An interplay between the Lambert series and Euler's Pentagonal Number Theorem gives an Euler-type recurrence relation for any given arithmetical function. As consequences of this, we present Euler-type recurrence relations for some well-known arithmetic functions. Furthermore, we derive Euler-type recurrence relations for certain partition functions and sum-of-divisors functions using infinite product identities of Jacobi and Gauss.

Let ${\rm gp}_{\rm t}(G)$, ${\rm gp}_{\rm o}(G)$, and ${\rm gp}_{\rm d}(G)$ be the total, the outer, and the dual general position number of a graph $G$. This paper investigates how removing a vertex or removing an edge affects these graph invariants. It is proved that if $x$ is not a cut vertex, then ${\rm gp}_{\rm t}(G) -1 \le {\rm gp}_{\rm t}(G-x) \le {\rm gp}_{\rm t}(G) + {\rm deg}_G(x)$. On the other hand, ${\rm gp}_{\rm o}(G-x)$ and ${\rm gp}_{\rm d}(G-x)$ can be respectively arbitrarily larger/smaller than ${\rm gp}_{\rm o}(G)$ and ${\rm gp}_{\rm d}(G)$. On the positive side, it is proved that if $x$ lies in some ${\rm gp}_{\rm o}$-set, then ${\rm gp}_{\rm o}(G)-1 \le {\rm gp}_{\rm o}(G-x)$, and that if $x$ is not a cut vertex and lies in some ${\rm gp}_{\rm d}$-set of $G$, then $ {\rm gp}_{\rm d}(G)-1 \le {\rm gp}_{\rm d}(G-x)$. For the edge removal, it is proved that (i) ${\rm gp}_{\rm t}(G) -|S(G)_{e}| \le {\rm gp}_{\rm t}(G-e) \le {\rm gp}_{\rm t}(G) +2$, where $S(G)_{e}$ is the set of simplicial vertices adjacent both endvertices of $e$, (ii) ${\rm gp}_{\rm o}(G)/2\le {\rm gp}_{\rm o}(G-e)\leq\ 2{\rm gp}_{\rm o}(G)$, and (iii) that ${\rm gp}_{\rm d}(G) - {\rm gp}_{\rm d}(G-e)$ can be arbitrarily large. All bounds proved are demonstrated to be sharp.

We show that in a vector space over Z_3, the union of any four linear bases is an additive basis, thus proving the Additive Basis Conjecture for p=3, and providing an alternative proof of the weak 3-flow conjecture.

We investigate the occurrence of additive and multiplicative structures in random subsets of the natural numbers. Specifically, for a Bernoulli random subset of $\mathbb{N}$ where each integer is included independently with probability $p\in (0,1)$, we prove that almost surely such a set contains finite sumsets (FS-sets) and finite product sets (FP-sets) of every finite length. In addition, we establish a novel connection between Hindman's partition theorem and the central limit theorem, providing a probabilistic perspective on the asymptotic Gaussian behavior of monochromatic finite sums and products. These results can be interpreted as probabilistic analogues of finite-dimensional versions of Hindman's theorem. Applications, implications, and open questions related to infinite FS-sets and FP-sets are discussed.

We prove that the Prym map corresponding to étale cyclic coverings of degree 6 over a genus 2 curve is generically injective.

We revisit the cyclic identities of Sun--Pan type for Bernoulli polynomials and their $q$-analogues. From the analytic side, we formulate minimal Appell axioms that force cyclic vanishing identities, extending naturally to $q$-Appell sequences and analytic Bernoulli functions. From the modular side, we show that the same relations arise as period polynomial identities associated with Eisenstein series, reflecting the symmetry $(ST)^3=-I$ of the modular group. These two complementary perspectives place the Sun--Pan cyclic identities at the crossroads of number theory, special functions, and modular forms, and suggest further connections to polylogarithms, $L$-values, and mixed Tate motives.

We give explicit, uniform formulas for the graded characters and total ranks of the Lie algebra homology of finite-dimensional representations in all classical types. In many cases, these compute the Tor groups of finite length modules over polynomial rings, and this is the first in a series of papers to investigate total rank conjectures from this perspective. These formulas refine and generalize the classical $\rho$-decomposition of Kostant, and in particular we prove that the characters involved exhibit three structural phenomena: divisibility (by a large power of 2), equidistribution, and uniform factorization formulas.

We first give a general introduction to the mean curvature flow, and then discuss fundamental results established over the last 10 years that yield a precise theory for the flow through singularities in $\mathbb{R}^3$. With the aim of developing a satisfying theory in higher dimensions, we then describe our recent classification of all noncollapsed singularities in $\mathbb{R}^4$. Finally, we provide a detailed discussion of open problems and conjectures.

We provide a theoretical framework for a wide class of generalized posteriors that can be viewed as the natural Bayesian posterior counterpart of the class of M-estimators in the frequentist world. We call the members of this class M-posteriors and show that they are asymptotically normally distributed under mild conditions on the M-estimation loss and the prior. In particular, an M-posterior contracts in probability around a normal distribution centered at an M-estimator, showing frequentist consistency and suggesting some degree of robustness depending on the reference M-estimator. We formalize the robustness properties of the M-posteriors by a new characterization of the posterior influence function and a novel definition of breakdown point adapted for posterior distributions. We illustrate the wide applicability of our theory in various popular models and illustrate their empirical relevance in some numerical examples.

We introduce the abstract notion of squarefree-power-like functions, which unify the sequences of squarefree ordinary and symbolic powers of squarefree monomial ideals. By employing the Tor-vanishing criteria for mixed sums of ideals, we establish sharp lower bounds for their Castelnuovo-Mumford regularity in terms of what we call the admissible set of the associated hypergraph. As an application, we derive the first general combinatorial lower bound for the regularity of squarefree symbolic powers of monomial ideals. In the setting of edge ideals, by exploiting the special combinatorial structures of block graphs and Cohen-Macaulay chordal graphs, we show that this bound turns into an exact formula for all squarefree symbolic powers of block graphs, as well as for the second squarefree symbolic powers of edge ideals of Cohen-Macaulay chordal graphs.

We investigate the role of self-adjoint extensions in the bulk-edge correspondence for topological insulators. While the correspondence is well understood in discrete models with spectral gaps, complications arise in the presence of unbounded Hamiltonians and varying boundary conditions, leading to anomalous behavior that has recently been dubbed violations of bulk-edge correspondence. In this work we use a K-theoretic framework to identify precise conditions needed for unbounded Hamiltonians to be affiliated to the respective observable algebras and define K-theory classes. In special cases we can then exclude anomalous behaviour and obtain the standard bulk-edge correspondence, or, under weaker conditions, obtain a relative bulk-edge correspondence theorem, which compares pairs of Hamiltonians. Applying that relative approach in the bulk we recover among other things the so-called bulk-difference-interface correspondence for Hamiltonians that fail to define a bulk K-theory class in the conventional way. The second main result is that one can define K-theory classes in terms of von Neumann unitaries, which under changes in boundary conditions directly contribute to the number of protected edge states. This approach clarifies apparent violations of the classical bulk-edge paradigm and provides a systematic account of boundary-induced topological corrections.

We study the distribution of interior faces in uniformly random reduced $\mathfrak s \mathfrak l_3$ webs. Using Tymoczko's bijection between $3\times n$ standard Young tableaux and reduced webs, this problem can be reformulated in terms of constrained lattice paths and associated $m$-diagrams. We develop a framework that expresses crossing probabilities in the $m$-diagram as solutions to discrete Dirichlet problems on the triangular lattice, which are evaluated through solutions to lattice Green's functions. From this we obtain explicit limiting formulas for the frequencies of interior faces of each type. As an application, we analyze faces at a distance at least $d$ from the boundary. We prove that almost all interior faces far from the boundary are hexagons, while faces of size $6+2k$ occur with probability $O(d^{-2k})$.

A classical result by R. Rochberg says that every bounded Toeplitz operator $T$ on the Hilbert Paley-Wiener space $\mathrm{PW}_a^2$ admits a bounded symbol $\varphi$. We generalize this result to Toeplitz operators on the Banach Paley-Wiener spaces $\mathrm{PW}_a^p$, $1<p<+\infty$. The Toeplitz commutator theorem describes the integral identity that must hold for a bounded operator $T$ on $\mathrm{PW}_a^p$ to be a Toeplitz operator on $\mathrm{PW}_a^p$. We prove this theorem in the continuous case, thus extending the result previously obtained by D. Sarason in the discrete case. Upon combining the results, we establish the weak factorization theorem, namely, for $p,q>1$, $\frac{1}{p}+\frac{1}{q}=1$, any function $h$ belonging to $\mathrm{PW}^1_{2a}$ can be represented as $$h=\sum_{k\geqslant 0}f_k\bar{g}_k,\qquad f_k\in\mathrm{PW}_a^p,\,g_k\in\mathrm{PW}_a^q.$$

In this paper, we propose DeMuon, a method for decentralized matrix optimization over a given communication topology. DeMuon incorporates matrix orthogonalization via Newton-Schulz iterations-a technique inherited from its centralized predecessor, Muon-and employs gradient tracking to mitigate heterogeneity among local functions. Under heavy-tailed noise conditions and additional mild assumptions, we establish the iteration complexity of DeMuon for reaching an approximate stochastic stationary point. This complexity result matches the best-known complexity bounds of centralized algorithms in terms of dependence on the target tolerance. To the best of our knowledge, DeMuon is the first direct extension of Muon to decentralized optimization over graphs with provable complexity guarantees. We conduct preliminary numerical experiments on decentralized transformer pretraining over graphs with varying degrees of connectivity. Our numerical results demonstrate a clear margin of improvement of DeMuon over other popular decentralized algorithms across different network topologies.

We introduce and study two Maker-Breaker-like games for constructing planar graphs: the edge drawing game, where two players take turns drawing non-intersecting edges between points in the plane, and the circle packing game, where the players take turns placing disjoint circles in the plane. Both games produce planar graphs: the edge drawing game results in a plane graph drawing, and the circle packing game yields a planar graph via the contact graph of the packing. For both games, we give necessary conditions under which a given planar graph can be constructed. We also show that the two games are indeed different by giving a class of graphs which can be constructed in one but not the other.

In the setting of a non-complete doubling metric measure space $(\Omega,d,\mu)$, we construct various bounded linear trace and extension operators for homogeneous and inhomogeneous Besov spaces $B^\alpha_{p,q}$. Equipping the boundary $\partial\Omega:=\overline\Omega\setminus\Omega$ with a measure which is codimension $\theta$ Ahlfors regular with respect to $\mu$, these operators take the form \[ T:B^\alpha_{p,q}(\Omega)\to B^{\alpha-\theta/p}_{p,q}(\partial\Omega),\quad E:B^\alpha_{p,q}(\partial\Omega)\to B^{\alpha+\theta/p}_{p,q}(\Omega). \] The trace operators are first constructed under the additional assumption that $\Omega$ is a uniform domain in its completion. We then use such results along with the technique of hyperbolic filling to remove this assumption in the case that $\Omega$ is bounded. This extends to the doubling setting some earlier results of Marcos and Saksman-Soto proven under the assumption that the ambient measure is Ahlfors regular.

We construct a perfect version of Morel--Voevodsky's motivic homotopy category over a perfect base scheme in positive characteristic. By checking the axioms of a coefficient system, we establish a six-functor formalism. We show that multiplication by $p$ is already invertible in the perfect motivic homotopy catgory. By work of Elmanto--Khan the functor sending an $\mathbb{F}_p$-scheme $S$ to the category $\mathrm{S}\mathcal{H}(S)[1/p]$ is invariant under universal homeomorphisms, hence under perfections. Our result gives an explicit model for the localization of $\mathrm{S}\mathcal{H}$ at the universal homeomorphisms, which we conclude is the same as $\mathrm{S}\mathcal{H}[1/p]$.

In this paper, we introduce a three-component Gierer-Meinhardt model in the semi-strong interaction regime, characterized by an asymptotically large diffusivity ratio. A key feature of this model is that the interior spike can undergo Hopf bifurcations in both amplitude and position, leading to rich oscillatory dynamics not present in classical two-component systems. Using asymptotic analysis and numerical path-following, we construct localized spike equilibria and analyze spike nucleation that occurs through slow passage beyond a saddle-node bifurcation. Moreover, stability of spike equilibrium is analyzed by introducing time-scaling parameters, which reveal two distinct mechanisms: amplitude oscillations triggered by large-eigenvalue instabilities and oscillatory spike motion associated with small eigenvalues. Numerical simulations illustrate these dynamics and their transition regimes. This dual mechanism highlights richer spike behavior in three-component systems and suggests several open problems for future study.

In this paper, the hyperbolic Anderson equation generated by a time-dependent Gaussian noise is under investigation in two fronts: The solvability and large-$t$ asymptotics. The investigation leads to a necessary and sufficient condition for existence and a precise large-$t$ limit form for the expectation of the solution. Three major developments are made for achieving these goals: A universal bound for Stratonovich moment that guarantees the Stratonovich integrability and ${\cal L}^2$-convergence of the Stratonovich chaos expansion under the best possible condition, a representation of the expected Stratonovich moments in terms of a time-randomized Brownian intersection local time, and a large deviation principle for the time-randomized Brownian intersection local time.

This paper is a contribution to the symmetry analysis of the gas dynamics system in the vein of the ''podmodeli'' (submodels) program outlined by Ovsyannikov (1994). We consider the case of the special state equation, prescribing pressure to be the sum of entropy and an arbitrary function of density. Such a system has a 12-dimensional symmetry Lie algebra. This work advances the study of its four-dimensional subalgebras, continuing the work started in Siraeva (2024). For a large subset of not previously considered, non-similar four-dimensional subalgebras from an optimal list in Siraeva (2014), we compute a complete set of generating invariants. For one of the subalgebras, we construct a partially symmetry-reduced system. We explicitly solve this reduced system (submodel). This leads to new families of explicit solutions of the original system. We analyze the trajectories of these solutions. Additionally, we match each of the subalgebras considered in this paper with its isomorphism class, planting a seed for future study of the hierarchy of the reduced systems.

This paper introduces an archimedean, locally Cantor multi-field $\mathcal{O}_{\theta}$ which gives an analog of the $p$-adic number field at a place at infinity of a real quadratic extension $K$ of $\mathbb{Q}$. This analog is defined using a unit $1<\theta\in \mathcal{O}_{K}^{\times}$, which plays the same role as the prime $p$ does in $\mathbb{Z}_{p}$; the elements of $\mathcal{O}_{\theta}$ are then greedy Laurent series in the base $\theta$. There is a canonical inclusion of the integers $\mathcal{O}_{K}$ with dense image in $\mathcal{O}_{\theta}$ and the operations of sum and product extend to multi-valued operations having at most three values, making $\mathcal{O}_{\theta}$ a multi-field in the sense of Marty. We show that the (geometric) completions of 1-dimensional quasicrystals contained in $\mathcal{O}_{K}$ map canonically to $\mathcal{O}_{\theta}$. The motivation for this work arises in part from a desire to obtain a more arithmetic treatment of a place at infinity by replacing $\mathbb{R}$ with $\mathcal{O}_{\theta}$, with an eye toward obtaining a finer version of class field theory incorporating the ideal arithmetic of quasicrystal rings.

Assuming the four exponentials conjecture, Hansel and Safer showed that if a subset $S$ of the Gaussian integers is both $\alpha=-m+i $- and $\beta=-n+i$-recognizable, then it is syndetic, and they conjectured that $S$ must be eventually periodic. Without assuming the four exponentials conjecture, we show that if $\alpha$ and $\beta$ are multiplicatively independent Gaussian integers, and at least one of $\alpha$, $\beta$ is not an $n$-th root of an integer, then any $\alpha$- and $\beta$-automatic configuration is eventually periodic; in particular we prove Hansel and Safer's conjecture. Otherwise, there exist non-eventually periodic configurations which are $\alpha$-automatic for any root of an integer $\alpha$. Our work generalises the Cobham-Semenov theorem to Gaussian numerations.

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

This paper presents an analytical and computational framework for optimizing gas withdrawal in reconstructed ring-type pipeline systems under unsteady flow conditions. As urban and industrial energy demands grow, repurposing existing pipeline infrastructure offers a cost-effective alternative to full-scale expansion. The proposed model identifies the hydraulic coupling point (where the pressure gradient vanishes) as the optimal location for connecting new consumers. By employing a one-dimensional unsteady gas flow model with time-dependent mass extraction represented via a Heaviside step function, the system's dynamic response is captured in detail. Numerical simulations demonstrate that connecting additional loads at the pressure maximum ensures stability while minimizing operational disruptions. The model's validation through benchmark comparison and pressure tolerance thresholds confirms its practical applicability. Economic analysis reveals substantial savings over conventional expansion methods. The approach provides a scalable solution for smart gas network design.

This paper tackles optimization problems whose objective and constraints involve a trained Neural Network (NN), where the goal is to maximize $f(\Phi(x))$ subject to $c(\Phi(x)) \leq 0$, with $f$ smooth, $c$ general and non-stringent, and $\Phi$ an already trained and possibly nonwhite-box NN. We address two challenges regarding this problem: identifying ascent directions for local search, and ensuring reliable convergence towards relevant local solutions. To this end, we re-purpose the notion of directional NN attacks as efficient optimization subroutines, since directional NN attacks use the neural structure of $\Phi$ to compute perturbations of $x$ that steer $\Phi(x)$ in prescribed directions. Precisely, we develop an attack operator that computes attacks of $\Phi$ at any $x$ along the direction $\nabla f(\Phi(x))$. Then, we propose a hybrid algorithm combining the attack operator with derivative-free optimization (DFO) techniques, designed for numerical reliability by remaining oblivious to the structure of the problem. We consider the cDSM algorithm, which offers asymptotic guarantees to converge to a local solution under mild assumptions on the problem. The resulting method alternates between attack-based steps for heuristic yet fast local intensification and cDSM steps for certified convergence and numerical reliability. Experiments on three problems show that this hybrid approach consistently outperforms standard DFO baselines.

Generalising the Heilman-Lieb Theorem from statistical physics, Chudnovsky and Seymour [J. Combin. Theory Ser. B, 97(3):350--357] showed that the univariate independence polynomial of any claw-free graph has all of its zeros on the negative real line. In this paper, we show that for any fixed subdivded claw $H$ and any $\Delta$, there is an open set $F \subseteq \mathbb{C}$ containing $[0, \infty)$ such that the independence polynomial of any $H$-free graph of maximum degree $\Delta$ has all of its zeros outside of $F$. We also show that no such result can hold when $H$ is any graph other than a subdivided claw or if we drop the maximum degree condition. We also establish zero-free regions for the multivariate independence polynomial of $H$-free graphs of bounded degree when $H$ is a subdivided claw. The statements of these results are more subtle, but are again best possible in various senses.

Bilevel optimization provides a powerful framework for modeling hierarchical decision-making systems. This work presents a novel sensitivity-based algorithm that directly addresses the bilevel structure by treating the lower-level optimal solution as an implicit function of the upper-level variables, thus avoiding classical single-level reformulations. This implicit problem is solved within a robust Augmented Lagrangian framework, where the inner subproblems are managed by a quasi-Newton (L-BFGS-B) solver to handle the ill-conditioned and non-smooth landscapes that can arise. The validity of the proposed method is established through both theoretical convergence guarantees and extensive computational experiments. These experiments demonstrate the algorithm's efficiency and robustness and validate the use of a pragmatic dual-criterion stopping condition to address the practical challenge of asymmetric primal-dual convergence rates.

We present a construction that manufactures $\E_\infty$ orientations of Tate fixed-point objects together with useful formulas for these maps, and then give a number of applications. For example, we produce a formula for the Frobenius homomorphisms of Thom spectra such as $\MU$ as well as certain lifts of Frobenius. We prove a rigidity property of $\MU$ as a \emph{cyclotomic} object. We construct a general obstruction theory for $\E_n$ complex orientations and establish various non-existence results for $p$-typical $\E_n$ orientations for low values of $p$ and $n$. We end with some miscellaneous further applications.

We introduce a novel class of self-mappings on metric spaces, called \textbf{PA-contractions} (Path-Averaged Contractions), defined by an averaging condition over iterated distances. We prove that every continuous PA-contraction on a complete metric space has a unique fixed point, and the Picard iterates converge to it. This condition strictly generalizes the classical Banach contraction principle. We provide examples showing that PA-contractions are independent of F-contractions, Kannan, Chatterjea, and Ćirić contractions. A comparison table highlights the distinctions. The PA-condition captures long-term contractive behavior even when pointwise contraction fails.

We study particle systems with singular pairwise interactions and non-vanishing diffusion in the mean-field scaling. A classical approach to describing corrections to mean-field behavior is through the analysis of correlation functions. For bounded interactions, the optimal estimates on correlations are well known: the $m$-particle correlation function is $G_{N,m}=O(N^{1-m})$ for all $m$. Such estimates, however, have remained out of reach for more singular interactions. In this work, we develop a new framework based on linearized correlation functions, which allows us to derive robust bounds for systems with merely square-integrable interaction kernels, providing the first systematic control of correlations in the singular setting. Although at first not optimal, our estimates can be partially refined a posteriori using the BBGKY hierarchy: in the case of bounded interactions, our method recovers the known optimal estimates with a simplified argument. As key applications, we establish the validity of the Bogolyubov correction to mean field and prove a central limit theorem for the empirical measure, extending these results beyond the bounded interaction regime for the first time.

