Bitcoin derives a verifiable temporal order from probabilistic block discovery and cumulative proof-of-work rather than from a trusted global clock. We show that block arrivals exhibit stable exponential behavior across difficulty epochs, and that the proof-of-work process maintains a high-entropy search state that collapses discretely upon the discovery of a valid block. This entropy-based interpretation provides a mechanistic account of Bitcoin's non-continuous temporal structure. In a distributed network, however, entropy collapse is not completed instantaneously across all participants. Using empirical observations of temporary forks, we show that collapse completion unfolds over a finite propagation-bounded interval, while remaining rapid in practice.

This paper reformulates Fermat's Last Theorem as an embedding problem of information-geometric structures. We reinterpret the Fermat equation as an $n$-th moment constraint, constructing a statistical manifold $\mathcal{M}_n$ of generalized normal distributions via the Maximum Entropy Principle. By Chentsov's Theorem, the natural metric is the Fisher information metric ($L^2$); however, the global structure is governed by the $L^n$ moment constraint. This reveals a discrepancy between the local quadratic metric and the global $L^n$ structure. We axiomatically define an "Information-Geometric Fermat Solution," postulating that the lattice structure must maintain "dual lattice consistency" under the Legendre transform. We prove the non-existence of such structures for $n \ge 3$. Through the Poisson Summation Formula and Hausdorff-Young Inequality, we demonstrate that the Fourier transform induces an alteration of the function family ($L^n \to L^q$, where $1/n + 1/q = 1$), rendering dual lattice consistency analytically impossible. This identifies a geometric obstruction where integer and energy structures are incompatible within a dually flat space. We conclude by discussing the correspondence between this model and elliptic curves.

We study the Jensen--Shannon divergence (JSD) between transcript distributions induced by neighboring datasets in the shuffle model when each user applies a fixed local randomizer and a trusted shuffler releases the output histogram. Under a mild positivity assumption, we prove an explicit two-term asymptotic expansion where the leading term is chi-squared divergence divided by 8n. Binary randomized response and k-ary randomized response follow as corollaries. For multi-message protocols based on independent repetition, the leading coefficient becomes (1 + chi-squared)^m - 1. A fully explicit remainder control is provided in the appendix.

Let $q=p^k$ be a prime power, let $n\geq2$ be an integer and let $\mathbb{F}_{q^n}$ be a finite field. It is shown that the set of primitive normal elements is a Salem set. Furthermore, it is proved that this set is strongly equidistributed in the finite field. Similar results are proved for the set of quadratic residues and the set of primitive roots modulo a large prime $p\geq 3$.

This paper considers the scattering of a time-harmonic acoustic plane wave by an elastic body with an unbounded periodic surface. The original problem can be confined to the analysis of the fields in one periodic cell. With the help of the perfectly matched layer (PML) technique, we can truncate the unbounded physical domain into a bounded computational domain. By respectively constructing the equivalent transparent boundary conditions of acoustic and elastic waves simultaneously, the well-posedness and exponential convergence of the solution to the associated truncated PML problem are established. The finite element method is applied to solve the PML problem of acoustic-elastic interaction. To address the singularity caused by the non-smooth surface of the elastic body, we establish a residual-type a posteriori error estimate and develop an adaptive PML finite element algorithm. Several numerical examples are presented to demonstrate the effectiveness of the proposed adaptive algorithm.

This paper introduces a novel optimization algorithm designed for nonlinear least-squares problems. The method is derived by preconditioning the gradient descent direction using the Singular Value Decomposition (SVD) of the Jacobian. This SVD-based preconditioner is then integrated with the first- and second-moment adaptive learning rate mechanism of the Adam optimizer. We establish the local linear convergence of the proposed method under standard regularity assumptions and prove global convergence for a modified version of the algorithm under suitable conditions. The effectiveness of the approach is demonstrated experimentally across a range of tasks, including function approximation, partial differential equation (PDE) solving, and image classification on the CIFAR-10 dataset. Results show that the proposed method consistently outperforms standard Adam, achieving faster convergence and lower error in both regression and classification settings.

When the sequence of regular polygons with consecutively increasing numbers of sides is joined edge-to-edge in a single direction while minimizing bending, the resulting structure assumes the shape of a logarithmic spiral. This paper proves that this spiral takes the form r=exp(4{\theta}/{\pi}). Specifically, it is derived that the distances between the curve and the centers of the even-sided and odd-sided regular polygons converge to 5/6 and 7/12, respectively, with the centers extending outward along the inner side of the spiral. A similar analysis applied to the sequence of regular polygons with consecutively increasing odd numbers of sides reveals that it forms the same type of spiral, establishing that the distances to the centers converge to 7/24.

In this article, we introduce the notion of circular k-superpatterns, defined as permutations that contain all length-k patterns up to rotation equivalence. We present a construction of a circular superpattern from a linear (k-1)-superpattern and explicitly derive an upper bound on its length. Motivated by the zigzag framework of Engen and Vatter, we adapt and simplify their score function to the circular setting and analyze its parity properties. For odd k, we propose a candidate zigzag construction for circular superpatterns, supported by computational evidence for small values of k.

We investigate the relationship between the sizes of the sum and difference sets of the Dicyclic Group $\mathrm{Dic}_{4n}$. We first determine the exact numbers of MSTD (more sums than differences), MDTS (more differences than sums), and balanced subsets of size two. As a consequence, we show that the numbers of MSTD and balanced subsets of size two are asymptotically equal as $n \to \infty$. For odd $n$, we then obtain exact counts of MSTD, MDTS, and balanced subsets of size three, with the results depending on whether $n$ is divisible by $3$. In this case, we establish that asymptotically the number of MSTD subsets of size three is six times the number of MDTS subsets and also six times the number of balanced subsets. Finally, we establish a lower bound for the number of MSTD, MDTS, and balanced subsets of $\mathrm{Dic}_{4n}$ corresponding to the boundary case of size $2n$.

We introduce and investigate a quantitative version of Steinhaus' property $(S)$ for Banach spaces, called the uniform property $(S)$. A Banach space $X$ is said to have uniform $(S)$ if for every pair of distinct unit vectors $x,y\in X$ and every $a>0$, the difference of the perturbed norms \[ \sup_{\|z\|\le a}\big|\|x+z\|-\|y+z\|\big| \] is bounded below by a positive function of $a$ and $\|x-y\|$. We compute this modulus exactly for the spaces $L_1(\mu)$ with atomless measure $\mu$, \[ U_{L_1(\mu)}(d;a)=\Big(\tfrac{4a}{2+d}\wedge 1\Big)d, \] The class of spaces with uniform $(S)$ is stable under ultrapowers, Bochner-$L_1$ constructions, and contains all Gurari\uı spaces as well as Banach lattices of almost universal disposition. In particular, every Banach space embeds isometrically into a non-strictly convex Banach space of the same density having uniform $(S)$. We further exhibit an explicit equivalent renorming of $\ell_1(\Gamma)$, \[ \|x\|_S=\big(\|x\|_1^2+\|x\|_2^2\big)^{1/2}, \] which endows $\ell_1(\Gamma)$ and all its ultrapowers with uniform $(S)$. These results settle, in ZFC, several open questions about the quantitative geometry of property $(S)$ posed by Kochanek and the second-named author.

The subject is partial desingularization preserving the normal crossings singularities of an algebraic or analytic variety X (over the complex field or over an uncountable algebraically closed field of characteristic zero, in the algebraic case). Our approach has three parts involving distinct techniques: (1) a formal splitting theorem for regular or analytic functions which satisfy a generic splitting hypothesis; (2) a study of singularities in the closure of the normal crossings locus, based on the combinatorics of G-circulant matrices, where G is a finite abelian group, leading to a theorem on reduction to group-circulant normal form; (3) a partial desingularization theorem, proved using (1) and (2) together with weighted blowings-up of group-circulant singularities. Previous results were for partial desingularization preserving simple normal crossings, or preserving general normal crossings when dim X < 5.

We study $\mathbb{S}_n$-equivariant topological and weight-graded compactly-supported Euler characteristics of the universal Picard stack $\mathrm{Pic}_{g, n}^d \to \mathcal{M}_{g, n}$ of degree-$d$ line bundles over $\mathcal{M}_{g, n}$. We prove that in the weight-zero and topological cases, the generating function for Euler characteristics of $\mathrm{Pic}_{g, n}^d$ is obtained from the corresponding one for $\mathcal{M}_{g, n}$ by an extremely simple combinatorial transformation. This lets us deduce closed formulas for the two generating functions, taking as input the Chan--Faber--Galatius--Payne formula in the weight-zero case and Gorsky's formula in the topological case. As a corollary, we also obtain a closed formula for the topological Euler characteristic of $\mathrm{Pic}^d_g$. Our weight-zero calculation is a corollary of a general result passing from the weight-graded Euler characteristics of $\mathcal{M}_{g, n}$ to those of $\mathrm{Pic}_{g,n}^d$.

In this paper we construct examples of spherically symmetric Dirac-Yang-Mills pairs on Riemannian 3-manifolds with the structure group SU(2). This approach yields coupled solutions (i.e. the connection is not a Yang-Mills connection) and among them are solutions on S^1(r_1) x S^2(r_2) for certain radii r_1 and r_2. These are, to the authors' best knowledge, the first examples of coupled Dirac-Yang-Mills pairs on a closed Riemannian spin manifold.

Iteration limited model predictive control (MPC) can stabilize a feedback control system under sufficient conditions; this work explores combining a low iteration limit MPC with a high iteration limit MPC for mixed-integer quadratic programs (MIQPs) where the suboptimality is due to solver iteration limits. To combine the two MPCs a hybrid systems controller is developed that ``unites'' two MIQP-MPC solvers where the iteration limits of interest are the branch-and-bound and quadratic programming iteration limits. Asymptotic stability and robustness of the hybrid feedback control system are theoretically derived. Then an interpretable branch-and-bound algorithm and implementable uniting controller algorithm are developed. Finally, the developed algorithms and varying iteration limits are empirically evaluated in simulation for the switching thruster and minimum thrust spacecraft rendezvous problems.

We present a modular function-based approach to explaining, for primes larger than 3, the exponents that appear in the prime decomposition of the order of the monster finite simple group.

We investigate the following general problem, closely related to the problem of isomorphic classification of Banach spaces $C(K)$ of continuous real-valued functions on a compact space $K$, equipped with the standard supremum norm:Let $\mathcal{K}$ be a class of compact spaces. How many isomorphism types of Banach spaces $C(K)$ of real-valued continuous functions on $K$ are there, for $K\in \mathcal{K}$? We prove that for any uncountable regular cardinal number $\kappa$, there exist exactly $2^\kappa$ isomorphism types of spaces $C(K)$ for compact spaces of weight $\kappa$. We show that, for the class $\mathcal{L}_{\omega_1}$ of separable compact linearly ordered spaces of weight $\omega_1$, the answer to the above question depends on additional set-theoretic axioms. In particular, assuming the continuum hypothesis, there are $2^{2^\omega}$ isomorphism types of $C(L)$, for $L\in \mathcal{L_{\omega_1}}$, and assuming a certain axiom proposed by Baumgartner, there is only one type.

We survey several non-absolutely convergent integrals, including the Henstock-Kurzweil and Pfeffer integrals, and use ideas from these theories to investigate the problem of multidimensional Young integration. We further present results on Young geometric integration, namely the integration of certain generalized differential forms over $m$-dimensional subsets of $\mathbb{R}^d$. This is achieved by introducing appropriate notions of chains and cochains, in the spirit of Whitney's geometric integration theory.

We study the extension theory for the two-dimensional first-order system $Ju' +qu = wf$ of differential equations on the real interval $(a,b)$ where $J$ is a constant, invertible, skew-hermitian matrix and $q$ and $w$ are matrices whose entries are real distributions of order $0$ with $q$ hermitian and $w$ non-negative. Specifically, we characterize the boundary conditions for solutions $u$ in the closure of the minimal relation, as well as describe the properties of quasi-boundary conditions which yield self-adjoint extensions. We then apply these ideas to a popular extension of non-negative minimal relations: the Krein-von Neumann extension. For more context on how the Krein-von Neumann is defined, an appendix shows a construction of the Friedrichs extension from which the Krein-von Neumann is traditionally defined.

We consider finite blocklength lossy compression of information sources whose components are independent but non-identically distributed. Crucially, Gaussian sources with memory and quadratic distortion can be cast in this form. We show that under the operational constraint of exceeding distortion $d$ with probability at most $\epsilon$, the minimum achievable rate at blocklength $n$ satisfies $R(n, d, \epsilon)=\mathbb{R}_n(d)+\sqrt{\frac{\mathbb{V}_n(d)}{n}}Q^{-1}(\epsilon)+O \left(\frac{\log n}{n}\right)$, where $Q^{-1}(\cdot)$ is the inverse $Q$-function, while $\mathbb{R}_n(d)$ and $\mathbb{V}_n(d)$ are fundamental characteristics of the source computed using its $n$-letter joint distribution and the distortion measure, called the $n$th-order informational rate-distortion function and the source dispersion, respectively. Our result generalizes the existing dispersion result for abstract sources with i.i.d. components. It also sharpens and extends the only known dispersion result for a source with memory, namely, the scalar Gauss-Markov source. The key novel technical tool in our analysis is the point-mass product proxy measure, which enables the construction of typical sets. This proxy generalizes the empirical distribution beyond the i.i.d. setting by preserving additivity across coordinates and facilitating a typicality analysis for sums of independent, non-identical terms.

In this paper we study nodal deformations of singular surfaces $S\subset \mathbb P^3$. In particular we consider the case in which $S$ has an isolated singularity of multiplicity $m$ and the case in which $S$ has only ordinary singularities along a line.

In this paper, we present a thorough von Neumann stability analysis of explicit and implicit Arbitrary-Lagrangian-Eulerian (ALE) ADER discontinuous Galerkin (DG) methods on classical and degenerate spacetime geometries for hyperbolic equations. First, we rigorously study the CFL stability conditions for the explicit ADER-DG method, confirming results widely used in the literature while specifying their limitations. Moreover, we highlight under which conditions on the mesh velocity the ALE methods, constrained to a given CFL, are actually stable. Next, we extend the stability study to ADER-DG in the presence of degenerate spacetime elements, with zero size at the beginning and the end of the time step, but with a non zero spacetime volume. This kind of elements has been introduced in a series of articles on direct ALE methods by Gaburro et al. to connect via spacetime control volumes regenerated Voronoi tessellations after a topology change. Here, we imitate this behavior in 1d by fictitiously inserting degenerate elements in between two cells. Then, we show that over this degenerate spacetime geometry, both for the explicit and implicit ADER-DG, the CFL stability constraints remain the same as those for classical geometries, laying the theoretical foundations for their use in the context of ALE methods.

We study the second cohomology group with coefficients in the adjoint module for a class of solvable Lie algebras $\mathcal{R}_{\mathcal{T}}$ that arise as maximal solvable extensions of nilpotent Lie algebras $\mathcal{N}$ of maximal rank. Under suitable structural assumptions on the root system determined by the action of a maximal torus $\mathcal{T}$ on $\mathcal{N}$, we obtain sufficient conditions for the cohomological rigidity of $\mathcal{R}_{\mathcal{T}}$. Conversely, we identify explicit configurations of roots that force the second cohomology group to be non-trivial, thereby producing broad families of solvable Lie algebras that are not cohomologically rigid. Our results extend the classical sufficient conditions of Leger and Luks, and they provide a unified and computationally effective framework for determining the cohomological rigidity of a wide class of solvable Lie algebras, including several known results.

Given a set of $m$ distinct points in projective space over a field, and $t$ a positive integer, a classical question asks for the least degree of a hypersurface that passes through each point with multiplicity at least $t$. Related to this, it remains unresolved whether there exists a dense open set of $m$-tuples of points for which this least degree is constant for each $t\ge 1$. We formulate a conjecture regarding the generic freeness of certain local cohomology modules that would answer this in the affirmative; we then prove this conjecture for up to $n+2$ points in projective $n$-space over a field. In a related direction, we prove a generic flatness result for the cohomology of thickenings of a closed subscheme of projective space over an integral domain.

This research delves into the exact controllability of semilinear measure-driven integrodifferential systems in nonlocal settings. We provide sufficient controllability requirements using the measure of noncompactness and the Mönch fixed point theorem without making any assumptions about how compact the evolution system is in relation to the linear part of the measure system. Here, we obtain results that both generalize and improve upon many prior findings.

The etale cycle class map links Chow groups with etale cohomology. Within the framework of motivic cohomology, the motivic cycle class map appears as a comparison morphism relating motivic and Lichtenbaum cohomology. By examining the comparison morphism constructed by Geisser and Levine, we show that the motivic cycle class map is induced by the etale cycle class map. We then study the behavior of cycle class maps under Weil restriction. Extending Karpenko's construction of the Weil restriction for algebraic cycles and Chow groups, we introduce a corresponding Weil restriction for l-adic cohomology and prove its compatibility with the motivic cycle class map. This provides a conceptual explanation for the descent of cycle classes under finite Galois extensions.

This work is concerned with nonparametric goodness-of-fit testing in the context of nonlinear inverse problems with random observations. Bayesian posterior distributions based upon a Gaussian process prior distribution are proven to contract at a certain rate uniformly over a set of true parameters. The corresponding posterior mean is shown to converge uniformly at the posterior contraction rate in the sense of satisfying a concentration inequality. Distinguishability for bounded alternatives separated from a composite null hypothesis at the posterior contraction rate is established using infimum plug-in tests based on the posterior mean and also on maximum a posteriori estimators. The results are applied to a class of inverse problems governed by ordinary differential equation initial value problems that is widely used in pharmacokinetics. For this class, uniform posterior contraction rates are proven and then used to establish distinguishability.

The Lagrange spectrum $L$ is the set of finite values of the best approximation constants $k(\alpha)=\limsup_{|p|,|q|\to \infty}|q(q\alpha-p)|^{-1}$, where $\alpha\in \mathbb{R}\setminus \mathbb{Q}$. It is a classical result that the pairs $(p,q)$ attaining these approximation constants arise from the convergents $(p_n,q_n)$ of the continued fraction of $\alpha$. Consequently, $k(\alpha)=\limsup_{n\to\infty}|q_n(q_n\alpha-p_n)|^{-1}$. Moreira proved that the function $d(t)=HD(L\cap(-\infty,t))$ where $HD$ denotes Hausdorff dimension, is continuous. Second Lagrange spectra are defined analogously to the classical Lagrange spectrum, but are associated with the problem of approximating an irrational number $\alpha$ by rational numbers $\frac{p}{q}$ that are not convergents of its continued fraction expansion. Two natural definitions arise depending on whether rational multiples $(p,q)=(kp_n,kq_n),k\geq 2$ which represent the same rational numbers as convergents, are allowed or excluded. Based on this distinction, Moshchevitin introduced two second Lagrange spectra, denoted $L_2$ and $L_2^*$. We prove that the function $d_2(t)=HD(L_2\cap (-\infty,t))$ is continuous, whereas $d_2^*(t)=HD(L_2^*\cap (-\infty,t))$ is discontinuous and assumes only the values 0 and 1.

This paper introduces two fundamentally new concepts to game theory: multilateral Nash equilibria and families of games. Starting with non-cooperative games, we show how these notions together seamlessly integrate into and naturally extend the classical theory, and simultaneously enable us to prove a powerful (multilateral) Nash equilibrium existence result with minimal assumptions on the game. Classically, a Nash equilibrium is a global strategy such that whichever player unilaterally deviates from the equilibrium, also reduces his own profit. For a k-lateral Nash equilibrium we now require that whichever group of k players collectively changes their strategies, also reduces all of the deviating players' profits. In this way, we obtain a filtration of equilibria, where the higher-lateral equilibria are less frequent. Furthermore, we derive an existence criterion for multilateral Nash equilibria and demonstrate how it reflects the increasing rarity of higher-lateral equilibria. Additionally, we show that some classical games have higher-lateral Nash equilibria, which in every case reveal the structure of these games from a new point of view. A family of games is a parameterized collection of non-cooperative games, where the parameter affects every aspect of the game. Typically, we assume that this dependence is continuous, thereby introducing a new structure. That way, we can avoid analyzing the games one at a time, and instead treat the family as a whole. This allows the parameter to take a central role in our theory, and shifts our attention from seeking a special strategy to searching for a special game with preferred strategies. Our main result proves the existence of a multilateral equilibrium in a family of games, maintaining minimalistic assumptions on the games individually. Surprisingly, the clique covering number of the Kneser graph makes a central appearance.

We consider the problem of parameter estimation from a generalized linear model with a random design matrix that is orthogonally invariant in law. Such a model allows the design have an arbitrary distribution of singular values and only assumes that its singular vectors are generic. It is a vast generalization of the i.i.d. Gaussian design typically considered in the theoretical literature, and is motivated by the fact that real data often have a complex correlation structure so that methods relying on i.i.d. assumptions can be highly suboptimal. Building on the paradigm of spectrally-initialized iterative optimization, this paper proposes optimal spectral estimators and combines them with an approximate message passing (AMP) algorithm, establishing rigorous performance guarantees for these two algorithmic steps. Both the spectral initialization and the subsequent AMP meet existing conjectures on the fundamental limits to estimation -- the former on the optimal sample complexity for efficient weak recovery, and the latter on the optimal errors. Numerical experiments suggest the effectiveness of our methods and accuracy of our theory beyond orthogonally invariant data.

We show that the category of V-groups, where V is a cartesian quantale, so in particular the category of preordered groups, is locally algebraically cartesian closed with respect to the class of points underlying the product V-category structure. We obtain this by observing that such points correspond to (V-Cat)-enriched functors from a V-group, seen as a one-object V-category, to the category V-Grp of V-groups. Moreover, we show that the actions corresponding to points underlying the product V-category structure are representable.

The widespread availability of generative artificial intelligence tools poses new challenges for school mathematics education, particularly regarding the formative role of traditional mathematical tasks. In post-AI educational contexts, many activities can be solved automatically, without engaging students in interpretation, decision-making, or mathematical validation processes. This study analyzes a secondary school classroom experience in which open mathematical tasks are implemented as a didactic response to this scenario, aiming to sustain students' mathematical activity. Adopting a qualitative and descriptive-interpretative approach, the study examines the forms of mathematical work that emerge during task resolution, mediated by the didactic regulation device COMPAS. The analysis is structured around four analytical axes: open task design in post-AI contexts, students' mathematical agency, human-AI complementarity, and modeling and validation practices. The findings suggest that, under explicit didactic regulation, students retain epistemic control over mathematical activity, even in the presence of generative artificial intelligence.

An equation $f(x)=a$, where $f$ is a complex meromorphic function and $a\in\mathbb{C}$ is a parameter, is solvable in elementary functions if the inverse map $x=f^{-1}(a)$ can be expressed as a finite composition of arithmetic operations (addition, subtraction, multiplication, and division), the exponential function, the complex logarithm, and constants. Specific functions such as $\tan x - x$, $\exp x + x$, $x^x$ have been proven to be unsolvable by Kanel-Belov, Malistov, Zaytsev, while almost all entire surjective functions of at most exponential growth have been covered by Zelenko. All these rely on one-dimensional topological Galois theory, developed by Khovanskii. We generalize to provide a proof for the unsolvability of all elementary meromorphic functions $f$ such that the derivative of $f$ has infinitely many roots $x_i$ and the set of distinct values $f(x_i)$ is infinite.

We investigate how enhanced Koszul properties of Galois cohomology behave under composita of fields. Given fields $K_1$ and $K_2$ containing $\mu_p$, with intersection $k$ and compositum $K = K_1K_2$, we formulate an abstract composita stability theorem: under a pro-$p$ amalgam decomposition $G_K \cong G_{K_1} *_{G_k} G_{K_2}$ of maximal pro-$p$ Galois groups, and natural Mayer-Vietoris compatibility assumptions on the mod-$p$ cohomology rings $H^\bullet(G_{K_1},\mathbb F_p)$, $H^\bullet(G_{K_2},\mathbb F_p)$, and $H^\bullet(G_k,\mathbb F_p)$, the quadratic presentation of $H^\bullet(G_K,\mathbb F_p)$ arises from a fiber-product construction on degree-$1$ generators and quadratic relations. Assuming stability of universal Koszulity under this quadratic gluing, we obtain that universal Koszulity of $H^\bullet(G_{K_1},\mathbb F_p)$ and $H^\bullet(G_{K_2},\mathbb F_p)$ implies universal Koszulity of $H^\bullet(G_K,\mathbb F_p)$. As a concrete application, we prove a composita stability theorem for certain Pythagorean fields whose maximal pro-$2$ Galois groups decompose as free pro-$2$ products of Demuškin groups and free factors. For suitable composita $K = K_1K_2$ of such fields, the mod-$2$ Galois cohomology ring $H^\bullet(G_K(2),\mathbb F_2)$ remains quadratic and universally Koszul. This provides large classes of fields, built from local, global, and Pythagorean base fields by admissible extensions and composita, whose maximal pro-$p$ Galois groups have universally Koszul cohomology, and yields inverse Galois obstructions: any finitely generated pro-$p$ group with nonquadratic or non-universally Koszul mod-$p$ cohomology cannot occur as the maximal pro-$p$ Galois group of a field in these families.

We present a Galerkin boundary element method for clamped Kirchhoff--Love plates with piecewise smooth boundary. It is a direct method based on the representation formula and requires the inversion of the single-layer operator and an application of the double-layer operator to the Dirichlet data. We present trace approximation spaces of arbitrary order, required for both the Dirichlet data and the unknown Neumann trace. Our boundary element method is quasi-optimal with respect to the natural trace norm and achieves optimal convergence order under minimal regularity assumptions. We provide explicit representations of both boundary integral operators and discuss the implementation of the appearing integrals. Numerical experiments for smooth and non-smooth domains confirm predicted convergence rates.

We exhibit a geometric morphism from the Grothendieck topos representing the Solovay model to the $\kappa$-pyknotic sets of Barwick-Haine and Clausen-Scholze. We then use the properties of this morphism and automatic continuity in the Solovay model to prove Clausen-Scholze's resolution of the Whitehead problem for discrete condensed abelian groups. We also exhibit an analogous internal $Ext$ computation between locally compact abelian groups in the Solovay model.

We consider infinite-dimensional linear Gaussian Bayesian inverse problems with uncorrelated sensor data, and focus on the problem of finding sensor placements that maximize the expected information gain (EIG). This study is motivated by optimal sensor placement for linear inverse problems constrained by partial differential equations (PDEs). We consider measurement models where each sensor collects a single-snapshot measurement. This covers sensor placement for inverse problems governed by linear steady PDEs or evolution equations with final-in-time observations. It is well-known that in the finite-dimensional (discretized) formulations of such inverse problems, EIG is a monotone submodular function. This also entails a theoretical guarantee for greedy sensor placement in the discretized setting. We extend the result on submodularity of the EIG to the infinite-dimensional setting, proving that the approximation guarantee of greedy sensor placement remains valid in the infinite-dimensional limit. We also discuss computational considerations and present strategies that exploit problem structure and submodularity to yield an efficient implementation of the greedy procedure.

We establish both local and global bifurcation results for traveling periodic solutions of the one-dimensional two-species Vlasov-Poisson equation. These solutions consist of strip-like regions of ions and electrons in phase space that propagate coherently and emerge from spatially homogeneous, velocity-dependent equilibrium layers. Depending on the geometry of the underlying equilibrium and on the selected Fourier mode, the bifurcation diagram exhibits either two or four solution branches. In all cases, the bifurcation is of pitchfork type; in symmetric configurations, the local structure near the equilibrium has a hyperbolic geometry. We further show that these locally constructed branches extend globally. This work extends the previous study by the same author of the purely electronic case, where the ions were modeled as an immobile neutralizing background. Allowing both species to evolve dynamically leads to a more intricate, higher-dimensional analysis. Finally, by means of an affine change of variables, we reveal a connection with the one-dimensional two-component Euler-Poisson system, which in turn enables the construction of traveling periodic waves of both small and large amplitude for that model as well.

The Penrose triangle, staircase, and related ``impossible objects'' have long been understood as related to first cohomology $H^1$: the obstruction to extending locally consistent interpretations around a loop. This paper develops a cohomological hierarchy for a class of visual paradoxes. Restricting to systems built from \emph{bistable} elements -- components admitting exactly two local states, such as the Necker cube's forward/backward orientations, a gear's clockwise/counterclockwise spin, or a rhombic tiling corner's convex/concave interpretation -- allows the use of $\mathbb{Z}_2$ coefficients throughout, reducing obstruction theory to parity arithmetic. This reveals a hierarchy of paradox classes from $H^0$ through $H^2$, refined at each degree by the relative/absolute distinction, ranging from ambiguity through impossibility to inaccessibility. A discrete Stokes theorem emerges as the central tool: at each degree, the connecting homomorphism of relative cohomology promotes boundary data to interior obstruction, providing the uniform mechanism by which paradoxes ascend the hierarchy. Three paradigmatic systems -- Necker cube fields, gear meshes, and rhombic tilings -- are studied in detail. Throughout, we pair cohomology with imagery and animation. To illuminate the underlying structure, we introduce the \emph{Method of Monodromic Apertures}, an animation technique that reveals monodromy through a configuration space of local sections.

A skew brace $A = (A,\cdot,\circ)$ is said to be \textit{left-simple} if $A\neq1$ and it has no left ideal other than $1$ and $A$. The purpose of this paper is to give a partial classification of the finite left-simple skew braces. A result of Stefanello and Trappeniers implies that finite left-simple skew braces correspond precisely to minimal Hopf--Galois structures on finite Galois extensions of fields.

We establish a strong unique continuation property for the subelliptic Baouendi operator under the presence of zero-order perturbations satisfying an almost Hardy-type growth condition. In particular, the admissible class includes both $L^\infty_{\mathrm{loc}}$ and singular potentials. We prove that any solution vanishing to infinite order at a point of the degeneracy manifold of the operator must be identically zero. The result holds extends to variable-coefficient operators with intrinsic Lipschitz regularity. A notable feature of the proof is that it relies exclusively on $L^2$ Carleman estimates combined with the classical Hardy inequality.

We consider the compressible Navier-Stokes-Poisson equations in $\mathbb{R}^d$ ($d\geq2$), a classical model for barotropic compressible flows coupled with a self-consistent electrostatic potential. We show that the electrostatic coupling has a significant impact on the long-time dynamics of solutions due to its underlying Klein-Gordon structure. As a first result, we prove the global well-posedness of the Cauchy problem with initial data near equilibrium in the full-frequency $L^{p}$-type critical Besov space \emph{without relying on hyperbolic symmetrization}. Compared with the Poisson-free case studied in several milestone works [Charve and Danchin, Arch. Rational Mech. Anal., 198 (2010), 233-271; Chen, Miao and Zhang, Commun. Pure Appl. Math., 63 (2010), 1173-1224; Haspot, Arch. Rational Mech. Anal., 202 (2011), 427-460], we remove the extra $L^{2}$ assumption in low frequencies and extend the admissible choice of $p$ to the sharp range $1\leq p<2d$. This is, to the best of our knowledge, the first result in compressible fluids that allows the initial velocity field to be highly oscillatory across all frequencies. Furthermore, stemming from the Poisson coupling, the density and velocity exhibit distinct low-frequency behaviors. Motivated by this feature, we propose a general $L^p$-type low-frequency assumption and establish the optimal convergence rates of global solutions toward equilibrium. For a broad class of indices, this assumption yields faster decay than those obtained under the classical $L^1$ framework. To this end, we develop a time-weighted energy method, which is of interest and enables us to capture maximal decay estimates without additional smallness of initial data.

For a one-dimensional Lévy process, we derive an explicit formula for the probability of first hitting a specified point among a fixed finite set. Moreover, using this formula, we obtain an explicit expression for each entry of the $Q$-matrix of the trace process on the finite set. These formulas involve solely the renormalized zero resolvent.

This paper develops stable canonical rules for intuitionistic modal logics, which were first introduced for superintuitionistic logics and transitive nor mal modal logics in [1] and [2] respectively. We first prove that every in tuitionistic modal multi-conclusion consequence relation is axiomatizable by stable canonical rules. This allows us to assume, without loss of generality, that rules considered by us are stable canonical ones. The idea turns out to be useful. In particular, using stable canonical rules, we get an alterna tive proof of the Blok-Esakia theorem for intuitionistic modal logics which was first proved in [3] and generalize it to multi-conclusion consequence re lations. We also prove the Dummett-Lemmon conjecture for intuitionistic modal multi-conclusion consequence relations, which, as far as we know, is a new result.

Let $G$ be a Kac-Moody group, split over $\mathbb R$. The totally nonnegative part of $G$ and its (ordinary) flag variety $G/B^+$ was introduced by Lusztig. It is known that the totally nonnegative parts of $G$ and $G/B^+$ have remarkable combinatorial and topological properties. In this paper, we consider the totally nonnegative part of the $J$-twisted flag variety $G/{}^J B^+$, where ${}^J B^+$ is the Borel subgroup opposite to $B^+$ in the standard parabolic subgroup $P_J^+$ of $G$. The $J$-twisted flag varieties include the ordinary flag variety $G/B^+$ as a special case. Our main result show that the totally nonnegative part of $G/{}^J B^+$ decomposes into cells, and the closure of each cell is a regular CW complex. This generalizes the work of Galashin-Karp-Lam \cite{GKL22} and the joint work of Bao with the first author \cite{BH24} for ordinary flag varieties. As an application, we deduce that the totally nonnegative part of the double flag variety $G/B^+ \times G/B^-$ with respect to the diagonal $G$-action has similar nice properties. We also establish some connections between the totally nonnegative part of the double flag with the canonical basis of the tensor product of a lowest weight module with a highest weight module of $G$. As another application, we show that the link of identity in a totally nonnegative reduced double Bruhat cell of $G$ is a regular CW complex. This generalizes the work of Hersh \cite{Her14} on the link of $U_{\geq0}^-$ and gives a positive answer to an open question of Fomin and Zelevinsky.

About 40 years ago Jonathan and Peter Borwein discovered the series identity $$ \sum_{n=0}^\infty \frac{(-1)^n(6n)!}{(3n)!(n!)^3} \frac{(A+nB)}{C^{n+1/2}} = \frac{1}{12\pi} $$ where \begin{align*} A&=1657145277365+212175710912\sqrt{61},\cr B&=107578229802750+13773980892672\sqrt{61},\cr C&=\left(5280(236674+30303\sqrt{61})\right)^3 \end{align*} which adds roughly 25 digits of accuracy per term. They noted that if each of the quadratic irrationals $A$, $B$ and $C$ is replaced by their conjugates, that is, each number $a+b\sqrt{61}$ is changed to $a-b\sqrt{61}$, then the resulting series also converges to a rational multiple of $1/\pi$. They gave several other examples of quadratic irrational series for $1/\pi$, and noted that the conjugate series converges to another rational multiple of $1/\pi$ or in some cases the conjugate series diverges. The purpose of this work is to provide an explanation and classification of such series. Our classification includes Ramanujan's 17 original series, as well as series of the Borweins, Chudnovskys, Sato and others. We extend the classification to genus-zero subgroups $\Gamma_0(\ell)+$, that is, for each $\ell \in \big\{1,2,3,\ldots,36,38,39,41,42,44,45,46,47,49,50,51,54,55,56,59,60,62,66,69,70, 71,78,87,92,94,95,105,110,119\big\}$ we calculate the Hauptmoduls, associated weight two modular forms, and the corresponding rational and real quadratic irrational series for $1/\pi$. The classification reveals many interrelations among the different series. For example, we show that the Borweins' series above, and its conjugate, are equivalent by hypergeometric transformation formulas to the level~7 rational series $$ \sum_{n=0}^\infty \left\{\sum_{j=0}^n {n \choose j}^2{2j \choose n} {n+j \choose j}\right\} (11895n+1286) \frac{(-1)^n}{22^{3n+3}} = \frac{1}{\pi\sqrt{7}}. $$

Geometric quantiles are popular location functionals to build rank-based statistical procedures in multivariate settings. They are obtained through the minimization of a non-smooth convex objective function. As a result, the singularity of the directional derivatives leads to numerical instabilities and poor sample properties as well as surprising `phase transitions' from empirical to population distributions. To solve these issues, we introduce a regularized version of geometric distribution functions and quantiles that are provably close to the usual geometric concepts and share their qualitative properties, both in the empirical and continuous case, while allowing for a much broader applicability of asymptotic results without any moment condition. We also show that any linear assignment of probability measures (such as the univariate distribution function), that is also translation- and orthogonal-equivariant, necessarily coincides with one of our regularized geometric distribution functions.

In our previous paper with Tudor Pădurariu, we introduced the notion of limit categories for moduli stacks of Higgs bundles and formulated the Dolbeault geometric Langlands correspondence. These limit categories are expected to provide an effective ``classical limit'' of the categories of D-modules on the moduli stack of bundles, and our formulation links categorical Donaldson-Thomas theory with the geometric Langlands correspondence. In this paper, we prove the above Dolbeault geometric Langlands correspondence for $\GL_2$ over the locus in the Hitchin base where the spectral curves are reduced. This is the first non-trivial case in which the relevant moduli stacks are not quasi-compact, and the use of limit categories is essential to the formulation and proof of the correspondence. Our approach also outlines a strategy for proving the correspondence in greater generality and explains the current obstructions to such an extension.

The main goal of this paper is to extend three important Schur positivity results to key positivity, replacing all Schur polynomials in relevant expressions with flagged Schur polynomials. Namely, we first show that the Temperley-Lieb immanants of (many) flagged Jacobi-Trudi matrices are key positive. Using this result, we give a combinatorial rule for the key expansion of (most) products of flagged skew Schur polynomials, and also give a log concavity result inspired by that of Lam-Postnikov-Pylyavskyy. The main tools in our proofs are Demazure crystals, and the recently defined shuffle tableaux of Nguyen and Pylyavskyy. In order to prove our main results, we must develop a new characterization of Demazure crystals, which builds off of prior work of Assaf and Gonzalez. This characterization may be useful in other contexts.

We present a new method for combining two cotorsion pairs to obtain an abelian model structure and we apply it to construct and study a new model structure on left $R$-modules over a left coherent ring $R$. Its class of fibrant objects is generated by the weakly Ding injective $R$-modules, a class of modules recently studied by Iacob. We give several characterizations of the fibrant modules, one being that they are the cycle modules of certain acyclic complexes of FP-injective (i.e., absolutely pure) $R$-modules. In the case that $R$ is a Ding-Chen ring, we show that they are precisely the modules appearing as cycles of acyclic complexes of FP-injectives. This leads to a new description of the stable module category of a Ding-Chen ring $R$, by way of modules we call Gorenstein FP-pro-injective. These are modules that appear as a cycle module of a totally acyclic complex of FP-projective-injective modules. As a completely separate application of the new model category method, we show that all complete cotorsion pairs, even non-hereditary ones, lift to abelian models for the derived category of a ring.

We study Schrödinger operators on $\mathbb R^3$ with finitely many concentric spherical $\delta$--shell interactions. The operators are defined by the quadratic form method and are described by continuity across each shell together with the usual jump condition for the radial derivative. Using a boundary integral approach based on the free Green kernel and single--layer potentials, we derive an explicit resolvent representation for an arbitrary number of shells with bounded coupling strengths. This yields a concrete Kre\uın--type formula and a boundary operator whose noninvertibility characterizes the discrete spectrum, and it is compatible with a partial--wave reduction under rotational symmetry. We then specialize to the two--shell case with constant couplings and obtain a detailed description of the negative spectrum. In particular, we show that the ground state (when it exists) lies in the $s$--wave sector and derive an explicit secular equation for bound states. For large shell separation, each bound level approaches the corresponding single--shell level with exponentially small corrections, while a genuine tunneling splitting appears when the single--shell levels are tuned to coincide. As a simple calibration, we relate the two--shell parameters to representative core--shell quantum dot scales. At the level of order--of--magnitude and qualitative trends, Type~I configurations yield a relatively strongly confined state, whereas Type~II configurations produce a comparatively shallow outer--shell state.

