}{2}\right)^2 \right\|, \end{eqnarray*} where $\nu(A)\geq 0$ is a real number involving the operator norm of the Cartesian decomposition of $A$. We also develop several new numerical radius inequalities for the products and sums of operators via Euclidean operator radius of $2$-tuples of operators. In addition, we deduce equality characterizations for the inequalities. As an application, we obtain numerical radius inequalities for the commutators of operators, which improves the Fong and Holbrook's inequality $w(AB\pm BA) \leq 2\sqrt{2} w(A) \|B\|$ [Canadian J. Math. 1983].

Let $\mathcal{G}=\{G_1, G_2, \ldots , G_k\}$ be a family of bipartite graphs on the same vertex set. A rainbow Hamilton path (cycle) in $\mathcal{G}$ is a path (cycle) that visits each vertex precisely once such that any two edges belong to different graphs of $\mathcal{G}.$ In this paper, by adopting the technique of bi-shifting, we present tight sufficient conditions in terms of the spectral radius for a family $\mathcal{G}$ to admit a rainbow Hamilton path and cycle, respectively. Meanwhile, we completely characterize the corresponding spectral extremal graphs.

We consider a sparse i.i.d.\ non-Hermitian random matrix model $X_n$ (with sparsity parameter $K_n$) and a deterministic finite-rank perturbation $E_n$. Assuming biorthogonality for $E_n$ and a growth condition on $K_n$, we outline a finite-rank resolvent reduction leading to asymptotics for the overlap between an outlier eigenvector of $Y_n:=X_n+E_n$ and the corresponding spike eigenspace. In particular, for an outlier spike $\mu$ with $|\mu|>1$, the squared projection of the associated (right) eigenvector onto the spike eigenspace converges in probability to $1-|\mu|^{-2}$. Our result generalizes Theorem 1.6 of [HLN26] to general finite rank case solving Open Problem 5.

This paper establishes strong and weak convergence rates for slow-fast systems driven by $\alpha$-stable processes with jump coefficients. Unlike existing studies on multiscale systems driven by additive Lévy white noise, our model incorporates multiplicative noise, which brings essential challenges in deriving the exponential ergodicity for the frozen process, particularly gradient estimates. We derive exponential ergodicity in two different ways: the coupling method and the spatial periodic method; then the gradient estimate is developed by heat kernel asymptotic expansion. Moreover, under sufficient Hölder regularity of the time-dependent coefficients of the slow process, we can yield an optimal strong convergence rate of order $1-\frac{1}{\alpha_{2}}$ and a weak convergence rate of order 1. Furthermore, explicit formulas for the tangent map between tangent spaces of $S^{d-1}$ as well as its Jacobian determinant are obtained, where the map is induced by a nonlinear immersion.

The graded Lie algebra associated with the Nottingham group over a field of prime characteristic serves as a fundamental example of Nottingham algebras, a class of infinite-dimensional, positively graded thin algebras. This paper completes the classification of Nottingham algebras initiated in earlier papers, proving both existence and uniqueness results that determine all such algebras up to isomorphism.

The Robinson Splitting Theorem states that a c.e. degree $\mathbf{b}$ splits over any low c.e. degree $\mathbf{c}<\mathbf{b}$. We prove that a weaker version of this theorem holds in models of $\mathrm{P}^-+\mathrm{I}\Sigma_1$, with lowness replaced by superlowness.

The rate-distortion-perception (RDP) tradeoff characterizes the fundamental limits of lossy compression by jointly considering bitrate, reconstruction fidelity, and perceptual quality. While recent neural compression methods have improved perceptual performance, they typically operate at fixed points on the RDP surface, requiring retraining to target different tradeoffs. In this work, we propose a training-free framework that leverages pre-trained diffusion models to traverse the entire RDP surface. Our approach integrates a reverse channel coding (RCC) module with a novel score-scaled probability flow ODE decoder. We theoretically prove that the proposed diffusion decoder is optimal for the distortion-perception tradeoff under AWGN observations and that the overall framework with the RCC module achieves the optimal RDP function in the Gaussian case. Empirical results across multiple datasets demonstrate the framework's flexibility and effectiveness in navigating the ternary RDP tradeoff using pre-trained diffusion models. Our results establish a practical and theoretically grounded approach to adaptive, perception-aware compression.

We provide sufficient conditions for the existence of a trace on the essential $\mathrm{C}^\ast$-algebra of a (not necessarily Hausdorff) étale groupoid $G$ which extends an invariant measure $\mu$ on the unit space of $G$. In particular, it suffices for the isotropy groups of $G$ to be amenable, or for $G$ to be essentially free with respect to $\mu$. We also show that $G$ is essentially free with respect to an invariant measure $\mu$ if and only if $\mu$ extends to a unique trace on the full $\mathrm{C}^\ast$-algebra of $G$. We work in the generality of possibly infinite measures and, accordingly, possibly unbounded traces. Moreover, whenever possible, we state our results for twisted groupoids. As an application, we show that gauge-invariant algebras of finite-state self-similar groups admit a unique tracial state.

Let $E/\mathbb{Q}_p$ be an elliptic curve whose mod $p$ Galois image is contained in the normaliser of a non-split Cartan. We classify the possible $p$-adic images of $E$ using tools from $p$-adic Hodge theory via a careful analysis of the local Galois structure of the $p$-power torsion of $E$. We pay special attention to the case where $E$ has potentially supersingular reduction, where we give an algorithm to determine the corresponding filtered $(\varphi,\operatorname{Gal}(K/\mathbb{Q}_p))$-module from a Weierstrass model (which appears to be novel), and introduce alternative division polynomials that may be of independent interest. We deduce global consequences for elliptic curves $E/\mathbb{Q}$: when the mod $p$ representation of $E$ has non-split Cartan image and $E$ doesn't have CM, the $p$-adic image must be the full preimage of the normaliser of a mod $p^n$ non-split Cartan for some $n \geq 1$. As an application, we sharpen existing bounds on the adelic image in terms of the Weil height of the $j$-invariant.

\cite{tsagris2025a} proposed the generalized circular projected Cauchy distribution, whose special case is the wrapped Cauchy distribution. In this paper we first derive the relationship with the wrapped Cauchy distribution and we propose a log-likelihood ratio test for the equality of two angular means, without assuming equality of thew concentration parameters. Simulation studies illustrate the performance of the test when one falsely assumes that the true underlying distribution is the wrapped Cauchy distribution.

We study the rapid stabilization of general linear systems, when the differential operator $\mathcal{A}$ has a Riesz basis of eigenvectors. We find simple sufficient conditions for the rapid stabilization and the construction of a relatively explicit feedback operator. We use an $F$-equivalence approach \textcolor{black}{relying on Fredholm transformation} to show a stronger result: under these sufficient conditions the system is equivalent to a simple exponentially stable system, with arbitrarily large decay rate. In particular, our conditions improve the existing conditions of rapid stabilization for non-parabolic operators such as skew-adjoint systems.

We present a new short proof of Van der Waerden's Theorem about the existence of arbitrarily long monochromatic arithmetic progressions. The proof uses algebra in the compact space of ultrafilters $\beta\N$, but contrarily to the other existing proofs, neither minimal nor idempotent ultrafilters are involved.

Rosenbloom and Tsfasman, in their foundational work on the $m$-metric, introduced algebraic-geometric codes defined by multiple points on a smooth projective curve $X$. This construction involves a divisor $G$ and another divisor $D=\sum n p_i$, where $p_i$ are distinct rational points with $p_i \notin \text{supp}(G)$ and $n\in\mathbb{N}$. Although these codes are significant, their formal development for arbitrary genus remains incomplete in the literature, as most studies have concentrated on the genus $0$ case. We present a rigorous treatment of this class of codes. Starting with a smooth projective curve $X$, an invertible sheaf $L$, and an effective divisor $D=\sum n_i p_i$ where the $n_i$ are not necessarily equal, as well as tuples of uniformizers $t_D$ at the points of $D$ and trivializations $\gamma_D$ for the localizations $L_{p_i}$, the associated differential Goppa code is defined. This code arises from the theory of $n$-jets of invertible sheaves on curves, which enables the description of codewords using Hasse-Schmidt derivatives of sections of $L$. The variation of the code under changes in the data $(t_D, \gamma_D)$ is examined, and the group acting on these parameters is described. The behavior of the minimum Hamming distance under such variations is analyzed, with explicit examples provided for curves of genus $0$ and $1$. A duality theorem is established, involving principal parts of meromorphic differential forms. It is demonstrated that Goppa codes constitute a proper subclass of differential Goppa codes, and that every linear code admits a differential Goppa code structure on $\mathbb P^1$ using only two rational points.

We introduce localization operators on weighted Bergman and Fock spaces and show that, under a natural scaling of symbols and window functions, localization operators on the weighted Bergman space $A_{\beta r^2}^2$ converge, in the weak sense, to localization operators on the Fock space $F_{\beta}^2$ as $r\to\infty$. From this we derive several applications, including one about sharp norm estimates for certain Toeplitz operators on Fock spaces, one about windowed Berezin transforms for weighted Bergman spaces, and another about Szegö-type theorems for localization operators on weighted Bergman spaces.

Performance indicators are essential tools for assessing the convergence behavior of multi-objective optimization algorithms, particularly when the true Pareto front is unknown or difficult to approximate. Classical reference-based metrics such as hypervolume and inverted generational distance are widely used, but may suffer from scalability limitations and sensitivity to parameter choices in many-objective scenarios. Indicators derived from Karush--Kuhn--Tucker (KKT) optimality conditions provide an intrinsic alternative by quantifying stationarity without relying on external reference sets. This paper revisits an entropy-inspired KKT-based convergence indicator and proposes a robust adaptive reformulation based on quantile normalization. The proposed indicator preserves the stationarity-based interpretation of the original formulation while improving robustness to heterogeneous distributions of stationarity residuals, a recurring issue in many-objective optimization.

In this paper, we present a novel semi-implicit numerical scheme for the stochastic Cahn--Hilliard equation driven by multiplicative noise. By reformulating the original equation into an equivalent stochastic scalar auxiliary variable (SSAV) system, our method enables an efficient and stable treatment of polynomial nonlinearities in a semi-implicit fashion. In order to accurately capture the impact of stochastic perturbations, we carefully incorporate Itô correction terms into the SSAV approximation. Leveraging the smoothing properties of the underlying semigroup and the $H^{-1}$-dissipative structure of the nonlinear term, we establish the optimal strong convergence order of one-half for the proposed scheme in the trace-class noise case. Moreover, we show that the modified SAV energy asymptotically preserves the energy evolution law. Finally, numerical experiments are provided to validate the theoretical results and to explore the influence of noise near the sharp-interface limit.

The small finitistic dimension fPD$(R)$ of a ring $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we show that a commutative ring $R$ has fPD$(R)\leq d$ if and only if for any finitely generated ideal $I$ of $R$, if $Ext_R^i(R/I,R)=0$ for each $i=0,\dots,d$, then $Ext_R^i(R/I,R)=0$ for all $i\geq 0.$ As applications, we obtain that, for any commutative ring $R$, fPD$(R)\leq \mbox{FP-}Id_RR$, the self-FP-injective dimension of $R$. We also give some applications of these results to (weak) $(n,d)$-rings, DW-rings and rings of Prufer type.

An eigenvalue $\lambda$ of a signed graph $S$ of order $n$ is called a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector $j$. Characterizing signed graphs with exactly $k$ $(1\le k\le n)$ distinct main eigenvalues is a problem in algebraic and graph theory that has been studied since 2020. Du et al. (2024, 2026) characterized a class of signed graphs with exactly two main eigenvalues by analyzing a type of multigraph whose base graph is a tree. In this paper, we extend this study to the case where the associated multigraph has a unicyclic base graph, and we conclude by proposing several open problems.

In modern e-commerce and service operations, firms must jointly manage inventory replenishment and real-time order fulfillment to maximize profit under demand uncertainty. While each component has been studied extensively in isolation, their interaction remains underexplored. This paper investigates a fundamental operational question: which lever plays a more decisive role in overall system performance, replenishment or fulfillment? We model the system as a one-location online order fulfillment problem with lost sales and stochastic customer arrivals, each offering heterogeneous rewards. Replenishment follows either a base-stock or constant-order policy, while real-time fulfillment decisions are made using online algorithms. Our core performance metric is the expected average profit per replenishment cycle, evaluated across all combinations of these policies and algorithms. Our main theoretical result shows that when the replenishment cycle is long, the cumulative regret of online fulfillment remains of the same order as in a corresponding single-cycle problem, even under repeated replenishment, revealing a form of regret stability. This phenomenon also extends to a multi-location setting. We further develop a regret-based framework that quantitatively compares the value of improving replenishment versus improving fulfillment, and we characterize regimes in which optimizing replenishment yields a larger revenue impact than refining the online fulfillment algorithm (and vice versa). Motivated by examples where myopic algorithms underperform, we introduce a novel look-ahead online algorithm that anticipates future replenishment and demand. Numerical experiments verify that this algorithm outperforms myopic baselines. Overall, our results provide both theoretical and managerial insights into situations where inventory replenishment policies are more influential and vice versa.

We investigate the relationship between the Eklof-Mekler-Shelah Construction Principle for a variety of algebras $\mathbf{V}$ and the question of superstability of the free objects in $\mathbf{V}$, denoted as $\mathcal{F}_\mathbf{V}$. We consider this question in the general setting of AEC-coverings of $\mathcal{F}_\mathbf{V}$, with applications to first-order logic and beyond. Our main result is that if a strong form of the Construction Principle is satisfied, then almost all AEC-covering of $\mathcal{F}_\mathbf{V}$ are unsuperstable. Concrete applications to $R$-modules and varieties of groups are also considered.

We introduce the bilinear Nevo-Thangavelu spherical means on the Heisenberg group $\mathbb{H}^n,$ and derive $L^{p_1}(\mathbb{H}^n) \times L^{p_2}(\mathbb{H}^n) \to L^{p}(\mathbb{H}^n)$ estimates for the single-scale bilinear averaging operators, the (full) bilinear Nevo-Thangavelu maximal operator and finally for the bilinear lacunary maximal operator on $\mathbb{H}^n; n \geq 2$. Our result for the full maximal operator is sharp. The principal tools in our analysis include newly developed estimates for single-scale bilinear averages, Hopf's maximal ergodic theorem, and a $T^*T$ argument adapted to this setting.

We study the empirical spectral distribution of the normalized Laplacian of linear preferential attachment graphs in the Barab{á}si-Albert regime with fixed out-degree. For the resulting sequence of random multigraphs, we prove that the empirical spectral distribution converges weakly in probability to a deterministic probability measure supported on the interval [0, 2]. The limit is characterized via the local weak limit of preferential attachment graphs (the P{ó}lya-point graph): the limiting Stieltjes transform is given by the expected diagonal Green function at the root of the normalized Laplacian operator on this infinite random graph. The proof combines a resolvent approach with a uniform Neumann-series expansion for the normalized Laplacian, a random-walk representation in terms of return probabilities on decorated neighborhoods, a truncation and Doob martingale-Azuma-Hoeffding concentration argument along the PA filtration, and an analytic continuation argument based on normal families.

In this paper, we consider the problem of minimizing a smooth function on a Riemannian manifold and present a Riemannian gradient method with momentum. The proposed algorithm represents a substantial and nontrivial extension of a recently introduced method for unconstrained optimization. We prove that the algorithm, supported by a safeguarding rule, produces an $\epsilon$-stationary point with a worst-case complexity bound of $\mathcal{O}(\epsilon^{-2})$. Extensive computational experiments on benchmark problems are carried out, comparing the proposed method with state-of-the-art solvers available in the Manopt package. The results demonstrate competitive and often superior performance. Overall, the numerical evidence confirms the effectiveness and robustness of the proposed approach, which provides a meaningful extension of the recently introduced momentum-based method to Riemannian optimization.

We establish unweighted Hardy-type inequalities on step-two Carnot groups with one-dimensional vertical layer, with explicit lower bounds for the optimal Hardy constant. The approach is based on a quantitative integration-by-parts mechanism that replaces the non-horizontal Euler vector field by a suitably constructed horizontal vector field with controlled norm. As applications, we obtain fully explicit bounds in the Heisenberg group for both the Kor{à}nyi gauge and the Carnot--Carath{é}odory distance, and we extend the results to non-isotropic step-two structures through a generalized Kor{à}nyi-type homogeneous norm.

We study the Wasserstein gradient flow of semi-discrete energies in the space of probability measures, that is functionals depending on two measures-one being an absolutely continuous density and the other an atomic measure. These energies appear naturally in the field of urban planning. This is done via the celebrated JKO scheme, for which we prove convergence to a limiting system composed of a parabolic PDE with singular advection coupled with an ODE, also presenting singular dynamics. This is first done under more general assumptions using classical tools, and in a second moment convergence is proven to hold in $L^2_tH^1_x$ for the cases of linear and Porous-Medium type diffusions. We then pass to the study of some qualitative properties of this system, such as the convergence of the atoms towards the baricenters of their corresponding Laguerre cells. We finish this work with extensive numerical simulations that aid in formulating conjectures for the qualitative behavior of this system; in the case of linear diffusion, for instance, we observe a dynamic crystallization phenomenon.

The use of stochastic differential equations in multi-objective optimization has been limited, in practice, by two persistent gaps: incomplete stability analyses and the absence of accessible implementations. We revisit a drift--diffusion model for unconstrained vector optimization in which the drift is induced by a common descent direction and the diffusion term preserves exploratory behavior. The main theoretical contribution is a self-contained Lyapunov analysis establishing global existence, pathwise uniqueness, and non-explosion under a dissipativity condition, together with positive recurrence under an additional coercivity assumption. We also derive an Euler--Maruyama discretization and implement the resulting iteration as a \emph{pymoo}-compatible algorithm -- \emph{pymoo} being an open-source Python framework for multi-objective optimization -- with an interactive \emph{PymooLab} front-end for reproducible experiments. Empirical results on DTLZ2 with objective counts from three to fifteen indicate a consistent trade-off: compared with established evolutionary baselines, the method is less competitive in low-dimensional regimes but remains a viable option under restricted evaluation budgets in higher-dimensional settings. Taken together, these observations suggest that stochastic drift--diffusion search occupies a mathematically tractable niche alongside population-based heuristics -- not as a replacement, but as an alternative whose favorable properties are amenable to rigorous analysis.

We study the orbits of the solutions to the Markoff-type equation $$X^2 + Y^2 + Z^2 = AX + BY + CZ + D$$ in $\mathbb{F}_p$ for fixed integers $A, B, C,$ and $D$ under the group of symmetries $\Gamma$ generated by $$V_1: (x, y, z)\mapsto (A + yz - x, y, z),$$ $$V_2: (x, y, z)\mapsto (x, B + xz - y, z),\text{ and}$$ $$V_3: (x, y, z)\mapsto (x, y, C + xy - z).$$ For most quadruples of parameters $(A, B, C, D)$, we show that there is a density one set of primes $p$ such that $\Gamma$ acts transitively on the bulk of the solutions mod $p$, with the remainder breaking up into a few small orbits which arise from finite orbits within the solutions over $\mathbb{C}$. For those ``degenerate'' quadruples of parameters $(A, B, C, D)$ to which this result does not apply, we show that there must be at least 2 large orbits, and in some cases 4 large orbits, under the action of this group. Our results become especially interesting when applied to two special subfamilies. The first is $$X^2 + Y^2 + Z^2 = XYZ + k$$ for $k \neq 4$, which arises in the study of the combinatorial group theory of $\text{SL}_2(\mathbb{F}_p)$. Our results very nearly prove the $Q$-classification conjecture of McCullough and Wanderley for density 1 of all primes, and thus by the work of Martin very nearly proves their Classification and $T$-Classification conjectures for density 1 of all primes. The second special family is $$x_1^2 + x_2^2 + x_3^2 + a_1x_2x_3 + a_2x_1x_3 + a_3x_1x_2 = (3+a_1+a_2+a_3)x_1x_2x_3,$$ which arises from certain generalized cluster algebras. Here, our notion of ``degenerate'' parameters $(A, B, C, D)$ specializes to the degeneracy condition of de Courcy-Ireland, Litman, and Mizuno. For this family, their results imply that our transitivity result applies to all sufficiently large primes $p$, independent of $a_1, a_2,$ and $a_3.$

In this short note, we present a trick to ensure that the Jacobian of a given smooth curve has strictly positive Mordell-Weil rank. More explicitly, we prove that a smooth curve with no rational non-trivial 2-torsion and no rational theta characteristic has non-zero Mordell-Weil rank assuming the existence of a rational degree 1 divisor class. This criteria is both of theoretical and computational interest as we show how to use it in practice. We also give refinements and explicit examples.

This paper establishes the well-posedness of stochastic partial differential equations with reflection in an infinite-dimensional ball, within the fully local monotone framework. We prove a key variational inequality under convergence of weak topologies. Our result is very general, including many important models such as the stochastic Allen-Cahn equations, stochastic p-Laplacian equations, as well as more complex systems like the stochastic Cahn-Hilliard equations and the stochastic 3D tamed Navier-Stokes equations.

We prove optimal regularity results for solutions to linear kinetic Fokker-Planck equations in bounded domains. Our contributions are two-fold. First, we establish the sharp $C^{1/2}$ regularity for either diffuse reflection or prescribed in-flow boundary conditions. Previously, in this setting, it was only known that solutions are $C^{\alpha}$ for some small $\alpha > 0$. Second, we provide a complete characterization of the solution behavior near the grazing set by proving higher order expansions beyond the critical regularity threshold of $\frac{1}{2}$. These results demonstrate for the first time that solutions maintain higher smoothness up to the grazing set near the incoming boundary.

We obtain some properties of the q-Narayana polynomials for q=-1 and compare them to corresponding properties for q=1.

We develop a non-abelian, gauge-theoretic framework for the Schwarzian derivative and for second-order differential equations on Riemann surfaces. As applications, we extend Dedekind's Schwarzian approach to elliptic periods to generic one-parameter families of curves of genus $g$ by replacing the non-canonical scalar Picard--Fuchs equation of order $2g$ with a canonical second-order equation with $g\times g$ matrix coefficients on the Hodge bundle. In higher dimensions, we discuss periods of a one-parameter family of cubic threefolds via the intermediate Jacobian. Finally, we discuss mass--spring systems in mechanics as a natural testing ground for the non-abelian Schwarzian viewpoint.

We obtain sharp estimates for functions harmonic with respect to $x$-dependent rectilinear stable processes in balls, under the assumption that the Dirichlet exterior data are radial about the center. The main idea of the proof is based on the construction of global barrier functions for the $x$-dependent rectilinear fractional Laplacian in balls.

One of the emerging problems in algebraic geometry is to characterize the affine $n$-space $\mathbb{A}^n$ among smooth affine schemes up to $\mathbb{A}^1$-contractibility. Recent efforts show that this characterization holds in dimensions $n<3$ over certain fields. In this thesis, we extend this observation to "reasonably" arbitrary base schemes in relative dimensions $d<3$, exploiting the Zariski local triviality and the triviality of the sheaf of relative differentials. From dimensions $n\geq 3$, the existence of smooth "exotic" affine schemes - those that are $\mathbb{A}^1$-contractible but not isomorphic to the affine $n$-space - has already been established. A well-studied family constitutes the Koras-Russell threefolds $\mathcal{K}$ and their higher-dimensional prototypes $\mathcal{X}_n$, whose $\mathbb{A}^1$-contractibility has been so far proven over fields of characteristic zero. Here, we extend the relative $\mathbb{A}^1$-contractibility of $\mathcal{K}$ and $\mathcal{X}_n$ over a Noetherian base scheme in arbitrary dimensions. Then, using these prototypes, we study the existence of "exotic spheres" - $n$-dimensional smooth schemes that are $\mathbb{A}^1$-homotopic, but not isomorphic to $\mathbb{A}^n \backslash \{0\}$ - in motivic homotopy theory. This result can be seen as the "compact" analog of the study of exotic affine schemes. Our main result shows that in all dimensions $n\geq 4$, the quasi-affine varieties $\mathcal{X}_n \backslash \{\bullet\}$ give a model for the exotic motivic spheres over infinite perfect fields. The novelty is that these constitute the first family of examples of smooth motivic spheres of dimension $n$, which are not isomorphic to $\mathbb{A}^n \backslash \{0\}$.

The Bayesian and Akaike information criteria aim at finding a good balance between under- and over-fitting. They are extensively used every day by practitioners. Yet we contend they suffer from at least two afflictions: their penalty parameter $\lambda=\log n$ and $\lambda=2$ are too small, leading to many false discoveries, and their inherent (best subset) discrete optimization is infeasible in high dimension. We alleviate these issues with the pivotal information criterion: PIC is defined as a continuous optimization problem, and the PIC penalty parameter $\lambda$ is selected at the detection boundary (under pure noise). PIC's choice of $\lambda$ is the quantile of a statistic that we prove to be (asymptotically) pivotal, provided the loss function is appropriately transformed. As a result, simulations show a phase transition in the probability of exact support recovery with PIC, a phenomenon studied with no noise in compressed sensing. Applied on real data, for similar predictive performances, PIC selects the least complex model among state-of-the-art learners.

A central objective in Ramsey theory is determining whether restricted families of discrete structures necessarily contain substantially larger homogeneous substructures, compared to the unrestricted structures. In the setting of tournaments, it is well known that every tournament contains a transitive subgraph of size $\log n$, and that this is best possible up to a constant factor. A restricted family of tournaments that has been extensively studied is the family of $k$-majority tournaments. They are obtained by taking $2k-1$ linear orders of a set $X$, and defining a tournament on $X$ which has an edge from $u$ to $v$ if $u$ precedes $v$ in at least $k$ of these orders. Milans, Schreiber, and West proved that such tournaments indeed have significantly larger transitive tournaments. More precisely, they proved that every $k$-majority tournament contains a transitive tournament of size $n^{2^{-\Theta(k)}}$. Our main goal in this paper is to give an exponential improvement in the dependence of the exponent on $k$ by showing that every $k$-majority tournament contains a transitive set of size $n^{\Omega(1/k)}$. Finally, we highlight several open problems and conjectural directions related to random $k$-majority tournaments.