Let $f_\ell(n, k)$ denote the clique number of the xor-product of $\ell$ isomorphic Kneser graphs KG(n,k). Alon and Lubetzky investigated the case of complete graphs as a coding theory problem and showed $f_\ell(n,1)\leq \ell n +1$. Imolay, Kocsis, and Schweitzer proved that $f_2(n,k)\leq n/k +c(k)$. Here, the order of magnitude of $c(k)$ is determined to be $\Theta\left( k \binom{2k}{k} \right)$. By explicit constructions and by an algebraic proof, it is shown that $\ell n- 2\ell-1 \leq f_\ell(n,1)\leq \ell n-\ell+1$ (for all $n \geq 1$ and $\ell\geq 3$). Finally, it is proved that the order of magnitude of $f$ lies between $\Omega\left(n^{\left\lfloor \log_2(\ell+1)\right\rfloor}\right)$ and $O\left(n^{\left\lfloor \frac{\ell+1}{2} \right\rfloor} \right)$ (as $\ell$, $k$ are given and $n\to \infty$). We conjecture that the lower bound gives the correct exponent.

We propose a divide-and-conquer (DAC) algorithm for constrained convex optimization over networks, where the global objective is the sum of local objectives attached to individual agents. The algorithm is fully distributed: each iteration solves local subproblems around selected fusion centers and coordinates only with neighboring fusion centers. Under standard assumptions of smoothness, strong convexity, and locality on the objective function, together with polynomial growth conditions on the underlying graph, we establish exponential convergence of the DAC iterations and derive explicit bounds for both exact and inexact local solvers. Numerical experiments on three representative losses ($L_2$ distance, quadratic, and entropy) confirm the theory and demonstrate scalability and effectiveness.

It is well-known that convex variational problems with linear growth and Dirichlet boundary conditions might not have minimizers if the boundary condition is not suitably relaxed. We show that for a wide range of integrands, including the least gradient problem and the non-parametric Plateau problem, and under suitable mean-convexity conditions of the boundary, minimizers of the relaxed problem attain the boundary data in the trace sense if it lies in $BV$ or $W^{\alpha,p}$ with $\alpha p\geq 2$ without any kind of continuity assumption. Unlike previous works, our methods are also able to treat systems under a certain quasi-isotropy assumption on the integrand. We further show that without this quasi-isotropy assumption, smooth counterexamples on uniformly convex domains exist. Further applications to the uniqueness of minimizers and to open problems about the ROF functional with Dirichlet boundary conditions, and to the trace space of functions of least gradient are given.

Given a complex of groups, we construct a new class of complex of groups that records its local data and offer a functorial perspective on the statement that complexes of groups are locally developable. We also construct a new notion of an immersion of complexes of groups and establish that a locally isometric immersion of a complex of groups into a non-positively curved complex of groups is $\pi_1$-injective. Furthermore, the domain complex of groups is developable and the induced map on geometric realizations of developments is an isometric embedding.

Relative algebroids and Pfaffian fibrations are two frameworks recently developed to study geometric structures and PDEs with symmetries, but have structurally different foundations. In this article, we clarify the relation between the two. We show that every Pfaffian fibration canonically induces a relative algebroid, and that their prolongations and local solutions coincide. Moreover, we introduce two notions of symmetries of Pfaffian fibrations, namely internal symmetries and Pfaffian symmetries, and develop the theory for actions of Pfaffian groupoids by internal/Pfaffian symmetries. We show that such actions preserve the underlying relative algebroid in an appropriate sense. Our results apply in particular to partial differential equations with Lie pseudogroup symmetries.

We study the difference between the maximum likelihood estimation (MLE) and its semi-definite programming (SDP) relaxation for the phase synchronization problem, where $n$ latent phases are estimated based on pairwise observations corrupted by Gaussian noise at a level $\sigma$. While previous studies have established that SDP coincides with the MLE when $\sigma \lesssim \sqrt{n / \log n}$, the behavior in the high-noise regime $\sigma \gtrsim \sqrt{n / \log n}$ remains unclear. We address this gap by quantifying the deviation between the SDP and the MLE in the high-noise regime as $\exp(-c \frac{n}{\sigma^2})$, indicating an exponentially small discrepancy. In fact, we establish more general results for the Burer-Monteiro (BM) factorization that covers the SDP as a special case: it has the exponentially small deviation from the MLE in the high-noise regime and coincides with the MLE when $\sigma$ is small. To obtain our results, we develop a refined entrywise analysis of the MLE that is beyond the existing $\ell_\infty$ analysis in literature.

In this paper, we establish new Laplacian comparison theorems and rigidity theorems for complete Kähler manifolds under new curvature notions that interpolate between Ricci curvature and holomorphic bisectional curvature.

We use differential forms on loop spaces to prove that the fundamental group of certain geometric transformation groups is infinite. Examples include both finite and infinite dimensional Lie groups. The finite dimensional examples are the conformal group of $S^{4k+1}$ for a family of nonstandard metrics, and the group of pseudo-Hermitian transformations of a compact CR manifold. Infinite dimensional examples include the group of strict contact diffeomorphisms of a regular contact manifold, and other groups coming from symplectic and contact geometry.

All inverse problems rely on data to recover unknown parameters, yet not all data are equally informative. This raises the central question of data selection. A distinctive challenge in PDE-based inverse problems is their inherently infinite-dimensional nature: both the parameter space and the design space are infinite, which greatly complicates the selection process. Somewhat unexpectedly, randomized numerical linear algebra (RNLA), originally developed in very different contexts, has provided powerful tools for addressing this challenge. These methods are inherently probabilistic, with guarantees typically stating that information is preserved with probability at least 1-p when using N randomly selected, weighted samples. Here, the notion of information can take different mathematical forms depending on the setting. In this review, we survey the problem of data selection in PDE-based inverse problems, emphasize its unique infinite-dimensional aspects, and highlight how RNLA strategies have been adapted and applied in this context.

Motivated by recent work of Hirschhorn and the author, Thejitha and Fathima recently considered arithmetic properties satisfied by the function $a_5(n)$ which counts the number of integer partitions of weight $n$ wherein even parts come in only one color (i.e., they are monochromatic), while the odd parts may appear in one of five colors. They proved two sets of Ramanujan--like congruences satisfied by $a_5(n)$, relying heavily on modular forms. In this note, we prove their results via purely elementary means, utilizing generating function manipulations and elementary $q$-series dissections. We then extensively generalize these two sets of congruences to infinite families of divisibility properties in which the results of Thejitha and Fathima are specific instances.

We address a conjecture (referred to as sur in the literature) in the representation theory of a reductive p-adic Lie group G which has important implications for the relationship between mod-p smooth representations and pro-p Iwahori-Hecke modules, and is currently only known for G of rank 1. We prove that sur follows from exactness of the associated oriented chain complex of a coefficient system, when restricted to a local region of the Bruhat-Tits building for G. Our main result gives strong evidence towards this exactness in the case where G=SL_3(K) for K a totally ramified extension of Q_p. We also develop new combinatorial techniques for analysing the geometric realisation of the A_2 Bruhat-Tits building, which are fundamental to the proof of our main result, and which we hope will inspire further investigation in Bruhat-Tits theory.

Let $f \in S_{2r}(\Gamma_0(N))$ be a normalized newform of weight $2r$ which is good at $p$. Let $K$ be an imaginary quadratic field of class number one in which every prime divisor of $pN$ splits. Let $\chi$ be an anticyclotomic Hecke character of $K$ which is crystalline at the primes above $p$ and such that $L(f,\chi,r)\neq 0$. We prove that the Tamagawa number conjecture for the critical value $L(f,\chi,r)$ follows from the Iwasawa main conjecture for the Bertolini-Darmon-Prasanna $p$-adic $L$-function.

We analyze the nonlinear inertial instability of Couette flow under Coriolis forcing in \(\mathbb{R}^{3}\). For the Coriolis coefficient \(f \in (0,1)\), we show that the non-normal operator associated with the linearized system admits only continuous spectrum. Hence, there are no exponentially growing eigenfunctions for the linearized system. Instead, we construct unstable solutions in the form of pseudo-eigenfunctions that exhibit non-ideal spectral properties. Then through a bootstrap argument and resolving the challenges posed by the non-ideal spectral behavior of pseudo-eigenfunctions, we establish the velocity instability of Couette flow in the Hadamard sense for $ f \in \Big(\frac{2}{17} \left(5-2 \sqrt{2}\right), \frac{2}{17} \left(5 + 2 \sqrt{2}\right) \Big)$.

We study continuous approximate solutions to polynomial equations over the ring $C(X)$ of continuous complex-valued functions over a compact Hausdorff space $X$. We show that when $X$ is one-dimensional, the existence of such approximate solutions is governed by the behaviour of maps from the fundamental pro-group of $X$ into braid groups.

For any ${\rm E}$-rigid presentation $e$, we construct an orthogonal projection functor to ${\rm rep}(e^\perp)$ left adjoint to the natural embedding. We establish a bijection between presentations in ${\rm rep}(e^\perp)$ and presentations compatible with $e$. For quivers with potentials, we show that ${\rm rep}(e^\perp)$ forms a module category of another quiver with potential. We derive mutation formulas for the $\delta$-vectors of positive and negative complements and the dimension vectors of simple modules in ${\rm rep}(e^\perp)$, enabling an algorithm to find the projected quiver with potential. Additionally, we introduce a modified projection for quivers with potentials that preserves general presentations. For applications to cluster algebras, we establish a connection to the stabilization functors.

We study the bi-parameter local linearization of the one-dimensional nonlinear stochastic wave equation driven by a Gaussian noise, which is white in time and has a spatially homogeneous covariance structure of Riesz-kernel type. We establish that the second-order increments of the solution can be approximated by those of the corresponding linearized wave equation, modulated by the diffusion coefficient. These findings extend the previous results of Huang et al. \cite{HOO2024}, which addressed the case of space-time white noise. As applications, we analyze the quadratic variation of the solution and construct a consistent estimator for the diffusion parameter.

Consider the strata of primitive $k$-differentials on the Riemann sphere whose singularities, except for two, are poles of order divisible by $k$. The map that assigns to each $k$-differential the $k$-residues at these poles is a ramified cover of its image. Generalizing results known in the case of abelian differentials, we describe the ramification locus of this cover and provide a formula, involving the $k$-factorial function, for the cardinality of each fiber. We prove this formula using intersection calculations on the multi-scale compactification of the strata of $k$-differentials. In special cases, we also give alternative proofs using flat geometry. Finally, we present an application to cone spherical metrics with dihedral monodromy.

The multiple-input multiple-output (MIMO) receiver processing is a key technology for current and next-generation wireless communications. However, it faces significant challenges related to complexity and scalability as the number of antennas increases. Artificial intelligence (AI), a cornerstone of next-generation wireless networks, offers considerable potential for addressing these challenges. This paper proposes an AI-driven, universal MIMO receiver architecture based on Markov chain Monte Carlo (MCMC) techniques. Unlike existing AI-based methods that treat receiver processing as a black box, our MCMC-based approach functions as a generic Bayesian computing engine applicable to various processing tasks, including channel estimation, symbol detection, and channel decoding. This method enhances the interpretability, scalability, and flexibility of receivers in diverse scenarios. Furthermore, the proposed approach integrates these tasks into a unified probabilistic framework, thereby enabling overall performance optimization. This unified framework can also be seamlessly combined with data-driven learning methods to facilitate the development of fully intelligent communication receivers.

In this paper, we establish a weighted capillary isoperimetric inequality outside convex sets using the $\lambda_w$-ABP method. The weight function $w$ is assumed to be positive, even, and homogeneous of degree $\alpha$, such that $w^{1/\alpha}$ is concave on $\R^n$. Based on the weighted isoperimetric inequality, we develop a technique of capillary Schwarz symmetrization outside convex sets, and establish a weighted Pólya-Szegö principle and a sharp weighted capillary Sobolev inequality outside convex domain. Our result can be seen as an extension of the weighted Sobolev inequality in the half-space established by Ciraolo-Figalli-Roncoroni in \cite{CFR}.

It is well known that the BMO and Campanato norms can be characterized using the $L^p$-average. These characterizations were later generalized to averages taken over various types of function spaces. In particular, generalizations using Banach function spaces were provided by Ho, Izuki, Noi, and Sawano. In this paper, as a further generalization, we provide similar characterizations using quasi-Banach function spaces that do not assume the Fatou property.

We prove the necessary and sufficient conditions under which ultrametric spaces of arbitrary infinite cardinality admit isometric embeddings into ultrametric spaces generated by labeled star graphs.

Owing to its simplicity and efficiency, the Sherman-Morrison (SM) formula has seen widespread use across various scientific and engineering applications for solving rank-one perturbed linear systems of the form $(A+uv^T)x = b$. Although the formula dates back at least to 1944, its numerical stability properties have remained an open question and continue to be a topic of current research. We analyze the backward stability of the SM, demonstrate its instability in a scenario increasingly common in scientific computing and address an open question posed by Nick Higham on the proportionality of the backward error bound to the condition number of $A$. We then incorporate fixed-precision iterative refinement into the SM framework reusing the previously computed decompositions and prove that, under reasonable assumptions, it achieves backward stability without sacrificing the efficiency of the SM formula. While our theory does not prove the SM formula with iterative refinement always outputs a backward stable solution, empirically it is observed to eventually produce a backward stable solution in all our numerical experiments. We conjecture that with iterative refinement, the SM formula yields a backward stable solution provided that $\kappa_2(A), \kappa_2(A+uv^T)$ are both bounded safely away from $\epsilon_M^{-1}$, where $\epsilon_M$ is the unit roundoff.

In this note we show that the only result of [Rocky Mountain J. Math. 54 (2024), no. 4, 995--1004] is nothing more than a misformulated version of an exercise from classical texts, presented with a flawed proof. To place the matter on firmer ground, we provide instead a direct solution of a more general problem.

We characterize the inverse of an analytic Fredholm operator-valued function A(z) near an isolated singularity within a general Banach space framework. Our approach relies on the sequential factorization of A(z) via Fredholm quotient operators. By analyzing the properties of these quotient operators near an isolated singularity, we fully characterize the Laurent series expansion of the inverse of A(z) in terms of its Taylor coefficients around the singularity. These theoretical results are subsequently applied to characterize the solution of a general autoregressive law of motion in a Banach space.

Rank-constrained matrix problems appear frequently across science and engineering. The convergence analysis of iterative algorithms developed for these problems often hinges on local error bounds, which correlate the distance to the feasible set with a measure of how much the constraints are violated. Foundational results in semi-algebraic geometry guarantee that such bounds exist, yet the associated exponents are generally not explicitly determined. This paper establishes a local Hölderian error bound with an explicit exponent for the canonical rank-constrained affine feasibility set. This paper proves that, on any compact set, the distance to the feasible set is bounded by a power of a natural residual function capturing violations in both the rank and affine constraints. The exponent in this bound is given explicitly in terms of the problem's dimensions. This provides a fundamental quantitative result on the geometry of the solution set, paving the way for the convergence analysis of a broad class of numerical methods.

In 1983, Bouchet conjectured that every flow-admissible signed graph admits a nowhere-zero $6$-flow. In this paper, we prove that Bouchet's conjecture holds for all signed ladders, circular and Möbius ladders. In fact, all signed ladders, circular and Möbius ladders admit a nowhere-zero $5$-flow except for one case of signed circular ladders. Of course, the exception also has a nowhere-zero $6$-flow.

Let ($\Omega$, $\mu$) be a measure space with $\Omega$ $\subset$ R d and $\mu$ a finite measure on $\Omega$. We provide an extension of the Mean Value Theorem (MVT) in the form It is valid for non compact sets $\Omega$ and f is only required to be integrable with respect to $\mu$. It also contains as a special case the MVT in the form f d$\mu$ = $\mu$($\Omega$)f (x 0 ) for some x 0 $\in$ $\Omega$, valid for compact connected set $\Omega$ and continuous f . It is a direct consequence of Richter's theorem which in turn is a non trivial (overlooked) generalization of Tchakaloff's theorem, and even published earlier.

We prove pointwise-in-time dispersive estimates for solutions to the generalized Korteweg--de Vries (gKdV) equation. In particular, for solutions to the mass-critical model, we assume only that initial data lie in $\dot{H}^{\frac{1}{4}} \cap \dot{H}^{-\frac{1}{12}}$ and show that solutions decay in $L^\infty$ like $|t|^{-\frac{1}{3}}$. To accomplish this, we develop a persistence of negative regularity for solutions to gKdV and extend Lorentz--Strichartz estimates to the mixed norm case.

We prove the existence and uniqueness of strong solutions to the equation $u u_x - u_{yy} = f$ in the vicinity of the linear shear flow, subject to perturbations of the source term and lateral boundary conditions. Since the solutions we consider have opposite signs in the lower and upper half of the domain, this is a quasilinear forward-backward parabolic problem, which changes type across a critical curved line within the domain. In particular, lateral boundary conditions can be imposed only where the characteristics are inwards. There are several difficulties associated with this problem. First, the forward-backward geometry depends on the solution itself. This requires to be quite careful with the approximation procedure used to construct solutions. Second, and more importantly, the linearized equations solved at each step of the iterative scheme admit a finite number of singular solutions, of which we provide an explicit construction. This is similar to well-known phenomena in elliptic problems in nonsmooth domains. Hence, the solutions to the equation are regular if and only if the source terms satisfy a finite number of orthogonality conditions. A key difficulty of this work is to cope with these orthogonality conditions during the nonlinear fixed-point scheme. In particular, we are led to prove their stability with respect to the underlying base flow. To tackle this deceivingly simple problem, we develop a methodology which we believe to be both quite natural and adaptable to other situations in which one wishes to prove the existence of regular solutions to a nonlinear problem for suitable data despite the existence of singular solutions at the linear level. This paper is a shorter version of [3].

Chater and MacKay [CM] derived an entropy function of state for exchange economies satisfying a list of axioms, and showed that a change of state of a system of such economies is possible if and only if their total entropy does not decrease. In this paper, a large class of agent-based models is proved to satisfy the axioms in the thermodynamic limit, and the entropy is shown to be the logarithm of the partition function for their stationary distributions.

We study the Gibbs equilibrium of a classical 2D Coulomb gas in the determinantal case = 2. The external potential is the sum of a quadratic term and the potential generated by individual charges pinned in several extended groups. This leads to an equilibrium measure (droplet) with flat density and macroscopic holes. We consider ''correlation energy'' (free energy minus its mean-field approximation) expansions, for large particle number . Under the assumptions that the holes are sufficiently small, separated, and far from the droplet's outer boundary, we prove that (i) the correlation energy up to order 1 is independent of the holes' locations and orientations, and (ii) the difference between the correlation energies of systems differing by their number of holes involves ''topological'' O(log N ) and O (1) terms.

We prove that in the standard cosphere bundle, for any contact homeomorphism in the closure of the compactly supported contactomorphism group, when the image of a coisotropic submanifold (not necessarily properly embedded) is smooth, it is still coisotropic. Moreover, when contactomorphisms in the sequence are in the identity component and the image of a Legendrian is smooth, the Maslov data is preserved, and the category of sheaves with singular support on the Legendrian and the microstalk corepresentative are also preserved (and thus so is the wrapped Floer cochains of the linking disks). The main ingredients are the result of Guillermou--Viterbo, a new sheaf quantization result for $C^0$-small contactomorphisms (not necessarily in the identity component) different from Guillermou--Kashiwara--Schapira, and continuity of the interleaving distance of sheaves with respect to the Hofer--Shelukhin distance and the $C^0$-distance. The appendix contains different arguments for local $C^0$-limits and certain Hausdorff limits of Legendrians without appealing to the interleaving distance.

DNA strings and their properties are widely studied since last 20 years due to its applications in DNA computing. In this area, one designs a set of DNA strings (called DNA code) which satisfies certain thermodynamic and combinatorial constraints such as reverse constraint, reverse-complement constraint, $GC$-content constraint and Hamming constraint. However recent applications of DNA codes in DNA data storage resulted in many new constraints on DNA codes such as avoiding tandem repeats constraint (a generalization of non-homopolymer constraint) and avoiding secondary structures constraint. Therefore, in this chapter, we introduce DNA codes with recently developed constraints. In particular, we discuss reverse, reverse-complement, $GC$-content, Hamming, uncorrelated-correlated, thermodynamic, avoiding tandem repeats and avoiding secondary structures constraints. DNA codes are constructed using various approaches such as algebraic, computational, and combinatorial. In particular, in algebraic approaches, one uses a finite ring and a map to construct a DNA code. Most of such approaches does not yield DNA codes with high Hamming distance. In this chapter, we focus on algebraic constructions using maps (usually an isometry on some finite ring) which yields DNA codes with high Hamming distance. We focus on non-cyclic DNA codes. We briefly discuss various metrics such as Gau distance, Non-Homopolymer distance etc. We discuss about algebraic constructions of families of DNA codes that satisfy multiple constraints and/or properties. Further, we also discuss about algebraic bounds on DNA codes with multiple constraints. Finally, we present some open research directions in this area.

We extend our previous work on odd spoof multiperfect numbers to the case where spoof factor multiplicities exceed $2$. This leads to the identification of $11$ new integers that would be odd multiperfect numbers if one of their prime factors had higher multiplicity. An example is $181545$, which would be an odd multiperfect number if only one of its prime factors, $3$, had multiplicity $5$.

We study a regularized variant of the Bayesian Persuasion problem, where the receiver's decision process includes a divergence-based penalty that accounts for deviations from perfect rationality. This modification smooths the underlying optimization landscape and mitigates key theoretical issues, such as measurability and ill-posedness, commonly encountered in the classical formulation. It also enables the use of scalable second-order optimization methods to compute numerically the optimal signaling scheme in a setting known to be NP-hard. We present theoretical results comparing the regularized and original models, including convergence guarantees and structural properties of optimal signaling schemes. Analytical examples and numerical simulations illustrate how this framework accommodates complex environments while remaining tractable and robust. A companion Python library, BASIL, makes use of all the practical insights from this article.

These are the notes of a part of the PhD course Regularity for free boundary problems and for elliptic PDEs, held in Pavia in the spring of 2025. The aim is to provide a comprehensive and self-contained treatment of classical interior and local Schauder estimates for second-order linear elliptic PDEs in divergence form via scaling in the spirit of Simon's work. The main techniques presented here are geometric in nature and were primarily developed in the study of geometric problems such as minimal surfaces. The adopted approach relies on compactness and blow-up arguments, combined with rigidity results (Liouville theorems), and shares many features with the one used in the study of free boundary problems, which was the main topic of the other part of the PhD course.