This note analyzes a linear recursion that arises in the computation of the subpacketization level for the multi-message PIR scheme of Banawan and Ulukus. We derive an explicit representation for the normalized subpacketization level $L$, whose smallest integer multiple yields the subpacketization level of the scheme, in terms of the number of servers $N$, the total number of messages $K$, and the number of demand messages $D$. The resulting formula shows that $L$ is a polynomial in $N$ with nonnegative coefficients, and its leading term is $N^{K-D+1}/D$.

We investigate the role of q-rationals in the context of link invariants. This leads us to introduce and study web categories that q-deform Deligne's categories.

This paper investigates the principal spectral theory and the asymptotic behavior of the principal spectrum point for a class of time-periodic cooperative systems with nonlocal dispersal operators, incorporating both coupled and uncoupled nonlocal terms. By applying the theory of resolvent positive operators and their perturbations, we first establish criteria for the existence of the principal eigenvalue. We then construct sequences of smooth upper and lower approximating matrix-valued functions, each of whose corresponding operators satisfies the principal eigenvalue existence condition. This approximation framework allows the principal spectrum point to effectively substitute for the principal eigenvalue in characterizing the global dynamics of the nonlinear system. Moreover, it facilitates the study of the asymptotic behavior of the principal spectrum point with respect to parameters under fairly general assumptions. Subsequently, for systems with both coupled and uncoupled nonlocal terms, we analyze the asymptotic behavior of the principal spectrum point in terms of the dispersal rate, dispersal range, and frequency. Finally, we illustrate the applicability of our theoretical results through a Zika virus model and a stem cell model.

Suppose $Y$ is a compact, connected, oriented 3-manifold possibly with boundary, such that $\pi_1(Y)$ is infinite. Let $\operatorname{Diff}_\partial(I\times Y)$ denote the group of self-diffeomorphisms of $I\times Y$ that are equal to the identity near the boundary. Let $\operatorname{Diff}_{PI}(I\times Y)$ denote the subgroup of $\operatorname{Diff}_\partial(I\times Y)$ consisting of elements pseudo-isotopic to the identity. Define $\operatorname{Homeo}_\partial(I\times Y)$, $\operatorname{Homeo}_{PI}(I\times Y)$ similarly for homeomorphisms. We show that the canonical map $\pi_0\operatorname{Diff}_{PI}(I\times Y) \to \pi_0\operatorname{Homeo}_{PI}(I\times Y)$ is of infinite rank. As a consequence, $\pi_0\operatorname{Diff}_{PI}(I\times Y)$, $\pi_0\operatorname{Diff}_{\partial}(I\times Y)$, $\pi_0\operatorname{Homeo}_{PI}(I\times Y)$, $\pi_0\operatorname{Homeo}_{\partial}(I\times Y)$ are all abelian groups of infinite rank. We also prove that $\pi_0\,C(Y)$ contains an abelian subgroup of infinite rank, and $\pi_0\,C(I\times Y)$ admits a surjection to an abelian group of infinite rank, where $C(X)$ denotes the concordance automorphism group $\operatorname{Diff}(I\times X, \{0\}\times X\cup I\times \partial X)$ or $\operatorname{Homeo}(I\times X, \{0\}\times X\cup I\times \partial X)$. These results are proved by studying the actions of barbell diffeomorphisms on the spaces of embedded arcs and configuration spaces.

For an ordinal $\lambda>0$, we use the Erdős--Rado partition theorem to prove the failure of strong completeness for modal languages of cardinality $(2^{|\lambda|+\aleph_0})^{+}$ with respect to models on ordinals equipped with the generalized Icard topologies $\mathcal{I}_{\lambda}$ and ${\tau_{c}}_{+\lambda}$. Specifically, we show that for such languages there exists a consistent set of formulas having neither $(\Theta, \mathcal{I}_{\lambda})$-model nor $(\Theta, {\tau_{c}}_{+\lambda})$-model. We also introduce a natural class of topological spaces, called $\lambda$-bouquet spaces, and prove that they yield strong completeness of $\mathsf{GL}$ for languages of cardinality $\lambda$.

We investigate the numerical ranges of weighted composition operators on weighted Dirichlet spaces, focusing on the properties of the inducing functions. We identify conditions on these functions under which the origin lies in the interior of the numerical range. The geometric structure of the numerical range is also analyzed, determining when it contains a circular or elliptical disc and computing the corresponding radius. Next, we introduce a class of Weyl-type weighted composition operators and obtain their Berezin range and Berezin number. Finally, we characterize the convexity of the Berezin range for weighted composition operators on these spaces.

Cayley's formula is a fundamental result in combinatorics that counts the number of labeled trees on n vertices. While existing proofs use approaches such as Prufer sequences and the Matrix-Tree Theorem, we give a combinatorial proof that highlights the role of degree sequences and structural properties of labeled trees. Our goal is to provide an accessible perspective and suggest connections to related enumeration problems.

Regularizing a linear ill-posed operator equation can be achieved by manipulating the spectrum of the operator's pseudo-inverse. Tikhonov regularization and spectral cutoff are well-known techniques within this category. This paper introduces an interpolating formula that defines a one-parameter family of regularizations, where Tikhonov and spectral cutoff methods are represented as limiting cases. By adjusting the interpolating parameter taking into account the specific operator equation under consideration, it is possible to mitigate the limitations associated with both Tikhonov and spectral cutoff regularizations. The proposed approach is demonstrated through numerical simulations in the fields of signal and image processing.

The present paper is devoted to study 2-local derivations on the Block-type Lie algebra which is an infinite-dimensional Lie algebra with some outer derivations. We prove that every 2-local derivation on the Block-type Lie algebra is a derivation.

Nous démontrons l'invariance Galoisienne de la propriété d'annulation en $1/2$ des fonctions L standard ou de Rankin-Selberg pour certaines représentations automorphes cuspidales algébriques régulières autoduales ou autoduales conjuguées de groupes linéaires sur un corps de nombres arbitraire. La démonstration repose sur l'utilisation de la cohomologie pondérée de Goresky-Harder-MacPherson et sur la construction de certaines représentations automorphes discrètes pour les groupes classiques comme résidus de séries d'Eisenstein. L'abandon de l'hypothèse ``$F$ totalement réel'' introduit de nouvelles difficultés concernant certains opérateurs d'entrelacement. Celles-ci sont résolues grâce à l'appendice, rédigé par J.-L. Waldspurger et l'un d'entre nous, démontrant l'holomorphie et la non-annulation de certains opérateurs d'entrelacement normalisés. Nous démontrons également l'invariance Galoisienne des facteurs epsilon correspondants, impliquant l'invariance Galoisienne de la parité de l'ordre d'annulation en $1/2$ de ces fonctions $L$. -- We prove the invariance under the Galois group of the vanishing at $1/2$ of standard and Rankin-Selberg L-functions for certain self-dual or conjugate self-dual algebraic cuspidal automorphic representations for general linear groups over an arbitrary number field. The proof uses Goresky-Harder-MacPherson weighted cohomology and the construction of certain discrete automorphic representations for classical groups as residues of Eisenstein series. New difficulties appear concerning certain intertwining operators. These are solved in the appendix by J.-L. Waldspurger and O. Taïbi proving the holomorphy and non-vanishing of these operators. We also prove the Galois invariance of epsilon factors, implying Galois invariance of the parity of the order at $1/2$ of L-functions.

Many modern iterative solvers for large-scale tomographic reconstruction incur two major computational costs per iteration: expensive forward/adjoint projections to update the data fidelity term and costly proximal computations for the regulariser, often done via inner iterations. This paper studies for the first time the application of methods that couple randomised skipping of the proximal with variance-reduced subset-based optimisation of data-fit term, to simultaneously reduce both costs in challenging tomographic reconstruction tasks. We provide a series of experiments using both synthetic and real data, demonstrating striking speed-ups of the order 5x--20x compared to the non-skipped counterparts which have been so far the standard approach for efficiently solving these problems. Our work lays the groundwork for broader adoption of these methods in inverse problems.

The factorization of three-dimensional data continues to gain attention due to its relevance in representing and compressing large-scale datasets. The linear-map-based tensor-tensor multiplication is a matrix-mimetic operation that extends the notion of matrix multiplication to higher order tensors, and which is a generalization of the T-product. Under this framework, we introduce the tensor CUR decomposition, show its performance in video foreground-background separation for different linear maps and compare it to a robust matrix CUR decomposition, another tensor approximation and the slice-based singular value decomposition (SS-SVD). We also provide a theoretical analysis of our tensor CUR decomposition, extending classical matrix results to establish exactness conditions and perturbation bounds.

We consider a zero-range process $\eta^N_t(x)$ with superlinear local jump rate, which in a hydrodynamic-small particle rescaling converges to the porous medium equation $\partial_t u=\frac12\Delta u^\alpha, \alpha>1$. As a main result we obtain a large deviation principle in any scaling regime of vanishing particle size $\chi_N\to 0$. The key challenge is to develop uniform integrability estimate on the nonlinearity $(\eta^N(x))^\alpha$ in a situation where neither pathwise regularity nor Dirichlet-form based regularity is readily available. We resolve this by introducing a novel multiscale argument exploiting the appearance of pathwise regularity across scales.

A graph $G$ is called {\em$F$-saturated} if $G$ does not contain $F$ as a subgraph but adding any missing edge to $G$ creates a copy of $F$. In this paper, we consider the spectral saturation problem for the linear forest $tP_4$, proving that every $n$-vertex $tP_4$-saturated graph $G$ with $t\geq 2$ and $n\ge 4t$ satisfies $\rho(G)\ge \frac{1+\sqrt{17}}{2}$, and characterizing all $tP_4$-saturated graphs for which equality holds. Moreover, we obtain that, for $t=2$ with odd $n\ge 13 $, and for $t\ge 3$ with $n\ge 6t+4$, the set of $n$-vertex $tP_4$-saturated graphs minimizing the spectral radius is disjoint from that minimizing the number of edges.

Sufficient and necessary conditions for an extension of a skew-derivation $(\delta_R,\alpha_R)$ of an associative $\mathbb{F}$-algebra $R$ to a skew derivation $(\delta_S,\alpha_S)$ on an extension $S$ of $R$ by $\mathbb{F}$ or a {\em homothetic extension $S$ of $R$ by $\mathbb{F}$} are derived. It is then shown that this yields the unique extension of the Ore extension $R[x;\alpha_R,\delta_R]$ of $R$ by $\mathbb{F}$ that embeds in the Ore extension $S[x;\alpha_S,\delta_S]$ of $S$ by the extended skew-derivation.

To every nearly convex optimization problem, that is a minimization problem with a nearly convex objective function and a nearly convex constraint set, we associate a uniquely defined convex optimization problem with a lower semicontinuous objective function and a closed constraint set. Interesting relationships between the original nearly convex problem and the associated convex problem are established. Optimality conditions in the form of Fermat's rules are obtained for both problems. We then get a Lagrange multiplier rule for a nearly convex optimization problem under a geometrical constraint and functional constraints from the Kuhn-Tucker conditions for the associated convex optimization problem. The obtained results are illustrated by concrete examples.

The classical Arazy's decomposition theorem provides a powerful tool in the study of sequences in (and isomorphisms on) a separable operator ideal $\mathcal C_E$ of the algebra $\mathcal B(H)$ of all bounded linear operators on the separable infinite-dimensional Hilbert space $H$. In this paper, we extend and strengthen Arazy's decomposition theorem to the setting of general bounded linear operators on a separable (quasi-Banach) operator ideal $\mathcal C_E$ of $\mathcal B(H)$. Several applications are given to the study of $\mathcal C_E$-strictly singular operators, largest proper ideals in the algebra $\mathcal B(\mathcal C_E)$ of all bounded linear operators on $\mathcal C_E$ and complementably homogeneous Banach spaces among others. Our versions of decomposition theorems supply tools for a noncommutative generalization of deep commutator theorems for operators on $\ell_p$ and $L_p$, $1\le p <\infty $, due to Brown and Pearcy, Apostol, and Dosev, Johnson and Schechtman. We are able to characterize commutators on the Schatten-von Neumann class $\mathcal C_p$, $1\le p<\infty $. For the crucial case, $p=1$, we establish that any operator $T\in\mathcal B(\mathcal C_1)$ is a commutator if and only if $T$ is not of the form $\lambda I+K$ for some $\lambda\neq 0$ and $\mathcal C_1$-strictly singular operator $K$.

The paper deals with homogenization and higher order approximations of solutions to nonlocal evolution equations of convolution type whose coefficients are periodic in the spatial variables and random stationary in time. We assume that the convolution kernel has finite moments up to order three. Under proper mixing assumptions, we study the limit behavior of the normalized difference between solutions of the original and the homogenized problems and show that this difference converges to the solution of a linear stochastic partial differential equation.

Given a simple graph $G$, its line graph, denoted by $L(G)$, is obtained by representing each edge of $G$ as a vertex, with two vertices in $L(G)$ adjacent whenever the corresponding edges in $G$ share a common endpoint. By applying the line graph operation repeatedly, we obtain higher order line graphs, denoted by $L^{r}(G)$. In other words, $L^{0}(G) = G$, and for any integer $r \ge 1$, $L^{r}(G) = L(L^{r-1}(G))$. Given a graph $G$ on $n$ vertices, we wish to efficiently find out (i) if $L^k(G)$ has an Euler path, (ii) the value of $\Delta(L^k(G))$. Note that the size of a higher order line graph could be much larger than that of $G$. For the first question, we show that for a graph $G$ with $n$ vertices and $m$ edges the largest $k$ where $L^k(G)$ has an Euler path satisfies $k = \mathcal O(nm)$. We also design an $\mathcal{O}(n^2m)$-time algorithm to output all $k$ such that $L^k(G)$ has an Euler path. For the second question, we study the growth of maximum degree of $L^k(G)$, $k \ge 0$. It is easy to calculate $\Delta(L^k(G))$ when $G$ is a path, cycle or a claw. Any other connected graph is called a prolific graph and we denote the set of all prolific graphs by $\mathcal G$. We extend the works of Hartke and Higgins to show that for any prolific graph $G$, there exists a constant rational number $dgc(G)$ and an integer $k_0$ such that for all $k \ge k_0$, $\Delta(L^k(G)) = dgc(G) \cdot 2^{k-4} + 2$. We show that $\{dgc(G) \mid G \in \mathcal G\}$ has first, second, third, fourth and fifth minimums, namely, $c_1 = 3$, $c_2 = 4$, $c_3 = 5.5$, $c_4 = 6$ and $c_5=7$; the third minimum stands out surprisingly from the other four. Moreover, for $i \in \{1, 2, 3, 4\}$, we provide a complete characterization of $\mathcal G_i = \{dgc(G) = c_i \mid G \in \mathcal G \}$. Apart from this, we show that the set $\{dgc(G) \mid G \in \mathcal G, 7 < dgc(G) < 8\}$ is countably infinite.

Let $ \kappa , \theta < \lambda$ be cardinals, with $\lambda$ and $\kappa$ regular. Concentrating on a simple case, we say that the triple $(\lambda,\kappa,\theta)$ has a Super Black Box when the following holds. For some stationary $S \subseteq \{\delta < \lambda : cf(\delta) = \kappa\}$ and $\overline C = \langle C_\delta : \delta \in S \rangle$, where $C_\delta$ is a club of $\delta$ of order type $\kappa$, for every coloring $\overline F = \langle F_\delta : \delta \in S \rangle$ with $F_\delta : {}^{C_\delta}\lambda \to \theta$, there exists $\langle c_\delta : \delta \in S\rangle \in {}^S\!\theta$ such that for every $f : \lambda \to \theta$, for stationarily many $\delta \in S$, we have $F_\delta(f \upharpoonright C_\delta) = c_\delta$. In an earlier work, it was proved (along with much more) that for a class of cardinals $\lambda$ this holds for many pairs $(\kappa,\theta)$. E.g.~$\kappa < \aleph_\omega$ is large enough, and $\beth_\omega(\theta) < \lambda$. However, the most interesting cases (at least with regards to Abelian groups) are $\kappa = \aleph_0,\aleph_1$ (which have not been covered yet). Here we restrict ourselves to the case where $\overline F$ is a {so-called} \emph{continuous coloring}, which includes the case where $F_\delta$ is computed from some $$ \big\langle F_{\delta,\beta}'(f \upharpoonright (C_\delta \cap \beta)) : \beta \in C_\delta \big\rangle. $$ This covers the cases we have in mind. We mainly prove results without any other caveats: e.g. For every regular $\kappa$ and $\theta$ there exists such a $\lambda$. We also deal with having multiple {$\bar C$-s}, and the existence of quite free subsets of ${}^\kappa\mu$.

We construct a family of points on the Lagrangian cone of a partial flag bundle associated to a (possibly non-split) vector bundle from any Weyl-invariant $I$-function of a prequotient. This result can be seen as the nonabelian analogue of the mirror theorem for projective bundles in arXiv:2307.03696, and generalizes Oh's mirror theorem for split partial flag bundles in arXiv:1607.08326.

We analyse the homogeneous parts of Clifford and meson algebras and point out that for the Clifford algebra it is related to fermionic statistics, that is, to fermionic parastatistics of order 1 while for the meson algebra it is related to fermionic parastatistics of order 2. We extend these homogeneous algebras into corresponding algebras related to fermionic parastatistics of all orders. We then define correspondingly higher order generalizations of Clifford and meson algebras.

Let $n\geq m$ be two positive integers, $S_{n,m}=K[x_1,\ldots,x_n,y_1,\ldots,y_m]$ and $I_{n,m}=(x_iy_j\;:\;1\leq i\leq n,1\leq j\leq m)\subset S_{n,m}$ the edge ideal of a complete bipartite graph. Denote $h(n,m)=\operatorname{hdepth}(S_{n,m}/I_{n,m})$. We prove that $h(n,m)\geq \left\lceil \frac{n}{2} \right\rceil$ and the equality holds if $m$ belong to a certain interval centered in $\left\lceil \frac{n} {2} \right\rceil$. Also, we find some tight bounds for $h(n,n)$ and we prove several inequalities between $h(n,m)$ and $h(n,m')$.

We investigate finite-time Lyapunov exponents (FTLEs), a measure for exponential separation of input perturbations, of deep neural networks within the framework of continuous-depth neural ODEs. We demonstrate that FTLEs are powerful organizers for input-output dynamics, allowing for better interpretability and the comparison of distinct model architectures. We establish a direct connection between Lyapunov exponents and adversarial vulnerability, and propose a novel training algorithm that improves robustness by FTLE regularization. The key idea is to suppress exponents far from zero in the early stage of the input dynamics. This approach enhances robustness and reduces computational cost compared to full-interval regularization, as it avoids a full ``double'' backpropagation.

We study discrete-time, discrete-state multistate Markov models from the perspective of algebraic statistics. These models are widely studied in event history analysis, and are characterized by the state space, the initial distribution and the transition probabilities. A finite path under the multistate Markov model is a particular set of states occupied at finite time instances $\{1, \dots, n\}$. The main goal of this paper is to establish a bridge between event history analysis and algebraic statistics. The joint probabilities of finite paths in these models have a natural monomial parametrization in terms of the initial distribution and the transition probabilities. We study the polynomial relations among joint path probabilities. When the statistical constraints on the parameters are disregarded, nonhomogeneous multistate Markov models of arbitrary order can be viewed as slices of decomposable hierarchical models. This yields a complete description of their vanishing ideals as toric ideals generated by explicit families of binomials. Moreover, the variety of this vanishing ideal equals the nonhomogeneous multistate Markov model on the probability simplex. In contrast, homogeneous multistate Markov models exhibit different algebraic behavior, as time homogeneity imposes additional polynomial relations, leading to vanishing ideals that are strictly larger than in the nonhomogeneous case. We also derive families of binomial relations that vanish on homogeneous multistate Markov models. We investigate maximum likelihood estimation from statistical and algebraic perspectives. For nonhomogeneous models, classical and algebraic formulas agree; in the homogeneous case, the algebraic approach is more complex. Lastly, we provide data applications where we demonstrate the statistical theory to obtain the maximum likelihood estimates of the parameters under specific multistate Markov models.

We study combinatorial and algebraic properties of $t$-clique-free complexes, a family of simplicial complexes associated with finite simple graphs that generalize the classical independence complex. For a graph $G$ and an integer $t \ge 2$, the $t$-clique-free complex $\mathsf{CF}_t(G)$ is the simplicial complex on the vertex set of $G$ whose faces are the subsets inducing no cliques of size $t$. Our main results provide sufficient conditions for shellability and related decomposability properties of $t$-clique-free complexes. In particular, we show that if $G$ is a $t$-diamond-free chordal graph (in particular, a block graph), then $\mathsf{CF}_t(G)$ is $(t-2)$-decomposable and hence shellable. We also investigate how graph modifications via clique attachments influence shellability. Generalizing earlier constructions involving whiskers and clique extensions, we introduce the following operation: given a graph $H$, a subset $S \subseteq V(H)$, and an integer $t \ge 2$, we form a graph $\operatorname{Cl}(H,S,t)$ by attaching to each vertex in $S$ a clique of size at least $t$. We prove that $\mathsf{CF}_t(H\setminus S)$ is shellable if and only if $\mathsf{CF}_t(\operatorname{Cl}(H,S,t))$ is shellable. This yields a flexible method for constructing shellable complexes, particularly when $S$ is a cycle cover. In addition, we extend the notion of clique whiskering and show that for any graph admitting a clique vertex-partition, the resulting $t$-clique whiskering produces a pure and shellable, and hence Cohen-Macaulay, $t$-clique-free complex. Finally, we establish a Fröberg-type result linking chordality and linear resolutions. We show that for any chordal graph $G$, the edge ideal of the complement $t$-clique clutter $\overline{\mathcal{CH}_t(G)}$ admits a $t$-linear resolution over any field.

We propose and analyze a new method for the unsteady incompressible magnetohydrodynamics equations on convex domains with hybrid approximations of both vector-valued and scalar-valued fields. The proposed method is convection-semirobust, meaning that, for sufficiently smooth solutions, one can derive a priori estimates for the velocity and the magnetic field that do not depend on the inverse of the diffusion coefficients. This is achieved while at the same time providing relevant additional features, namely an improved order of convergence for the (asymptotic) diffusion-dominated regime, a small stencil (owing to the absence of inter-element penalty terms), and the possibility to significantly reduce the size of the algebraic problems through static condensation. The theoretical results are confirmed by a complete panel of numerical experiments.

That Dirichlet processes are discrete with probability 1 is demonstrated once more. And yes, these two pages spent fifty years in Norwegian.

This study investigates how the interaction between gene expression time delay and domain size governs spatio-temporal pattern formation in a reaction-diffusion system. To investigate these phenomena, we utilize a modified version of the Schnakenberg model called the ligand internalisation (LI) model. In a one-dimensional domain, a linear relationship has been observed between the gene expression time delay and the time it takes for patterns to form. We extend the model to the two-dimensional domain and confirm that a similar relationship holds there as well. However, our exploration reveals a non-monotonic correlation between domain size and the time required for pattern emergence. To unravel these dynamics, we consider a range of initial conditions, including random perturbations of the spatially homogeneous steady state and initial conditions from its unstable manifold. We compute a two-parameter chart of patterns with respect to time delay and domain size.

We study the global solvability of a class of differential complexes on the product manifold $\mathbb{T}^m \times \mathbb{R}^n$ associated with systems of evolution operators of the form $L_r = \partial_{t_r} + ia_r(t)P(x,D_x), r=1,\ldots,m,$ where the coefficients $a_r$ are real-valued Gevrey functions on the torus and $P(x,D_x)$ is a globally elliptic normal differential operator on $\mathbb{R}^n$. Within the framework of time-periodic Gelfand--Shilov spaces, we introduce a natural differential complex generated by these operators and investigate its solvability in both functional and ultradistributional settings. We provide a complete characterization of global solvability in terms of a Diophantine condition involving the constant part of the associated $1$-form and the spectrum of $P$. We also analyze global hypoellipticity of the complex. These results extend previous works on scalar operators and constant coefficient systems to the setting of differential complexes with time-dependent real coefficients.

This paper focuses on a nonlinear convection-diffusion equation with space and time-fractional Laplacian operators of orders $1<\beta<2$ and $0<\alpha\leq1$, respectively. We develop local discontinuous Galerkin methods, including Legendre basis functions, for a solution to this class of fractional diffusion problem, and prove stability and optimal order of convergence $O(h^{k+1}+(\Delta t)^{1+\frac{p}{2}}+p^2)$. This technique turns the equation into a system of first-order equations and approximates the solution by selecting the appropriate basis functions. Regarding accuracy and stability, the basis functions greatly improve the method. According to the numerical results, the proposed scheme performs efficiently and accurately in various conditions and meets the optimal order of convergence.

The paper studies multifractal random measures on the sphere $\mathbb{S}^d$ constructed via multifractal products of random fields. It presents new limit theorems for multifractal products of spherical fields and conditions for the non-degeneracy of the limiting measure. The multifractal properties of the limiting measure are investigated, and its Rényi function is derived. Compared to earlier results on multifractal products of spherical fields, the obtained limit theorems hold under general mixing conditions, enabling the consideration of multifractal products of fields from a broad class and the construction of random measures with flexible multifractal properties.

In \cite{Lions}, J. L. Lions considered a reproducing kernel Hilbert space (RKHS) of harmonic functions on a regular domain with Sobolev traces and obtained a formula that expresses the kernel of this space as an integral on the boundary of some derivatives of the Green function associated with the Laplace operator and the homogeneous Dirichlet boundary condition. This result was simplified and extended later by Englis, Lukkassen, Peetre, and Persson in \cite{ELPL} to more general elliptic systems of even orders. In particular, they emphasized that the resemblance between Lions' type formula and the Hadamard variational formula only appears when the operator is of order $2$. In this paper, we investigate some RKHS of $a$-harmonic functions, where $a$ in $(0,1)$ refers to a fractional exponent of the Laplace operator. For such fractional order pseudo-differential operators, the local nonhomogeneous Dirichlet problem can be addressed by means of some $a$-transmission Sobolev spaces, which were introduced by Hörmander in the sixties and recently developed by Grubb in a series of papers. We deduce from these works a fractional Poisson formula, which is applied to obtain a Lions' type formula. We observe, in particular, that despite the order of the operator not being $2$, this formula resembles the Hadamard variational formula that we prove in the companion paper \cite{sidy-franck_1}. As a complementary remark, we observe that for a family of RKHS associated with the steady Stokes system, a second order system, there is also a Lions' type formula for their two-point kernels, which turn out not to be similar to the corresponding Hadamard variation formula.

This paper focuses on the input-to-state stabilization problem for an ordinary differential equation (ODE) cascaded by parabolic partial differential equation (PDE) in the presence of Dirichlet-Robin boundary disturbances, as well as in-domain disturbances. For the cascaded system with a Dirichlet pointwise interconnection, the ODE takes the value of a Robin boundary condition at the ODE-PDE interface as its direct input, and the PDE is driven by a Dirichlet boundary input at the opposite end. We first employ the backstepping method to design a boundary controller and to decouple the cascaded system. This decoupling facilitates independent stability analysis of the PDE and ODE systems sequentially. Then, to address the challenges posed by Dirichlet boundary disturbances to the application of the classical Lyapunov method, we utilize the generalized Lyapunov method to establish the ISS in the max-norm for the cascaded system involving Dirichlet boundary disturbances and two other types of disturbances. The obtained result indicates that even in the presence of different types of disturbances, ISS analysis can still be conducted within the framework of Lyapunov stability theory. For the well-posedness of the target system, it is conducted by using the technique of lifting and the semigroup method. Finally, numerical simulations are conducted to illustrate the effectiveness of the proposed control scheme and ISS properties for a cascaded system with different disturbances.

We prove that for any test category $A$, in the sense of Grothendieck, satisfying a compatibility condition between homology equivalences and weak equivalences of presheaves, the homotopy category of abelian presheaves on $A$ is equivalent to the non-negative derived category of abelian groups. This provides a homotopical generalization of the Dold-Kan correspondence for presheaves of abelian groups over a wide range of test categories. This equivalence of homotopy categories comes from a Quillen equivalence for a model structure on abelian presheaves that we introduce under these conditions. We then show that this result applies to Joyal's category $\Theta$.

This paper studies the workload distribution of a finite-capacity queue driven by a spectrally one-sided Markov additive process (MAP). Our main result provides the Laplace-Stieltjes transform of the workload at an exponentially distributed time, thereby uniquely characterizing its transient distribution. The proposed approach combines several decompositions with established fluctuation-theoretic results for spectrally one-sided Lévy processes. For the special case of Markov-modulated compound Poisson input, we additionally derive results for the idle time and the cumulative amount of lost work. We conclude this paper with a series of numerical experiments.

We establish Sobolev and Moser-Trudinger inequalities with best constants on noncompact Riemannan manifolds with Ricci curvature bounded below, and positive injectivity radius.

This paper presents a systematic review of recent advances in nonlinear filtering algorithms, structured into three principal categories: Kalman-type methods, Monte Carlo methods, and the Yau-Yau algorithm. For each category, we provide a comprehensive synthesis of theoretical developments, algorithmic variants, and practical applications that have emerged in recent years. Importantly, this review addresses both continuous-time and discrete-time system formulations, offering a unified review of filtering methodologies across different frameworks. Furthermore, our analysis reveals the transformative influence of artificial intelligence breakthroughs on the entire nonlinear filtering field, particularly in areas such as learning-based filters, neural network-augmented algorithms, and data-driven approaches.

Let $Q_1,...,Q_r\in \mathbb{Z}[x]$ be polynomials having $0$ as a root. Let $f(x,y)\in\mathbb{Z}[x,y]$ be a homogeneous polynomial with factorization $f(x,y)=f_1(x,y)^{e_1}\cdots f_u(x,y)^{e_u}$, where $f_i(x,y)$ are irreducible homogeneous polynomials of degree $d_i\geq 2$. Fix some positive integers $A_1,...,A_r$. We show that under certain conditions, the diophantine equation $\prod_{i=1}^rQ_i(A_i^{n_i}n_i!)=f(x,y)$ has finitely many integer solutions.

We study a generalized vanishing discount problem for Hamilton--Jacobi equations, removing the standard monotonicity assumption, either in a global sense or when integrated against all Mather measures. Specifically, we consider \[ \lambda a(x)u(x)+H(x,Du(x))-A\lambda=c_0, \] with a suitably chosen constant $A>0$. By appropriately changing the signs of the function $a(x)$ on different static classes associated with $H$, we show that the maximal viscosity solution converges uniformly as $\lambda\to 0^+$ and that all elementary solutions of the stationary equation \[ H(x,Du(x))=c_0 \] can be selected as limits. This provides the first result for selecting multiple viscosity solutions in vanishing discount problems beyond the usual monotonicity and integral assumptions, as long as $a(x)$ is positive on one static class. Our results highlight the crucial role of static classes in controlling the asymptotic behavior of viscosity solutions. Previously, under usual monotonicity assumptions, only a single solution could be selected (as discussed in \cite{GL}), whereas our approach allows controlled selection of multiple solutions via static class-guided discount coefficients.

This paper presents a unified theory for the power of a point with respect to generalized spheres (spheres, horospheres, and hyperspheres) in $n$-dimensional hyperbolic space $\mathbf{H}^n$. By extending the classical secant theorem, we derive a novel formula for hyperspheres and also prove that the radical surface of any two non-concentric generalized spheres is a hyperplane. These results provide tools for constructing power diagrams and studying hyperball packings.

We prove a refined version of Markov's theorem in Diophantine approximation. More precisely, we characterize completely the set of irrationals $x$ such that $\left|x-\frac{p}{q}\right|<\frac{1}{3q^2}$ has only finitely many rational solutions: their continued fraction is eventually a balanced sequence through a simple coding. As consequence, we show that all such numbers are either quadratic surds or transcendental numbers. In particular, for any algebraic real number $x$ of degree at least $3$ there are infinitely rational numbers $\frac{p}{q}$ such that $\left|x-\frac{p}{q}\right|<\frac{1}{3q^2}$.

Polyhedra and spectrahedra over the real numbers, or more generally their images under linear maps, are respectively the feasible sets of linear and semidefinite programming, and form the family of semidefinite-representable sets. This paper studies analogues of these sets, as well as the associated optimization problems, when the data are taken over a valued field $K$. For $K$-polyhedra and linear programming over $K$ we present an algorithm based on the computation of Smith normal forms. We prove that fundamental properties of semidefinite-representable sets extend to the valued setting. In particular, we exhibit examples of non-polyhedral $K$-spectrahedra, as well as sets that are semidefinite-representable over $K$ but are not $K$-spectrahedra.

This short note is devoted to the study of $G$-Higgs bundles twisted by a central gerbe. These objects arise naturally in the decomposition of the inertia stacks of $G$-Higgs bundles in terms of coendoscopic data. We establish that stabilised point-counts and cohomology are insensitive to the central twist. Along the way we show an analogue of Ngô's product formula for twisted Hitchin fibres.

Directed information (DI) is an information measure that attempts to capture directionality in the flow of information from one random process to another. It is closely related to other causal influence measures, such as transfer entropy, Granger causality, and Pearl's causal framework. This monograph provides an overview of DI and its main application in information theory, namely, characterizing the capacity of channels with feedback and memory. We begin by reviewing the definitions of DI, its basic properties, and its relation to Shannon's mutual information. Next, we provide a survey of DI estimation techniques, ranging from classic plug-in estimators to modern neural-network-based estimators. Considering the application of channel capacity estimation, we describe how such estimators numerically optimize DI rate over a class of joint distributions on input and output processes. A significant part of the monograph is devoted to techniques to compute the feedback capacity of finite-state channels (FSCs). The feedback capacity of a strongly connected FSC involves the maximization of the DI rate from the channel input process to the output process. This maximization is performed over the class of causal conditioned probability input distributions. When the FSC is also unifilar, i.e., the next state is given by a time-invariant function of the current state and the new input-output symbol pair, the feedback capacity is the optimal average reward of an appropriately formulated Markov decision process (MDP). This MDP formulation has been exploited to develop several methods to compute exactly, or at least estimate closely, the feedback capacity of a unifilar FSC. This monograph describes these methods, starting from the value iteration algorithm, to Q-graph methods, and reinforcement learning algorithms that can handle large input and output alphabets.

High-order moving-mesh methods can effectively reduce numerical diffusion, but their formal accuracy typically relies on the regularity of the mesh velocity. This dependency creates a fundamental conflict in the numerical solution of hyperbolic conservation laws, where solution-driven adaptation may induce nonsmooth mesh motion, thereby degrading convergence order. We introduce \emph{transport polynomial exactness} (TPE($k$)), a mesh-motion-independent criterion that generalizes classical free-stream preservation (TPE(0)) to the exact advection of degree-$k$ polynomials. We show that the classical geometric conservation law (GCL) is insufficient to ensure TPE($k$) for $k \ge 1$ due to mismatches in higher-order geometric moments. To resolve this, we propose \emph{evolved geometric moments} (EGMs), obtained by solving auxiliary transport equations discretized compatibly with the physical variables. We rigorously prove that second-degree EGMs evolved via the third-order strong stability preserving Runge--Kutta (SSPRK3) method coincide with the exact geometric moments. This exactness arises from a \emph{superconvergence} mechanism wherein SSPRK3 reduces to Simpson's rule for EGM evolution. Leveraging this result, we construct a third-order conservative finite-volume rezoning moving-mesh scheme. The scheme satisfies the TPE(2) property for \emph{arbitrary mesh motion} and \emph{any pseudo-time step size}, thereby naturally accommodating spatiotemporally discontinuous mesh velocity. Crucially, this \emph{breaks the efficiency bottleneck} in the conventional advection-based remapping step and reduces the required pseudo-time levels from $\mathcal{O}(h^{-1})$ to $\mathcal{O}(1)$ under bounded but discontinuous mesh velocity. Numerical experiments verify exact quadratic transport and stable third-order convergence under extreme mesh deformation, demonstrating substantial efficiency gains.

We show that the spectrum of the open-boundary limit of banded Toeplitz matrices is real whenever the associated symbol function is real-valued along a closed polar curve. Building on this result, we develop both analytical and numerical methods to symmetrise a class of banded non-Hermitian Toeplitz matrices whose asymptotic spectra are real. Finally, we provide a rigorous mathematical foundation for the generalised Brillouin zone, a concept widely used in non-Hermitian physics, by proving that it coincides with the polar curve on which the symbol function takes real values.

In this article, we first introduce the one-sided BLO space $\mathrm{BLO}^+(\mathbb{R})$ and characterize it, respectively, in terms of the one-sided Muckenhoupt class $A_1^+(\mathbb{R})$ and the one-sided John--Nirenberg inequality. Using these, we establish the Coifman--Rochberg type decomposition of $\mathrm{BLO}^+(\mathbb{R})$ functions and show that $\mathrm{BLO}^+(\mathbb{R})$ is independent of the distance between the two intervals, which further induces the characterization of this space in terms of the one-sided BMO space $\mathrm{BMO}^+(\mathbb{R})$ (the Bennett type lemma). As applications, we prove that any $\mathrm{BMO}^+(\mathbb{R})$ function can split into the sum of two $\mathrm{BLO}^+(\mathbb{R})$ functions and we provide an explicit description of the distance from $\mathrm{BLO}^+(\mathbb{R})$ functions to $L^\infty(\mathbb{R})$. Finally, as a higher-dimensional analogue we introduce the parabolic BLO space $\mathrm{PBLO}_\gamma^-(\mathbb{R}^{n+1})$ with time lag, and we extend all the above one-dimensional results to $\mathrm{PBLO}_\gamma^-(\mathbb{R}^{n+1})$; furthermore, as applications, we not only establish the relationships between $\mathrm{PBLO}_\gamma^-(\mathbb{R}^{n+1})$ and the solutions of doubly nonlinear parabolic equations, but also provide a necessary condition for the negative logarithm of the parabolic distance function to belong to $\mathrm{PBLO}_\gamma^-(\mathbb{R}^{n+1})$ in terms of the weak porosity of the set.

We study commutators of the Riesz potential $I_\alpha$ with functions $b$ in the capacitary space $\mathrm{BMO}^\beta(\mathbb{R}^n)$, defined through the Hausdorff content $\mathcal{H}^\beta_\infty$. We prove a Chanillo-type theorem characterising $\mathrm{BMO}^\beta(\mathbb{R}^n)$ via the boundedness of the commutator $[b,I_\alpha]$ on capacitary Lebesgue spaces. In addition, we obtain the endpoint estimate in the form of a capacitary modular weak-type inequality. These results follow from a pointwise estimate for the $\beta$-dimensional sharp maximal function of the commutator, together with a capacitary Fefferman-Stein inequality recently proved in [CC24].

In this paper, we characterize all polynomial Kolmogorov vector fields for which the standard $n$-sphere is invariant. We exhibit completely integrable Kolmogorov vector fields of degree $m$ on $\mathbb{S}^n$ for any $m >2$. Then, we show that there is no cubic Hamiltonian Kolmogorov vector field that makes an odd-dimensional sphere invariant. We examine the conditions under which a cubic Kolmogorov vector field has a Darboux first integral. In many cases, we determine whether they constitute necessary and sufficient conditions. Moreover, we study the complete integrability of cubic Kolmogorov vector fields having an invariant $n$-sphere.