In this paper, we study evolutive Hamilton Jacobi equations with Hamiltonians that are discontinuous in time, posed on a simple network consisting of two edges on the real line connected at a single junction. We introduce a notion of (flux-limited) viscosity solution for Hamiltonians H=H(t,x,p) that are assumed to be only measurable in time t. The flux limiter, A=A(t), acting at the junction, is not required to be continuous but only in L infinity. In the case of convex Hamiltonians, we prove a comparison principle and establish an existence result via the construction of an optimal control problem. Generalisations to the nonconvex case and to more general networks are also discussed.

Epistemic uncertainty arises in lack of complete knowledge about the state of a system. There are multiple mathematical frameworks for measuring such uncertainty quantitatively, often referred to as imprecise probability theories. Inspired by work of Opdan, we introduce a general category theoretic definition of epistemic calculi, which we use as a foundation for modelling and studying contradictions and synergies between several philosophical epistemological concepts. We further develop an enriched category theoretic process for changing calculi, and use this to study relationships between existing examples, like possibility theory and certainty factors. Finally, we introduce a general categorical form of belief updating based on change of enrichment, and prove that Bayesian updating and possibilistic conditioning arise as examples.

We prove a converse Lyapunov theorem for boundedness of reachability sets for a general class of control systems whose flow is Lipschitz continuous on compact intervals with respect to trajectory-dominated inputs. We show that this condition is satisfied by many semi-linear evolution equations. For ordinary differential equations, as a consequence of our results, we obtain a converse Lyapunov theorem for forward completeness, without a priori restrictions on the magnitude of inputs.

We exhibit two finitely generated residually finite groups $G$ and $H$ with isomorphic profinite completions $\widehat{G} \cong \widehat{H}$, such that $G$ is co-Hopfian while $H$ is not. The construction utilizes Wise's residually finite version of the Rips construction applied to a finitely presented acyclic group $U$ with trivial profinite completion and a strong universality property. A key feature of our approach is the construction of $H$ as a preimage subgroup of $G$ which is conjugate to a proper subgroup of itself. This renders the non-co-Hopfianity of $H$ immediate without requiring a detailed structural analysis of the Rips kernel.

Theoretical and applied research into privacy encompasses an incredibly broad swathe of differing approaches, emphasis and aims. This work introduces a new quantitative notion of privacy that is both contextual and specific. We argue that it provides a more meaningful notion of privacy than the widely utilised framework of differential privacy and a more explicit and rigorous formulation than what is commonly used in statistical disclosure theory. Our definition relies on concepts inherent to standard Bayesian decision theory, while departing from it in several important respects. In particular, the party controlling the release of sensitive information should make disclosure decisions from the prior viewpoint, rather than conditional on the data, even when the data is itself observed. Illuminating toy examples and computational methods are discussed in high detail in order to highlight the specificities of the method.

The classical Grunwald--Wang theorem asserts that, unless we are in the so-called special case, local cyclic Galois extensions at finitely many completions of a number field can be approximated by a global cyclic extension. In the special case the obstruction is measured by a group of order 2. It has been known for a long time that the Grunwald--Wang theorem extends to a very general context of valued fields. Therefore it is natural to ask whether in the special case the obstruction is always measured by a finite group and if so, is the order of this group bounded independently of the number of places under consideration. We show that the answer to both questions is negative in general, already for rational function fields and discrete valuations coming from points of the affine line. This has some interesting links to the arithmetic of function fields over Q or Q_p.

We exhibit planar, rational curves of large degree over ${\mathbb F}_2$ that have a unique singular point, which has multiplicity 2. In characteristic 0 such curves exist only for degrees up to $6$.

We develop a statistically robust framework for reconstructing metal--semiconductor contact regions using topological gradients. The inverse problem is formulated as the identification of an unknown contact region from boundary measurements governed by an elliptic model with piecewise coefficients. Deterministic stability of the topological gradient with respect to measurement noise is established, and the analysis is extended to a statistical setting with multiple independent observations. A central limit theorem in a separable Hilbert space is proved for the empirical topological gradient, yielding optimal $n^{-1/2}$ convergence and enabling the construction of confidence intervals and hypothesis tests for contact detection. To further refine the reconstruction, a shape optimization procedure is employed, where the free parameter $\beta$ in the CCBM formulation plays a crucial role in controlling interface sensitivity. While $\beta$ affects both topological and shape reconstructions, its influence is particularly pronounced in the shape optimization stage, allowing more accurate estimation of the size and geometry of the contact subregion. The proposed approach provides a rigorous criterion for distinguishing true structural features from noise-induced artifacts, and numerical experiments demonstrate the robustness, precision, and enhanced performance of the combined statistical, topological, and $\beta$-informed shape-based reconstruction.

Arrow patterns were introduced by Berman and Tenner as a generalization of vincular patterns. They observed that arrow patterns have the potential to bridge the divide between a permutation's cycle notation and its one-line notation; in support of this, they used arrow avoidance to enumerate shallow and cyclic shallow permutations. More recently, $321$-avoiding cyclic permutations were recharacterized entirely in terms of arrow avoidance. Motivated by these results, we initiate a systematic study of arrow avoidance. In this paper, we prove structural results about arrow patterns, including defining arrow-Wilf equivalence, and enumerate several arrow avoidance classes. Finally, we consider the avoidance of pairs of arrow patterns, focusing on cases that prohibit fixed points in the underlying permutation.

We study infinite-horizon Markov decision processes (MDPs) where the decision maker evaluates each of her strategies by aggregating the infinite stream of expected stage-rewards. The crucial feature of our approach is that the aggregation is performed by means of a given diffuse charge (a diffuse finitely additive probability measure) on the set of stages. The results of Neyman [2023] imply that in this setting, in every MDP with finite state and action spaces, the decision maker has a pure optimal strategy as long as the diffuse charge satisfies the time value of money principle. His result raises the question of existence of an optimal strategy without additional assumptions on the aggregation charge. We answer this question in the negative with a counterexample. With a delicately constructed aggregation charge, the MDP has no optimal strategy at all, neither pure nor randomized.

We study paths in the p-Bratteli diagram associated with hook partitions, where p is an odd prime. By comparing blocks along a path, we define inversions and descents. We prove that the sign balance derived from inversions vanishes at every vertex of the diagram. Using descents, we introduce the p^(k)-Fibonacci numbers and derive recurrence relations for them. For k=0, we recover the OEIS sequence A391520, while for k>=1 we obtain new families of Fibonacci-type sequences.

In this note, we test the performance of six algorithms from the family of graph-based splitting methods [SIAM J. Optim., 34 (2024), pp. 1569-1594] specialized to normal cones of linear subspaces. To do this, we first implement some numerical experiments to determine the best relaxation parameter for each algorithm. Then, we compare the number of iterations each algorithm requires to reach a given stopping criterion, using the previously identified best relaxation parameter. The numerical results allow us to identify some relevant patterns and provide numerical evidence that may guide further theoretical analysis.

This work introduces a data-driven, non-intrusive reduced-order modeling (ROM) framework that leverages Optimal Transport (OT) for multi-fidelity and parametric problems. Building upon the success of displacement interpolation for data augmentation in handling nonlinear dynamics, we extend its application to more complex and practical scenarios. The framework is designed to correct a computationally inexpensive low-fidelity (LF) model to match an accurate high-fidelity (HF) one by capturing its temporal evolution via displacement interpolation while preserving the problem's physical consistency. The framework is further extended to address systems dependent on a physical parameter, for which we construct a surrogate model using a hierarchical, two-level interpolation strategy. First, it creates synthetic HF checkpoints via displacement interpolation in the parameter space. Second, the residual between these synthetic HF checkpoints and a true LF solution is interpolated in the time domain using the multi-fidelity OT-based methodology. This strategy provides a robust and efficient way to explore the parameter space and to obtain a refined description of the dynamical system. The potential of the method is discussed in the context of complex and computationally expensive diffuse interface methods for two-phase flow simulations, which are characterized by moving interfaces and nonlinear evolution, and challenging to be dealt with traditional ROM techniques.

We prove that the geodesic disks are the unique maximisers of the first non-trivial Neumann eigenvalue among all simply connected domains of the sphere $\mathbb S^2$ with fixed area.

In this paper we introduce conformal heat flow of (extrinsic) biharmonic maps on $4$-manifold, simply called bi-conformal heat flow (bi-CHF), and study its properties. Similar to other CHF of harmonic maps and regularized $n$-harmonic maps, (CHF and regularized $n$-CHF respectively), we obtain global smoothness and no finite time singularity.

We characterize the extreme and exposed points of the unit ball (with respect to the $L^1$-norm) in the shift-invariant space generated by the Gaussian function, as well as in the quasi shift-invariant space generated by the hyperbolic secant.

Magnitude is an isometric invariant of metric spaces introduced by Leinster. Although magnitude is nowhere continuous on the Gromov-Hausdorff space of finite metric spaces, continuity results are possible if we restrict the ambient space. In this paper, we focus on $\ell_1^N$ and prove that magnitude is continuous at every skew finite subset of $\ell_1^N$, that is, at every finite set whose coordinate projections are injective. For such sets, we analyze cubical thickenings and derive an explicit formula for their weight measures. This yields a formula for the magnitude of these thickenings, which we use to prove that their magnitude converges to that of the underlying finite set. Since skew finite subsets of $\ell_1^N$ form an open and dense subset of the space of all finite subsets, magnitude is continuous on an open dense subset of the space of finite subsets of $\ell_1^N$.

In this paper, we consider sums of four generalized polygonal numbers whose parameters are restricted to integers with a bounded number of prime divisors. With some restriction on m modulo 30, we show that for n sufficiently large, it can be represented as such a sum, where the parameters are restricted to have at most 988 prime factors.

This paper studies maintenance optimization for a two-component system under mixed observability. Component~$U_1$ is fully monitored, whereas component~$U_2$ is only partially observable due to sensing limitations. The system exhibits unidirectional positive degradation dependence, in which the health state of component~$U_1$ influences the degradation process of component~$U_2$, but not vice versa. We propose a novel framework for modeling and optimizing maintenance decisions for such systems using a partially observable Markov decision process (POMDP). Under mild conditions, we analytically establish structural properties of the optimal maintenance policy. Baum-Welch algorithm with multiple sample paths is developed to estimate the unknown system parameters in the context of a covariate-dependent Hidden Markov Model. %from observational data with multiple trajectories. Numerical experiments demonstrate the effectiveness of the proposed parameter estimation and the maintenance policy. Across 64 instances, we show that it consistently outperforms classical threshold-based policies. Specifically, when the degradation of component $U_1$ is faster, it achieves maximal cost reductions of up to approximately $6\%$

We prove that, over an arbitrary imaginary quadratic field, every symmetric formal Fourier-Jacobi series of Hermitian modular forms converges and equals the Fourier-Jacobi expansion of a genuine Hermitian modular form. As an application, we show that the Chow-valued Kudla generating series of special cycles on unitary Shimura varieties of signature $(p,1)$ is modular of weight $p+1$ for a Weil representation, establishing the geometric unitary Kudla conjecture in arbitrary codimension. This removes the modularity hypothesis from the arithmetic inner product formula of Li-Liu.

We study the geometry of $q$-rational numbers, introduced by Morier-Genoud and Ovsienko, for positive real $q$. In particular, we construct and analyse the deformed Farey triangulation and the deformed modular surface. We interpret every $q$-rational geometrically as a circle, similar to the famous Ford circles. Further, we define and study new operations on $q$-rationals, the Springborn operations, which can be seen as a quadratic version of the Farey addition. Geometrically, the Springborn operations correspond to taking the homothety centers of a pair of two circles.

We provide an explicit classification of all simple $\mathfrak{sl}_2$-modules that are torsion free of rank $1$ over the Cartan subalgebra. We also establish a similar result for the first Weyl algebra and for the Lie superalgebra $\mathfrak{osp}(1|2)$.

Coupled cell systems model interacting dynamical units and provide a natural framework for studying synchrony phenomena arising from collective behavior. Graph symmetries often induce such patterns, but certain networks exhibit additional synchronies not associated with automorphisms, commonly referred to as exotic synchronies. In undirected asymmetric graphs, any synchrony, if present, must be non-symmetry-induced, and determining when such exotic patterns occur remains a challenging structural problem. In this work, we address this question for networks whose underlying coupling graph is a tree, a class of graphs that naturally models hierarchical interactions among elements. We prove that exotic synchronizations do not arise in tree-type networks, showing that every balanced coloring is a fixed-point coloration determined by graph automorphisms. Furthermore, we identify the importance of the role played by the leaves of a graph in this context. Beyond existence results, we investigate the dynamical consequences of these structures by analyzing the linear stability of equilibria and the Lyapunov stability of synchrony subspaces for admissible vector fields defined on tree networks. Particular attention is devoted to cherry- type configurations, where local symmetries generated by leaves attached to a common vertex influence the stability properties of the associated synchronous states, thereby clarifying how the combinatorial architecture of trees constrains both the emergence and the stability of synchrony.

Integral-equation-based fast direct solvers for electromagnetic scattering can substantially reduce computational costs, especially in the presence of multiple excitations. We recently proposed a new high-frequency fast direct solver strategy that combines preconditioning techniques with acceleration algorithms. However, the validity of this approach applied to non-canonical geometries requires further justification. In this contribution, we collect relevant semiclassical microlocal results and use them to assess the legitimacy and effectiveness of the proposed fast direct solver in the high-frequency regime.

We derive a hydrodynamics formulation for a modified Dirac equation with a nonlinear mass term that preserves the homogeneity of the original Dirac equation. The nonlinear Dirac equation admits a symmetric split into the left and right-handed spinor components. It is formulated using Clifford algebra tools. We prove global existence for a regularized equation.

In this paper, we consider an age-structured mechanical model for tumor growth. This model takes into account the life-cycle of tumor cells by including an age variable. The underlying process for tumor growth is the same as classical tumor models, where growth is driven by pressure-limited cell proliferation, and cell movement away from regions of high pressure. The main contribution of this paper is establishing the convergence of solutions of the age-structured model to a limit satisfying a Hele-Shaw free boundary problem. This limiting problem describes the geometric motion of the tumor as it grows according to a nonlinear Darcy's law.

We study packet routing in the Kautz digraph K(d,D), where every ordered pair of distinct vertices is connected by a unique shortest directed path. The regular routing introduced in earlier work schedules all ordered pairs in tau(d,D) = (D-1)d^(D-2) + D d^(D-1) steps. We show that, for every fixed outdegree d at least 2 and all sufficiently large diameters D, no shortest-path routing scheme can match this makespan. More precisely, we prove that K(d,D) contains an edge whose shortest-path congestion strictly exceeds tau(d,D) when D is sufficiently large. Our construction uses edge-words drawn from a subset of ternary unbordered square-free words, together with a trimming inequality that propagates large congestion at distance D down to shorter distances. Computations for d=2 and small D show that for all D at least 4 there is an edge in K(2,D) with congestion greater than tau(2,D).

Geometric (also known as spatial) quantiles, introduced by Chaudhury and representing one of the three principal approaches to defining multivariate quantiles, have been well studied in the literature. In this work, we focus on the extremal behaviour of these quantiles. We establish new extremal properties, namely general lower and upper bounds for the norm of extreme geometric quantiles, free of any moment conditions. We discuss the impact of such results on the characterization of distribution behaviour. Importantly, the lower bound can be directly linked to univariate quantiles and to halfspace (Tukey) depth central regions, highlighting a novel connection between these two fundamental notions of multivariate quantiles.

This paper introduces Exp-ParaDiag, a novel time-parallel method that combines the strength of exponential integrators into the ParaDiag framework. We develop and analyze Exp-ParaDiag based on first and second order accurate exponential integrators. We establish the convergence of the proposed methods both as preconditioned fixed-point iterations and as precon- ditioners within the GMRES framework. Furthermore, we extend the Exp-ParaDiag formulation to achieve sixth-order temporal accuracy using exponential integrators. The proposed approach is also generalized to nonlinear problems, for which convergence is rigorously demonstrated. A series of numerical experiments is presented to validate the theoretical results and to illustrate the robustness and efficiency of the developed methods.

This paper establishes a comparison theorem for the maximum eigenvalue of a sum of independent random symmetric matrices. The theorem states that the maximum eigenvalue of the matrix sum is dominated by the maximum eigenvalue of a Gaussian random matrix that inherits its statistics from the sum, and it strengthens previous results of this type. Corollaries address the minimum eigenvalue and the spectral norm. The comparison methodology is powerful because of the vast arsenal of tools for treating Gaussian random matrices. As applications, the paper improves on existing eigenvalue bounds for random matrices arising in spectral graph theory, quantum information theory, high-dimensional statistics, and numerical linear algebra. In particular, these techniques deliver the first complete proof that a sparse random dimension reduction map has the injectivity properties conjectured by Nelson & Nguyen in 2013.

Skewed distributions are fundamental in modelling asymmetric data on the d-dimensional torus. In this context, asymmetry is introduced through the sine-skewing mechanism, which is the only skewing mechanism that has been proposed on the hyper-torus in the literature. Some sine-skewed models are known to suffer from a singular Fisher information matrix in the vicinity of symmetry, which poses a significant issue for inferential purposes. It is an open question to determine for which sine-skewed models Fisher information singularity occurs. In this paper, a general characterization of the class of models that exhibit this singularity is given in the general d-dimensional setting.

We formalize in Lean the following foundational result in commutative algebra: Let $R \to S$ be a faithfully flat map of (not necessarily noetherian) commutative rings, and let $P$ be an arbitrary $R$-module. Then $P$ is projective over $R$ if and only if $S\otimes_R P$ is projective over $S$. This formalizes and verifies Perry's fix of a subtle gap in the classical work of Raynaud and Gruson, a result which is a key ingredient in the study of finitistic dimension of commutative noetherian rings.

We show that for any infinite tree of finite cone type satisfying a mild expansion condition, the only typical process on its vertices with covariance induced by the Green's function is the Gaussian wave. This generalizes a result of Backhausz and Szegedy, who proved this for the infinite regular tree of degree $d\geq 3$. We do this by giving a reduction to a statement concerning the distribution of the inner product of our process with columns of the Green's function, which in turn are straightforward to calculate. As a consequence, for random bipartite biregular graphs, the distribution of local neighborhoods of eigenvectors must approximate the Gaussian wave. Moreover, for generic configuration models including random lifts, the local distribution of a uniformly chosen eigenvector from any arbitrarily small spectral window likewise converges to the Gaussian wave.

Existence of superdecomposable pure-injective modules reflects complexity in the category of finite-dimensional representations over an algebra. Such an existence occurs when an algebra is non-domestic; a conjecture due to M. Prest. G. Puniski confirms the conjecture for non-domestic string algebras. Geiß, Labardini-Fragoso and Schröer show that every Jacobian algebra associated with a triangulation of a closed surface with marked points is finite-dimensional and tame. We show that, excluding only the case of a sphere with four (or fewer) punctures, there exists a special family of pointed modules, called an independent pair of dense chains of pointed modules. In the process, we show the existence of such an independent pair in a non-domestic skew-gentle algebra and (skew) Brauer graph algebras by showing that the Galois semi-covering functor and trivial extension preserve such pairs. Then it follows from a result of M. Ziegler that there exists a superdecomposable pure-injective module if the algebraically closed field is countable.

This paper is devoted to the classification and studying properties of complex unital $3$-dimensional structurable algebras. We provide a complete list of non-isomorphic classes, identifying five algebras for type $(2, 1)$ and two algebras for type $(1, 2).$ For each obtained algebra, we describe the derivation algebra, the automorphism group, the lattice of subalgebras and ideals, and functional identities of degree $2$. Furthermore, we investigate the Allison-Kantor construction for the classified algebras. We determine the structure of the resulting $\mathbb{Z}$-graded Lie algebras, providing their dimensions and Levi decompositions.

Many research works have concerned normality-preserving selection rules and operations on the sequence of digits of a given normal number that maintain or violate normality. This leads us to introduce rearrangement operations on finite subwords appearing within the digit expansions of normal numbers, and this is inspired by the concept of an abelian complexity function in the field of combinatorics on words. We introduce the concept of an abelian-normal number, with respect to a given base and a given weighting/counting function on subwords, by analogy with normal numbers and with the use of the equivalence classes associated with abelian complexity functions. We then construct a non-normal analogue $D_{10}$ of Champernowne's constant $C_{10}$ and prove that $D_{10}$ is abelian-normal with respect to a given weighting function. We conclude with two open problems concerning our Champernowne-like constant $D_{10}$.

In this study, we investigate the ISS of impulsive switched systems that have modes with both stable and unstable flows. We assume that the switching signal satisfies mode-dependent average dwell and leave time conditions. To establish ISS conditions, we propose two types of time-varying ISS-Lyapunov functions: one that is non-decreasing and another one that is decreasing. Our research proves that the existence of either of these ISS-Lyapunov functions is a necessary and sufficient condition for ISS. We also present a technique for constructing a decreasing ISS-Lyapunov function from a non-decreasing one, which is useful for its own sake. Our findings also have added value to previous research that only studied sufficient conditions for ISS, as our results apply to a broader class of systems. This is because we impose less restrictive dwell and leave time constraints on the switching signal and our ISS-Lyapunov functions are time-varying with general nonlinear conditions imposed on them. Moreover, we provide a method to guarantee the ISS of a particular class of impulsive switched systems when the switching signal is unknown.

In 2018, McInnes et al. introduced a dimensionality reduction algorithm called UMAP, which enjoys wide popularity among data scientists. Their work introduces a finite variant of a functor called the metric realization, based on an unpublished draft by Spivak. This draft contains many errors, most of which are reproduced by McInnes et al. and subsequent publications. This article aims to repair these errors and provide a self-contained document with the full derivation of Spivak's functors and McInnes et al.'s finite variant. We contribute an explicit description of the metric realization and related functors. At the end, we discuss the UMAP algorithm, as well as claims about properties of the algorithm and the correspondence of McInnes et al.'s finite variant to the UMAP algorithm.

Most of the known non-invertible symmetries of quantum field theories in three and four spacetime dimensions act invertibly on local operators. An exception is coset symmetries, which can be constructed from gauging a non-normal subgroup of an invertible symmetry. In this paper, we study the action of a general finite non-invertible symmetry on local operators in four dimensions. We show that non-invertible symmetries without topological line operators necessarily act invertibly on local operators. Using this result, we argue that the action of a general non-invertible symmetry in 3+1d on local operators can be decomposed into the invertible action of some operators composed with the action of a gauging interface. We use this result to study when such a symmetry is anomaly-free. We find a necessary condition for a finite non-invertible symmetry in 3+1d to be anomaly-free, and show that anomaly-free non-invertible symmetries without topological line operators are non-intrinsically non-invertible.

We introduce a spectral approach to characterizing the three-dimensional Edwards-Anderson spin glass. By analyzing the eigenvalue statistics of overlap matrices constructed from two-dimensional cross-sections, we identify a crossover from the Wigner semicircle law at high temperatures towards a Gaussian distribution, which is consistently attained near the spin-glass critical point. Visible for different distributions of the random coupling, the Gaussian distribution can potentially serve as a robust spectral indicator of criticality. Remarkably, the spectral density is well-described by Tsallis statistics, with the entropic index $q$ evolving from $q = -1$ (semicircle, $T=\infty$) to $q = 1$ (Gaussian) at $T_c$, revealing a statistical structure inside the paramagnetic phase. We find $q\le 1$ within numerical precision. While the local level statistics remain consistent with GOE statistics, reflecting standard level repulsion, the temperature dependence appears mainly in the global spectral density. Our results present spectral statistics as a computationally efficient complement to multi-replica correlator methods and provide a new perspective on cooperative and critical phenomena in disordered systems.

We study reinforcement learning in infinite-horizon discounted Markov decision processes with continuous state spaces, where data are generated online from a single trajectory under a Markovian behavior policy. To avoid maintaining an infinite-dimensional, function-valued estimate, we propose the novel Q-Measure-Learning, which learns a signed empirical measure supported on visited state-action pairs and reconstructs an action-value estimate via kernel integration. The method jointly estimates the stationary distribution of the behavior chain and the Q-measure through coupled stochastic approximation, leading to an efficient weight-based implementation with $O(n)$ memory and $O(n)$ computation cost per iteration. Under uniform ergodicity of the behavior chain, we prove almost sure sup-norm convergence of the induced Q-function to the fixed point of a kernel-smoothed Bellman operator. We also bound the approximation error between this limit and the optimal $Q^*$ as a function of the kernel bandwidth. To assess the performance of our proposed algorithm, we conduct RL experiments in a two-item inventory control setting.

Inspired by the formalism that relates the star-exponential with the quantum propagator for bosonic systems, in this work we introduce the analogous extension for the fermionic case. In particular, we analyse the problem of calculating the star-exponential (i.e., the symbol of the evolution operator) for Fermi systems within the deformation quantization program. Grassmann variables and coherent states are considered in order to obtain a closed-form expression for the fermionic star-exponential in terms of its associated propagator. As a primary application, a fermionic version of the Feynman-Kac formula is derived within this formalism, thus allowing a straightforward calculation of the ground state energy in phase space. Finally, the method is validated by successfully applying it to the simple harmonic and driven Fermi oscillators, for which the results developed here provide a powerful alternative computational tool for the study of fermionic systems.

The No Free Lunch (NFL) theorem guarantees equal average performance only under uniform sampling of a function space closed under permutation (c.u.p.). We ask when this averaging ceases to reflect what benchmarking actually reports. We study an iterative-search setting with sampling without replacement, where algorithms differ only in evaluation order. Binary objectives allow exhaustive evaluation in the fully enumerable case, and efficiency is defined by the first time the global minimum is reached. We then construct two additional benchmarks by algebraically recombining the same baseline functions through sums and differences. Function-algorithm relations are examined via correlation structure, hierarchical clustering, delta heatmaps, and PCA. A one-way ANOVA with Tukey contrasts confirms that algebraic reformulations induce statistically meaningful shifts in performance patterns. The uniformly sampled baseline remains consistent with the global NFL symmetry. In contrast, the algebraically modified benchmarks yield stable re-rankings and coherent clusters of functions and sampling policies. Composite objectives can also exhibit non-additive search effort despite being built from simpler components. Monte Carlo experiments indicate that order effects persist in larger spaces and depend on function class. Taken together, the results show how objective reformulation and benchmark design can generate structured local departures from NFL intuition. They motivate algorithm choice that is aware of both the problem class and the objective representation. This message applies to evolutionary computation as well as to statistical procedures based on relabeling, resampling, and permutation tests.