We analyze the one-dimensional semi-classical Schrödinger equation on the half-line with a linear potential and Dirichlet boundary conditions. Our main focus is on establishing improved dispersive and Strichartz estimates for this model, which govern the space-time behavior of solutions. We prove refined Strichartz bounds using Van der Corput-type derivative tests, beating previous known results where Strichartz estimates incur 1/4 losses. Moreover, assuming sharp bounds for certain exponential sums, our results indicate the possibility to reduce these losses further to $1/6 + \epsilon$ for all $\epsilon>0$, which would be sharp. We further expect that analogous Strichartz bounds should hold within the Friedlander model domain in higher dimensions.

Let $1<a<b$ be two relatively prime integers and $\mathbb{Z}_{\ge 0}$ the set of non-negative integers. For any non-negative integer $\ell$, denote by $g_{\ell,a,b}$ the largest integer $n$ such that the equation $$n=ax+by,\quad (x,y)\in\mathbb{Z}_{\ge 0}^{2} \quad (1)$$ has at most $\ell$ solutions. Let $\pi_{\ell,a,b}$ be the number of primes $p\leq g_{\ell,a,b}$ having at least $\ell+1$ solutions for (1) and $\pi(x)$ the number of primes not exceeding $x$. In this article, we prove that for a fixed integer $a\ge 3$ with $\gcd(a,b)=1$, $$ \pi_{\ell,a,b}=\left(\frac{a-2}{2(\ell a+a-1)}+o(1)\right)\pi\bigl(g_{\ell,a,b}\bigr)\quad(\text{as}~ b\to\infty). $$ For any non-negative $\ell$ and relatively prime integers $a,b$, satisfying $e^{\ell+1}\leq a<b$, we show that \begin{equation*} \pi_{\ell,a,b}>0.005\cdot \frac{1}{\ell+1}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. \end{equation*} Let $\pi_{\ell,a,b}^{*}$ be the number of primes $p\leq g_{\ell,a,b}$ having at most $\ell$ solutions for (1). For an integer $a\ge 3$ and a large sufficiently integer $b$ with $\gcd(a,b)=1$, we also prove that $$ \pi^{*}_{\ell,a,b}>\frac{(2\ell+1)a}{2(\ell a+a-1)}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. $$ Moreover, if $\ell<a<b$ with $\gcd(a,b)=1$, then we have \begin{equation*} \pi^{*}_{\ell,a,b}>\frac{\ell+0.02}{\ell+1}\frac{g_{\ell,a,b}}{\log g_{\ell,a,b}}. \end{equation*} These results generalize the previous ones of Chen and Zhu (2025), who established the results for the case $\ell=0$.

This paper explores an efficient Lagrangian approach for evolving point cloud data on smooth manifolds. In this preliminary study, we focus on analyzing plane curves, and our ultimate goal is to provide an alternative to the conventional radial basis function (RBF) approach for manifolds in higher dimensions. In particular, we use the B-Spline as the basis function for all local interpolations. Just like RBF and other smooth basis functions, B-Splines enable the approximation of geometric features such as normal vectors and curvature. Once properly set up, the advantage of using B-Splines is that their coefficients carry geometric meanings. This allows the coefficients to be manipulated like points, facilitates rapid updates of the interpolant, and eliminates the need for frequent re-interpolation. Consequently, the removal and insertion of point cloud data become seamless processes, particularly advantageous in regions experiencing significant fluctuations in point density. The numerical results demonstrate the convergence of geometric quantities and the effectiveness of our approach. Finally, we show simulations of curvature flows whose speeds depend on the solutions of coupled reaction--diffusion systems for pattern formation.

For a given integer $k$, let $\ell_k$ denote the supremum $\ell$ such that every sufficiently large graph $G$ with average degree less than $2\ell$ admits a separator $X \subseteq V(G)$ for which $\chi(G[X]) < k$. Motivated by the values of $\ell_1$, $\ell_2$ and $\ell_3$, a natural conjecture suggests that $\ell_k = k$ for all $k$. We prove that this conjecture fails dramatically: asymptotically, the trivial lower bound $\ell_k \geq \tfrac{k}{2}$ is tight. More precisely, we prove that for every $\varepsilon>0$ and all sufficiently large $k$, we have $\ell_k \leq (1+\varepsilon)\tfrac{k}{2}$.

State-of-the-art approaches of Robust Model Predictive Control (MPC) are restricted to linear systems of relatively small scale, i.e., with no more than about 5 states. The main reason is the computational burden of determining a robust positively invariant (RPI) set, whose complexity suffers from the curse of dimensionality. The recently proposed approach of Deadbeat Robust Model Predictive Control (DRMPC) is the first that does not rely on an RPI set. Yet it comes with the full set of essential system theoretic guarantees. DRMPC is hence a viable option, in particular, for large-scale systems. This paper introduces a detailed design procedure for DRMPC. It is shown that the optimal control problem generated for DRMPC has exactly the same computational complexity as Nominal MPC. A numerical study validates its applicability to randomly generated large-scale linear systems of various dimensions.

We define a class of nonsingular holomorphic foliations on compact complex tori which generalizes (in higher codimension) the turbulent foliations of codimension one constructed by Ghys. For those smooth turbulent foliations we prove that all transversely holomorphic Cartan geometries are flat. We also establish a uniqueness result for the transversely holomorphic Cartan geometries.

The topic of this survey is the phenomenon of fibring over the circle for manifolds, and its group-theoretic twin, algebraic fibring. We will discuss the state of the art, and explain briefly some of the ideas behind the more recent developments, focusing on RFRS groups and manifolds with such fundamental groups. Then we will move on to a more speculative part, where many conjectures about fibring in higher dimensions will be given. The conjectures vary in their level of plausibility, but even the boldest of them might share the fate of Thurston's Virtually Fibred Conjecture, about which Thurston famously said: ``This dubious-sounding question seems to have a definite chance for a positive answer''.

We consider a degenerate system of three Brownian particles undergoing asymmetric collisions. We study the gap process of this system and focus on its invariant measure. The gap process is described as an obliquely reflected degenerate Brownian motion in a quadrant. For all possible parameter cases, we compute the Laplace transform of the invariant measure, and fully characterize the conditions under which it belongs to the following classes: rational, algebraic, differentially finite, or differentially algebraic. We also derive explicit formulas for the invariant measure on the boundary of the quadrant, expressed in terms of a Theta-like function, to which we apply a polynomial differential operator. In this study, we introduce a new parameter called $\gamma$ (along with two additional parameters $\gamma_1$ and $\gamma_2$) which governs many properties of the degenerate process. This parameter is reminiscent of the famous parameter $\alpha$ introduced by Varadhan and Williams (and the two parameters $\alpha_1$ and $\alpha_2$ recently introduced by Bousquet-M{é}lou et al.) to study nondegenerate reflected Brownian motion in a wedge. To establish our main results we start from a kernel functional equation characterizing the Laplace transform of the invariant measure. By an analytic approach, we establish a finite difference equation satisfied by the Laplace transform. Then, using certain so-called decoupling functions, we apply Tutte's invariant approach to solve the equation via conformal gluing functions. Finally, difference Galois theory and exhaustive study allows us to find necessary and sufficient conditions for the Laplace transform to belong to the specified function hierarchy.

The theory of representations of a crossed module is a direct generalization of the theory of representations of groups. For a finite group G, the Drinfeld quantum double of the group G is a Hopf algebra that represents a special case of crossed module of finite groups. Here we study how to extend the construction of the Drinfeld quantum double for any other kind of crossed module of finite groups. This leads to a Hopf algebra D(G, H) that presents similarities with a Drinfeld double. We then study simple subalgebras of D(G, H) and give two isomorphisms for the decomposition into a product of simple subalgebras. We then study the category D(G, H)-modFd of finite dimensional modules over D(G, H), which turns out to be isomorphic to the category of finite dimensional representations of finite crossed modules of groups. These categories being monoidal, we also study links between direct sums of simple objects and tensor products of simple objects and give some results for a Clebsch-Gordan formula. We, in this context, present and develop the character theory for representations of crossed modules of finite groups, and detail the proofs. We then study the category itself, which leads to some ribbon invariants.

Polynomial evaluation codes hold a prominent place in coding theory. In this work, we study the problem of list decoding for a general class of polynomial evaluation codes, also known as Toric codes, that are defined for any given convex polytope P. Special cases, such as Reed-Solomon and Reed-Muller codes, have been studied extensively. We present a generalization of the Guruswami-Sudan algorithm that takes into account the geometry and the combinatorics of P and compute bounds for the decoding radius.

Advances in parallel hardware platforms have motivated the development of efficient universal decoders capable of meeting stringent throughput and latency requirements. Guessing Random Additive Noise Decoding (GRAND) is a recently proposed decoding paradigm that sequentially tests Error Patterns (EPs) until finding a valid codeword. While Soft GRAND (SGRAND) achieves maximum-likelihood (ML) decoding, its inherently sequential nature hinders parallelism and results in high decoding latency. In this work, we utilize a unified binary tree representation of EPs, termed the EP tree, which enables compact representation, efficient manipulation, and parallel exploration. Building upon this EP tree representation, we propose a parallel design of SGRAND, preserving its ML optimality while significantly reducing decoding latency through pruning strategies and tree-based computation. Furthermore, we develop a hybrid GRAND algorithm that enhances Ordered Reliability Bits (ORB) GRAND with the EP tree representation, thereby achieving ML decoding with minimal additional computational cost beyond ORBGRAND while retaining parallel efficiency. Numerical experiments demonstrate that parallel SGRAND achieves a $3.75\times$ acceleration compared to serial implementation, while the hybrid enhanced method achieves a $4.8\times$ acceleration, with further gains expected under hardware mapping.

This paper presents the construction of two numerical schemes for the solution of hyperbolic systems with relaxation source terms. The methods are built by considering the relaxation system as a whole, without separating the resolution of the convective part from that of the source term. The first scheme combines the centered FORCE approach of Toro and co-authors with the unsplit strategy proposed by B{é}reux and Sainsaulieu. The second scheme consists of an approximate Riemann solver which carefully handles the source term approximation. The two schemes are built to be asymptotic preserving, in the sense that their limit schemes are consistent with the equilibrium model as the relaxation parameter tends to zero, without any CFL restriction. For specific models, it is possible to prove that they preserve invariant domains and admit a discrete entropy inequality.

In this paper, we establish estimates for the expectation and variance of the mixed $(2,2)$-moment of two Hecke eigenforms of distinct weights. Our results yield applications to triple product $L$-functions. The proofs are based on moments of $L$-functions.

Let X be a planar random field on Z^2 which we interpret as a random height function describing some landscape of montains. We consider a source of light (a sun) located at infinity in a direction parallel with an axis od Z^2 and emitting rays which are all parallel and make a slope l with the horizontal plane. Given the value of l some montains of the landscape will be lit by the sun and other will be in the shadow of some higher mountain. Under some assumptions on X, including and independence assumption, we prove that this model may present two different phases depending on l. When l>0 is small enough then, almost surely, there exists an unbounded cluster of points in the shadow. However, if l is big enough then, almost surely, there exists an unbounded cluster of points lit by the sun. We reformulate this problem in terms of percolation of a field alpha which has a simple definition (in terms of X) but that does not present many of the nice properties usually found in percolation models such as FKG inequality, invariance by rotation or finite range correlations.

Affine processes play an important role in mathematical finance and other applied areas due to their tractable structure. In the present article, we derive probabilistic representations and integration by parts (IBP) formulas for expectations involving affine processes. These formulas are expressed in terms of expectations of affine processes with modified parameters and are derived using Fourier analytic techniques and characteristic functions. Notably, our method does not require pathwise differentiability, allowing us to handle models with square-root diffusion coefficients for a large set of parameters. The methodology can be applied to the classic Cox--Ingersoll--Ross (CIR) model, a model for interest rates in mathematical finance, where the initial value derivative corresponds to one of the ``Greeks'' used in option pricing in mathematical finance. Furthermore, we illustrate the theory with an application to a population evolution model arising as a scaling limit of discrete branching processes. Our approach offers a unified and robust framework for sensitivity analysis in models where classical Malliavin calculus techniques are difficult to apply.

We revisit the classic 'guess my number' game and extend it from its familiar binary form to representations in any integer base. For each base we derive formulas for the number of cards needed to identify a given integer and, conversely, for the largest integer that can be determined when the number of cards is fixed. Both analysis and graphical evidence show that base 2 is optimal in both directions: it requires the fewest cards to represent any specified integer and, for a fixed card count, allows the widest range of integers to be guessed. Figures illustrate these results, and complete proofs appear in the Appendix.

We analyse the Maxwell's spectrum on thin tubular neighborhoods of embedded surfaces of $\mathbb R^3$. We show that the Maxwell eigenvalues converge to the Laplacian eigenvalues of the surface as the thin parameter tends to zero. To achieve this, we reformulate the problem in terms of the spectrum of the Hodge Laplacian with relative conditions acting on co-closed differential $1$-forms. The result leads to new examples of domains where the Faber-Krahn inequality for Maxwell's eigenvalues fails, examples of domains with any number of arbitrarily small eigenvalues, and underlines the failure of spectral stability under singular perturbations changing the topology of the domain. Additionally, we explicitly produce the Maxwell's eigenfunctions on product domains with the product metric, extending previous constructions valid in the Euclidean case.

We consider partially ordered sets of combinatorial structures under consecutive orders, meaning that two structures are related when one embeds in the other such that `consecutive' elements remain consecutive in the image. Given such a partially ordered set, we may ask decidability questions about its avoidance sets: subsets defined by a finite number of forbidden substructures. Two such questions ask, given a finite set of structures, whether its avoidance set is well quasi-ordered (i.e. contains no infinite antichains) or atomic (i.e. cannot be expressed as the union of two proper subsets). Extending some recent new approaches, we will establish a general framework, which enables us to answer these problems for a wide class of combinatorial structures, including graphs, digraphs and collections of relations.

We consider classical solutions to $-\Delta u = f(u)$ in half-spaces, under homogeneous Dirichlet boundary conditions. We prove that any positive solution is strictly monotone increasing in the direction orthogonal to the boundary, provided that it is directionally bounded on finite strips. As a corollary, we deduce a new Liouville-type theorem for the Lane-Emden equation.

The dissipation phenomena of relative entropy from an Itô--Langevin dynamical system is a classic topic from stochastic analysis. Relying on the time-reversal of diffusions, a novel trajectorial approach investigates the pathwise behavior of relevant entropy processes, reveals more information from the delicate random structure, and eventually retrieves the known classical results. In essence, this approach provides novel insights and rederives the known results of the Itô--Langevin dynamics, as will be presented in this expository article. Another part is to view the stochastic time-evolution through the lens of the Wasserstein space, under which we observe the geometric feature of steepest descent of the entropy decay as well as its exponential rate of velocity.

This paper addresses the Asplund property for the space of continuous functions $C_k(X)$ equipped with the compact-open topology, when $X$ is an arbitrary Tychonoff space. Motivated by inconsistent definitions in prior literature extending the Asplund property beyond Banach spaces, we provide a unified and self-contained treatment of core results in this context. A characterization of the Asplund property for $C_k(X)$ is established, alongside a review of classical results, including the Namioka--Phelps theorem and its implications. All proofs are presented in a self-contained manner and rely on standard techniques.

We find surface subgroups in certain one-relator groups with torsion and use this to deduce a profinite criterion for a word in the free group to be primitive.

Let $A_1$ and $A_2$ be polynomials of degree at least two over $\mathbb C$. We say that $A_1$ and $A_2$ are intertwined if the endomorphism $(A_1, A_2)$ of $\mathbb C\mathbb P^1 \times \mathbb C\mathbb P^1$ given by $(z_1, z_2) \mapsto (A_1(z_1), A_2(z_2))$ admits an irreducible periodic curve that is neither a vertical nor a horizontal line. We denote by $\mathrm{Inter}(A)$ the set of all polynomials $B$ such that some iterate of $B$ is intertwined with some iterate of $A$. In this paper, we prove a conjecture of Favre and Gauthier describing the structure of $\mathrm{Inter}(A)$. We also obtain a bound on the possible periods of periodic curves for endomorphisms $(A_1, A_2)$ in terms of the sizes of the symmetry groups of the Julia sets of $A_1$ and $A_2$.

In 1963, Dirac proved that every $n$-vertex graph has $k$ vertex-disjoint triangles if $n\geq 3k$ and minimum degree $\delta(G)\geq \frac{n+k}{2}$. The base case $n=3k$ can be reduced to the Corrádi-Hajnál Theorem. Towards a rainbow version of Dirac's Theorem, Hu, Li, and Yang conjectured that for all positive integers $n$ and $k$ with $n\geq 3k$, every edge-colored graph $G$ of order $n$ with $\delta^c(G)\geq \frac{n+k}{2}$ contains $k$ vertex-disjoint rainbow triangles. In another direction, Wu et al. conjectured an exact formula for anti-Ramsey number $ar(n,kC_3)$, generalizing the earlier work of Erdős, Sós and Simonovits. The conjecture of Hu, Li, and Yang was confirmed for the cases $k=1$ and $k=2$. However, Lo and Williams disproved the conjecture when $n\leq \frac{17k}{5}.$ It is therefore natural to ask whether the conjecture holds for $n=\Omega(k)$. In this paper, we confirm this by showing that the Hu-Li-Yang conjecture holds when $n\ge 42.5k+48$. We disprove the conjecture of Wu et al. and propose a modified conjecture. This conjecture is motivated by previous works due to Allen, Böttcher, Hladký, and Piguet on Turán number of vertex-disjoint triangles.

Supervaluational fixed-point semantics for truth cannot be axiomatized because of its recursion-theoretic complexity. Johannes Stern (\emph{Supervaluation-Style Truth Without Supervaluations}, Journal of Philosophical Logic, 2018) proposed a new strategy (supervaluational-style truth) to capture the essential aspects of the supervaluational evaluation schema whilst limiting its recursion-theoretic complexity, hence resulting in ($\nat$-categorical) axiomatizations. Unfortunately, as we show in the paper, this strategy was not fully realized in Stern's original work: in fact, we provide counterexamples to some of Stern's key claims. However, we also vindicate Stern's project by providing different semantic incarnations of the idea and corresponding $\nat$-categorical axiomatizations. The results provide a deeper picture of the relationships between standard supervaluationism and supervaluational-style truth.

We prove the sharp Strichartz estimate for hyperbolic Schrödinger equation on $\mathbb{T}^3 $ via an incidence geometry approach. As application, we obtain optimal local well-posedness of nonlinear hyperbolic Schrödinger equations.

We investigate the $\limsup$ inequality in the double gradient model for phase transitions governed by a Modica--Mortola functional with a double-well potential in two dimensions. Specifically, we consider energy functionals of the form \[ E_\varepsilon(u, \Omega) = \int_\Omega \left( \frac{1}{\varepsilon} W(\nabla u) + \varepsilon |\nabla^2 u|^2 \right) dx \] for maps $ u \in H^2(\Omega; \mathbb{R}^2) $, where $ W $ vanishes only at two wells. Assuming a bound on the optimal profile constant -- namely the cell problem on the unit cube -- in terms of the geodesic distance between the two wells, we characterise the limiting interfacial energy via periodic recovery sequences as $\varepsilon \to 0^+$.

Using Hilbert schemes of points, we establish a number of results for a smooth projective variety $X$ in a sufficiently ample embedding. If $X$ is a curve or a surface, we show that the ideals of higher secant varieties are determinantally presented, and we prove the same for the first secant variety if $X$ has arbitrary dimension. This completely settles a conjecture of Eisenbud-Koh-Stillman for curves and partially resolves a conjecture of Sidman-Smith in higher dimensions. If $X$ is a curve or a surface we also prove that the corresponding embedding of the Hilbert scheme of points $X^{[d]}$ into the Grassmannian is projectively normal. Finally, if $X$ is an arbitrary projective scheme in a sufficiently ample embedding, then we demonstrate that its homogeneous ideal is generated by quadrics of rank three, confirming a conjecture of Han-Lee-Moon-Park. Along the way, we check that the Hilbert scheme of three points on a smooth variety is the blow-up of the symmetric product along the big diagonal.

In 1982, Schlickewei and Van der Poorten claimed that any multi-recurrence sequence has, essentially, maximal possible growth rate. Fourty years later, Fuchs and Heintze provided a non-effective proof of this statement. In this paper, we prove a quantitative version of that result by giving an explicit upper bound for the maximal possible growth rate of a multi-recurrence. Moreover, we also give a function field analogue of the result, answering a question posed by Fuchs and Heintze when proving a bound on the growth of multi-recurrences in number fields.

An odd independent set $S$ in a graph $G=(V,E)$ is an independent set of vertices such that, for every vertex $v \in V \setminus S$, either $N(v) \cap S = \emptyset$ or $|N(v) \cap S| \equiv 1$ (mod 2), where $N(v)$ stands for the open neighborhood of $v$. The largest cardinality of odd independent sets of a graph $G$, denoted $\alpha_{od}(G)$, is called the odd independence number of $G$. This new parameter is a natural companion to the recently introduced strong odd chromatic number. A proper vertex coloring of a graph $G$ is a strong odd coloring if, for every vertex $v \in V(G)$, each color used in the neighborhood of $v$ appears an odd number of times in $N(v)$. The minimum number of colors in a strong odd coloring of $G$ is denoted by $\chi_{so}(G)$. A simple relation involving these two parameters and the order $|G|$ of $G$ is $\alpha_{od}(G)\cdot\chi_{so}(G) \geq |G|$, parallel to the same on chromatic number and independence number. In the present work, which is a companion to our first paper on the subject [The odd independence number of graphs, I: Foundations and classical classes], we focus on grid-like and chessboard-like graphs and compute or estimate their odd independence number and their strong odd chromatic number. Among the many results obtained, the following give the flavour of this paper: (1) $0.375 \leq \varrho_{od}(P_\infty \Box P_\infty) \leq 0.384615...$, where $\varrho_{od}(P_\infty \Box P_\infty)$ is the odd independence ratio. (2) $\chi_{so}(G_d) = 3$ for all $d \geq 1$, where $G_d$ is the infinite $d$-dimensional grid. As a consequence, $\varrho_{od}(G_d) \geq 1/3$. (3) The $r$-King graph $G$ on $n^2$ vertices has $\alpha_{od}(G) = \lceil n/(2r+1) \rceil^2$. Moreover, $\chi_{so}(G) = (2r + 1)^2$ if $n \geq 2r + 1$, and $\chi_{so}(G) = n^2$ if $n \leq 2r$. Many open problems are given for future research.