In model extraction attacks, the goal is to reveal the parameters of a black-box machine learning model by querying the model for a selected set of data points. Due to an increasing demand for explanations, this may involve counterfactual queries besides the typically considered factual queries. In this work, we consider linear models and three types of queries: factual, counterfactual, and robust counterfactual. First, for an arbitrary set of queries, we derive novel mathematical formulations for the classification regions for which the decision of the unknown model is known, without recovering any of the model parameters. Second, we derive bounds on the number of queries needed to extract the model's parameters for (robust) counterfactual queries under arbitrary norm-based distances. We show that the full model can be recovered using just a single counterfactual query when differentiable distance measures are employed. In contrast, when using polyhedral distances for instance, the number of required queries grows linearly with the dimension of the data space. For robust counterfactuals, the latter number of queries doubles. Consequently, the applied distance function and robustness of counterfactuals have a significant impact on the model's security.

In the previous decades, the size of level sets of functions have been extensively studied in various setups involving different regularity properties and size notions. In the case of Hölder functions, the authors have provided various bounds, but to date no explicit formulae have been found for any studied dimension and the results were valid only about very specific fractals. In this paper, for the first time, we have a result valid for a large class of self-similar sets, namely we prove that for these fractals Lebesgue almost every level set of the generic 1-Hölder-$\alpha$ function defined on $F\subseteq \mathbb{R}^p$ has upper box dimension $\dim_H F - \alpha$.

Let $\Tei_{g,n}$ be the Teichmüller space of Riemann surfaces of genus $g$ with $n$ punctures. It is conjectured that the Teichmüller and Carathéodory metrics agree on a Teichmüller disk if and only if all the zeros of the corresponding holomorphic quadratic differential are of even order. The conjecture was proved by Gekhtman and Markovic for $\Tei_{0,5}\cong \Tei_{1,2}$. We confirm the conjecture for $\Tei_{2,0}\cong\Tei_{0,6}$.

The Gaussian kernel is one of the most important kernels, applicable to many research fields, including scientific computing and data science. In this paper, we present asymptotic analysis of the Gaussian kernel matrix in high dimension under a statistical model of noisy data. The main result is a nice combination of Karoui's asymptotic analysis with procedures of constrained low rank matrix approximations. More specifically, Karouli clarified an important asymptotic structure of the Gaussian kernel matrix, leading to strong consistency of the eigenvectors, though the eigenvalues are inconsistent. This paper focuses on the above results and presents a consistent estimator with the use of the smallest eigenvalue, whenever the target kernel matrix tends to low rank in the asymptotic regime. Importantly, asymptotic analysis is given under a statistical model representing partial noise. Although a naive estimator is inconsistent, applying an optimization method for low rank approximations with constraints, we overcome the difficulty caused by the inconsistency, resulting in a new estimator with strong consistency in rank deficient cases.

We study frequency moments of partition statistics arising from Euler products $A(q)=\prod_{r\ge1}(1-q^r)^{-c(r)}$ via a transform that expresses the moment generating functions as $B(q)$ times explicit divisor--sum series determined by $c(r)$. When $A(q)$ is modular (typically an $\eta$--quotient), this yields (quasi)modular forms whose coefficients can be projected to arithmetic progressions and certified modulo primes by a Sturm bound, giving an effective pipeline for detecting and proving Ramanujan--type congruences for frequency moments. For ordinary partitions we recover and certify several congruences for odd moments in nonzero residue classes (e.g.\ $M_3(7n+5)\equiv 0\pmod7$ and $M_3(11n+6)\equiv 0\pmod{11}$). As a second input, we apply the same pipeline to overpartitions and certify a family of zero--class congruences $M_m^{\overline{\ }}(\ell n)\equiv 0\pmod{\ell}$ (including $m=5,7,11,13$), exhibiting a sharp contrast with the ordinary partition case: no nonzero residue--class congruences are observed for overpartition moments in our scan range. We also demonstrate that filtering the statistic via the Glaisher--character dictionary can itself create new Ramanujan--type progressions, e.g.\ a quadratic twist yields the certified congruence $\widehat{M}^{\chi_5}_3(5n+4)\equiv 0\pmod{5}$.

We analyze the dynamical (in)stability of nematic liquid crystals in the presence of external magnetic fields and Rapini-Papoular surface potential. The P-HAN transition is investigated using a simplified 3D Ericksen-Leslie system. We find the thickness threshold of the P-HAN transition. If the thickness of the nematic layer exceeds this threshold, there is a global-in-time suitable weak solution converging exponentially to a nontrivial equilibrium state as time tends to infinity. If the thickness is no more than the threshold, the global-in-time suitable weak solution has a trivial long-time asymptotic limit. Our results rigorously justify the P-HAN transition discussed in the physics literature.

We introduce the notion of 'centre' for pomonoid-graded strong monads which generalizes some previous work that describes the centre of (not graded) strong monads. We show that, whenever the centre exists, this determines a pomonoid-graded commutative submonad of the original one. We also discuss how this relates to duoidally-graded strong monads.

We study the Muskat problem, which describes the motion of two immiscible, incompressible fluids in a homogeneous porous medium occupying the full space ${\mathbb{R}^{N+1}}$, $N \geq 2$, driven by gravity. The interface between the fluids is given as graph of a function over $\mathbb{R}^N$. The problem is reformulated as a nonlinear, nonlocal evolution problem for this function, involving singular integrals arising from potential representations of the velocity and pressure fields. Using results from harmonic analysis, we demonstrate that the evolution is of parabolic type in the open set identified by the Rayleigh-Taylor condition. We use the abstract theory of such problems to establish that the Muskat problem defines a semiflow on this set in all subcritical Sobolev spaces $H^s(\mathbb{R}^N)$, $s>s_c$, where ${s_c=1+N/2}$ is the critical exponent. We additionally obtain parabolic smoothing up to ${\rm C}^\infty$.

We introduce a Toeplitz-based framework for data-driven spectral estimation of linear evolution operators in dynamical systems. Focusing on transfer and Koopman operators from equilibrium trajectories without access to the underlying equations of motion, our method applies Toeplitz filters to the infinitesimal generator to extract eigenvalues, eigenfunctions, and spectral measures. Structural prior knowledge, such as self-adjointness or skew-symmetry, can be incorporated by design. The approach is statistically consistent and computationally efficient, leveraging both primal and dual algorithms commonly used in statistical learning. Numerical experiments on deterministic and chaotic systems demonstrate that the framework can recover spectral properties beyond the reach of standard data-driven methods.

We perform the quantization of Teukolsky scalars of spin $0$, $\pm 1$, and $\pm 2$ within the algebraic approach to quantum field theory. We first discuss the classical phase space, from which we subsequently construct the algebra. This sheds light on which fields are conjugates of each other. Further, we construct the Unruh state for this theory on Kerr and show that it is Hadamard on the black hole exterior and the interior up to the inner horizon. This shows not only that Hadamard states exist for this theory, but also extends the existence and Hadamard property of the Unruh state to (bosonic) Teukolsky fields on Kerr, where such a result was previously missing.

Here we show that the Weil abscissa of the procyclic groups $\prod_{p \in S} \mathbb{Z}_p$ equals $2$ for three sets $S$: (i) the set of primes $p \equiv 1 \bmod 3$, (ii) the set of primes $p \equiv 1 \bmod 4$ and (iii) the set of primes $p \equiv 1,3 \bmod 8$. Our argument is based on the observation that integers all of whose prime factors lie in $S$ can be represented by a suitable binary quadratic form, which allows us to use a theorem of Iwaniec to exhibit a minorant for the Weil representation zeta function.

This article introduces a novel framework for developing quantum algorithms for the Lattice Boltzmann Method (LBM) applied to the advection-diffusion equation. We formulate the collision-streaming evolution of the LBM as a compact time-marching scheme and rigorously establish its stability under low Mach number conditions. This unified formulation eliminates the need for classical measurement at each time step, enabling a systematic and fully quantum implementation. Building upon this representation, we investigate two distinct quantum algorithmic approaches. The first is a time-marching quantum algorithm realized through sequential evolution operators, for which we provide a detailed implementation-including block-encoding and dilating unitarization-along with a full complexity analysis. The second employs a quantum linear systems algorithm, which encodes the entire time evolution into a single global linear system. We demonstrate that both methods achieve comparable asymptotic time complexities. The proposed algorithms are validated through numerical simulations of benchmark problems in one and two dimensions. This work provides a systematic, measurement-free pathway for the quantum simulation of advection-diffusion processes via the lattice Boltzmann paradigm.

Fix an integer $r\ge2$. For each $n$ we consider families $\mathcal F\subseteq 2^{[n]}$ that form an antichain and have the property that, for every $t$, if there exists $A\in\mathcal F$ with $|A|=t$ then there exist at least $r$ members of $\mathcal F$ of size $t$. A problem of Erdős and Trotter asserts that, for each fixed $r$, there exists a threshold $n_0(r)$ such that whenever $n>n_0(r)$ one can achieve $n-3$ distinct set sizes in such a family, and asks for estimates on $n_0(r)$. We compute that $n_0(2)=3$ and $n_0(3)=8$. For all $r\ge4$ we prove matching linear bounds up to lower-order terms, namely $$ 2r+2 \le n_0(r) \le 2r+2\log_2 r + O(\log_2\log_2 r). $$

Consider the motion of a viscous incompressible fluid filling a 3D exterior domain $\Omega$ subject to the Navier slip-with-friction boundary condition as well as outflow at infinity. For the Oseen system as the linearization, we discuss the resolvent set under a certain relationship among the geometry of the boundary $\partial\Omega$, friction coefficient $\alpha(x)$ and the outflow $u_\infty$. We then study the regularity of the resolvent near the origin in the complex plane to develop $L^q$-$L^r$ decay estimates of the Oseen semigroup provided that $\alpha(x)+u_\infty\cdot\nu(x)/2\geq 0$ for every $x\in\partial\Omega$, where $\nu(x)$ stands for the outward unit normal to the boundary $\partial\Omega$.

We study the behavior of solutions of a monostable reaction-diffusion equation $u_t=\Delta_x u +u_{yy} +f(u)$ ($x \in \mathbb{R}^{n-1}$, $y \in \mathbb{R}$, $t>0$), with the unstable equilibrium point $0$ and the stable equilibrium point $1$. Under the condition that the corresponding one-dimensional equation has a pushed front $\Phi_{c^*}(z)$ with $\Phi_{c^*}(-\infty)=1$, $\Phi_{c^*}(\infty)=0$, we show that the solution $u(x,y,t)$ approaches $\Phi_{c^*}(y-\gamma(x,t))$ for some $\gamma(x,t)$ as $t \to \infty$, if initially $u(x,y,0)$ decays sufficiently fast as $y \to \infty$ and is bounded below by some positive constant near $y=-\infty$. It is also shown that $\gamma(x,t)$ is approximated by the mean curvature flow with a drift term.

Existence of solutions to doubly nonlinear equations in reflexive Banach spaces is established by resorting to a global-in-time variational approach inspired by De Giorgi's principle, which characterizes the associated flows as null-minimizers of a suitable energy-dissipation functional defined on trajectories. In contrast to the celebrated minimizing movements scheme, the proposed strategy does not rely on any time-discretization or iterative constructions. Instead, it provides a direct method based on the relaxation of the problem in spaces of measures, constrained by the continuity equation: in this procedure, no gap is introduced due to the Ambrosio's superposition principle. Within this weak convex framework, the validity of the null-minimization property is recovered through two further steps. First, a careful application of the Von Neumann minimax theorem yields an identification of the dual problem as a supremum over the set of smooth and bounded cylinder functions, solving an Hamilton-Jacobi-type inequality. Secondly, a suitable "backward boundedness" property of solutions to such Hamilton-Jacobi system gives a proper bound of the dual problem, ensuring that the minimum value of the original functional is actually zero. The proposed strategy naturally extends to non-autonomous equations, encompassing time- and space-dependent dissipation potentials and time-dependent potential energies.

We characterise all vertex-transitive finite connected graphs as essentially 5-connected or on a short list of explicit graph-classes. Our proof heavily uses Tutte-type canonical decompositions.

The space of all finite non-empty subsets of a topological space $X$, also known as the Ran space of $X$, is weakly contractible for $X$ path connected. We consider subspaces $\mathrm{Ran}_{\leqslant n}(X)$ of the Ran space given by all subsets of $X$ of size at most $n$, and present results on their first homotopy groups. In particular, we show that the induced map $\pi_1(\mathrm{Ran}_{\leqslant n}(X)) \to \pi_1(\mathrm{Ran}_{\leqslant n+2}(X))$ is trivial for all positive integers $n$, and even more, show that $\pi_1(\mathrm{Ran}_{\leqslant n}(X)) = 0$ for all $n\geqslant 4$, by explicitly drawing the path homotopies that contract any loop to a point.

We study Blaschke--Santal{ó}-type inequalities for $N \ge 2$ sets (functions) and a special class of cost functions. In particular, we prove new results about reduction of the maximization problem for the Blaschke--Santal{ó}-type functional to homogeneous case (functional inequalities on the sphere) and extend the symmetrization argument to the case of $N > 2$ sets. We also discuss links to the multimagrinal optimal transportation problem and the related sharp transportation-information inequalities.

We study the well-quasi-order (wqo) consisting of the set of finite trees with leaf labels coming from an arbitrary wqo $Q$, ordered by tree homomorphisms which respect the order on the labels. This is a variant of the usual Kruskal tree ordering without infima preservation. We calculate the precise maximal order types of this class of wqos as a function of the maximal order type of the labels $Q$. In the process, we sharpen some recent results of Friedman and Weiermann. Furthermore, we show a correspondence with indecomposable transfinite sequences with finite range, over elements of the wqo $Q$, of length less than $\omega^\omega$. Nash-Williams proved that arbitrary transfinite sequences with finite range are also well-quasi-ordered, but there are no known methods to extract bounds on the maximal order type from the proof. More concrete proofs for sequences of length less than $\alpha$ for some $\alpha < \omega^\omega$ were given by Erdős and Rado. Using the correspondence, we obtain precise bounds for the entire collection of transfinite sequences with finite range of length less than $\omega^\omega$.

Consider a unitary group $G(\mathbb{A}_{F^+})=U_{2r}(\mathbb{A}_{F^+})$ over a CM extension $F/F^+$ with $G(\mathbb{A}_\infty)$ compact. In this article, we study the Beilinson--Bloch--Kato conjecture for motives associated to irreducible cuspidal automorphic representations $\pi$ of $G(\mathbb{A}_{F^+}).$ We prove that if $\pi$ is distinguished by the unitary Friedberg--Jacquet period, then the Bloch--Kato Selmer group (with coefficients in a favorable field) of the motive of $\Pi=\mathrm{BC}(\pi)$ vanishes.

The broken random sample problem was first introduced by DeGroot, Feder, and Gole (1971, Ann. Math. Statist.): in each observation (batch), a random sample of $M$ i.i.d. point pairs $ ((X_i,Y_i))_{i=1}^M$ is drawn from a joint distribution with density $p(x,y)$, but we can observe only the unordered multisets $(X_i)_{i=1}^M$ and $(Y_i)_{i=1}^M$ separately; that is, the pairing information is lost. For large $M$, inferring $p$ from a single observation has been shown to be essentially impossible. In this paper, we propose a parametric method based on a pseudo-log-likelihood to estimate $p$ from $N$ i.i.d. broken sample batches, and we prove a fast convergence rate in $N$ for our estimator that is uniform in $M$, under mild assumptions.

Stochastic minimax optimization on Riemannian manifolds has recently attracted significant attention due to its broad range of applications, such as robust training of neural networks and robust maximum likelihood estimation. Existing optimization methods for these problems typically require selecting stepsizes based on prior knowledge of specific problem parameters, such as Lipschitz-type constants and (geodesic) strong concavity constants. Unfortunately, these parameters are often unknown in practice. To overcome this issue, we develop single-loop adaptive methods that automatically adjust stepsizes using cumulative Riemannian (stochastic) gradient norms. We first propose a deterministic single-loop Riemannian adaptive gradient descent ascent method and show that it attains an $\epsilon$-stationary point within $O(\epsilon^{-2})$ iterations. This deterministic method is of independent interest and lays the foundation for our subsequent stochastic method. In particular, we propose the Riemannian stochastic adaptive gradient descent ascent method, which finds an $\epsilon$-stationary point in $O(\epsilon^{-6})$ iterations. Under additional second-order smoothness, this iteration complexity is further improved to $O(\epsilon^{-4})$, which even outperforms the corresponding complexity result in Euclidean space. Some numerical experiments on real-world applications are conducted, including the regularized robust maximum likelihood estimation problem, and the robust training of neural networks with orthonormal weights. The results are encouraging and demonstrate the effectiveness of adaptivity in practice.

We present a theoretical analysis of stochastic optimization methods in terms of their sensitivity with respect to the step size. We identify a key quantity that, for each method, describes how the performance degrades as the step size becomes too large. For convex problems, we show that this quantity directly impacts the suboptimality bound of the method. Most importantly, our analysis provides direct theoretical evidence that adaptive step-size methods, such as SPS or NGN, are more robust than SGD. This allows us to quantify the advantage of these adaptive methods beyond empirical evaluation. Finally, we show through experiments that our theoretical bound qualitatively mirrors the actual performance as a function of the step size, even for nonconvex problems.

In this paper, we present the Brouwer-Schauder-Tychonoff fixed point theorem on locally convex spaces as the following extension and improvement: Suppose that S is a compact star-shaped subset with respect to p in S with its convexity index alpha(p)>0. Then every continuous self-mapping f has one of the following two properties: (a) The point p is a fixed point of f; (b) f has uncountably many different eigenvalues and eigenvectors. Note that a closed bounded star-shaped set in a locally convex space is convex if and only if alpha=1, and we extend a Brouwer's type fixed-point theorem on compact star-shaped sets in Banach spaces in a more concise manner to locally convex spaces, thereby this is a simplification and an improvement of the Tychonoff fixed-point theorem to compact star-shaped sets.

This paper establishes the asymptotic error distribution of the tamed Euler method for stochastic differential equations (SDEs) with a coupled monotonicity condition, that is, the limit distribution of the corresponding normalized error process. Specifically, for SDEs driven by multiplicative noise, we first propose a tamed Euler method parameterized by $\alpha\in (0, 1]$ and establish that its strong convergence rate is $\alpha\wedge\frac{1}{2}$. Notably, $\alpha $ can take arbitrary positive values by adjusting the regularization coefficient without altering the strong convergence rate. We then derive the asymptotic error distribution for this tamed Euler method. Further, we infer from the limit equation that among the tamed Euler method of strong order $\frac{1}{2}$, the one with $\alpha = \frac{1}{2}$ yields the largest mean-square error after a long time, while those of $\alpha>\frac{1}{2}$ share a unified asymptotic error distribution. In addition, our analysis is also extended to SDEs with additive noise and similar conclusions are obtained. Additional treatments are required to accommodate super-linearly growing coefficients, a feature that distinguishes our analysis on the asymptotic error distribution from established results.

The directed landscape is a random directed metric on the plane that arises as the scaling limit of metric models in the KPZ universality class. For a pair of points p, q, the disjointness gap G(p; q) measures the shortfall when we optimize length over pairs of disjoint paths from p to q versus optimizing over all pairs of paths. Any spatial marginal of G is simply the gap between the top two lines in an Airy line ensemble. In this paper, we show that when the start and end time are fixed, the disjointness gap fully encodes the set of exceptional geodesic networks. The correspondence uses simple features of the disjointness gap, e.g. zeroes, local minima. We give a similar correspondence relating semi-infinite geodesic networks to a Busemann gap function. The proofs are deterministic given a list of soft properties related to the coalescent geometry of the directed landscape.

Aboulker, Aubian, Charbit, and Lopes (2023) defined the clique number of a tournament to be the minimum clique number of one of its backedge graphs. Here we show that if $T$ is a tournament of sufficiently large clique number, then $T$ contains a subtournament of large clique number from one of two simple families of tournaments. In particular, large clique number is always certified by a bounded-size set. This answers a question of Aboulker, Aubian, Charbit, and Lopes (2023), and gives new insight into a line of research initiated by Kim and Kim (2018) into unavoidable subtournaments in tournaments with large dichromatic number.

In this paper, we study the GENERIC structures of multi-species spatially inhomogeneous Boltzmann and Landau equations with Bose-Einstein, Maxwell-Boltzmann, and Fermi-Dirac statistics. In addition, under suitable assumptions on the collision kernels, we show that the Fisher information for the multi-species spatially homogeneous Boltzmann equation is non-increasing in time.

We introduce a new biharmonic Steklov problem on differential forms with Dirichlet-type boundary conditions and show that it is elliptic. We prove the existence of a discrete spectrum for this problem and give variational characterizations for eigenvalues associated to it. We establish eigenvalue estimates known as Kuttler-Sigillito inequalities, that connect the eigenvalues of different problems on differential forms with curvature quantities on the manifold.

Let $k$ be an algebraically closed field with characteristic $2$, and let $X$ be a smooth projective algebraic curve of genus $g \geqslant 2$ over $k$. Let $\mathcal{M}^s_X(2,\mathcal{L})$ be the moduli space of rank $2$ stable vector bundles with determinant $\mathcal{L}$ on $X$. The Frobenius stratification measures the instability of bundles in $\mathcal{M}^s_X(r,\mathcal{L})$ under pullback by the Frobenius map. We show that there exists a Frobenius stratum in $\mathcal{M}^s_X(2,\mathcal{L})$ which is covered by Hecke curves.

Given a graph $G$, a configuration space of $G$ can be thought of as the set of all possible configurations of "robots" which can move throughout $G$, subject to some constraints. We introduce a type of configuration space which we call Grouped Stirling complexes, denoted by $S_{\vec r}(G)$, in which we place robots in groups subject to two constraints. First, there must be at least one robot on each vertex of $G$, and second, any two robots from the same group must be "separated by at least one full open edge" of $G$. The space $S_{\vec r}(G)$ has a closed cell structure, which means it can be built out of cells of various dimensions. Our main results show $S_{\vec r}(G)$ is path-connected, provided there are at least three groups, and determine the number of cells of $S_{\vec r}(G)$ in certain cases.

We provide a complete geometric solution to the differentiation problem for simplicial manifolds, extending classical Lie theory and subsuming existing homotopical and formal approaches within a unified framework. First, we establish a normal form theorem setting a system of compatible tubular neighborhoods. Building on this description, we identify a differentiating ideal in the algebra of cochains, prove that the quotient is semi-free, and interpret it as the Chevalley-Eilenberg algebra of the thus defined higher Lie algebroid. As an application, we introduce a higher version of the van Est map and prove a van Est isomorphism theorem in cohomology, under natural connectivity assumptions. Finally, we identify the algebraic mechanism underlying geometric differentiation as a monoidal refinement of the dual Dold-Kan correspondence, providing a conceptual explanation of the construction and relating it to earlier homotopical and functor-of-points approaches.

For any imaginary quadratic field $K$, the Galois group $G_K$ of its maximal unramified pro-$3$-extension is a Schur $\sigma$-group. If this has Zassenhaus type $(3,3)$, there are 13 possibilities for the isomorphism class of the finite quotient $G_K/D_4(G_K)$. We prove that for 10 of these 13 cases $G_K$ is either finite or isomorphic to an open subgroup of a form of $\mathop{\rm PGL}_2$ over $\mathbb{Q}_3$. Combined with the Fontaine-Mazur conjecture, or with earlier work on an analogue of the Cohen--Lenstra heuristic for Schur $\sigma$-groups, this lends credence to the "if" part of a conjecture of McLeman. Using explicit computations of triple Massey products, we also test the heuristic for all imaginary quadratic fields $K$ with $d(G_K)=2$ and discriminant $-10^8 < d_K < 0$ and find a reasonably good agreement.

The fine curve complex of a surface is a simplicial complex whose vertices are essential simple closed curves and whose $k$-simplices are collections of $k+1$ disjoint curves. We prove that the fine curve complex is homotopy equivalent to the curve complex. We also prove that the fine arc complex is contractible.

This paper is a sequel to arXiv:2501.14444, in which we shall give proofs of several results stated in arXiv:2501.14444 (Theorems D--L) which, for brevity and clarity, we postponed to this sequel paper. These results were the following: for any prime number $p$, first we show that every $2$-by-$2$ symmetric matrix with coefficients in $\mathbb{Q}_p$ can be reduced to a canonical form, and we give the exact numbers of families of normal forms with one parameter and of isolated normal forms, which depend on $p$. Then we make the same analysis for $4$-by-$4$ matrices. We also prove that, for higher size, the number of families of normal forms of matrices, even in the non-degenerate case, grows almost exponentially with the size. The paper can be read independently of arXiv:2501.14444 as we recall the statements of arXiv:2501.14444 that we shall prove here. The statements and proofs of the present paper are of an algebraic and arithmetical nature, and rely mainly on Galois theory of $p$-adic extension fields.

We investigate homological and depth-theoretic properties of finitely generated modules of finite projective dimension over Noetherian local rings. A central theme is the study of criteria for freeness and reflexivity derived from the torsion-freeness or reflexivity of tensor products of the form \( M \otimes_R M \) and \( M \otimes_R M^* \). Under mild homological assumptions, we prove that such properties of these tensor products impose strong structural constraints on \( M \), often forcing it to be free. These results generalize classical theorems of Auslander beyond the regular case and contribute to the broader understanding of rigidity phenomena in commutative algebra. The second part of the paper is devoted to the dimension and support of Ext-modules, particularly \( {Ext}^i_R(M, R) \) for critical values of i, when \( M \) has finite projective dimension. We establish sharp bounds on their Krull dimensions, analyze their behavior for prime and equidimensional modules, and relate these findings to the grade conjecture and other homological conjectures. Applications include new cases of a question of Jorgensen, which asks whether \( {pd}(M) < i \) whenever \({Ext}^i_R(M, M) = 0 \) and M has finite projective dimension over a complete intersection ring. Finally, we examine the projective dimension of prime ideals in rings that fail chain conditions. We show that in non-catenary or pathological saturated chain settings, such prime ideals often have infinite projective dimension.

We study the local Hölder continuity of nonnegative solutions to doubly nonlinear equations by introducing a new technique that allows us to treat the cases where the equation is both singular and degenerate, up to specific Barenblatt numbers. Our argument relies on a new integral $L^1$-$L^1$ Harnack estimate, of independent interest.

It is shown that if $F$ denotes the number of filled cells in a superimposed pair of maximal orthogonal partial Latin squares of order $n$, then $F\ge n^2/3$. This resolves a conjecture raised in an earlier paper by the current authors. It is also shown that, for $n\ge 21$, the least possible number of filled cells in a pair of maximal orthogonal partial Latin squares is $\lceil n^2/3 \rceil$, and that the structure that achieves this bound is unique up to permutations of rows, columns and entries.

We explain how the Johnson homomorphism and the Enomoto-Satoh trace, as well as higher-loop-order generalizations, can be obtained from graph complexes originating in the Goodwillie-Weiss calculus. This paper can be seen as an addendum to our earlier work. It contains little new mathematical content, but is intended to give an overview of a different viewpoint on the Johnson homomorphism, for experts working mainly in the latter area.

We derive unique Banach-valued solutions to stochastic Volterra equations with random coefficients that may depend on pure chance and involve singular kernels. In particular, for controlled and distribution-dependent coefficients these solutions become strong, as a measurability analysis of the Wasserstein metric confirms. The presented novel approach is based on the proof that a stochastic Volterra integral admits a progressively measurable modification in a weak sense and on sharp moment estimates for non-negative product measurable processes.

For each link type $K$ in the 3-sphere, we show that there is a polynomial $p_K$ such that any two diagrams of $K$ with $c_1$ and $c_2$ crossings differ by at most $p_K(c_1) + p_K(c_2)$ Reidemeister moves. As a consequence, the problem of recognising whether a given link diagram represents $K$ is in the complexity class NP and hence can be completed deterministically in exponential time. We calculate this polynomial $p_K$ explicitly for various classes of links.

We study a representation of a problem that appears in numerous transport systems: $N$ servers distributed over a given space (e.g., cars on an urban network), receive random requests from arriving users who get assigned to the closest server, after which this server is replaced by a new one at a random location. We show that this creates a negative feedback loop, which we call \textit{Increasing Gap Dynamics} (IGD): when a server is assigned a spatial gap forms, which is more likely to attract new users that further widen the gap. The simplest version of our model is a one-dimensional circle, for which we derive analytical results showing that the system converges to an inefficient equilibrium, worse than both balanced and fully random distributions of servers. We prove that an optimal assignment policy always matches the user to one of its two neighbouring servers so that long gaps tend to widen. Hence, the IGD persists even when assigning optimally rather than greedily. In two dimensions, the appearance of the IGD is illustrated through simulations on a square region. Finally, simulations of a proper ride-hailing system using real data from Manhattan confirms that the IGD arises and that it is responsible for the appearance of the well-known Wild Goose Chase.

Feedback optimization refers to a class of methods that steer a control system to a steady state that solves an optimization problem. Despite tremendous progress on the topic, an important problem remains open: enforcing state constraints at all times. The difficulty in addressing it lies on mediating between the safety enforcement and the closed-loop stability, and ensuring the equivalence between closed-loop equilibria and the optimization problem's critical points. In this work, we present a feedback-optimization method that enforces state constraints at all times employing high-order control-barrier functions. We provide several results on the proposed controller dynamics, including well-posedness, safety guarantees, equivalence between equilibria and critical points, and local and global (in certain convex cases) asymptotic stability of optima. Various simulations illustrate our results.

We study the almost complete intersection ring $R$ defined by $n+1$ general quadrics in a polynomial ring in $n$ variables over a field $\sf{k}$ and a corresponding linked Gorenstein ring $A$. The overarching theme is that, while not Koszul (except for some small values of $n$), these rings have homological properties that extend those of Koszul rings. We establish that finitely generated modules over these rings have rational Poincaré series and we give concrete formulas for the Poincaré series of $\sf{k}$ over both $A$ and $R$. We also show that $A$ has minimal rate and its Yoneda algebra $\text{Ext}_A(\sf{k},\sf{k})$ is generated by its elements of degrees $1$ and $2$. While the graded Betti numbers of $R$ and $A$ over the polynomial ring are not known when $n$ is odd, our approach provides bounds and yields values for two of these Betti numbers, showing in particular that $R$ is level.

In this work we introduce a viscosity-based notion of solution for general approximation schemes associated with partial differential equations, such as dynamic programming principles~(DPPs). A key feature of our approach is that it bypasses any measurability requirement on solutions of the DPP, an assumption that is often difficult to verify and may even fail in relevant examples. We establish a comparison principle between classical strict supersolutions and viscosity subsolutions of the DPP, which yields stability results under minimal and natural hypotheses. As a consequence, we prove existence of viscosity solutions of the DPP and their convergence to viscosity solutions of a PDE that is consistent with the underlying approximation scheme. Moreover, we show that solutions of the limiting PDE admit an asymptotic expansion encoded by the approximation operator. Finally, we demonstrate that a broad class of local, nonlocal, and nonlinear partial differential equations fits into our framework, recovering known examples in the literature and completing gaps in the existing literature.

Recent concurrent work by Dupré la Tour and Fujii and by Hollender, Manurangsi, Meka, and Suksompong [ITCS'26] introduced a generalization of classical discrepancy theory to non-additive functions, motivated by applications in fair division. As many classical techniques from discrepancy theory seem to fail in this setting, including linear algebraic methods like the Beck-Fiala Theorem [Discrete Appl. Math '81], it remains widely open whether comparable non-additive bounds can be achieved. Towards a better understanding of non-additive discrepancy, we study coverage functions in a sparse setting comparable to the classical Beck-Fiala Theorem. Our setting generalizes the additive Beck-Fiala setting, rank functions of partition matroids, and edge coverage in graphs. More precisely, assuming each of the $n$ items covers only $t$ elements across all functions, we prove a constructive discrepancy bound that is polynomial in $t$, the number of colors $k$, and $\log n$.

In this paper, we prove a localisation of a slightly supercritical (Orlicz) regularity criterion for the 3D incompressible Navier-Stokes equations. This is a refinement to the recent partial positive answer to Tao's conjecture [Tao21] as given in [BP21b]. The proof requires new quantitative estimates for critically bounded solutions of the forced Navier-Stokes equations, where the forcing is induced by the localisation. A by-product of these new estimates is an application to the Boussinesq equations, where we prove a quantitative blow-up rate for the critical $L^3$ norm of the velocity. We prove these quantitative estimates using Carleman inequalities as in [Tao21], and subsequently in [BP21a], with an additional forcing term. An obstacle to doing this is that, in the Carleman inequalities, the forcing term is amplified on large scales. Additionally, the low regularity of the forcing requires the addition of Caccioppoli-type estimates to deal with the Carleman inequalities appropriately.

We show that any uniformly escaping and wandering dynamics of a holomorphic function on a compact subset of the plane can be realised by a transcendental meromorphic function on $\mathbb{C}$. More precisely, let $\varphi$ be a holomorphic function on an open subset of the complex plane, and suppose that $K$ is a compact set such that $\varphi$ and all its iterates $\varphi^n$ are defined on $K$, and $\varphi^n(K)\to\infty$ as $n\to\infty$. We prove that there exist a transcendental meromorphic function $f\colon\mathbb{C}\to\widehat{\mathbb{C}}$ and a compact set $\widetilde{K}$ such that the dynamics of $f$ on the orbit of $\widetilde{K}$ is conjugate, via a smooth change of coordinate close to the identity, to that of $\varphi$ on the orbit of $K$. If $K$ does not separate the plane, the function $f$ may be chosen to be entire. If all iterates of $\varphi$ are univalent on $K$, we can take $\widetilde{K}=K$. We also prove a similar theorem for oscillating dynamics. Finally, we use our results to answer a number of questions of Benini et al. concerning wandering domains of entire functions.

In calculus, l'Hopital's rule provides a simple way to evaluate the limits of quotient functions when both the numerator and denominator vanish. But what happens when we move beyond real functions on a real interval? In this article, we study when the quotient of two complex-valued functions in higher dimension can be defined continuously at the points where both functions vanish. Surprisingly, the answer is far subtler than in the real-valued setting. We provide a complete characterization for the continuity of the quotient function. We also point out why extending this result to smoother quotients remains an intriguing challenge.

We study the problem of learning multi-index models (MIMs), where the label depends on the input $\boldsymbol{x} \in \mathbb{R}^d$ only through an unknown $\mathsf{s}$-dimensional projection $\boldsymbol{W}_*^\mathsf{T} \boldsymbol{x} \in \mathbb{R}^\mathsf{s}$. Exploiting the equivariance of this problem under the orthogonal group $\mathcal{O}_d$, we obtain a sharp harmonic-analytic characterization of the learning complexity for MIMs with spherically symmetric inputs -- which refines and generalizes previous Gaussian-specific analyses. Specifically, we derive statistical and computational complexity lower bounds within the Statistical Query (SQ) and Low-Degree Polynomial (LDP) frameworks. These bounds decompose naturally across spherical harmonic subspaces. Guided by this decomposition, we construct a family of spectral algorithms based on harmonic tensor unfolding that sequentially recover the latent directions and (nearly) achieve these SQ and LDP lower bounds. Depending on the choice of harmonic degree sequence, these estimators can realize a broad range of trade-offs between sample and runtime complexity. From a technical standpoint, our results build on the semisimple decomposition of the $\mathcal{O}_d$-action on $L^2 (\mathbb{S}^{d-1})$ and the intertwining isomorphism between spherical harmonics and traceless symmetric tensors.

Let $k$ and $\ell$ be positive integers. The multiset star transposition graph ST$_k^\ell$ has as vertices the $k\ell$-strings $v_0\cdots v_{k\ell-1}$ on $k$ symbols, each symbol repeated $\ell$ times, and edges given by the transpositions $(v_0\;v_i)$ with $v_i\ne v_0$ ($01$ and $\ell>2$ that ST$_k^\ell$ is $(\ell-1)$-choosable and that, as a result, admits total colorings. In order to prove such assertions, the notion of efficient domination set (or E-set) of a graph is generalized for $\ell>1$ to that of an efficient dominating$\,^\ell$-set and applied to the graphs ST$_k^\ell$\,, showing they admit vertex partitions that generalize the Dejter-Serra partitions of ST$_k^1$ into E-sets, but not efficiently in the sense that the distance of each E$^\ell$-set be 3. Efficiently in such sense however, $ST^2_k$ and the related 2-set pancake permutation graph PC$^2_k$, among other intermediate permutation graphs, are shown to admit total colorings with $2k-1$ colors that determine partitions into $2k-1$ E-sets, each with distance 3. Furthermore, associated E-chains are examined.

We compute an explicit closed formula for the Hilbert polynomial of the Jacobian algebra $M(f)$ of a reduced surface $X:f=0$ in $\mathbb P^3$ in terms of the graded Betti numbers of the algebra $M(f)$. When $X$ has only isolated singularities, a result by A. du Plessis and C. T. C. Wall yields new necessary condition for a set of positive integers to be the graded Betti numbers of the Jacobian algebra of such a surface. The comparison with the plane curve case is discussed in detail and additional information is given in the case of nodal surfaces.

We prove that the category of profinite cosheaves valued in a pro-regular category (satisfying mild assumptions) is itself a pro-regular category. As a corollary, we extend Wilkes's cosheaf-bundle equivalence from profinite modules to profinite groups.

A homogeneous bivariate $d$-form defines an $(i+1)$-rowed Toeplitz matrix for each $i$ between $0$ and $d$. We use Hodge theory and Schur polynomials to prove that if the $(i+1)$-rowed Toeplitz matrix of a form is totally nonnegative, then so is the $i$-rowed one. This fixes a gap in the main result of paper above.

We consider a fractional linear differential equation with successive derivatives given by $ \mathbb{D}_\alpha^{n}y+ p_{n-1}(x) \mathbb{D}_\alpha^{n-1}y+ \dots +p_{1}(x)\mathbb{D}_\alpha y+p_0(x)y=0$, where $\mathbb{D}_\alpha^{j}$ is the $j$th iteration of the Caputo-Djrbashian fractional derivative of order $\alpha>0$, $p_j$ are $\alpha$-analytic functions for $0

Using a new approach involving embedding spaces in II$_1$ factors with plenty of freely independent Haar unitaries, we prove that $C^\ast_r(\mathbb{F}_n)\ncong C^\ast_r(\mathbb{F}_m)$ for $n \neq m$. This recovers the seminal result of Pimsner and Voiculescu with a short new proof.

Outer-approximation-based branch-and-bound is a common algorithmic framework for solving MINLPs (mixed-integer nonlinear programs) to global optimality, with branching variable selection critically influencing overall performance. In modern global MINLP solvers, it is unclear whether branching on fractional integer variables should be prioritized over spatial branching on variables, potentially continuous, that show constraint violations, with different solvers following different defaults. We address this question using a data-driven approach. Based on a test set of hundreds of heterogeneous public and industrial MINLP instances, we train linear and random forest regression models to predict the relative speedup of the FICO(R) Xpress Global solver when using a branching rule that always prioritizes variables with violated integralities versus a mixed rule, allowing for early spatial branches. We introduce a practical evaluation methodology that measures the effect of the learned model directly in terms of the shifted geometric mean runtime. Using only four features derived from strong branching and the nonlinear structure, our linear regression model achieves an 8-9% reduction in geometric-mean solving time for the Xpress solver, with over 10% improvement on hard instances. We also analyze a random regression forest model. Experiments across solver versions show that a model trained on Xpress 9.6 still yields significant improvements on Xpress 9.8 without retraining. Our results demonstrate how regression models can successfully guide the branching-rule selection and improve the performance of a state-of-the-art commercial MINLP solver.