We develop a rigorous framework for extending neural operators to handle out-of-distribution input functions. We leverage kernel approximation techniques and provide theory for characterizing the input-output function spaces in terms of Reproducing Kernel Hilbert Spaces (RKHSs). We provide theorems on the requirements for reliable extensions and their predicted approximation accuracy. We also establish formal relationships between specific kernel choices and their corresponding Sobolev Native Spaces. This connection further allows the extended neural operators to reliably capture not only function values but also their derivatives. Our methods are empirically validated through the solution of elliptic partial differential equations (PDEs) involving operators on manifolds having point-cloud representations and handling geometric contributions. We report results on key factors impacting the accuracy and computational performance of the extension approaches.

Low-dimensional structure in real-world data plays an important role in the success of generative models, which motivates diffusion models defined on intrinsic data manifolds. Such models are driven by stochastic differential equations (SDEs) on manifolds, which raises the need for convergence theory of numerical schemes for manifold-valued SDEs. In Euclidean space, the Euler--Maruyama (EM) scheme achieves strong convergence with order $1/2$, but an analogous result for manifold discretizations is less understood in general settings. In this work, we study a geometric version of the EM scheme for SDEs on Riemannian manifolds and prove strong convergence with order $1/2$ under geometric and regularity conditions. As an application, we obtain a Wasserstein bound for sampling on manifolds via the geometric EM discretization of Riemannian Langevin dynamics.

The HOMFLY-PT and Kauffman polynomials are related to each other for special classes of knots constructed by full twists and Jucys-Murphy twists. The conditions for this relation are articulated in terms of characters of the Birman-Murakami-Wenzl algebra. The latter are the coefficients in the expansion of the Kauffman polynomial involving the quantum dimensions of SO(N + 1). This expansion allows to prove the conjectural 1-1 correspondence between the HOMFLY-PT/Kauffman relation and the Harer-Zagier (HZ) factorisability for a large family of 3-strand knots. However, explicit counterexamples with 4-strands negate one side of the conjecture, i.e. the HOMFLY-PT/Kauffman relation only implies HZ factorisability for knots with braid index four or higher.

Safety filters based on Control Barrier Functions (CBFs) provide formal guarantees of forward invariance, but are often difficult to implement in networked dynamical systems. This is due to global coupling and communication requirements. This paper develops locally implementable approximations of networked CBF safety filters that require no coordination across subsystems. The proposed approach is based on a two-time-scale dynamic implementation inspired by singular perturbation theory, where a small parameter $\epsilon$ separates fast filter dynamics from the plant dynamics; then, a local implementation is enabled via derivative estimation. Explicit bounds are derived to quantify the mismatch between trajectories of the systems with dynamic filter and with the ideal centralized safety filter. These results characterize how safety degradation depends on the time-scale parameter $\epsilon$, estimation errors, and filter activation time, thereby quantifying trade-offs between safety guarantees and local implementability.

Learning-to-communicate (LTC) in partially observable environments has received increasing attention in deep multi-agent reinforcement learning, where the control and communication strategies are jointly learned. Meanwhile, the impact of communication on decision-making has been extensively studied in control theory. In this paper, we seek to formalize and better understand LTC by bridging these two lines of work, through the lens of information structures (ISs). To this end, we formalize LTC in decentralized partially observable Markov decision processes (Dec-POMDPs) under the common-information-based framework from decentralized stochastic control, and classify LTC problems based on the ISs before (additional) information sharing. We first show that non-classical LTCs are computationally intractable in general, and thus focus on quasi-classical (QC) LTCs. We then propose a series of conditions for QC LTCs, under which LTCs preserve the QC IS after information sharing, whereas violating which can cause computational hardness in general. Further, we develop provable planning and learning algorithms for QC LTCs, and establish quasi-polynomial time and sample complexities for several QC LTC examples that satisfy the above conditions. Along the way, we also establish results on the relationship between (strictly) QC IS and the condition of having strategy-independent common-information-based beliefs (SI-CIBs), as well as on solving Dec-POMDPs without computationally intractable oracles but beyond those with SI-CIBs, which may be of independent interest.

Despite the remarkable empirical success of score-based diffusion models, their statistical guarantees remain underdeveloped. Existing analyses often provide pessimistic convergence rates that do not reflect the intrinsic low-dimensional structure common in real data, such as that arising in natural images. In this work, we study the statistical convergence of score-based diffusion models for learning an unknown distribution $\mu$ from finitely many samples. Under mild regularity conditions on the forward diffusion process and the data distribution, we derive finite-sample error bounds on the learned generative distribution, measured in the Wasserstein-$p$ distance. Unlike prior results, our guarantees hold for all $p \ge 1$ and require only a finite-moment assumption on $\mu$, without compact-support, manifold, or smooth-density conditions. Specifically, given $n$ i.i.d.\ samples from $\mu$ with finite $q$-th moment and appropriately chosen network architectures, hyperparameters, and discretization schemes, we show that the expected Wasserstein-$p$ error between the learned distribution $\hat{\mu}$ and $\mu$ scales as $\mathbb{E}\, \mathbb{W}_p(\hat{\mu},\mu) = \widetilde{O}\!\left(n^{-1 / d^\ast_{p,q}(\mu)}\right),$ where $d^\ast_{p,q}(\mu)$ is the $(p,q)$-Wasserstein dimension of $\mu$. Our results demonstrate that diffusion models naturally adapt to the intrinsic geometry of data and mitigate the curse of dimensionality, since the convergence rate depends on $d^\ast_{p,q}(\mu)$ rather than the ambient dimension. Moreover, our theory conceptually bridges the analysis of diffusion models with that of GANs and the sharp minimax rates established in optimal transport. The proposed $(p,q)$-Wasserstein dimension also extends classical Wasserstein dimension notions to distributions with unbounded support, which may be of independent theoretical interest.

Malware attacks in today's vast digital ecosystem pose a serious threat. Understanding malware propagation dynamics and designing effective control strategies are therefore essential. In this work, we propose a generic SEIRV model formulated using ordinary differential equations to study malware spread. We establish the positivity and boundedness of the system, derive the malware propagation threshold, and analyze the local and global stability of the malware-free equilibrium. The separatrix defining epidemic regions in the control space is identified, and the existence of a forward bifurcation is demonstrated. Using normalized forward sensitivity indices, we determine the parameters most influential to the propagation threshold. We further examine the nonlinear dependence of key epidemic characteristics on the transmission rate, including the maximum number of infected, time to peak infection, and total number of infected. We propose a hybrid gradient-based global optimization framework using simulated annealing approach to identify effective and cost-efficient control strategies. Finally, we calibrate the proposed model using infection data from the "Windows Malware Dataset with PE API Calls" and investigated the effect of intervention onset time on averted cases, revealing an exponential decay relationship between delayed intervention and averted cases.

Many large-scale platforms and networked control systems have a centralized decision maker interacting with a massive population of agents under strict observability constraints. Motivated by such applications, we study a cooperative Markov game with a global agent and $n$ homogeneous local agents in a communication-constrained regime, where the global agent only observes a subset of $k$ local agent states per time step. We propose an alternating learning framework $(\texttt{ALTERNATING-MARL})$, where the global agent performs subsampled mean-field $Q$-learning against a fixed local policy, and local agents update by optimizing in an induced MDP. We prove that these approximate best-response dynamics converge to an $\widetilde{O}(1/\sqrt{k})$-approximate Nash Equilibrium, while yielding a separation in the sample complexities between the joint state space and action space. Finally, we validate our results in numerical simulations for multi-robot control and federated optimization.

This paper is motivated by a cutting-edge application in neuroscience: the analysis of electroencephalogram (EEG) signals recorded under flash stimulation. Under commonly used signal-processing assumptions, only the phase angle of the EEG is required for the analysis of such applications. We demonstrate that these assumptions imply that the phase has a projected isotropic normal distribution. We revisit this distribution and derive several new properties, including closed-form expressions for its trigonometric moments. We then examine the distribution of the mean resultant and its square -- a statistic of central importance in phase-based EEG studies. The distribution of the resultant is analytically intricate; to make it practically useful, we develop two approximations based on the well-known resultant distribution for the von Mises distribution. We then study inference problems for this projected isotropic normal distribution. The method is illustrated with an application to EEG data from flash-stimulation experiments.

State-of-the-art ensemble Kalman filtering (EnKF) algorithms require incorporating localization techniques to cope with the rank deficiency and the inherited spurious correlations in their error covariance matrices. Localization techniques are mostly ad-hoc, based on some distances between the state and observation variables, requiring demanding manual tuning. This work introduces a new ensemble filtering approach, which is inherently localized, avoiding the need for any auxiliary localization technique. Instead of explicitly applying localization on ensembles, the idea is to first localize the continuous analysis probability density function (pdf) before ensemble sampling. The localization of the analysis pdf is performed through an approximation by a product of independent marginal pdfs corresponding to small partitions of the state vector, using the variational Bayesian optimization. These marginals are then sampled following stochastic EnKF and deterministic ensemble transform Kalman filtering (ETKF) procedures, using ensembles larger than the partitions' size. The resulting filters involve the same forecast steps as their standard EnKF and ETKF counterparts but different analysis steps, iteratively adjusting the EnKF and ETKF updates of each partition based on the ensemble means of the other partitions. Numerical experiments are conducted with the Lorenz-96 model under different scenarios to demonstrate the potential of the proposed filters. The new filters' performances are comparable to those of the EnKF and ETKF with already tuned localization, both in terms of computational burden and estimation accuracy.

Quantum Singular Value Transformation (QSVT) provides a unified framework for applying polynomial functions to the singular values of a block-encoded matrix. QSVT prepares a state proportional to $\bA^{-1}\bb$ with circuit depth $O(d\cdot\mathrm{polylog}(N))$, where $d$ is the polynomial degree of the $1/x$ approximation and $N$ is the size of $\bA$. Current polynomial approximation methods are over the continuous interval $[a,1]$, giving $d = O(\sqrt{\kap}\log(1/\varepsilon))$, and make no use of any properties of $\bA$. We observe here that QSVT solution accuracy depends only on the polynomial accuracy at the eigenvalues of $\bA$. When all $N$ eigenvalues are known exactly, a pure spectral polynomial $p_{S}$ can interpolate $1/x$ at these eigenvalues and achieve unit fidelity at reduced degree. But its practical applicability is limited. To address this, we propose a spectral correction that exploits prior knowledge of $K$ eigenvalues of $\bA$. Given any base polynomial $p_0$, such as Remez, of degree $d_0$, a $K\times K$ linear system enforces exact interpolation of $1/x$ only at these $K$ eigenvalues without increasing $d_0$. The spectrally corrected polynomial $p_{SC}$ preserves the continuous error profile between eigenvalues and inherits the parity of $p_0$. QSVT experiments on the 1D Poisson equation demonstrate up to a $5\times$ reduction in circuit depth relative to the base polynomial, at unit fidelity and improved compliance error. The correction is agnostic to the choice of base polynomial and robust to eigenvalue perturbations up to $10\%$ relative error. Extension to the 2D Poisson equation suggests that correcting a small fraction of the spectrum may suffice to achieve fidelity above $0.999$.

In the pure Calculus of Constructions (CC) one can define data types and function over these, and there is a powerful higher order logic to reason over these functions and data types. This is due to the combination of impredicativity and dependent types, and most of these features can already be observed in polymorphic (second order) dependent type theory $\lambda$P2. The impredicative encoding of data types (in $\lambda$P2 or CC) is powerful but not fully satisfactory: for example, the induction principle is not provable. As a matter of fact, it can be shown that induction is not provable for whatever possible representation of data types. In a recent paper, Awodey, Frey and Speight show that in an extension of $\lambda$P2 with Sigma-types, identity types with uniqueness of identity proofs and function extensionality, it is possible to define data types for which the induction principle is provable. More recently it has been shown that in this extension of $\lambda$P2, also quotient types can be defined with the proper induction principle, and, using quotient types, coinductive types can be defined with the proper coinduction principle. This leaves various questions open: Are quotient types with induction principle not definable in the original $\lambda$P2? And how about coinductive types, is it impossible to get a strong coinduction principle in $\lambda$P2? Looking at it from the other side: which of the extensions used are really needed to make induction and coinduction work? In this paper, we contribute partial answers to these questions: parametric quotient types are not definable in $\lambda$P2 and the well-known definable stream type does not have a coinduction principle. For the latter question we show that, if we just extend $\lambda$P2 with Sigma-types and identity types with uniqueness of identity proofs, we still cannot prove an induction principle for the natural numbers. So function extensionality is crucial in making induction provable. We show these results by studying models of $\lambda$P2 where the types representing these principles are empty, so these models act as counter models to the derivability of the principles.

Noisy labels in distributed datasets induce severe local overfitting and consequently compromise the global model in federated learning (FL). Most existing solutions rely on selecting clean devices or aligning with public clean datasets, rather than endowing the model itself with robustness. In this paper, we propose FedCova, a dependency-free federated covariance learning framework that eliminates such external reliances by enhancing the model's intrinsic robustness via a new perspective on feature covariances. Specifically, FedCova encodes data into a discriminative but resilient feature space to tolerate label noise. Built on mutual information maximization, we design a novel objective for federated lossy feature encoding that relies solely on class feature covariances with an error tolerance term. Leveraging feature subspaces characterized by covariances, we construct a subspace-augmented federated classifier. FedCova unifies three key processes through the covariance: (1) training the network for feature encoding, (2) constructing a classifier directly from the learned features, and (3) correcting noisy labels based on feature subspaces. We implement FedCova across both symmetric and asymmetric noisy settings under heterogeneous data distribution. Experimental results on CIFAR-10/100 and real-world noisy dataset Clothing1M demonstrate the superior robustness of FedCova compared with the state-of-the-art methods.

In stellarator design, the coil winding surfaces $\Sigma\subset\mathbb R^3$ support current distributions $j$ that shape the magnetic field. This work provides a theoretical framework explaining the emergence of centre and saddle point regions, a key feature in coil optimisation. For coil winding surfaces with a toroidal shape, we prove a dichotomy principle: the current distribution has both centre and saddle point regions or is no-where vanishing. For coil winding surfaces that consist of piecewise cylinders, we show that if $j$ is oppositely oriented on the two boundary circles, centre and saddle points appear, and all but finitely many field lines of $j$ are periodic. When $j$ admits a harmonic potential, all field lines are closed poloidal orbits. These results offer insights into current patterns on winding surfaces, with implications for coil design strategies and their simplification.

We construct nonlinear wave operators and prove small-data asymptotic completeness for the Maxwell--Higgs system on the domain of outer communications of every four-dimensional subextremal Kerr black hole $(\mathcal D_{M,a},g_{M,a})$ with $M>0$ and $|a|<M$, for gauge-invariant nonnegative scalar potentials $P$ satisfying Assumption~\ref{asumsiP} with mass parameter $m^{2}\ge0$. The massless case $m=0$ is unconditional on the full subextremal range. For $m^{2}>0$ the same conclusions follow assuming the massive linear package $\Lin_{k}^{(m)}$ for the linear comparison system (in particular, no exponentially growing modes); this fails for an open set of masses due to superradiant instability \cite{ShlapentokhRothmanKGKerr}. We work in the radiative (charge-free) regime; stationary Coulomb (Kerr--Newman) modes are treated separately. Asymptotic states are described by gauge-covariant radiation fields on $\mathcal I^{\pm}\cup\mathcal H^{\pm}$ (and, when $m>0$, an additional timelike/Dollard channel), yielding a gauge-invariant nonlinear scattering map on the residual-gauge quotient. The scattering map is a small-data bijection, is Fréchet differentiable at $0$ with derivative equal to linear Kerr scattering, admits a quadratic (Born) expansion with an $O(\|U\|^{3})$ remainder in the natural asymptotic topology, and is real-analytic for analytic $P$. The nonlinear argument is presented as a transfer principle from a black-box linear estimate package for inhomogeneous Klein--Gordon and charge-free Maxwell fields, verified here in the massless Kerr case (and proved self-contained in Schwarzschild).

Density aggregation is a central problem in machine learning, for instance when combining predictions from a Deep Ensemble. The choice of aggregation remains an open question with two commonly proposed approaches being linear pooling (probability averaging) and geometric pooling (logit averaging). In this work, we address this question by studying the normalized generalized mean of order $r \in \mathbb{R} \cup \{-\infty,+\infty\}$ through the lens of log-likelihood, the standard evaluation criterion in machine learning. This provides a unifying aggregation formalism and shows different optimal configurations for different situations. We show that the regime $r \in [0,1]$ is the only range ensuring systematic improvements relative to individual distributions, thereby providing a principled justification for the reliability and widespread practical use of linear ($r=1$) and geometric ($r=0$) pooling. In contrast, we show that aggregation rules with $r \notin [0,1]$ may fail to provide consistent gains with explicit counterexamples. Finally, we corroborate our theoretical findings with empirical evaluations using Deep Ensembles on image and text classification benchmarks.

A symbolic determinant under rank-one restriction computes a polynomial of the form $\det(A_0+A_1y_1+\ldots+A_ny_n)$, where $A_0,A_1,\ldots,A_n$ are square matrices over a field $\mathbb{F}$ and $rank(A_i)=1$ for each $i\in[n]$. This class of polynomials has been studied extensively, since the work of Edmonds (1967), in the context of linear matroids, matching, matrix completion and polynomial identity testing. We study the following learning problem for this class: Given black-box access to an $n$-variate polynomial $f=\det(A_0+A_1y_1+ \ldots+A_ny_n)$, where $A_0,A_1,\ldots,A_n$ are unknown square matrices over $\mathbb{F}$ and rank$(A_i)=1$ for each $i\in[n]$, find a square matrix $B_0$ and rank-one square matrices $B_1,\ldots,B_n$ over $\mathbb{F}$ such that $f=\det(B_0+B_1y_1+\ldots+B_ny_n)$. In this work, we give a randomized poly(n) time algorithm to solve this problem. As the above-mentioned class is known to be equivalent to the class of read-once determinants (RODs), we will refer to the problem as learning RODs. The algorithm for learning RODs is obtained by connecting with a well-known open problem in linear algebra, namely the Principal Minor Assignment Problem (PMAP), which asks to find (if possible) a matrix having prescribed principal minors. PMAP has also been studied in machine learning to learn the kernel matrix of a determinantal point process. Here, we study a natural black-box version of PMAP: Given black-box access to an $n$-variate polynomial $f = \det(A + Y)$, where $A \in \mathbb{F}^{n \times n}$ is unknown and $Y = diag(y_1,\ldots,y_n)$, find a $B\in\mathbb{F}^{n\times n}$ such that $f=det(B+Y)$. We show that black-box PMAP can be solved in randomized poly(n) time, and further, it is randomized polynomial-time equivalent to learning RODs. We resolve black-box PMAP by investigating a property of dense matrices that we call the rank-one extension property.

The interfacial diffusion associated with finite volume method (FVM) discretizations of multiphase flows creates the need for an interface sharpening mechanism. Such solutions for structured quadrilateral grids are well documented, but various engineering applications require mesh designs specific to the irregular geometry of the physical system it is modeling. Therefore this study casts interface sharpening as an ant-idiffusive volumetric body force whose calculation procedure is generalizable to an arbitrarily constructed grid. The force magnitude is derived at cell centers as a function of the local compressible flow characteristics and the geometry of the cell neighborhood. The flow model uses an AUSM+up based method for flux evaluation and imposes a stiffened equation of state onto each of the fluids in order to close the linear system and extract auxiliary variables. Validation tests show good agreement with the Young-Laplace condition whereby the interface converges to the analytical solution corresponding to a balance between a pressure jump and interfacial forces. Further results show the recovery of a circle starting from a shape with highly variational curvature through the combined effects of surface tension and interface sharpening. Lastly shear-driven droplet pinchoff results show good agreement with droplet shapes provided by the surrounding literature at various Weber-Ohnesorge number combinations.

We present a new method for computation of the index of completely mixed equilibria in finite games, based on the work of Eisenbud et al.(1977). We apply this method to solving two questions about the relation of the index of equilibria and the index of fixed points, and the index of equilibria and payoff-robustness: any integer can be the index of an isolated completely mixed equilibrium of a finite game. In a particular class of isolated completely mixed equilibria, called monogenic, the index can be $0$, $+1$ or $-1$ only. In this class non-zero index is equivalent to payoff-robustness. We also discuss extensions of the method of computation to extensive-form games, and cases where the equilibria might be located on the boundary of the strategy set.

To characterize the community structure in network data, researchers have developed various block-type models, including the stochastic block model, the degree-corrected stochastic block model, the mixed membership block model, the degree-corrected mixed membership block model, and others. A critical step in applying these models effectively is determining the number of communities in the network. However, to the best of our knowledge, existing methods for estimating the number of network communities either rely on explicit model fitting or fail to simultaneously accommodate network sparsity and a diverging number of communities. In this paper, we propose a model-free spectral inference method based on eigengap ratios that addresses these challenges. The inference procedure is straightforward to compute, requires no parameter tuning, and can be applied to a wide range of block models without the need to estimate network distribution parameters. Furthermore, it is effective for both dense and sparse networks with a divergent number of communities. Technically, we show that the proposed spectral test statistic converges to a {function of the type-I Tracy-Widom distribution via the Airy kernel} under the null hypothesis, and that the test is asymptotically powerful under weak alternatives. Simulation studies on both dense and sparse networks demonstrate the efficacy of the proposed method. Three real-world examples are presented to illustrate the usefulness of the proposed test.

Federated learning (FL) faces two structural tensions: gradient sharing enables data-reconstruction attacks, while non-IID client distributions degrade aggregation quality. We introduce PTOPOFL, a framework that addresses both challenges simultaneously by replacing gradient communication with topological descriptors derived from persistent homology (PH). Clients transmit only 48-dimensional PH feature vectors-compact shape summaries whose many-to-one structure makes inversion provably ill-posed-rather than model gradients. The server performs topology-guided personalised aggregation: clients are clustered by Wasserstein similarity between their PH diagrams, intra-cluster models are topology-weighted,and clusters are blended with a global consensus. We prove an information-contraction theorem showing that PH descriptors leak strictly less mutual information per sample than gradients under strongly convex loss functions, and we establish linear convergence of the Wasserstein-weighted aggregation scheme with an error floor strictly smaller than FedAvg. Evaluated against FedAvg, FedProx, SCAFFOLD, and pFedMe on a non-IID healthcare scenario (8 hospitals, 2 adversarial) and a pathological benchmark (10 clients), PTOPOFL achieves AUC 0.841 and 0.910 respectively-the highest in both settings-while reducing reconstruction risk by a factor of 4.5 relative to gradient sharing. Code is publicly available at this https URL and data at this https URL.

This paper focuses on error thresholds for Pauli channels. We numerically compute lower bounds for the thresholds using the analytic framework of coset weight enumerators pioneered by DiVincenzo, Shor and Smolin in 1998. In particular, we study potential non-additivity of a variety of small stabilizer codes and their concatenations, and report several new concatenated stabilizer codes of small length that show significant non-additivity. We also give a closed form expression of coset weight enumerators of concatenated phase and bit flip repetition codes. Using insights from this formalism, we estimate the threshold for concatenated repetition codes of large lengths. Finally, for several concatenations of small stabilizer codes we optimize for channels which lead to maximal non-additivity at the hashing point of the corresponding channel. We supplement these results with a discussion on the performance of various stabilizer codes from the perspective of the non-additivity and threshold problem. We report both positive and negative results, and highlight some counterintuitive observations, to support subsequent work on lower bounds for error thresholds.

In this paper we present a method for constructing multiple-sum $q$-series for what is known as Mixed Mock Modular forms. We also present some multi-sum analogues of the Durfee identity, and discuss a construction of its combinatorial interpretation in terms of partitions.

A hyperbolic transcendental entire function with connected Fatou set is said to be "of disjoint type". It is known that a disjoint-type function provides a model for the dynamics near infinity of all maps in the same parameter space; hence a good understanding of these functions has implications in wider generality. Our goal is to study the topological properties of the Julia sets of entire functions of disjoint type. In particular, we give a detailed description of the topology of their connected components. More precisely, consider a "Julia continuum" C of such a function, i.e. the closure in the Riemann sphere of a component of the Julia set. We show that infinity is a terminal point of C, and that C has span zero in the sense of Lelek; under a mild geometric assumption on the function C is arc-like. (Whether every span zero continuum is also arc-like was a famous question in continuum theory, only recently resolved in the negative.) Conversely, we construct a single disjoint-type entire function with the remarkable property that each arc-like continuum with at least one terminal point is realised as a Julia continuum. The class of arc-like continua with terminal points is uncountable. It includes, in particular, the sin(1/x)-curve, the Knaster buckethandle and the pseudo-arc, so these can all occur as Julia continua of a disjoint-type entire function. We give similar descriptions of the possible topology of Julia continua that contain periodic points or points with bounded orbits, and answer a question of Barański and Karpińska by showing that Julia continua need not contain points that are accessible from the Fatou set. Furthermore, we construct an entire function whose Julia set has connected components on which the iterates tend to infinity pointwise, but not uniformly. This is related to a famous conjecture of Eremenko concerning escaping sets of entire functions.

For planar self-affine sets satisfying the strong separation condition, recent work of Bárány, Hochman, and Rapaport gives mild assumptions under which the Hausdorff dimension equals the affinity dimension. In this paper, we study dominated systems in that regime and ask which finer geometric properties can be characterized. In the range $\dim_{\mathrm{H}}(X) < 1$, we characterize Ahlfors regularity by equivalent conditions involving positivity of $\mathcal{H}^s(X)$, control of projection fibers, and the identity $\dim_{\mathrm{L}}(X)=\dim_{\mathrm{H}}(X)=\dim_{\mathrm{A}}(X)$. In the range $\dim_{\mathrm{H}}(X) \ge 1$, we identify the maximal slice dimension as $\dim_{\mathrm{A}}(X)-1$ in Furstenberg directions and provide examples showing that Marstrand-type all-slice bounds cannot hold in general. We also derive projection consequences for Assouad dimension and exhibit dominated irreducible examples with $\dim_{\mathrm{aff}}(X)<\dim_{\mathrm{A}}(X)$.