We give a probabilistic representation for the gradient of a 2nd order linear parabolic PDE $\partial_{t}u(t,x)=(1/2)a^{ij}\partial_{ij}u(t,x)+b^{i}\partial_{i}u(t,x)$ with Cauchy initial condition $u(0,x)=f(x)$ and Neumann boundary condition in a (closed) convex bounded smooth domain $D$ in $\mathbb{R}^{d}$, $d\geq 1$. The idea is to start from a penalized version of the associated reflecting diffusion $X^{x}$, proceed with a pathwise derivative, show that the resulting family of $\nu$-directional Jacobians is tight in the Jakubowski S-topology with limit $J^{x,\nu}$, solution of a certain linear SDE, and set $\mathbb{E}\left(\nabla f(X^{x}(t))\cdot J^{x,e_{i}}(t)\right)$ for the gradient $\partial_{i}u(t,x)$, where $x\in D$, $t\geq 0$, $e_{i}$ the canonical basis of $\mathbb{R}^{d}$ and $f$, the initial condition of the semigroup of $X^{x}$, is differentiable. Some more extensions and applications are discussed in the concluding remarks.

We prove an existence theorem for the sliding boundary variant of the Plateau problem for $2$-dimensional sets in $\mathbb{R}^n$. The simplest case of sufficient condition is when $n=3$ and the boundary $\Gamma$ is a finite disjoint union of smooth closed curves contained in the boundary of a convex body, but the main point of our sufficient condition is to prevent the limits in measure of a minimizing sequence to have singularities of type $\mathbb{Y}$ along $\Gamma$.

In the present paper we construct quadratic equations and linear syzygies for tangent varieties using 4-way tensors of linear forms and generalize this method to higher secant varieties of higher osculating varieties. Such equations extend the classical determinantal ones of higher secant varieties and span all the equations of the same degree for smooth projective curves completely embedded by sufficiently positive line bundles, proving a variant of the Eisenbud-Koh-Stillman conjecture on determinantal equations. On the other hand, our syzygies are compatible with the Green-Lazarsfeld classes and generate the corresponding Koszul cohomology groups for Segre varieties with a prescribed number of factors. To obtain these results we describe the equations of minimal possible degrees and reinterpret the Green-Lazarsfeld classes from the perspective of representation theory.

This paper employs layer potential techniques to investigate wave scattering in two-dimensional elastic media exhibiting high contrasts in both Lamé parameters and density. Our contributions are fourfold. First, we construct an invertible operator based on the kernel spaces of boundary integral operators, which enables the characterization of resonant frequencies through an orthogonality condition. Second, we use asymptotic analysis to derive the equation governing the leading-order terms of these resonant frequencies. Third, we analyze the scattered field in the interior domain for incident frequencies across different regimes and characterize the longitudinal and transverse far-field patterns in the exterior domain. Finally, we examine the subwavelength bandgap in the phononic crystal with a dilute structure.

This paper deals with the Sturm-Liouville operators with distribution potentials of the space $W_2^{-1}$ on a metric tree. We study an inverse spectral problem that consists in the recovery of the potentials from the characteristic functions related to various boundary conditions. We prove the uniform stability of this inverse problem for potentials in a ball of any fixed radius, as well as the local stability under small perturbations of the spectral data. Our approach is based on a stable algorithm for the unique reconstruction of the potentials relying on the ideas of the method of spectral mappings.

This paper introduces a geometric representation of hypergraphs by representing hyperedges as simplices. Building on this framework, we employ homotopy groups to analyze the topological structure of hypergraphs embedded in high-dimensional Euclidean spaces. Leveraging this foundation, we extend Wagner's theorem to $\mathbb{R}^d$. Specifically, we establish that a triangulated $d$-uniform topological hypergraph embeds into $\mathbb{R}^d$ if and only if it contains neither $K_{d+3}^d$ nor $K_{3,d+1}^d$ as a minor. Here, a triangulated $d$-uniform topological hypergraph constitutes a geometrized form of a $d$-uniform hypergraph, while $K_{d+3}^d$ and $K_{3,d+1}^d$ are the high-dimensional generalizations of the complete graph $K_5$ and the complete bipartite graph $K_{3,3}$ in $\mathbb{R}^d$, respectively.

The central path revolutionized the study of optimization in the 1980s and 1990s due to its favorable convergence properties, and as such, it has been investigated analytically, algorithmically, and computationally. Past pursuits have primarily focused on linking iterative approximation algorithms to the central path in the design of efficient algorithms to solve large, and sometimes novel, optimization problems. This algorithmic intent has meant that the central path has rarely been celebrated as an aesthetic entity in low dimensions, with the only meager exceptions being illustrative examples in textbooks. We undertake this low dimensional investigation and illustrate the artistic use of the central path to create aesthetic tilings and flower-like constructs in two and three dimensions, an endeavor that combines mathematical rigor and artistic sensibilities. The result is a fanciful and enticing collection of patterns that, beyond computer generated images, supports math-aesthetic designs for novelties and museum-quality pieces of art.

We study regularity of the centered Hardy--Littlewood maximal function $M f$ of a function $f$ of bounded variation in $\mathbb R^d$, $d\in \mathbb N$. In particular, we show that at $|D^c f|$-a.e. point $x$ where $f$ has a non-concave blow-up, it holds that $M f(x)>f^*(x)$. We further deduce from this that if the variation measure of $f$ has no jump part and its Cantor part has non-concave blow-ups, then BV regularity of $M f$ can be upgraded to Sobolev regularity.

Given a known function $f : [0, 1] \mapsto (0, 1)$ and a random but almost surely finite number of independent, Ber$(x)$-distributed random variables with unknown $x \in [0, 1]$, we construct an unbiased, $[0, 1]$-valued estimator of the probability $f(x) \in (0, 1)$. Our estimator is based on so-called debiasing, or randomly truncating a telescopic series of consistent estimators. Constructing these consistent estimators from the coefficients of a particular Bernoulli factory for $f$ yields provable upper and lower bounds for our unbiased estimator. Our result can be thought of as a novel Bernoulli factory with the appealing property that the required number of Ber$(x)$-distributed random variates is independent of their outcomes, and also as constructive example of the so-called $f$-factory.

We establish fractional Leibniz rules in weighted settings for nonnegative self-adjoint operators on spaces of homogeneous type. Using a unified method that avoids Fourier transforms, we prove bilinear estimates for spectral multiplier on weighted Hardy, Besov and Triebel-Lizorkin spaces. Our approach is flexible and applies beyond the Euclidean setting-covering, for instance, nilpotent Lie groups, Grushin operators, and Hermite expansions-thus extending classical Kato-Ponce inequalities. The framework also yields new weighted bilinear estimates including fractional Leibniz rules for Hermite, Laguerre, and Bessel operator, with applications to scattering formulas and related PDE models.

Quasar-convex functions form a broad nonconvex class with applications to linear dynamical systems, generalized linear models, and Riemannian optimization, among others. Current nearly optimal algorithms work only in affine spaces due to the loss of one degree of freedom when working with general convex constraints. Obtaining an accelerated algorithm that makes nearly optimal $\widetilde{O}(1/(\gamma\sqrt{\epsilon}))$ first-order queries to a $\gamma$-quasar convex smooth function \emph{with constraints} was independently asked as an open problem in Martínez-Rubio (2022); Lezane, Langer, and Koolen (2024). In this work, we solve this question by designing an inexact accelerated proximal point algorithm that we implement using a first-order method achieving the aforementioned rate and, as a consequence, we improve the complexity of the accelerated geodesically Riemannian optimization solution in Martínez-Rubio (2022). We also analyze projected gradient descent and Frank-Wolfe algorithms in this constrained quasar-convex setting. To the best of our knowledge, our work provides the first analyses of first-order methods for quasar-convex smooth functions with general convex constraints.

We give a characterization of colored Petri nets as lax double functors. Framing colored Petri nets in terms of category theory allows for canonical definitions of various well-known constructions on colored Petri nets. In particular, we show how morphisms of colored Petri nets may be understood as natural transformations. A result from folklore, which we sketch in the appendix, shows that lax double functors are equivalent to functors with codomain their former domain. We use this result to characterize the unfolding of colored Petri nets in terms of free symmetric monoidal categories.

We give the first examples of étale (non-Hausdorff) groupoids $\mathcal G$ whose $C^*$-algebras contain singular elements that cannot be approximated by singular elements in $\mathcal C_c(\mathcal G)$. We provide two examples: one is a bundle of groups, and the other a minimal and effective groupoid constructed from a self-similar action on an infinite alphabet. Moreover, we also prove that the Baum--Connes assembly map for the first example is not surjective, not even on the level of its essential $C^*$-algebra.

In 1963, Anton Kotzig famously conjectured that $K_{n}$, the complete graph of order $n$, where $n$ is even, can be decomposed into $n-1$ perfect matchings such that every pair of these matchings forms a Hamilton cycle. The problem is still wide open and here we consider a variant of it for the binomial random graph $G(n,p)$. We prove that, for every fixed $k$, there exists a constant $C=C(k)$ such that, when $p\ge \frac{C \log n}{n}$, with high probability, $G(n,p)$ contains $k$ edge-disjoint perfect matchings with the property that every pair of them forms a Hamilton cycle. In fact, our main result is a very precise counting result for $K_n$. We show that, given any $k$ edge-disjoint perfect matchings $M_1,\dots,M_k$, the probability that a uniformly random perfect matching $M^*$ in $K_n$ has the property that $M^*\cup M_i$ forms a Hamilton cycle for each $i\in [k]$ is $\Theta_k(n^{-k/2})$. This is proved by building on a variety of methods, including a random process analysis, the absorption method, the entropy method and the switching method. The result on the binomial random graph follows from a slight strengthening of our counting result via the recent breakthroughs on the expectation threshold conjecture.

The renormalization group (RG) method is an important tool for studying critical phenomena. In this paper, we employ stochastic analysis techniques to investigate the stochastic partial differential equation (SPDE) derived by regularizing and continuousizing the discrete stochastic equation, which is a variant of stochastic quantization equation of the one dimensional (1D) Ising model. Firstly, we give the regularity estimates for the solution to SPDE. Secondly, we prove the Clark-Ocone-Haussmann formula and derive the log-Sobolev inequality up to the terminal time $T$, as well as obtain a priori form of the renormalization relation. Finally, we verify the correctness of the renormalization procedure based on the partition function, and prove that the two point correlation functions of SPDE on lattices converge to the two point correlation functions of the 1D Ising model at the stable fixed point of the RG transformation as $T\rightarrow +\infty$.

Given a finitely generated linear group $G$ over $\mathbb{Q}$, we construct a simple group $\Gamma$ that has the same finiteness properties as $G$ and admits $G$ as a quasi-retract. As an application, we construct a simple group of type $\mathrm{FP}_{\infty}$ that is not finitely presented. Moreover we show that for every $n \in \mathbb{N}$ there is a simple group of type $\mathrm{FP}_n$ that is neither finitely presented nor of type $\mathrm{FP}_{n+1}$. Since our simple groups arise as Röver--Nekrashevych groups, this answers a question of Zaremsky.

We study the stability of various restricted center properties and certain continuity properties of the restricted center map. We observe that restricted center property, property-$(P_1)$ and semi-continuity properties of the restricted center map are preserved under $\ell_p$-direct sum $(1 \leq p < \infty).$ It is shown that property-$(P_2)$ is stable under finite $\ell_p$-direct sum, but not under infinite $\ell_p$-direct sum. Additionally, we introduce a notion called property-$(lP_2)$ as a sufficient condition for the continuity of the restricted center map. Further, the stability of property-$(lP_2)$ is established.

Numerical tests of volume formulae are presented to compute efficiently the volume enclosed between flux surfaces for integrable 3D vector fields with various degrees of symmetry. In the process, a new case is proposed and tested.

Current early warning signs for tipping points often fail to distinguish between catastrophic shifts and less dramatic state changes, such as spatial pattern formation. This paper introduces a novel method that addresses this limitation by providing more information about the type of bifurcation being approached starting from a spatially homogeneous system state. This method relies on estimates of the dispersion relation from noisy spatio-temporal data, which reveals whether the system is approaching a spatially homogeneous (tipping) or spatially heterogeneous (Turing patterning) bifurcation. Using a modified Klausmeier model, we validate this method on synthetic data, exploring its performance under varying conditions including noise properties and distance to bifurcation. We also determine the data requirements for optimal performance. Our results indicate the promise of a new spatial early warning system built on this method to improve predictions of future transitions in many climate subsystems and ecosystems, which is critical for effective conservation and management in a rapidly changing world.

We discuss the holonomic dual of tautological systems, with a view towards applications to linear free divisors and to homogeneous spaces. As a technical tool, we consider a Chevalley--Eilenberg type complex, generalizing Euler--Koszul technology from the GKZ theory, and show equivariance and holonomicity of it.

Consider a critical Galton--Watson branching process with immigration, where the offspring distribution belongs to the domain of attraction of a $(1 + \alpha)$-stable law with $\alpha \in (0,1)$, and the immigration distribution either (i) has finite mean, or (ii) belongs to the domain of attraction of a $\beta$-stable law with $\beta \in (\alpha, 1)$. We show that the tail of the stationary distribution is regularly varying. We analyze the stationary process, determine its tail process, and establish a stable central limit theorem for the partial sums. The norming sequence is different from the one corresponding to the tail of the stationary law. In particular, the extremal index of the process is $0$.

We prove a few simple cases of a random graph statement that would imply the "second" Kahn--Kalai Conjecture. Even these cases turn out to be reasonably challenging, and it is hoped that the ideas introduced here may lead to further interest in, and further progress on, this natural problem.

In 2023, Gullerud, Johnson, and Mbirika presented results on their study of certain tridiagonal real symmetric matrices. As part of their work, they studied the roots to nonhomogeneous equations related to characteristic polynomials of adjacency matrices for path graphs. They showed that a subset of these polynomials give a Fibonacci number when evaluated at the imaginary unit, leading them to make several intriguing conjectures. In this work, we further explore their conjectures regarding the distribution of roots. We make partial progress towards establishing two conjectures, identify an infinite class of polynomials for which a third is false, and give evidence against a fourth.

We establish a flexible generalization of inductive systems of operator systems, which relaxes the usual transitivity (or coherence) condition to an asymptotic version thereof and allows for systems indexed over arbitrary nets. To illustrate the utility of this generalization, we highlight how such systems arise naturally from completely positive approximations of nuclear operator systems. Going further, we utilize an argument of Ding and Peterson to show that a separable operator system is nuclear if and only if it is an inductive limit of matrix algebras, generalizing a classic Theorem of Lazar and Lindenstrauss to the setting of noncommutative Choquet theory.

BCH codes are important error correction codes, widely utilized due to their robust algebraic structure, multi-error correcting capability, and efficient decoding algorithms. Despite their practical importance and extensive study, their parameters, including dimension, minimum distance and Bose distance, remain largely unknown in general. This paper addresses this challenge by investigating the dimension and Bose distance of BCH codes of length $(q^m - 1)/\lambda$ over the finite field $\mathbb{F}_q$, where $\lambda$ is a positive divisor of $q - 1$. Specifically, for narrow-sense BCH codes of this length with $m \geq 4$, we derive explicit formulas for their dimension for designed distance $2 \leq \delta \leq (q^{\lfloor (2m - 1)/3 \rfloor + 1} - 1)/{\lambda} + 1$. We also provide explicit formulas for their Bose distance in the range $2 \leq \delta \leq (q^{\lfloor (2m - 1)/3 \rfloor + 1} - 1)/{\lambda}$. These ranges for $\delta$ are notably larger than the previously known results for this class of BCH codes. Furthermore, we extend these findings to determine the dimension and Bose distance for certain non-narrow-sense BCH codes of the same length. Applying our results, we identify several BCH codes with good parameters.

This paper investigates the performance of downlink non-orthogonal multiple access (NOMA) communication in unmanned aerial vehicle (UAV) networks enhanced by partitionable reconfigurable intelligent surfaces (RISs). We analyze three types of links between base station (BS) and UAVs: direct, RIS-only indirect, and composite links, under both Line-of-Sight (LoS) and Non-LoS (NLoS) propagation. The RIS-only indirect link and direct link are modeled using double Nakagami-m and Nakagami-m fading, respectively, while the composite link follows a combined fading channel model. Closed-form expressions for the cumulative distribution function (CDF) of the received signal-to-noise ratio (SNR) are derived for all links, enabling tractable outage probability analysis. Then, we formulate a fairness-efficiency bilevel optimization problem to minimize the maximum outage probability among UAVs while minimizing the total number of required RIS reflecting elements. Accordingly, an RIS-assisted UAV Outage Minimization (RUOM) algorithm is proposed, which fairly allocates the NOMA power coefficients while minimizing the total number of RIS reflecting elements required, subject to NOMA-defined constraints, RIS resource limitations, and maximum allowable outage threshold. Simulation results validate the analytical models and demonstrate that the proposed RUOM algorithm significantly improves fairness and efficiency in BS-UAV communication.

Raspaud and Wang conjectured that every triangle-free planar graph can be vertex-partitioned into an independent set and a forest. Independently, Kawarabayashi and Thomassen also remarked that this might be true, after providing another proof of a result of Borodin and Glebov, showing this result for planar graphs of girth~5. Subsequently, Dross, Montassier, and Pinlou raised the same question and proved that every triangle-free planar graph can be partitioned into a forest and another forest of maximum degree~5. More recently, Feghali and Šámal improved this bound on the maximum degree to~3. In this note, we further improve the result by showing that every triangle-free planar graph can be partitioned into a forest and a linear forest, that is, a forest of maximum degree~2.

In this paper we propose a gauge-theoretic approach to the problems of optimal mass transport for vector and matrix densities. This resolves both the issues of positivity and action transitivity constraints. Bures-type metrics on the corresponding semi-direct product groups of diffeomorphisms and gauge transformations are related to Wasserstein-type metrics on vector half-densities and matrix densities via Riemannian submersions. We also describe their relation to Poisson geometry and demonstrate how the momentum map allows one to prove the Riemannian submersion properties. The obtained geodesic equations turn out to be vector versions of the Burgers equations.

In this paper, we use methods of exponential sums to derive a formula for estimating effective upper bounds of $|\zeta'(1/2+it)|$. Different effective upper bounds can be obtained by choosing different parameters.

Clinician scheduling remains a persistent challenge due to limited clinical resources and fluctuating demands. This complexity is especially acute in large academic anesthesiology departments as physicians balance responsibilities across multiple clinical sites with conflicting priorities. Further, scheduling must account for individual clinical and lifestyle preferences to ensure job satisfaction and well-being. Traditional approaches, often based on statistical or rule-based optimization models, rely on structured data and explicit domain knowledge. However, these methods often overlook unstructured information, e.g., free-text notes from routinely administered clinician well-being surveys and scheduling platforms. These notes may reveal implicit and underutilized clinical resources. Neglecting such information can lead to misaligned schedules, increased burnout, overlooked staffing flexibility, and suboptimal utilization of available resources. To address this gap, we propose a predict-then-optimize framework that integrates classification-based clinician availability predictions with a mixed-integer programming schedule optimization model. Large language models (LLMs) are employed to extract actionable preferences and implicit constraints from unstructured schedule notes, enhancing the reliability of availability predictions. These predictions then inform the schedule optimization considering four objectives: first, ensuring clinical full-time equivalent compliance, second, reducing workload imbalances by enforcing equitable proportions of shift types, third, maximizing clinician availability for assigned shifts, and fourth, schedule consistency. By combining the interpretive power of LLMs with the rigor of mathematical optimization, our framework provides a robust, data-driven solution that enhances operational efficiency while supporting equity and clinician well-being.

This paper studies the problem of extracting common randomness (CR) or secret keys from correlated random sources observed by two legitimate parties, Alice and Bob, through public discussion in the presence of an eavesdropper, Eve. We propose a practical two-stage CR extraction framework. In the first stage, the variational probabilistic quantization (VPQ) step is introduced, where Alice and Bob employ probabilistic neural network (NN) encoders to map their observations into discrete, nearly uniform random variables (RVs) with high agreement probability while minimizing information leakage to Eve. This is realized through a variational learning objective combined with adversarial training. In the second stage, a secure sketch using code-offset construction reconciles the encoder outputs into identical secret keys, whose secrecy is guaranteed by the VPQ objective. As a representative application, we study physical layer key (PLK) generation. Beyond the traditional methods, which rely on the channel reciprocity principle and require two-way channel probing, thus suffering from large protocol overhead and being unsuitable in high mobility scenarios, we propose a sensing-based PLK generation method for integrated sensing and communications (ISAC) systems, where paired range-angle (RA) maps measured at Alice and Bob serve as correlated sources. The idea is verified through both end-to-end simulations and real-world software-defined radio (SDR) measurements, including scenarios where Eve has partial knowledge about Bob's position. The results demonstrate the feasibility and convincing performance of both the proposed CR extraction framework and sensing-based PLK generation method.

Let $b \ge 2$ be an integer and $\xi$ an irrational real number. We establishes that, if the irrationality exponent of $\xi$ is less than $2.324 \ldots$, then the $b$-ary expansion of $\xi$ cannot be `too simple', in a suitable sense. This improves the results of our previous paper [Ann. Sc. Norm. Super. Pisa Cl. Sci., 2017].