We analyze the stability and bifurcation structure of steady states in a mechanochemical model of pattern formation in regenerating tissue spheroids. The model couples morphogen dynamics with tissue mechanics via a positive feedback loop: mechanical stretching enhances morphogen production, while morphogen concentration modulates tissue elasticity. Global strain conservation implements a nonlocal inhibitory effect, realizing a mechanochemical variant of the local activation--long-range inhibition mechanism. For exponential elasticity-morphogen coupling, the system admits a variational formulation. We prove existence of nonconstant steady states for small diffusion and uniqueness of the homogeneous state for large diffusion. Linear stability analysis shows that only unimodal patterns are stable, while multimodal solutions are unstable. Bifurcation analysis reveals subcritical and supercritical pitchforks, with fold bifurcations generating bistable regimes. Our results demonstrate that mechanochemical feedback provides a robust mechanism for single-peaked pattern formation without requiring a second diffusible inhibitor.

This paper introduces and studies split two-sided 2-fibrations and locally discrete split two-sided 2-fibrations, using a formal categorical approach. We generalise Street's notion of split two-sided fibration internal to a 2-category to one internal to a sesquicategory. Given a sesquicategory we construct a virtual double category whose horizontal (loose) morphisms are its internal split two-sided fibrations. Specialising to the sesquicategory of lax natural transformations we obtain the virtual double category of split two-sided 2-fibrations, which we study in detail. We then restrict to the sub-virtual double category of locally discrete split two-sided 2-fibrations and show that therein the usual Yoneda 2-functors satisfy a double-categorical formal notion of Yoneda morphism, which formally captures universal properties similar to those satisfied by the morphisms comprising a Yoneda structure on a 2-category. As a consequence we obtain a 'two-sided Grothendieck correspondence' of locally discrete split two-sided 2-fibrations $A \nrightarrow B$ and 2-functors $B \to Cat^{A^{op}}$. Restricting to $A = 1$, the terminal 2-category, we improve Buckley and Lambert's 'Grothendieck correspondence' for locally discrete split op-2-fibrations by extending the sense in which it is functorial.

We give recurrences, generating functions and explicit exact expressions for the enumeration of fundamental quantities involving runs in binary strings. We first focus on enumerations concerning runs of ones, and we then analyse the same enumerations when runs of ones and runs of zeros are jointly considered. We give the connections between these two types of run enumeration, and with the problem of compositions. We also analyse the same enumerations with a Hamming weight constraint. We discuss which of the many number sequences that emerge from these problems are already known and listed in the OEIS. Additionally, we extend our main enumerative results to the probabilistic scenario in which binary strings are outcomes of independent and identically distributed Bernoulli variables.

In this work, we solve a long-standing open problem in coding theory with broad applications in networking and systems: designing an erasure code that simultaneously satisfies three requirements: (1) high coding efficiency, (2) low coding complexity, and (3) being a streaming code (defined as one with low decoding latency). We propose METTLE (Multi-Edge Type with Touch-less Leading Edge), the first erasure code to meet all three requirements. Compared to "streaming RaptorQ" (RaptorQ configured with a small source block size to ensure a low decoding latency), METTLE is only slightly worse in coding efficiency, but 47.7 to 84.6 times faster to decode.

It is known that when minimizing smooth Polyak-Łojasiewicz (PL) functions, momentum algorithms cannot significantly improve the convergence bound of gradient descent, contrasting with the acceleration phenomenon occurring in the strongly convex case. To bridge this gap, the literature has proposed strongly quasar-convex functions as an intermediate non-convex class, for which accelerated bounds have been suggested to persist. We show that this is not true in general: the additional structure of strong quasar-convexity does not suffice to guaranty better worst-case bounds for momentum compared to gradient descent. As an alternative, we study PL functions under an aiming condition that measures how well the descent direction points toward a minimizer. This perspective clarifies the geometric ingredient enabling provable acceleration by momentum when minimizing PL functions.

A matrix $M$ over the finite field $ \mathbb{F}_q $ is called \emph{maximum distance separable} (MDS) if all of its square submatrices are non-singular. These MDS matrices are very important in cryptography and coding theory because they provide strong data protection and help spread information efficiently. In this paper, we introduce a new type of matrix called a \emph{consta-$g$-circulant matrix}, which extends the idea of $g$-circulant matrices. These matrices come from a linear transformation defined by the polynomial $ h(x) = x^m - \lambda + \sum_{i=0}^{m-1} h_i x^i $ over $ \mathbb{F}_q $. We find the upper bound of such matrices exist and give conditions to check when they are invertible. This helps us know when they are MDS matrices. If the polynomial $ x^m - \lambda $ factors as $ x^m - \lambda = \prod_{i=1}^{t} f_i(x)^{e_i}, $ where each \( f_i(x) \) is irreducible, then the number of invertible consta-$g$-circulant matrices is $ N \cdot \prod_{i=1}^{t} \left( q^{°f_i} - 1 \right), $ where $r$ is the multiplicative order of $\lambda$, and \( N \) is the number of integers \( k \) such that $ 0 \leq k < \left\lfloor \frac{m - 1}{r} \right\rfloor + 1 \quad \text{and} \quad \gcd(1 + rk, m) = 1. $ This formula help us to reduce the number of cases to check whether such matrices is MDS. Moreover, we give complete characterization of $g$-circulant MDS matrices of order 3 and 4. Additionally, inspired by skew polynomial rings, we construct a new variant of $g$-circulant matrix. In the last, we provide some examples related to our findings.

Let $f$ be a $C^r$ ($r>1$) diffeomorphism on a compact surface $M$ with $h_{\rm top}(f)\geq\frac{\lambda^{+}(f)}{r}$ where $\lambda^{+}(f):=\lim_{n\to+\infty}\frac{1}{n}\max_{x\in M}\log \left\|Df^{n}_{x}\right\|$. We establish an equivalent formula for the topological entropy: $$h_{\rm top}(f)=\lim_{n\to+\infty}\frac{1}{n}\log\int_{M}\left\|Df^{n}_{x}\right\|\,dx.$$ Our approach builds on the key ideas developed in the works of Buzzi-Crovisier-Sarig (\emph{Invent. Math.}, 2022) and Burguet (\emph{Ann. Henri Poincaré}, 2024) concerning the continuity of the Lyapunov exponents.

We study holomorphic vector fields whose singular locus contains a local complete intersection smooth positive-dimensional component. We prove global and local formulas expressing the limiting Milnor/Poincare-Hopf contribution along such a component in terms of its embedded scheme structure, and we obtain sharp lower bounds for this contribution under holomorphic perturbations. We provide explicit families show optimality and illustrate how singularities may redistribute between a fixed neighborhood of the component and the part at infinity in projective compactifications.

We show that any ruled surface $X$ with $\chi(X) < 0$ admits infinitely many inequivalent Lefschetz pencils of fixed genus and number of base points. Our proof proceeds by building infinitely many inequivalent Lefschetz fibrations on a blow-up $X \# 4 \overline{\mathbb{CP}^2}$ of $X$ with constant fiber class, via a mechanism known as partial conjugation. Furthermore, there exists a symplectic form on $X$ compatible with all such pencils, and similarly for the fibrations in $X\#4\overline{\mathbb{CP}^2}$. This provides the first example of this phenomenon and makes progress on Problem 4.98 of the K3 list of problems in low-dimensional topology in the case of ruled surfaces.

The friendship paradox index is a network summary statistic used to quantify the friendship paradox, which describes the tendency for an individual's friends to have more friends than the individual. In this paper, we utilize Markov's inequality to derive the weak law of large numbers for the friendship paradox index in a random geometric graph, a widely-used model for networks with spatial dependence and geometry. For uniform random geometric graph, where the nodes are uniformly distributed in a space, the friendship paradox index is asymptotically equal to $1/4$. On the contrary, in nonuniform random geometric graphs, the nonuniform node distribution leads to distinct limiting properties for the index. In the relatively sparse regime, the friendship paradox index is still asymptotically equal to $1/4$, the same as in the uniform case. In the intermediate sparse regime, however, the index converges in probability to $1/4$ plus a constant that is explicitly dependent on the node distribution. Finally, in the relatively dense case, the index diverges to infinity as the graph size increases. Our results highlight the sharp contrast between the uniform case and its nonuniform counterpart.

We prove convergence to equilibrium for a class of coagulation-fragmentation equations that do not satisfy a detailed balance condition. More precisely, we consider perturbations of constant rate kernels. Our result provides in particular explicit convergence rates. Uniqueness of the stationary states is proven as well for the considered class of kernels.

We study the Euler equation on the rotating sphere in the case where the absolute vorticity is initially sharply concentrated around several points. We follow the literature already concerning vorticity confinement for the planar Euler equations, and obtain similar results on the rotating sphere, with new challenges due to the geometry. More precisely, we show the improbability of collisions for point-vortices, logarithmic in time absolute vorticity confinement for general configurations, the optimality of this last result in general, and the existence of configurations with power-law long confinement. We take this opportunity to write a unified, self-contained, and improved version of all the proofs, previously scattered across multiple papers on the planar case, with detailed exposition for pedagogical clarity.

In this paper, we consider a generalized polyharmonic eigenvalue problem of the form $A(u)= \lambda h(u)$ in a bounded smooth domain with Dirichlet boundary conditions in the setting of higher-order Orlicz-Sobolev spaces. Here, $A$ is a very general operator depending on $u$ and arbitrary higher-order derivatives of $u$, whose growth is governed by an Orlicz function, and $h$ is a lower order term. Combining the theories of pseudomonotone operators with complementary systems, we prove that this eigenvalue problem has an infinite number of eigenfunctions and that the corresponding sequence of eigenvalues tends to infinity. We point out that the $\Delta_2$-condition is not assumed for the involved Orlicz functions. Finally, we prove a first regularity result for eigenfunctions by following a De Giorgi's iteration scheme.

This paper considers the asymptotic behavior in $\beta$-Hölder spaces, and under $L^p$ loss, of the gamma kernel density estimator introduced by Chen [Ann. Inst. Statist. Math. 52 (2000), 471-480] for the analysis of nonnegative data, when the target's support is assumed to be upper bounded. It is shown that this estimator can achieve the minimax rate asymptotically for a suitable choice of bandwidth whenever $(p,\beta)\in [1,3)\times(0,2]$ or $(p,\beta)\in [3,4)\times ((p-3)/(p-2),2]$. It is also shown that this estimator cannot be minimax when either $p\in [4,\infty)$ or $\beta\in (2,\infty)$.

Imaging systems are commonly described using resolution, contrast, and signal-to-noise ratio, but these quantities do not provide a general account of how physical transformations affect the flow of information. This paper introduces an operator-based formulation of information theory for imaging. The approach models the imaging chain as a composition of bounded operators acting on functions, and characterises information redistribution using the spectral properties of these operators. Three measures are developed. Operator entropy quantifies how an operator distributes energy across its singular spectrum. Operator information capacity describes the number of modes that remain recoverable above a noise-dependent threshold. An irreversibility index measures the information lost through suppression or elimination of modes and captures the accumulation of information loss under operator composition. The framework applies to linear, nonlinear, and stochastic operators and does not depend on the specific imaging modality. Analytical examples show how attenuation, blur, and sampling affect entropy, capacity, and irreversibility in different ways. The results provide a general structure for analysing the physical limits of imaging and form the basis for subsequent work on information geometry, spatiotemporal budgets, nonlinear channels, and reconstruction algorithms.

Answering a question of Claudet, we prove that the uniformly random graph $G\sim \mathbb G(n, 1/2)$ is $\Omega(\sqrt n)$-vertex-minor universal with high probability. That is, for some constant $\alpha\approx 0.911$, any graph on any $\alpha\sqrt n$ specified vertices of $G$ can be obtained as a vertex-minor of $G$. This has direct implications for quantum communications networks: an $n$-vertex $k$-vertex-minor universal graph corresponds to an $n$-qubit $k$-stabilizer universal graph state, which has the property that one can induce any stabilizer state on any $k$ qubits using only local operations and classical communications. We further employ our methods in two other contexts. We obtain a bipartite pivot-minor version of our main result, and we use it to derive a universality statement for minors in random binary matroids. We also introduce the vertex-minor Ramsey number $R_{\mathrm{vm}}(k)$ to be the smallest value $n$ such that every $n$-vertex graph contains an independent set of size $k$ as a vertex-minor. Supported by our main result, we conjecture that $R_{\mathrm{vm}}(k)$ is polynomial in $k$. We prove $\Omega(k^2) \leq R_{\mathrm{vm}}(k) \leq 2^k - 1$.

Persistent entropy (PE) is an information-theoretic summary statistic of persistence barcodes that has been widely used to detect regime changes in complex systems. Despite its empirical success, a general theoretical understanding of when and why persistent entropy reliably detects phase transitions has remained limited, particularly in stochastic and data-driven settings. In this work, we establish a general, model-independent theorem providing sufficient conditions under which persistent entropy provably separates two phases. We show that persistent entropy exhibits an asymptotically non-vanishing gap across phases. The result relies only on continuity of persistent entropy along the convergent diagram sequence, or under mild regularization, and is therefore broadly applicable across data modalities, filtrations, and homological degrees. To connect asymptotic theory with finite-time computations, we introduce an operational framework based on topological stabilization, defining a topological transition time by stabilizing a chosen topological statistic over sliding windows, and a probability-based estimator of critical parameters within a finite observation horizon. We validate the framework on the Kuramoto synchronization transition, the Vicsek order-to-disorder transition in collective motion, and neural network training dynamics across multiple datasets and architectures. Across all experiments, stabilization of persistent entropy and collapse of variability across realizations provide robust numerical signatures consistent with the theoretical mechanism.

Estimating delay tail probabilities in scheduling and load balancing systems is a critical but computationally prohibitive task due to the rarity of violation events. Quantum Amplitude Estimation (QAE) offers a generic quadratic reduction in sample complexity 1/sqrt(p) vs 1/p, but applying it to steady-state queueing networks in challenging: classical simulations involve unbounded state spaces and random regeneration cycles, whereas quantum circuits have fixed depth and finite registers. In this paper, we develop a framework for quantum simulation of delay tail probabilities based on truncated regenerative simulation. We show that regenerative rare-event estimators can be reformulated as deterministic, reversible functions of finite random seeds by truncating regeneration cycles. To control the resulting bias, we use Lyapunov drift and concentration arguments to derive exponential tail bounds on regeneration times. This allows the truncation horizon--and hence the quantum circuit depth--to be chosen such that the bias is provably negligible compared to the statistical error. The proposed framework enables quantum estimation in models with countably infinite state spaces, avoiding the challenge of determining the sufficient mixing time required for direct finite-horizon simulation. We provide bounds on qubit and circuit complexity for a GI-GI-1 queue, a wireless network under MaxWeight scheduling, and a multi-server system with Join-the-Shortest-Queue (JSQ) routing.

In this same journal, Arnold Zellner published a seminal paper on Bayes' theorem as an optimal information processing rule. This result led to the variational formulation of Bayes' theorem, which is the central idea in generalized variational inference. Almost 40 years later, we revisit these ideas, but from the perspective of information deletion. We investigate rules which update a posterior distribution into an antedata distribution when a portion of data is removed. In such context, a rule which does not destroy or create information is called the optimal information deletion rule and we prove that it coincides with the traditional use of Bayes' theorem.

Many metal manufacturing processes involve phase change phenomena, which include melting, boiling, and vaporization. These phenomena often occur concurrently. A prototypical 1D model for understanding the phase change phenomena is the Stefan problem. There is a large body of literature discussing the analytical solution to the two-phase Stefan problem that describes only the melting or boiling of phase change materials (PCMs) with one moving interface. Density-change effects that induce additional fluid flow during phase change are generally neglected in the literature to simplify the math of the Stefan problem. In our recent work [1], we provide analytical and numerical solutions to the three-phase Stefan problem with simultaneous occurrences of melting, solidification, boiling, and condensation in Cartesian coordinates. Our current work builds on our previous work to solve a more challenging problem: the three-phase Stefan problem in spherical coordinates for finite-sized particles. There are three moving interfaces in this system: the melt front, the boiling front, and the outer boundary which is in contact with the atmosphere. Although an analytical solution could not be found for this problem, we solved the governing equations using a fixed-grid sharp-interface method with second-order spatio-temporal accuracy. Using a small-time analytical solution, we predict a reasonably accurate estimate of temperature (in the three phases) and interface positions and velocities at the start of the simulation. Our numerical method is validated by reproducing the two-phase nanoparticle melting results of Font et al. [2]. Lastly, we solve the three-phase Stefan problems numerically to demonstrate the importance of kinetic energy terms during phase change of smaller (nano) particles. In contrast, these effects diminish for large particles (microns and larger).

In this paper, we derive an accelerated continuous-time formulation of Adam by modeling it as a second-order integro-differential dynamical system. We relate this inertial nonlocal model to an existing first-order nonlocal Adam flow through an $\alpha$-refinement limit, and we provide Lyapunov-based stability and convergence analyses. We also introduce an Adam-inspired nonlocal Lagrangian formulation, offering a variational viewpoint. Numerical simulations on Rosenbrock-type examples show agreement between the proposed dynamics and discrete Adam.

A flat connection on a Riemann surface with values in an infinite dimensional Lie algebra provides a systematic and effective tool for generating an infinite family of polylogarithms via iterated integrals. The recent literature offers different types of connections, in one or several variables, on compact Riemann surfaces with or without punctures, and in the meromorphic or single-valued categories. In this work, we show that the flatness conditions for the single-valued and modular DHS connection in multiple variables, which was introduced in the companion paper arXiv:2602.01461, are equivalent to the union of all the interchange and Fay identities among DHS integration kernels that were proven in arXiv:2407.11476. Based on the same combinatorial techniques, the flatness conditions on the multivariable Enriquez connection is shown to imply the union of all the interchange and Fay identities for Enriquez kernels.

Recent generative and tool-using AI systems can surface a large volume of candidates at low marginal cost, yet only a small fraction can be checked carefully. This creates a decoder-side bottleneck: downstream decision-makers must form reliable posteriors from many public records under scarce attention. We formalize this regime via Attention-Constrained Inference (ACI), in which a cheap screening stage processes $K$ records and an expensive verification stage can follow up on at most $B$ of them. Under Bayes log-loss, we study the maximum achievable reduction in posterior uncertainty per window, which we call \emph{epistemic throughput}. Our main result is a ``JaKoB'' scaling law showing that epistemic throughput has a baseline term that grows linearly with verification and prevalence, and an additional \emph{information-leverage} term that scales as $\sqrt{JKB}$, where $J$ summarizes screening quality. Thus, expanding cheap screening can nonlinearly amplify scarce verification, even when informative records are rare. We further show that this scaling is tight in a weak-screening limit, and that in the sparse-verification regime ($B \ll K$), substantial leverage requires heavy-tailed score distributions; for light-tailed scores the amplification is only logarithmic.

We develop a perturbation-based frequency-response framework for analyzing amplification mechanisms that are central to subcritical routes to transition in wall-bounded shear flows. By systematically expanding the input-output dynamics of fluctuations about the laminar base flow with respect to forcing amplitude, we establish a rigorous correspondence between linear resolvent analysis and higher-order nonlinear interactions. At second order, quadratic interactions of unsteady oblique waves generate steady streamwise streaks via the lift-up mechanism. We demonstrate that the spatial structure of these streaks is captured by the second output singular function of the streamwise-constant resolvent operator. At higher orders, nonlinear coupling between oblique waves and induced streaks acts as structured forcing of the laminar linearized dynamics, yielding additional streak components whose relative phase governs reinforcement or attenuation of the leading-order streak response. Our analysis identifies a critical forcing amplitude marking the breakdown of the weakly nonlinear regime, beyond which direct numerical simulations exhibit sustained unsteadiness. We show that this breakdown coincides with the onset of secondary instability, revealing that the nonlinear interactions responsible for streak formation also drive the modal growth central to classical transition theory. The resulting framework provides a mechanistically transparent and computationally efficient description of transition that unifies non-modal amplification, streak formation, and modal instability within a single formulation derived directly from the Navier-Stokes equations.

We study correlation functions with fractional-mode excitations of the R-symmetry currents in D1-D5 CFT. We show how fractional-mode excitations lift to the covering surface associated with correlation functions as a specific sum of integer-mode excitations, with coefficients that can be determined exactly from the covering map in terms of Bell polynomials. We consider the four-point functions of fractional excitations of two chiral/anti-chiral NS fields, Ramond ground states and the twist-two scalar modulus deformation operator that drives the CFT away from the free point. We derive explicit formulas for classes of these functions with twist structures $(n)$-$(2)$-$(2)$-$(n)$ and $(n_1)(n_2)$-$(2)$-$(2)$-$(n_1)(n_2)$, the latter involving double-cycle fields. The final answer for the four-point functions always depends only on the lift of the base-space cross-ratio. We discuss how this relates to Hurwitz blocks associated with different conjugacy classes of permutations, the corresponding OPE channels and fusion rules.

Analyzing overdispersed, zero-inflated, longitudinal count data poses significant modeling and computational challenges, which standard count models (e.g., Poisson or negative binomial mixed effects models) fail to adequately address. We propose a Zero-Inflated Beta-Binomial Mixed Effects Regression (ZIBBMR) model that augments a beta-binomial count model with a zero-inflation component, fixed effects for covariates, and subject-specific random effects, accommodating excessive zeros, overdispersion, and within-subject correlation. Maximum likelihood estimation is performed via a Stochastic Approximation EM (SAEM) algorithm with latent variable augmentation, which circumvents the model's intractable likelihood and enables efficient computation. Simulation studies show that ZIBBMR achieves accuracy comparable to leading mixed-model approaches in the literature and surpasses simpler zero-inflated count formulations, particularly in small-sample scenarios. As a case study, we analyze longitudinal microbiome data, comparing ZIBBMR with an external Zero-Inflated Beta Regression (ZIBR) benchmark; the results indicate that applying both count- and proportion-based models in parallel can enhance inference robustness when both data types are available.

We use a difference Lax form to construct simultaneous integrals of motion of the fourth Painlevé equation and the difference second Painlevé equation over fields with finite characteristic $p>0$. For $p\neq 3$, we show that the integrals can be normalised to be completely invariant under the corresponding extended affine Weyl group action. We show that components of reducible fibres of integrals correspond to reductions to Riccatti equations. We further describe a method to construct non-rational algebraic solutions in a given positive characteristic. We also discuss a projective reduction of the integrals.

In this work, we propose a new bounded arithmetic theory, denoted $APX_1$, designed to formalize a broad class of probabilistic arguments commonly used in theoretical computer science. Under plausible assumptions, $APX_1$ is strictly weaker than previously proposed frameworks, such as the theory $APC_1$ introduced in the seminal work of Jerabek (2007). From a computational standpoint, $APX_1$ is closely tied to approximate counting and to the central question in derandomization, the prBPP versus prP problem, whereas $APC_1$ is linked to the dual weak pigeonhole principle and to the existence of Boolean functions with exponential circuit complexity. A key motivation for introducing $APX_1$ is that its weaker axioms expose finer proof-theoretic structure, making it a natural setting for several lines of research, including unprovability of complexity conjectures and reverse mathematics of randomized lower bounds. In particular, the framework we develop for $APX_1$ enables the formulation of precise questions concerning the provability of prBPP=prP in deterministic feasible mathematics. Since the (un)provability of P versus NP in bounded arithmetic has long served as a central theme in the field, we expect this line of investigation to be of particular interest. Our technical contributions include developing a comprehensive foundation for probabilistic reasoning from weaker axioms, formalizing non-trivial results from theoretical computer science in $APX_1$, and establishing a tailored witnessing theorem for its provably total TFNP problems. As a byproduct of our analysis of the minimal proof-theoretic strength required to formalize statements arising in theoretical computer science, we resolve an open problem regarding the provability of $AC^0$ lower bounds in $PV_1$, which was considered in earlier works by Razborov (1995), Krajicek (1995), and Muller and Pich (2020).

We propose a physics-informed consistency modeling framework for solving partial differential equations (PDEs) via fast, few-step generative inference. We identify a key stability challenge in physics-constrained consistency training, where PDE residuals can drive the model toward trivial or degenerate solutions, degrading the learned data distribution. To address this, we introduce a structure-preserving two-stage training strategy that decouples distribution learning from physics enforcement by freezing the coefficient decoder during physics-informed fine-tuning. We further propose a two-step residual objective that enforces physical consistency on refined, structurally valid generative trajectories rather than noisy single-step predictions. The resulting framework enables stable, high-fidelity inference for both unconditional generation and forward problems. We demonstrate that forward solutions can be obtained via a projection-based zero-shot inpainting procedure, achieving consistent accuracy of diffusion baselines with orders of magnitude reduction in computational cost.

Manufacturing lines, service journeys, supply chains, and AI reasoning chains share a common challenge: attributing a terminal outcome to the intermediate stage that caused it. We establish an information-theoretic barrier to this credit assignment problem: the signal connecting early steps to final outcomes decays exponentially with depth, creating a critical horizon beyond which no algorithm can learn from endpoint data alone. We prove four results. First, a Signal Decay Bound: sample complexity for attributing outcomes to early stages grows exponentially in the number of intervening steps. Second, Width Limits: parallel rollouts provide only logarithmic relief, with correlation capping the effective number of independent samples. Third, an Objective Mismatch: additive reward aggregation optimizes the wrong quantity when sequential validity requires all steps to be correct. Fourth, Optimal Inspection Design: uniform checkpoint spacing is minimax-optimal under homogeneous signal attenuation, while a greedy algorithm yields optimal non-uniform schedules under heterogeneous attenuation. Together, these results provide a common analytical foundation for inspection design in operations and supervision design in AI.

We examine the connection between training error and generalization error for arbitrary estimating procedures, working in an overparameterized linear model under general priors in a Bayesian setup. We find determining factors inherent to the prior distribution $\pi$, giving explicit conditions under which optimal generalization necessitates that the training error be (i) near interpolating relative to the noise size (i.e., memorization is necessary), or (ii) close to the noise level (i.e., overfitting is harmful). Remarkably, these phenomena occur when the noise reaches thresholds determined by the Fisher information and the variance parameters of the prior $\pi$.

Since the early 2000s, control of multiagent systems has attracted significant research interest, with applications ranging from natural collective behaviors and social dynamics to engineered systems such as autonomous vehicles, sensor networks, and smart grids. Although research on multi-agent systems has diversified into numerous specialized directions, textbooks -- including those in English -- that provide a systematic treatment of the fundamental principles of multi-agent system control remain scarce. The material presented in this book has been developed and used in teaching since 2021, initially as a concise Vietnamese-language reference for the courses Networked Control Systems and Control of Multi-Agent Systems at Hanoi University of Science and Technology. The book focuses on a selection of fundamental topics of broad and continuing interest in the field. The complexity of several topics is asymptotic to that encountered in research-level studies, however, the analysis is presented in a step-by-step manner to facilitate access to commonly used methods and tools. The material is divided into three main parts. Part I introduces multiagent systems and basic graph-theoretic concepts. Part II addresses the design and analysis of linear consensus algorithms. Part III covers selected applications and research directions, including formation control, network localization, distributed optimization, opinion dynamics, and matrix-weighted networks. Each chapter concludes with notes on notable researchers in this field, further reading, and exercises. This book cannot be completed without the encouragement, support and suggestions from families, colleagues and friends. The authors appreciate feedback from readers to further improve the content of the book.

We develop an operator-theoretic framework for stability and statistical concentration in nonlinear inverse problems with block-structured parameters. Under a unified set of assumptions combining blockwise Lipschitz geometry, local identifiability, and sub-Gaussian noise, we establish deterministic stability inequalities, global Lipschitz bounds for least-squares misfit functionals, and nonasymptotic concentration estimates. These results yield high-probability parameter error bounds that are intrinsic to the forward operator and independent of any specific reconstruction algorithm. As a concrete instantiation, we verify that the Gaussian Splatting rendering operator satisfies the proposed assumptions and derive explicit constants governing its Lipschitz continuity and resolution-dependent observability. This leads to a fundamental stability--resolution tradeoff, showing that estimation error is inherently constrained by the ratio between image resolution and model complexity. Overall, the analysis characterizes operator-level limits for a broad class of high-dimensional nonlinear inverse problems arising in modern imaging and differentiable rendering.

Reaction-diffusion (RD) equations underpin pattern formation across chemistry, biology, and physics, yet learning stable operators that forecast their long-term dynamics from data remains challenging. Neural-operator surrogates provide resolution-robust prediction, but autoregressive rollouts can drift due to the accumulation of error, and out-of-distribution (OOD) initial conditions often degrade accuracy. Physics-based numerical residual objectives can regularize operator learning, although they introduce additional assumptions, sensitivity to discretization and loss design, and higher training cost. Here we develop a purely data-driven operator learner with adaptive recurrent training (DDOL-ART) using a robust recurrent strategy with lightweight validation milestones that early-exit unproductive rollout segments and redirect optimization. Trained only on a single in-distribution toroidal Gaussian family over short horizons, DDOL-ART learns one-step operators that remain stable under long rollouts and generalize zero-shot to strong morphology shifts across FitzHugh-Nagumo (FN), Gray-Scott (GS), and Lambda-Omega (LO) systems. Across these benchmarks, DDOL-ART delivers a strong accuracy and cost trade-off. It is several-fold faster than a physics-based numerical-loss operator learner (NLOL) under matched settings, and it remains competitive on both in-distribution stability and OOD robustness. Training-dynamics diagnostics show that adaptivity strengthens the correlation between validation error and OOD test error performance, acting as a feedback controller that limits optimization drift. Our results indicate that feedback-controlled recurrent training of DDOL-ART generates robust flow-map surrogates without PDE residuals, while simultaneously maintaining competitiveness with NLOL at significantly reduced training costs.

We introduce AutoSpec, a neural network framework for discovering iterative spectral algorithms for large-scale numerical linear algebra and numerical optimization. Our self-supervised models adapt to input operators using coarse spectral information (e.g., eigenvalue estimates and residual norms), and they predict recurrence coefficients for computing or applying a matrix polynomial tailored to a downstream task. The effectiveness of AutoSpec relies on three ingredients: an architecture whose inference pass implements short, executable numerical linear algebra recurrences; efficient training on small synthetic problems with transfer to large-scale real-world operators; and task-defined objectives that enforce the desired approximation or preconditioning behavior across the range of spectral profiles represented in the training set. We apply AutoSpec to discovering algorithms for representative numerical linear algebra tasks: accelerating matrix-function approximation; accelerating sparse linear solvers; and spectral filtering/preconditioning for eigenvalue computations. On real-world matrices, the learned procedures deliver orders-of-magnitude improvements in accuracy and/or reductions in iteration count, relative to basic baselines. We also find clear connections to classical theory: the induced polynomials often exhibit near-equiripple, near-minimax behavior characteristic of Chebyshev polynomials.

Trajectory tracking for microswimmers remains a key challenge in microrobotics, where low-Reynolds-number dynamics make control design particularly complex. In this work, we formulate the trajectory tracking problem as an optimal control problem and solve it using a combination of B-spline parametrization with Bayesian optimization, allowing the treatment of high computational costs without requiring complex gradient computations. Applied to a flagellated magnetic swimmer, the proposed method reproduces a variety of target trajectories, including biologically inspired paths observed in experimental studies. We further evaluate the approach on a three-sphere swimmer model, demonstrating that it can adapt to and partially compensate for wall-induced hydrodynamic effects. The proposed optimization strategy can be applied consistently across models of different fidelity, from low-dimensional ODE-based models to high-fidelity PDE-based simulations, showing its robustness and generality. These results highlight the potential of Bayesian optimization as a versatile tool for optimal control strategies in microscale locomotion under complex fluid-structure interactions.

Quantum simulation of the linear non-unitary dynamics is crucial in scientific computing. In this work, we establish a generic framework, referred to as the Amplitude-Phase Separation (APS) methods, which formulates any non-unitary evolution into separate simulation of a unitary operator and a Hermitian operator, thus allow one to take best advantage of, and to even improve existing algorithms, developed for unitary or Hermitian evolution respectively. We utilize two techniques: the first achieves a provably optimal query complexity via a shifted Dyson series; the second breaks the conventional linear dependency, achieving fast-forwarding by exhibiting a square-root dependence on the norm of the dissipative part. Furthermore, one can derive existing methods such as the LCHS (Linear Combination of Hamiltonian Simulation) and the NDME (Non-Diagonal Density Matrix Encoding) methods from APS. The APS provides an effective and generic pathway for developing efficient quantum algorithms for general non-unitary dynamics to achieve either optimal query complexity or fast-forwarding property, outperforming the existing algorithms for the same problems.

We present a structural classification of constraint satisfaction problems (CSP) described by reflexive complete $2$-edge-coloured graphs. In particular, this classification extends the structural dichotomy for graph homomorphism problems known as the Hell--Nešetřil theorem (1990). Our classification is also efficient: we can check in polynomial time whether the CSP of a reflexive complete $2$-edge-coloured graph is in P or NP-complete, whereas for arbitrary $2$-edge-coloured graphs, this task is NP-complete. We then apply our main result in the context of matrix partition problems and sandwich problems. Firstly, we obtain one of the few algorithmic solutions to general classes of matrix partition problems. And secondly, we present a P vs. NP-complete classification of sandwich problems for matrix partitions.

We propose a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of parametric partial differential equations (PDEs) conditioned on partial observations, which includes, in particular, forward and inverse PDE problems. We learn the joint distribution of PDE parameters and solutions via a diffusion process in a latent space of scaled spectral representations, where Gaussian noise corresponds to functions with controlled regularity. This spectral formulation enables significant dimensionality reduction compared to grid-based diffusion models and ensures that the induced process in function space remains within a class of functions for which the PDE operators are well defined. Building on diffusion posterior sampling, we enforce physics-informed constraints and measurement conditions during inference, applying Adam-based updates at each diffusion step. We evaluate the proposed approach on Poisson, Helmholtz, and incompressible Navier--Stokes equations, demonstrating improved accuracy and computational efficiency compared with existing diffusion-based PDE solvers, which are state of the art for sparse observations. Code is available at this https URL.

Higher-order networks encode the many-body interactions of complex systems ranging from the brain to biological transportation networks. Simplicial and cell complexes are ideal higher-order network representations for investigating higher-order topological dynamics where dynamical variables are not only associated with nodes, but also with edges, triangles, and higher-order simplices and cells. Global Topological Synchronization (GTS) refers to the dynamical state in which identical oscillators associated with higher-dimensional simplices and cells oscillate in unison. On standard unweighted and undirected complexes this dynamical state can be achieved only under strict topological and combinatorial conditions on the underlying discrete support. In this work we consider generalized higher-order network representations including directed and hollow complexes. Based on an in depth investigation of their topology defined by their associated algebraic topology operators and Betti numbers, we determine under which conditions GTS can be observed. We show that directed complexes always admit a global topological synchronization state independently of their topology and structure. However, we demonstrate that for directed complexes this dynamical state cannot be asymptotically stable. While hollow complexes require more stringent topological conditions to sustain global topological synchronization, these topologies can favor both the existence and the stability of global topological synchronization with respect to undirected and unweighted complexes.

Recent advances in imaging technologies, deep learning and numerical performance have enabled non-invasive detailed analysis of artworks, supporting their documentation and conservation. In particular, automated detection of craquelure in digitized paintings is crucial for assessing degradation and guiding restoration, yet remains challenging due to the possibly complex scenery and the visual similarity between cracks and crack-like artistic features such as brush strokes or hair. We propose a hybrid approach that models crack detection as an inverse problem, decomposing an observed image into a crack-free painting and a crack component. A deep generative model is employed as powerful prior for the underlying artwork, while crack structures are captured using a Mumford--Shah-type variational functional together with a crack prior. Joint optimization yields a pixel-level map of crack localizations in the painting.

In this article, we further explore the construction and computation of expectation values for Wilson loops in higher-rank 5d $\mathcal{N} = 1$ gauge theories on $\mathbb{C}_2 \times S_1$, by explicitly computing the Wilson loops via Chern-character insertion and qq-characters, including cases with the exceptional gauge group $G_2$. In particular, we propose a systematic way to write down the general blowup equations for Wilson loops by using the constraints from the one-form symmetry and low-instanton data from the instanton partition function. In addition, for one-instanton contributions in a large family of Wilson loop representations, we observe that they admit a $q_1q_2$-expansion, similar to the Hilbert-series structure of instanton partitions in pure gauge theories.

We investigate the structure of $k$-positivity and Schmidt numbers for classes of linear maps and bipartite quantum states exhibiting symplectic group symmetry. Specifically, we consider (1) linear maps on $M_d(\mathbb{C})$ which are covariant under conjugation by unitary symplectic matrices $S$, and (2) $d\otimes d$ bipartite states which are invariant under $S\otimes S$ or $S\otimes \overline{S}$ actions, each parametrized by two real variables. We provide a complete characterization of all $k$-positivity and decomposability conditions for these maps and explicitly compute the Schmidt numbers for the corresponding bipartite states. In particular, our analysis yields a broad class of PPT states with Schmidt number $d/2$ and the first explicit constructions of (optimal) $k$-positive indecomposable linear maps for arbitrary $k=1,\ldots, d/2-1$, achieving the best-known bounds. Overall, our results offer a natural and analytically tractable framework in which both strong forms of positive indecomposability and high degrees of PPT entanglement can be studied systematically. We present two further applications of symplectic group symmetry. First, we show that the PPT-squared conjecture holds within the class of PPT linear maps that are either symplectic-covariant or conjugate-symplectic-covariant. Second, we resolve a conjecture of Pal and Vertesi concerning the optimal lower bound of the Sindici-Piani semidefinite program for PPT entanglement.

The advent of quantum computation compels the cryptographic community to design digital signature schemes whose security extends beyond the classical hardness assumptions. In this work, we introduce Spinel, a post-quantum digital signature scheme that combines the proven security of SPHINCS+ (CCS 2019) with a new family of algebraic hash functions (Adv. Math. Commun. 2025) derived from the Tillich-Zemor paradigm (Eurocrypt 2008) with security rooted in the hardness of navigating expander graphs over SL_n(F_p), a problem believed to be hard even for quantum adversaries. We first provide empirical evidence of the security of this hash function, complementing the original theoretical analysis. We then show how the hash function can be integrated within the SPHINCS+ framework to give a secure signature scheme. We then model and analyze the security degradation of the proposed scheme, which informs the parameter selection we discuss next. Finally, we provide an implementation of the hash function and the proposed signature scheme Spinel as well as detailed empirical results for the performance of Spinel showing its feasibility in practice. Our approach lays the foundations for the design of algebraic hash-based signature schemes, expanding the toolkit of post-quantum cryptography.

This paper presents a quantum generalization of the multinomial distribution for the transition probabilities of $m$ identical photons in a $k$-port linear optical interferometer: two multinomial coefficients (one for the input configuration, one for the output) times the squared modulus of a coherent sum over routing matrices, weighted by the multivariate hypergeometric distribution; no Hilbert space formalism is needed to state or evaluate it. The classical multinomial is recovered when all photons enter through a single port, the coherent sum degenerating to a single term with no interference; the quantum family is not a generalization in the Askey sense but a parallel family that departs from classical statistics through the coherence of the amplitude summation. The $r$-th factorial moment carries a squared multinomial coefficient in place of the classical single one, the extra factor arising from the two copies of the amplitude expansion whose indices the Fock state forces to agree; for the beam splitter, the third cumulant is invariant under bosonic interference and the quantum departure first appears in the fourth cumulant as negative excess kurtosis; for multiport interferometers, however, three-body interference breaks this invariance and the departure enters already at the third cumulant. Cross-mode covariances involve the phases of the scattering matrix through coherence terms that strengthen output anti-correlations beyond the classical value; together with the squared-coefficient signature in the single-mode moments, these provide low-order statistical witnesses for boson sampling verification without requiring the full permanent computation.