We construct a series of finite-dimensional quantum groups as braided Drinfeld doubles of Nichols algebras of type Super A, for an even root of unity, and classify ribbon structures for these quantum groups. Ribbon structures exist if and only if the rank is even and all simple roots are odd. In this case, the quantum groups have a unique ribbon structure which comes from a non-semisimple spherical structure on the negative Borel Hopf subalgebra. Hence, the categories of finite-dimensional modules over these quantum groups provide examples of non-semisimple modular categories. In the rank-two case, we explicitly describe all simple modules of these quantum groups. We finish by computing link invariants, based on generalized traces, associated to a four-dimensional simple module of the rank-two quantum group. These knot invariants distinguish certain knots indistinguishable by the Jones or HOMFLYPT polynomials and are related to a specialization of the Links-Gould invariant.

We prove that the tail probabilities of sums of independent uniform random variables, up to a multiplicative constant, are dominated by the Gaussian tail with matching variance and find the sharp constant for such stochastic domination.

Newton's method may exhibit slower convergence than vanilla Gradient Descent in its initial phase on strongly convex problems. Classical Newton-type multilevel methods mitigate this but, like Gradient Descent, achieve only linear convergence near the minimizer. We introduce an adaptive multilevel Newton-type method with a principled automatic switch to full Newton once its quadratic phase is reached. The local quadratic convergence for strongly convex functions with Lipschitz continuous Hessians and for self-concordant functions is established and confirmed empirically. Although per-iteration cost can exceed that of classical multilevel schemes, the method is efficient and consistently outperforms Newton's method, Gradient Descent, and the multilevel Newton method, indicating that second-order methods can outperform first-order methods even when Newton's method is initially slow. The promising empirical results open new avenues for designing reduced-cost second- and high-order methods with extremely fast convergence rates.

We investigate minimality and stability of periodic brake orbits in natural Lagrangian systems on smooth Riemannian manifolds. We prove that every non-constant periodic brake orbit is not a minimizer of the fixed-time action, for any conormal boundary condition. Under an orbit-cylinder hypothesis, its Morse index strictly increases in the free-time setting. As a consequence, strongly nondegenerate brake orbits fail to be linearly stable under a dimensional condition; in dimension at least three, nondegenerate mountain-pass brake orbits are spectrally unstable when the monodromy is semisimple. The key ingredient is a local index contribution at each brake instant. Using Seifert collar coordinates near the Hill boundary, we reduce the normal dynamics to a one-dimensional model, exhibiting a degeneracy inherent to brake symmetry. We illustrate the results by explicit Morse index computations for the planar anisotropic oscillator, the planar pendulum, and the planar Kepler problem; in the Kepler case, the ejection--collision orbit is treated via cotangent-lift Levi--Civita--Lissajous regularization.

We provide criteria for Itô integration to behave continuously with respect to Skorokhod's J1 and M1 topologies, when the integrands and integrators converge weakly or in probability. The results are novel in the M1 setting and unify existing theories in the J1 case. Beyond sufficient criteria, we present an example of uniformly convergent martingale integrators for which the continuity breaks down. Moreover, we show that, for families of local martingales, M1 tightness in fact implies J1 tightness under a mild localised uniform integrability condition. Finally, we apply our results to study scaling limits of models of anomalous diffusion driven by continuous-time random walks. This yields new results on weak M1 and J1 convergence to stochastic integrals against subordinated stable processes. In the case of superdiffusive scaling, an interesting counterexample is obtained.

We show that every locally flat topological embedding of a 3-manifold in a smooth 5-manifold is homotopic, by a small homotopy, to a smooth embedding. We deduce that topologically locally flat concordance implies smooth concordance for smooth surfaces in smooth 4-manifolds.

We study relative differential and integral forms on families of supermanifolds and their cohomology. We prove a relative Poincaré--Verdier duality and show that it relates the cohomology of differential and integral forms, admitting a concrete geometric realization via Berezin fiber integration. We further introduce the Poincaré--dual integral form associated to an embedded even family and prove that it satisfies the correct localization property. We then apply these results to supergravity, focusing on the $3d$ case. In this setting, we show that relative Poincaré duality provides the natural framework for encoding the data needed to relate a superspace formulation to the physical spacetime, thereby yielding a rigorous definition of picture changing operators used in the physics literature. Building on this, after a careful analysis of the space of fields and the relevant constraints, we prove that the component, superspace, and geometric formulation of the theory are all equivalent. Finally, under suitable hypotheses, we argue that our construction illustrates a general principle governing the mathematical formulation of classical field theories on supermanifolds.

Let $A$ be an abelian variety with totally degenerate reduction over a non-Archimedean field. We describe the moduli space of semihomogeneous vector bundles on $A$ from the perspective of non-Archimedean uniformization and show that the essential skeleton may be identified with a tropical analogue of this moduli space. For $H=0$ our moduli space may be identified with the moduli space $M_{0,r}(A)$ of semistable vector bundles with vanishing Chern classes on $A$. In this case we construct a surjective analytic morphism from the character variety of the analytic fundamental group of $A$ onto $M_{0,r}(A)$, which naturally tropicalizes. One may view this construction as a non-Archimedean uniformization of $M_{0,r}(A)$.

In this paper we construct $n$-valued maps on $k$-dimensional tori, where $n,k\geq 2$, that are not homotopic to affine $n$-valued maps. This is in high contrast with the single valued case, where any such map is homotopic to an affine (even linear) map. We do this by investigating necessary and sufficient algebraic conditions on certain induced morphisms.

Fix an integer $p\geq 1$ and refer to it as the number of growing domains. For each $i\in\{1,\ldots,p\}$, fix a compact subset $D_i\subseteq\mathbb R^{d_i}$ where $d_1,\ldots,d_p\ge 1$. Let $d= d_1+\dots+d_{p}$ be the total underlying dimension. Consider a continuous, stationary, centered Gaussian field $B=(B_x)_{x\in \mathbb R^d}$ with unit variance. Finally, let $\varphi:\mathbb R \rightarrow \mathbb R$ be a measurable function such that $\mathrm E[\varphi(N)^2]<\infty$ for $N\sim N(0,1)$. In this paper, we investigate central and non-central limit theorems as $t_1,\ldots,t_p\to\infty$ for functionals of the form \[ Y(t_1,\dots,t_p):=\int_{t_1D_1\times\dots \times t_pD_p}\varphi(B_{x})dx. \] Firstly, we assume that the covariance function $C$ of $B$ is {\it separable} (that is, $C=C_1\otimes\ldots\otimes C_{p}$ with $C_i:\mathbb R^{d_i}\to\mathbb R$), and thoroughly investigate under what condition $Y(t_1,\dots,t_p)$ satisfies a central or non-central limit theorem when the same holds for $\int_{t_iD_i}\varphi(B^{(i)}_{x_i})dx_i$ for at least one (resp. for all) $i\in \{1,\ldots,p\}$, where $B^{(i)}$ stands for a stationary, centered, Gaussian field on $\mathbb R^{d_i}$ admitting $C_i$ for covariance function. When $\varphi$ is an Hermite polynomial, we also provide a quantitative version of the previous result, which improves some bounds from A. Reveillac, M. Stauch, and C. A. Tudor, Hermite variations of the fractional brownian sheet, Stochastics and Dynamics 12 (2012). Secondly, we extend our study beyond the separable case, examining what can be inferred when the covariance function is either in the Gneiting class or is additively separable.

We consider the inverse mean curvature flow by parallel hypersurfaces in space forms. We show that such a flow exists if and only if the initial hypersurface is isoparametric. The flow is characterized by an algebraic equation satisfied by the distance function of the parallel hypersurfaces. The solutions to the flow are obtained explicitly when the distinct principal curvatures have the same multiplicity. This is an additional assumption only for isoparametric hypersurfaces of the hyperbolic space or of the sphere with two or four distinct principal curvatures. The boundaries of the maximal interval of definition, when finite, are determined in terms of the number $g$ of distinct principal curvatures, their multiplicities $m$ and the mean curvature $H$ of the initial hypersurface. We describe the collapsing submanifolds of the flow at the boundaries of the interval. In particular, we show in the Euclidean space the solutions are eternal, while in the hyperbolic space there are eternal and immortal solutions. Starting with a connected isoparametric submanifold of the sphere, we show that the flow is an ancient solution, that collapses into a minimal hypersurface whose square length of its second fundamental form and its scalar curvature are constants given in terms of $g$ and $n$. The minimal hypersurface is totally geodesic when $g=1$, it is a Clifford minimal hypersurface of the sphere when $g=2$ and it is a Cartan type minimal submanifold when $g\in\{3,4,6\}$.

We prove an ergodic theorem for Markov chains indexed by the Ulam-Harris-Neveu tree over large subsets with arbitrary shape under two assumptions: with high probability, two vertices in the large subset are far from each other and have their common ancestor close to the root. The assumption on the common ancestor can be replaced by some regularity assumption on the Markov transition kernel. We verify that those assumptions are satisfied for some usual trees. Finally, with Markov-Chain Monte-Carlo considerations in mind, we prove when the underlying Markov chain is stationary and reversible that the Markov chain, that is the line graph, yields minimal variance for the empirical average estimator among trees with a given number of nodes.

In this work we develop an algorithmic procedure for associating a function defined on the Riemann surface of the $\log$ to given asymptotic data from a function at an essential singularity. We do this by means of rational approximations (Padé approximants) used in tandem with Borel-Écalle summation. Our method is capable of handling situations where classical methods either do not work or converge very slowly eg. We provide a general outline of the procedure and then apply it to generating approximate tritronquée solutions to Painlevé's first equation ($\text{P}_\text{I}$). Our approximations (including $\text{P}_\text{I}$) are written as a finite linear combination of exponential integrals $\text{Ei}^+$. Furthermore, we have explicit rational approximations for each $\text{Ei}^+$ and thus for the approximation as a whole. In addition to rational approximations of $\text{P}_\text{I}$, we provide the first hundred or so poles of a tritronquée solution with essentially arbitrary accuracy which is dependent upon the order of Padé used.

The following refinement of the Higman embedding theorem is proved: A finitely generated group $R$ is recursively presented if and only if there exists a quasi-isometric malnormal embedding of $R$ into a finitely presented group $H$ such that the image of the embedding enjoys the congruence extension property. Moreover, it is shown that the finitely presented group $H$ can be constructed to have decidable Word Problem if and only if the Word Problem for $R$ is decidable, yielding a refinement of a theorem of Clapham. Finally, given a countable group $G$ and a computable function $\ell:G\to\mathbb{N}$ satisfying some necessary requirements, it is proved that there exists a malnormal embedding of $G$ into a finitely presented group $H$ such that the restriction of $|\cdot|_H$ to $G$ is equivalent to $\ell$, producing a refinement of a theorem of Ol'shanskii.

Polynomials commute under composition are referred to as commuting polynomials. In this paper, we study division properties for commuting polynomials with rational (and integer) coefficients. As a consequence, we show an algebraic particularity of the commuting polynomials coming from weighted sums for cycle graphs with pendant edges (arXiv:2402.07209v1.). We also discuss a set of commuting polynomials over a field of positive characteristic.

In [8,9], the authors developed a nice formula to compute the Nielsen number of a self-map on an infra-nilmanifold. For the case of nilmanifolds this formula was extended to $n$-valued maps in [4]. In this paper, we extend these results further and establish the averaging formula to compute the Nielsen number of any $n$-valued affine map on an infra-nilmanifold.

In their seminal paper introducing the theory of random graphs, Erdős and Rényi considered the evolution of the structure of a random subgraph of $K_n$ as the density increases from $0$ to $1$, identifying two key points in this evolution -- the \emph{percolation threshold}, where the order of the largest component seemingly jumps from logarithmic to linear in size, and the \emph{connectivity threshold}, where the subgraph becomes connected. Similar phenomena have been observed in many other random graph models, and in particular, works of Ajtai, Komlós and Szemerédi and of Spencer and Erdős determine corresponding thresholds for random subgraphs of the hypercube. We study similar questions on the \emph{permutahedron}. The permutahedron, like the hypercube, has many different equivalent representations, and arises as a natural object of study in many areas of combinatorics. In particular, as a highly-symmetric simple polytope, like the $n$-simplex and $n$-cube, this percolation model naturally generalises the Erdős-Rényi random graph and the percolated hypercube. We determine the percolation threshold and the connectivity threshold for random subgraphs of the permutahedron. Along the way we develop a novel graph exploration technique which can be used to find exponentially large clusters after percolation in high-dimensional geometric graphs and we initiate the study of the isoperimetric properties of the permutahedron.

Our goal is to study $p$-adic local systems on a rigid-analytic variety with semistable formal model. We prove that such a local system is semistable if and only if so are its restrictions to the points corresponding to the irreducible components of the special fiber. For this, the main body of the paper concerns analytic prismatic $F$-crystals on the absolute logarithmic prismatic site of a semistable $p$-adic log formal scheme. Analyzing Breuil-Kisin log prisms, we obtain a prismatic purity theorem and deduce the above purity theorem for semistable local systems.

Spherical spin random fields are used to model the Cosmic Microwave Background polarization, the study of which is at the heart of modern Cosmology and will be the subject of the LITEBIRD mission, in the 2030s. Its scope is to collect datas to test the theoretical predictions of the Cosmic Inflation model. In particular, the Minkowski functionals, or the Lipschitz-Killing curvatures, of excursion sets can be used to detect deviations from Gaussianity and anisotropies of random fields, being fine descriptors of their geometry and topology. In this paper we give an explicit, non-asymptotic, formula for the expectation of the Lipshitz-Killing curvatures of the excursion set of the real part of an arbitrary left-invariant Gaussian spin spherical random field, seen as a field on $SO(3)$. Our findings are coherent with the asymptotic ones presented in Carrón Duque, J. et al. "Minkowski Functionals in $SO(3)$ for the spin-2 CMB polarisation field", Journal of Cosmology and Astroparticle Physics (2024). We also give explicit expressions for the Adler-Taylor metric, and its curvature. We obtain such result as an application of a general formula that applies to any nondegenerate Gaussian random field defined on an arbitrary three dimensional compact Riemannian manifold. The novelty is that the Lipschitz-Killing curvatures are computed with respect to an arbitrary metric, possibly different than the Adler-Taylor metric of the field.

Space-time finite-element discretizations are well-developed in many areas of science and engineering, but much work remains within the development of specialized solvers for the resulting linear and nonlinear systems. In this work, we consider the all-at-once solution of the discretized Navier-Stokes equations over a space-time domain using waveform relaxation multigrid methods. In particular, we show how to extend the efficient spatial multigrid relaxation methods from [37] to a waveform relaxation method, and demonstrate the efficiency of the resulting monolithic Newton-Krylov-multigrid solver. Numerical results demonstrate the scalability of the solver for varying discretization order and physical parameters.

We classify deformation quantizations of the symplectic supervarieties that are smooth and admissible. This generalizes the corresponding result of Bezrukavnikov and Kaledin to the super case. We relate the equivalence classes of quantizations of supervarieties with that of their even reduced symplectic varieties. Finally, we prove that certain nilpotent orbits of basic Lie superalgebras are admissible and split, and classify their deformation quantizations.

Score-driven (SD) models are a standard tool in statistics and econometrics, with applications in hundreds of published articles in the past decade. We provide an information-theoretic characterization of SD updates based on reductions in the expected Kullback-Leibler (EKL) divergence relative to the true -- but unknown -- data-generating density. EKL reductions occur if and only if the expected update direction aligns with the expected score; i.e., their inner product should be positive. This equivalence condition uniquely identifies SD updates (including scaled or clipped variants) as being EKL reducing, even in non-concave, multivariate, and misspecified settings. We further derive explicit bounds on admissible learning rates in terms of score moments, linking SD methods to adaptive optimization techniques. By contrast, alternative performance measures in the literature impose stronger conditions (e.g., concave logarithmic densities) and do not characterize SD updates: other updating rules may improve these measures, while SD updates need not. Our results provide a rigorous justification for SD models and establish EKL as their natural information-theoretic foundation.

BGK (Bhatnagar-Gross-Krook) model is a relaxation-type model of the Boltzmann equation, which is popularly used in place of the Boltzmann equation in physics and engineering. In this paper, we address the ill-posedness problem for the BGK model, in which the solution instantly escapes the initial solution space. For this, we propose two ill-posedness scenarios, namely, the homogeneous and the inhomogeneous ill-posedness mechanisms. In the former case, we find a class of spatially homogeneous solutions to the BGK model, where removing the small velocity part of the initial data triggers ill-posedness by increasing temperature. For the latter, we construct a spatially inhomogeneous solution to the BGK model such that the local temperature constructed from the solution has a polynomial growth in spatial variable. These ill-posedness properties for the BGK model pose a stark contrast with the Boltzmann equation for which the solution map is, at least for a finite time, stable in the corresponding solution spaces.

The classical trisecant lemma says that a general chord of a non-degenerate space curve is not a trisecant; that is, the chord only meets the curve in two points. The generalized trisecant lemma extends the result to higher-dimensional varieties. It states that the linear space spanned by general points on a projective variety intersects the variety in exactly these points, provided the dimension of the linear space is smaller than the codimension of the variety and that the variety is irreducible, reduced, and non-degenerate. We prove a real analogue of the generalized trisecant lemma, which takes the form of a trichotomy. Along the way, we characterize the possible numbers of real intersection points between a real projective variety and a complimentary dimension real linear space. We show that any integer of correct parity between a minimum and a maximum number can be achieved. We then specialize to Segre-Veronese varieties, where our results apply to the identifiability of independent component analysis, tensor decomposition and to typical tensor ranks.

For an $n$-valued self-map $f$ of a closed manifold $X$, we prove an averaging formula for the Reidemeister trace of $f$ in terms of the Reidemeister coincidence traces of single-valued maps between finite orientable covering spaces of $X$. We then derive analogous formulas for the Lefschetz and Nielsen numbers of $f$. In the special case where $X$ is an infra-nilmanifold, we obtain explicit formulas for the Lefschetz and Nielsen numbers of any $n$-valued map on $X$.

We study the p-adic analogue of the arithmetic Gan-Gross-Prasad (GGP) conjectures for unitary groups. Let $\Pi$ be a conjugate-selfdual cuspidal automorphic representation of GL_{n} x GL_{n+1} over a CM field, which is algebraic of minimal regular weight at infinity. We first show the rationality of twists of the ratio of L-values of $\Pi$ appearing in the GGP conjectures. Then, when $\Pi$ is p-ordinary at a prime p, we construct a cyclotomic p-adic L-function $L_p(M_\Pi)$ interpolating those twists. Finally, under some local assumptions, we prove a precise formula relating the first derivative of $L_p(M_\Pi)$ to the p-adic heights of Selmer classes arising from arithmetic diagonal cycles on unitary Shimura varieties. We deduce applications to the p-adic Beilinson-Bloch-Kato conjecture for the motive attached to $\Pi$. All proofs are based on some relative-trace formulas in p-adic coefficients.

This paper presents a discrete homotopy theory and a discrete homology theory for finite posets. In particular, the discrete and classical homotopy groups of finite posets are always isomorphic. Moreover, this discrete homology theory is related to the discrete homotopy theory through a discrete analogue of the Hurewicz map.

This paper presents a study of the generalized Davis-Wielandt radius of Hilbert space operators. New lower bounds for the generalized Davis-Wielandt radius and numerical radius are provided. An alternative of the triangular inequality for operators is also derived.

In this paper we provide a detailed study on effective versions of the celebrated Bilu's equidistribution theorem for Galois orbits of sequences of points of small height in the $N$-dimensional algebraic torus, identifying the quantitative dependence of the convergence in terms of the regularity of the test functions considered. We develop a general Fourier analysis framework that extends previous results obtained by Petsche (2005), and by D'Andrea, Narváez-Clauss and Sombra (2017).

We produce canonical sets of right coset representatives for the congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$ and $\Gamma(N)$, and prove that the corresponding fundamental domains are connected. Key to our construction is a study of the projective line $P^1({\mathbb Z}/N{\mathbb Z})$ using a function $M: {\mathbb Z}/N{\mathbb Z}\to {\mathbb Z}_{\geq 0}$, representing multiplicities. We further study this function and show that it is simply one less than another much more computable function $W:{\mathbb Z}/N{\mathbb Z}\to {\mathbb N}$, of possible independent interest. We present some examples and pictures at the end.

We study the property $P_{\text {naive }}$ of mapping class groups of surfaces of infinite type, that is, for any finite collection of non-trivial elements $h_{1},h_{2}, \cdots, h_{n}$, there exists another element $g\neq 1$ of infinite order such that for all $i$, $\langle g, h_{i}\rangle \cong \langle g \rangle * \langle h_{i} \rangle$.

We analyze Szöllősi's recent construction of a conjecturally optimal five-dimensional kissing configuration and produce a new such configuration, the fourth to be discovered. We construct five-dimensional sphere packings from these configurations, which augment Conway and Sloane's list of conjecturally optimal packings. We also construct a new kissing configuration in nine dimensions. None of these constructions improves on the known records, but they provide geometrically distinct constructions achieving these records.

Semi-topological Galois theory associates a canonical finite splitting covering to a monic Weierstrass polynomial. The inverse limit of the corresponding deck groups defines the absolute semi-topological Galois group, $\PiST(X,x)$. This paper develops a cohomology theory for $\PiST(X,x)$ with discrete torsion coefficients, establishing its fundamental properties and canonical comparison maps to singular cohomology. A Lyndon-Hochschild-Serre spectral sequence is used to yield an obstruction theory for semi-topological embedding problems. We prove several structural and vanishing results, including ST-fullness for free fundamental groups and triviality for finite fundamental groups. As applications, we provide a criterion for lifting finite projective monodromy to linear monodromy, formulate the $\pi_1$-detectable Weierstrass realizability conjecture for divisor classes and show that this conjecture is true for abelian varieties, smooth complex projective curves and ruled surfaces over positive-genus curves.

We extend the classical coding of measured $\mathbb R$-trees by continuous excursion-type functions to càdlàg excursion-type functions through the notion of parametric representations. The main feature of this extension is its continuity properties with respect to the Gromov-Hausdorff-Prokhorov topology for $\mathbb R$-trees and Skorokhod's $M_1$ topology for càdlàg functions. As a first application, we study the $\mathbb R$-trees $\mathcal T_{x^{(\alpha)}}$ encoded by excursions of spectrally positive $\alpha$-stable Lévy processes for $\alpha \in (1,2]$. In a second time, we use this setting to study the large-scale effects of a well-known bijection between plane trees and binary trees, the so-called rotation. Marckert has proved that the rotation acts as a dilation on large uniform trees, and we show that this remains true when the rotation is applied to large critical Bienaymé trees with offspring distribution attracted to a Gaussian distribution. However, this does not hold anymore when the offspring distribution falls in the domain of attraction of an $\alpha$-stable law with $\alpha \in (1,2)$, and instead we prove that the scaling limit of the rotated trees is $\mathcal T_{x^{(\alpha)}}$.

We study the rationality of some geometrically rational three-dimensional conic and quadric surface bundles, defined over the reals and more general real closed fields, for which the real locus is connected and the intermediate Jacobian obstructions to rationality vanish. We obtain both negative and positive results, using unramified cohomology and birational rigidity techniques, as well as concrete rationality constructions.

We introduce a notion of stratification for rigidly-compactly generated tensor-triangulated categories relative to the homological spectrum and develop the fundamental features of this theory. In particular, we demonstrate that it exhibits excellent descent properties. In conjunction with Balmer's Nerves of Steel conjecture, we conclude that stratification admits a general form of descent. This gives a uniform treatment of several recent stratification results and provides a complete answer to the question: When does stratification descend? As a new application, we extend earlier work on the tensor triangular geometry of equivariant module spectra from finite groups to compact Lie groups.

We generalize Hopf's theorem to thermostats: the total thermostat curvature of a thermostat without conjugate points is non-positive and vanishes only if the thermostat curvature is identically zero. We further show that, if the thermostat curvature is zero, then the flow has no conjugate points and the Green bundles collapse almost everywhere. Given a thermostat without conjugate points, we prove that the Green bundles are transverse everywhere if and only if it is projectively Anosov. Finally, we provide an example showing that Hopf's rigidity theorem on the 2-torus cannot be extended to thermostats. It is also the first example of a projectively Anosov thermostat which is not Anosov.

In this paper we propose a framework to analyze iterative first-order optimization algorithms for time-varying convex optimization. We assume that the temporal variability is caused by a time-varying parameter entering the objective, which can be measured at the time of decision but whose future values are unknown. We consider the case of strongly convex objective functions with Lipschitz continuous gradients under a convex constraint set. We model the algorithms as discrete-time linear parameter varying (LPV) systems in feedback with monotone operators such as the time-varying gradient. We leverage the approach of analyzing algorithms as uncertain control interconnections with integral quadratic constraints (IQCs) and generalize that framework to the time-varying case. We propose novel IQCs that are capable of capturing the behavior of time-varying nonlinearities and leverage techniques from the LPV literature to establish novel bounds on the tracking error. Quantitative bounds can be computed by solving a semi-definite program and can be interpreted as an input-to-state stability result with respect to a disturbance signal which increases with the temporal variability of the problem. As a departure from results in this research area, our bounds introduce a dependence on different additional measures of temporal variations, such as the function value and gradient rate of change. We exemplify our main results with numerical experiments that showcase how our analysis framework is able to capture convergence rates of different first-order algorithms for time-varying optimization through the choice of IQC and rate bounds.

Sparse optimization seeks an optimal solution with few nonzero entries. To achieve this, it is common to add to the criterion a penalty term proportional to the $\ell_1$-norm, which is recognized as the archetype of sparsity-inducing norms. In this approach, the number of nonzero entries is not controlled a priori. By contrast, in this paper, our motivation is to find an optimal solution with at most~$k$ nonzero coordinates (or for short, $k$-sparse vectors), where $k$ is a given sparsity threshold (or ``sparsity budget''). For this purpose, we study the class of generalized $k$-support dual~norms that arise from any given so-called source norm. When added as a penalty term, we provide conditions under which such generalized $k$-support dual~norms promote $k$-sparse solutions. The result follows from an analysis of the exposed faces of closed convex sets generated by $k$-sparse vectors, and of how primal support identification can be deduced from dual information. Finally, we study some of the geometric properties of the unit balls for the $k$-support dual~norms and their dual norms when the source norm belongs to the family of $\ell_p$-norms. In particular, we show a striking structural property: every proper face of the unit balls for the $k$-support dual~norms is a hypersimplex, i.e., the convex hull of $0/1$-valued points with the same $\ell_0$-norm.