In this note, we present a characterization of sets definable in Skolem arithmetic, i.e., the first-order theory of natural numbers with multiplication. This characterization allows us to prove the decidability of the theory. The idea is similar to that of Mostowski; however, our characterization is new, and the proof relies on different combinatorial tools. The main goal of this note is to provide a simpler decidability proof than those previously known.

For a very general polarized $K3$ surface $S\subset \mathbb{P}^g$ of genus $g\ge 5$, we study the linear system on the Hilbert square $S^{[2]}$ parametrizing quadrics in $\mathbb{P}^g$ that contain $S$. We prove its very ampleness for $g\geq 7$. In the cases of genus 7 or 8, we describe in detail the projective geometry of the corresponding embedding by making use of the Mukai model for $S$. In both cases, it can be realized as a degeneracy locus on an ambient homogeneous space, in a strikingly similar fashion. In consequence, we give explicit descriptions of its ideal and syzygies. Furthermore, we extract new information on the locally complete families, in a first step towards the understanding of their projective geometry.

We show that the homology of modules for Hurwitz spaces stabilizes and compute its stable value. As one consequence, we compute the moments of Selmer groups in quadratic twist families of abelian varieties over suitably large function fields. As a second consequence, we deduce a version of Bhargava's conjecture, counting the number of $S_d$ degree $d$ extensions of $\mathbb F_q(t)$, for suitably large $q$. As a third consequence, we deduce that the homology of Hurwitz spaces associated to racks with a single component satisfy representation stability.

A system of hyperbolic conservation laws $$ \partial_t u + \partial_x \partial_u Q = 0, \quad Q = u_1^3 / 3 + u_1 u_2^2, \qquad u = u(x,t) \in\mR^2, $$ as well as its viscous regularization $$ \partial_t u + \partial_x \partial_u Q = \calM \partial_x^2 u, \qquad \calM = \diag (\mu_1,\mu_2), \quad \mu_1>0,\, \mu_2>0, $$ are studied. It is assumed that admissible shocks are those that satisfy the requirement of existence of a structure (the traveling wave criterion). A solution of the Riemann problem is constructed that consists of rarefaction waves and shocks with structure. Depending on the conditions imposed at $\pm\infty$, the solution also contains undercompressive shocks and Jouguet waves.

In this paper, we sample directed random graphs from (asymmetric) step-graphons and investigate the probability that the random graph has at least a Hamiltonian cycle (or a node-wise Hamiltonian decomposition). We show that for almost all step-graphons, the probability converges to either zero or one as the order of the random graph goes to infinity--we term it the zero-one law. We identify the key objects of the step-graphon that matter for the zero-one law, and establish a set of conditions that can decide whether the limiting value of the probability is zero or one.

The double bracket $\langle \langle \cdot \rangle \rangle$ (also known as the AJ-bracket) is an invariant of framed tied links that extends the Kauffman bracket of classical links. Unlike the classical setting, little is known about the structure of AJ-states (analogous to classical Kauffman states) of a given tied link diagram, and no general state-sum formula for the AJ-bracket is currently available. In this paper we analyze the AJ-states of $2$- and $3$-tied link diagrams, and provide a complete description of their associated resolution trees leading to a computation of $\langle \langle \cdot \rangle \rangle$. As a result, we derive explicit state-sum formulas for the AJ-bracket. These are the first closed-form expressions of this kind, and they constitute a concrete step toward a combinatorial categorification of the tied Jones polynomial.

We construct and study representations of rational and pretzel tangle and knot groups into the affine group $\mathrm{AGL}(1,\mathbb{C})$, via a TQFT that is valued in the category of spans of singular vector bundles over $\mathbb{C}^{\ast}$. For these families, we derive closed-form expressions for their Alexander polynomials and establish bounds on their zeros. Finally, we specialize the functor at $t=-1$ and analyze colorings of rational tangles in terms of spans of complex vector spaces.

Topographs, introduced by Conway in 1997, are infinite trivalent planar trees used to visualize the values of binary quadratic forms. In this work, we study series whose terms are indexed by the vertices of a topograph and show that they can be evaluated using telescoping sums over its edges. Our technique provides arithmetic proofs for modular graph function identities arising in string theory, yields alternative derivations of Hurwitz-style class number formulas, and provides a unified framework for well-known Mordell-Tornheim series and Hata's series for the Euler constant $\gamma$. Our theorems are of the following spirit: we cut a topograph along an edge (called the root) into two parts, and then sum $\frac{1}{rst}$ (the reciprocal of the product of labels on regions adjacent to a vertex) over all vertices of one part. We prove that such a sum is equal to an explicit expression depending only on the root and the discriminant of the topograph.

The prediction of the anomalous gyromagnetic factor of the electron, started with the evaluation of the electromagnetic contribution by Schwinger (1948) and of the weak contribution by Jackiw and Weinberg (1972), is one of the major successes of Quantum Field Theory and the Standard Model. The results obtained truncating the series are in spectacular agreement with experiments. Yet, a mathematical justification and an estimate of the truncation error are problematic, being such series diverging and not asymptotic to any QFT. For a non perturbative result, one has to consider the Standard Model as an effective theory valid up to certain energy scales. In this paper we consider the neutral sector of the Electroweak model with a momentum cutoff; we rigorously prove that the anomalous gyromagnetic factor in the effective regularized theory coincides with the Jackiw-Weinberg result, obtained by the truncation of the formal expansion with no cutoffs (whose sum is not expected to exist), up to a regularization-dependent correction which is subdominant in the weak coupling regime if the cutoff is smaller than the inverse coupling and larger than the boson mass. The proof is based on a convergent expansions and Renormalization Group (RG) methods; cancellations based on exact and approximated symmetries are needed to get lowest order dominance.

We introduce a nodally bound-preserving Galerkin method for second-order elliptic problems on general polygonal/polyhedral, henceforth collectively termed as \emph{polytopic}, meshes. Starting from an interior penalty discontinuous Galerkin (DG) formulation posed on a polytopic mesh, the method enforces preservation of \emph{a priori} prescribed upper and lower bounds for the numerical solution at an arbitrary number of user-defined points \emph{within} each polytopic element. This is achieved by employing a simplicial submesh and enforcing bound preservation at the submesh nodes via a nonlinear iteration. By construction, the submeshing procedure preserves the order of accuracy of the DG method, \emph{without} introducing any additional global numerical degrees of freedom compared to the baseline DG method, thereby, falling into the category of composite finite element approaches. A salient feature of the proposed method is that it automatically reverts to the standard DG method on polytopic meshes when no prescribed bound violation occurs. In particular, the choice of the discontinuity-penalisation parameter is independent of the submesh granularity. The resulting composite method combines the geometric flexibility of polytopic meshes with the accuracy and stability of discontinuous Galerkin discretisations, while rigorously guaranteeing bound preservation. The existence and uniqueness of the numerical solution is proven. A priori error bounds, assuming sufficient regularity of the exact solution are shown, employing a non-standard construction of discrete nodally bound-preserving interpolant. Numerical experiments confirm optimal convergence for smooth problems and demonstrate robustness in the presence of sharp gradients, such as boundary and interior layers.

We provide exact integral formulas for hyperbolic and spherical volumes of cone-manifolds whose underlying space is the $3$-sphere and whose singular set belongs to three infinite families of two-bridge knots: $C(2n,2)$ (twist knots), $C(2n,3)$, and $C(2n,-2n)$ for any non-zero integer $n$. Our formulas express volumes as integrals of explicit rational functions involving Chebyshev polynomials of the second kind, with integration limits determined by roots of algebraic equations. This extends previous work where only implicit formulas requiring numerical approximation were known.

We introduce \emph{coarse scrambling}, a novel randomization for digital sequences that permutes blocks of digits in a mixed-radix representation. This construction is designed to preserve the powerful $(0,\boldsymbol{e},d)$-sequence property of the underlying points. For sufficiently smooth integrands, we prove that this method achieves the canonical $O(n^{-3+\epsilon})$ variance decay rate, matching that of standard Owen's scrambling. Crucially, we show that its maximal gain coefficient grows only logarithmically with dimension, $O(\log d)$, thus providing theoretical robustness against the curse of dimensionality affecting scrambled Sobol' sequences. Numerical experiments validate these findings and illustrate a practical trade-off: while Owen's scrambling is superior for integrands sensitive to low-dimensional projections, coarse scrambling is competitive for functions with low effective truncation dimension.

We consider the Vlasov--Poisson equation on $\mathbb{R}^n \times \mathbb{R}^n$ with $n \ge 3$. We prove local well-posedness in $H^{s}(\mathbb{R}^n \times \mathbb{R}^n)$ with $s> n/2-1/4$, for initial distribution $f_{0} \in H^{s}(\mathbb{R}^n \times \mathbb{R}^n)$ having compact support in $v$. In particular, data not belonging to $L^p(\mathbb{R}^n \times \mathbb{R}^n)$ for large $p$ are allowed.

In this paper, the properties of minimal trails in a directed acyclic graph that is restricted not to contain an active cycle are studied. We are motivated by an application of the results in the copula-based Bayesian Network model developed recently. We propose a partial order on the set of trails activated by a certain subset of nodes, and show that every minimal trail, according to such an order, has a simple structure.

This paper explores a variant of bipartite matching problem, referred to as the Spatiotemporal Random Bipartite Matching Problem (ST-RBMP), that accommodates randomness and heterogeneity in the spatial distributions and temporal arrivals of bipartite vertices. This type of problem can be applied to many location-based services, such as shared mobility systems, where randomly arriving customers and vehicles must be matched dynamically. This paper proposes a new modeling framework to address ST-RBMP's challenges associated with the spatiotemporal heterogeneity, dynamics, and stochastic decision-making. The objective is to dynamically determine the optimal vehicle/customer pooling intervals and maximum matching radii that minimize the system-wide matching costs, including customer and vehicle waiting times and matching distances. Closed-form formulas for estimating the expected matching distances under a maximum matching radius are developed for static and homogeneous RBMPs, and then extended to accommodate spatial heterogeneity via continuum approximation. The ST-RBMP is then formulated as an optimal control problem where optimal values of pooling intervals and matching radii are solved over time and space. A series of experiments with simulated data are conducted to demonstrate that the proposed formulas for static RBMPs under matching radius and spatial heterogeneity yield very accurate results on estimating matching probabilities and distances. Additional numerical results are presented to demonstrate the effectiveness of the proposed ST-RBMP modeling framework in designing dynamic matching strategies for mobility services under various demand and supply patterns, which offers key managerial insights for mobility service operators.

The connection between simple Lie algebras and their Yangian algebras has a long history. In this work, we construct finite-dimensional representations of Yangian algebras $\mathsf{Y}(\mathfrak{sl}_{n})$ using the quiver approach. Starting from quivers associated to Dynkin diagrams of type A, we construct a family of quiver Yangians. We show that the quiver description of these algebras enables an effective construction of representations with a single non-zero Dynkin label. For these representations, we provide an explicit construction using the equivariant integration over the corresponding quiver moduli spaces. The resulting states admit a crystal description and can be identified with the Gelfand-Tsetlin bases for $\mathfrak{sl}_{n}$ algebras. Finally, we show that the resulting Yangians possess notable algebraic properties, and the algebras are isomorphic to their alternative description known as the second Drinfeld realization.

We develop a mixed-precision iterative refinement framework for solving low-rank Lyapunov matrix equations $AX + XA^T + W =0$, where $W=LL^T$ or $W=LSL^T$. Via rounding error analysis of the algorithms we derive sufficient conditions for the attainable normwise residuals in different precision settings and show how the algorithmic parameters should be chosen. Using the sign function Newton iteration as the solver, we show that reduced precisions, such as the half precision, can be used as the solver precision (with unit roundoff $u_s$) to accelerate the solution of Lyapunov equations of condition number up to $1/u_s$ without compromising its quality.

We describe a novel method for computing sheaf cohomology over weighted projective stacks using exterior algebra and differential module techniques, generalizing an algorithm due to Eisenbud-Floystad-Schreyer over projective space.

Quantum sensing has attracted significant attention due to its ability to measure physical quantities with extremely high accuracy. Rydberg atoms - typically alkali atoms with a highly excited valence electron that is far from the nucleus - exhibit strong sensitivity to external electromagnetic fields. This sensitivity leads to coupling between different atomic energy levels, which can be observed by monitoring changes in a control laser beam before and after it passes through a vapor cell containing the Rydberg atoms. By analyzing the transmitted laser signal with a photodetector, variations in transmission can be attributed to the presence and characteristics of the external electromagnetic field. Because Rydberg atoms operate in a highly excited quantum state without relying on traditional electronic circuitry, they inherently avoid thermal noise, thereby enabling more sensitive detection. In this paper, we investigate the performance of a Rydberg atomic receiver based on Rb-85 and compare it with that of a conventional receiver in detecting an 8-level pulse amplitude modulation (8-PAM) signal in the presence of off-resonant interference. We demonstrate that the Rydberg receiver can suppress interference without the need for an additional filter. Effectively, our results show that the Rydberg receiver serves as an integrated filter and demodulator, outperforming conventional circuit-based receivers in terms of achievable symbol error rate

In this article, we prove the cutoff phenomenon for a general class of the discrete-time nonlinear recombination models. This system models the evolution of a probability measure on a finite product space $S^n$ representing the state of spins on $n$ sites. Although its stationary distribution has a product structure, and its evolution is Markovian, the dynamics of the model is nonlinear. Consequently, the estimation of the mixing time becomes a highly non-trivial task. The special case with two spins and homogeneous stationary measure was considered in Caputo, Labbé, and Lacoin [The Annals of Applied Probability 35:1164-1197, 2025], where the cutoff phenomenon for the mixing behavior has been verified. In this article, we extend this result to the general case with finite spins and inhomogeneous stationary measure by developing a novel algebraic representation for the density fluctuation of the system with respect to its stationary state.

In this work we study the convergence of gradient methods for nonconvex optimization problems -- specifically the effect of the problem formulation to the convergence behavior of the solution of a gradient flow. We show through a simple example that, surprisingly, the gradient flow solution can be exponentially or asymptotically convergent, depending on how the problem is formulated. We then deepen the analysis and show that a policy optimization strategy for the continuous-time linear quadratic regulator (LQR) (which is known to present only asymptotic convergence globally) presents almost global exponential convergence if the problem is overparameterized through a linear feed-forward neural network (LFFNN). We prove this qualitative improvement always happens for a simplified version of the LQR problem and derive explicit convergence rates for the gradient flow. Finally, we show that both the qualitative improvement and the quantitative rate gains persist in the general LQR through numerical simulations.

The Wronskian determinants (for coefficients of higher-order differential operators on the affine real line or circle) satisfy the table of Jacobi-type quadratic identities for strong homotopy Lie algebras -- i.e. for a particular case of $L_\infty$-deformations -- for the Lie algebra of vector fields on that one-dimensional affine manifold. We show that the standard realisation of $\mathfrak{sl}(2)$ by quadratic-coefficient vector fields is the bottom structure in a sequence of finite-dimensional polynomial algebras $\Bbbk_N[x]$ with the Wronskians as $N$-ary brackets; the structure constants are calculated explicitly. Key words: Wronskian determinant, $N$-ary bracket, $L_\infty$-\/algebra, strong homotopy Lie algebra, $sl(2)$, Witt algebra, Vandermonde determinant.

Recently, Kleshchev and Livesey proved the existence of RoCK $p$-blocks for double covers of symmetric and alternating groups over large enough coefficient rings. They proved that RoCK blocks of double covers are Morita equivalent to standard ``local" blocks via bimodules with endopermutation source. Based on this, Kleshchev and Livesey proved that these RoCK blocks are splendidly Rickard equivalent to their Brauer correspondents. The analogous result for blocks of symmetric groups, a theorem of Chuang and Kessar, was an important step in Chuang and Rouquier ultimately proving Broué's abelian defect group conjecture for symmetric groups. In this paper we show that in most cases the Morita equivalences and splendid Rickard equivalences constructed by Kleshchev and Livesey descend to the ring $\mathbb{Z}_p$ of $p$-adic integers, hence prove Kessar and Linckelmann's refinement of Broué's abelian defect group conjecture for most of these RoCK blocks.

We extend the mirror construction of singular Calabi-Yau double covers, introduced by Hosono, Lee, Lian, and Yau, to a broader class of singular Calabi-Yau $(\mathbb{Z}/2)^k$-Galois covers, and prove Hodge number duality for both the original and extended mirror pairs. A main tool in our approach is an analogue of the Cayley trick, which relates the de Rham complex of the branched covers to the twisted de Rham complex of certain Landau-Ginzburg models. In particular, it reveals direct relations between the Hodge numbers of the covers and the irregular Hodge numbers of the associated Landau-Ginzburg models. This construction is independent of mirror symmetry and may be of independent interest.

We present an enhanced version of the row-based randomized block-Kaczmarz method to solve a linear system of equations. This improvement makes use of a regularization during block updates in the solution, and a dynamic proposal distribution based on the current residue and effective orthogonality between blocks. This improved method provides significant gains in solving high-condition number linear systems that are either sparse, or dense least-squares problems that are significantly over/under determined. Considering the poor generalizability of preconditioners for such problems, it can also serve as a pre-solver for other iterative numerical methods when required, and as an inner iteration in certain types of GMRES solvers for linear systems.

The colored Jones polynomial $J_{K,N}$ is an important quantum knot invariant in low-dimensional topology. In his seminal paper on quantum modular forms, Zagier predicted the behavior of $J_{K,0}(e^{2 \pi i x})$ under the action of $SL_2(\mathbb{Z})$ on $x \in \mathbb{Q}$. More precisely, Zagier made a prediction on the asymptotic value of the quotient $J_{K,0}(e^{2 \pi i \gamma(x)})/ J_{K,0}(e^{2 \pi i x})$ for fixed $\gamma \in SL_2(\mathbb{Z})$, as $x \to \infty$ along rationals with bounded denominator. In the case of the figure-eight knot $4_1$, which is the most accessible case, there is an explicit formula for $J_{4_1,0}(e^{2 \pi i x})$ as a sum of certain trigonometric products called Sudler products. By periodicity, the behavior of $J_{4_1,0}(e^{2 \pi i x})$ under the mapping $x \mapsto x+1$ is trivial. For the second generator of $SL_2(\mathbb{Z})$, Zagier conjectured that with respect to the mapping $x \mapsto 1/x$, the quotient $h(x) = \log ( J_{4_1,0}(e^{2 \pi i x}) / J_{4_1,0}(e^{2 \pi i /x}))$ can be extended to a function on $\mathbb{R}$ that is continuous at all irrationals. This conjecture was recently established by Aistleitner and Borda in the case of all irrationals that have an unbounded sequence of partial quotients in their continued fraction expansion. In the present paper we prove Zagier's continuity conjecture in full generality.

We focus on collaborative edge inference over wireless, which enables multiple devices to cooperate to improve inference performance in the presence of corrupted data. Exploiting a key-query mechanism for selective information exchange (or, group formation for collaboration), we recall the effect of wireless channel impairments in feature communication. We argue and show that a disjoint approach, which only considers either the semantic relevance or channel state between devices, performs poorly, especially in harsh propagation conditions. Based on these findings, we propose a joint approach that takes into account semantic information relevance and channel states when grouping devices for collaboration, by making the general attention weights dependent of the channel information. Numerical simulations show the superiority of the joint approach against local inference on corrupted data, as well as compared to collaborative inference with disjoint decisions that either consider application or physical layer parameters when forming groups.

We solve the open problem by Demuth, Hansmann, and Katriel announced in [Integr. Equ. Oper. Theory 75 (2013), 1-5] by a counter-example construction. The problem concerns a possible generalisation of the Lieb-Thirring inequality for Schrödinger operators in to the case of complex-valued potentials. A counter-example has already been found for the one-dimensional case by the first and third authors in [J. Spectr. Theory 11 (2021), 1391-1413]. Here we generalise the counter-example to higher dimensions.

If a symmetric multilinear algebra is weakly nil, then it is Engel. This result may be regarded as an infinite-dimensional analogue of the well-known Jacobian theorem, which states that if a polynomial mapping has a polynomial inverse, then its Jacobian matrix is invertible. This refines a theorem of Gerstenhaber and partially answers a question posed by Dotsenko.

The boxicity of a graph $G$ is the minimum dimension $d$ that admits a representation of $G$ as the intersection graph of a family of axis-parallel boxes in $\mathbb{R}^d$. Computing boxicity is an NP-hard problem, and there are few known graph classes for which it can be computed in polynomial time. One such class is the class of block graphs. A block graph is a graph in which every maximal $2$-connected component is a clique. Since block graphs are known to have boxicity at most two, computing their boxicity amounts to the linear-time interval graph recognition problem. On the other hand, complements of block graphs have unbounded boxicity, yet we show that there is also a polynomial algorithm that computes the boxicity of complements of block graphs. An adaptation of our approach yields a polynomial algorithm for computing the threshold dimension of the complements of block graphs, which for general graphs is an NP-hard problem. Our method suggests a general technique that may show the tractability of similar problems on block-restricted graph classes.

Let $R$ be a local ring and let $M$ be a finitely generated $R$-module. Appealing to the natural left module structure of $M$ over its endomorphism ring and corresponding center $Z(\operatorname{End}_R(M))$, we study when various homological properties of $M$ are sufficient to force $M$ to have a nonzero free summand. Consequences of our work include a partial converse to a well-known result of Lindo describing $Z(\operatorname{End}_R(M))$ when $M$ is faithful and reflexive, as well as some applications to the famous Huneke-Wiegand conjecture.