Applying Nearest Convex Hull Classification (NCHC) to blind user identification in a massive Multiple Input Multiple Output (MIMO) communications system is proposed. The method is blind in the way that the Base Station (BS) only requires a training sequence containing unknown data symbols obtained from the user without further knowledge on the channel, modulation, coding or even noise power. We evaluate the algorithm under the assumption of gaussian transmit signals using the non-rigorous replica method. To facilitate the computations the existence of an Operator Valued Free Fourier Transform is postulated, which is verified by Monte Carlo simulation. The replica computations are conducted in the large but finite system by applying saddle-point integration with inverse temperature $\beta$ as the large parameter. The classifier accuracy is estimated by gaussian approximation through moment-matching.

Lloyd's k-means algorithm is one of the most widely used clustering methods. We prove that in high-dimensional, high-noise settings, the algorithm exhibits catastrophic failure: with high probability, essentially every partition of the data is a fixed point. Consequently, Lloyd's algorithm simply returns its initial partition - even when the underlying clusters are trivially recoverable by other methods. In contrast, we prove that Hartigan's k-means algorithm does not exhibit this pathology. Our results show the stark difference between these algorithms and offer a theoretical explanation for the empirical difficulties often observed with k-means in high dimensions.

Motivated by recent results on the dual formulation of optimal stopping problems, we investigate in this short paper how the knowledge of an approximating dual martingale can improve the efficiency of primal methods. In particular, we show on numerical examples that accurate approximations of a dual martingale efficiently reduce the variance for the primal optimal stopping problem.

We present an algorithm for the numerical evaluation of elliptic multiple polylogarithms for arbitrary arguments and to arbitrary precision. The cornerstone of our approach is a procedure to obtain a convergent $q$-series representation of elliptic multiple polylogarithms. Its coefficients are expressed in terms of ordinary multiple polylogarithms, which can be evaluated efficiently using existing libraries. In a series of preparation steps the elliptic polylogarithms are mapped into a region where the $q$-series converges rapidly. We also present an implementation of our algorithm into the \texttt{GiNaC} framework. This release constitutes the first public package capable of evaluating elliptic multiple polylogarithms to high precision and for arbitrary values of the arguments.

Selective conformal prediction aims to construct prediction sets with valid coverage for a test unit conditional on it being selected by a data-driven mechanism. While existing methods in the offline setting handle any selection mechanism that is permutation invariant to the labeled data, their extension to the online setting -- where data arrives sequentially and later decisions depend on earlier ones -- is challenged by the fact that the selection mechanism is naturally asymmetric. As such, existing methods only address a limited collection of selection mechanisms. In this paper, we propose PErmutation-based Mondrian Conformal Inference (PEMI), a general permutation-based framework for selective conformal prediction with arbitrary asymmetric selection rules. Motivated by full and Mondrian conformal prediction, PEMI identifies all permutations of the observed data (or a Monte-Carlo subset thereof) that lead to the same selection event, and calibrates a prediction set using conformity scores over this selection-preserving reference set. Under standard exchangeability conditions, our prediction sets achieve finite-sample exact selection-conditional coverage for any asymmetric selection mechanism and any prediction model. PEMI naturally incorporates additional offline labeled data, extends to selection mechanisms with multiple test samples, and achieves FCR control with fine-grained selection taxonomies. We further work out several efficient instantiations for commonly-used online selection rules, including covariate-based rules, conformal p/e-values-based procedures, and selection based on earlier outcomes. Finally, we demonstrate the efficacy of our methods across various selection rules on a real drug discovery dataset and investigate their performance via simulations.

We introduce a novel observable associated to Abelian monopole currents defined in the Maximal Abelian Projection of SU(3) Yang-Mills theory that captures the topology of the current loop. This observable, referred to as the $\textit{simplicity}$, is defined as the ratio of the zeroth over the first Betti number of the current graph for a given field configuration. A numerical study of the expectation value of the simplicity performed in the framework of Lattice Gauge Theories enables us to determine the deconfinement temperature to a higher degree of accuracy than that reached by conventional methods at a comparable computational effort. Our results suggest that Abelian current loops are strongly correlated with the degrees of freedoms of the theory that determine confinement. Our investigation opens new perspectives for the definition of an order parameter for deconfinement in Quantum Chromodynamics able to expose the potentially rich phase structure of the theory.

The article is devoted to the formulation and proof of the theorem on convergence with probability 1 of expansion of iterated Ito stochastic integrals of arbitrary multiplicity based on generalized multiple Fourier series converging in the sense of norm in Hilbert space. The cases of multiple Fourier-Legendre series and multiple trigonomertic Fourier series are considered in detail. The proof of the mentioned theorem is based on the general properties of multiple Fourier series as well as on the estimate for the fourth moment of approximation error in the method of expansion of iterated Ito stochastic integrals based on generalized multiple Fourier series.

We show that a mixed characteristic valuation ring with a value group $\Gamma$, $\val$ its valuation and a residue field of characteristic $p>0$, is a filtered colimit of complete intersection $\bf Z$-algebras if $\Gamma/{\bf Z}\val(p)$ has no $p$-torsion and $V$ is Henselian.

A book, concerning the classical restricted three body problem, and the approach to this old conundrum coming from the modern methods of symplectic and contact geometry. It is split into Part I (theoretical aspects), and Part II (practical aspects). The main themes are Floer theory, contact topology, symplectic dynamics, and astrodynamics (with a view towards space mission design).

In this article, we prove that sequences generated by the functional calculus $(f(T)(e_n))_{n \in \mathbb{N}}$ can be equivalently written as function sequences $(f_n(T) g)_{n \in \mathbb{N}}$, when $T$ is normal and $g$ a cyclic vector for $T$. Here, $(e_n)_{n \in \mathbb{N}}$ is a sequence of vectors, $T$ is a bounded normal operator, $f$ and $(f_n)_{n \in \mathbb{N}}$ are functions defined on a neighborhood of the spectrum $\sigma(T)$, and $g$ is a cyclic vector for $T$. After that, we characterize the frame property of such sequences in terms of the approximate point spectrum of $T^*$. Examples include certain operators (normal operators, compact operators, unilateral shifts, multiplication operators on Hardy spaces, etc.) that either generate only Riesz bases or allow redundancy. Our bridge theorem makes explicit the structural equivalence between frames generated by the functional calculus and function frames.

We explicitly construct new subgroups of the mapping class groups of an uncountable collection of infinite-type surfaces, including, but not limited to, free groups, Baumslag-Solitar groups, mapping class groups of other surfaces, and a large collection of wreath products. For each such subgroup $H$ and surface $S$, we show that there are countably many non-conjugate embeddings of $H$ into $\textrm{Map}(S)$; in certain cases, there are uncountably many such embeddings. The images of each of these embeddings cannot lie in the isometry group of $S$ for any hyperbolic metric and are not contained in the closure of the compactly supported subgroup of $\textrm{Map}(S)$. In this sense, our construction is new and does not rely on previously known techniques for constructing subgroups of mapping class groups. Notably, our embeddings of $\textrm{Map}(S')$ into $\textrm{Map}(S)$ are not induced by embeddings of $S'$ into $S$. Our main tool for all of these constructions is the utilization of special homeomorphisms of $S$ called shift maps, and more generally, multipush maps.

For any pair $(n,p)$, $n\in\mathbb{N}$ and $01$ such that $\inf_{0\le r<1} \frac{\int_{r}^1\omega(s)\,ds}{\int_{1-\frac{1-r}{k}}^1 \omega(s)\,ds}>1$. In this paper we extend this result to the setting of fractional derivatives. Being precise, for an analytic function $f(z)=\sum_{n=0}^\infty \widehat{f}(n) z^n$ we consider the fractional derivative $ D^{\mu}(f)(z)=\sum\limits_{n=0}^{\infty} \frac{\widehat{f}(n)}{\mu_{2n+1}} z^n $ induced by a radial weight $\mu \in \mathcal{D}$, where $\mu_{2n+1}=\int_0^1 r^{2n+1}\mu(r)\,dr$. Then, we prove that for any $p\in (0,\infty)$, the Littlewood-Paley equivalence $$\int_{\mathbb{D}} |f(z)|^p \omega(z)\,dA(z)\asymp \int_{\mathbb{D}}|D^{\mu}(f)(z)|^p\left[\int_{|z|}^1\mu(s)\,ds\right]^p\omega(z)\,dA(z)$$ holds for any analytic function $f$ in $\mathbb{D}$ if and only if $\omega\in\mathcal{D}$. We also prove that for any $p\in (0,\infty)$, the inequality $$\int_{\mathbb{D}} |D^{\mu}(f)(z)|^p \left[\int_{|z|}^1\mu(s)\,ds\right]^p\omega(z)\,dA(z) \lesssim \int_{\mathbb{D}} |f(z)|^p \omega(z)\,dA(z) $$ holds for any analytic function $f$ in $\mathbb{D}$ if and only if $\omega\in \widehat{\mathcal{D}}$.

We give a definition of (refined) Bloch groups of general commutative rings which agrees with the standard definition in the case of local rings whose residue field has at least $4$ elements. Under appropriate conditions on a ring $A$, satisfied by any field or local ring, these groups are closely related to third homology of $\mathrm{SL}_2(A)$ and to indecomposable $K_3$ of $A$. We analyze these conditions. We calculate the Bloch groups of $\mathbb{F}_2,\mathbb{F}_3,\mathbb{Z}$ and $\mathbb{Z}[\frac{1}{2}]$.

The goal of this work is to introduce and investigate a generalised Fisher Information in the setting of linear Fokker-Planck equations. This functional, which depends on two functions instead of one, exhibits the same decay behaviour as the standard Fisher information, and allows us to investigate different parts of the Fokker-Planck solution via an appropriate decomposition. Focusing almost exclusively on Fokker-Planck equations with constant drift and diffusion matrices, we will use a modification of the well established Bakry-Emery method with this newly defined functional to provide an alternative proof to the sharp long time behaviour of relative entropies of solutions to such equations when the diffusion matrix is positive definite and the drift matrix is defective. This novel approach is different to previous techniques and relies on minimal spectral information on the Fokker-Planck operator, unlike the one presented the authors' previous work, where powerful tools from spectral theory were needed.

We study universal solutions to reflection equations with a spectral parameter, so-called K-operators, within a general framework of universal K-matrices - an extended version of the approach introduced by Appel-Vlaar. Here, the input data is a quasi-triangular Hopf algebra $H$, its comodule algebra $B$ and a pair of consistent twists. In our setting, the universal K-matrix is an element of $B\otimes H$ satisfying certain axioms, and we consider the case $H$ is the quantum loop algebra for $sl_2$, and $B={\cal A}_q$ is the alternating central extension of the $q$-Onsager algebra. Considering tensor products of evaluation representations of $H$ in "non-semisimple" cases, the new set of axioms allows us to introduce and study fused K-operators of spin-$j$; in particular, to prove that for all $j\in\frac{1}{2}\mathbb{N}$ they satisfy the spectral-parameter dependent reflection equation. We provide their explicit expression in terms of elements of the algebra ${\cal A}_q$ for small values of spin-$j$. The precise relation between the fused K-operators of spin-$j$ and evaluations of a universal K-matrix for ${\cal A}_q$ is conjectured based on supporting evidences. We finally discuss implications of our results on the K-operators for quantum integrable systems.

In the setting of stochastic Volterra equations, and in particular rough volatility models, we show that conditional expectations are the unique classical solutions to path-dependent PDEs. The latter arise from the functional Itô formula developed by [Viens, F., & Zhang, J. (2019). A martingale approach for fractional Brownian motions and related path dependent PDEs. Ann. Appl. Probab.]. We then leverage these tools to study weak rates of convergence for discretised stochastic integrals of smooth functions of a Riemann-Liouville fractional Brownian motion with Hurst parameter $H \in (0,\frac{1}{2})$. These integrals approximate log-stock prices in rough volatility models. We obtain the optimal weak error rates of order $1$ if the test function is quadratic and of order $(3H+\frac{1}{2})\wedge1$ if the test function is five times differentiable; in particular these conditions are independent of the value of $H$.

In this paper, we consider the problem of constructing new optimal explicit and implicit Adams-type difference formulas for finding an approximate solution to the Cauchy problem for an ordinary differential equation in a Hilbert space. In this work, I minimize the norm of the error functional of the difference formula with respect to the coefficients, we obtain a system of linear algebraic equations for the coefficients of the difference formulas. This system of equations is reduced to a system of equations in convolution and the system of equations is completely solved using a discrete analog of a differential operator $d^2/dx^2-1$. Here we present an algorithm for constructing optimal explicit and implicit difference formulas in a specific Hilbert space. In addition, comparing the Euler method with optimal explicit and implicit difference formulas, numerical experiments are given. Experiments show that the optimal formulas give a good approximation compared to the Euler method.

We consider a well posed SPDE$\colon dZ=(AZ+b(Z)) dt+dW(t),\,Z_0=x, $ on a separable Hilbert space $H$, where $A\colon H\to H$ is self-adjoint, negative and such that $A^{-1+\beta}$ is of trace class for some $\beta>0$, $b\colon H\to H$ is Lipschitz continuous and $W$ is a cylindrical Wiener process on $H$. We denote by $W_A(t)=\int_0^te^{(t-s)A}\,dW(s),\,t\in[0,T],$ the stochastic convolution. We prove, with the help of a formula for nonlinear transformations of Gaussian integrals due to R. Ramer, the following identity $$(P\circ Z_x^{-1})(\Phi) =\int_X\Phi(h+e^{\cdot A}x)\, \exp\left\{ -\tfrac12|\gamma_x(h)|^2_{ H_{Q_T}} + I(\gamma_x)(h)\right\} N_{Q_T}(dh), $$ where $ N_{Q_T}$ is the law of $W_A$ in $C([0,T],H)$, $ H_{Q_T}$ its Cameron--Martin space, $$ [\gamma_x(k)](t)=\int_0^t e^{(t-s)A}b(k(s)+e^{sA}x) ds,\quad t\in[0,T], \; k \in C([0,T],H) $$ and $I(\gamma_x) $ is the Itô integral of $\gamma_x$. Some applications are discussed; in particular, when $b$ is dissipative we provide an explicit formula for the law of the stationary process and the invariant measure $\nu$ of the Markov semigroup $(P_t)$. Some concluding remarks are devoted to a similar problem with colored noise.

The goal of this work is to find the sharp rate of convergence to equilibrium under the quadratic Fisher information functional for solutions to Fokker-Planck equations governed by a constant drift term and a constant, yet possibly degenerate, diffusion matrix. A key ingredient in our investigation is a recent work of Arnold, Signorello, and Schmeiser, where the $L^2$-propagator norm of such Fokker-Planck equations was shown to be identical to the propagator norm of a finite dimensional ODE which is determined by matrices that are intimately connected to those appearing in the associated Fokker-Planck equations.

We study e-values for quantifying evidence against exchangeability and general invariance of a random variable under a compact group. We start by characterizing such e-values, and explaining how they nest traditional group invariance tests as a special case. We show they can be easily designed for an arbitrary test statistic, and computed through Monte Carlo sampling. We prove a result that characterizes optimal e-values for group invariance against optimality targets that satisfy a mild orbit-wise decomposition property. We apply this to design expected-utility-optimal e-values for group invariance, which include both Neyman-Pearson-optimal tests and log-optimal e-values. Moreover, we generalize the notion of rank- and sign-based testing to compact groups, by using a representative inversion kernel. In addition, we characterize e-processes for group invariance for arbitrary filtrations, and provide tools to construct them. We also describe test martingales under a natural filtration, which are simpler to construct. Peeking beyond compact groups, we encounter e-values and e-processes based on ergodic theorems. These nest e-processes based on de Finetti's theorem for testing exchangeability.

During 2022--2023 Z.-W. Sun posed many conjectures on infinite series with summands involving generalized harmonic numbers. Motivated by this, we deduce $58$ series identities involving harmonic numbers, eight of which were previously conjectured by the second author. For example, we obtain that \[ \sum_{k=1}^{\infty} \frac{(-1)^k}{k^2{2k \choose k}{3k \choose k}} \left( \frac{7 k-2}{2 k-1} H_{k-1}^{(2)}-\frac{3}{4 k^2} \right) = \frac{\pi^4}{720}. \] and \[ \sum_{k=1}^\infty \frac{1}{k^2 {2k \choose k}^2} \left( \frac{30k-11}{k(2k-1)} (H_{2k-1}^{(3)} + 2 H_{k-1}^{(3)}) + \frac{27}{8k^4} \right) = 4 \zeta(3)^2, \] where $H_n^{(m)}$ denotes $\sum_{0

We introduce a new method to study mixed characteristic deformation of line bundles. In particular, for sufficiently large smooth projective families $f : \mathscr{X} \to \mathscr{S}$ defined over the ring of $N$-integers $\mathscr{O}_{L}[1/N]$ of a number field $L$, we produce a proper closed subscheme $\mathscr{E} \subsetneq \mathscr{S}$ outside of which all line bundles appearing in positive characteristic fibres of $f$ admit characteristic zero lifts. This in particular applies to elliptic surfaces over $\mathbb{P}^1$ and projective hypersurfaces in $\mathbb{P}^3$ of degree $d \geq 5$. We also study the locus in $\mathscr{E}$ in more detail in the $h^{0, 2} = 2$ case.

Keller-Segel systems are a set of nonlinear partial differential equations used to model chemotaxis in biology. In this paper, we propose two alternating direction implicit (ADI) schemes to solve the 2D Keller-Segel systems directly with minimal computational cost, while preserving positivity, energy dissipation law and mass conservation. One scheme unconditionally preserves positivity, while the other does so conditionally. Both schemes achieve second-order accuracy in space, with the former being first-order accuracy in time and the latter second-order accuracy in time. Besides, the former scheme preserves the energy dissipation law asymptotically. We validate these results through numerical experiments, and also compare the efficiency of our schemes with the standard five-point scheme, demonstrating that our approaches effectively reduce computational costs.

For all $r\ge1,$ we verify the following conjecture of Hironaka: for a $p$-adic field $F$ with $p$ odd, the space of spherical functions of $\mathrm{Sym}_{r\times r}(F)\cap\mathrm{GL}_r(F)$ is free of rank $4^r$ over the Hecke algebra.

We give explicit models for spherical functions on $p$-adic symmetric spaces $X=H\backslash G$ for pairs of $p$-adic groups $(G,H)$ of the form $(\mathrm{U}(2r),\mathrm{U}(r)\times \mathrm{U}(r)),$ $(\mathrm{O}(2r),\mathrm{O}(r)\times \mathrm{O}(r)),$ $(\mathrm{Sp}(4r),\mathrm{Sp}(2r)\times\mathrm{Sp}(2r))),$ $(\mathrm{U}(2r+1),\mathrm{U}(r+1)\times \mathrm{U}(r)),$ and $ (\mathrm{O}(2r+1),\mathrm{O}(r+1)\times \mathrm{O}(r)).$ As an application, we completely describe their Hecke module structure.

We study a multidimensional stochastic differential equation with additive noise: \[ d X_t=b(t, X_t) dt +d \xi_t, \] where the drift $b$ is integrable in space and time, and $\xi$ is either a fractional Brownian motion or a Lévy process. We show weak existence of solutions to this equation under the optimal condition on integrability indices of $b$, going beyond the subcritical Krylov--Röckner (Prodi--Serrin--Ladyzhenskaya) regime. This extends the recent results of Krylov (2020) to the fractional Brownian and Lévy cases. We also construct a counterexample to demonstrate the optimality of this condition. In the one-dimensional case, we show the existence of a strong solution under the same condition. Our methods are built upon a version of the stochastic sewing lemma of Lê and the John--Nirenberg inequality.

Districting-and-routing is a strategic problem aiming to aggregate basic geographical units (e.g., zip codes) into delivery districts. Its goal is to minimize the expected long-term routing cost of performing deliveries in each district separately. Solving this stochastic problem poses critical challenges since repeatedly evaluating routing costs on a set of scenarios while searching for optimal districts takes considerable time. Consequently, solution approaches usually replace the true cost estimation with continuous cost approximation formulas extending Beardwood-Halton-Hammersley and Daganzo's work. These formulas commit errors that can be magnified during the optimization step. To reconcile speed and solution quality, we introduce a supervised learning and optimization methodology leveraging a graph neural network for delivery-cost estimation. This network is trained to imitate known costs generated on a limited subset of training districts. It is used within an iterated local search procedure to produce high-quality districting plans. Our computational experiments, conducted on five metropolitan areas in the United Kingdom, demonstrate that the graph neural network predicts long-term district cost operations more accurately, and that optimizing over this oracle permits large economic gains (10.12% on average) over baseline methods that use continuous approximation formulas or shallow neural networks. Finally, we observe that having compact districts alone does not guarantee high-quality solutions and that other learnable geometrical features of the districts play an essential role.

We study a type of forced discrete mechanical system $(Q,L_d,f_d)$ -- that we name of Routh type -- whose (discrete) time-flow preserves a symplectic structure on $Q\times Q$. That structure arises as the pullback via the forced discrete Legendre transform of the canonical symplectic structure on $T^*Q$ modified by a "magnetic term". One example of this type of system is provided by the Lagrangian reduction of a symmetric (unforced) discrete mechanical system in the Routh style. In this particular case, we do not reduce by the full symmetry group but, rather, by an appropriate isotropy subgroup. In this context, the preserved symplectic structure can be alternatively seen as the Marsden-Weinstein reduction of the canonical symplectic structure $\omega_{L_d}$ on $Q\times Q$.

Besides classical feed-forward neural networks such as multilayer perceptrons, also neural ordinary differential equations (neural ODEs) have gained particular interest in recent years. Neural ODEs can be interpreted as an infinite depth limit of feed-forward or residual neural networks. We study the input-output dynamics of finite and infinite depth neural networks with scalar output. In the finite depth case, the input is a state associated with a finite number of nodes, which maps under multiple non-linear transformations to the state of one output node. In analogy, a neural ODE maps an affine linear transformation of the input to an affine linear transformation of its time-$T$ map. We show that, depending on the specific structure of the network, the input-output map has different properties regarding the existence and regularity of critical points. These properties can be characterized via Morse functions, which are scalar functions where every critical point is non-degenerate. We prove that critical points cannot exist if the dimension of the hidden layer is monotonically decreasing or the dimension of the phase space is smaller than or equal to the input dimension. In the case that critical points exist, we classify their regularity depending on the specific architecture of the network. We show that except for a Lebesgue measure zero set in the weight space, each critical point is non-degenerate if for finite depth neural networks the underlying graph has no bottleneck, and if for neural ODEs, the affine linear transformations used have full rank. For each type of architecture, the proven properties are comparable in the finite and infinite depth cases. The established theorems allow us to formulate results on universal embedding and universal approximation, i.e., on the exact and approximate representation of maps by neural networks and neural ODEs.

We will see how to define the metric $\beta$, which turns the topological space of continuous functions whose domains are open subsets of a locally compact and second countable space $X$ to values in a polish space $Y$, called $(C_{od}(X,Y),\tau_{\iota,D})$ into a polish space. In particular, we will present a metric for the inverse semigroup of homeomorphisms of a locally compact, Hausdorff, and second-countable space.

This article establishes alcove walk models for intersections of Schubert varieties and partially semi-infinite orbits in the affine Grassmannian of a split reductive group (we call such intersections parabolic Mirković-Vilonen intersections). More precisely, we describe explicit cellular pavings of these intersections, indexed by certain positively-folded alcove walks. We prove a parametrization of the irreducible components of maximal possible dimension, in terms of alcove walks of maximal possible dimension. We then deduce a new combinatorial description of branching to Levi subgroups of irreducible highest weight representations, and in particular we give a new algorithm for computing the characters of such representations.

We consider the question whether a real threefold X fibred into quadric surfaces over the real projective line is stably rational (over R) if the topological space X(R) is connected. We give a counterexample. When all geometric fibres are irreducible, the question is open. We investigate a family of such fibrations for which the intermediate jacobian technique is not available. We produce two independent methods which in many cases enable one to prove decomposition of the diagonal.

Chung's Lemma is a classical tool for establishing asymptotic convergence rates of (stochastic) optimization methods under strong convexity-type assumptions and appropriate polynomial diminishing step sizes. In this work, we develop a generalized version of Chung's Lemma, which provides a simple non-asymptotic convergence framework for a more general family of step size rules. We demonstrate broad applicability of the proposed generalized lemma by deriving tight non-asymptotic convergence rates for a large variety of stochastic methods. In particular, we obtain partially new non-asymptotic complexity results for stochastic optimization methods, such as Stochastic Gradient Descent (SGD) and Random Reshuffling (RR), under a general $(\theta,\mu)$-Polyak-Lojasiewicz (PL) condition and for various step sizes strategies, including polynomial, constant, exponential, and cosine step sizes rules. Notably, as a by-product of our analysis, we observe that exponential step sizes exhibit superior adaptivity to both landscape geometry and gradient noise; specifically, they achieve optimal convergence rates without requiring exact knowledge of the underlying landscape or separate parameter selection strategies for noisy and noise-free regimes. Our results demonstrate that the developed variant of Chung's Lemma offers a versatile, systematic, and streamlined approach to establish non-asymptotic convergence rates under general step size rules.

We consider maps with tight boundaries, i.e. maps whose boundaries have minimal length in their homotopy class, and discuss the properties of their generating functions $T^{(g)}_{\ell_1,\ldots,\ell_n}$ for fixed genus $g$ and prescribed boundary lengths $\ell_1,\ldots,\ell_n$, with a control on the degrees of inner faces. We find that these series appear as coefficients in the expansion of $\omega^{(g)}_n(z_1,\ldots,z_n)$, a fundamental quantity in the Eynard-Orantin theory of topological recursion, thereby providing a combinatorial interpretation of the Zhukovsky transformation used in this context. This interpretation results from the so-called trumpet decomposition of maps with arbitrary boundaries. In the planar bipartite case, we obtain a fully explicit formula for $T^{(0)}_{2\ell_1,\ldots,2\ell_n}$ from the Collet-Fusy formula. We also find recursion relations satisfied by $T^{(g)}_{\ell_1,\ldots,\ell_n}$, which consist in adding an extra tight boundary, keeping the genus $g$ fixed. Building on a result of Norbury and Scott, we show that $T^{(g)}_{\ell_1,\ldots,\ell_n}$ is equal to a parity-dependent quasi-polynomial in $\ell_1^2,\ldots,\ell_n^2$ times a simple power of the basic generating function $R$. In passing, we provide a bijective derivation in the case $(g,n)=(0,3)$, generalizing a recent construction of ours to the non bipartite case.

Let $k$ be an arbitrary field. We classify the maximal reductive subgroups of maximal rank in any classical simple algebraic $k$-group in terms of combinatorial data associated to their indices. This result complements [S, 2022], which does the same for the exceptional groups. We determine which of these subgroups may be realised over a finite field, the real numbers, or over a $\mathfrak{p}$-adic field. We also look at the asymptotics of the number of such subgroups as the rank grows large.

We develop a functional-analytic characterization of output tracking controllability for finite-dimensional linear systems. By formulating tracking as the surjectivity of the control-to-output map on suitable trajectory spaces, we show that exact tracking is equivalent to a nonstandard observability inequality for the adjoint dynamics. This characterization enables a Hilbert Uniqueness Method (HUM) type variational construction of minimum-norm tracking controls and makes explicit the intrinsic regularity requirements on reference trajectories induced by the system dynamics and the output operator. The same framework also yields a natural notion of approximate tracking when exact tracking fails. We provide explicit formulas in the scalar case and report numerical experiments for ODEs and semi-discretized PDEs, demonstrating the method for both smooth and non-smooth targets.

We investigate the properties of Hecke operator for sesquiharmonic Maass forms. We begin by proving Hecke equivariance of the divisor lifting with respect to sesquiharmonic Mass functions, which maps an integral weight meromorphic modular form to the holomorphic part of the Fourier expansion of a weight 2 sesquiharmonic Maass form. Using this Hecke equivariance, we show that the sesquiharmonic Maass functions, whose images under the hyperbolic Laplace operator are the Faber polynomials $J_n$ of the $j$-function, form a Hecke system analogous to $J_n$. By combining the Hecke equivariance of the divisor lifting with that of the Borcherds isomorphism, we extend Matsusaka's finding on the twisted traces of sesquiharmonic Maass functions.

This paper presents a new mixed finite element method for the Cahn-Hilliard equation. The well-posedness of the mixed formulation is established and the error estimates for its linearized fully discrete scheme are provided. The new mixed finite element method provides a unified construction in two and three dimensions allowing for arbitrary polynomial degrees. Numerical experiments are given to validate the efficiency and accuracy of the theoretical results.

We give new characterizations of growth conditions at strict local minimizers. The main characterizations are a variant of the so-called tilt stability property and an analog of the classical Polyak--Łojasiewicz condition, where the gradient is replaced by linear perturbations.

We introduce and study the Separation Problem for infinite graphs, which involves determining whether a connected graph splits into at least two infinite connected components after the removal of a given finite set of edges. We prove that this problem is decidable for every highly computable graph with finitely many ends. Using this result, we demonstrate that König's Infinity Lemma is effective for such graphs. We also apply it to analyze the complexity of the Eulerian Path Problem for infinite graphs, showing that much of its complexity arises from counting ends. Indeed, the Eulerian Path Problem becomes strictly easier when restricted to graphs with a fixed number of ends. Under this restriction, we provide a complete characterization of the problem. Finally, we study the Separation Problem in a uniform setting (i.e., where the graph is also part of the input) and offer a nearly complete characterization of its complexity and its relationship to counting the number of ends.

We establish a geometric criterion for local microlocal holonomies to be globally regular on the moduli space of Lagrangian fillings. This local-to-global regularity result holds for arbitrary Legendrian links and it is a key input for the study of cluster structures on such moduli spaces. Specifically, we construct regular functions on derived moduli stacks of sheaves with Legendrian microsupport by studying the Hochschild homology of the associated dg-categories via relative Lagrangian skeleta. In this construction, a key geometric result is that local microlocal merodromies along positive relative cycles in Lagrangian fillings yield global Hochschild 0-cycles for these dg-categories.

In this paper, we propose a new conjecture describing the structure of the unitary dual in terms of Arthur representations for connected reductive algebraic groups defined over any non-Archimedean local field of characteristic zero. This conjecture provides a candidate set for the unitary dual, constructed from Arthur representations. For classical groups, we develop an explicit algorithm to generate this candidate set. Evidence for its exhaustiveness includes compatibility with the known generic unitary dual, unramified unitary dual, and low-corank representations. As further support, we verify the conjecture for the unitary dual of the exceptional group of type $G_2$.

Given a variety with a suitable Brauer class, we present a general pullback construction that produces varieties that has Brauer--Manin obstruction to the existence of rational points. We then study Severi--Brauer fibrations and their Brauer groups without relying on explicit defining equations. As a key application, we show that there exist Severi--Brauer fibrations with index one that fails Hasse principle.

We prove Vojta's abc conjecture for projective space ${\Bbb P}^n({\Bbb C})$, assuming that the entire curves in ${\Bbb P}^n({\Bbb C})$ are highly ramified over the coordinate hyperplanes. This extends the results of Guo Ji and the second-named author for the case $n=2$ (see \cite{GW22}). We also explore the corresponding results for projective toric varieties. Consequently, we establish a version of Campana's orbifold conjecture for finite coverings of projective toric varieties.

The population control problem is a parameterised problem where a controller sends messages to a whole population of identical finite-state agents, aiming to eventually move them all into a target state. The decision problem asks whether this can be achieved for arbitrarily large finite populations. We focus on the randomised version of this problem, where every agent is a copy of the same finite Markov Decision Process and non-determinism in the {global} action chosen by the controller is resolved independently and uniformly at random. Colcombet, Fijalkow and Ohlmann showed that this problem is decidable, but without any complexity upper bound. We show that the random population control problem is in fact EXPTIME-complete.

We systematically investigate the preservation of differential privacy in functional data analysis, beginning with functional mean estimation and extending to varying coefficient model estimation. Our work introduces a distributed learning framework involving multiple servers, each responsible for collecting several sparsely observed functions. This hierarchical setup introduces a mixed notion of privacy. Within each function, user-level differential privacy is applied to $m$ discrete observations. At the server level, central differential privacy is deployed to account for the centralised nature of data collection. Across servers, only private information is exchanged, adhering to federated differential privacy constraints. To address this complex hierarchy, we employ minimax theory to reveal several fundamental phenomena: from sparse to dense functional data analysis, from user-level to central and federated differential privacy costs, and the intricate interplay between different regimes of functional data analysis and privacy preservation. To the best of our knowledge, this is the first study to rigorously examine functional data estimation under multiple privacy constraints. Our theoretical findings are complemented by efficient private algorithms and extensive numerical evidence, providing a comprehensive exploration of this challenging problem.

We study propagation in a system consisting of two topological insulators without a magnetic field, whose interface is a non-compact, smooth, and connected curve without boundary. The dynamics are governed by an adiabatic modulation of a Dirac operator with a smooth, effective variable mass. We determine the evolution of the semiclassical measure of the solution using a two-scale Wigner measure method, after reducing the Hamiltonian to a normal form.

This paper develops a graphical calculus to determine the $n$-shifted Poisson structures on finitely generated semi-free commutative differential graded algebras. When applied to the Chevalley-Eilenberg algebra of an ordinary Lie algebra, we recover Safronov's result that the $(n=1)$- and $(n=2)$-shifted Poisson structures in this case are given by quasi-Lie bialgebra structures and, respectively, invariant symmetric tensors. We generalize these results to the Chevalley-Eilenberg algebra of a Lie $2$-algebra and obtain $n\in\{1,2,3,4\}$ shifted Poisson structures in this case, which we interpret as semi-classical data of `higher quantum groups'.

In this paper, we first establish the quantitative properties for positive solutions to the Moser-Trudinger equations in the two-dimensional Poincaré disk $\mathbb{B}^2$: \begin{equation*}\label{mt1} \left\{ \begin{aligned} &-\Delta_{\mathbb{B}^2}u=\lambda ue^{u^2},\ x\in\mathbb{B}^2, &u\to0,\ \text{when}\ \rho(x)\to\infty, &||\nabla_{\mathbb{B}^2} u||_{L^2(\mathbb{B}^2)}^2\leq M_0, \end{aligned} \right. \end{equation*} where $0<\lambda<\frac{1}{4}=\inf\limits_{u\in W^{1,2}(\mathbb{B}^2)\backslash\{0\}}\frac{\|\nabla_{\mathbb{B}^2}u\|_{L^2(\mathbb{B}^2)}^2}{\|u\|_{L^2(\mathbb{B}^2)}^2}$, $\rho(x)$ denotes the geodesic distance between $x$ and the origin and $M_0$ is a fixed large positive constant (see Theorem 1.1). Furthermore, by doing a delicate expansion for Dirichlet energy $\|\nabla_{\mathbb{B}^2}u\|_{L^2(\mathbb{B}^2)}^2$ when $\lambda$ approaches to $0,$ we prove that there exists $\Lambda^\ast>4\pi$ such that the Moser-Trudinger functional $F(u)=\int_{\mathbb{B}^2}\left(e^{u^2}-1\right) dV_{\mathbb{B}^2}$ under the constraint $\int_{\mathbb{B}^2}|\nabla_{\mathbb{B}^2}u|^2 dV_{\mathbb{B}^2}=\Lambda$ has at least one positive critical point for $\Lambda\in(4\pi,\Lambda^{\ast})$ up to some Möbius transformation. Finally, when $\lambda\rightarrow 0$, by doing a more accurate expansion for $u$ near the origin and away from the origin, applying a local Pohozaev identity around the origin and the uniqueness of the Cauchy initial value problem for ODE,Cauchy-initial uniqueness for ODE, we prove that the Moser-Trudinger equation only has one positive solution when $\lambda$ is close to $0.$ During the process of the proofs, we overcome some new difficulties which involves the decay properties of the positive solutions, as well as some precise expansions for the solutions both near the origin and away from the origin.

In this paper, we consider the following Hardy-type mean field equation \[ \left\{ {\begin{array}{*{20}{c}} { - \Delta u-\frac{1}{(1-|x|^2)^2} u = \lambda e^u}, & {\rm in} \ \ B_1,\\ {\ \ \ \ u = 0,} &\ {\rm on}\ \partial B_1, \end{array}} \right. \] \[\] where $\lambda>0$ is small and $B_1$ is the standard unit disc of $\mathbb{R}^2$. Applying the moving plane method of hyperbolic space and the accurate expansion of heat kernel on hyperbolic space, we establish the radial symmetry and Brezis-Merle lemma for solutions of Hardy-type mean field equation. Meanwhile, we also derive the quantitative results for solutions of Hardy-type mean field equation, which improves significantly the compactness results for classical mean-field equation obtained by Brezis-Merle and Li-Shafrir. Furthermore, applying the local Pohozaev identity from scaling, blow-up analysis and a contradiction argument, we prove that the solutions are unique when $\lambda$ is sufficiently close to 0.

In this paper, we explore functional identities with central values in gr-prime rings involving pairs of homogeneous derivations. We establish commutativity conditions that extend classical results from prime rings to the graded setting. In particular, we show that under certain conditions on homogeneous derivations, the ring must be commutative. Furthermore, we demonstrate that these results cannot be extended to gr-semiprime rings.

In this paper, we focus on Hamilton's pinching conjecture formulated in Hamilton's paper "Three-manifolds with positive Ricci curvature". Let $(M, g)$ be a complete, connected, noncompact Riemannian $3$-manifold satisfying the Ricci-pinching condition. Then, it is flat. Here, we give an alternative proof, based on nonlinear potential theory, under the extra hypothesis of superquadratic volume growth.

We propose a new approach to constructing semistable integral models of hypersurfaces over a discretely valued complete field K. For each stable hypersurface X over K we define a continuous stability function on the Bruhat-Tits building of PGL_{n+1}(K); its global minima control semistable hypersurface models after finite extensions of K. In particular, in residue characteristic zero the problem reduces to minimizing this function on the original building and then passing to a finite extension that turns a rational minimizer into a vertex. This extends work of Kollar and of Elsenhans-Stoll on minimal hypersurface models. We implement the resulting strategy for plane curves over p-adic number fields. In a follow-up article we use our results to compute the semistable reduction of smooth plane quartics.

The stochastic three points (STP) algorithm is a derivative-free optimization technique designed for unconstrained optimization problems in $\mathbb{R}^d$. In this paper, we analyze this algorithm for three classes of functions: smooth functions that may lack convexity, smooth convex functions, and smooth functions that are strongly convex. Our work provides the first almost sure convergence results of the STP algorithm, alongside some convergence results in expectation. For the class of smooth functions, we establish that the best gradient iterate of the STP algorithm converges almost surely to zero at a rate of $o(1/{T^{\frac{1}{2}-\epsilon}})$ for any $\epsilon\in (0,\frac{1}{2})$, where $T$ is the number of iterations. Furthermore, within the same class of functions, we establish both almost sure convergence and convergence in expectation of the final gradient iterate towards zero. For the class of smooth convex functions, we establish that $f(\theta^T)$ converges to $\inf_{\theta \in \mathbb{R}^d} f(\theta)$ almost surely at a rate of $o(1/{T^{1-\epsilon}})$ for any $\epsilon\in (0,1)$, and in expectation at a rate of $O(\frac{d}{T})$ where $d$ is the dimension of the space. Finally, for the class of smooth functions that are strongly convex, we establish that when step sizes are obtained by approximating the directional derivatives of the function, $f(\theta^T)$ converges to $\inf_{\theta \in \mathbb{R}^d} f(\theta)$ in expectation at a rate of $O((1-\frac{\mu}{2\pi dL})^T)$, and almost surely at a rate of $o((1-s\frac{\mu}{2\pi dL})^T)$ for any $s\in (0,1)$, where $\mu$ and $L$ are the strong convexity and smoothness parameters of the function.