This is the first article of a series of our recent works, addressing an open question of Bonk-Heinonen-Koskela [5], to study the relationship between (inner) uniformality and Gromov hyperbolicity in infinite dimensional spaces. Our main focus of this paper is to establish a dimension-free Gehring-Hayman inequality for quasigeodesics. A well-known theorem of J. Heinonen and S. Rohde in 1993 states that if $D\subset \mathbb{R}^n$ is quasiconformally equivalently to an uniform domain, then the Gehring-Hayman inequality holds in $D$: quasihyperbolic geodesics in $D$ minimizes the Euclidean length among all curves in $D$ with the same end points, up to a universal dimension-dependent multiplicative constant. In this paper, we develop a new approach to strengthen the above result in the following three aspects: 1) obtain a dimension-free multiplicative constant in the Gehring-Hayman inequality; 2) relax the class of quasihyperbolic geodesics to more general quasigeodesics; 3) relax the quasiconformal equivalence to more general coarsely quasihyperbolic equivalence. As a byproduct of our general approach, we are able to prove that the above improved Gehring-Hayman inequality indeed holds in Banach spaces. This answers affirmatively an open problem raised by J. Heinonen and S. Rohde in 1993 and reformulated by J. Väisälä in 2005.

Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\|<1/2$ for $1\leq i\leq m$. For ${\bf a}=(a_1,\ldots, a_m)\in {\Bbb R}^{md}$, let $\pi^{\bf a}\colon \Sigma=\{1,\ldots, m\}^{\Bbb N}\to {\Bbb R}^d$ denote the coding map associated with the affine IFS $\{T_ix+a_i\}_{i=1}^m$, and let $K^{\bf a}$ denote the attractor of this IFS. Let $W$ be a linear subspace of ${\Bbb R}^d$ and $P_W$ the orthogonal projection onto $W$. We show that for $\mathcal L^{md}$-a.e.~${\bf a}\in {\Bbb R}^{md}$, the Hausdorff and box-counting dimensions of $P_W(K^{\bf a})$ coincide and are determined by the zero point of a certain pressure function associated with $T_1,\ldots, T_m$ and $W$. Moreover, for every ergodic $\sigma$-invariant measure $\mu$ on $\Sigma$ and for $\mathcal L^{md}$-a.e.~${\bf a}\in {\Bbb R}^{md}$, the local dimensions of $(P_W\pi^{\bf a})_*\mu$ exist almost everywhere, here $(P_W\pi^{\bf a})_*\mu$ stands for the push-forward of $\mu$ by $P_W\pi^{\bf a}$. However, as illustrated by examples, $(P_W\pi^{\bf a})_*\mu$ may not be exact dimensional for $\mathcal L^{md}$-a.e.~${\bf a}\in {\Bbb R}^{md}$. Nevertheless, when $\mu$ is a Bernoulli product measure, or more generally, a supermultiplicative ergodic $\sigma$-invariant measure, $(P_W\pi^{\bf a})_*\mu$ is exact dimensional for $\mathcal L^{md}$-a.e.~${\bf a}\in {\Bbb R}^{md}$.

We introduce the Morse parametric qualification condition for bilevel programming. Generic semi-algebraic functions are Morse parametric in a piecewise sense. Thus, bilevel programs with a Morse parametric lower level constitute a relevant intermediate class between strongly convex and fully generic lower levels. In this framework, we study bilevel gradient algorithms with two strategies: the single-step multi-step strategy, which involves a sequence of steps on the lower-level problems followed by one step on the upper-level problem, and a differentiable programming strategy that optimizes a smooth approximation of the bilevel problem. While the first is shown to be a biased gradient method on the problem with rich properties, the second, inspired by meta-learning applications, is less stable but offers simplicity and ease of implementation.

Double ramification loci parametrise marked curves where a weighted sum of the markings is linearly trivial; higher-rank loci are obtained by imposing several such conditions simultaneously. We obtain closed formulae for the orbifold Euler characteristics of double ramification loci, and their higher-rank generalisations, in genus one. The rank-one formula is a polynomial, while the higher-rank formula involves greatest common divisors of matrix minors. The proof is based on a recurrence relation, which allows for induction on the rank and number of markings.

In this paper, we give a geometric interpretation of virtual knotoids as arcs in thickened surfaces. Then we show that virtual knotoid theory is a generalization of classical knotoid theory. This gives a proof of a conjecture of Kauffman and the first author.

This paper explores the critical behavior of the semilinear heat equation $u_t+\mathcal{L}_{a, b}u=|u|^p+f(x)$, considering both the presence and absence of a forcing term $f(x).$ The mixed local-nonlocal operator $\mathcal{L}_{a, b}=-a\Delta+b(-\Delta)^s,\,a,\,b \in \mathbb{R}_+,$ incorporates both local and nonlocal Laplacians. We determine the Fujita-type critical exponents by considering the existence or nonexistence of global solutions. Interestingly, the critical exponent is determined by the nonlocal component of the operator and, as a result, coincides with that of the fractional Laplacian. In the case without a forcing term, our results improve upon recent findings by Biagi et al. [Bull. London Math. Soc. 57 (2025), 265-284] and Del Pezzo et al. [Nonlinear Analysis 255 (2025), 113761]. When a forcing term is included, our results refine those of Wang et al. [J. Math. Anal. Appl., 488 (1) (2020), 124067] and complement the work of Majdoub [La Matematica, 2 (2023), 340-361].

We investigate the use of low-precision first-order methods (FOMs) within a fix-and-propagate (FP) framework for solving mixed-integer programming problems (MIPs). We employ GPU-accelerated PDLP, a variant of the Primal-Dual Hybrid Gradient (PDHG) method specialized to LP problems, to solve the \LP-relaxation of our MIPs to low accuracy. This solution is used to motivate fixings within our FP framework. We evaluate the performance of our heuristic on MIPLIB 2017, demonstrating that low-accuracy LP solutions do not lead to a loss in the quality of the FP heuristic solutions. Further, we use our FP framework to produce high-accuracy solutions for large-scale (up to 243 million nonzeros and 8 million decision variables) unit commitment-based dispatch and expansion planning problems created with the modeling framework REMix. For the largest problems, we can generate solutions with a primal-dual gap of under 2% in less than 4 hours, whereas state-of-the-art commercial solvers cannot produce feasible solutions within 2 days of runtime.

For $N>1$, we constructed a canonical connected fundamental domain for $\Gamma_0(N)$ in [Nie, Parent], utilizing an interesting function $W: {\mathbb Z}/N\to {\mathbb N}$. In this paper, we further study the function $W$, prove some identities, and use it to match the cusps, with widths, produced by our connected fundamental domain with the known cusp classes of $\Gamma_0(N)$. Furthermore, we list the boundary arcs and the gluing patterns of our connected fundamental domain, a key step in understanding the modular curve $X_0(N)$ by this approach.

Let $\mathfrak b_n(\mathbb F_q)$ denote the Lie algebra of upper triangular $n \times n$ matrices over $\mathbb F_q$, and let $\mathfrak u_n(\mathbb F_q)$ be the subalgebra of strictly upper triangular matrices. For every $\mathfrak b_n(\mathbb F_q)$-stable ideal $\mathfrak a \subseteq \mathfrak u_n(\mathbb F_q)$ and partition $\mu$ of $n$, we prove two explicit formulas for the number of elements in $\mathfrak a$ of Jordan type $\mu$. The first one is given by the Hall scalar product of a modified Hall-Littlewood function indexed by $\mu$ and a chromatic quasisymmetric function associated to $\mathfrak a$. The second one is given in terms of a sum of products of $q$-integers over certain standard tableaux that are compatible with a partial order associated to $\mathfrak a$. In the special case $\mathfrak a = \mathfrak u_\Lambda(\mathbb F_q)$, the nilradical of the standard parabolic subalgebra of $\mathfrak{gl}_n(\mathbb F_q)$ corresponding to a composition $\Lambda$ of $n$, our formula specializes to a result of Karp and Thomas: up to a polynomial in $q$, the number of elements in $\mathfrak u_\Lambda(\mathbb F_q)$ of Jordan type $\mu$ equals the coefficient of $\mathbf x^\Lambda$ in the specialization of the dual Macdonald symmetric function $\mathrm Q_{\mu'}(\mathbf x; q^{-1}, t)$ at $t = 0$. We give a new and shorter proof using a parabolic version of Borodin's division algorithm. We present three applications: (1) a formula for the number of points on a nilpotent Hessenberg variety; (2) a new proof of the Kirillov-Melnikov-Ekhad-Zeilberger formula for the number of strictly upper triangular matrices $X$ over $\mathbb F_q$ such that $X^2=0$; (3) a formula for the number of double cosets $\mathsf U_1 \backslash \mathsf{GL}_n(\mathbb F_q) / \mathsf U_2$, where $\mathsf U_1$ and $\mathsf U_2$ are unipotent subgroups corresponding to $\mathfrak b_n(\mathbb F_q)$-stable ideals.

This article is concerned with a new filtered two-step variational integrator for solving the charged-particle dynamics in a mildly non-uniform moderate or strong magnetic field with a dimensionless parameter $\varepsilon$ inversely proportional to the strength of the magnetic field. In the case of a moderate magnetic field ($\varepsilon=1$), second-order error bounds and long-time near-conservation of energy and momentum are obtained. Moreover, the proof of the long-term analysis is accomplished by the backward error analysis. For $0<\varepsilon \ll 1$, the proposed integrator achieves uniform second-order accuracy in the position and the parallel velocity for large step sizes, while attaining first-order accuracy with respect to the small parameter $\varepsilon$ for smaller step sizes. The error bounds are derived from a comparison of the modulated Fourier expansions of the exact and numerical solutions. Moreover, long-time near-conservation of the energy and the magnetic moment is established using modulated Fourier expansion and backward error analysis. All the theoretical results of the error behavior and long-time near-conservation are numerically demonstrated by four numerical experiments.

For $ 0 < r < 1 $, let $ \mathbb{A}_r = \{ z \in \mathbb{C} : r < |z| < 1 \} $ be the annulus with boundary $ \partial \overline{\mathbb{A}}_r = \mathbb{T} \cup r\mathbb{T} $, where $ \mathbb{T} $ is the unit circle in the complex plane $\mathbb C$. We study the class of operators \[ C_{1,r} = \{ T : T \text{ is invertible and } \|T\|, \|rT^{-1}\| \leq 1 \}, \] introduced by Bello and Yakubovich. Any operator $T$ for which the closed annulus $\overline{\mathbb{A}}_r$ is a spectral set is in $C_{1,r}$. The class $C_{1, r}$ is closely related to the \textit{quantum annulus} which is given by \[ QA_r = \{ T : T \text{ is invertible and } \|rT\|, \|rT^{-1}\| \leq 1 \}. \] McCullough and Pascoe proved that an operator in $ QA_r $ admits a dilation to an operator $ S $ satisfying $(r^{-2} + r^2)I - S^*S - S^{-1}S^{-*} = 0$. An analogous dilation result holds for operators in $ C_{1,r}$ class. We extend these dilation results to doubly commuting tuples of operators in quantum annulus as well as in $C_{1,r}$ class. We also provide characterizations and decomposition results for such tuples.

We study community recovery in the planted partition model in regimes where the number and sizes of communities may vary arbitrarily with the number of vertices. In such highly unbalanced settings, standard accuracy or overlap-based metrics become inadequate for assessing recovery performance. Instead, we propose the correlation coefficient between partitions as a recovery metric, which remains meaningful even when the number or sizes of communities differ substantially. We then analyze a simple common-neighbor-based clustering rule which groups two adjacent vertices if they share more than one common neighbor. We establish explicit recovery conditions under sparse inter-community connectivity, without requiring prior knowledge of the model parameters. In particular, in graphs of size $n$, this algorithm achieves exact recovery for communities with sizes $\Omega(\log n)$, almost exact recovery for sizes $\omega(1)$ and weak recovery for sizes $\Omega(1)$. In contrast to most existing results, which assume (nearly) balanced communities, our method successfully recovers small and heterogeneously-sized communities, and improves existing guarantees even in some balanced settings. Finally, our results apply to community sizes that follow a power-law distribution, a characteristic frequently found in real-world networks.

An $n$-vertex graph is degree 3-critical if it has $2n - 2$ edges and no proper induced subgraph with minimum degree at least 3. In 1988, Erdős, Faudree, Gyárfás, and Schelp asked whether one can always find cycles of all short lengths in these graphs, which was disproven by Narins, Pokrovskiy, and Szabó through a construction based on leaf-to-leaf paths in trees whose vertices have degree either 1 or 3. They went on to suggest several weaker conjectures about cycle lengths in degree 3-critical graphs and leaf-to-leaf path lengths in these so-called 1-3 trees. We resolve three of their questions either fully or up to a constant factor. Our main results are the following: - every $n$-vertex degree 3-critical graph has $\Omega(\log n)$ distinct cycle lengths; -every tree with maximum degree $\Delta \ge 3$ and $\ell$ leaves has at least $\log_{\Delta-1}\, ((\Delta-2)\ell)$ distinct leaf-to-leaf path lengths; - for every integer $N\geq 1$, there exist arbitrarily large 1-3 trees which have $O(N^{0.91})$ distinct leaf-to-leaf path lengths smaller than $N$, and, conversely, every 1-3 tree on at least $2^N$ vertices has $\Omega(N^{2/3})$ distinct leaf-to-leaf path lengths smaller than $N$. Several of our proofs rely on purely combinatorial means, while others exploit a connection to an additive problem that might be of independent interest.

Given an integer $n \ge 2$, its prime factorization is expressed as $n= \prod_{i=1}^s p_i^{a_i}$. We define the function $f(n)$ as the smallest positive integer such that $f(n)!$ is divisible by $n$. The main objective of this paper is to derive an asymptotic formula for both sums $\sum_{n \le x} f(n)$ and $\sum_{n \le x, n \in S_k} f(n)$, where $S_k$ denotes the set of all $k$-free integers.

Simplex slicing (Webb, 1996) is a sharp upper bound on the volume of central hyperplane sections of the regular simplex. We extend this to sharp bounds in the probabilistic framework of negative moments, and beyond, of centred log-concave random variables, establishing a curious phase transition of the extremising distribution for new sharp reverse Hölder-type inequalities.

A novel framework is introduced to formalize identifiability in well-specified but ill-posed linear regression models. The framework is distribution-free and accommodates highly correlated features that may or may not relate to the response, reflecting typical real-data structures. First, the identifiable parameter is defined as the least-squares solution obtained by regressing the response on the largest subset of relevant features whose condition number does not exceed a specified threshold, and the relative risk incurred by using this predictor instead of the optimal one is quantified. Second, simple, verifiable conditions are provided under which a broad class of linear dimensionality reduction algorithms can estimate identifiable parameters; algorithms satisfying these conditions are termed statistically interpretable. Third, sharp high-probability error bounds are derived for these algorithms, with rates explicitly reflecting the degree of ill-posedness. With heavy-tailed features and sufficiently low effective rank, these algorithms achieve convergence rates that improve upon both the minimax least-squares rate and lower bounds for sparse estimation under sub-Gaussian features. Results are illustrated via simulations and a real-data application, in which effective rank grows logarithmically with dimension. The framework may extend to algorithms modeling nonlinear response-feature dependence.

In his foundational study of $p$-adic Hodge theory, Faltings introduced the method of almost étale extensions to establish fundamental comparison results of various $p$-adic cohomology theories. Scholze introduced the tilting operations to study algebraic objects arising from $p$-adic Hodge theory in mixed characteristic via the Frobenius map. In this article, we prove a few results which clarify certain ring-theoretic or homological properties of the tilt of an extension between perfectoid rings treated in the construction of big Cohen-Macaulay algebras.

In this paper, we show that for a graph $\Gamma$ from a class named H-rigid graphs, its subgraph ${\rm Int}(\Gamma)$, named the internal graph of $\Gamma$, is an isomorphism invariant of the graph product of hyperfinite II$_1$-factors $R_{\Gamma}$. In particular, we can classify $R_{\Gamma}$ for some typical types of graphs, such as lines, cyclic graphs and infinite regular trees. As an application, we also show that for two isomorphic graph products of hyperfinite II$_1$-factors over H-rigid graphs, the difference of the radius between the two graphs will not be larger than 1. Our proof is based on the recent resolution of the Peterson-Thom conjecture.

The concept of statistical depth extends the notions of the median and quantiles to other statistical models. These procedures aim to formalize the idea of identifying deeply embedded fits to a model that are less influenced by contamination. In the multivariate case, Tukey's median was a groundbreaking concept for multivariate location estimation, and its counterpart for scatter matrices has recently attracted considerable interest. The breakdown point and the maximum asymptotic bias are key concepts used to summarize an estimator's behavior under contamination. In the multivariate and regression setting we analyze recently introduced concentration inequalities that provide a unified framework for studying both the statistical convergence rate and robustness of Tukey's median, depth-based scatter matrices and depth-based multivariate regression estimators. We observe that slight variations in these inequalities allow us to visualize the maximum bias behavior of the deepest estimators. We explicitly obtain the maximum bias curve and breakdown point of the deepest scatter matrices. For the location and scale model, we consider two closely related depth formulations, whose deepest estimators display significantly different behavior in terms of breakdown point. A numerical study is performed to compare the finite sample bias performance of several robust estimators in the multivariate setting.

We prove that unital graph C*-algebras often admit a convenient decomposition into amalgamated free products. We use this to give a complete characterization of when a unital graph C*-algebra is residually finite-dimensional and when it is operator norm stable (that is, matricially semiprojective).

Volume comparison theorem is a type of fundamental results in Riemannian geometry. In this article, we extend the volume comparison result in \cite{Besse2008} to the comparison of total $\sigma_l$-curvature with respect to $\sigma_k$-curvature ($l<k$). In particular, we prove the comparison holds for metrics close to strictly stable positive Einstein metric with $l<\frac{n}{2}$. As for negative Einstein metrics, we prove a similar comparison result provided certain assumptions on sectional curvature holds for the manifold.

This paper introduces two variational inference approaches for infinite-dimensional inverse problems, developed through gradient descent with a constant learning rate. The proposed methods enable efficient approximate sampling from the target posterior distribution using a constant-rate stochastic gradient descent (cSGD) iteration. Specifically, we introduce a randomization strategy that incorporates stochastic gradient noise, allowing the cSGD iteration to be viewed as a discrete-time process. This transformation establishes key relationships between the covariance operators of the approximate and true posterior distributions, thereby validating cSGD as a variational inference method. We also investigate the regularization properties of the cSGD iteration and provide a theoretical analysis of the discretization error between the approximated posterior mean and the true background function. Building on this framework, we develop a preconditioned version of cSGD to further improve sampling efficiency. Finally, we apply the proposed methods to two practical inverse problems: one governed by a simple smooth equation and the other by the steady-state Darcy flow equation. Numerical results confirm our theoretical findings and compare the sampling performance of the two approaches for solving linear and non-linear inverse problems.

We consider a game with two players, consisting of a number of rounds, where the first player to win $n$ rounds becomes the overall winner. Who wins each individual round is governed by a certain urn having two types of balls (type 1 and type 2). At each round, we randomly pick a ball from the urn, and its type determines which of the two players wins. We study the game under three regimes. In the first and the third regimes, a ball is taken without replacement, whilst in the second regime, it is returned to the urn with one more ball of the same colour. We study the properties of the random variables equal to the properly defined overall net profits of the players, and the results are drastically different in all three regimes.

We derive an integral expression $G(z)$ for the reciprocal gamma function, $1/\Gamma(z)=G(z)/\pi$, that is valid for all $z\in\mathbb{C}$, without the need for analytic continuation. The same integral avoids the singularities of the gamma function and satisfies $G(1-z)=\Gamma(z)\sin(\pi z)$ for all $z\in\mathbb{C}$.

In this paper we consider the numerical approximation of a semilinear reaction-diffusion model problem (PDEs) by means of reduced order methods (ROMs) based on proper orthogonal decomposition (POD). We focus on the time integration of the fully discrete reduced order model. Most of the analysis in the literature has been carried out for the implicit Euler method as time integrator. We integrate in time the reduced order model with the BDF-q time stepping ($1\le q\le 5$) and prove optimal rate of convergence of order $q$ in time. Our set of snapshots is obtained from finite element approximations to the original model problem computed at different times. These finite element approximations can be obtained with any time integrator. The POD method is based on first order difference quotients of the snapshots. The reason for doing this is twofold. On the one hand, the use of difference quotients allow us to provide pointwise-in-time error bounds. On the other, the use of difference quotients is essential to get the expected rate $q$ in time since we apply that the BDF-q time stepping, $1\le q\le 5$, can be written as a linear combination of first order difference quotients.

Spanning 7-24 GHz, frequency range 3 (FR3), is a key enabler for next-generation wireless networks by bridging the coverage of sub-6 GHz and the capacity of millimeter-wave bands. Its unique propagation characteristics, such as extended near-field regions and spatially nonstationary fading, enable new transmission strategies. This article explores the potential of FR3 for integrated sensing and communication (ISAC), which unifies wireless communication and environmental sensing. We show that FR3's bandwidth and multiple-input multiple-output (MIMO) capabilities enable high-resolution sensing, multi-target tracking, and fast data transmission. We emphasize the importance of ultra-massive MIMO with extremely large aperture arrays (ELAAs) and the need for unified near-field and far-field channel models to support efficient ISAC. Finally, we outline challenges and future research directions for ELAA-based ISAC in 6G FR3.

In this paper, we develop algorithms for computing the recurrence coefficients corresponding to multiple orthogonal polynomials on the step-line. We reformulate the problem as an inverse eigenvalue problem, which can be solved using numerical linear algebra techniques. We consider two approaches: the first is based on the link with block Krylov subspaces and results in a biorthogonal Lanczos process with multiple starting vectors; the second consists of applying a sequence of Gaussian eliminations on a diagonal matrix to construct the banded Hessenberg matrix containing the recurrence coefficients. We analyze the accuracy and stability of the algorithms with numerical experiments on the ill-conditioned inverse eigenvalue problemshave related to Kravchuk and Hahn polynomials, as well as on other better conditioned examples.

We sharpen the moment comparison inequalities with sharp constants for sums of random vectors uniform on Euclidean spheres, providing a deficit term (optimal in high dimensions).

This paper introduces a new method for solving the planar heat equation based on the Lightning Method. The lightning method is a recent development in the numerical solution of linear PDEs which expresses solutions using sums of polynomials and rational functions, or more generally as sums of fundamental solutions. The method is particularly well suited to handle domains with sharp corners where solution singularities are present. Boundary conditions are formed on a set of collocation points which is then solved as an overdetermined linear system. The approach of the present work is to utilize the Laplace transform to obtain a modified Helmholtz equation which is solved by an application of the lightning method. The numerical inversion of the Laplace transform is then performed by means of Talbot integration. Our validation of the method against existing results and multiple challenging test problems shows the method attains spectral accuracy with root-exponential convergence while being robust across a wide range of time intervals and adaptable to a variety of geometric scenarios.

The problem of bistatic integrated sensing and communications over memoryless relay channels is considered, where destination concurrently decodes the message sent by the source and estimates unknown parameters from received signals with the help of a relay. A state-dependent discrete memoryless relay channel is considered to model this setup, and the fundamental limits of the communication-sensing performance tradeoff are characterized by the capacity-distortion function. An upper bound on the capacity-distortion function is derived, extending the cut-set bound results to address the sensing operation at the destination. A hybrid-partial-decode-and-compress-forward coding scheme is also proposed to facilitate source-relay cooperation for both message transmission and sensing, establishing a lower bound on the capacity-distortion function. It is found that the hybrid-partial-decode-and-compress-forward scheme achieves optimal sensing performance when the communication task is ignored. Furthermore, the upper and lower bounds are shown to coincide for three specific classes of relay channels. Numerical examples are provided to illustrate the communication-sensing tradeoff and demonstrate the benefits of integrated design.

We consider an operator-differential expression of the form $$ \ell y=\frac{d^m}{dx^m}\Big(By^{(n)}+Cy\Big), \quad 0<x<1, $$ where $B$ is a linear bounded invertible operator, while $C$ is some finite-dimensional linear operator relatively bounded to the operator of $n$-fold differentiation. To such a form, we can reduce, in particular, various singular differential expressions with the coefficients in negative Sobolev spaces, which creates an alternative to their regularization. In the case when $B$ is an integral Volterra operator of the second kind with a continuous kernel vanishing at the diagonal, we establish completeness of the root functions of an operator generated by the expression $\ell y$ and irregular semi-separated boundary conditions.

We extend the classical Kolmogorov-Riesz compactness theorem to the setting of asymptotic $L_p$ spaces on $\mathbb{R}^n$. These are nonlocally convex $\mathrm{F}$-spaces that contain the standard $L_p$ spaces as dense subspaces and include all measurable functions supported on sets of finite measure. In contrast with the classical $L_p$ setting, an additional almost equiboundedness condition is needed, and we prove that together with the natural tail and translation conditions it characterizes relative compactness. We conclude with illustrative examples.

We study a bulk-surface Cahn--Hilliard model with non-degenerate mobility and singular potentials in two dimensions. Following the ideas of the recent work by Conti, Galimberti, Gatti, and Giorgini [Calc. Var. Partial Differential Equations, 64(3):Paper No. 87, 32, 2025] for the Cahn--Hilliard equation with homogeneous Neumann boundary conditions, we show the uniqueness of weak solutions together with a continuous dependence estimate for sufficiently regular mobility functions. Next, under weaker assumptions on the mobility functions, we show the existence of a weak solution that exhibits the propagation of uniform-in-time regularity and satisfies the instantaneous separation property. Lastly, we consider the long-time behavior and prove that the unique weak solution converges to a solution of the stationary bulk-surface Cahn--Hilliard equation. Our approach for the uniqueness proof relies on a new well-posedness and regularity theory for a bulk-surface elliptic system with non-constant coefficients, which may be of independent interest.

We consider an inverse initial-data problem for the compressible anisotropic Navier--Stokes equations, in which the goal is to reconstruct the initial velocity field from noisy lateral boundary observations. In the formulation studied here, the density, pressure, anisotropic viscosity tensor, and body force are assumed known, while the initial velocity is the quantity to be recovered. We introduce a new computational framework based on Legendre time-dimensional reduction, in which the velocity field is projected onto an exponentially weighted Legendre basis in time. This transformation reduces the original time-dependent inverse problem to a coupled system of time-independent elliptic equations for the Fourier coefficients of the velocity field. The resulting reduced model is solved using a combination of quasi-reversibility and a damped Picard iteration. Numerical experiments in two dimensions show that the proposed method accurately and robustly reconstructs initial velocity fields, even in the presence of significant measurement noise, geometrically complex structures, and anisotropic effects. The method provides a flexible and computationally tractable approach for inverse fluid problems in anisotropic media.