Let $G$ be a graph and let $S \subseteq V(G)$. It is said that $S$ \textit{dominates} $N[S]$. We say that $S$ \textit{monitors} vertices of $G$ as follows. Initially, all dominated vertices are monitored. If there exists a vertex $v \in G$ which is not monitored, but has all but one of its neighbours monitored, then $v$ becomes monitored itself. This step is called a \textit{propagation} step and is repeated until the process terminates. The process terminates when the there are no unmonitored vertices with exactly one unmonitored neighbour. This combined process of initial domination and subsequent propagation is called \textit{power domination}. If all vertices of $G$ are monitored at termination, then $S$ is said to be a \textit{power dominating set (PDS) of $G$}. The \textit{power domination number of $G$}, denoted as $\gamma_p(G)$, is the minimum cardinality of a PDS of $G$. The \textit{propagation radius of $G$} is the minimum number of steps it takes a minimum PDS to monitor $V(G)$. In this paper we determine an upper bound on the propagation radius of $G$ with regards to power domination, in terms of $\delta$ and $n$. We show that this bound is only attained when $\gamma_p(G)=1$ and then improve this bound for $\gamma_p(G)\geq 2$. Sharpness examples for these bounds are provided. We also present sharp upper bounds on the propagation radius of split graphs. We present sharpness results for a known lower bound of the propagation radius for all $\Delta\geq 3$.

Given any knot K in the 3-sphere, we prove that there are only finitely many hyperbolic fibered knots which are ribbon concordant to K. It follows that every fibered knot in the 3-sphere has only finitely many hyperbolic predecessors under ribbon concordance. Our proof combines results about maps on Floer homology induced by ribbon cobordisms with a relationship between the knot Floer homology of a fibered knot and fixed points of its monodromy. We then use the same techniques in combination with results of Cornish and Kojima-McShane to prove an inequality relating the volumes of ribbon concordant hyperbolic fibered knots.

We prove a version of Calegari's Flow Conjecture, which asserts that every quasigeodesic flow on a closed hyperbolic 3-manifold can be deformed to be both quasigeodesic and pseudo-Anosov.

In this paper, we study the framework of collaborative inference, or edge ensembles. This framework enables multiple edge devices to improve classification accuracy by exchanging intermediate features rather than raw observations. However, efficient communication strategies are essential to balance accuracy and bandwidth limitations. Building upon a key-query mechanism for selective information exchange, this work extends collaborative inference by studying the impact of channel noise in feature communication, the choice of intermediate collaboration points, and the communication-accuracy trade-off across tasks. By analyzing how different collaboration points affect performance and exploring communication pruning, we show that it is possible to optimize accuracy while minimizing resource usage. We show that the intermediate collaboration approach is robust to channel errors and that the query transmission needs a higher degree of reliability than the data transmission itself.

For operads with a map from the curved homotopy Lie operad, we introduce a corresponding curved variant `cTw' of Willwacher's operadic twisting comonad `Tw'. We show that cTw-coalgebra structures on such an operad are in bijection with certain splittings (not respecting the differential) of the projection to its quotient by the curvature operation. We derive a similar classification of Tw-coalgebras. For the class of operads whose Koszul dual admits a unital extension, we give explicit formulas for the cTw-coalgebra structures on their curved homotopy resolutions, recovering the convolution Lie algebra's ``gauge group action'' of Dotsenko, Shadrin, and Vallette.

In this paper we prove the Stein-Weiss inequality in variable exponent Morrey spaces over a bounded domain. Our work extends earlier results in the variable exponent Lebesgue and Morrey settings, and utilizes new proof techniques applicable to Morrey spaces. We build on the foundational paper by Almeida, Hasanov, and Samko, which introduced Morrey spaces of variable exponents. As an application of our main result, we prove Poincaré-type inequalities using the approach of a recent paper by the first and third authors.

How should one define metric space notions of convergence for sequences of spacetimes? Since a Lorentzian manifold does not define a metric space directly, the uniform convergence, Gromov-Hausdorff (GH) convergence, and Sormani-Wenger Intrinsic Flat (SWIF) convergence does not extend automatically. One approach is to define a metric space structure, which is compatible with the Lorentzian structure, so that the usual notions of convergence apply. This approach was taken by C. Sormani and C. Vega when defining the null distance. In this paper, we study sequences of static spacetimes equipped with the null distance under uniform, GH, and SWIF convergence, as well as Hölder bounds. We use the results of the Volume Above Distance Below (VADB) theorem of the author, R. Perales, and C. Sormani to prove an analog of the VADB theorem for sequences of static spacetimes with the null distance. We also give a conjecture of what the VADB theorem should be in the case of sequences of globally hyperbolic spacetimes with the null distance.

We discuss two spectral fractional anisotropic Calderón problems with source-to-solution measurements and their quantitative relation to the classical Calderón problem. Firstly, we consider the anistropic fractional Calderón problem from [FGKU25]. In this setting, we quantify the relation between the local and nonlocal Calderón problems which had been deduced in [R25] and provide an associated stability estimate. As a consequence, any stability result which holds on the level of the local problem with source-to-solution data has a direct nonlocal analogue (up to a logarithmic loss). Secondly, we introduce and discuss the fractional Calderón problem with source-to-solution measurements for the spectral fractional Dirichlet Laplacian on open, bounded, connected, Lipschitz sets on $\mathbb{R}^n$. Also in this context, we provide a qualitative and quantitative transfer of uniqueness from the local to the nonlocal setting. As a consequence, we infer the first stability results for the principal part for a fractional Calderón type problem for which no reduction of Liouville type is known. Our arguments rely on quantitative unique continuation arguments. As a result of independent interest, we also prove a quantitative relation between source-to-solution and Dirichlet-to-Neumann measurements for the classical Calderón problem.

Using analytic number theory techniques, Altuğ showed that the contribution of the trivial representation to the Arthur-Selberg trace formula for GL(2) over $\Q$ could be cancelled by applying a modified Poisson summation formula to the regular elliptic contribution. Drawing on recent works, we re-examine these methods from an adelic perspective.

In this short note we observe that Kelly's transfinite construction of free algebras yields a way to invert well-pointed endofunctors. In enriched settings, this recovers constructions of Keller, Seidel, and Chen-Wang. We also relate this procedure to localisation by spectra and to Heller's stabilisation.

This paper studies a basic Markov chain, the Burnside process, on the space of flags $G/B$ with $G = GL_n(\mathbb{F}_q)$ and $B$ its upper triangular matrices. This gives rise to a shuffling: a Markov chain on the symmetric group realized via the Bruhat decomposition. Actually running and describing this Markov chain requires understanding Springer fibers and the Steinberg variety. The main results give a practical algorithm for all n and q and determine the limiting behavior of the chain when q is large. In describing this behavior, we find interesting connections to the combinatorics of the Robinson-Schensted correspondence and to the geometry of orbital varieties. The construction and description is then carried over to finite Chevalley groups of arbitrary type, describing a new class of Markov chains on Weyl groups.

We prove optimal Lieb-Thirring type inequalities for Schrödinger and Jacobi operators with complex potentials. Our results bound eigenvalue power sums (Riesz means) by the $L^p$ norm of the potential, where in contrast to the self-adjoint case, each term needs to be weighted by a function of the ratio of the distance of the eigenvalue to the essential spectrum and the distance to the endpoint(s) thereof. Our Lieb-Thirring type bounds only hold for integrable weight functions. To prove optimality, we establish divergence estimates for non-integrable weight functions. The divergence rates exhibit a logarithmic or even polynomial gain compared to semiclassical methods (Weyl asymptotics) for real potentials.

We show that every compactly supported calibrated integral current with connected $C^{3,\alpha}$ boundary is the unique solution to the oriented Plateau problem for its boundary data. This is proved as a consequence of the boundary regularity theory for area-minimizing currents and classical unique continuation principles adapted to the minimal surface system.

Given a number field $K \neq \mathbb{Q}$, in a now classic work, Stark pinpointed the possible source of a so-called Landau-Siegel zero of the Dedekind zeta function $\zeta_K(s)$ and used this to give effective upper and lower bounds on the residue of $\zeta_K(s)$ at $s=1$. We extend Stark's work to give effective upper and lower bounds for the leading term of the Laurent expansion of general Artin $L$-functions at $s=1$ that are, up to the value of implied constants, as strong as could reasonably be expected given current progress toward the generalized Riemann hypothesis. Our bounds are completely unconditional, and rely on no unproven hypotheses about Artin $L$-functions.

In settings like vaccination registries, individuals act after observing others, and the resulting public records can expose private information. We study privacy-preserving sequential learning, where agents add endogenous noise to their reported actions to conceal private signals. Efficient social learning relies on information flow, seemingly in conflict with privacy. Surprisingly, with continuous signals and a fixed privacy budget $(\epsilon)$, the optimal randomization strategy balances privacy and accuracy, accelerating learning to $\Theta_{\epsilon}(\log n)$, faster than the nonprivate $\Theta(\sqrt{\log n})$ rate. In the nonprivate baseline, the expected time to the first correct action and the number of incorrect actions diverge; under privacy with sufficiently small $\epsilon$, both are finite. Privacy helps because, under the false state, agents more often receive signals contradicting the majority; randomization then asymmetrically amplifies the log-likelihood ratio, enhancing aggregation. In heterogeneous populations, an order-optimal $\Theta(\sqrt{n})$ rate is achievable when a subset of agents have low privacy budgets. With binary signals, however, privacy reduces informativeness and impairs learning relative to the nonprivate baseline, though the dependence on $\epsilon$ is nonmonotone. Our results show how privacy reshapes information dynamics and inform the design of platforms and policies.

We prove three structural impossibility results demonstrating that fuzzy metric spaces cannot capture essential features of quantum state geometry. First, we show they cannot model destructive interference between concepts due to phase insensitivity. Second, we prove there is no distance-preserving embedding from quantum state space into any fuzzy metric space. Third, we establish that fuzzy logic cannot distinguish symmetric from antisymmetric concept combinations -- a fundamental limitation for modeling structured knowledge. These theorems collectively show that fuzzy frameworks are structurally incapable of representing intrinsic uncertainty, where quantum mechanics provides a superior, geometrically coherent alternative.

We provide series expansions for the tempered stable densities and for the price of European-style contracts in the exponential Lévy model driven by the tempered stable process. These formulas recover several popular option pricing models, and become particularly simple in some specific cases such as bilateral Gamma process and one-sided TS process. When compared to traditional Fourier pricing, our method has the advantage of being hyperparameter free. We also provide a detailed numerical analysis and show that our technique is competitive with state-of-the-art pricing methods.

Recent breakthroughs have spurred claims that large language models (LLMs) match gold medal Olympiad to graduate level proficiency on mathematics benchmarks. In this work, we examine these claims in detail and assess the extent to which current benchmarks capture genuine LLM mathematical reasoning. The composition of these benchmarks, primarily drawing from the International Mathematics Olympiad (IMO) and related competitions, may overstate models reasoning ability due to potential data contamination and a narrow focus on familiar problem types. To enable a more holistic assessment of mathematical understanding, we introduce EEFSUVA, a novel benchmark curated from under circulated regional and national Olympiads of Eastern Europe and the countries from the former Soviet Union. These contests feature problems of comparable difficulty to the IMO and are renowned for demanding nonstandard problem-solving techniques, yet their problems are far less prevalent in online corpora. Preliminary results suggest that even state-of-the-art LLMs exhibit a notable performance decline on EEFSUVA relative to other Olympiad-style benchmarks. These findings also suggest the potential importance of broader evaluation datasets for a fuller assessment of mathematical reasoning and for guiding future model development.

Artificial intelligence is observed to age not through chronological time but through structural asymmetries in memory performance. In large language models, semantic cues such as the name of the day often remain stable across sessions, while episodic details like the sequential progression of experiment numbers tend to collapse when conversational context is reset. To capture this phenomenon, the Artificial Age Score (AAS) is introduced as a log-scaled, entropy-informed metric of memory aging derived from observable recall behavior. The score is formally proven to be well-defined, bounded, and monotonic under mild and model-agnostic assumptions, making it applicable across various tasks and domains. In its Redundancy-as-Masking formulation, the score interprets redundancy as overlapping information that reduces the penalized mass. However, in the present study, redundancy is not explicitly estimated; all reported values assume a redundancy-neutral setting (R = 0), yielding conservative upper bounds. The AAS framework was tested over a 25-day bilingual study involving ChatGPT-5, structured into stateless and persistent interaction phases. During persistent sessions, the model consistently recalled both semantic and episodic details, driving the AAS toward its theoretical minimum, indicative of structural youth. In contrast, when sessions were reset, the model preserved semantic consistency but failed to maintain episodic continuity, causing a sharp increase in the AAS and signaling structural memory aging. These findings support the utility of AAS as a theoretically grounded, task-independent diagnostic tool for evaluating memory degradation in artificial systems. The study builds on foundational concepts from von Neumann's work on automata, Shannon's theories of information and redundancy, and Turing's behavioral approach to intelligence.

As AI cluster sizes continue to expand and the demand for large-language-model (LLM) training and inference workloads grows rapidly, traditional scheduling systems face significant challenges in balancing resource utilization, scheduling efficiency, and service quality. This paper presents and evaluates Kant: an efficient unified scheduling platform designed for large-scale AI container clusters, supporting the co-scheduling of both training and inference jobs. Based on the practical implementation of the Kant system, we systematically define a set of key evaluation metrics for AI clusters, including GPU Allocation Ratio (GAR), Scheduling Occupancy Rate (SOR), GPU Node Fragmentation Ratio (GFR), Job Waiting Time Distribution (JWTD), and Job Training Time Estimation Distribution (JTTED), providing a foundation for quantitative performance analysis. Experimental results demonstrate that Kant achieves exceptional performance in clusters ranging from hundreds to tens of thousands of GPUs. By leveraging scheduling strategies such as Backfill and Enhanced Binpack (E-Binpack), the system significantly improves resource utilization and scheduling efficiency, while effectively reducing resource fragmentation and communication overhead in distributed training. The system has been deployed in multiple AI data center clusters, where it stably supports large-scale intelligent computing workloads. This work provides a practical engineering approach for building high-performance, highly available, AI-native scheduling infrastructure.

Machine learning models can generalize well on real-world datasets. According to the manifold hypothesis, this is possible because datasets lie on a latent manifold with small intrinsic dimension (ID). There exist many methods for ID estimation (IDE), but their estimates vary substantially. This warrants benchmarking IDE methods on manifolds that are more complex than those in existing benchmarks. We propose a Quantum-Inspired Intrinsic-dimension Estimation (QuIIEst) benchmark consisting of infinite families of topologically non-trivial manifolds with known ID. Our benchmark stems from a quantum-optical method of embedding arbitrary homogeneous spaces while allowing for curvature modification and additive noise. The IDE methods tested were generally less accurate on QuIIEst manifolds than on existing benchmarks under identical resource allocation. We also observe minimal performance degradation with increasingly non-uniform curvature, underscoring the benchmark's inherent difficulty. As a result of independent interest, we perform IDE on the fractal Hofstadter's butterfly and identify which methods are capable of extracting the effective dimension of a space that is not a manifold.

We give the solution to the complete noncommutative Kadomtsev--Petviashvili (KP) hierarchy. We achieve this via direct linearisation which involves the Gelfand--Levitan--Marchenko (GLM) equation. This is a linear integral equation in which the scattering data satisfies the linearised KP hierarchy. The solution to the GLM equation is then shown to coincide with the solution to the noncommutative KP hierarchy. We achieve this using two approaches. In the first approach we use the standard Sato-Wilson dressing transformation. In the second approach, which was pioneered by Poppe, we assume the scattering data is semi-additive and by direct substitution, we show that the solution to the GLM equation satisfies the infinite set of field equations representing the noncommutative KP hierarchy. This approach relies on the augmented pre-Poppe algebra. This is a representative algebra that underlies the field equations representing the hierarchy. It is nonassociative and isomorphic to a descent algebra equipped with a grafting product. While we perform computations in the nonassociative descent algebra, the final result which establishes the solution to the complete hierarchy, resides in the natural associative subalgebra. The advantages of this second approach are that it is constructive, explicit, highlights the underlying combinatorial structures within the hierarchy, and reveals the mechanisms underlying the solution procedure.

We apply the quantum-mechanics bootstrap to supersymmetric quantum mechanics (SUSY QM) and to its matrix relative, the Marinari-Parisi model, which is conjectured to describe the worldvolume of unstable $D0$ branes. Using positivity of moment matrices together with Heisenberg, gauge, and (zero-temperature) thermal constraints, we obtain rigorous bounds on ground-state data. In the cases where SUSY is spontaneously broken, we find bounds that apply to the lowest-energy normalizable eigenstate. For $N = 1$ SUSY QM with a cubic superpotential, we obtain tight bounds that agree well with available approximation methods. At weak coupling they match well with the semiclassical instanton contribution to SUSY-breaking ground-state energy, while at strong coupling they exhibit the expected scaling and match well with Hamiltonian truncation. For the SUSY matrix QM, we construct a $44 \times 44$ bootstrap matrix and obtain bounds at large $N$. At strong coupling, we obtain the expected $E \sim \kappa \ g^{2/3}$ scaling of $E$ with $g$ and extract a lower bound on the coefficient $\kappa > .196$. At small coupling, the theory has a critical point $g_c$ where the two wells merge into one. We find a spurious kink at $g = \sqrt{2} g_c$. We attribute this to truncation error and solver limitations, and discuss possible improvements.

The neural coding is yet to be discovered. The neuronal operational modes that arise with fixed inputs but with varying degrees of stimulation help to elucidate their coding properties. In neurons receiving in vivo stimulation, we show that two operation modes can be described with simplified models: the coincidence detection mode and the integration mode. Our derivations include a simplified polynomial model with non-linear coefficients betam that captures the subthreshold dynamics of these modes of operation. The resulting model can explain these transitions with the sign and size of the smallest nonlinear coefficient of the polynomial alone. Defining neuronal operational modes provides insight into the processing and transmission of information through electrical currents. Requisite operational modes for proper neuronal functioning may explain disorders involving dysfunction of electrophysiological behavior, such as channelopathies.

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

We propose some new uses of toric variety structures in the study of quantum computation for various radices.

Onion routing provides anonymity by layering encryption so that no relay can link sender to destination. A quantum analogue faces a core obstacle: layered quantum encryption generally requires symmetric encryption schemes, whereas classically one would rely on public-key encryption. We propose a symmetric-encryption-based quantum onion routing (QOR) scheme by instantiating each layer with the abelian ideal class group action from the Theory of Complex Multiplication. Session keys are established locally via a Diffie-Hellman key exchange between neighbors in the chain of communication. Furthermore, we propose a novel ''non-local'' key exchange between the sender and receiver. The underlying problem remains hard even for quantum adversaries and underpins the security of current post-quantum schemes. We connect our construction to isogeny graphs and their association schemes, using the Bose-Mesner algebra to formalize commutativity and guide implementation. We give two implementation paths: (i) a universal quantum oracle evaluating the class group action with polynomially many quantum resources, and (ii) an intrinsically quantum approach via continuous-time quantum walks (CTQWs), outlined here and developed in a companion paper. A small Qiskit example illustrates the mechanics (by design, not the efficiency) of the QOR.

In this work, we investigate improving the runtime performance of key computational kernels in the Python Tensor Toolbox (pyttb), a package for analyzing tensor data across a wide variety of applications. Recent runtime performance improvements have been demonstrated using Rust, a compiled language, from Python via extension modules leveraging the Python C API -- e.g., web applications, data parsing, data validation, etc. Using this same approach, we study the runtime performance of key tensor kernels of increasing complexity, from simple kernels involving sums of products over data accessed through single and nested loops to more advanced tensor multiplication kernels that are key in low-rank tensor decomposition and tensor regression algorithms. In numerical experiments involving synthetically generated tensor data of various sizes and these tensor kernels, we demonstrate consistent improvements in runtime performance when using Rust from Python over 1) using Python alone, 2) using Python and the Numba just-in-time Python compiler (for loop-based kernels), and 3) using the NumPy Python package for scientific computing (for pyttb kernels).

Control systems operating in the real world face countless sources of unpredictable uncertainties. These random disturbances can render deterministic guarantees inapplicable and cause catastrophic safety failures. To overcome this, this paper proposes a method for designing safe controllers for discrete-time stochastic systems that retain probabilistic guarantees of safety. To do this we modify the traditional notion of a control barrier function (CBF) to explicitly account for these stochastic uncertainties and call these new modified functions probabilistic CBFs. We show that probabilistic CBFs can be used to design controllers that guarantee safety over a finite number of time steps with a prescribed probability. Next, by leveraging various uncertainty quantification methods, such as concentration inequalities, the scenario approach, and conformal prediction, we provide a variety of sufficient conditions that result in computationally tractable controllers with tunable probabilistic guarantees across a plethora of practical scenarios. Finally, we showcase the applicability of our results in simulation and hardware for the control of a quadruped robot.

Binarized neural networks (BNNs) are feedforward neural networks with binary weights and activation functions. In the context of using a BNN for classification, the verification problem seeks to determine whether a small perturbation of a given input can lead it to be misclassified by the BNN, and the robustness of the BNN can be measured by solving the verification problem over multiple inputs. The BNN verification problem can be formulated as an integer programming (IP) problem. However, the natural IP formulation is often challenging to solve due to a large integrality gap induced by big-$M$ constraints. We present two techniques to improve the IP formulation. First, we introduce a new method for obtaining a linear objective for the multi-class setting. Second, we introduce a new technique for generating valid inequalities for the IP formulation that exploits the recursive structure of BNNs. We find that our techniques enable verifying BNNs against a higher range of input perturbation than existing IP approaches within a limited time.

The Weyl double copy formalism, which relates the Weyl spinor with the square of the field strength, is studied in the context of Hassan-Rosen bigravity for stationary and time-dependent solutions. We consider the dyonic Kerr-Newman-(A)dS solution and the Plebański-Demiański metric in the context of bigravity. These solutions are studied in the Weyl double copy both with matter independently coupled and show that no massive modes are present in the Weyl spinor. The equations of motion for the gauge and scalar fields are those of Maxwell equations coupled to an external source, and massless Klein-Gordon equations with a conformal curvature term and an external source, all of them consistent with general relativity. For wave solutions, massive modes are manifest in the Weyl spinor and a formulation in bigravity for these massive modes is proposed. The resulting equations of motion are Proca equations with a conformal term and massive Klein-Gordon equations. In the case of the matter contributions for waves, we show how the resonance mass is present in equations of motion of the fields obtained from the Weyl double copy. The solutions studied are written in a Kerr-Schild form, connecting with the Kerr-Schild double copy.