Inspired by a chessboard puzzle of Dudeney, the general position problem in graph theory asks for a largest set $S$ of vertices in a graph such that no three elements of $S$ lie on a common shortest path. The number of vertices in such a largest set is the \emph{general position number} of the graph. This paper provides a survey of this rapidly growing problem, which now has an extensive literature. We cover exact results for various graph classes and the behaviour of the general position number under graph products and operations. We also discuss interesting variations of the general position problem, including those corresponding to different graph convexities, as well as dynamic, fractional, colouring and game versions of the problem.

Smoothness is crucial for attaining fast rates in first-order optimization. However, many optimization problems in modern machine learning involve non-smooth objectives. Recent studies relax the smoothness assumption by allowing the Lipschitz constant of the gradient to grow with respect to the gradient norm, which accommodates a broad range of objectives in practice. Despite this progress, existing generalizations of smoothness are restricted to Euclidean geometry with $\ell_2$-norm and only have theoretical guarantees for optimization in the Euclidean space. In this paper, we address this limitation by introducing a new $\ell*$-smoothness concept that measures the norm of Hessians in terms of a general norm and its dual, and establish convergence for mirror-descent-type algorithms, matching the rates under the classic smoothness. Notably, we propose a generalized self-bounding property that facilitates bounding the gradients via controlling suboptimality gaps, serving as a principal component for convergence analysis. Beyond deterministic optimization, we establish sharp convergence for stochastic mirror descent, matching state-of-the-art under classic smoothness. Our theory also extends to non-convex and composite optimization, which may shed light on practical usages of mirror descent, including pre-training and post-training of LLMs.

We review the main achievements regarding the interactions between gem theory (which makes use of edge-colored graphs to represent PL-manifolds of arbitrary dimension) and both the classical representation of PL 4-manifolds via Kirby diagrams and the more recent one via trisections. Original results also appear (in particular, about gems representing closed 4-manifolds which need 3-handles in their handle decomposition, as well as about trisection diagrams), together with open problems and further possible applications to the study of compact PL 4-manifolds.

We study an individual-based stochastic SIR epidemic model with infection-age dependent infectivity on a large random graph, capturing individual heterogeneity and non-homogeneous connectivity. Each individual is associated with particular characteristics (for example, spatial location and age structure), which may not be i.i.d., and is represented by a particular node. The connectivities among the individuals are given by a non-homogeneous random graph, whose connecting probabilities may depend on the individual characteristics of the edge. To each individual is associated a random infectivity function of its infection age, which is allowed to depend upon the individual characteristics. We use measure-valued processes to describe the epidemic evolution dynamics, tracking the infection age of all individuals, and their associated characteristics. We consider the epidemic dynamics as the population size grows to infinity under a specific scaling of the connectivity graph related to the convergence to a graphon. In the limit, we obtain a system of measure-valued equations, which can be also represented as a PDE model on graphon, and reflects the heterogeneities in individual characteristics and social connectivity.

The Nonlinear GMRES (NGMRES) proposed by Washio and Oosterlee [Electron. Trans. Numer. Anal, 6(271-290), 1997] is an acceleration method for fixed point iterations. It has been demonstrated to be effective, but its convergence properties have not been extensively studied in the literature so far. In this work we aim to close some of this gap, by offering a convergence analysis for NGMRES applied to linear systems. A central part of our analysis focuses on identifying equivalences between NGMRES and the classical Krylov subspace GMRES method.

This paper establishes explicit structural descriptions of $K$-theory for three specific classes of algebras. For cyclic $p$-group rings and truncated polynomial rings over $\mathbb{Z}/p^s\mathbb{Z}$, we determine their $\widetilde{K}_2$-structures by deriving explicit formulas, obtained by relating $(\mathbb{Z}/p^s\mathbb{Z})[x]/(x^n)$ and $(\mathbb{Z}/p^s\mathbb{Z})[C_{p^n}]$ to a common algebraic framework. For abelian $p$-group rings with coefficients in $\widehat{\mathbb{Z}}_p$, we extend the known isomorphism between reduced continuous $K_2$-groups and the first cyclic homology group to arbitrary finite abelian $p$-groups. By employing a generalized Artin-Hasse map, we provide a constructive proof of this isomorphism, establishing an explicit splitting of the relevant exact sequence. Furthermore, this isomorphism is explicitly realized by extending Oliver's $p$-adic logarithm of continuous $K_2$-groups. A corresponding description is also provided for the map from continuous $K_2$ to linearized $K_2$. These results further clarify the structural relationships between continuous $K_2$, the first cyclic homology group, the module of Kähler differentials, and linearized $K_2$.

This paper investigates quasi-Monte Carlo (QMC) integration of Lebesgue integrable functions with respect to a density function over $\mathbb{R}^s$. We extend the construction-free median QMC rule proposed by Goda and L'ecuyer (SIAM J. Sci. Comput., 2022) to the weighted unanchored Sobolev space of functions defined over $\mathbb{R}^s$ introduced by Nichols and Kuo (J. Complexity, 2014). By taking the median of $k = \mathcal{O}(\log N)$ independent randomized QMC estimators, we prove that for any $\epsilon\in (0,r-\frac{1}{2}]$, our method achieves a mean absolute error bound of $\mathcal{O}(N^{-r+\epsilon})$, where $N$ is the number of points and $r>\frac{1}{2}$ is a parameter determined by the function space. This rate matches the rate of randomly shifted lattice rules obtained via a component-by-component (CBC) construction, while our approach requires no specific CBC constructions or prior knowledge of the space's weight structure. Numerical experiments demonstrate that our method attains an accuracy comparable to the CBC construction based method, and outperforms the Monte Carlo method.

In 2020, Dahlberg, She, and van Willigenburg conjectured that the chromatic symmetric function of any tree with maximum degree at least 4 is not e-positive. Zheng and Tom verified this conjecture for all trees with maximum degree at least 5 and spiders with maximum degree 4, and in their proofs the following necessary condition given by Wolfgang plays an important role: every connected graph having e-positive chromatic symmetric function must contain a connected partition of every type. In order to make further progress on this conjecture, we refine Wolfgang's result in a quantitative way. At first, we give an explicit formula for the e-coefficients of trees in terms of their connected partitions, by which e-positivity is equivalent to a series of inequalities for the numbers of connected partitions. Based on this formula, we present several necessary conditions on the numbers of connected partitions or acyclic orientations for trees to be e-positive. These necessary conditions turn out to be characterizations on the structure of e-positive trees, and as sample applications we prove the non-e-positivity of several classes of trees with maximum degree 3 or 4. We further make more discussions and calculations on trees with maximum degree 4 and having a connected partition of every type, which inspire us to come up with a list of open problems towards the final resolution of the above conjecture.

Consider the underdamped Langevin process $(q(t),p(t))_{t\geq0}$ in $\R^d\times\R^d$. We derive the low-temperature asymptotic of its mean-transition time between basins of attraction for a double-well potential. This asymptotic is called Eyring-Kramers law and often relies in the literature on Potential theory tools which are ill-defined for hypoelliptic processes like the underdamped Langevin process. In this work, we implement a novel approach which circumvents the use of these traditional methods.

We disprove a conjecture of Kuznetsov--Shinder, which posits that $D$-equivalent simply connected varieties are $L$-equivalent, by constructing a counterexample using moduli spaces of sheaves on K3 surfaces.

Importance Sampling (IS) is a widely used variance reduction technique for enhancing the efficiency of Monte Carlo methods, particularly in rare-event simulation and related applications. Despite its effectiveness, the performance of IS is highly sensitive to the choice of the proposal distribution and often requires stochastic calibration. While the design and analysis of IS have been extensively studied in estimation settings, applying IS within stochastic optimization introduces a lesser-known fundamental challenge: the decision variable and the importance sampling distribution are mutually dependent, creating a circular optimization structure. This interdependence complicates both convergence analysis and variance control. In this paper, we consider the generic setting of convex stochastic optimization with linear constraints. We propose a single-loop stochastic approximation algorithm, based on a variant of Nesterov's dual averaging, that jointly updates the decision variable and the importance sampling distribution, notably without time-scale separation or nested optimization. The method is globally convergent and achieves the minimal asymptotic variance among stochastic gradient schemes, which moreover matches the performance of an oracle sampler adapted to the optimal solution and thus effectively resolves the circular optimization challenge.

We examine the ideal $I=(x_1^2, \dots, x_n^2, (x_1+\dots+x_n)^2)$ in the polynomial ring $Q=k[x_1, \dots, x_n]$, where $k$ is a field of characteristic zero or greater than $n$. We also study the Gorenstein ideal $G$ linked to $I$ via the complete intersection ideal $(x_1^2, \dots, x_n^2)$. We compute the Betti numbers of $I$ and $G$ over $Q$ when $n$ is odd and extend known computations when $n$ is even. A consequence is that the socle of $Q/I$ is generated in a single degree (thus $Q/I$ is level) and its dimension is a Catalan number. We also describe the generators and the initial ideal with respect to reverse lexicographic order for the Gorenstein ideal $G$.

We investigate the nonlinear time-asymptotic stability of the composite wave consisting of two viscous shocks and a viscous contact discontinuity for the one-dimensional compressible Navier-Stokes-Fourier (NSF) equations. Specifically, we establish that if the composite wave strength and the perturbations are sufficiently small, the NSF system admits a unique global-in-time strong solution, which converges uniformly in space as time tends to infinity, towards the corresponding composite wave, up to dynamical shifts in the positions of the two viscous shocks. Notably, the strengths of the two viscous shocks can be chosen independently. Our proof relies upon the $a$-contraction method with time-dependent shifts and suitable weight functions.

In this paper, we study the $L^{p_1}(G) \times L^{p_2}(G)$ to $L^{p}(G)$ boundedness of the bilinear Bochner-Riesz means associated with the sub-Laplacian on Métivier group $G$ under the Hölder's relation $1/p = 1/p_1 + 1/p_2$, $1\leq p_1, p_2 \leq \infty$. Our objective is to obtain boundedness results, analogous to the Euclidean setting, where the Euclidean dimension in the smoothness threshold is possibly replaced by the topological dimension of the underlying Métivier group $G$.

We study a class of signomials whose positive support is the set of vertices of a simplex and which may have several negative support points in the simplex. Various groups of authors have provided an exact characterization for the global nonnegativity of a signomial in this class in terms of circuit signomials and that characterization provides a tractable nonnegativity test. We generalize this characterization to the constrained nonnegativity over a set $X$ under an additional convexity precondition in the exponential moment space. This provides a tractable nonnegativity test over $X$ for the class in terms of a power cone program. Our proof methods rely on a variant of the convex cone of constrained SAGE signomials (sums of arithmetic-geometric exponentials) and the duality theory.

We give a new proof of the Hodge conjecture for abelian fourfolds of Weil type with discriminant 1 and all of their powers. The Hodge conjecture for these abelian fourfolds was proven by Markman using hyperholomorphic sheaves on hyper-Kähler varieties of generalized Kummer type, and by constructing semiregular sheaves on abelian varieties. Our proof instead relies on a direct geometric relation between abelian fourfolds of Weil type with discriminant 1 and the six-dimensional hyper-Kähler varieties $\widetilde{K}$ of O'Grady type arising as crepant resolutions $\widetilde{K}\to K$ of a locally trivial deformation of a singular moduli space of sheaves on an abelian surface. As applications, we establish the Hodge conjecture and the Tate conjecture for any variety $\widetilde{K}$ of OG6-type as above, and all of its powers.

We explore the implications of the finiteness of homological dimensions for Ext modules, focusing on projective dimension, injective dimension, and their Gorenstein counterpart. In this direction, we establish several finiteness criteria for homological dimensions. Under such finiteness conditions, our main result is a new duality for certain Ext modules when tensored by canonical modules.

This articles investigates physics-based passive imaging problem, wherein one infers an unknown medium using ambient noise and correlation of the noise signal. We develop a general backpropagation framework via the so-called extended adjoint state, suitable for any elliptic PDE; crucially, this approach reduces by half the number of required PDE solves. Applications to several different PDE models demonstrate the universality of our method. In addition, we analyze the nonlinearity of the correlated model, revealing a surprising tangential cone condition-like structure, thereby advancing the state of the art towards a convergence guarantee for regularized reconstruction in passive imaging.

In this article, we study the continuous-discrete projection filter for exponential-family manifolds with conjugate likelihoods. We first derive the local projection error of the prediction step of the continuous-discrete projection filter. We then derive the exact Bayesian update algorithm for a class of discrete measurement processes with additive Gaussian noise. To control the stiffness of the natural parameters' ordinary differential equations, we introduce a regularization method via projection to the Fisher information metric's eigenspace. Lastly, we apply the proposed method to approximate the filtering density of a modified Van der Pol oscillator problem and a coupled stochastic FitzHugh--Nagumo system. The proposed projection filter shows superior performance compared to several state-of-the-art parametric continuous-discrete filtering methods.

In this paper we study a family of polynomials $$S_n^{(m)}(x):=\sum_{i,j=0}^n\binom ni^m\binom nj^m\binom{i+j}ix^{i+j}\ \ (m,n=0,1,2,\ldots).$$ For example, we show that $$\sum_{k=0}^{p-1}S_k^{(0)}(x)\equiv\frac x{2x-1}\left(1+\left(\frac{1-4x^2}p\right)\right)\pmod p $$ for any odd prime $p$ and integer $x\not\equiv1/2\pmod p$, where $(\frac{\cdot}p)$ denotes the Legendre symbol. We also formulate some open conjectures on related congruences and series for $1/\pi$. For example, we conjecture that $$\sum_{k=0}^\infty(7k+1)\frac{S_k^{(2)}(1/11)}{9^k}=\frac{5445}{104\sqrt{39}\,\pi}$$ and $$\sum_{k=0}^\infty(1365k+181)\frac{S_k^{(2)}(1/18)}{16^k}=\frac{1377}{\sqrt2\,\pi}.$$

This paper addresses the question: What is the de Rham theory for general differentiable spaces? We identify two potential answers and study them. In the first part, we show that the de Rham cohomology calculated using (the completion of) the exterior algebra of the cotangent complex yields non-trivial local invariants for singular differentiable spaces. In particular, in some cases, it differs from the constant sheaf cohomology, which provides an obstruction for the de Rham comparison map to be an equivalence. Moreover, we provide conditions under which this local invariant trivializes, yielding a de Rham-type isomorphism. In the second part, we show that for a suitably defined de Rham stack, there is always an isomorphism between functions on it and constant sheaf cohomology of the underlying topological space. Consequently, there exists a version of the de Rham theorem for singular differentiable spaces which holds with almost no restrictions. Finally, we sketch a generalization of this result to other theories of smooth functions, such as holomorphic or analytic functions. The last part is thus related to analytic de Rham stacks of Rodriguez Camargo used by Scholze to geometrize the local Langlands correspondence.

Benthic algae as a primary producer in riverine ecosystems develop biofilms on the riverbed. Their population dynamics involve growth and decay processes, the former owing to the balance between biological proliferation and mortality, while the latter to mechanical abrasion because of the transport of sediment particles. Contrary to the assumptions of previous studies, the decay has experimentally been found to exhibit long-memory behavior, where the population decreases at an algebraic rate. However, the origin and mathematical theory of this phenomenon remain unresolved. The objective of this study is to introduce a novel mathematical model employing spin processes to describe microscopic biofilm dynamics. A spin process is a continuous-time jump process transitioning between states 0 and 1, and the continuum limit of these processes captures the long-memory decay and generates generic growth. The proposed framework leverages heterogeneous spin rates, achieved by appropriately superposing spin processes with distinct rates, to reproduce the long-memory decay. Computational simulations demonstrate the behavior of the model, particularly emphasizing rate-induced tipping phenomena. This mathematical model provides a computationally tractable interpretation of benthic algae dynamics and their long-term prediction, relevant to river-engineering applications.

Our main goal is to explicitly compute the best constant for the Sobolev-type inequality involving the polyharmonic operator obtained in (Analysis and Applications 22, pp. 1417-1446, 2024). To achieve this goal, we also establish both regularity and classification results for a generalized critical polyharmonic equation in the radial setting.

We study the boundedness of a mutation class for quivers with real weights. The main result is a characterization of bounded mutation classes for real quivers of rank 3.

An automorphism of the free group $F_n$ is called pure symmetric if it sends each generator to a conjugate of itself. The group $\mathrm{PSA}_n$ of all pure symmetric automorphisms and its quotient $\mathrm{PSO}_n$ by the group of inner automorphisms are called the McCool groups. In this paper we prove that every BNSR-invariant $\Sigma^m$ of a McCool group is either dense or empty in the character sphere, and we characterize precisely when each situation occurs. Our techniques involve understanding higher generation properties of abelian subgroups of McCool groups, coming from the McCullough-Miller space. We also investigate further properties of the second invariant $\Sigma^2$ for McCool groups using a general criterion due to Meinert for a character to lie in $\Sigma^2$.

We show that the class of Kalmár elementary functions can be inductively generated from the addition, the integer remainder, and the base-two exponentiation, hence improving previous results by Marchenkov and Mazzanti. We also prove that the substitution basis defined by these three operations is minimal. Furthermore, we discuss alternative substitution bases under arity constraints.

The conjugator length function of a finitely generated group $\Gamma$ gives the optimal upper bound on the length of a shortest conjugator for any pair of conjugate elements in the ball of radius $n$ in the Cayley graph of $\Gamma$. We prove that polynomials of arbitrary degree arise as conjugator length functions of finitely presented groups. To establish this, we analyse the geometry of conjugation in the discrete model filiform groups $\Gamma_d = \mathbb{Z}^d\rtimes_\phi\mathbb{Z}$ where is $\phi$ is the automorphism of $\mathbb{Z}^d$ that fixes the last element of a basis $a_1,\dots,a_d$ and sends $a_i$ to $a_ia_{i+1}$ for $i

We establish upper bounds on the lengths of minimal conjugators in 2-step nilpotent groups. These bounds exploit the existence of small integral solutions to systems of linear Diophantine equations. We prove that in some cases these bounds are sharp. This enables us to construct a family of finitely generated 2-step nilpotent groups $(G_m)_{m\in\mathbb{N}}$ such that the conjugator length function of $G_m$ grows like a polynomial of degree $m+1$.

Learning stabilizing controllers from data is an important task in engineering applications; however, collecting informative data is challenging because unstable systems often lead to rapidly growing or erratic trajectories. In this work, we propose an adaptive sampling scheme that generates data while simultaneously stabilizing the system to avoid instabilities during the data collection. Under mild assumptions, the approach provably generates data sets that are informative for stabilization and have minimal size. The numerical experiments demonstrate that controller inference with the novel adaptive sampling approach learns controllers with up to one order of magnitude fewer data samples than unguided data generation. The results show that the proposed approach opens the door to stabilizing systems in edge cases and limit states where instabilities often occur and data collection is inherently difficult.

Since the work of Mirzakhani and Petri on random hyperbolic surfaces of large genus, length statistics of closed geodesics have been studied extensively. We focus on the case of random hyperbolic surfaces with cusps, the number of which grows with the genus. We prove that if the number of cusps grows fast enough and we restrict attention to special geodesics that are tight, we recover upon proper normalization the same Poisson point process in the large genus limit for the length statistics. The proof relies on a recursion formula for tight Weil-Petersson volumes obtained recently by Budd and Zonneveld and on a generalization of Mirzakhani's integration formula to the tight setting.

A partition of a (hyper)graph is $\varepsilon$-homogenous if the edge densities between almost all clusters are either at most $\varepsilon$ or at least $1-\varepsilon$. Suppose a $3$-graph has the property that the link of every vertex has an $\varepsilon$-homogenous partition of size $\text{poly}(1/\varepsilon)$. Does this guarantee that the $3$-graph also has a small homogenous partition? Terry and Wolf proved that such a $3$-graph has an $\varepsilon$-homogenous partition of size given by a wowzer-type function. Terry recently improved this to a double exponential bound, and conjectured that this bound is tight. Our first result in this paper disproves this conjecture by giving an improved (single) exponential bound, which is best possible. We further obtain an analogous result for $k$-graphs of all uniformities $k \geq 3$. The above problem is part of a much broader programme which seeks to understand the conditions under which a (hyper)graph has small $\varepsilon$-regular partitions. While this problem is fairly well understood for graphs, the situation is (as always) much more involved already for $3$-graphs. For example, it is natural to ask if one can strengthen our first result by only requiring each link to have $\varepsilon$-regular partitions of size $\text{poly}(1/\varepsilon)$. Our second result shows that surprisingly the answer is `no', namely, a $3$-graph might only have regular partitions of tower-type size, even though the link of every vertex has an $\varepsilon$-regular partition of polynomial size.

Monolithic preconditioners applied to the linear systems arising during the solution of the discretized incompressible Navier-Stokes equations are typically more robust than preconditioners based on incomplete block factorizations. Lower number of iterations and a reduced sensitivity to parameters like velocity and viscosity can significantly outweigh the additional cost for their setup. Different monolithic preconditioning techniques are introduced and compared to a selection of block preconditioners. In particular, two-level additive overlapping Schwarz methods (OSM) are used to set up monolithic preconditioners and to approximate the inverses arising in the block preconditioners. GDSW-type (Generalized Dryja-Smith-Widlund) coarse spaces are used for the second level. These highly scalable, parallel preconditioners have been implemented in the solver framework FROSch (Fast and Robust Overlapping Schwarz), which is part of the software library Trilinos. The new GDSW-type coarse space GDSW* is introduced; combining it with other techniques results in a robust algorithm. The block preconditioners PCD (Pressure Convection-Diffusion), SIMPLE (Semi-Implicit Method for Pressure Linked Equations), and LSC (Least-Squares Commutator) are considered to various degrees. The OSM for the monolithic as well as the block approach allows the optimized combination of different coarse spaces for the velocity and pressure components, enabling the use of tailored coarse spaces. The numerical and parallel performance of the different preconditioning methods for finite element discretizations of stationary as well as time-dependent incompressible fluid flow problems is investigated and compared. Their robustness is analyzed for a range of Reynolds and Courant-Friedrichs-Lewy (CFL) numbers with respect to a realistic problem setting.

We study the long-time dynamics of the nonlinear processes modeled by diffusion-transport partial differential equations in non-divergence form with drifts. The solutions are subject to some inhomogeneous Dirichlet boundary condition. Starting with the reduced linear problem, we obtain the asymptotic estimates for the solutions, as time $t\to\infty$, depending on the asymptotic behavior of the forcing term and boundary data. These are established in both cases when the drifts are uniformly bounded, and unbounded as $t\to\infty$. For the nonlinear problem, we prove the convergence of the solutions under suitable conditions that balance the growth of the nonlinear term with the decay of the data. To take advantage of the diffusion in the non-divergence form, we prove an inhomogeneous version of the Landis-typed Growth Lemma and apply it to successive time-intervals. At each time step, the center for the barrier function is selected carefully to optimize the contracting factor. Our rigorous results show the robustness of the model.

We prove an equivalence between the actions of the Gaudin algebras with irregular singularities for $\mathfrak{gl}_d$ and $\mathfrak{gl}_{p+m|q+n}$ on the Fock space of $d(p+m)$ bosonic and $d(q+n)$ fermionic oscillators. This establishes a duality of $(\mathfrak{gl}_d, \mathfrak{gl}_{p+m|q+n})$ for Gaudin models. As an application, we show that the Gaudin algebra with irregular singularities for $\mathfrak{gl}_{p+m|q+n}$ acts cyclically on each weight space of a certain class of infinite-dimensional modules over a direct sum of Takiff superalgebras over $\mathfrak{gl}_{p+m|q+n}$ and that the action is diagonalizable with a simple spectrum under a generic condition. We also study the classical versions of Gaudin algebras with irregular singularities and demonstrate a duality of $(\mathfrak{gl}_d, \mathfrak{gl}_{p+m|q+n})$ for classical Gaudin models.

We determine completely the analytic functions $\varphi$ in the unit disk $\mathbb D$ such that for all (normalized) orientation-preserving harmonic mappings $f=h+\overline g$ produced by the shear construction with $h+g=\varphi$, the condition that each $f$ maps $\mathbb D$ onto a convex domain holds. As a consequence, we obtain the following more general result: for a given complex number $\eta$, with $|\eta|=1$, we characterize those holomorphic mappings $\varphi$ in $\mathbb D$ such that every harmonic function $f=h+\overline g$ as above with $h-\eta g=\varphi$ maps $\mathbb D$ onto a convex domain. The resulting functions are mappings onto a half-plane and mappings onto a strip, and the shear direction, determined by the parameter $\eta$ above, is parallel to the linear boundaries of the half-planes and strips.

This paper investigates the detection and estimation of a single change in high-dimensional linear models. We derive minimax lower bounds for the detection boundary and the estimation rate, which uncover a phase transition governed by the sparsity of the covariance-weighted differential parameter. This form of "inherent sparsity" captures a delicate interplay between the covariance structure of the regressors and the change in regression coefficients on the detectability of a change point. Complementing the lower bounds, we introduce two covariance scanning-based methods, McScan and QcSan, which achieve minimax optimal performance (up to possible logarithmic factors) in the sparse and the dense regimes, respectively. In particular, QcScan is the first method shown to achieve consistency in the dense regime and further, we devise a combined procedure which is adaptively minimax optimal across sparse and dense regimes without the knowledge of the sparsity. Computationally, covariance scanning-based methods avoid costly computation of Lasso-type estimators and attain worst-case computation complexity that is linear in the dimension and sample size. Additionally, we consider the post-detection estimation of the differential parameter and the refinement of the change point estimator. Simulation studies support the theoretical findings and demonstrate the computational and statistical efficiency of the proposed covariance scanning methods.

We analyze the spectral properties of the high-dimensional random geometric graph $G(n, d, p)$, formed by sampling $n$ i.i.d vectors $\{v_i\}_{i=1}^{n}$ uniformly on a $d$-dimensional unit sphere and connecting each pair $\{i,j\}$ whenever $\langle v_i, v_j \rangle \geq \tau$ so that $p=\mathbb P(\langle v_i,v_j\rangle \geq \tau)$. This model defines a nonlinear random matrix ensemble with dependent entries. We show that if $d =\omega( np\log^{2}(1/p))$ and $np\to\infty$, the limiting spectral distribution of the normalized adjacency matrix $\frac{A}{\sqrt{np(1-p)}}$ is the semicircle law. To our knowledge, this is the first such result for $G(n, d, p)$ in the sparse regime. In the constant sparsity case $p=\alpha/n$, we further show that if $d=\omega(\log^2(n))$ the limiting spectral distribution of $A$ in $G(n,\alpha/n)$ coincides with that of the Erdős-Rényi graph $G(n,\alpha/n)$. Our approach combines the classical moment method in random matrix theory with a novel recursive decomposition of closed-walk graphs, leveraging block-cut trees and ear decompositions, to control the moments of the empirical spectral distribution. A refined high trace analysis further yields a near-optimal bound on the second eigenvalue when $np=\Omega(\log^4 (n))$, removing technical conditions previously imposed in (Liu et al. 2023). As an application, we demonstrate that this improved eigenvalue bound sharpens the parameter requirements on $d$ and $p$ for spontaneous synchronization on random geometric graphs in (Abdalla et al. 2024) under the homogeneous Kuramoto model.

We show that a strong version of the Brascamp--Lieb inequality for symmetric log-concave measure with $\alpha$-homogeneous potential $V$ is equivalent to a $p$-Brunn--Minkowski inequality for level sets of $V$ with some $p(\alpha,n)<0$. We establish links between several inequalities of this type on the sphere and the Euclidean space. Exploiting these observations, we prove new sufficient conditions for symmetric $p$-Brunn--Minkowski inequality with $p<1$. In particular, we prove the local log-Brunn--Minkowski for $L_q$-balls for all $q\geq 1$ in all dimensions, which was previously known only for $q\geq 2$.

Topology optimization has emerged as a powerful and increasingly relevant strategy for enhancing the flexibility and efficiency of power system operations. However, solving these problems is computationally demanding due to their combinatorial nature and the use of big-M formulations. Optimization-based bound tightening (OBBT) is a well-known strategy to improve the solution of mixed-integer linear programs (MILPs) by computing tighter bounds for continuous variables. Yet, existing OBBT approaches in topology optimization typically relax all switching decisions in the bounding subproblems, leading to excessively loose feasible regions and limited bound improvements. In this work, we propose a topology-aware bound tightening method that uses network structure to determine which switching variables to relax. Through extensive computational experiments on the IEEE 118-bus system, we find that keeping a small subset of switching variables as binary, while relaxing the rest, strikes a sweet spot between the computational effort required to solve the bounding problems and the tightness of the resulting bounds.

We show that every conformal vector field on a Damek-Ricci space is necessarily Killing, establishing a strong form of infinitesimal conformal rigidity. Although this rigidity phenomenon is classically known in the Einstein setting, our proof follows a completely different approach. We formulate the conformal Killing condition as an explicit system of partial differential equations on the solvable Lie group model and analyze it directly. This local and analytic method yields a constructive proof of rigidity without relying on global arguments or transformation groups.

Laurent Lafforgue's presentation of a Grassmannian Gr$^{d, E}$ naturally comes equipped with the induced action of a subtorus $\mathbb{T}_\bullet$ of PGL$(E)$. By investigating the defining ideals of $\mathbb{T}_\bullet$-orbit closures through general points of Gr$^{d,E}$ and studying their degenerations, we obtain a morphsim $\mathfrak{q}: \mathbb{F}^{d, E_\bullet} \to \mathbb{H}^{d, E_{\bullet}}$ such that $\mathbb{H}^{d, E_\bullet}$, termed the $\mathfrak{G}$-quotient of Gr$^{d,E}$ by $\mathbb{T}_\bullet$, is birational to $[{\rm Gr}^{d, E}/\mathbb{T}_\bullet]$, and $\mathfrak{q}$, termed $\mathfrak{G}$-family of Gr$^{d,E}$ by $\mathbb{T}_\bullet$, is a family of general $\mathbb{T}_\bullet$-orbit closures and their degenerations. We obtain a series of new results on $\mathbb{H}^{d, E_{\bullet}}$ and $\mathbb{F}^{d, E_\bullet}$.

Quick response is a widely adopted strategy to mitigate overproduction in the manufacturing industry, yet recent research reveals a counter-intuitive paradox: while it reduces waste from unsold finished goods, it may incentivize firms to procure more raw materials, potentially increasing total system waste. Additionally, existing models that guide quick response strategies rely on the assumption of a known demand distribution, whereas in practice, demand patterns are often ambiguous and historical data are scarce. To address these challenges, we develop a distributionally robust optimization (DRO) framework for the quick response model that builds robust policies even with limited data. We further integrate a novel waste-to-consumption ratio constraint into this framework, empowering firms to explicitly control the environmental impact of their operations. Our numerical experiments demonstrate that policies optimized for specific demand assumptions suffer severe performance degradation under distributional shifts, whereas our data-driven DRO approach consistently delivers superior robustness. Moreover, we find that the constrained quick response model resolves the central paradox: it can achieve higher profits with verifiably less total waste than a traditional, non-flexible alternative. These results resolve the `quick response or not' debate by showing that the question is not \emph{whether} to use quick response, but \emph{how} to manage it. By incorporating socially responsible metrics as constraints, the quick response system delivers a `win-win' outcome for both profitability and the environment compared to traditional systems.

We apply mixed-precision to the low-rank Lyapunov ADI (LR-ADI) by performing certain aspects of the algorithm in a lower working precision. Namely, we accumulate the overall solution, solve the linear systems comprising the ADI iteration, and store the inner low-rank factors of the residuals in various combinations of IEEE 754 single and double precision. We empirically test our implementation on Lyapunov equations arising from first- and second-order descriptor systems. For the first-order examples, accumulating the solution in single-precision yields an almost-as-small residual as for the double-precision solution. For certain applications, like computing the H2 norm of a descriptor system, low- or mixed-precision variants of the ADI can be quite competitive

A new type of conjectures on characters of finite groups, related to the McKay conjecture, have recently been proposed. In this paper, we study these conjectures for symmetric groups.

We study offline scheduling for large language model (LLM) serving under a fixed KV-cache memory budget, where requests have heterogeneous prompt (prefill) and response (decode) lengths. Prompt tokens determine initial KV usage, and each generated token increases memory by one unit. Given a backlog of n requests arriving together, we schedule mixed prefill and decode batches to minimize total end-to-end latency. We show that heterogeneity in prompt lengths makes the problem computationally intractable and that widely used heuristics such as first-come-first-served and shortest-first can be arbitrarily suboptimal. We propose Sorted-F, which repeatedly forms feasible batches using a new selection metric that balances batch size against downstream decode cost, and prove it achieves a constant-factor guarantee on total latency. We further develop practical variants -- an exact solver for small instances and fast heuristics for larger ones -- and evaluate them on a public workload spanning short conversations and long-document summarization, where they consistently reduce average latency relative to standard baselines. Our results highlight that during peak-hour tidal backlogs, greedy GPU packing or short-request prioritization can perform poorly when prompt lengths vary widely, and provide a principled, tunable framework for designing production batch schedulers and planning capacity in memory-constrained LLM serving systems.

We extend a previous result [Sutter et al., J. Phys. A: Math. Theor. 57, 475202 (2024)] to give an explicit form of the set of $v$-representable densities on the one-dimensional torus with any fixed number of particles in contact with a heat bath at finite temperature. The particle interaction has to satisfy some mild assumptions but is kept entirely general otherwise. For densities, we consider the Sobolev space $H^1$ and exploit the convexity of the functionals. This leads to a broader set of potentials than the usual $L^p$ spaces and encompasses distributions. By including temperature and thus considering all excited states in the Gibbs ensemble, Gâteaux differentiability of the thermal universal functional is guaranteed. This yields $v$-representability and it is demonstrated that the given set of $v$-representable densities is even maximal.

We introduce the notions of local realizability at a prime and algebraic realizability of an integer sequence. After discussing this notion in general we consider it for the Euler numbers, the Bernoulli denominators, and the Bernoulli numerators. This gives, for example, a dynamical characterization of the Bernoulli regular primes. Algebraic realizability of the Bernoulli denominators is shown at every prime, giving a different perspective on the great diversity of congruences satisfied by this sequence. We show that the sequence of Euler numbers cannot be realized on a nilpotent group, which may explain why it is less hospitable to congruence hunting.

In this paper, we introduce a new type degenerate Simsek numbers and their generating function, which are different from degenerate Simsek number studied so far. We establish the explicit formula, recurrence relation and other identities for these numbers. We also derive several interesting expressions and relations between these numbers and certain other special numbers in the literature. In addition, several numerical examples and graphical illustrations are provided to support the theoretical results and to illustrate the behavior of the introduced numbers.

For coprime $p,q\in\mathbb{Z}_{\geq 2}$, the triplet vertex operator algebra $W_{p,q}$ is a non-simple extension of the universal Virasoro vertex operator algebra of central charge $c_{p,q}=1-\frac{6(p-q)^2}{pq}$, and it is a basic example of a vertex operator algebra appearing in logarithmic conformal field theory. Here, we give a new construction of $W_{p,q}$ different from the original screening operator definition of Feigin-Gainutdinov-Semikhatov-Tipunin. Using our earlier work on the tensor category structure of modules for the Virasoro algebra at central charge $c_{p,q}$, we show that the simple modules appearing in the decomposition of $W_{p,q}$ as a module for the Virasoro algebra have $\mathrm{PSL}_2$-fusion rules and generate a symmetric tensor category equivalent to $\operatorname{Rep}\mathrm{PSL}_2$. Then we use the theory of commutative algebras in braided tensor categories to construct $W_{p,q}$ as an appropriate non-simple modification of the canonical algebra in the Deligne tensor product of $\operatorname{Rep}\mathrm{PSL}_2$ with this Virasoro subcategory. As a consequence, we show that the automorphism group of $W_{p,q}$ is $\mathrm{PSL}_2(\mathbb{C})$. We also define a braided tensor category $\mathcal{O}_{c_{p,q}}^0$ consisting of modules for the Virasoro algebra at central charge $c_{p,q}$ that induce to untwisted modules of $W_{p,q}$. We show that $\mathcal{O}_{c_{p,q}}^0$ tensor embeds into the $\mathrm{PSL}_2(\mathbb{C})$-equivariantization of the category of $W_{p,q}$-modules and is closed under contragredient modules. We conjecture that $\mathcal{O}_{c_{p,q}}^0$ has enough projective objects and is the correct category of Virasoro modules for constructing logarithmic minimal models in conformal field theory.

We will study the $1$-weighted Ricci curvature in view of the extrinsic geometric analysis. We derive several geometric consequences concerning stable weighted minimal hypersurfaces in weighted manifolds under a lower $1$-weighted Ricci curvature bound. We prove a Schoen-Yau type criterion, and conclude a structure theorem for three-dimensional weighted manifolds of non-negative $1$-weighted Ricci curvature. We also show non-existence results under volume growth conditions, and conclude smooth compactness theorems.

We present and analyze in a unified setting two schemes for the numerical discretization of a Darcy-Forchheimer fluid flow model coupled with an advection-diffusion equation modeling the temperature distribution in the fluid. The first approach is based on fully discontinuous Galerkin discretization spaces. In contrast, in the second approach, the velocity is approximated in the Raviart-Thomas space, and the pressure and temperature are still piecewise discontinuous. A fixed-point linearization strategy, naturally inducing an iterative splitting solution, is proposed for treating the nonlinearities of the problem. We present a unified stability analysis and prove the convergence of the iterative algorithm under mild requirements on the problem data. A wide set of two- and three-dimensional simulations is presented to assess the error decay and demonstrate the practical performance of the proposed approaches in physically sound test cases.

We establish new Euclidean Sobolev logarithmic inequalities in the framework of fractional Sobolev spaces and their weighted version. Our approach relies on a interpolation inequality, which can be viewed as a fractional Caffarelli-Kohn-Nirenberg type inequality. We further relate the optimal constant in this interpolation inequality to a corresponding variational problem. These results extend classical Sobolev logarithmic inequalities to the nonlocal Euclidean framework and provide new tools for analysis in fractional Sobolev spaces.

Systems of N = 1, 2, . . . first-order hyperbolic conservation laws feature N undamped waves propagating at finite speeds. On their own hand, multi-step Finite Difference and lattice Boltzmann schemes with q = N + 1, N + 2, . . . unknowns involve N ''physical'' waves, which are aimed at being as closely-looking as possible to the ones of the PDEs, and q-N ''numerical-spurious-parasitic'' waves, which are subject to their own speed of propagation, and either damped or undamped. The whole picture is even more complicated in the discrete setting-as numerical schemes act as dispersive media, thus propagate different harmonics at different phase (and group) velocities. For compelling practical reasons, simulations must always be conducted on bounded domains, even when the target problem is unbounded in space. The importance of transparent boundary conditions, preventing artificial boundaries from acting as mirrors producing polluting ricochets, naturally follows. This work presents, building on Besse, Coulombel, and Noble [ESAIM: M2AN, 55 (2021)], a systematic way of developing perfectly transparent boundary conditions for lattice Boltzmann schemes tackling linear problems in one and two space dimensions. Our boundary conditions are ''perfectly'' transparent, at least for 1D problems, as they absorb both physical and spurious waves regardless of their frequency. After presenting, in a simple framework, several approaches to handle the fact that q > N , we elect the so-called ''scalar'' approach (which despite its name, also works when N > 1) as method of choice for more involved problems. This method solely relies on computing the coefficients of the Laurent series at infinity of the roots of the dispersion relation of the bulk scheme. We insist on asymptotics for these coefficients in the spirit of analytic combinatorics. The reason is two-fold: asymptotics guide truncation of boundary conditions to make them depending on a fixed number of past time-steps, and make it clearduring the process of computing coefficients-whether intermediate quantities can be safely stored using floating-point arithmetic or not. Numerous numerical investigations in 1D and 2D with N = 1 and 2 are carried out, and show the effectiveness of the proposed boundary conditions.