We consider the filtering problem with the partially observed Lorenz 96 model. Although the accuracy of the 3DVar filter in this problem has been established, the theoretical guarantee for the ensemble Kalman filter (EnKF) remains limited due to the analytical difficulty of handling non-symmetric matrices that emerge in the partial observation setting. This study establishes uniform-in-time error bounds of a stochastic variant of the EnKF, known as the perturbed observation (PO) method. By utilizing additive covariance inflation, we successfully obtain the bounds both with and without projecting the background covariance onto the observation space. Our analysis with the projection complements existing results for the deterministic variant of the EnKF, while our approach without the projection offers an extended mathematical framework to handle the non-symmetric matrix products directly. A numerical example validates the theoretical findings and shows comparable accuracies between the two settings.

We study the problem of continuity of derivations over Banach algebras. More specifically, we consider a class of Banach algebras that contain a dense '$C^*$-like' subalgebra. We discuss applications to $L^p$-crossed products and symmetrized $L^p$-crossed products. As an example, our results imply that every derivation over the $L^p$-crossed product $F^p(G,X,\alpha)$ is continuous, provided that $G$ is infinite, finitely generated, has polynomial growth, and acts freely on the compact Hausdorff space $X$.

In this note, we demonstrate how determinant representations for correlation functions in conformal field theory can be used to derive explicit determinant formulas for powers of the classical $\eta$-function, expressed via deformed elliptic functions with parameters. In particular, we obtain counterparts of Garvan's formulas for the modular discriminant corresponding to the genus two Riemann surface case.

We introduce the notion of 0-shifted cosymplectic structure on differentiable stacks and develop a corresponding moment map theory for Hamiltonian cosymplectic actions. We present a reduction procedure, establish a version of the Kirwan convexity theorem, and obtain examples of Morse-Bott Lie groupoid morphisms.

The problem of constrained stabilization on the n-sphere under star-shaped constraints is considered. We propose a control strategy that allows to almost globally steer the state to a desired location while avoiding star-shaped constraints on the n-sphere. Depending on the state's proximity to the unsafe regions, the state is either guided towards the target location along the geodesic connecting the target to the state or steered towards the antipode of a predefined point lying in the interior of the nearest unsafe region. We prove that the target location is almost globally asymptotically stable under the proposed continuous, time-invariant feedback control law. Nontrivial simulation results on the 2-sphere and the 3-sphere demonstrate the effectiveness of the theoretical results.

In previous works, we introduced the notion of dominant vertices in the context of dynamical systems on networks. This is a set of nodes in the underlying network whose evolution determines the whole network's dynamics after a transient time. In this paper, we focus on the case of Boolean networks. We define a reduced graph on the dominant vertices and an induced dynamics on this graph, which we prove is asymptotically equivalent to the original Boolean dynamics. Asymptotic conjugacy ensures that the systems, restricted to their respective attractors, are dynamically equivalent. For a significant class of networks, the induced dynamics is indeed a reduction of the original system. In these cases, the reduction, which is obtained from the structure of dominant vertices, supplies a more tractable system with the same structure of attractors as the original one. Furthermore, the structure of the induced system allows us to establish bounds on the number and period of the attractors, as well as on the reduction of the basin's sizes and transient lengths. We illustrate this reduction by considering a class of networks, which we call clover networks, whose dominant set is a singleton. To get insight into the structure of the basins of attraction of Boolean networks with a single dominant vertex, we complement this work with a numerical exploration of the behavior of a parametrized ensemble of systems of this kind.

We study a system of fully nonlinear elliptic equations, depending on a small parameter $\eps$, that models long-range segregation of populations. The diffusion is governed by the negative Pucci operator. In the linear case, this system was previously investigated by Caffarelli, the second author, and Quitalo in \cite{CL2} as a model in population dynamics. We establish the existence of solutions and prove convergence as $\eps\to0^+$ to a free boundary problem in which populations remain segregated at a positive distance. In addition, we show that the supports of the limiting functions are sets of finite perimeter and satisfy a semi-convexity property.

Let $I$ be a monomial ideal in a polynomial ring $S=K[x_1,\ldots,x_n]$ over a field $K$ with $n=2$ or $3$, and let $\overline{I}$ be its integral closure. We will show that $\text{reg} (\overline{I}) \le \text{reg} (I)$. Furthermore, if $I$ is generated by elements of degree $d$, then $\text{reg} (I)=d$ if and only if $I$ has linear quotients.

We investigate the existence of generalised densities for the $\Phi^4_d$ $(d=1,2,3)$ measures, in finite volume, through the lens of Onsager-Machlup (OM) functionals. The latter are rigorously defined for measures on metric spaces as limiting ratios of small ball probabilities. In one dimension, we show that the standard OM functional of the $\Phi^4_1$ measure coincides with the $\Phi^4$ action as expected. In two dimensions, we show that OM functionals of the $P(\Phi)_2$ measures agree with the corresponding actions, by considering ``enhanced" distances, defined with respect to Wick powers of the Gaussian Free Field, which are analogous to rough path metrics. In dimension $3$, two natural generalisations of the OM functional are proved to be degenerate. Finally, we recover the $\Phi^4_3$ action, under appropriate regularity conditions, by considering joint small radius-large frequency limits.

We investigate the persistence of spatial analyticity for solutions to the Majda-Biello and Hirota-Satsuma systems with analytic initial data. This result is the first to establish analyticity persistence in such coupled KdV systems.

For a Galois number field $K$, the Galois group $\text{Gal}(K/\mathbb{Q})$ acts on the class group $Cl_K$ in a very natural way: $\sigma\cdot[I]=[\sigma(I)]$ for any $\sigma \in \text{Gal}(K/\mathbb{Q})$, $[I]\in Cl_K$. In this paper, we will explore how the unique properties of this group action work together to elucidate the relationship between these two groups. While previous work on this problem has focused on representation theory, we take a direct approach to some classical and new problems. The paper then turns to an exploration of the class groups of overrings of the ring of integers $\mathcal{O}_K$. These turn out to be powerful tools for understanding $Cl_K$ and overrings of $\mathcal{O}_K$. We conclude with an analysis of the arithmetic of normsets of a rings of integers which is closely tied to this action.

Two confluent rewriting systems in noncommutatives polynomials are constructed using the equations allowing the identification of the local coordinates (of second kind) of the graphs of the $\zeta$ polymorphism as being (shuffle or quasi-shuffle) characters and bridging two algebraic structures of polyzetas. In each system, the left side of each rewriting rule corresponds to the leading monomial of the associated homogeneous in weight polynomial while the right side is canonically represented on the Q-algebra generated by irreducible terms which encode an algebraic basis of the Q-algebra of polyzetas. These polynomials are totally lexicographically ordered and generate the kernels of the $\zeta$ polymorphism meaning that the Q-free algebra of polyzetas is graded and the irreducible polyzetas are transcendent numbers, Q-algebraically independent, and then $\pi$ 2 is Q-algebraically independent on odd zeta values (so does $\pi$).

An action of a finite group $G$ is a pair $(S,\hat{G})$, where $S$ is a compact Riemann surface of genus $g \geqslant 2$ and $\hat{G} \leqslant {\rm Aut}(S)$ is isomorphic to $G$. To each action $(S,\hat{G})$ there is associated a signature $(\gamma;k_{1},\ldots,k_{r})$ that codifies the orbifold structure of $S/\hat{G}$. Two actions of $G$, say $(S_{1},G_{1})$ and $(S_{2},G_{2})$, are topologically equivalent if there is an orientation-preserving homeomorphism $\varphi:S_{1} \to S_{2}$ such that $\varphi G_{1} \varphi^{-1}=G_{2}$. Topologically equivalent actions necessarily must have the same signature. The problem of determining the number of different topological actions of $G$ for a given signature is in general a difficult task. In this article, we describe, up to topological equivalence, those actions when $G$ is an abelian group and quotient genus $\gamma=0$. We are particularly interested in the case $G={\mathbb Z}_{k}^{m}$ and the quotient signature of the action to be of the form $(0;k,\stackrel{n+1}{\ldots},k)$.

In this paper, we consider high-dimensional Lp-quantile regression which only requires a low order moment of the error and is also a natural generalization of the above methods and Lp-regression as well. The loss function of Lp-quantile regression circumvents the non-differentiability of the absolute loss function and the difficulty of the squares loss function requiring the finiteness of error's variance and thus promises excellent properties of Lp-quantile regression. Specifically, we first develop a new method called composite Lp-quantile regression(CLpQR). We study the oracle model selection theory based on CLpQR (call the estimator CLpQR-oracle) and show in some cases of p CLpQR-oracle behaves better than CQR-oracle (based on composite quantile regression) when error's variance is infinite. Moreover, CLpQR has high efficiency and can be sometimes arbitrarily more efficient than both CQR and the least squares regression. Second, we propose another new regression method,i.e. near quantile regression and prove the asymptotic normality of the estimator when p converges to 1 and the sample size infinity simultaneously. As its applications, a new thought of smoothing quantile objective functions and a new estimation are provided for the asymptotic covariance matrix of quantile regression. Third, we develop a unified efficient algorithm for fitting high-dimensional Lp-quantile regression by combining the cyclic coordinate descent and an augmented proximal gradient algorithm. Remarkably, the algorithm turns out to be a favourable alternative of the commonly used liner programming and interior point algorithm when fitting quantile regression.

Carrollian manifolds offer an intrinsic geometric framework for the physics in the ultra-relativistic limit. The recently introduced Carrollian Lie algebroids are generalised to the setting of $\rho$-commutative geometry, (also known as almost commutative geometry), where the underlying algebras commute up to a numerical factor. Via $\rho$-Lie-Rinehart pairs, it is shown that the foundational tenets of Carrollian geometry have analogous statements in the almost commutative world. We explicitly build two toy examples: we equip the extended quantum plane and the noncommutative $2$-torus with Carrollian structures. This opens up the rigorous study of noncommutative Carrollian geometry via almost commutative geometry.

In relation to Fuglede's conjecture, we establish several Plancherel-type identities and demonstrate the surjectivity of the Fourier transform between certain unbounded tiling sets of $\mathbb{R}$ that are in duality. In the terminology commonly used in the context of Fuglede's conjecture, our result states that an open set tiles $\mathbb{R}$ by the finite set $\{0,1,\dots,p-1\}$ if and only if it admits a spectrum (or, equivalently, a dual pair measure) given by the Lebesgue measure on $\left[-\tfrac{1}{2p}, \tfrac{1}{2p}\right] + \mathbb{Z}$.

In modeling biological systems and other applications, an important recurring question is whether those systems maintain healthy regimes not just at dynamic attractors but also during transient excursions away from those attractors. For ODE models, these excursions are not due only to nonlinearities. Even in a linear, autonomous system with a global attractor at the origin, some trajectories will move transiently away from the origin before eventually being attracted asymptotically. Reactivity, defined by Neubert and Caswell in 1997, captures this idea of transient amplification of perturbations by measuring the maximum instantaneous rate of radial growth. In this paper, we introduce a novel framework for analyzing reactivity and transient dynamics in two-dimensional linear ODEs using a radial and tangential decomposition of the vector field. We establish how to view the eigen-structure of the system through this lens and introduce a matching structure of orthovectors and orthovalues that characterize where the radial velocity becomes positive. From this perspective, we gain geometric insight about the regions of state space where positive radial growth occurs and how solutions trajectories traverse through these regions. Additionally, we propose four standard matrix forms that characterize both transient and asymptotic behavior and which highlight reactivity features more directly. Finally, we apply our framework to explore the limits of reactivity and maximal amplification and to characterize how transient reactivity can accumulate in nonautonomous linear systems to result in asymptotically unstable behavior.

A new operator-level necessity result for the Chapman--Enskog expansion is established: in closed and unforced kinetic systems, the $O(\varepsilon)$ deviatoric stress arises if and only if the first Chapman--Enskog correction $f^{(1)}$ is nonzero. This resolves a gap in the classical kinetic-to-continuum literature, where the presence of first-order deviatoric stress is typically assumed or derived formally but not shown to be necessary under explicit functional-analytic hypotheses. Under precise nullspace structure, coercivity or quantitative hypocoercivity, and Fredholm solvability of the linearized collision operator--together with uniform $O(\varepsilon^2)$ remainder control--a sharp necessity theorem (Theorem 6.1) is proved: if $f^{(1)}\equiv 0$, then no $O(\varepsilon)$ deviatoric stress can appear in the hydrodynamic limit. The argument identifies the bounded mapping \[ f^{(0)} \mapsto f^{(1)} = -L^{-1}(\partial_t^{(0)} + v \cdot \nabla_x) f^{(0)}, \] and the induced moment-to-stress operator, and shows how remainder bounds preclude hidden $O(\varepsilon)$ contributions. A worked BGK example verifies the construction, transport coefficients, and operator constants. Detailed assumptions and analytic estimates are provided in Section 4 and Appendix A. The discussion concludes by describing how microscopic seeds (deterministic or finite-$N$) can project into macroscopic amplification channels relevant for transition and turbulence.

As is known, every finite-dimensional algebra over a field is isomorphic to the centralizer algebra of \textbf{two} matrices. So it is fundamental to study first the centralizer algebra of a single matrix, called a centralizer matrix algebra. In this article, stable equivalences between centralizer matrix algebras over arbitrary fields are completely characterized in terms of a new type of equivalence relation on matrices. Moreover, stable equivalences of centralizer matrix algebras over any fields induce stable equivalences of Morita type, thus preserve dominant, finitistic and global dimensions. Our methods also show that the Alperin--Auslander/Auslander--Reiten conjecture holds true for stable equivalences between an arbitrary algebra and a centralizer matrix algebra over a common field.

We classify the irreducible anyon sectors of Levin-Wen models over an arbitrary unitary fusion category $\mathcal{C}$, showing that they are in one-to-one correspondence with equivalence classes of simple objects of the Drinfeld center $Z(\mathcal{C})$. We achieve this by making explicit how the Levin-Wen Hamiltonian stabilizes subspaces isomorphic to state spaces of the corresponding Turaev-Viro TQFT, and developing a detailed understanding of these state spaces on punctured disks. In particular, we construct Drinfeld insertion operators on such spaces which can move anyons between the punctures, and can change their fusion channels. Using these Drinfeld insertions, we construct explicit string operators that excite anyons above the ground state. The fusion and braiding properties of these anyons will be analysed in a companion paper.

We extend the complete census of orientable cusped hyperbolic $3$-manifolds to $10$ tetrahedra, giving the next $150730$ manifolds and their $496638$ minimal ideal triangulations. As applications, we find the precisely $439898$ exceptional Dehn fillings on them, revealing the next $1849$ simplest hyperbolic knot exteriors in $S^3$. We also give the simplest example of an orientable cusped hyperbolic $3$-manifold containing a closed totally geodesic surface.

We couple projective limits of probability measures to direct limits of their symmetry groups. We show that the direct limit group is the group of symmetries of the projective limit probability measure. If projective systems of probability measures represent point processes in increasingly larger finite regions of the same infinite space, then we show that under some additional niceness and consistency assumptions, an extension of the direct limit group is the symmetry group of the projective limit point process in the whole infinite space. The application of these results to random graph limits provides ``shortest paths'' to graphons and graphexes as it recovers these random graph limits as trivial corollaries. Another application example encompasses a broad class of limits of random graphs with bounded average degrees. This class includes a representative collection of paradigmatic random graph models that have attracted significant research attention in diverse areas of science. Our approach thus provides a general unified framework to study limits of very different types of random graphs.

Based on entropy and symmetrical uncertainty (SU), we define a metric for categorical random variables and show that this metric can be promoted into an appropriate quotient space of categorical random variables. Moreover, we also show that there is a natural commutative monoid structure in the same quotient space, which is compatible with the topology induced by the metric, in the sense that the monoid operation is continuous.

This paper investigates the (fractional) heat equation with a nonlocal nonlinearity involving a Riesz potential: \begin{equation*} u_{t}+(-\Delta)^{\frac{\beta}{2}} u= I_\alpha(|u|^{p}),\qquad x\in \mathbb{R}^n,\,\,\,t>0, \end{equation*} where $\alpha\in(0,n)$, $\beta\in(0,2]$, $n\geq1$, $p>1.$ We introduce the Fujita-type critical exponent $p_{\mathrm{Fuj}}(n,\beta,\alpha)=1+(\beta+\alpha)/(n-\alpha)$, which characterizes the global behavior of solutions: global existence for small initial data when $p>p_{\mathrm{Fuj}}(n,\beta,\alpha),$ and finite-time blow-up when $p\leq p_{\mathrm{Fuj}}(n,\beta,\alpha)$. It is remarkable that the critical Fujita exponent is not determined by the usual scaling argument that yields $p_{sc}=1+(\beta+\alpha)/n$, but instead arises in an unconventional manner, similar to the results of Cazenave et al. [Nonlinear Analysis, 68 (2008), 862-874] for the heat equation with a nonlocal nonlinearity of the form $\int_0^t(t-s)^{-\gamma}|u(s)|^{p-1}u(s)ds,\,0\leq \gamma<1.$ The result on global existence for $p>p_{\mathrm{Fuj}}(n,2,\alpha),$ provides a positive answer to the hypothesis proposed by Mitidieri and Pohozaev in [Proc. Steklov Inst. Math., 248 (2005) 164-185]. We further establish global nonexistence results for the above heat equation, where the Riesz potential term $I_\alpha(|u|^{p})$ is replaced by a more general convolution operator $(\mathcal{K}\ast |u|^p),\,\mathcal{K}\in L^1_{loc}$, thereby extending the Mitidieri-Pohozaev's results established in the aforementioned work. Proofs of the blow-up results are obtained using a nonlinear capacity method specifically adapted to the structure of the problem, while global existence is established via a fixed-point argument combined with the Hardy-Littlewood-Sobolev inequality.

An exact channel impulse response (CIR) for the three-dimensional point-to-sphere absorbing channel under drift has remained unavailable due to symmetry breaking. This letter closes this gap by deriving an exact analytical CIR for a fully absorbing spherical receiver under uniform drift with arbitrary direction. By formulating the problem in terms of joint first-hitting time-location statistics and applying a Girsanov-based measure change, drift effects are isolated into an explicit multiplicative factor, yielding an exact series representation. The resulting CIR provides a rigorous reference model and enables efficient, noise-free evaluation of key channel metrics without relying on Monte Carlo simulations.

A multiscale optimization framework for problems over a space of Lipschitz continuous functions is developed. The method solves a coarse-grid discretization followed by linear interpolation to warm-start project gradient descent on progressively finer grids. Greedy and lazy variants are analyzed and convergence guarantees are derived that show the multiscale approach achieves provably tighter error bounds at lower computational cost than single-scale optimization. The analysis extends to any base algorithm with iterate convergence at a fixed rate. Constraint modification techniques preserve feasibility across scales. Numerical experiments on probability density estimation problems, including geological data, demonstrate speedups of an order of magnitude or better.

We derive error bounds for CUR matrix approximation using determinant-based methods that relate local projection errors to global approximation quality. For general matrices, we establish determinant identities for bordered Gramian matrices that decompose CUR approximation errors into interpretable local components. These identities connect projection errors onto submatrix column spaces directly to determinants, providing geometric insight into approximation degradation. We develop a probabilistic framework based on volume sampling that yields interpolation-type error bounds quantifying the benefits of oversampling: when $r > k$ rows are selected for $k$ columns, the expected error factor transitions linearly from $(k+1)^2$ (no oversampling) to $(k+1)$ (full oversampling). Our analysis establishes that the expected squared error is bounded by this interpolation factor times the squared error of the best rank-$k$ approximation, directly connecting CUR approximation quality to the optimal low-rank approximation. The framework applies to both CUR decomposition for general matrices and the Nyström method for symmetric positive semi-definite matrices, providing a unified theoretical foundation for determinant-based low-rank approximation analysis.

This paper analyzes a service system modeled as a single-server queue, in which the service provider aims to dynamically maximize the expected revenue per unit of time. This is achieved by constructing a stochastic gradient descent algorithm that dynamically adjusts the price. A key feature of our modeling framework is that customers may choose to balk - that is, decide not to join - when facing high congestion. A notable strength of our approach is that the revenue-maximizing algorithm relies solely on information about effective arrivals, meaning that only the behavior of customers who choose not to balk is observable and used in decision-making. This results in an elaborate interplay between the pricing policy and the effective arrival process, yielding a non-standard state dependent queueing process. An important contribution of our work concerns a novel Infinitesimal Perturbation Analysis (IPA) procedure that is able to consistently estimate the stationary effective arrival rate. This is further leveraged to construct an iterative algorithm that converges, under mild regularity conditions, to the optimal price with provable asymptotic guarantees.

The established equivalence between 2-crossed modules and Gray 3-groups [M. Sarikaya and E. Ulualan, 2024] serves as a benchmark for higher-dimensional algebraic models. However, to the best of our knowledge, the established definitions of 3-crossed modules [Z. Arvasi, T. S. Kuzpinari, and E. Ö. Uslu, 2009] are not clearly suited for extending this equivalence. In this paper, we propose an alternative formulation of a 3-crossed module, equipped with a new type of lifting, which is specifically designed to serve as a foundation for this higher-order categorical correspondence. As the primary results of this paper, we validate this new structure. We prove that the simplicial set induced by our 3-crossed module forms a quasi-category. Furthermore, we show that the Moore complex of length 3 associated with a simplicial group naturally admits the structure of our 3-crossed module. This work establishes our definition as a robust candidate for modeling the next level in this algebraic-categorical program.

Let $C\subset\mathbb{R}^2$ be a convex body, and for a positive integer $N$, let $\mathcal{P}$ be a configuration of $N$ points in $[0,1)^2$. The discrepancy of $\mathcal{P}$ with respect to $C$ is defined by \begin{equation*} \mathcal{D}(\mathcal{P},\, C)=\sum_{\mathbf{p}\in\mathcal{P}}\sum_{\mathbf{n}\in\mathbb{Z}^2}\mathbf{1}_C(\mathbf{p}+\mathbf{n})-N|C|, \end{equation*} and one may estimate how $\mathcal{P}$ deviates from uniformity by averaging the latter quantity over a family of sets. When considering quadratic averages over translated and dilated copies of $C$, one gets the \textit{homothetic quadratic discrepancy} \begin{equation*} \mathcal{D}_2(\mathcal{P},\, C)=\int_{0}^{1}\int_{[0,1)^2}\left|\mathcal{D}( \mathcal{P},\,\boldsymbol{\tau}+\delta C)\right|^2\,{\rm d}\boldsymbol{\tau}\,{\rm d} \delta. \end{equation*} We investigate the behaviour of the optimal \textit{homothetic quadratic discrepancy}, that is \begin{equation*} \inf_{\# \mathcal{P}=N} \mathcal{D}_2(\mathcal{P},\, C)\quad\text{as}\quad N\to+\infty. \end{equation*} Beck~\cite{MR915529} and Beck and Chen~\cite{MR1489133} showed that the optimal \textit{h.q.d.} of convex polygons has an order of growth of $\log N$, and more recently, Brandolini and Travaglini~\cite{MR4358540} proved that the optimal \textit{h.q.d.} of planar convex bodies with a $\mathcal{C}^2$ boundary has an order of growth of $N^{1/2}$. We show that, in general, a single order of growth for the optimal \textit{h.q.d.} need not exist. First, by an implicit geometric construction of $C$, we obtain prescribed oscillations between $\log N$ and $N^{1/2}$. Second, by a subtler design of $\partial C$ and via Fourier-analytic methods, we obtain prescribed polynomial-order oscillations in the range $N^\alpha$ with $\alpha\in(2/5,1/2)$.

We study the Collatz function famously related to the Collatz Conjecture under the topological and ergodic perspectives, including an approach with thermodynamic formalism. By introducing a key topology and its Borel $\sigma$-algebra we show that recurrence implies periodicity. Moreover, we establish that the set of periodic orbits is finite if, and only if, every continuous potential possesses some equilibrium state. The uniqueness of periodic orbits is equivalent to the uniqueness of equilibrium state for every bounded and continuous potential. Additionally, by using the Alexandroff compactification of the established topology, we prove finiteness of cycles, which is a significant advance to the conjecture itself. We also present a proof of uniqueness of cycle and show that there is no divergent orbits. Finally, we apply our technique to the Baker and Syracuse maps, obtaining a similar result on the finiteness of orbits and no divergence for a general class of important maps.

We present a universal concentration bound for sums of random variables under arbitrary dependence, and we prove that it is asymptotically optimal for broad families of marginals admitting a uniform integrable tail-quantile envelope. The bound follows directly from the subadditivity of expected shortfall, a property well known in the risk-measure literature. Our sharpness result relies on an explicit construction of asymptotically extremal couplings. We furthermore provide practical sufficient conditions -- based on convex transformation order comparisons with exponential and power-law envelopes -- under which the bound admits simple, explicit tail profiles.

We revisit a fundamental question in hypothesis testing: given two sets of probability measures $\mathcal{P}$ and $\mathcal{Q}$, when does a nontrivial (i.e. strictly unbiased) test for $\mathcal{P}$ against $\mathcal{Q}$ exist? Le Cam showed that, when $\mathcal{P}$ and $\mathcal{Q}$ have a common dominating measure, a test that has power exceeding its level by more than $\varepsilon$ exists if and only if the convex hulls of $\mathcal{P}$ and $\mathcal{Q}$ are separated in total-variation distance by more than $\varepsilon$. The requirement of a dominating measure is frequently violated in nonparametric statistics. In a passing remark, Le Cam described an approach to address more general scenarios, but he stopped short of stating a formal theorem. This work completes Le Cam's program, by presenting a matching necessary and sufficient condition for testability: for the aforementioned theorem to hold without assumptions, one must take the closures of the convex hulls of $\mathcal{P}$ and $\mathcal{Q}$ in the space of bounded finitely additive measures. We provide simple elucidating examples, and elaborate on various subtle measure theoretic and topological points regarding compactness and achievability.