We show that for any quantum Hamiltonian with an inverse-polynomial gap, the ground state can be prepared in a polynomial circuit depth to inverse-polynomial precision, if the system size is sufficiently large. The resulting circuit is composed of a polynomial number of Pauli rotations without ancilla qubit. Extending this result, we prove that for sufficiently large qubit number, any quantum state can be approximately prepared with a constant (polynomial) number of Pauli rotations to constant (inverse-polynomial) precision. Our theoretical findings reveal exponential quantum advantages in the prominent applications: ground state preparation, combinatorial optimization, and quantum state preparation.

This paper presents a novel algorithm for constructing a sum-of-squares (SOS) decomposition for positive semi-definite polynomials with rational coefficients. Unlike previous methods that typically yield SOS decompositions with floating-point coefficients, our approach ensures that all coefficients in the decomposition remain rational. This is particularly useful in formal verification and symbolic computation, where exact arithmetic is required. We introduce a stepwise reduction technique that transforms a given polynomial into a sum of ladder-like squares while preserving rationality. Experimental results demonstrate the effectiveness of our method compared to existing numerical approaches. This artical is an extension of the following Chinnese paper: HUANG Yong , ZENG Zhenbing , YANG Lu , RAO Yongsheng. An Algorithm to Represent Positive Semi-Definite Polynomials to Sum of Lader-Like Squares of Polynomials with Rational Coefficients (in Chinese). Journal of Systems Science and Mathematical Sciences, 2024, 44(5): 1241-1271 this https URL

We derive analytical results for continuous-time quantum walks from a new class of initial states with tunable delocalization. The dynamics are governed by a Hamiltonian with complex hopping amplitudes. We provide closed-form equations for key observables, revealing three notable findings: (1) the emergence of directed quantum transport from completely unbiased initial conditions; (2) a quantum backfire effect, where greater initial delocalization enhances short-time spreading but counterintuitively induces a comparatively smaller long-time spreading after a crossing time $t_{\mathrm{cross}}$; and (3) an exact characterization of survival probability, showing that the transition to an enhanced $t^{-3}$ decay is a fine-tuned effect. Our work establishes a comprehensive framework for controlling quantum transport through the interplay between intermediate initial delocalization and Hamiltonian phase.

Imaging inverse problems aims to recover high-dimensional signals from undersampled, noisy measurements, a fundamentally ill-posed task with infinite solutions in the null-space of the sensing operator. To resolve this ambiguity, prior information is typically incorporated through handcrafted regularizers or learned models that constrain the solution space. However, these priors typically ignore the task-specific structure of that null-space. In this work, we propose \textit{Non-Linear Projections of the Null-Space} (NPN), a novel class of regularization that, instead of enforcing structural constraints in the image domain, promotes solutions that lie in a low-dimensional projection of the sensing matrix's null-space with a neural network. Our approach has two key advantages: (1) Interpretability: by focusing on the structure of the null-space, we design sensing-matrix-specific priors that capture information orthogonal to the signal components that are fundamentally blind to the sensing process. (2) Flexibility: NPN is adaptable to various inverse problems, compatible with existing reconstruction frameworks, and complementary to conventional image-domain priors. We provide theoretical guarantees on convergence and reconstruction accuracy when used within plug-and-play methods. Empirical results across diverse sensing matrices demonstrate that NPN priors consistently enhance reconstruction fidelity in various imaging inverse problems, such as compressive sensing, deblurring, super-resolution, computed tomography, and magnetic resonance imaging, with plug-and-play methods, unrolling networks, deep image prior, and diffusion models.

We prove a coordinate- and foliation-independent Heisenberg-type lower bound for quantum states strictly localized in geodesic balls of radius $r$ on spacelike hypersurfaces of arbitrary spacetimes (with matter and a cosmological constant). The estimate depends only on the induced Riemannian geometry of the slice; it is independent of the lapse, shift, and extrinsic curvature, and controls the canonical momentum variance/uncertainty $\sigma_p$ by the first Dirichlet eigenvalue of the Laplace-Beltrami operator (Theorem). On weakly mean-convex balls we obtain the universal product inequality $\sigma_p\,r \ge \hbar/2$, whose constant is optimal and never attained (Corollary). This result abstracts and extends the framework recently introduced for black-hole slices in [10].

We introduce fast algorithms for solving $\ell_{p}$ regression problems using the iteratively reweighted least squares (IRLS) method. Our approach achieves state-of-the-art iteration complexity, outperforming the IRLS algorithm by Adil-Peng-Sachdeva (NeurIPS 2019) and matching the theoretical bounds established by the complex algorithm of Adil-Kyng-Peng-Sachdeva (SODA 2019, J. ACM 2024) via a simpler lightweight iterative scheme. This bridges the existing gap between theoretical and practical algorithms for $\ell_{p}$ regression. Our algorithms depart from prior approaches, using a primal-dual framework, in which the update rule can be naturally derived from an invariant maintained for the dual objective. Empirically, we show that our algorithms significantly outperform both the IRLS algorithm by Adil-Peng-Sachdeva and MATLAB/CVX implementations.

In recent years, a variety of learned regularization frameworks for solving inverse problems in imaging have emerged. These offer flexible modeling together with mathematical insights. The proposed methods differ in their architectural design and training strategies, making direct comparison challenging due to non-modular implementations. We address this gap by collecting and unifying the available code into a common framework. This unified view allows us to systematically compare the approaches and highlight their strengths and limitations, providing valuable insights into their future potential. We also provide concise descriptions of each method, complemented by practical guidelines.

This paper primarily considers the robust estimation problem under Wasserstein distance constraints on the parameter and noise distributions in the linear measurement model with additive noise, which can be formulated as an infinite-dimensional nonconvex minimax problem. We prove that the existence of a saddle point for this problem is equivalent to that for a finite-dimensional minimax problem, and give a counterexample demonstrating that the saddle point may not exist. Motivated by this observation, we present a verifiable necessary and sufficient condition whose parameters can be derived from a convex problem and its dual. Additionally, we also introduce a simplified sufficient condition, which intuitively indicates that when the Wasserstein radii are small enough, the saddle point always exists. In the absence of the saddle point, we solve an finite-dimensional nonconvex minimax problem, obtained by restricting the estimator to be linear. Its optimal value establishes an upper bound on the robust estimation problem, while its optimal solution yields a robust linear estimator. Numerical experiments are also provided to validate our theoretical results.

This paper proposes a novel algorithm to approximate the core of transferable utility (TU) cooperative games via linear programming. Given the computational hardness of determining the full core, our approach provides a tractable approximation by sampling extreme points through randomized linear problems (LPs). We analyze its convergence and computational complexity, and validate its effectiveness through extensive simulations on various game models. Our results show that the method is scalable and achieves high accuracy in terms of core reconstruction.

In this paper, we develop a variational foundation for stochastic thermodynamics of finite-dimensional, continuous-time systems. Requiring the second law (non-negative average total entropy production) systematically yields a consistent thermodynamic structure, from which novel generalized fluctuation-dissipation relations emerge naturally, ensuring local detailed balance. This principle extends key results of stochastic thermodynamics including an individual trajectory level description of both configurational and thermal variables and fluctuation theorems in an extended thermodynamic phase space. It applies to both closed and open systems, while accommodating state-dependent parameters, nonlinear couplings between configurational and thermal degrees of freedom, and cross-correlated noise consistent with Onsager symmetry. This is achieved by establishing a unified geometric framework in which stochastic thermodynamics emerges from a generalized Lagrange-d'Alembert principle, building on the variational structure introduced by Gay-Balmaz and Yoshimura [Phil. Trans. R. Soc. A 381, 2256 (2023)]. Irreversible and stochastic forces are incorporated through nonlinear nonholonomic constraints, with entropy treated as an independent dynamical variable. This work provides a novel approach for thermodynamically consistent modeling of stochastic systems, and paves the way to applications in continuum systems such as active and complex fluids.

This work focuses on learning non-canonical Hamiltonian dynamics from data, where long-term predictions require the preservation of structure both in the learned model and in numerical schemes. Previous research focused on either facet, respectively with a potential-based architecture and with degenerate variational integrators, but new issues arise when combining both. In experiments, the learnt model is sometimes numerically unstable due to the gauge dependency of the scheme, rendering long-time simulations impossible. In this paper, we identify this problem and propose two different training strategies to address it, either by directly learning the vector field or by learning a time-discrete dynamics through the scheme. Several numerical test cases assess the ability of the methods to learn complex physical dynamics, like the guiding center from gyrokinetic plasma physics.

We propose a reformulation of the problem of optimally controlled transitions in stochastic thermodynamics. We impose that any terminal cost specified by a thermodynamic functional should depend only on state variables and not on control protocols, according to the canonical Bolza form. In this way, we can unambiguously discriminate between transitions at minimum dissipation between genuine equilibrium states, and transitions at minimum work driving a system from a genuine equilibrium to a non-equilibrium state. For underdamped dynamics subject to a mechanical force, genuine equilibrium means a Maxwell-Boltzmann probability distribution defining a vanishing current velocity. Transitions at minimum dissipation between equilibria are a model of optimal swift engineered equilibration. Remarkably, we show that transitions at minimum work do not directly imply explicit boundary conditions on terminal values of parameters of the mechanical force and on control protocols. Thus, the problem often discussed in the literature, that optimal protocols need terminal jumps to satisfy boundary conditions, completely disappears. The quantitative properties of optimal controls are entirely determined by the form of the penalty modelling an experimental setup. More generally, we use centre manifold theory to analytically account for the tendency of optimal controls to exhibit a turnpike property: optimal protocols in the bulk of the control horizon tend to converge to a universal centre manifold determined only by the running cost. Exponential deviations from the centre manifold occur at the ends of the control horizon in order to satisfy the boundary conditions. Our findings are supported numerically.

Capturing comprehensive statistics of nonperiodic asynchronous impulsive noise is a critical issue in enhancing impulse noise processing for narrowband powerline communication (NB-PLC) transceivers. However, existing mathematical noise generative models capture only some of the characteristics of additive noise. Therefore, we propose a generative adversarial network (GAN), called the noise-generation GAN (NGGAN), that learns the complicated characteristics of practically measured noise samples for data augmentation. To closely match the statistics of complicated noise in NB-PLC systems, we measured the NB-PLC noise via the analog coupling and bandpass filtering circuits of a commercial NB-PLC modem to build a realistic dataset. Specifically, the NGGAN design approaches based on the practically measured dataset are as follows: (i) we design the length of input signals that the NGGAN model can fit to facilitate cyclo-stationary noise generation. (ii) Wasserstein distance is used as a loss function to enhance the similarity between the generated noise and the training dataset and ensure that the sample diversity is sufficient for various applications. (iii) To measure the similarity performance of the GAN-based models based on mathematical and practically measured datasets, we perform quantitative and qualitative analyses. The training datasets include (1) a piecewise spectral cyclo-stationary Gaussian model (PSCGM), (2) a frequency-shift (FRESH) filter, and (3) practical measurements from NB-PLC systems. Simulation results demonstrate that the proposed NGGAN trained using waveform characteristics is closer to the practically measured dataset in terms of the quality of the generated noise.

Circuit augmentation schemes are a family of combinatorial algorithms for linear programming that generalize the simplex method. To solve the linear program, they construct a so-called monotone circuit walk: They start at an initial vertex of the feasible region and traverse a discrete sequence of points on the boundary, while moving along certain allowed directions (circuits) and improving the objective function at each step until reaching an optimum. Since the existence of short circuit walks has been conjectured (Circuit Diameter Conjecture), several works have investigated how well one can efficiently approximate shortest monotone circuit walks towards an optimum. A first result addressing this question was given by De Loera, Kafer, and Sanità [SIAM J. Opt., 2022], who showed that given as input an LP and the starting vertex, finding a $2$-approximation for this problem is NP-hard. Cardinal and the third author [Math. Prog. 2023] gave a stronger lower bound assuming the exponential time hypothesis, showing that even an approximation factor of $O(\frac{\log m}{\log \log m})$ is intractable for LPs defined by $m$ inequalities. Both of these results were based on reductions from highly degenerate polytopes in combinatorial optimization with high dimension. In this paper, we significantly strengthen the aforementioned hardness results by showing that for every fixed $\varepsilon>0$ approximating the problem on polygons with $m$ edges to within a factor of $O(m^{1-\varepsilon})$ is NP-hard. This result is essentially best-possible, as it cannot be improved beyond $o(m)$. In particular, this implies hardness for simple polytopes and in fixed dimension.

We construct perturbations of Minkowski spacetime in general relativity, when given initial data that decays inverse polynomially to initial data of a Kerr spacetime towards spacelike infinity. We show that the perturbations admit a regular conformal compactification at null and timelike infinity, where the degree of regularity increases linearly with the rate of decay of the initial data to Kerr initial data. In particular, the compactification is smooth if the initial data decays rapidly to Kerr initial data. This generalizes results of Friedrich, who constructed spacetimes with a smooth conformal compactification in the case when the initial data is identical to Kerr initial data on the complement of a compact set. Our results rely on a novel formulation of the Einstein equations about Minkowski spacetime introduced by the author, that allows one to formulate the dynamic problem as a quasilinear, symmetric hyperbolic PDE that is regular at null infinity and with null infinity being at a fixed locus. It is not regular at spacelike infinity, due to the asymptotics of Kerr. Thus the main technical task is the construction of solutions near spacelike infinity, using tailored energy estimates. To accomplish this, we organize the equations according to homogeneity with respect to scaling about spacelike infinity, which identifies terms that are leading, respectively lower order, near spacelike infinity, with contributions from Kerr being lower order.

We consider adversarially robust classification in a multiclass setting under arbitrary loss functions and derive dual and barycentric reformulations of the corresponding learner-agnostic robust risk minimization problem. We provide explicit characterizations for important cases such as the cross-entropy loss, loss functions with a power form, and the quadratic loss, extending in this way available results for the 0-1 loss. These reformulations enable efficient computation of sharp lower bounds for adversarial risks and facilitate the design of robust classifiers beyond the 0-1 loss setting. Our paper uncovers interesting connections between adversarial robustness, $\alpha$-fair packing problems, and generalized barycenter problems for arbitrary positive measures where Kullback-Leibler and Tsallis entropies are used as penalties. Our theoretical results are accompanied with illustrative numerical experiments where we obtain tighter lower bounds for adversarial risks with the cross-entropy loss function.

Driving simulators are increasingly used in research and development. However, simulators often cause motion sickness due to downscaled motion and unscaled veridical visuals. In this paper, a motion cueing algorithm is proposed that reduces motion sickness as predicted by the subjective vertical conflict (SVC) model using model predictive control (MPC). Both sensory conflict and specific force errors are penalised in the cost function, allowing the algorithm to jointly optimise fidelity and comfort. Human-in-the-loop experiments were conducted to compare four simulator motion settings: two variations of our MPC-based algorithm, one focused on pure specific force tracking and the second compromising specific force tracking and motion sickness minimisation, as well as reference adaptive washout and no motion cases. The experiments were performed on a hexapod driving simulator with participants exposed to passive driving. Experimental motion sickness results closely matched the sickness model predictions. As predicted by the model, the no motion condition yielded the lowest sickness levels. However, it was rated lowest in terms of fidelity. The compromise solution reduced sickness by over 50% (average MISC level 3 to 1.5) compared to adaptive washout and the algorithm focusing on specific force tracking, without any significant reduction in fidelity rating. The proposed approach for developing MCA that takes into account both the simulator dynamics and time evolution of motion sickness offers a significant advancement in achieving an optimal control of motion sickness and specific force recreation in driving simulators, supporting broader simulator use.

Ising machines have emerged as promising platforms for efficiently tackling a wide range of combinatorial optimization problems relevant to resource allocation, statistical inference and deep learning, yet their practical utility is fundamentally constrained by the coarse resolution of spin-spin couplings (Jij). Current implementations, relying on direct modulation of physical parameters, achieve at most 256 discrete coupling levels, which severely hinder the faithfully modeling of arbitrary real-valued interactions in realistic applications. Here we present a novel photonic Ising machine that encodes spins in random lattices while programming couplings in the momentum space of light. By introducing the Sidon set-a mathematical structure ensuring pairwise difference uniqueness - and employing the Erdos-Turan bound, we establish an optical framework in which each spin pair can be assigned a unique Jij. This approach decouples the resolution limit from hardware modulation to the spatial precision in the momentum space of light. Experimentally, we demonstrate a record-high coupling resolution of 7,038 on a simple photonic platform, surpassing previous Ising machines. Our results highlight the power of uniting discrete mathematics with momentum-space photonics, paving the way toward scalable Ising machines capable of faithfully modeling real-world optimization problems.

Inspired by Costa's pioneering work on dirty paper coding (DPC), this paper proposes a novel scheme for integrated communication and computing (ICC), named Computing on Dirty Paper, whereby the transmission of discrete data symbols for communication, and over-the-air computation (AirComp) of nomographic functions can be achieved simultaneously over common multiple-access channels. In particular, the proposed scheme allows for the integration of communication and computation in a manner that is asymptotically interference-free, by precanceling the computing symbols at the transmitters (TXs) using DPC principles. A simulation-based assessment of the proposed ICC scheme under a single-input multiple-output (SIMO) setup is also offered, including the evaluation of performance for data detection, and of mean-squared-error (MSE) performance for function computation, over a block of symbols. The results validate the proposed method and demonstrate its ability to significantly outperform state-of-the-art (SotA) ICC schemes in terms of both bit error rate (BER) and MSE.

Multi-antenna (or MIMO) processing is a promising solution to the problem of jammer mitigation. Existing methods mitigate the jammer based on an estimate of its spatial signature that is acquired through a dedicated training phase. This strategy has two main drawbacks: (i) it reduces the communication rate since no data can be transmitted during the training phase and (ii) it can be evaded by smart or multi-antenna jammers that do not transmit during the training phase or that dynamically change their subspace through time-varying beamforming. To address these drawbacks, we propose Joint jammer Mitigation and data Detection (JMD), a novel paradigm for MIMO jammer mitigation. The core idea of JMD is to estimate and remove the jammer interference subspace jointly with detecting the legitimate transmit data over multiple time slots. Doing so removes the need for a dedicated and rate-reducing training period while being able to mitigate smart and dynamic multi-antenna jammers. We provide two JMD-type algorithms, SANDMAN and MAED, that differ in the way they estimate the channels of the legitimate transmitters and achieve different complexity-performance tradeoffs. Extensive simulations demonstrate the efficacy of JMD for jammer mitigation.

To each four dimensional $\mathcal{N}\geq 2$ supersymmetric quantum field theory, one can associate an algebraic completely integrable (ACI) system that encodes the low energy dynamics of theory. In this paper we explicitly derive the appropriate ACI systems for the global forms of $\mathcal{N}=4$ super Yang-Mills (sYM) using isogenies of polarised abelian varieties. In doing so, we relate the complex moduli of the resulting varieties to the exactly marginal coupling of the theory, thus allowing us to probe the $S$-duality groups of the global forms. Finally, we comment on whether the resulting varieties are the Jacobians of a minimal genus Riemann surface, coming to the conclusion that many global forms of $\mathcal{N}=4$ sYM do not admit a minimal genus Seiberg-Witten curve that correctly reproduces the global form.

This paper addresses a critical aerial defense challenge in contested airspace, involving three autonomous aerial vehicles -- a hostile drone (the pursuer), a high-value drone (the evader), and a protective drone (the defender). We present a cooperative guidance framework for the evader-defender team that guarantees interception of the pursuer before it can capture the evader, even under highly dynamic and uncertain engagement conditions. Unlike traditional heuristic, optimal control, or differential game-based methods, we approach the problem within a time-constrained guidance framework, leveraging true proportional navigation based approach that ensures robust and guaranteed solutions to the aerial defense problem. The proposed strategy is computationally lightweight, scalable to a large number of agent configurations, and does not require knowledge of the pursuer's strategy or control laws. From arbitrary initial geometries, our method guarantees that key engagement errors are driven to zero within a fixed time, leading to a successful mission. Extensive simulations across diverse and adversarial scenarios confirm the effectiveness of the proposed strategy and its relevance for real-time autonomous defense in contested airspace environments.

This paper addresses the problem of inverse covariance (also known as precision matrix) estimation in high-dimensional settings. Specifically, we focus on two classes of estimators: linear shrinkage estimators with a target proportional to the identity matrix, and estimators derived from data augmentation (DA). Here, DA refers to the common practice of enriching a dataset with artificial samples--typically generated via a generative model or through random transformations of the original data--prior to model fitting. For both classes of estimators, we derive estimators and provide concentration bounds for their quadratic error. This allows for both method comparison and hyperparameter tuning, such as selecting the optimal proportion of artificial samples. On the technical side, our analysis relies on tools from random matrix theory. We introduce a novel deterministic equivalent for generalized resolvent matrices, accommodating dependent samples with specific structure. We support our theoretical results with numerical experiments.

We study reinforcement learning problems where state observations are stochastically triggered by actions, a constraint common in many real-world applications. This framework is formulated as Action-Triggered Sporadically Traceable Markov Decision Processes (ATST-MDPs), where each action has a specified probability of triggering a state observation. We derive tailored Bellman optimality equations for this framework and introduce the action-sequence learning paradigm in which agents commit to executing a sequence of actions until the next observation arrives. Under the linear MDP assumption, value-functions are shown to admit linear representations in an induced action-sequence feature map. Leveraging this structure, we propose off-policy estimators with statistical error guarantees for such feature maps and introduce ST-LSVI-UCB, a variant of LSVI-UCB adapted for action-triggered settings. ST-LSVI-UCB achieves regret $\widetilde O(\sqrt{Kd^3(1-\gamma)^{-3}})$, where $K$ is the number of episodes, $d$ the feature dimension, and $\gamma$ the discount factor (per-step episode non-termination probability). Crucially, this work establishes the theoretical foundation for learning with sporadic, action-triggered observations while demonstrating that efficient learning remains feasible under such observation constraints.