We prove a logarithm law-type result for the spiraling of geodesics around certain types of compact subsets (e.g. quotients of periodic Morse flats) in quotients of rank one CAT(0) spaces.

In this article, we construct infinitely many (small Seifert fibred, hyperbolic and toroidal) rational homology $3$-spheres that admit co-orientable taut foliations, but none with vanishing Euler class. In the context of the $L$-space conjecture, these examples provide rational homology $3$-spheres that admit co-orientable taut foliations (and hence are not $L$-spaces) and have left-orderable fundamental groups, yet none of the left orders arise directly from the universal circle actions associated to co-orientable taut foliations. The hyperbolic and non-Seifert toroidal examples are obtained from Dehn surgeries on knots in the $3$-sphere and use Heegaard Floer homology to obstruct the existence of a co-orientable foliation with vanishing Euler class. For the Seifert fibred case, we establish necessary and sufficient conditions for the Euler class of the normal bundle of the Seifert fibration to vanish. Moreover, when the base orbifold is hyperbolic, we also provide a second proof of this condition from the viewpoint of discrete faithful representations of Fuchsian groups.

The complete list of connected components of the set of Morse functions in the deformations of function singularities of class $J_{10}$ is given. Thus, the isotopy classification of Morse perturbations of parabolic real function singularities is finished.

The series of papers is devoted to the study of convergence for pairs of surfaces and smooth functions thereon. We model such pairs with varifolds and multiple-valued functions to capture their limits. In the present paper, we study Young functions, a measure-theoretic approach to multiple-valued functions, and the graph measures associated with pairs of measures (in particular, varifolds) and Young functions. This setting allows us to model the convergence of pairs of surfaces and functions thereon via the weak convergence of their associated graph measures, and a compactness theorem follows immediately. As a prerequisite for the concepts of differentiability for Young functions in the upcoming papers, we introduce and investigate several test function spaces.

This work presents problems of constructing finite-difference formulas in the Hilbert space, i.e., setting problems of constructing finite-difference formulas using functional methods. The work presents a functional statement of the problem of optimizing finite-difference formulas in the space $W_{2}^{\left(m,m-1\right)} \left(0,1\right)$. Here, representations of optimal coefficients of explicit finite-difference formulas of the Adams type on classes $W_{2}^{\left(m,m-1\right)} \left(0,1\right)$ for any $m\ge 3$ will be found.

We consider a problem of covariance estimation from a sample of i.i.d. high-dimensional random vectors. To avoid the curse of dimensionality, we impose an additional assumption on the structure of the covariance matrix $\Sigma$. To be more precise, we study the case when $\Sigma$ can be approximated by a sum of double Kronecker products of smaller matrices in a tensor train (TT) format. Our setup naturally extends widely known Kronecker sum and CANDECOMP/PARAFAC models but admits richer interaction across modes. We suggest an iterative polynomial time algorithm based on TT-SVD and higher-order orthogonal iteration (HOOI) adapted to Tucker-2 hybrid structure. We derive non-asymptotic dimension-free bounds on the accuracy of covariance estimation taking into account hidden Kronecker product and tensor train structures. The efficiency of our approach is illustrated with numerical experiments.

We analyze stochastic partial differential equations (SPDEs) with quadratic nonlinearities close to a change of stability. To this aim we compute finite-time Lyapunov exponents (FTLEs), observing a change of sign based on the interplay between the distance towards the bifurcation and the noise intensity. A technical challenge is to provide a suitable control of the nonlinear terms coupling the dominant and stable modes of the SPDE and of the corresponding linearization. In order to illustrate our results we apply them to the stochastic Burgers equation.

Qualification conditions (also termed constraint qualifications) help avoid pathological behavior at domain boundaries in convex analysis. By generalizing facial reduction from conic programming to general convex programs of the form $f(x) + g(Ax)$, we provide qualification-free generalizations of several key results: an exact Fenchel-Rockafellar dual, KKT optimality conditions, an attained infimal convolution for the conjugate of a sum, subdifferential sum and chain rules, and normal cones of intersections. All our results reduce seamlessly to their original formulations when qualification conditions hold. The core insight is that for a sum of two convex functions, there is an affine subspace$\unicode{x2014}$the joint supporting subspace$\unicode{x2014}$that contains the feasible region, and such that qualification conditions hold when restricting the effective domain of each function to it. We offer a number of characterizations for the joint supporting subspace, including one that obtains the affine subspace via iterative, bilateral reduction between the two domains. In our proofs, which are self-contained, we develop a structured induction on faces where inductive steps are associated with normal vectors nested in supporting subspaces (a generalization of supporting hyperplanes). With this tool, we characterize the facial structure of the difference of two convex sets from the facial structures of the individual convex sets.

We prove that the jump quasi-seminorm of order $\varrho= 2$ for a general Ornstein--Uhlenbeck semigroup $\left(\mathcal H_t\right)_{t>0}$ in $\mathbb R^n$ defines an operator of weak type $(1,1)$ with respect to the invariant measure. This provides an example of a weak-type jump inequality for a nonsymmetric semigroup in a nondoubling measure space. Our result may be seen as an endpoint refinement of the weak type $(1,1)$ inequality for the $\varrho$-th order variation seminorm of $\left(\mathcal H_t\right)_{t>0}$, recently proved by the authors when $\varrho>2$, and disproved for $\varrho=2$.

In this paper, we introduce a model-free, real-time, dynamic optimization and control method for a class of rigid body dynamics. Our method is based on a recent extremum seeking control for vibrational stabilization (ESC-VS) approach that is applicable to a class of second-order mechanical systems. The new ESC-VS method is able to stabilize a rigid body dynamic system about the optimal state of an objective function that can be unknown expression-wise, but assessable through measurements; the ESC-VS is operable by using only one perturbation/vibrational signal. We demonstrate the effectiveness and the applicability of our ESC-VS approach via three rigid-body systems: (1) satellite attitude dynamics, (2) quadcopter attitude dynamics, and (3) acceleration-controlled unicycle dynamics. The results, including simulations with and without measurement delays/noise, illustrate the ability of our ESC-VS to operate successfully as a new methodology of optimization and control for rigid body dynamics.

If a rational homology 3-sphere $M$ bounds a rational homology 4-ball $W$, then the kernel of the inclusion-induced homomorphism $H_1(M;\mathbb{Z})\to H_1(W;\mathbb{Z})$ is a Lagrangian for the $\mathbb{Q}/\mathbb{Z}$-valued torsion linking form $\lambda_2$ on $H_1(M;\mathbb{Z})$. In this short paper, we prove that the Freedman-Krushkal triple torsion linking form $\lambda_3$ (arXiv:2506.11941v3) vanishes on this Lagrangian under the assumption that $H_2(W;\mathbb{Z})=0$. We then pose several questions about topological rational homology cobordism.

Automated proof assistants are a technology pre-empting mistakes in mathematics. In our practice we have seen that reasoning about planar diagrams is difficult to both humans and computers. One example that has led to wrong statements in publications is that an orientation-preserving mapping is not always defined by how it acts on triples of elements. In this paper we formalise orientation-preserving mappings in proof assistant software Lean and report on our take-aways.

In this paper, we develop a unified approach to study the intersection Betti numbers of moduli spaces of one-dimensional semistable sheaves on smooth projective surfaces. Assuming the irreducibility of such moduli spaces, we prove that their intersection Betti numbers in a certain range of degrees coincide with the stable Betti numbers of Hilbert schemes of points. As an application, for minimal surfaces of Kodaira dimension 0, we show that these intersection Betti numbers stabilize in each fixed degree, which fits into the broader context of stable cohomology for moduli spaces of sheaves. In the case of Enriques surfaces, we also prove a refined stabilization result related to Oberdieck's conjecture on perverse Hodge numbers.

We consider the Landau-Lifshitz-Gilbert equation (LLG) that models time-dependent micromagnetic phenomena. We propose a full discretization that employs first-order finite elements in space and a BDF2-type two-step method in time. In each time step, only one linear system of equations has to be solved. We employ linear interpolation in time to reconstruct the discrete space-time magnetization. We prove that the integrator is unconditionally stable and thus guarantees that a subsequence of the reconstructed magnetization converges weakly in $H^1$ towards a weak solution of LLG in the space-time domain. Numerical experiments verify that the proposed integrator is indeed first-order in space and second-order in time.

In this paper, we develop a novel numerical framework, namely the stochastic interacting particle-field method with particle-in-cell acceleration (SIPF-PIC), for the efficient simulation of the three-dimensional (3D) parabolic-parabolic Keller-Segel (KS) systems. The SIPF-PIC method integrates Lagrangian particle dynamics with spectral field solvers by leveraging localized particle-grid interpolations and fast Fourier transform (FFT) techniques. For $P$ particles and $H$ Fourier modes per spatial dimension, the SIPF-PIC method achieves a computational complexity of $O(P + H^3 \log H)$ per time step, a significant improvement over the original SIPF method (proposed in \cite{SIPF1}), which has a computational complexity of $O(PH^3)$, while preserving numerical accuracy. Moreover, we carry out a rigorous error analysis for the proposed method and establish the corresponding error estimates. Finally, we present numerical experiments to validate the convergence order and demonstrate the computational efficiency of SIPF-PIC. Further numerical experiments show the method's capability of capturing complex blowup dynamics beyond single-point collapse, including ring-shaped singularities.

In this paper, we investigate dead-core problems for fully nonlinear degenerate parabolic equations with strong absorption, \begin{equation*} |Du|^{p} F(D^{2}u) - u_{t} = \lambda_{0}(x,t)\, u^{\mu}\, \chi_{\{u>0\}}(x,t) \qquad \text{in } \quad Q_{T} := Q \times (0,T), \end{equation*} where $0 \leq p < \infty$ and $0 < \mu < 1$. We establish a sharp and improved parabolic $C^{\alpha}$-regularity estimate along the free boundary $\partial \{ u > 0 \}$, where \[ \alpha := \frac{2+p}{1+p-\mu} > 1 + \frac{1}{1+p}. \] Moreover, we establish weak geometric properties of solutions, such as non-degeneracy and uniform positive density. As an application, we obtain a Liouville-type theorem for entire solutions and gradient bounds. Finally, as a byproduct of our approach, we derive a novel $L^{\delta}$-average estimate for fully nonlinear singular elliptic equations and present a new formulation of the gradient decay property. It is worth noting that the results presented here extend those in da Silva {\it et al.} ({\it Pacific J. Math}., \textbf{300} (2019), 179--213) and ({\it J. Differential Equations}., \textbf{264} (2018), 7270--7293) to the degenerate setting, and can be viewed as a parabolic analogue of da Silva {\it et al.} ({\it Math. Nachr}., \textbf{294} (2021), 38--55) and Teixeira ({\it Math. Ann}., \textbf{364} (2016), 1121--1134). Additionally, of independent mathematical interest, we emphasize that our manuscript establishes a comparison principle result and the compactness of viscosity solutions to fully nonlinear degenerate parabolic models with continuous and bounded forcing terms. These compactness and comparison properties serve as key ingredients in deriving enhanced regularity estimates along free boundary points for our model problem with strong absorption.

In classical factorization theory, an integral domain is called \emph{atomic} if every nonzero nonunit element can be written as a finite product of irreducible elements. Here, we introduce and study a weaker notion of atomicity, which relaxes the requirement that all elements admit a factorization into irreducibles. Namely, we say that an integral domain is \emph{sub-atomic} if every nonunit divisor of an atomic element is also atomic. We further consider several factorization properties associated with this notion. Then, we investigate the basic properties of such domains, provide examples, and explore the behavior of the sub-atomic property under standard constructions such as localization, polynomial rings, and $D+M$ constructions. Our results highlight the independence of the sub-atomic property from other classical factorization properties and introduce an important class of integral domains that lies between atomic and non-atomic domains.

Deciding when to stop stochastic gradient descent (SGD) has long remained unresolved in a statistically rigorous sense. While SGD is routinely monitored as it runs, the classical theory of SGD provides guarantees only at pre-specified iteration horizons and offers no valid way to decide, based on the observed trajectory, when further computation is justified. We address this gap by developing anytime-valid confidence sequences for stochastic gradient methods, which remain valid under continuous monitoring and directly induce statistically valid, trajectory-dependent stopping rules: stop as soon as the current upper confidence bound on an appropriate performance measure falls below a user-specified tolerance. The confidence sequences are constructed using nonnegative supermartingales, are time-uniform, and depend only on observable quantities along the SGD trajectory, without requiring prior knowledge of the optimization horizon. In convex optimization, this yields anytime-valid certificates for weighted suboptimality of projected SGD under general stepsize schedules, without assuming smoothness or strong convexity. In nonconvex optimization, it yields time-uniform certificates for weighted first-order stationarity under smoothness assumptions. We further characterize the stopping-time complexity of the resulting stopping rules under standard stepsize schedules. To the best of our knowledge, this is the first framework that provides statistically valid, time-uniform stopping rules for SGD across both convex and nonconvex settings based solely on its observed trajectory.

Uniform interpolation is a strengthening of interpolation that holds for certain propositional logics. The starting point of this chapter is a theorem of A. Pitts, which shows that uniform interpolation holds for intuitionistic propositional logic. We outline how this theorem may be proved semantically via the definability of bisimulation quantifiers, and how it generalizes to an open mapping theorem between Esakia spaces. We also discuss connections between uniform interpolation and research in categorical logic, algebra, and model theory.

We introduce a new, relatively simple, line-breaking construction of the $\alpha$-stable tree which realises its random finite-dimensional distributions. This is a direct analogue of Aldous' line-breaking construction of the Brownian continuum random tree, which is based on an inhomogeneous Poisson process. Here, we replace the deterministic rate function from the Brownian setting by a random rate process, given by a certain measure-changed $(\alpha-1)$-stable subordinator. Rather than attaching uniformly, the line-segments now connect to locations chosen with probability proportional to the sizes of the jumps of the rate process. We also give a new proof of an invariance principle originally due to Duquesne, which states that the family tree of a Bienaymé branching process with critical offspring distribution in the domain of attraction of an $\alpha$-stable law (for $\alpha \in (1,2))$, conditioned to have $n$ vertices, converges on rescaling distances appropriately to the $\alpha$-stable tree. Our proof makes use of a discrete line-breaking construction of the branching process tree, which we show converges to our continuous line-breaking construction.

We introduce a quantitative version of polynomial cohomology for discrete groups and show that it coincides with usual group cohomology when combinatorial filling functions are polynomially bounded. As an application, we show that Betti numbers of nilpotent groups are invariant by mutually cobounded $\textrm L^p$-measure equivalence. We also use this to obtain new vanishing results for non-cocompact lattices in rank 1 simple Lie groups.

We establish a local topological obstruction to the simultaneous flattening of Berry curvature in spin--orbit-coupled Bose--Einstein condensates (SOC BECs), which remains valid even when the global Chern number vanishes. For a generic two-component SOC BEC, the extended parameter space is the total space $M$ of a principal $U(1)_+ \times U(1)_-$ bundle over the Brillouin torus $T^{2}_{\mathrm{BZ}}$. On $M$, we construct a Kaluza--Klein metric and a natural metric connection $\nabla^{C}$ whose torsion 3-form encodes the synthetic gauge fields. Under the physically relevant assumption of constant Berry curvatures, the harmonic part of this torsion defines a mixed cohomology class $[\omega] \in \bigl(H^{2}(T^{2}_{\mathrm{BZ}}) \otimes H^{1}(S^{1}_{\phi_{+}})\bigr) \oplus \bigl(H^{2}(T^{2}_{\mathrm{BZ}}) \otimes H^{1}(S^{1}_{\phi_{-}})\bigr) $ with mixed tensor rank $r=1$. By adapting the Pigazzini--Toda (PT) lower bound to the Kaluza--Klein setting through explicit pointwise curvature analysis, we demonstrate that the obstruction kernel $\mathcal{K}$ vanishes for the physical metric, yielding the sharp inequality $\dim \mathfrak{hol}^{\mathrm{off}}(\nabla^{C}) \geq 1$. This bound forces the existence of at least one off-diagonal curvature operator, preventing the complete gauging-away of Berry phases even in regimes with zero net topological charge. This work provides the first cohomological lower bound, based on the PT framework, certifying locally irremovable curvature in SOC BECs beyond the Chern-number paradigm.

We determine the Picard number and the Ulrich complexity of general bidouble covers of the projective plane, providing the first systematic study of Ulrich bundles on non-cyclic abelian covers. For a bidouble plane branched along three smooth curves of degrees $n_1,n_2,n_3$, we show that $\rho(S)=1$ unless $(n_1,n_2,n_3)$ belongs to an explicit list, thereby extending Buium's classical results on double planes to the non-cyclic case. As an application, we determine the range of branch degrees for which Ulrich line bundles could exist. Our method combines the invariant-theoretic decomposition of $H^2(S,\mathbb{Q})$ under the Galois group with cohomological criteria for Ulrich bundles.

We study the sine-Gordon measure defined on each homotopy class. The energy space decomposes into infinitely many such classes indexed by the topological degree $Q \in \mathbf{Z}$. Even though the sine-Gordon action admits no minimizer in homotopy classes with $|Q| \ge 2$, we prove that the Gibbs measure on each class nevertheless concentrates and exhibits Ornstein-Uhlenbeck fluctuations near the multi-soliton manifold in the joint low-temperature and infinite-volume limit. Moreover, we show that soliton collisions are unlikely events, so that typical states consist of solitons separated at an appropriate scale. Finally, we identify the joint distribution of the multi-soliton centers as the ordered statistics of independent uniform random variables, so that each soliton's location follows a Beta distribution.

In this paper, we systematically investigate the nilpotentizer and nilpotent graph for a Lie superalgebra over the field of characteristic not equal to 2. First, we establish some fundamental properties of the nilpotentizer. Next, we show that the nilpotent graph is one of the isomorphic invariants of Lie superalgebras. Furthermore, we introduce the nilpotency measure which provides a quantitative assessment of nilpotency for a Lie superalgebra. Finally, we use category theory to establish connections between Lie super?algebras and their nilpotent substructures, based on the construction of the nilpotentizer.

We construct compact hyperbolic 3-manifolds with totally geodesic boundary, such that the closed 3-pseudomanifolds obtained by coning off the boundary components are negatively curved and contain locally convex subspaces whose fundamental groups have property (T). In particular, the fundamental groups of these 3-pseudomanifolds are word hyperbolic but not cubulable. We deduce that in any relative cubulation of one of these hyperbolic 3-manifold groups some hyperplane stabilizer has infinite intersection with the fundamental group of some boundary component.

This paper develops a bridge between bi-Hamiltonian structures of Poisson-Lie type, contact Hamiltonian dynamics, and the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) formalism for quantum open systems. On the classical side, we consider bi-Hamiltonian systems defined by a Poisson pencil with non-trivial invariants. Using an exact symplectic realization, these invariants are lifted and projected onto a contact manifold, yielding a completely integrable contact Hamiltonian system in terms of dissipated quantities and a Jacobi-commutative algebra of observables. On the quantum side, we introduce a class of contact-compatible Lindblad generators: GKSL evolutions whose dissipative part preserves a commutative $C^\ast$-subalgebra generated by the quantizations of the classical dissipated quantities, and whose Hamiltonian part admits an Egorov-type semiclassical limit to the contact dynamics. This construction provides a mathematical mechanism compatible with the semiclassical limit for pure dephasing, compatible with integrability and contact dissipation. An explicit Euler-top-type Poisson-Lie pencil, inspired by deformed Euler top models, is developed as a fully worked-out example illustrating the resulting bi-Lindblad structure and its semiclassical behavior.

Let $K$ be a number field of degree $d$ so that $K/\mathbb Q$ is a Galois extension. The {\it normal basis theorem} states that $K$ has a $\mathbb Q$-basis consisting of algebraic conjugates, in fact $K$ contains infinitely many such bases. We prove an effective version of this theorem, obtaining a normal basis for $K/\mathbb Q$ of bounded Weil height with an explicit bound in terms of the degree and discriminant of $K$. In the case when $d$ is prime, we obtain a particularly good bound using a different method.

We construct an irreducible rational curve of degree 10 in $CP^2$ which has 12 triple points and a union of three rational quartics with 19 triple points. This gives counter-examples to a conjecture by Dimca, Harbourne, and Sticlaru. We also prove that there exists an analytic family $C_u$ of curves of degree 10 with 12 triple points which tends as $u\to 0$ to the union of the dual Hesse arrangement of lines (9 lines with 12 triple points) with an additional line. We hope that our approach to the proof of the latter fact could be of independent interest.

In this work, we construct examples of holomorphic functions in $D_2(\B_2)$, the Dirichlet space on $\B_2$, for which there exists an index $\alpha_c \in [\frac12,2]$ such that the function is cyclic in $D_\alpha(\B_2)$ if and only if $\alpha \leq \alpha_c$. To this end, we use the notion of \emph{interpolation sets} in smooth ball algebras, as studied by Bruna, Ortega, Chaumat, and Chollet.

By the work of Hernandez-Leclerc, Leclerc-Plamondon, and Keller-Scherotzke, affine graded Nakajima quiver varieties associated with a Dynkin quiver $Q$ admit an algebraic description in terms of modules over the singular Nakajima category $\mathcal{S}$ and a stratification functor to the derived category of $Q$. In this paper, we extend this framework to Nakajima's $n$-fold affine graded tensor product varieties, which allow one to geometrically realize $n$-fold tensor products of standard modules over the quantum affine algebra. We introduce a category of filtrations with splitting of length $n$ of modules over a category and show that it is equivalent to the module category of a triangular matrix category. Applied to the singular Nakajima category, this yields a category $\mathcal{S}^{n\operatorname{-filt}}$ whose modules are parametrized by the points of the $n$-fold tensor product varieties. Generalizing the results of Keller-Scherotzke from $\mathcal{S}$ to $\mathcal{S}^{n\operatorname{-filt}}$, we prove that the stable category of pseudo-coherent Gorenstein projective $\mathcal{S}^{n\operatorname{-filt}}$-modules is triangle equivalent to the derived category of the algebra of $n \times n$ upper triangular matrices over the path algebra of $Q$, and we obtain a corresponding stratification functor.

We determine an infinite family of linear identities for the number $A_4(n)$ of partition pairs of $n$ with $4$-cores by employing elementary $q$-series techniques and certain $3$-dissection formulas. We then discover an infinite family of congruences for $A_4(n)$ as a consequence of these linear identities.

We construct explicit examples of really perverse central configurations in the spatial Newtonian $N$-body problem. A central configuration is called really perverse if it satisfies the central configuration equations for two distinct mass distributions having the same total mass. While such configurations were previously known only in the planar case for large $N$, we prove the existence of spatial really perverse central configurations for $N=27,\dots,55$.

We prove quantum ergodicity and quantum mixing for sequences of finite Schreier graphs converging to an infinite Cayley graph whose adjacency operator has absolutely continuous spectrum. Under Benjamini-Schramm convergence (or strong convergence in distribution), we show that correlations between eigenvectors at distinct energies vanish asymptotically when tested against a broad class of local observables. Our results apply to all orthonormal eigenbases and do not require tree-like structure or periodicity of the limiting graph, unlike previous approaches based on non-backtracking operators or Floquet theory. The proof introduces a new framework for quantum ergodicity, based on trace identities, resolvent approximations and representation-theoretic techniques and extends to certain families of non-regular graphs. We illustrate the assumptions and consequences of our theorems on Schreier graphs arising from free products of groups, right-angled Coxeter groups and lifts of a fixed base graph.

Recently, some window functions have been introduced into the nonuniform fast Fourier transform and the regularized Shannon sampling. Inspired by these works, we utilize a sinh-type function to accelerate the convergence of the Lagrangian nonuniform sampling series. Our theoretical error estimates and numerical experiments demonstrate that the sinh regularized nonuniform sampling series achieves a superior convergence rate compared to the fastest existing Gaussian regularized nonuniform sampling series.

We consider the 2-limited packing problem: for a graph $G=(V,E)$ one seeks to find a maximum cardinality subset $B\subseteq V$, such that, for all $v\in V$, the closed neighbourhood of $v$ contains at most two vertices in $B$. We compare this packing problem to the well-known Roman domination problem by pointing out some similarities and differences in the behaviour of the optimal solutions of both problems and show that these two problems are weakly dual. We show that for trees, the two problems are strongly dual, letting us solve the Roman domination problem by computing an optimal solution to the 2-limited packing problem.

We study concentration inequalities for structured weighted sums of random data, including (i) tensor inner products and (ii) sequential matrix sums. We are interested in tail bounds and concentration inequalities for those structured weighted sums under exchangeability, extending beyond the classical framework of independent terms. We develop Hoeffding and Bernstein bounds provided with structure-dependent exchangeability. Along the way, we recover known results in weighted sum of exchangeable random variables and i.i.d. sums of random matrices to the optimal constants. Notably, we develop a sharper concentration bound for combinatorial sum of matrix arrays than the results previously derived from Chatterjee's method of exchangeable pairs. For applications, the richer structures provide us with novel analytical tools for estimating the average effect of multi-factor response models and studying fixed-design sketching methods in federated averaging. We apply our results to these problems, and find that our theoretical predictions are corroborated by numerical evidence.

In this work we describe a new model for the evolution of a diploid structured population backwards in time that allows for large migrations and uneven offspring distributions. The model generalizes both the mean-field model of Birkner et al. [\textit{Electron. J. Probab.} 23: 1-44 (2018)] and the haploid structured model of Möhle [\textit{Theor. Popul. Biol.} 2024 Apr:156:103-116]. We show convergence, with mild conditions on the joint distribution of offspring frequencies and migrations, of gene genealogies conditional on the pedigree to a time-inhomogeneous coalescent process driven by a Poisson point process $\Psi$ that records the timing and scale of large migrations and uneven offspring distributions. This quenched scaling limit demonstrates a significant difference in the predictions of the classical annealed theory of structured coalescent processes. In particular, the annealed and quenched scaling limits coincide if and only if these large migrations and uneven offspring distributions are absent. The proof proceeds by the method of moments and utilizes coupling techniques from the theory of random walks in random environments. Several examples are given and their quenched scaling limits established.

We provide a new construction of a topological group model for the string group of a compact, simple, and simply-connected Lie group, by solving the obstruction realization problem for compact group $G$-kernels on full factors. Furthermore, we introduce the Rohlin property for actions and cocycle actions of compact groups in order to establish cohomology vanishing theorems.

The paper considers a global version of the notion of log canonical threshold for plurisubharmonic functions $u$ of logarithmic growth in $\mathbb{C}^n$, aiming at description of the range of all $p>0$ such that $e^{-u}\in L^p(\mathbb{C}^n)$. Explicit formulas are obtained in the toric case. By considering Bergman functions of corresponding weighted Hilbert spaces, a new polynomial approximation of plurisubharmonic functions of logarithmic growth with control over its singularities and behavior at infinity (a global version of Demailly's approximation theorem) is established. Some application to tame polynomial maps are given.

The numerical computation of shortest paths or geodesics on surfaces, along with the associated geodesic distance, has a wide range of applications. Compared to Euclidean distance computation, these tasks are more complex due to the influence of surface geometry on the behavior of shortest paths. This paper introduces a primal-dual level set method for computing geodesic distances. A key insight is that the underlying surface can be implicitly represented as a zero level set, allowing us to formulate a constraint minimization problem. We employ the primal-dual methodology, along with regularization and acceleration techniques, to develop our algorithm. This approach is robust, efficient, and easy to implement. We establish a convergence result for the high-resolution PDE system, and numerical evidence suggests that the method converges to a geodesic in the limit of refinement.

In this paper, extending the work of Gal'perin (Comm. Math. Phys. 154: 63-84, 1993), we investigate generalizations of the concepts of centroids and static equilibrium points of a convex body in spherical, hyperbolic and normed spaces. In addition, we examine the minimum number of equilibrium points a $2$- or $3$-dimensional convex body can have in these spaces. In particular, we show that every plane convex body in any of these spaces has at least four equilibrium points, and that there are mono-monostatic convex bodies in $3$-dimensional spherical, hyperbolic, and certain normed spaces. Our results are generalizations of results of Domokos, Papadopoulos and Ruina (J. Elasticity 36: 59-66, 1994), and Várkonyi and Domokos (J. Nonlinear Sci. 16: 255-281, 2006) for convex bodies in Euclidean space.

In this work, we propose a decoding method of Golay codes from the perspective of Polarization Adjusted Convolutional (PAC) codes. By invoking Forney's cubing construction of Golay codes and their generators $G^*(8,7)/(8,4)$, we found different construction methods of Golay codes from PAC codes, which result in an efficient parallel list decoding algorithm with near-maximum likelihood performance. Compared with existing methods, our method can get rid of index permutation and codeword puncturing. Using the new decoding method, some related lattices, such as Leech lattice $\Lambda_{24}$ and its principal sublattice $H_{24}$, can be also decoded efficiently.

We prove a conjecture of Peter Neumann from 1966, predicting that every finite non-regular primitive permutation group of degree $n$ contains an element fixing at least one point and at most $n^{1/2}$ points. In fact, we prove a stronger version, where $n^{1/2}$ is replaced by $n^{1/3}$, and this is best possible. The case where $G$ is affine was proved by Guralnick and Malle; in this paper we address the case where $G$ is non-affine.

We study the nef cones and fundamental domains of Hilbert schemes of points on the Cayley K3 surface $S$ and its generalizations $S_a$. For the Hilbert square $S^{[2]}$, we explicitly compute the nef cone and describe a fundamental domain using the automorphisms of $S^{[2]}$ and lattice-theoretic methods. For higher Hilbert schemes $S_a^{[n]}$, we determine the nef cones using Bridgeland stability methods that identify the contracted curves defining walls and the divisors generating the extremal rays.

We represent the Euler alternating series (sometimes called the "Dirichlet eta function"), and generally $(b^s-b)\zeta(s)/b^s$ for $b>1$ an integer, in the half-plane $\Re s>0$, via series dominated by geometric series, with arbitrarily small convergence ratio (up to the prize of a longer first approximation). Due to the underlying recurrence, the cost for each new term is at first sight linearly increasing, so the cost appears to be quadratic in the number of terms kept. And the number of terms needed to achieve a given target precision increases linearly with the imaginary part of $s$.

In this paper, a new class of \emph{Taylor-accelerated neural network interpolation operators} is introduced on quasi-uniform irregular grids. These operators improve existing neural network interpolation operators by incorporating Taylor polynomials at the sampling nodes, thereby exploiting higher smoothness of the target function. The proposed operators are shown to be well defined, uniformly bounded, and to satisfy an exact interpolation property at the grid points. In addition, polynomial reproduction up to a prescribed degree is established. Jackson-type approximation estimates are derived in terms of higher-order moduli of smoothness, yielding enhanced convergence rates for sufficiently smooth functions. Numerical experiments are presented to support the theoretical analysis and to demonstrate the significant accuracy improvement achieved through the Taylor-accelerated construction. In particular, higher-order convergence on irregular grids is obtained, and the proposed approach outperforms existing neural network interpolation operators on irregular grids, including Lagrange-based schemes.

In this paper, we introduce, for the first time, the fractional--logarithmic Laplacian \( (-\Delta)^{s+\log} \), defined as the derivative of the fractional Laplacian \( (-\Delta)^t \) at \( t=s \). It is a singular integral operator with Fourier symbol \( |\xi|^{2s}(2\ln|\xi|) \), and we prove the pointwise integral representation \[ (-\Delta)^{s+\log}u(x) = c_{n,s}\,\mathrm{PV}\!\int_{\mathbb{R}^n} \frac{u(x)-u(y)}{|x-y|^{n+2s}}\bigl(-2\ln|x-y|\bigr)\,dy + b_{n,s}(-\Delta)^s u(x), \] where \( c_{n,s} \) is the normalization constant of the fractional Laplacian and \( b_{n,s}:=\frac{d}{ds}c_{n,s}.\) We also establish several equivalent formulations of \( (-\Delta)^{s+\log} \), including the singular-integral representation, the Fourier-multiplier representation, the spectral-calculus definition, and an extension characterization. We develop the associated functional framework on both \( \mathbb{R}^n \) and bounded Lipschitz domains, introducing the natural energy spaces and proving embedding results. In particular, we obtain a compact embedding at the critical exponent \( 2_s^*=\frac{2n}{n-2s},\) a phenomenon that differs from the classical Sobolev and fractional Sobolev settings. We further study the Poisson problem, proving existence and \( L^\infty \)-regularity results. We then investigate the Dirichlet eigenvalue problem and establish qualitative spectral properties. Finally, we derive a Weyl-type asymptotic law for the eigenvalue counting function and for the \( k \)-th Dirichlet eigenvalue, showing that the high-frequency behavior combines the fractional Weyl scaling with a logarithmic growth factor, thereby interpolating between the fractional Laplacian and the logarithmic Laplacian.

We give explicit matrix presentations of the Blanchfield pairing and certain twisted Blanchfield pairings of the $(m,n)$-torus knot $T(m,n)$. Our method uses a taut identity realizing a genus-two Heegaard splitting of the manifold $X_{T(m,n)}$ obtained from $S^3$ by $0$-surgery along $T(m,n)$. The taut identity allows us to construct a chain complex of $X_{T(m,n)}$ with few generators. As a result, we obtain explicit matrix presentations of the Blanchfield pairing of $T(m,n)$. Moreover, for each Casson-Gordon type metabelian representation and for suitable roots of unity $\xi$ depending on the representation, we describe the $(t-\xi)$-primary part of the associated twisted Alexander module and give an explicit description of the restriction of the twisted Blanchfield pairing to this primary summand.

Movable antenna (MA) has emerged as a promising technology for future wireless systems. Compared with traditional fixed-position antennas, MA improves system performance by antenna movement to optimize channel conditions. For multiuser wideband MA systems, this paper proposes deep learning-based framework integrating channel estimation (CE), antenna position optimization, and beamforming, with a clear workflow and enhanced efficiency. Specifically, to obtain accurate channel state information (CSI), we design a two-stage CE mechanism: first reconstructing the channel matrix from limited measurements via compressive sensing, then introducing a Swin-Transformer-based denoising network to refine CE accuracy for subsequent optimization. Building on this, we address the joint optimization challenge by proposing a Transformer-based network that intelligently maps CSI sequences of candidate positions to optimal MA positions while combining a model-driven weighted minimum mean square error (WMMSE) beamforming approach to achieve better performance. Simulation results demonstrate that the proposed methods achieve superior performance compared with existing counterparts under various conditions. The codes about this work are available at this https URL.

In this paper, we propose a novel class of change of measure inequalities via a unified framework based on the data processing inequality for $f$-divergences, which is surprisingly elementary yet powerful enough to yield tighter inequalities. We provide change of measure inequalities in terms of a broad family of information measures, including $f$-divergences (with Kullback-Leibler divergence and $\chi^2$-divergence as special cases), Rényi divergence, and $\alpha$-mutual information (with maximal leakage as a special case). We then embed these inequalities into the analysis of generalization error for stochastic learning algorithms, yielding novel and tighter high-probability information-theoretic generalization bounds, while also recovering several best-known results via simplified analyses. A key advantage of our framework is its flexibility: it readily adapts to a range of settings, including the conditional mutual information framework, PAC-Bayesian theory, and differential privacy mechanisms, for which we derive new generalization bounds.

We study the nonlocal Kuramoto-Sivashinsky equation on the one-dimensional torus, \[ u_t+u u_x=\Lambda^{r}u-\varepsilon \Lambda^{s}u,\qquad x\in\mathbb T, \] where $\varepsilon>0$, $s>1$, $r\in[-1,s)$. We first prove local and global well-posedness for initial data in $H^{3}(\mathbb T)$. We then investigate the steady-state problem and show that the trivial branch undergoes bifurcation at the critical values $\varepsilon_k=k^{\,r-s}$, $k\in\mathbb N$. Using the Crandall-Rabinowitz theorem we obtain smooth local curves of nontrivial equilibria emanating from each $(\varepsilon_k,0)$ and compute the bifurcation direction. To address the global continuation of these branches we derive global a priori bounds and apply a global alternative based on the Fitzpatrick-Pejsachowicz-Rabier degree for Fredholm maps of index zero. In particular, for the component bifurcating from the first critical point we prove that its $\varepsilon$-projection contains the interval $(2^{r-s},1)$, yielding the existence of nontrivial steady states for that parameter range. We complement the theory with numerical continuation results illustrating the bifurcation diagram and solution profiles.

The classical Stein--Tomas theorem extends the theory of linear Fourier restriction estimates from smooth manifolds to fractal measures exhibiting Fourier decay. In the multilinear setting, transversality allows for Fourier extension estimates that go beyond those implied by the linear theory to hold. We establish a multilinear Fourier extension estimate for measures whose convolution belongs to an $L^p$ space, applicable to known results by Shmerkin and Solomyak that exploit `transversality' between self-similar measures. Moreover, we generalise work by Hambrook--Łaba and Chen from the linear setting to obtain Knapp-type examples for multilinear estimates; we obtain two necessary conditions: one in terms of the upper box dimension of the measures' supports, and another one in terms of their Fourier decay and a ball condition. In particular, these conditions give a more restrictive range compared with previously known results whenever the convolution of the measures at play is singular.

This document is a slightly expanded version of a series of talks given by J. Giansiracusa at the workshop `Geometry over semirings' at Universitat Autònoma de Barcelona in July 2025. In the first lecture we introduce tropical polynomials, ideals, congruences, and how the connection with tropical geometry is made via congruences of bend relations. Tropical geometry and matroid theory are telling us that we should focus attention on a narrow slice of the world of tropical algebra, and this leads to the theory of tropical ideals (as developed by Maclagan and Rincón) and an abundance of interesting open questions. In the second lecture we examine the relationship between Berkovich analytification and tropicalization from the perspective of bend relations, giving a refinement of Payne's influential limit theorem. In the third lecture we set aside geometry and focus on tropicalization via bend relations as a construction in commutative and non-commutative algebra. Constructions such as symmetric algebras, exterior algebras, matrix algebras, and Clifford algebras can be tropicalized. In the case of exterior algebras, the resulting tropical notion beautifully completes the picture of the Plücker embedding and gives a new perspective on the tropical Plücker relations. For matrix algebras and Clifford algebras, Morita theory becomes an interesting topic.

Branch-price-and-cut is the state-of-the-art exact method for solving many types of vehicle routing problems, and is particularly effective for vehicle routing problems with time windows. A well-known challenge in branch-price-and-cut is that the generation of columns is guided by information from the linear relaxation of the master problem, with no guarantee that they will be useful from an integer perspective. As a consequence, high-quality primal solutions are often found only after significant cutting and branching or the use of primal heuristics. In this work, based on the ideas of pricing for integrality, we propose a new primal heuristic for vehicle routing problems. The heuristic is designed to generate columns that are more likely to be part of high-quality integer solutions. It begins by constructing a partial integer solution from a given column pool and then iteratively searches for columns that complement this solution. The search is done by modifying the pricing problem with respect to the partial solution, linear program dual information as well as previously generated columns in the heuristic. Computational tests are performed on the electrical vehicle routing problem with time windows extended with charging time slots, a problem that has both scheduling and routing aspects, making it well-suited to evaluate the performance of the proposed heuristic. The results show that the proposed heuristic closes 30% - 40% of the root node gap on average in comparison to a restricted master heuristic.