From any graph $G$ arises a flat space wavefunction, obtained by integrating a product of propagators associated to the vertices and edges of $G$. This function is a key ingredient in the computation of cosmological correlators, and several representations for it have been proposed. We formulate three such representations and prove their correctness. In particular, we show that the flat space wavefunction can be read off from the canonical form of the cosmological polytope, and we settle a conjecture of Fevola, Pimentel, Sattelberger, and Westerdijk regarding a partial fraction decomposition for the flat space wavefunction. The terms of the decomposition correspond to certain collections of connected subgraphs associated to $G$ and its spanning subgraphs, reflecting the fact that the flat space wavefunction contains information about how $G$ is connected.

In this paper, we prove that biorthogonal ensembles on the real line with a specific derivative structure admit an explicit correlation kernel of double contour integral form. We will demonstrate that this expression is a valuable starting point for asymptotic analysis and that our class of biorthogonal ensembles admits a large variety of limit kernels, by proving that two new classes of limit kernels can occur. The first type is a deformation of the hard edge Bessel kernel which arises in polynomial ensembles describing the eigenvalues of the sum of two random matrices, while the second type arises for Muttalib-Borodin type deformations of polynomial ensembles.

The Reynolds equation is derived from the incompressible Navier Stokes equations under the lubrication assumptions of a long and thin domain geometry and a small scaled Reynolds number. The Reynolds equation is an elliptic differential equation and a dramatic simplification from the governing equations. When the fluid domain is piecewise linear, the Reynolds equation has an exact solution that we formulate by coupling the exact solutions of each piecewise component. We consider a formulation specifically for piecewise constant heights, and a more general formulation for piecewise linear heights; in both cases the linear system is inverted using the Schur complement. These methods can also be applied in the case of non-linear heights by approximating the height as piecewise constant or piecewise linear, in which case the methods achieve second order accuracy. We assess the time complexity of the two methods, and determine that the method for piecewise linear heights is linear time for the number of piecewise components. As an application of these methods, we explore the limits of validity for lubrication theory by comparing the solutions of the Reynolds and the Stokes equations for a variety of linear and non-linear textured slider geometries.

In "Bipartite minors" [Journal of Combinatorial Theory, Series B, 2016], Chudnovsky et al. introduced the bipartite minor relation, a quasi-order on the class of bipartite graphs somewhat analogous the minor relation on general graphs and asked whether it is a well-quasi-order. We answer this question negatively by giving an infinite set of 2-connected bipartite graphs that are pairwise incomparable with respect to the bipartite minor relation. We additionally give two sets of infinitely many pairs of bipartite graphs: one set of pairs G, H such that H is a bipartite minor, but not a minor, of G, and one set of pairs G, H such that H is a minor, but not a bipartite minor, of G.

We propose a novel computational method for unit commitment (UC), which does not require linearized approximation and provides several orders of magnitude performance improvement over current state-of-the-art. The performance improvement is achieved by introducing a heuristic tailored for UC problems. The method can be implemented using existing continuous optimization solvers and adapted for different applications. We demonstrate value of the new method in examples of advanced UC analyses at the scale where use of current state-of-the-art tools is infeasible. We expect that the capability demonstrated in this paper will be critical to address emerging power systems challenges with more volatile large loads, such as data centers, and generation that is composed of larger number of smaller units, including significant behind-the-meter generation.

We study homology of ample groupoids via the compactly supported Moore complex of the nerve. Let $A$ be a topological abelian group. For $n\ge 0$ set $C_n(\mathcal G;A) := C_c(\mathcal G_n,A)$ and define $\partial_n^A=\sum_{i=0}^n(-1)^i(d_i)_*$. This defines $H_n(\mathcal G;A)$. The theory is functorial for continuous étale homomorphisms. It is compatible with standard reductions, including restriction to saturated clopen subsets. In the ample setting it is invariant under Kakutani equivalence. We reprove Matui type long exact sequences and identify the comparison maps at chain level. For discrete $A$ we prove a natural universal coefficient short exact sequence $$0\to H_n(\mathcal G)\otimes_{\mathbb Z}A\xrightarrow{\ \iota_n^{\mathcal G}\ }H_n(\mathcal G;A)\xrightarrow{\ \kappa_n^{\mathcal G}\ }\operatorname{Tor}_1^{\mathbb Z}\bigl(H_{n-1}(\mathcal G),A\bigr)\to 0.$$ The key input is the chain level isomorphism $C_c(\mathcal G_n,\mathbb Z)\otimes_{\mathbb Z}A\cong C_c(\mathcal G_n,A)$, which reduces the groupoid statement to the classical algebraic UCT for the free complex $C_c(\mathcal G_\bullet,\mathbb Z)$. We also isolate the obstruction for non-discrete coefficients. For a locally compact totally disconnected Hausdorff space $X$ with a basis of compact open sets, the image of $\Phi_X:C_c(X,\mathbb Z)\otimes_{\mathbb Z}A\to C_c(X,A)$ is exactly the compactly supported functions with finite image. Thus $\Phi_X$ is surjective if and only if every $f\in C_c(X,A)$ has finite image, and for suitable $X$ one can produce compactly supported continuous maps $X\to A$ with infinite image. Finally, for a clopen saturated cover $\mathcal G_0=U_1\cup U_2$ we construct a short exact sequence of Moore complexes and derive a Mayer-Vietoris long exact sequence for $H_\bullet(\mathcal G;A)$ for explicit computations.

We present FastLSQ, a framework for fast PDE solving and inverse problems built on sinusoidal random Fourier features with exact analytical derivatives. Sinusoids are eigenfunctions of differentiation: derivatives of any order admit closed-form evaluation in $O(1)$ operations, enabling graph-free operator assembly without automatic differentiation. Linear PDEs are solved in a single least-squares call; nonlinear PDEs via Newton--Raphson iteration where each step reuses the analytical assembly. On 17 PDEs spanning 1 to 6 dimensions, FastLSQ achieves $10^{-7}$ in 0.07\,s on linear problems and $10^{-8}$ to $10^{-9}$ on nonlinear problems in under 9\,s -- orders of magnitude faster and more accurate than iterative PINN solvers. The framework extends to inverse problems (heat-source localisation, coil recovery from sparse sensors) and PDE discovery via analytical derivative dictionaries. Code is publicly available at this http URL and via pip install fastlsq.

For a connected orientable closed surface $(\Sigma,g)$ of genus $G$ with Anosov geodesic flow, we show the existence of an open subset $U_g$ of finite-dimensional irreducible representations of the fundamental group of its unit tangent bundle, whose complement has complex codimension at least one and such that for any $\rho \in U_g$, the twisted Ruelle zeta function $\zeta_{g,\rho}(s)$ vanishes at $s=0$ to order ${\rm dim}(\rho)(2G-2)$ if $\rho$ factors through $\pi_1(\Sigma)$, and does not vanish otherwise. In the second case, we show that $\zeta_{g,\rho}(0)$ is given by the Reidemeister--Turaev torsion, thus extending Fried's conjecture to a generic set of acyclic (but not necessarily unitary) representations. We also show that the order of vanishing of the untwisted zeta function is constant for an open and dense subset of Anosov metrics in the connected component of a hyperbolic $3$-metric. Our proofs rely on computing the dimensions of the spaces of generalized twisted Pollicott--Ruelle resonant states at zero.

We compute minimal zero-free regions for the Riemann zeta-function of the Littlewood form which ensure there is always a prime between consecutive perfect $k$th powers. Our computations cover powers $k\geq 65$ and quantify how far we are away from proving certain milestones toward an infamous open problem (Legendre's conjecture). In addition, we prove there is always a prime between consecutive perfect $86$th powers, and identify an integer sequence (that is a subset of the positive integers) for which there is always a prime between consecutive $70$th powers.

A.~Einstein considered a nonsymmetric (0,2)-tensor $G=g+F$, where $g$ is a pseudo-Riemannian metric and $F\ne0$ is skew-symmetric, and a linear connection $\nabla$ with torsion $T$ such that $(\nabla_X\,G)(Y,Z)=-G(T(X,Y),Z)$. M. Prvanović (1995) obtained the explicit form of the Einstein connection of an almost Hermitian manifold. In this paper, first, we present the result above in coordinate-free form, and then extend it to almost contact metric mani\-folds satisfying the $f^2$-torsion condition $T(f^2X, Y) = T(X,f^2Y) = f^2 T(X,Y)$, where $F(X,Y)=g(X,fY)$. We then derive the Einstein connection of nonsymmetric pseudo-Riemannian, in particular, weak almost Hermitian manifolds $(M,f,g)$, satisfying the $f^2$-torsion condition, give explicit formulas for the torsion in terms of $\nabla^g F$, $dF$ and a new (1,1)-tensor $\widetilde Q=-f^2-{I}$, and show that in the almost Hermitian case, our results reduce to the coordinate-free form of Prvanović's solution. Finally, we discuss special Einstein connections, and indicate the Gray-Hervella classes. Illustrative examples are given, including the construction of a weighted product.

By work of Davison and Meinhardt, the cohomological Hall algebra of a symmetric quiver with potential admits a geometrically defined filtration (the perverse filtration) whose associated graded is a supercommutative algebra. In the case of the triple quiver of a quiver with the canonical cubic potential, which corresponds to the preprojective algebra of the quiver via dimensional reduction, there is an additional filtration (the less perverse filtration), which is defined more generally for cohomological Hall algebras of suitably geometric $2$-Calabi-Yau categories in work of Davison. In this paper, we show that the degenerations of the cohomological Hall algebras of preprojective algebras and more generally $2$-Calabi-Yau categories with respect to the less perverse filtration is isomorphic to the enveloping algebra of the current Lie algebra of the BPS Lie algebra. This result applies in particular to CoHAs of local systems on Riemann surfaces and Higgs bundles on smooth projective curves. We extend this description to deformations of the cohomological Hall algebra obtained via torus actions on the arrows of the quiver and deformed canonical cubic potentials via the deformed dimensional reduction of Davison-P\u adurariu. We prove all our results at the level of sheafified CoHAs, which allows us to deduce similar statements for all versions of nilpotent CoHAs. Last, we use our results to compare the less perverse filtration on CoHAs of preprojective algebras with the order filtration on the Maulik-Okounkov Yangian, via the comparison isomorphism of Botta-Davison and Schiffmann-Vasserot.

We study Bakry-Emery curvature for fractional Laplacian generators using a Fourier representation of the carré du champ operator. For the stable generator of order gamma, the associated kernel on same-sign frequencies coincides with the covariance kernel of fractional Brownian motion with Hurst parameter equal to gamma divided by two. This observation allows the curvature inequality to be reformulated as a generalized eigenvalue problem for covariance matrices. On the one dimensional torus we analyze this matrix formulation for trigonometric polynomials. In the Cauchy case (gamma equal to one), corresponding to Brownian covariance, the eigenstructure can be computed explicitly and yields a Bakry-Emery curvature bound on the corresponding Fourier subspaces. We also study the effect of adding a confining drift to the Cauchy generator and show that the curvature spectrum undergoes a simple scalar shift. These results provide a matrix formulation of Bakry-Emery curvature for certain nonlocal operators and highlight a structural connection between fractional Laplacians and fractional Brownian covariance kernels.

We study visibility from a fixed point in the presence of a Poisson process of $\lambda$--geodesic hyperplanes in a $d$-dimensional hyperbolic space. The family of $\lambda$--geodesic hyperplanes interpolates between totally geodesic hyperplanes and horospheres. Our main result establishes a universality principle for this model: we prove that the fundamental visibility properties are invariant with respect to the parameter $\lambda\in[0,1]$. Namely, there is a critical intensity $\gamma_{\mathrm{crit}}>0$ such that the visible region is unbounded with positive probability for $\gamma < \gamma_{\mathrm{crit}}$ and almost surely bounded for $\gamma > \gamma_{\mathrm{crit}}$. For $d=2$ we establish almost sure boundedness also at criticality. The value for $\gamma_{\mathrm{crit}}$ is explicit and does not depend on $\lambda$. In the bounded phase, we show that the mean visible volume is identical with the known formula for $\lambda=0$. The key integral-geometric step is an explicit computation showing that the measure of $\lambda$-geodesic hyperplanes hitting a geodesic segment is a linear function of the length of the segment, independent of~$\lambda$.

We prove concentration bounds for random Euclidean combinatorial optimization problems with $p$--costs. For bipartite matching and for the (mono- and bi-partite) traveling salesperson problem in dimension $d\ge 3$, we obtain concentration at the natural energy scale $n^{1-p/d}$ for $1\le p<d^2/2$. Our method combines a Poincaré inequality with a robust geometric mechanism providing uniform bounds on the edges of optimizers. We also formulate a conjectural $p\!\to\!q$ transfer principle for the $p$--optimal matching which, if true, would extend the concentration range to all $p\ge 1$.

We verify the local analogue of Jiang's conjecture for the upper bound of the geometric wavefront sets of Arthur type representations of split classical $p$-adic groups with $p\gg 0$, under a certain condition. As a consequence, we also obtain the upper bound conjecture of Kim and the second author, and Hazeltine--Liu--Lo--Shahidi, under the same assumptions. The proof uses Waldspurger's work on the endoscopic transfer supplemented by results of Konno and Varma, as well as the wavefront set computations in the unipotent case by Mason-Brown--Okada and the second author.

We prove the following generalization of a well-known result of Duffin and Schaeffer: For any given countable sets $Y \subset\mathbb{R}$ and $Z\subset\mathbb{R}\setminus\operatorname{span}_\mathbb{Q}(\{1\}\cup Y)$, there exist functions $\psi$ such that the set of inhomogeneously $\psi$-approximable numbers has zero measure or full measure, according as the inhomogeneous parameter lies in $Y$ or $Z$. The proof uses an analogue of residue systems where the residues can take arbitrary real values, and it also requires information about the distribution of primes lying in Bohr sets. We extend a theorem of Rogers to the more general real residues setting, and we extend Dirichlet's theorem for prime numbers lying in arithmetic progressions to prime numbers lying in Bohr sets. We also prove that circle rotations equidistribute when sampled along such primes, provided the rotation angle is rationally independent of the Bohr set parameter, generalizing a theorem of Vinogradov. An appendix by Manuel Hauke answers a combinatorial question that is posed in the introduction.

Let X be an irreducible complex affine algebraic variety defined over $\mathbb{R}$, equipped with a faithful action of a finite group G, and let Y = X // G denote the categorical quotient with projection $\pi$. We study the geometry of the real image $L = \pi(X(\mathbb{R})) \subset Y(\mathbb{R})$ and its consequences for G-invariant optimization. Equipping $Y(\mathbb{R})$ with the measure induced by a G-invariant metric on X, we prove that the relative volume of L in $Y(\mathbb{R})$ equals $(\#\mathrm{Inv}(G))^{-1}$, where $\mathrm{Inv}(G)$ is the set of involutions of G. For the symmetric group $S_n$ acting on $\mathbb{R}^n$, this ratio decays super-exponentially in n. In particular, L is metrically rare within the ambient real quotient. We apply this result to two phenomena observed in G-invariant optimization problems: Regime I (Rarity of asymmetric critical points). The super-exponential decay of the volume of L renders the interior $L^\circ$ statistically negligible as a locus for critical points. This geometric rarity provides a rationale for the observed prevalence of symmetry: generic critical points are constrained to the boundary strata of L, corresponding to orbits with non-trivial stabilizers. Regime II (Energetic ordering by symmetry). We formulate the Active Constraint hypothesis: due to the metric rarity of the real image L, the landscape is dominated by a global gradient that drives the deepest descent trajectories toward the boundary of L. This global gradient directs the global minimum into the high-codimension strata of the boundary -- corresponding to large stabilizers -- thereby establishing a structural link between low energy and non-trivial stabilizers. This mechanism rationalizes the funnel topography of Lennard-Jones clusters, where the system is funneled into a crystallized ground state.

Given a graph $F$, the expansion $F^{(r)}$ of $F$ is defined as the $r$-uniform hypergraph obtained from $F$ by adding a set of $(r-2)$ distinct new vertices to each edge of $F$. In this paper, we investigate spectral stability results for hypergraphs and their this http URL first establish a spectral stability property: for any $r$-uniform hypergraph containing no copy of the expansion $F^{(r)}$ of a $(k+1)$-chromatic graph $F$, if its $p$-spectral is close to the extremal value, then the hypergraph is structurally close to $T_r(n, k)$, the complete $k$-partite $r$-uniform hypergraph on $n$ vertices where sizes of any two parts differ by at most this http URL this spectral stability result, we determine the unique extremal hypergraph that maximizes the $p$-spectral radius among all $n$-vertex $r$-uniform hypergraphs without $t$ vertex-disjoint copies of the expansion $K_{k+1}^{(r)}$ of $K_{k+1}$. We prove that this extremal hypergraph is isomorphic to $K_{t-1}^{r} \,\vee\, T_r(n-t+1, k)$, the join of the complete $r$-uniform hypergraph $K_{t-1}^{r}$ and $T_r(n-t+1, k)$.As a corollary, we show that $K_{t-1}^{r} \,\vee\, T_r(n-t+1, k)$ is the unique extremal hypergraph for $tK_{k+1}^{(r)}$, which extends a result of Pikhurko [J. Combin. Theory Ser. B, 103 (2013) 220--225] for expanded complete graphs.

In this paper we give a construction for a linear quotient ordering of a class of products of two ideals which have linear quotients. We apply this construction to give a class of modified anticycle graphs whose square and cube have linear quotients.

Duflo--Serganova functors play an important role in the representation theory of Lie superalgebra. While it is desirable to understand the images of modules under DS, little is known beyond finite dimensional representations. For general linear Lie superalgebras, Brundan--Goodwin observed that certain parabolic induction functors are compatible with the principal Whittaker coinvariant functor $H_0$. In this paper we show that, for certain rank-one DS functors, we explicitly compute the image of these parabolically induced modules under DS explicitly. This class of module includes many infinite dimensional representations such as $\mathfrak b$-Verma supermodules (attached to a suitable Borel subal gebra $\mathfrak b$) and a family of modules whose $H_0$-images are tensor products of irreducible evaluation modules for the super Yangian.

We provide a unified way to calculate the Gromov norm of the Kähler class of all (compact manifolds uniformized by) bounded symmetric domains. This was done for three classical domains by Domin and Toledo and for the general case by Clerc and Ørsted. Here, the calculation is much simplified by a combination of the ideas in Domin-Toledo and a work of Toledo, with the help of the Polydisc Theorem. The equality is achieved if and only if the triangle is ideal with three vertices on the Shilov boundary.

The quasi-Monte Carlo method is widely used in computational finance, whose efficiency strongly depends on the smoothness and effective dimension of the integrand. In this work, we investigate the combination of importance sampling and the active subspace method under the quasi-Monte Carlo framework and propose a three-step approach, referred to as the IS-AS-preintegration method, which sequentially applies importance sampling, active subspace, and preintegration. The proposed method is applied to the option pricing and sensitivity analysis problems in finance, and its performance is evaluated through extensive numerical experiments. The results demonstrate that the proposed method is highly competitive compared with existing popular methods. In particular, for out-of-the-money and deep out-of-the-money options, the proposed approach overcomes the limitations of the preintegration via active subspace method and achieves superior variance reduction, while maintaining comparable performance for other moneyness cases.

Mathematical modeling plays a vital role in epidemiology, offering insights into the spread and control of infectious diseases. The compartmental models developed by Kermack and McKendrick, particularly the SI (Susceptible-Infected) and SIR (Susceptible-Infected-Recovered) models, form the basis of many epidemic studies. While some simple cases permit analytical solutions, most real-world models require numerical methods such as Euler's method, the fourth-order Runge-Kutta (RK4) method, and Predictor-Corrector (P-C) methods. These methods are typically implemented in scientific computing software like Python, MATLAB, and R. However, the computational efficiency and run-time performance of these software tools in solving epidemiological models have not been comprehensively compared in the literature. This study addresses this gap by solving the SI and SIR models using Euler's method, RK4, and P-C methods in Python, MATLAB, and R. Execution times are recorded for each implementation to evaluate computational efficiency. Additionally, for the SI model, where an exact analytical solution exists, R2 values are computed to assess numerical accuracy. For the SIR model, a high-accuracy reference solution is obtained by solving the system using MATLAB's ODE45 solver, and the SIR solutions computed via the RK4 method in MATLAB are compared against this reference. The results provide a comparative perspective on the accuracy and run-time performance across different software and numerical methods, offering practical guidance for researchers and practitioners in selecting suitable tools for epidemic modeling.

The integration of storage and renewable resources fundamentally alters resource-adequacy analysis. Because storage couples decisions across time, it invalidates the traditional reliability models that are based on time-independent capacity demand curves. Moreover, renewables introduce temporally correlated intermittency. To address this, we formulate the capacity procurement problem as a two-stage stochastic program, where the capacity decision is made in the first stage, while the expected unserved energy is evaluated by a second-stage dispatch problem that considers uncertainties such as generator failures via Markov chains, temporally correlated renewable output, and stochastic load. We implement the resulting stochastic capacity procurement (SCP) model on a New England system with 305 generators, including conventional, renewable, and storage units. Using the stochastic decomposition (SD) algorithm, we solve the SCP with up to 20,000 Monte Carlo samples, each representing a six-month trajectory of more than 4,300 hours of uncertainty across all units. We analyze the convergence behavior of SD and show that convergence for the stochastic program happens faster than reliable estimation of the reliability metrics, which require more samples than are used in typical stochastic programs. These results show that chronologically detailed Monte Carlo sampling can be integrated into capacity procurement optimization in a computationally tractable manner, enabling reliability evaluation with controlled statistical accuracy at realistic system scales.

Let $C$ be a smooth complex projective curve of genus $g>1$ and $A=J(C)$ its Jacobian with principal polarization $\Theta$. Starting from a semistable vector bundle $V$ on $C$ with $\mu(V)>2g-2$, we consider the Fourier--Mukai transform $E=\Phi_{\mathcal P}(a_*V)$. We prove that $E(\Theta)$ satisfies the $\mathrm{IT}_0$ property. As a consequence, for every $m\ge g+1$, the polarized Jacobian $(A,m\Theta)$ admits Ulrich bundles constructed functorially from $V$. Further we analyze stability and Chern classes of the resulting bundles. We show that the construction induces a natural morphism from the generically finite cover of moduli space of stable bundles on $C$ to the moduli space of stable bundles on $A$, producing positive-dimensional families of stable Ulrich bundles.

Despite the success of large language models (LLMs) across domains, their potential for efficient channel state information (CSI) compression and feedback in frequency division duplex (FDD) massive multiple-input multiple-output (mMIMO) systems remains largely unexplored yet increasingly important. In this paper, we propose a novel LLM-based framework for CSI feedback to exploit the potential of LLMs. We first reformulate the CSI compression feedback task as a masked token prediction task that aligns more closely with the functionality of LLMs. Subsequently, we design an information-theoretic mask selection strategy based on self-information, identifying and selecting CSI elements with the highest self-information at the user equipment (UE) for feedback. This ensures that masked tokens correspond to elements with lower self-information, while visible tokens correspond to elements with higher self-information, thus maximizing the accuracy of LLM predictions.

A fundamental challenge in diagnostic imaging is the phenomenon of topological equivalence, where benign and malignant structures share global topology but differ in critical geometric detail, leading to diagnostic errors in both conventional and deep learning models. We introduce GeoTop, a mathematically principled framework that unifies Topological Data Analysis (TDA) and Lipschitz-Killing Curvatures (LKCs) to resolve this ambiguity. Unlike hybrid deep learning approaches, GeoTop provides intrinsic interpretability by fusing the capacity of persistent homology to identify robust topological signatures with the precision of LKCs in quantifying local geometric features such as boundary complexity and surface regularity. The framework's clinical utility is demonstrated through its application to skin lesion classification, where it achieves a consistent accuracy improvement of 3.6% and reduces false positives and negatives by 15-18% compared to conventional single-modality methods. Crucially, GeoTop directly addresses the problem of topological equivalence by incorporating geometric differentiators, providing both theoretical guarantees (via a formal lemma) and empirical validation via controlled benchmarks. Beyond its predictive performance, GeoTop offers inherent mathematical interpretability through persistence diagrams and curvature-based descriptors, computational efficiency for large datasets (processing 224x224 pixel images in less or equal 0.5 s), and demonstrated generalisability to molecular-level data. By unifying topological invariance with geometric sensitivity, GeoTop provides a principled, interpretable solution for advanced shape discrimination in diagnostic imaging.

We study the problem of hypothesis selection under the constraint of local differential privacy. Given a class $\mathcal{F}$ of $k$ distributions and a set of i.i.d. samples from an unknown distribution $h$, the goal of hypothesis selection is to pick a distribution $\hat{f}$ whose total variation distance to $h$ is comparable with the best distribution in $\mathcal{F}$ (with high probability). We devise an $\varepsilon$-locally-differentially-private ($\varepsilon$-LDP) algorithm that uses $\Theta\left(\frac{k}{\alpha^2\min \{\varepsilon^2,1\}}\right)$ samples to guarantee that $d_{TV}(h,\hat{f})\leq \alpha + 9 \min_{f\in \mathcal{F}}d_{TV}(h,f)$ with high probability. This sample complexity is optimal for $\varepsilon<1$, matching the lower bound of Gopi et al. (2020). All previously known algorithms for this problem required $\Omega\left(\frac{k\log k}{\alpha^2\min \{ \varepsilon^2 ,1\}} \right)$ samples to work. Moreover, our result demonstrates the power of interaction for $\varepsilon$-LDP hypothesis selection. Namely, it breaks the known lower bound of $\Omega\left(\frac{k\log k}{\alpha^2\min \{ \varepsilon^2 ,1\}} \right)$ for the sample complexity of non-interactive hypothesis selection. Our algorithm breaks this barrier using only $\Theta(\log \log k)$ rounds of interaction. To prove our results, we define the notion of \emph{critical queries} for a Statistical Query Algorithm (SQA) which may be of independent interest. Informally, an SQA is said to use a small number of critical queries if its success relies on the accuracy of only a small number of queries it asks. We then design an LDP algorithm that uses a smaller number of critical queries.

The classical theory of efficient allocations of an aggregate endowment in a pure-exchange economy has hitherto primarily focused on the Pareto-efficiency of allocations, under the implicit assumption that transfers between agents are frictionless, and hence costless to the economy. In this paper, we argue that certain transfers cause frictions that result in costs to the economy. We show that these frictional costs are tantamount to a form of subadditivity of the cost of transferring endowments between agents. We suggest an axiomatic study of allocation mechanisms, that is, the mechanisms that transform feasible allocations into other feasible allocations, in the presence of such transfer costs. Among other results, we provide an axiomatic characterization of those allocation mechanisms that admit representations as robust (worst-case) linear allocation mechanisms, as well as those mechanisms that admit representations as worst-case conditional expectations. We call the latter Robust Conditional Mean Allocation mechanisms, and we relate our results to the literature on (decentralized) risk sharing within a pool of agents.