This paper defines a Mitchell rank for supercompact cardinals. If $\kappa$ is a $\theta$-supercompact cardinal then $o_{\theta-sc}(\kappa) = \sup \{ o_{\theta-sc}(\mu) + 1 \ | \ \mu \in m(\kappa)\}$, where $m(\kappa)$ is the collection of normal fine measures on $P_{\kappa}\theta$. We show how to force to kill the degree of a measurable cardinal $\kappa$ to any specified degree which is less than or equal to the degree of $\kappa$ in the ground model. We will also show how to softly kill the Mitchell rank for supercompactness of any supercompact cardinal so that in the forcing extension it is any desired degree less than or equal to its degree in the ground model, along with some results concerning strongly compact cardinals.

In this article we study domino tilings of a family of finite regions called Aztec diamonds. Every such tiling determines a partition of the Aztec diamond into five sub-regions; in the four outer sub-regions, every tile lines up with nearby tiles, while in the fifth, central sub-region, differently-oriented tiles co-exist side by side. We show that when n is sufficiently large, the shape of the central sub-region becomes arbitrarily close to a perfect circle of radius n/sqrt(2) for all but a negligible proportion of the tilings. Our proof uses techniques from the theory of interacting particle systems. In particular, we prove and make use of a classification of the stationary behaviors of a totally asymmetric one-dimensional exclusion process in discrete time.

Understanding the power of quantum data in machine learning is central to many proposed applications of quantum technologies. While access to quantum data can offer exponential advantages for carefully designed learning tasks and often under strong assumptions on the data distribution, it remains an open question whether such advantages persist in less structured settings and under more realistic, naturally occurring distributions. Motivated by these practical concerns, we introduce a systematic framework based on quantum lossy data compression to bound the power of quantum data in the context of probably approximately correct (PAC) learning. Specifically, we provide lower bounds on the sample complexity of quantum learners for arbitrary functions when data is drawn from Zipf's distribution, a widely used model for the empirical distributions of real-world data. We also establish lower bounds on the size of quantum input data required to learn linear functions, thereby proving the optimality of previous positive results. Beyond learning theory, we show that our framework has applications in secure delegated quantum computation within the measurement-based quantum computation (MBQC) model. In particular, we constrain the amount of private information the server can infer, strengthening the security guarantees of the delegation protocol proposed in (Mantri et al., PRX, 2017).

In this paper, we consider three variations on standard PageRank: Non-backtracking PageRank, $\mu$-PageRank, and $\infty$-PageRank, all of which alter the standard formula by adjusting the likelihood of backtracking in the algorithm's random walk. We show that in the case of regular and bipartite biregular graphs, standard PageRank and its variants are equivalent. We also compare each centrality measure and investigate their clustering capabilities.

Network data is ubiquitous in various scientific disciplines, including sociology, economics, and neuroscience. Latent space models are often employed in network data analysis, but the geometric effect of latent space curvature remains a significant, unresolved issue. In this work, we propose a hyperbolic network latent space model with a learnable curvature parameter. We theoretically justify that learning the optimal curvature is essential to minimizing the embedding error across all hyperbolic embedding methods beyond network latent space models. A maximum-likelihood estimation strategy, employing manifold gradient optimization, is developed, and we establish the consistency and convergence rates for the maximum-likelihood estimators, both of which are technically challenging due to the non-linearity and non-convexity of the hyperbolic distance metric. We further demonstrate the geometric effect of latent space curvature and the superior performance of the proposed model through extensive simulation studies and an application using a Facebook friendship network.

Rapid advancements in data science require us to have fundamentally new frameworks to tackle prevalent but highly non-trivial "irregular" inference problems, to which the large sample central limit theorem does not apply. Typical examples are those involving discrete or non-numerical parameters and those involving non-numerical data, etc. In this article, we present an innovative, wide-reaching, and effective approach, called "repro samples method," to conduct statistical inference for these irregular problems plus more. The development relates to but improves several existing simulation-inspired inference approaches, and we provide both exact and approximate theories to support our development. Moreover, the proposed approach is broadly applicable and subsumes the classical Neyman-Pearson framework as a special case. For the often-seen irregular inference problems that involve both discrete/non-numerical and continuous parameters, we propose an effective three-step procedure to make inferences for all parameters. We also develop a unique matching scheme that turns the discreteness of discrete/non-numerical parameters from an obstacle for forming inferential theories into a beneficial attribute for improving computational efficiency. We demonstrate the effectiveness of the proposed general methodology using various examples, including a case study example on a Gaussian mixture model with unknown number of components. This case study example provides a solution to a long-standing open inference question in statistics on how to quantify the estimation uncertainty for the unknown number of components and other associated parameters. Real data and simulation studies, with comparisons to existing approaches, demonstrate the far superior performance of the proposed method.

Shared Autonomous Vehicles (SAVs) enable transit agencies to design more agile and responsive services at lower operating costs. This study designs and evaluates a semi-on-demand hybrid route directional service in the public transit network, offering on-demand flexible route service in low-density areas and fixed route service in higher-density areas. We develop analytically tractable cost expressions that capture access, waiting, and riding costs for users, and distance-based operating and time-based vehicle costs for operators. Two formulations are presented for strategic and tactical decisions in flexible route portion, fleet size, headway, and vehicle size optimization, enabling the determination of route types between fixed, hybrid, and flexible routes based on demand, cost, and operational parameters. Analytical results demonstrate that the lower operating costs of SAVs favor more flexible route services. The practical applications and benefits of semi-on-demand feeders are presented with numerical examples and a large-scale case study in the Chicago metropolitan area, USA. Findings reveal scenarios in which flexible route portions serving passengers located further away reduce total costs, particularly user costs, whereas higher demand densities favor more traditional line-based operations. Current cost forecasts suggest smaller vehicles with fully flexible routes are optimal, but operating constraints or higher operating costs would favor larger vehicles with hybrid routes. The study provides an analytical tool to design SAVs as directional services and transit feeders, and tractable continuous approximation formulations for planning and research in transit network design.

We explore topological manipulations in one spatial dimension, which are defined for a system with a global symmetry and map the system to another one with a dual symmetry. In particular, we discuss fusion category symmetries enhanced by the invariance of the actions of topological manipulations, i.e., self-$G$-alities for topological manipulations. Based on the self-$G$-ality conditions, we provide LSM-type constraints on the ground states of many-body Hamiltonians. We clarify the relationship between different enhanced symmetries and introduce the notion of $\textit{codimension-two transitions}$. We explore concrete lattice models for such self-$G$-alities and find how the self-$G$-ality structures match the IR critical theories.

We establish a lower bound on the total mass of the time slices of (n + 1)-dimensional asymptotically flat standard static spacetimes under the timelike convergence condition. The inequality can be viewed equivalently as a Minkowski-type inequality in these spaces, i.e. as a lower bound on the total mean curvature of the boundary, and thus extends inequalities from [3], [22], [19], and [10]. Equality is achieved only by slices of Schwarzschild space and is related to the characterization of quasi-spherical static vacuum metrics from [10]. As a notable special case of the main inequality, we obtain the Riemannian Penrose inequality in all dimensions for static spaces under the TCC.

It is of great interest to understand the thermalization of open quantum many-body systems, and how quantum computers are able to efficiently simulate that process. A recently introduced disispative evolution, inspired by existing models of open system thermalization, has been shown to be efficiently implementable on a quantum computer. Here, we prove that, at high enough temperatures, this evolution reaches the Gibbs state in time scaling logarithmically with system size. The result holds for Hamiltonians that satisfy the Lieb-Robinson bound, such as local Hamiltonians on a lattice, and includes long-range systems. To the best of our knowledge, these are the first results rigorously establishing the rapid mixing property of high-temperature quantum Gibbs samplers, which is known to give the fastest possible speed for thermalization in the many-body setting. We then employ our result to the problem of estimating partition functions at high temperature, showing an improved performance over previous classical and quantum algorithms.

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

We propose a predator-prey model with an age-structured predator population that exhibits a functional role reversal. The structure of the predator population in our model embodies the ecological concept of an "ontogenetic niche shift," in which a species' functional role changes as it grows. This structure adds complexity to our model but increases its biological relevance. The time evolution of the age-structured predator population is motivated by the Kermack-McKendrick Renewal Equation (KMRE). Unlike KMRE, the predator population's birth and death rate functions depend on the prey population's size. We establish the existence, uniqueness, and positivity of the solutions to the proposed model's initial value problem. The dynamical properties of the proposed model are investigated via Latin Hypercube Sampling in the 15-dimensional space of its parameters. Our Linear Discriminant Analysis suggests that the most influential parameters are the maturation age of the predator and the rate of consumption of juvenile predators by the prey. We carry out a detailed study of the long-term behavior of the proposed model as a function of these two parameters. In addition, we reduce the proposed age-structured model to ordinary and delayed differential equation (ODE and DDE) models. The comparison of the long-term behavior of the ODE, DDE, and the age-structured models with matching parameter settings shows that the age structure promotes the instability of the Coexistence Equilibrium and the emergence of the Coexistence Periodic Attractor.

Avoided crossings are fundamental phenomena in quantum mechanics and photonics that originate from the interaction between coupled energy levels and have been extensively studied in linear dispersive dynamics. Their manifestation in open, driven-dissipative systems, however, where nonlinear dynamics of quantum fluctuations come into play, remains largely unexplored. In this work, we analyze the hitherto unexplored occurrence of avoided and genuine crossings in the spectrum of quantum noise. We demonstrate that avoided crossings arise naturally when a single parameter is varied, leading to hypersensitivity of the associated singular vectors and suggesting the presence of genuine crossings (diabolical points) in nearby systems. We show that these spectral features can be deliberately designed, highlighting the possibility of programming the quantum noise response of photonic systems. As a notable example, such control can be exploited to generate broad, flat-band squeezing spectra - a desirable feature for enhancing degaussification protocols. Our analysis is based on a detailed study of the Analytic Bloch-Messiah Decomposition (ABMD), which we use to characterize the parameter-dependent behavior of singular values and their corresponding vectors. This study provides new insights into the structure of multimode quantum correlations and offers a theoretical framework for the experimental exploitation of complex quantum optical systems.

Modular vehicles present a novel area of academic and industrial interest in the field of multi-agent systems. Modularity allows vehicles to connect and disconnect with each other mid-transit which provides a balance between efficiency and flexibility when solving complex and large scale tasks in urban or aerial transportation. This paper details a generalized scheme to route multiple modular agents on a graph to a predetermined set of target nodes. The objective is to visit all target nodes while incurring minimum resource expenditure. Agents that are joined together will incur the equivalent cost of a single agent, which is motivated by the logistical benefits of traffic reduction and increased fuel efficiency. To solve this problem, we introduce a novel algorithm that seeks to balance the optimality of the path that every single module takes and the cost benefit of joining modules. Our approach models the agents and targets as point charges, where the modules take the path of highest attractive force from its target node and neighboring agents. We validate our approach by simulating multiple modular agents along real-world transportation routes in the road network of Champaign-Urbana, Illinois, USA. The proposed method easily exceeds the available benchmarks and illustrates the benefits of modularity in multi-target planning problems.

This article studies the fundamental problem of using i.i.d. coin tosses from an entropy source to efficiently generate random variables $X_i \sim P_i$ $(i \ge 1)$, where $(P_1, P_2, \dots)$ is a random sequence of rational discrete probability distributions subject to an \textit{arbitrary} stochastic process. Our method achieves an amortized expected entropy cost within $\varepsilon > 0$ bits of the information-theoretically optimal Shannon lower bound using $O(\log(1/\varepsilon))$ space. This result holds both pointwise in terms of the Shannon information content conditioned on $X_i$ and $P_i$, and in expectation to obtain a rate of $\mathbb{E}[H(P_1) + \dots + H(P_n)]/n + \varepsilon$ bits per sample as $n \to \infty$ (where $H$ is the Shannon entropy). The combination of space, time, and entropy properties of our method improves upon the Knuth and Yao (1976) entropy-optimal algorithm and Han and Hoshi (1997) interval algorithm for online sampling, which require unbounded space. It also uses exponentially less space than the more specialized methods of Kozen and Soloviev (2022) and Shao and Wang (2025) that generate i.i.d. samples from a fixed distribution. Our online sampling algorithm rests on a powerful algorithmic technique called \textit{randomness recycling}, which reuses a fraction of the random information consumed by a probabilistic algorithm to reduce its amortized entropy cost. On the practical side, we develop randomness recycling techniques to accelerate a variety of prominent sampling algorithms. We show that randomness recycling enables state-of-the-art runtime performance on the Fisher-Yates shuffle when using a cryptographically secure pseudorandom number generator, and that it reduces the entropy cost of discrete Gaussian sampling. Accompanying the manuscript is a performant software library in the C programming language.

This article presents an input convex neural network architecture using Kolmogorov-Arnold networks (ICKAN). Two specific networks are presented: the first is based on a low-order, linear-by-part, representation of functions, and a universal approximation theorem is provided. The second is based on cubic splines, for which only numerical results support convergence. We demonstrate on simple tests that these networks perform competitively with classical input convex neural networks (ICNNs). In a second part, we use the networks to solve some optimal transport problems needing a convex approximation of functions and demonstrate their effectiveness. Comparisons with ICNNs show that cubic ICKANs produce results similar to those of classical ICNNs.

We consider the problem of sampling from a probability distribution $\pi$. It is well known that this can be written as an optimisation problem over the space of probability distribution in which we aim to minimise the Kullback--Leibler divergence from $\pi$. We consider several partial differential equations (PDEs) whose solution is a minimiser of the Kullback--Leibler divergence from $\pi$ and connect them to well-known Monte Carlo algorithms. We focus in particular on PDEs obtained by considering the Wasserstein--Fisher--Rao geometry over the space of probabilities and show that these lead to a natural implementation using importance sampling and sequential Monte Carlo. We propose a novel algorithm to approximate the Wasserstein--Fisher--Rao flow of the Kullback--Leibler divergence and conduct an extensive empirical study to identify when these algorithms outperforms other popular Monte Carlo algorithms.

This work presents a hardware-efficient and fully parallelizable decoder for quantum LDPC codes that leverages belief propagation (BP) with a speculative post-processing strategy inspired by classical Chase decoding algorithm. By monitoring bit-level oscillation patterns during BP, our method identifies unreliable bits and generates multiple candidate vectors to selectively flip syndromes. Each modified syndrome is then decoded independently using short-depth BP, a process we refer to as BP-SF (syndrome flip). This design eliminates the need for costly Gaussian elimination used in the current BP-OSD approaches. Our implementation achieves logical error rates comparable to or better than BP-OSD while offering significantly lower latency due to its high degree of parallelism for a variety of bivariate bicycle codes. Evaluation on the [[144,12,12]] bivariate bicycle code shows that the proposed decoder reduces average latency to approximately $70\%$ of BP-OSD. When post-processing is parallelized the average latency is reduced by $55\%$ compared to the single process implementation, with the maximum latency reaching as low as $18\%$. These advantages make it particularly well-suited for real-time and resource-constrained quantum error correction systems.

We study convergence in high-probability of SGD-type methods in non-convex optimization and the presence of heavy-tailed noise. To combat the heavy-tailed noise, a general black-box nonlinear framework is considered, subsuming nonlinearities like sign, clipping, normalization and their smooth counterparts. Our first result shows that nonlinear SGD (N-SGD) achieves the rate $\widetilde{\mathcal{O}}(t^{-1/2})$, for any noise with unbounded moments and a symmetric probability density function (PDF). Crucially, N-SGD has exponentially decaying tails, matching the performance of linear SGD under light-tailed noise. To handle non-symmetric noise, we propose two novel estimators, based on the idea of noise symmetrization. The first, dubbed Symmetrized Gradient Estimator (SGE), assumes a noiseless gradient at any reference point is available at the start of training, while the second, dubbed Mini-batch SGE (MSGE), uses mini-batches to estimate the noiseless gradient. Combined with the nonlinear framework, we get N-SGE and N-MSGE methods, respectively, both achieving the same convergence rate and exponentially decaying tails as N-SGD, while allowing for non-symmetric noise with unbounded moments and PDF satisfying a mild technical condition, with N-MSGE additionally requiring bounded noise moment of order $p \in (1,2]$. Compared to works assuming noise with bounded $p$-th moment, our results: 1) are based on a novel symmetrization approach; 2) provide a unified framework and relaxed moment conditions; 3) imply optimal oracle complexity of N-SGD and N-SGE, strictly better than existing works when $p < 2$, while the complexity of N-MSGE is close to existing works. Compared to works assuming symmetric noise with unbounded moments, we: 1) provide a sharper analysis and improved rates; 2) facilitate state-dependent symmetric noise; 3) extend the strong guarantees to non-symmetric noise.

In previous work arXiv:2407.13556, we proposed an extended phase space structure at null infinity accommodating large gauge symmetries for sub$^n$-leading soft theorems in Yang-Mills, via dressing fields arising in the Stueckelberg procedure. Here, we give an explicit boundary action controlling the dynamics of these fields. This allows for a derivation from first principles of the associated charges, together with an explicit renormalization procedure when taking the limit to null and spatial infinity, matching with charges proposed in previous work. Using the language of fibre bundles, we relate the existence of Stueckelberg fields to the notion of extension/reduction of the structure group of a principal bundle, thereby deriving their transformation rules in a natural way, thus realising them as Goldstone-like objects. Finally, this allows us to give a geometric picture of the gauge transformation structure at the boundary, via a loop group coming from formal expansions in the coordinate transversal to the boundary.

This paper addresses the lifting problem for the Černý conjecture: namely, whether the validity of the conjecture for a quotient automaton can always be transferred (or "lifted") to the original automaton. Although a complete solution remains open, we show that it is sufficient to verify the Černý conjecture for three specific subclasses of reset automata: radical, simple, and quasi-simple. Our approach relies on establishing a Galois connection between the lattices of congruences and ideals of the transition monoid. This connection not only serves as the main tool in our proofs but also provides a systematic method for computing the radical ideal and for deriving structural insights about these classes. In particular, we show that for every simple or quasi-simple automaton $\mathcal{A}$, the transition monoid $\text{M}(\mathcal{A})$ possesses a unique ideal covering the minimal ideal of constant (reset) maps; a result of similar flavor holds for the class of radical automata.

Accurate state estimation is critical for legged and aerial robots operating in dynamic, uncertain environments. A key challenge lies in specifying process and measurement noise covariances, which are typically unknown or manually tuned. In this work, we introduce a bi-level optimization framework that jointly calibrates covariance matrices and kinematic parameters in an estimator-in-the-loop manner. The upper level treats noise covariances and model parameters as optimization variables, while the lower level executes a full-information estimator. Differentiating through the estimator allows direct optimization of trajectory-level objectives, resulting in accurate and consistent state estimates. We validate our approach on quadrupedal and humanoid robots, demonstrating significantly improved estimation accuracy and uncertainty calibration compared to hand-tuned baselines. Our method unifies state estimation, sensor, and kinematics calibration into a principled, data-driven framework applicable across diverse robotic platforms.

The default Gaussian latent in flow-based generative models poses challenges when learning certain distributions such as heavy-tailed ones. We introduce a general framework for learning data-adaptive latent distributions using one-dimensional quantile functions, optimized via the Wasserstein distance between noise and data. The quantile-based parameterization naturally adapts to both heavy-tailed and compactly supported distributions and shortens transport paths. Numerical results confirm the method's flexibility and effectiveness achieved with negligible computational overhead.

The association between a continuous and an ordinal variable is commonly modeled through the polyserial correlation model. However, this model, which is based on a partially-latent normality assumption, may be misspecified in practice, due to, for example (but not limited to), outliers or careless responses. The typically used maximum likelihood (ML) estimator is highly susceptible to such misspecification: One single observation not generated by partially-latent normality can suffice to produce arbitrarily poor estimates. As a remedy, we propose a novel estimator of the polyserial correlation model designed to be robust against the adverse effects of observations discrepant to that model. The estimator leverages density power divergence estimation to achieve robustness by implicitly downweighting such observations; the ensuing weights constitute a useful tool for pinpointing potential sources of model misspecification. The proposed estimator generalizes ML and is consistent as well as asymptotically Gaussian. As price for robustness, some efficiency must be sacrificed, but substantial robustness can be gained while maintaining more than 98% of ML efficiency. We demonstrate our estimator's robustness and practical usefulness in simulation experiments and an empirical application in personality psychology where our estimator helps identify outliers. Finally, the proposed methodology is implemented in free open-source software.

In $d$-dimensional de Sitter spacetime, consistency of the perturbative expansion necessitates imposing all second-order gravitational constraints associated with the $SO(1,d)$ isometry group, rather than restricting to the $\R\times SO(d-1)$ subgroup, to address linearization instability. Since generic de Sitter isometries do not preserve a fixed static patch, these constraints cannot be implemented within a fixed local algebra. In this paper, we develop a framework that consistently imposes all $SO(1,d)$ constraints while incorporating multiple observers on arbitrary timelike geodesics. This is achieved by introducing the concept of covariant observer, whose geodesic transforms covariantly under the isometry group. Upon quantization, the observer is described by a superposition of geodesics, with the associated static patches fluctuating, providing a quantum reference frame $L^2(SO(1,d))$. We realize this structure in an action model in which a particle carries a set of conserved charges, each one corresponding to a generator of de Sitter isometry group, which parametrize its geodesic and upon quantization lead to a fluctuating geodesic. Inspired by the timelike tube theorem, we propose that the algebra of observables accessible to a covariant observer is generated by all degrees of freedom within its fluctuating static patch, including quantum field modes and other observers, which are treated as part of the matter system. Imposing the $SO(1,d)$ constraints yields a gauge-invariant algebra that takes the form of an averaged modular crossed product algebra over static patches and configurations of other geodesics, thereby generalizing the notion of a local algebra associated with a fixed region to that of a fluctuating region. We show this algebra is of type II by explicitly constructing a faithful normal trace, leading to an observer-dependent notion of von Neumann entropy. For semiclassical states, by imposing a UV cutoff in QFT and proposing a quantum generalization of the first law, we demonstrate the agreement between the algebraic and generalized entropies.

We consider combinatorial multi-item markets and propose the notion of a $\Delta$-regret Walras equilibrium, which is an allocation of items to players and a set of item prices that achieve the following goals: prices clear the market, the allocation is capacity-feasible, and the players' strategies lead to a total regret of $\Delta$. The regret is defined as the sum of individual player regrets measured by the utility gap with respect to the optimal item bundle given the prices. We derive a complete characterization for the existence of $\Delta$-regret equilibria by introducing the concept of a parameterized social welfare problem, where the right-hand side of the original social welfare problem is changed. Our characterization then relates the achievable regret value with the associated duality/integrality gap of the parameterized social welfare problem. For the special case of monotone valuations this translates to regret bounds recovering the duality/integrality gap of the original social welfare problem. We further establish an interesting connection to the area of sensitivity theory in linear optimization. We show that the sensitivity gap of the optimal-value function of two (configuration) linear programs with changed right-hand side can be used to establish a bound on the achievable regret. Finally, we use these general structural results to translate known approximation algorithms for the social welfare optimization problem into algorithms computing low-regret Walras equilibria. We also demonstrate how to derive strong lower bounds based on integrality and duality gaps but also based on NP-complexity theory.

Free-boundary diffeomorphism optimization, an important and widely occurring task in geometric modeling, computer graphics, and biological imaging, requires simultaneously determining a planar target domain and a locally bijective map with well-controlled distortion. We formulate this task through the least-squares quasiconformal (LSQC) operator and establish key structural properties of the LSQC minimizer, including well-posedness under mild conditions, invariance under similarity transformations, and resolution-independent behavior with stability under mesh refinement. We further analyze the sensitivity of the LSQC solution with respect to the Beltrami coefficient, establishing stability and differentiability properties that enable gradient-based optimization over the space of Beltrami coefficients. To make this differentiable formulation practical at scale and to facilitate the optimization process, we introduce the Spectral Beltrami Network (SBN), a multiscale mesh-spectral surrogate that approximates the LSQC solution operator in a single differentiable forward pass. This yields SBN-Opt, an optimization framework that searches over admissible Beltrami coefficients and pinning conditions to solve free-boundary diffeomorphism objectives with explicit distortion control. Extensive experiments on equiareal parameterization and inconsistent surface registration demonstrate consistent improvements over traditional numerical algorithms.

We introduce a novel analytical approach for studying free-fermion (XX) chains with smoothly varying, site-dependent hoppings and magnetic fields. Building on a discrete WKB-like approximation applied directly to the recurrence relation for the single-particle eigenfunctions, we derive a closed-form expression for the local fermion density profile as a function of the Fermi energy, which is valid for arbitrary fillings, hopping amplitudes and magnetic fields. This formula reproduces the depletion and saturation effects observed in previous studies of inhomogeneous free-fermion chains, and provides a theoretical framework to understand entanglement entropy suppression in these models. We demonstrate the accuracy of our asymptotic formula in several chains with different hopping and magnetic field profiles. Our findings are thus the first step towards an analytical treatment of entanglement in free-fermion chains beyond the reach of conventional field-theoretic techniques.

Partial Differential Equations (PDEs) are central to modeling complex systems across physical, biological, and engineering domains, yet traditional numerical methods often struggle with high-dimensional or complex problems. Physics-Informed Neural Networks (PINNs) have emerged as an efficient alternative by embedding physics-based constraints into deep learning frameworks, but they face challenges in achieving high accuracy and handling complex boundary conditions. In this work, we extend the Time-Evolving Natural Gradient (TENG) framework to address Dirichlet boundary conditions, integrating natural gradient optimization with numerical time-stepping schemes, including Euler and Heun methods, to ensure both stability and accuracy. By incorporating boundary condition penalty terms into the loss function, the proposed approach enables precise enforcement of Dirichlet constraints. Experiments on the heat equation demonstrate the superior accuracy of the Heun method due to its second-order corrections and the computational efficiency of the Euler method for simpler scenarios. This work establishes a foundation for extending the framework to Neumann and mixed boundary conditions, as well as broader classes of PDEs, advancing the applicability of neural network-based solvers for real-world problems.

We consider both unitary and nonunitary A-D-E minimal models on the cylinder with topological defects along the non-contractible cycle of the cylinder. We define the coset graph $A \otimes G/\mathbb{Z}_2$ and argue that it encodes not only the (i) coset graph fusion algebra, but also (ii) the Affleck-Ludwig boundary g-factors; (iii) the defect g-factors (quantum dimensions) and (iv) the relative symmetry resolved entanglement entropy. By studying A-D-E restricted solid-on-solid models, we find that these boundary conformal field theory structures are also present on the lattice: defects (seams) are implemented by face weights with special values of the spectral parameter. Integrability allows the study of lattice transfer matrix T- and Y-system functional equations to reproduce the fusion algebra of defect lines. The effective central charges and conformal weights are expressed in terms of dilogarithms of the braid and bulk asymptotics of the Y-system expressed in terms of the quantum dimensions.

Online eXp-concave Optimization (OXO) is a fundamental problem in online learning, where the goal is to minimize regret when loss functions are exponentially concave. The standard algorithm, Online Newton Step (ONS), guarantees an optimal $O(d \log T)$ regret, where $d$ is the dimension and $T$ is the time horizon. Despite its simplicity, ONS may face a computational bottleneck due to the Mahalanobis projection at each round. This step costs $\Omega(d^\omega)$ arithmetic operations for bounded domains, even for simple domains such as the unit ball, where $\omega \in (2,3]$ is the matrix-multiplication exponent. As a result, the total runtime can reach $\tilde{O}(d^\omega T)$, particularly when iterates frequently oscillate near the domain boundary. This paper proposes a simple variant of ONS, called LightONS, which reduces the total runtime to $O(d^2 T + d^\omega \sqrt{T \log T})$ while preserving the optimal regret. Deploying LightONS with the online-to-batch conversion implies a method for stochastic exp-concave optimization with runtime $\tilde{O}(d^3/\epsilon)$, thereby answering an open problem posed by Koren [2013]. The design leverages domain-conversion techniques from parameter-free online learning and defers expensive Mahalanobis projections until necessary, thereby preserving the elegant structure of ONS and enabling LightONS to act as an efficient plug-in replacement in broader scenarios, including gradient-norm adaptivity, parametric stochastic bandits, and memory-efficient OXO.

Estimating the Riesz representer is central to debiased machine learning for causal and structural parameter estimation. We propose generalized Riesz regression, a unified framework for estimating the Riesz representer by fitting a representer model via Bregman divergence minimization. This framework includes various divergences as special cases, such as the squared distance and the Kullback--Leibler (KL) divergence, where the former recovers Riesz regression and the latter recovers tailored loss minimization. Under suitable pairs of divergence and model specifications (link functions), the dual problems of the Riesz representer fitting problem correspond to covariate balancing, which we call automatic covariate balancing. Moreover, under the same specifications, the sample average of outcomes weighted by the estimated Riesz representer satisfies Neyman orthogonality even without estimating the regression function, a property we call automatic Neyman orthogonalization. This property not only reduces the estimation error of Neyman orthogonal scores but also clarifies a key distinction between debiased machine learning and targeted maximum likelihood estimation (TMLE). Our framework can also be viewed as a generalization of density ratio fitting under Bregman divergences to Riesz representer estimation, and it applies beyond density ratio estimation. We provide convergence analyses for both reproducing kernel Hilbert space (RKHS) and neural network model classes. A Python package for generalized Riesz regression is released as genriesz and is available at this https URL.

Effective field theories consistent with quantum gravity obey surprising finiteness constraints, appearing in several distinct but interconnected forms. In this work we develop a framework that unifies these observations by proposing that the defining data of such theories, as well as the landscape of effective field theories that are valid at least up to a fixed cutoff, admit descriptions with a uniform bound on complexity. To make this precise, we use tame geometry and work in sharply o-minimal structures, in which tame sets and functions come with two integer parameters that quantify their information content; we call this pair their tame complexity. Our Finite Complexity Conjectures are supported by controlled examples in which an infinite Wilsonian expansion nevertheless admits an equivalent finite-complexity description, typically through hidden rigidity conditions such as differential or recursion relations. We further assemble evidence from string compactifications, highlighting the constraining role of moduli space geometry and the importance of dualities. This perspective also yields mathematically well-defined notions of counting and volume measures on the space of effective theories, formulated in terms of effective field theory domains and coverings, whose finiteness is naturally enforced by the conjectures.

Characterizing errors in quantum circuits is essential for device calibration, yet detecting rare error events requires a large number of samples. This challenge is particularly severe in calibrating fault-tolerant, error-corrected circuits, where logical error probabilities are suppressed to higher order relative to physical noise and are therefore difficult to calibrate through direct logical measurements. Recently, Wagner et al. [PRL 130, 200601 (2023)] showed that, for phenomenological Pauli noise models, the logical channel can instead be inferred from syndrome measurement data generated during error correction. Here, we extend this framework to realistic circuit-level noise models. From a unified code-theoretic perspective and spacetime code formalism, we derive necessary and sufficient conditions for learning the logical channel from syndrome data alone and explicitly characterize the learnable degrees of freedom of circuit-level Pauli faults. Using Fourier analysis and compressed sensing, we develop efficient estimators with provable guarantees on sample complexity and computational cost. We further present an end-to-end protocol and demonstrate its performance on several syndrome-extraction circuits, achieving orders-of-magnitude sample-complexity savings over direct logical benchmarking. Our results establish syndrome-based learning as a practical approach to characterizing the logical channel in fault-tolerant quantum devices.

An important distinction in our understanding of capacities of classical versus quantum channels is marked by the following question: is there an algorithm which can compute (or even efficiently compute) the capacity? While there is overwhelming evidence suggesting that quantum channel capacities may be uncomputable, a formal proof of any such statement is elusive. We initiate the study of the hardness of computing quantum channel capacities. We show that, for a general quantum channel, it is QMA-hard to compute its quantum capacity, and that the maximal-entanglement-assisted zero-error one-shot classical capacity is uncomputable.

Barren plateaus in variational quantum algorithms are typically described by gradient concentration at random initialization. In contrast, rigorous results for the Hessian, even at the level of entry-wise variance, remain limited. In this work, we analyze the scaling of Hessian-entry variances at initialization. Using exact second-order parameter-shift identities, we write $H_{jk}$ as a constant-size linear combination of shifted cost evaluations, which reduces ${\rm Var}_\rho(H_{jk})$ to a finite-dimensional covariance--quadratic form. For global objectives, under an exponential concentration condition on the cost at initialization, ${\rm Var}_\rho(H_{jk})$ decays exponentially with the number of qubits $n$. For local averaged objectives in bounded-depth circuits, ${\rm Var}_\rho(H_{jk})$ admits polynomial bounds controlled by the growth of the backward lightcone on the interaction graph. As a consequence, the number of measurement shots required to estimate $H_{jk}$ to fixed accuracy inherits the same exponential (global) or polynomial (local) scaling. Extensive numerical experiments over system size, circuit depth, and interaction graphs validate the predicted variance scaling. Overall, the paper quantifies when Hessian entries can be resolved at initialization under finite sampling, providing a mathematically grounded basis for second-order information in variational optimization.

Inverse problems are fundamental to many scientific and engineering disciplines; they arise when one seeks to reconstruct hidden, underlying quantities from noisy measurements. Many applications demand not just point estimates but interpretable uncertainty. Providing fast inference alongside uncertainty estimates remains challenging yet desirable in numerous applications. We propose the Variational Sparse Paired Autoencoder (vsPAIR) to address this challenge. The architecture pairs a standard VAE encoding observations with a sparse VAE encoding quantities of interest, connected through a learned latent mapping. The variational structure enables uncertainty estimation, the paired architecture encourages interpretability by anchoring QoI representations to clean data, and sparse encodings provide structure by concentrating information into identifiable factors rather than diffusing across all dimensions. To validate the effectiveness of our proposed architecture, we conduct experiments on blind inpainting and computed tomography, demonstrating that vsPAIR is a capable inverse problem solver that can provide interpretable and structured uncertainty estimates.

Entropic optimal transport (EOT) via Sinkhorn iterations is widely used in modern machine learning, yet GPU solvers remain inefficient at scale. Tensorized implementations suffer quadratic HBM traffic from dense $n\times m$ interactions, while existing online backends avoid storing dense matrices but still rely on generic tiled map-reduce reduction kernels with limited fusion. We present \textbf{FlashSinkhorn}, an IO-aware EOT solver for squared Euclidean cost that rewrites stabilized log-domain Sinkhorn updates as row-wise LogSumExp reductions of biased dot-product scores, the same normalization as transformer attention. This enables FlashAttention-style fusion and tiling: fused Triton kernels stream tiles through on-chip SRAM and update dual potentials in a single pass, substantially reducing HBM IO per iteration while retaining linear-memory operations. We further provide streaming kernels for transport application, enabling scalable first- and second-order optimization. On A100 GPUs, FlashSinkhorn achieves up to $32\times$ forward-pass and $161\times$ end-to-end speedups over state-of-the-art online baselines on point-cloud OT, improves scalability on OT-based downstream tasks. For reproducibility, we release an open-source implementation at this https URL.

We study the statistical behaviour of reasoning probes in a stylized model of looped reasoning, given by Boolean circuits whose computational graph is a perfect $\nu$-ary tree ($\nu\ge 2$) and whose output is appended to the input and fed back iteratively for subsequent computation rounds. A reasoning probe has access to a sampled subset of internal computation nodes, possibly without covering the entire graph, and seeks to infer which $\nu$-ary Boolean gate is executed at each queried node, representing uncertainty via a probability distribution over a fixed collection of $\mathtt{m}$ admissible $\nu$-ary gates. This partial observability induces a generalization problem, which we analyze in a realizable, transductive setting. We show that, when the reasoning probe is parameterized by a graph convolutional network (GCN)-based hypothesis class and queries $N$ nodes, the worst-case generalization error attains the optimal rate $\mathcal{O}(\sqrt{\log(2/\delta)}/\sqrt{N})$ with probability at least $1-\delta$, for $\delta\in (0,1)$. Our analysis combines snowflake metric embedding techniques with tools from statistical optimal transport. A key insight is that this optimal rate is achievable independently of graph size, owing to the existence of a low-distortion one-dimensional snowflake embedding of the induced graph metric. As a consequence, our results provide a sharp characterization of how structural properties of the computational graph govern the statistical efficiency of reasoning under partial access.

In the era of large foundation models, the quality of embeddings has become a central determinant of downstream task performance and overall system capability. Yet widely used dense embeddings are often extremely high-dimensional, incurring substantial costs in storage, memory, and inference latency. To address these, Contrastive Sparse Representation (CSR) is recently proposed as a promising direction, mapping dense embeddings into high-dimensional but k-sparse vectors, in contrast to compact dense embeddings such as Matryoshka Representation Learning (MRL). Despite its promise, CSR suffers severe degradation in the ultra-sparse regime, where over 80% of neurons remain inactive, leaving much of its efficiency potential unrealized. In this paper, we introduce CSRv2, a principled training approach designed to make ultra-sparse embeddings viable. CSRv2 stabilizes sparsity learning through progressive k-annealing, enhances representational quality via supervised contrastive objectives, and ensures end-to-end adaptability with full backbone finetuning. CSRv2 reduces dead neurons from 80% to 20% and delivers a 14% accuracy gain at k=2, bringing ultra-sparse embeddings on par with CSR at k=8 and MRL at 32 dimensions, all with only two active features. While maintaining comparable performance, CSRv2 delivers a 7x speedup over MRL, and yields up to 300x improvements in compute and memory efficiency relative to dense embeddings in text representation. Extensive experiments across text and vision demonstrate that CSRv2 makes ultra-sparse embeddings practical without compromising performance, where CSRv2 achieves 7%/4% improvement over CSR when k=4 and further increases this gap to 14%/6% when k=2 in text/vision representation. By making extreme sparsity viable, CSRv2 broadens the design space for real-time and edge-deployable AI systems where both embedding quality and efficiency are critical.

Lewis' account of common knowledge in Convention describes the generation of higher-order expectations between agents as hinging upon agents' inductive standards and a shared witness. This paper attempts to draw from insights in learning theory to provide a formal account of common inductive knowledge and how it can be generated by a witness. Our language has a rather rich syntax in order to capture equally rich notions central to Lewis' account of common knowledge; for instance, we speak of an agent 'having some reason to believe' a proposition and one proposition 'indicating' to an agent that another proposition holds. A similar line of work was pursued by Cubitt & Sugden 2003; however, their account was left wanting for a corresponding semantics. Our syntax affords a novel topological semantics which, following Kelly 1996's approach in The Logic of Reliable Inquiry, takes as primitives agents' information bases. In particular, we endow each agent with a 'switching tolerance' meant to represent their personal inductive standards for learning. Curiously, when all agents are truly inductive learners (not choosing to believe only those propositions which are deductively verified), we show that the set of worlds where a proposition $P$ is common inductive knowledge is invariant of agents' switching tolerances. Contrarily, the question of whether a specific witness $W$ generates common inductive knowledge of $P$ is sensitive to changing agents' switching tolerances. After establishing soundness of our proof system with respect to this semantics, we conclude by applying our logic to solve an 'inductive' variant of the coordinated attack problem.