Insufficient reference data is ubiquitous in data-driven computational fluid dynamics, as it is usually too expensive to compute or impossible to observe over long enough times needed for data-driven methods. The lack of data can significantly compromise the fidelity of results computed with data-driven methods or render them inapplicable. To challenge this problem, we propose a probabilistic reconstruction method that enhances the hyper-parameterisation (HP) approach with ideas underlying the probabilistic-evolutionary approach. We offer to use the HP method ``Advection of the image point'' on data sampled from the joint probability distribution of the reference dataset. The HP method has been tested regionally on the sea surface temperature and surface relative vorticity computed with the global 1/4-deg and 1/12-deg resolution NEMO model. Our results show that the HP solution (the solution computed with the HP method) in the probabilistically-reconstructed and reduced (in terms of dimensionality) phase space at 1/4-deg resolution is more accurate than the 1/4-deg-solution computed with NEMO. Additionally, the HP solution is several orders of magnitude faster to compute than the 1/4-deg NEMO solution. The proposed method shows encouraging results for the NEMO model and the potential for the use in other operational ocean and ocean-atmospheric models for both deterministic and probabilistic predictions. Furthermore, the method can be used as a fast reanalysis tool allowing the complex dynamics of a comprehensive ocean model to be approximated by the HP solution. It can also function as a dynamic interpolation method to fill gaps in observational data.

Tracking the solution of time-varying variational inequalities is an important problem with applications in game theory, optimization, and machine learning. Existing work considers time-varying games or time-varying optimization problems. For strongly convex optimization problems or strongly monotone games, these results provide tracking guarantees under the assumption that the variation of the time-varying problem is restrained, that is, problems with a sublinear solution path. In this work we extend existing results in two ways: In our first result, we provide tracking bounds for (1) variational inequalities with a sublinear solution path but not necessarily monotone functions, and (2) for periodic time-varying variational inequalities that do not necessarily have a sublinear solution path-length. Our second main contribution is an extensive study of the convergence behavior and trajectory of discrete dynamical systems of periodic time-varying VI. We show that these systems can exhibit provably chaotic behavior or can converge to the solution. Finally, we illustrate our theoretical results with experiments.

We argue that a general system of scalar fields and gauge fields manifesting vacuum degeneracy induces a principal groupoid bundle over spacetime and that the pattern of spontaneous symmetry breaking and the Higgs mechanism are encoded by the singular foliation canonically induced on the moduli space of scalar vacuum expectation values by the Lie groupoid structure. Recent mathematical results in the classification of singular foliations then provide a qualitative classification of the possible patterns of vacuum degeneracy.

Quantum state exclusion is an operational task with application to ontological interpretations of quantum states. In such a task, one is given a system whose state is randomly selected from a finite set, and the goal is to identify a state from the set that is not the true state of the system. An error occurs if and only if the state identified is the true state. In this paper, we study the optimal error probability of quantum state exclusion and its error exponent from an information-theoretic perspective. Our main finding is a single-letter upper bound on the error exponent of state exclusion given by the multivariate log-Euclidean Chernoff divergence, and we prove that this improves upon the best previously known upper bound. We also extend our analysis to quantum channel exclusion, and we establish a single-letter and efficiently computable upper bound on its error exponent, admitting the use of adaptive strategies. We derive both upper bounds, for state and channel exclusion, based on one-shot analysis and formulate them as a type of multivariate divergence measure called a barycentric Chernoff divergence. Moreover, our result on channel exclusion has implications in two important special cases. First, when there are two hypotheses, our result provides the first known efficiently computable upper bound on the error exponent of symmetric binary channel discrimination. Second, when all channels are classical, we show that our upper bound is achievable by a parallel strategy, thus solving the exact error exponent of classical channel exclusion.

The Reynolds equation from lubrication theory and the Stokes equations for zero Reynolds number flows are distinct models for an incompressible fluid with negligible inertia. Here we investigate the sensitivity of the Reynolds equation to large surface gradients, and explore flow recirculation in corner geometries in comparison to the Stokes equation. We compare the solutions for the Reynolds and Stokes equations in the backward facing step (BFS), the regularized BFS, and the lid-driven triangular cavity. For the BFS variations listed above, we compute the error in terms of the average pressure drop through the channel and show how the error increases with increasing expansion ratio and with increasing magnitude of surface gradients. We further investigate the phenomenology of corner flow recirculation that arises in the Stokes solutions. In particular, we observe that occluding the corner separated region in the Stokes solution to the BFS does not disrupt the bulk flow characteristics.

This article introduces Levy-driven graph supOU processes, a parsimonious parametrisation for high-dimensional time series in which dependence between components is governed by a graph structure. Specifically, the model bridges short- and long-range dependence within a single parametric family while accommodating a wide range of marginal distributions. We further develop a generalised method of moments estimator, establish its consistency and asymptotic normality, and assess its finite-sample performance through a simulation study. Finally, we illustrate the practical relevance of our model and estimation method in an empirical study of wind capacity factors in a European electricity network context.

We study best-arm identification in stochastic multi-armed bandits under the fixed-confidence setting, focusing on instances with multiple optimal arms. Unlike prior work that addresses the unknown-number-of-optimal-arms case, we consider the setting where the number of optimal arms is known in advance. We derive a new information-theoretic lower bound on the expected sample complexity that leverages this structural knowledge and is strictly tighter than previous bounds. Building on the Track-and-Stop algorithm, we propose a modified, tie-aware stopping rule and prove that it achieves asymptotic instance-optimality, matching the new lower bound. Our results provide the first formal guarantee of optimality for Track-and-Stop in multi-optimal settings with known cardinality, offering both theoretical insights and practical guidance for efficiently identifying any optimal arm.

We study the convergence properties and escape dynamics of Stochastic Gradient Descent (SGD) in one-dimensional landscapes, separately considering infinite- and finite-variance noise. Our main focus is to identify the time scales on which SGD reliably moves from an initial point to the local minimum in the same ''basin''. Under suitable conditions on the noise distribution, we prove that SGD converges to the basin's minimum unless the initial point lies too close to a local maximum. In that near-maximum scenario, we show that SGD can linger for a long time in its neighborhood. For initial points near a ''sharp'' maximum, we show that SGD does not remain stuck there, and we provide results to estimate the probability that it will reach each of the two neighboring minima. Overall, our findings present a nuanced view of SGD's transitions between local maxima and minima, influenced by both noise characteristics and the underlying function geometry.

We study Nash equilibrium problems with mixed-integer variables in which each player solves a mixed-integer optimization problem parameterized by the rivals' strategies. We distinguish between standard Nash equilibrium problems (NEPs), where parameterization affects only the objective functions, and generalized Nash equilibrium problems (GNEPs), where strategy sets may additionally depend on rivals' strategies. We introduce a branch-and-cut (B&C) algorithm for such mixed-integer games that, upon termination, either computes a pure Nash equilibrium or decides their non-existence. Our approach reformulates the game as a bilevel problem using the Nikaido--Isoda function. We then use bilevel-optimization techniques to get a computationally tractable relaxation of this reformulation and embed it into a B&C framework. We derive sufficient conditions for the existence of suitable cuts and finite termination of our method depending on the setting. For GNEPs, we adapt the idea of intersection cuts from bilevel optimization and mixed-integer linear optimization. We can guarantee the existence of such cuts under suitable assumptions, which are particularly fulfilled for pure-integer GNEPs with decoupled concave objectives and linear coupling constraints. For NEPs, we show that suitable cuts always exist via best-response inequalities and prove that our B&C method terminates in finite time whenever the set of best-response sets is finite. We show that this condition is fulfilled for the important special cases of (i) players' cost functions being concave in their own continuous strategies and (ii) the players' cost functions only depending on their own strategy and the rivals' integer strategy components. Finally, we present preliminary numerical results for two different types of knapsack games, a game based on capacitated flow problems, and integer NEPs with quadratic objectives.

We propose a new Bayesian approach to generalize the federated Alternating Direction Method of Multipliers (ADMM). We show that the solutions of variational-Bayesian (VB) objectives are associated with a duality structure that not only resembles the structure of ADMM's fixed-points but also generalizes it. For example, ADMM-like updates are recovered when the VB objective is optimized over the isotropic-Gaussian family, and new non-trivial extensions are obtained for other exponential-family distributions. These extensions include a Newton-like variant that converges in one step on quadratic objectives and an Adam-like variant that yields up to 7% accuracy boosts for deep heterogeneous cases. Our work opens a new Bayesian way to generalize ADMM and other primal-dual methods.

In this work, we propose Fast Equivariant Imaging (FEI), a novel unsupervised learning framework to rapidly and efficiently train deep imaging networks without ground-truth data. From the perspective of reformulating the Equivariant Imaging based optimization problem via the method of Lagrange multipliers and utilizing plug-and-play denoisers, this novel unsupervised scheme shows superior efficiency and performance compared to the vanilla Equivariant Imaging paradigm. In particular, our FEI schemes achieve an order-of-magnitude (10x) acceleration over standard EI on training U-Net for X-ray CT reconstruction and image inpainting, with improved generalization performance. In addition, the proposed scheme enables efficient test-time adaptation of a pretrained model to individual samples to secure further performance improvements. Extensive experiments show that the proposed approach provides a noticeable efficiency and performance gain over existing unsupervised methods and model adaptation techniques.

We study the Finite-Dimensional Distributions (FDDs) of deep neural networks with randomly initialized weights that have finite-order moments. Specifically, we establish Gaussian approximation bounds in the Wasserstein-$1$ norm between the FDDs and their Gaussian limit assuming a Lipschitz activation function and allowing the layer widths to grow to infinity at arbitrary relative rates. In the special case where all widths are proportional to a common scale parameter $n$ and there are $L-1$ hidden layers, we obtain convergence rates of order $n^{-({1}/{6})^{L-1} + \epsilon}$, for any $\epsilon > 0$.

In this study, we propose a robust control strategy for a counter-current heat exchanger. The primary objective is to regulate the outlet temperature of one fluid stream by manipulating the flow rate of the second counter-current fluid stream. By leveraging the energy balance equations, we develop a structured bilinear system model derived by using a uniform spatial discretization of each stream into a cascade of homogeneous volumes and by considering the heat transfer and convective phenomena within the exchanger. We introduce two control strategies: (i) an output feedback controller incorporating a state observer and (ii) a purely integral control law. The effectiveness of the proposed control strategy is validated through real experiments on a real heat exchanger.

We study the resurgent structure of the topological string dual to 2d $U(N)$ Yang-Mills on torus. We find closed form formulas for instanton amplitudes up to arbitrarily high instanton orders, based on which we propose the non-perturbative partition function including contributions from all the real instantons, which is real for positive modulus and string coupling. We also explore complex instantons and find two infinite towers of them. We expect them to correspond to BPS states in type II string.

We propose an accurate, efficient, and low-memory sum-of-Gaussians tensor neural network (SOG-TNN) algorithm for solving the high-dimensional Schrödinger equation. The SOG-TNN utilizes a low-rank tensor product representation of the solution to overcome the curse of dimensionality associated with high-dimensional integration. To handle the Coulomb interaction, we introduce an SOG decomposition to approximate the interaction kernel such that it is dimensionally separable, leading to a tensor representation with rapid convergence. We further develop a range-splitting scheme that partitions the Gaussian terms into short-, long-, and mid-range components. They are treated with the asymptotic expansion, the low-rank Chebyshev expansion, and the model reduction with singular-value decomposition, respectively, significantly reducing the number of two-dimensional integrals in computing electron-electron interactions. The SOG decomposition well resolves the computational challenge due to the singularity of the Coulomb interaction, leading to an efficient algorithm for the high-dimensional problem under the TNN framework. Numerical results demonstrate the outstanding performance of the new method, revealing that the SOG-TNN is a promising way for accurately tackling quantum systems.

Quantum conditional entropies play a fundamental role in quantum information theory. In quantum key distribution, they are exploited to obtain reliable lower bounds on the secret-key rates in the finite-size regime, against collective attacks and coherent attacks under suitable assumptions. Here we consider continuous-variable communication protocols, where the sender Alice encodes information using a discrete modulation of phase-shifted coherent states, and the receiver Bob decodes by homodyne or heterodyne detection. We compute the Petz-Rényi and sandwiched Rényi conditional entropies associated with these setups, assuming either a passive eavesdropper or one that injects thermal photons into the channel, who gathers the quantum information leaked through a lossy communication line of known or bounded transmittance. Whereas our results do not directly provide reliable key-rate estimates, they do represent useful ball-park figures. We obtain analytical or semi-analytical expressions that do not require intensive numerical calculations. These expressions serve as bounds on the key rates that may be tight in certain scenarios. We compare different estimates, including known bounds that have already appeared in the literature and new bounds. The latter are found to be tighter for very short block sizes.

Clustering is a hard discrete optimization problem. Nonconvex approaches such as low-rank semidefinite programming (SDP) have recently demonstrated promising statistical and local algorithmic guarantees for cluster recovery. Due to the combinatorial structure of the $K$-means clustering problem, current relaxation algorithms struggle to balance their constraint feasibility and objective optimality, presenting tremendous challenges in computing the second-order critical points with rigorous guarantees. In this paper, we provide a new formulation of the $K$-means problem as a smooth unconstrained optimization over a submanifold and characterize its Riemannian structures to allow it to be solved using a second-order cubic-regularized Riemannian Newton algorithm. By factorizing the $K$-means manifold into a product manifold, we show how each Newton subproblem can be solved in linear time. Our numerical experiments show that the proposed method converges significantly faster than the state-of-the-art first-order nonnegative low-rank factorization method, while achieving similarly optimal statistical accuracy.

This paper studies the use of kernel density estimation (KDE) for linear algebraic tasks involving the kernel matrix of a collection of $n$ data points in $\mathbb R^d$. In particular, we improve upon existing algorithms for computing the following up to $(1+\varepsilon)$ relative error: matrix-vector products, matrix-matrix products, the spectral norm, and sum of all entries. The runtimes of our algorithms depend on the dimension $d$, the number of points $n$, and the target error $\varepsilon$. Importantly, the dependence on $n$ in each case is far lower when accessing the kernel matrix through KDE queries as opposed to reading individual entries. Our improvements over existing best algorithms (particularly those of Backurs, Indyk, Musco, and Wagner '21) for these tasks reduce the polynomial dependence on $\varepsilon$, and additionally decreases the dependence on $n$ in the case of computing the sum of all entries of the kernel matrix. We complement our upper bounds with several lower bounds for related problems, which provide (conditional) quadratic time hardness results and additionally hint at the limits of KDE based approaches for the problems we study.

Lubrication theory makes use of the assumptions of a long and thin fluid domain and a small scaled Reynolds number to formulate a linearized approximation to the Navier-Stokes equations. Extended lubrication theory aims to improve the model accuracy by relaxing these assumptions and including additional terms in the formulation. However, such models are sensitive to large surface gradients which lead the assumptions of the model to break down. In this paper, we present a formulation of extended lubrication theory, and compare our model with several existing models, along with the numerical solution to the Stokes equations. The error in pressure and velocity is characterized for a variety of fluid domain geometries. Our results indicate that the new solution is suitable for a wide range of geometries. The magnitude of surface variation and the length scale ratio are both important factors influencing the accuracy of the extended lubrication theory models.

Adam [Kingma & Ba, 2015] is the de facto optimizer in deep learning, yet its theoretical understanding remains limited. Prior analyses show that Adam favors solutions aligned with $\ell_\infty$-geometry, but these results are restricted to the full-batch regime. In this work, we study the implicit bias of incremental Adam (using one sample per step) for logistic regression on linearly separable data, and show that its bias can deviate from the full-batch behavior. As an extreme example, we construct datasets on which incremental Adam provably converges to the $\ell_2$-max-margin classifier, in contrast to the $\ell_\infty$-max-margin bias of full-batch Adam. For general datasets, we characterize its bias using a proxy algorithm for the $\beta_2 \to 1$ limit. This proxy maximizes a data-adaptive Mahalanobis-norm margin, whose associated covariance matrix is determined by a data-dependent dual fixed-point formulation. We further present concrete datasets where this bias reduces to the standard $\ell_2$- and $\ell_\infty$-max-margin classifiers. As a counterpoint, we prove that Signum [Bernstein et al., 2018] converges to the $\ell_\infty$-max-margin classifier for any batch size. Overall, our results highlight that the implicit bias of Adam crucially depends on both the batching scheme and the dataset, while Signum remains invariant.

The adiabatic theorem is one of the most interesting and significant theorems in quantum mechanics. However, the adiabatic theorem can fail for general non-Hermitian quantum systems. In this paper, by utilizing the complex geometric phase, the functional calculus for biorthogonal systems and the Grönwall inequality, we prove rigorously that the adiabatic theorem is still valid for diagonalizable non-Hermitian systems with real eigenvalues. The proof also justifies the definition of a complex Berry phase in non-Hermitian systems.

We establish the integrability of a family of Sachdev-Ye-Kitaev (SYK) models with uniform $p$-body interactions. We derive the R-matrix and mutually commuting transfer matrices that generate the Hamiltonians of these models, and obtain their exact eigenspectra and eigenstates. Remarkably, the R-matrix is that of the critical transverse-field Ising chain. This work reveals an unexpected connection between the SYK model, central to many-body quantum chaos, and the critical Ising chain, a cornerstone of statistical mechanics.

Wasserstein gradient flow provides a general framework for minimizing an energy functional $J$ over the space of probability measures on a Riemannian manifold $(M,g)$. Its canonical time-discretization, the Jordan-Kinderlehrer-Otto (JKO) scheme, produces for any step size $\eta>0$ a sequence of probability distributions $\rho_k^\eta$ that approximate to first order in $\eta$ Wasserstein gradient flow on $J$. But the JKO scheme also has many other remarkable properties not shared by other first order integrators, e.g. it preserves energy dissipation and exhibits unconditional stability for $\lambda$-geodesically convex functionals $J$. To better understand the JKO scheme we characterize its implicit bias at second order in $\eta$. We show that $\rho_k^\eta$ are approximated to order $\eta^2$ by Wasserstein gradient flow on a modified energy \[ J^{\eta}(\rho) = J(\rho) - \frac{\eta}{4}\int_M \Big\lVert \nabla_g \frac{\delta J}{\delta \rho} (\rho) \Big\rVert_{2}^{2} \,\rho(dx), \] obtained by subtracting from $J$ the squared metric curvature of $J$ times $\eta/4$. The JKO scheme therefore adds at second order in $\eta$ a deceleration in directions where the metric curvature of $J$ is rapidly changing. This corresponds to canonical implicit biases for common functionals: for entropy the implicit bias is the Fisher information, for KL-divergence it is the Fisher-Hyv{ä}rinen divergence, and for Riemannian gradient descent it is the kinetic energy in the metric $g$. To understand the differences between minimizing $J$ and $J^\eta$ we study JKO-Flow, Wasserstein gradient flow on $J^\eta$, in several simple numerical examples. These include exactly solvable Langevin dynamics on the Bures-Wasserstein space and Langevin sampling from a quartic potential in 1D.

Boundary-catalytic branching processes describe a broad class of natural phenomena where the population of diffusing particles grows due to their spontaneous binary branching (e.g., division, fission or splitting) on a catalytic boundary located in a complex environment. We investigate the possibility of the geometric control of the population growth by compensating the proliferation of particles due to catalytic branching events by their absorptions in the bulk or on absorbing regions of the boundary. We identify an appropriate Steklov spectral problem to obtain the phase diagram of this out-of-equilibrium stochastic process. The principal eigenvalue determines the critical line that separates an exponential growth of the population from its extinction in a bounded domain. In other words, we establish a powerful tool for calculating the growth-regulating absorption rate that equilibrates the opposite effects of branching and absorption events and thus results in steady-state behavior of this diffusion-reaction system. Moreover, we show the existence of a critical catalytic rate above which no compensation is possible, so that the population cannot be controlled and keeps growing exponentially. The proposed framework opens promising perspectives for better understanding, modeling and control of various boundary-catalytic branching processes, with applications in physics, chemistry, and life sciences.

There exist several different proposals for a measure in Quantum Gravity theories. Although sometimes being labelled as non covariant, the measure derived in [7] for GR has the particularity that, in the extremal, the volume divergences cancel out. The analogous for Quadratic Gravity [1]-[2] was considered in [44]-[45]. However, as far as the author knows, the issue of volume divergences was not considered for this last measure. The present work fills this gap and presents an analysis showing that, in the extremal, these divergences cancel as well. This is up to some subtleties related to superdeterminants. The possibility of employing non invariant measures may be accepted if the anomaly in the measure is compensated by counter term redefinitions of the model under analysis. This makes difficult to disprove, at the present times, some choices of measures. Quadratic Gravity[1]-[2], is known to be renormalizable in flat space, and there are a finite number of counter terms needed in order to renormalize its effective action. However, around a curved space this is not known, and this complicates considerably the analysis. These issues are reviewed in the text, together with an analysis of covariant measures. In particular, it is shown how these measures [47]-[49] can be found if one condition in [7] is relaxed.

PANDA is a powerful generic algorithm for answering conjunctive queries (CQs) and disjunctive datalog rules (DDRs) given input degree constraints. In the special case where degree constraints are cardinality constraints and the query is Boolean, PANDA runs in $\tilde O (N^{subw})$-time, where $N$ is the input size, and $subw$ is the submodular width of the query, a notion introduced by Daniel Marx (JACM 2013). When specialized to certain classes of sub-graph pattern finding problems, the $\tilde O(N^{subw})$ runtime matches the optimal runtime possible, modulo some conjectures in fine-grained complexity (Bringmann and Gorbachev (STOC 25)). The PANDA framework is much more general, as it handles arbitrary input degree constraints, which capture common statistics and integrity constraints used in relational database management systems, it works for queries with free variables, and for both CQs and DDRs. The key weakness of PANDA is the large $polylog(N)$-factor hidden in the $\tilde O(\cdot)$ notation. This makes PANDA completely impractical, and fall short of what is achievable with specialized algorithms. This paper resolves this weakness with two novel ideas. First, we prove a new probabilistic inequality that upper-bounds the output size of DDRs under arbitrary degree constraints. Second, the proof of this inequality directly leads to a new algorithm named PANDAExpress that is both simpler and faster than PANDA. The novel feature of PANDAExpress is a new partitioning scheme that uses arbitrary hyperplane cuts instead of axis-parallel hyperplanes used in PANDA. These hyperplanes are dynamically constructed based on data-skewness statistics carefully tracked throughout the algorithm's execution. As a result, PANDAExpress removes the $polylog(N)$-factor from the runtime of PANDA, matching the runtimes of intricate specialized algorithms, while retaining all its generality and power.

Building on the earlier work by Araki and Tanii, Aschieri et al., and Buratti et al., we demonstrate that every model for self-dual nonlinear electrodynamics in four dimensions has a $\mathsf{U}(1)$ duality-invariant extension to $4p>4$ dimensions and construct new self-dual nonlinear theories for a gauge $(2p-1)$-form. We present a family of models for self-dual $(2p-1)$-form electrodynamics in which the trace of the energy-momentum tensor determines the flow with respect to a duality-invariant deformation parameter.

Partial Differential Equations are precise in modelling the physical, biological and graphical phenomena. However, the numerical methods suffer from the curse of dimensionality, high computation costs and domain-specific discretization. We aim to explore pros and cons of different PDE solvers, and apply them to specific scientific simulation problems, including forwarding solution, inverse problems and equations discovery. In particular, we extend the recent CNF (NeurIPS 2023) framework solver to coupled and non-linear settings, together with down-stream applications. The outcomes include implementation of selected methods, self-tuning techniques, evaluation on benchmark problems and a comprehensive survey of neural PDE solvers and scientific simulation applications.

The Lorentz mirror model provides a clean setting to study macroscopic transport generated solely by quenched environmental randomness. We introduce a hierarchical version whose distribution of left--right crossings satisfies an exact recursion. In dimensions $d\geq 3$, we prove normal transport: the mean conductance scales as (cross-section)/(length) on all length scales. A Gaussian closure, supported by numerics, predicts that the variance-to-mean ratio of the conductance converges to the universal value $2/3$ for all $d\geq 2$ (the ``$2/3$ law''). We provide numerical evidence for the $2/3$ law in the original (non-hierarchical) Lorentz mirror model in $d=3$, and conjecture that it is a universal signature of normal transport induced by random current matching. In the marginal case $d=2$, our hierarchical recursion reproduces the known scaling of the mean conductance and its variance. A YouTube video discussing the background and the main results of the paper is available: this https URL

We present the exact analytic solution of the Makeenko-Migdal loop equations, solving planar QCD ($N_c \to \infty$) in the continuum limit. Quantizing internal Majorana fermions (elves) on a rigid Hodge-dual minimal surface provides the algebraic mechanism satisfying the unintegrated vector loop equations. The Pauli principle exactly cancels non-planar intersections to reproduce planar factorization, while iterating the equation generates the planar graphs of asymptotically free QCD. Holographically fixing the bulk geometry by the boundary loop strictly avoids Liouville instability. Momentum loop space integrates out coordinate-space cusp singularities, yielding a finite local limit. Gauge-fixing the Virasoro constraint parametrizes the reduced phase-space measure by boundary twistors. The theory reduces to a confining analytic twistor string: a boundary sigma model $S^1 \to (S^3 \times S^3)/S^1$ coupled to a holographically determined Liouville field. The meson spectrum becomes a 1D functional integral over boundary twistor trajectories. Complexified action monodromies reveal a discrete mass spectrum governed by Catastrophe Theory, classified by the topological number of twistor poles inside the unit circle. This geometric localization is an infinite-dimensional realization of Exact WKB analysis and Picard-Lefschetz resurgence, where twistor poles act identically to Seiberg-Witten branch points. The 1-pole sector yields the exact Regge spectrum $m^2 = \frac{\pi\sigma}{2}(n + 1/24)$, matching experimental $\pi, K, \rho$ trajectories within $95\%$ confidence. The empirically correct open-string intercepts emerge directly from the conformal anomaly of the microscopic elves rather than macroscopic string vibrations. Ultimately, this explicitly realizes Witten's Master Field as a critical classical trajectory in twistor space.