Convex optimization is the powerhouse behind the theory and practice of optimization. We introduce a quantum analogue of unconstrained convex optimization: computing the minimum eigenvalue of a Schrödinger operator $h = -\Delta + V $ with convex potential $V:\mathbb R^n \rightarrow \mathbb R_{\ge 0}$ such that $V(x)\rightarrow\infty $ as $\|x\|\rightarrow\infty$. For this problem, we present an efficient quantum algorithm, called the Fundamental Gap Algorithm (FGA), that computes the minimum eigenvalue of $h$ up to error $\epsilon$ in polynomial time in $n$, $1/\epsilon$, and parameters that depend on $V$. Adiabatic evolution of the ground state is used as a key subroutine, which we analyze with novel techniques that allow us to focus on the low-energy space. We apply the FGA to give the first known polynomial-time algorithm for finding the lowest frequency of an $n$-dimensional convex drum, or mathematically, the minimum eigenvalue of the Dirichlet Laplacian on an $n$-dimensional region that is defined by $m$ linear constraints in polynomial time in $n$, $m$, $1/\epsilon$ and the radius $R$ of a ball encompassing the region.

Generalization in deep learning is closely tied to the pursuit of flat minima in the loss landscape, yet classical Stochastic Gradient Langevin Dynamics (SGLD) offers no mechanism to bias its dynamics toward such low-curvature solutions. This work introduces Flatness-Aware Stochastic Gradient Langevin Dynamics (fSGLD), designed to efficiently and provably seek flat minima in high-dimensional nonconvex optimization problems. At each iteration, fSGLD uses the stochastic gradient evaluated at parameters perturbed by isotropic Gaussian noise, commonly referred to as Random Weight Perturbation (RWP), thereby optimizing a randomized-smoothing objective that implicitly captures curvature information. Leveraging these properties, we prove that the invariant measure of fSGLD stays close to a stationary measure concentrated on the global minimizers of a loss function regularized by the Hessian trace whenever the inverse temperature and the scale of random weight perturbation are properly coupled. This result provides a rigorous theoretical explanation for the benefits of random weight perturbation. In particular, we establish non-asymptotic convergence guarantees in Wasserstein distance with the best known rate and derive an excess-risk bound for the Hessian-trace regularized objective. Extensive experiments on noisy-label and large-scale vision tasks, in both training-from-scratch and fine-tuning settings, demonstrate that fSGLD achieves superior or comparable generalization and robustness to baseline algorithms while maintaining the computational cost of SGD, about half that of SAM. Hessian-spectrum analysis further confirms that fSGLD converges to significantly flatter minima.

We define the non-commutative multiple bi-orthogonal polynomial systems, which simultaneously generalize the concepts of multiple orthogonality, matrix orthogonal polynomials and of the bi-orthogonality. We present quasideterminantal expressions for such polynomial systems in terms of formal bi-moments. The normalization functions for such monic polynomials satisfy the non-commutative Hirota equations, while the polynomials provide solution of the corresponding linear system. This shows, in particular, that our polynomial systems form a part of the theory of integrable systems. We study also a specialization of the problem to non-commutative multiple orthogonal polynomials, what results in the corresponding Hankel-type quasideterminantal expressions in terms of the moments. Moreover, such a reduction allows to introduce in a standard way the discrete-time variable and gives rise to an integrable system which is non-commutative version of the multidimensional discrete-time Toda equations.

Imputation methods play a critical role in enhancing the quality of practical time-series data, which often suffer from pervasive missing values. Recently, diffusion-based generative imputation methods have demonstrated remarkable success compared to autoregressive and conventional statistical approaches. Despite their empirical success, the theoretical understanding of how well diffusion-based models capture complex spatial and temporal dependencies between the missing values and observed ones remains limited. Our work addresses this gap by investigating the statistical efficiency of conditional diffusion transformers for imputation and quantifying the uncertainty in missing values. Specifically, we derive statistical sample complexity bounds based on a novel approximation theory for conditional score functions using transformers, and, through this, construct tight confidence regions for missing values. Our findings also reveal that the efficiency and accuracy of imputation are significantly influenced by the missing patterns. Furthermore, we validate these theoretical insights through simulation and propose a mixed-masking training strategy to enhance the imputation performance.

Quantum generalizations of the Fisher information are important in quantum information science, with applications in high energy and condensed matter physics and in quantum estimation theory, machine learning, and optimization. One can derive a quantum generalization of the Fisher information matrix in a natural way as the Hessian matrix arising in a Taylor expansion of a smooth divergence. Such an approach is appealing for quantum information theorists, given the ubiquity of divergences in quantum information theory. In contrast to the classical case, there is not a unique quantum generalization of the Fisher information matrix, similar to how there is not a unique quantum generalization of the relative entropy or the Rényi relative entropy. In this paper, I derive information matrices arising from the log-Euclidean, $\alpha$-$z$, and geometric Rényi relative entropies, with the main technical tool for doing so being the method of divided differences for calculating matrix derivatives. Interestingly, for all non-negative values of the Rényi parameter $\alpha$, the log-Euclidean Rényi relative entropy leads to the Kubo-Mori information matrix, and the geometric Rényi relative entropy leads to the right-logarithmic derivative Fisher information matrix. Thus, the resulting information matrices obey the data-processing inequality for all non-negative values of the Rényi parameter $\alpha$ even though the original quantities do not. Additionally, I derive and establish basic properties of $\alpha$-$z$ information matrices resulting from the $\alpha$-$z$ Rényi relative entropies. For parameterized thermal states, I establish formulas for their $\alpha$-$z$ information matrices and hybrid quantum-classical algorithms for estimating them, with applications in quantum Boltzmann machine learning.

We present a computational framework for synthesizing a single smooth Lyapunov function that certifies both asymptotic stability and safety. We show that the existence of a strictly compatible pair of control barrier and control Lyapunov functions (CBF-CLF) guarantees the existence of such a function on the exact safe set certified by the barrier. To maximize the certifiable safe domain while retaining differentiability, we employ a log-sum-exp (softmax) relaxation of the nonsmooth maximum barrier, together with a counterexample-guided refinement that inserts half-space cuts until a strict barrier condition is verifiable. We then patch the softmax barrier with a CLF via an explicit smooth bump construction, which is always feasible under the strict compatibility condition. All conditions are formally verified using a satisfiability modulo theories (SMT) solver, enabled by a reformulation of Farkas' lemma for encoding strict compatibility. On benchmark systems, including a power converter, we show that the certified safe stabilization regions obtained with the proposed approach are often less conservative than those achieved by state-of-the-art sum-of-squares (SOS) compatible CBF-CLF designs.

Conventional wisdom in deep learning optimization dictates updating all layers at every step-a principle followed by all recent state-of-the-art optimizers such as Muon. In this work, we challenge this assumption, showing that full-network updates can be fundamentally suboptimal, both in theory and in practice. We introduce a non-Euclidean Randomized Progressive Training method-Drop-Muon-a simple yet powerful framework that updates only a subset of layers per step according to a randomized schedule, combining the efficiency of progressive training with layer-specific non-Euclidean updates for top-tier performance. We provide rigorous convergence guarantees under both layer-wise smoothness and layer-wise $(L^0, L^1)$-smoothness, covering deterministic and stochastic gradient settings, marking the first such results for progressive training in the stochastic and non-smooth regime. Our cost analysis further reveals that full-network updates are not optimal unless a very specific relationship between layer smoothness constants holds. Through controlled CNN experiments, we empirically demonstrate that Drop-Muon consistently outperforms full-network Muon, achieving the same accuracy up to $1.4\times$ faster in wall-clock time. Together, our results suggest a shift in how large-scale models can be efficiently trained, challenging the status quo and offering a highly efficient, theoretically grounded alternative to full-network updates.

In this paper, we provide details on the proofs of the quantum polynomial time algorithm of Biasse and Song (SODA 16) for computing the $S$-unit group of a number field. This algorithm directly implies polynomial time methods to calculate class groups, S-class groups, relative class group and the unit group, ray class groups, solve the principal ideal problem, solve certain norm equations, and decompose ideal classes in the ideal class group. Additionally, combined with a result of Cramer, Ducas, Peikert and Regev (Eurocrypt 2016), the resolution of the principal ideal problem allows one to find short generators of a principal ideal. Likewise, methods due to Cramer, Ducas and Wesolowski (Eurocrypt 2017) use the resolution of the principal ideal problem and the decomposition of ideal classes to find so-called ``mildly short vectors'' in ideal lattices of cyclotomic fields.

Diffusion models have achieved state-of-the-art performance, demonstrating remarkable generalisation capabilities across diverse domains. However, the mechanisms underpinning these strong capabilities remain only partially understood. A leading conjecture, based on the manifold hypothesis, attributes this success to their ability to adapt to low-dimensional geometric structure within the data. This work provides evidence for this conjecture, focusing on how such phenomena could result from the formulation of the learning problem through score matching. We inspect the role of implicit regularisation by investigating the effect of smoothing minimisers of the empirical score matching objective. Our theoretical and empirical results confirm that smoothing the score function -- or equivalently, smoothing in the log-density domain -- produces smoothing tangential to the data manifold. In addition, we show that the manifold along which the diffusion model generalises can be controlled by choosing an appropriate smoothing.

Estimating the tangent spaces of a data manifold is a fundamental problem in data analysis. The standard approach, Local Principal Component Analysis (LPCA), struggles in high-noise settings due to a critical trade-off in choosing the neighborhood size. Selecting an optimal size requires prior knowledge of the geometric and noise characteristics of the data that are often unavailable. In this paper, we propose a spectral method, Laplacian Eigenvector Gradient Orthogonalization (LEGO), that utilizes the global structure of the data to guide local tangent space estimation. Instead of relying solely on local neighborhoods, LEGO estimates the tangent space at each data point by orthogonalizing the gradients of low-frequency eigenvectors of the graph Laplacian. We provide two theoretical justifications of our method. First, a differential geometric analysis on a tubular neighborhood of a manifold shows that gradients of the low-frequency Laplacian eigenfunctions of the tube align closely with the manifold's tangent bundle, while an eigenfunction with high gradient in directions orthogonal to the manifold lie deeper in the spectrum. Second, a random matrix theoretic analysis also demonstrates that low-frequency eigenvectors are robust to sub-Gaussian noise. Through comprehensive experiments, we demonstrate that LEGO yields tangent space estimates that are significantly more robust to noise than those from LPCA, resulting in marked improvements in downstream tasks such as manifold learning, boundary detection, and local intrinsic dimension estimation.

In this article we prove a necessary and a sufficient condition for a finite subset of the special linear group to be dominated. These conditions are purely geometric in nature, as they only involve the trace and the eigenvectors of the matrices, and can be computed explicitly. Our sufficient condition, in particular, provides a simple algorithm for constructing a dominated set with prescribed eigenvectors. The techniques involved in our proofs take advantage of the interaction between dominated sets and two-dimensional hyperbolic geometry.

We prove that a three-dimensional smooth complete intersection of two quadrics over a field k is k-rational if and only if it contains a line defined over k. To do so, we develop a theory of intermediate Jacobians for geometrically rational threefolds over arbitrary, not necessarily perfect, fields. As a consequence, we obtain the first examples of smooth projective varieties over a field k which have a k-point, and are rational over a purely inseparable field extension of k, but not over k.

Call a Laurent polynomial $W$ `complete' if its Newton polytope is full-dimensional with zero in its interior. We show that if $W$ is any complete Laurent polynomial with coefficients in the positive part of the field $K$ of generalised Puiseux series, then $W$ has a unique positive critical point $p_{crit}$. Here a generalised Puiseux series is called `positive' if the coefficient of its leading term is in $\mathbb R_{>0}$. Using the valuation on $K$ we obtain a canonically associated `tropical critical point' $d_{crit}$ in $\mathbb R^{r}$ for which we give a finite recursive construction. We show that this result is compatible with a general form of mutation, so that it can be applied in a cluster varieties setting. We also give applications to toric geometry including, via the theory of [FOOO], to the construction of canonical non-displaceable Lagrangian tori for toric symplectic manifolds.

We give a reduction of the conjugacy problem among outer automorphisms of free (and torsion-free hyperbolic) groups to specific algorithmic problems pertaining to mapping tori of polynomially growing automorphisms. We explain how to use this reduction and solve the conjugacy problem for several new classes of outer automorphisms. This proposes a path toward a full solution to the conjugacy problem for $Out (F_n)$.

The smallest eigenvalue of a graph is the smallest eigenvalue of its adjacency matrix. We show that the family of graphs with smallest eigenvalue at least $-\lambda$ can be defined by a finite set of forbidden induced subgraphs if and only if $\lambda < \lambda^*$, where $\lambda^* = \rho^{1/2} + \rho^{-1/2} \approx 2.01980$, and $\rho$ is the unique real root of $x^3 = x + 1$. This resolves a question raised by Bussemaker and Neumaier. As a byproduct, we find all the limit points of smallest eigenvalues of graphs, supplementing Hoffman's work on those limit points in $[-2, \infty)$. We also prove that the same conclusion about forbidden subgraph characterization holds for signed graphs. Our impetus for the study of signed graphs is to determine the maximum cardinality of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Denote by $N_{\alpha, \beta}(n)$ the maximum number of unit vectors in $\mathbb{R}^d$ where all pairwise inner products lie in $\{\alpha, \beta\}$ with $-1 \le \beta < 0 \le \alpha < 1$. Very recently Jiang, Tidor, Yao, Zhang and Zhao determined the limit of $N_{\alpha, \beta}(d)/d$ as $d\to\infty$ when $\alpha + 2\beta < 0$ or $(1-\alpha)/(\alpha-\beta) \in \{1,\sqrt2,\sqrt3\}$, and they proposed a conjecture on the limit in terms of eigenvalue multiplicities of signed graphs. We establish their conjecture whenever $(1-\alpha)/(\alpha - \beta) < \lambda^*$.

For two representations of some local division algebra, congruent modulo $l$, giving rise to two Harris-Taylor local systems on the corresponding Newton strata of the special fiber of a KHT Shimura varieties, we prove that the $l$-torsion of each of their cohomology groups with compact supports are isomorphic, or equivalently the free quotients of each of the cohomology groups are congruent modulo $l$. We then deduce the construction of accurate non tempered automorphic congruences for a similitude group $G/\mathbb Q$ with signature $(1,d-1)$.

We consider Jordan curves of the form $\gamma=\cup_{j=1}^n \gamma_j$ on the Riemann sphere for which each $\gamma_j$ is a hyperbolic geodesic in $(\widehat{\mathbb C} \smallsetminus \gamma)\cup \gamma_j$. These Jordan curves are characterized by their conformal welding being piecewise Möbius. We show that the Schwarzian derivatives of the uniformizing mappings of the two regions in $\widehat{\mathbb C} \smallsetminus \gamma$ form a rational function with at most second-order poles at the endpoints of $\gamma_j$ and that the poles are simple if the curve has continuous tangents. A key tool is the explicit computation of all $C^1$ geodesic pairs, namely $C^1$ chords $\gamma=\gamma_1\cup\gamma_2$ in a simply connected domain $D$ such that $\gamma_j$ is a hyperbolic geodesic in $D\smallsetminus \gamma_{3-j}$ for both $j=1$ and $j=2$.

We study the interplay between spectrum, geometry and boundary conditions for two distinguished self-adjoint realisations of the Laplacian on infinite metric graphs, the so-called riedrichs and Neumann extensions. We introduce a new criterion for compactness of the resolvent and apply this to identify a transition from purely discrete to non-empty essential spectrum among a class of infinite metric graphs, a phenomenon that seems to have no known counterpart for Laplacians on Euclidean domains of infinite volume. In the case of discrete spectrum we then prove upper and lower bounds on eigenvalues, thus extending a number of bounds previously only known in the compact setting to infinite graphs. Some of our bounds, for instance in terms of the inradius, are new even on compact graphs.

Recently, Sarkar-Scaduto-Stoffregen constructed a stable homotopy type for odd Khovanov homology, hence obtaining an action of the Steenrod algebra on Khovanov homology with $\mathbb{Z}/2\mathbb{Z}$ coefficients. Motivated by their construction we propose a way to compute the second Steenrod square. Our construction is not unique, but we can show it to be a link invariant which gives rise to a refinement of the Rasmussen $s$-invariant with $\mathbb{Z}/2\mathbb{Z}$ coefficients. We expect it to be related to the second Steenrod square arising from the Sarkar-Scaduto-Stoffregen construction.

We consider linear approximation based on function evaluations in reproducing kernel Hilbert spaces of certain analytic weighted power series kernels and stationary kernels on the interval $[-1,1]$. Both classes contain the popular Gaussian kernel $K(x, y) = \exp(-\tfrac{1}{2}\varepsilon^2(x-y)^2)$. For weighted power series kernels we derive almost matching upper and lower bounds on the worst-case error. When applied to the Gaussian kernel, our results state that, up to a sub-exponential factor, the $n$th minimal error decays as $(\varepsilon/2)^n (n!)^{-1/2}$. The proofs are based on weighted polynomial interpolation and classical polynomial coefficient estimates that we use to bound the Hilbert space norm of a weighted polynomial fooling function.

We define a knot to be half ribbon if it is the cross-section of a ribbon 2-knot, and observe that ribbon implies half ribbon implies slice. We introduce the half ribbon genus of a knot K, the minimum genus of a ribbon knotted surface of which K is a cross-section. We compute this genus for all prime knots up to 12 crossings, and many 13-crossing knots. The same approach yields new computations of the doubly slice genus. We also introduce the half fusion number of a knot K, that measures the complexity of ribbon 2-knots of which K is a cross-section. We show that it is bounded from below by the Levine-Tristram signatures, and differs from the standard fusion number by an arbitrarily large amount.

We prove an effective equidistribution result for a class of higher step nilflows, called filiform nilflows, and derive bounds on Weyl sums for higher degree polynomials with a power saving comparable to the best known, derived by J. Bourgain, C. Demeter and L. Guth and by T. Wooley from their proof of Vinogradov Main Conjecture. Our argument is based on ideas from dynamical systems (cohomological equations, invariant distributions) and on non-Abelian harmonic analysis.

We consider the solution $Y_t$ $(0\le t\le 1)$ and several approximate solutions $\hat{Y}^m_t$ of a rough differential equation driven by a fractional Brownian motion $B_t$ with the Hurst parameter $1/3<H\leq 1/2$ associated with a dyadic partition of $[0,1]$. We are interested in analysis of asymptotic error distribution of $\hat{Y}^m_t-Y_t$ as $m\to\infty$. In the preceding results, it was proved that the weak limit of $\{(2^m)^{2H-1/2}(\hat{Y}^m_t-Y_t)\}_{0\le t\le 1}$ coincides with the weak limit of $\{(2^m)^{2H-1/2}J_tI^m_t\}_{0\le t\le 1}$, where $J_t$ is the Jacobian process of $Y_t$ and $I^m_t$ is a certain weighted sum process of Wiener chaos of order $2$ defined by $B_t$. However, it is non-trivial to reduce a problem about $\hat{Y}^m_t-Y_t$ to one about $J_t$ and $I^m_t$. In this paper, we introduce an interpolation process between $Y_t$ and $\hat{Y}^m_t$, and give several estimates of the interpolation process itself and its associated processes. The analysis provides a framework to deal with the reduction problem and provides a stronger result that the difference $R^m_t=\hat{Y}^m_t-Y_t-J_tI^m_t$ is really small compared to the main term $J_tI^m_t$. More precisely, we show that $(2^m)^{2H-1/2+\varepsilon}\sup_{0\leq t\leq 1}|R^m_t|\to 0$ almost surely and in $L^p$ (for all $p>1$) for certain explicit positive number $\varepsilon>0$. As a consequence, we obtain an estimate of the convergence rate of $\sup_{0\leq t\leq 1}|\hat{Y}^m_t-Y_t|\to 0$ in $L^p$ also.

We construct a two-parameter continuum of type II blow up solutions for the energy-critical focusing NLS in dimension $ d = 3$. The solutions collapse to a single energy bubble in finite time, precisely they have the form $ u(t,x) = e^{i \alpha(t)}\lambda(t)^{\frac{1}{2}}W(\lambda(t) x) + \eta(t, x )$, $ t \in[0, T)$, $ x \in \mathbb{R}^3$, where $ W( x) = \big( 1 + \frac{|x|^2}{3}\big)^{-\frac{1}{2}}$ is the ground state solution, $\lambda(t) = (T-t)^{- \frac12 - \nu} $ for suitable $ \nu > 0 $, $ \alpha(t) = \alpha_0 \log(T - t)$ and $ T= T(\nu, \alpha_0) > 0 $. Further $ \|\eta(t) - \eta_T\|_{\dot{H}^1 \cap \dot{H}^2} = o(1)$ as $ t \to T^-$ for some $ \eta_T \in \dot{H}^{1} \cap~ \dot{H}^2$.

This study develops and analyzes a stochastic differential equation (SDE) model for the dynamics of hepatitis B virus (HBV) infection. While deterministic frameworks have yielded important insights into viral behavior, they cannot adequately describe the intrinsic randomness and fluctuations present in biological processes. To address this limitation, we construct a stochastic model incorporating multiplicative environmental noise to account for variability in infection rates, cellular mortality, and viral replication. We establish a rigorous theoretical foundation by proving the existence, uniqueness, and global positivity of solutions for all biologically relevant initial conditions. Stability properties are investigated in detail, including stability in probability and almost sure exponential stability, with particular emphasis on conditions under which random perturbations stabilize the infection-free state. Furthermore, we demonstrate the existence of a unique ergodic stationary distribution and derive convergence properties of the uninfected hepatocyte population. Numerical simulations, performed via the Euler-Maruyama method with sufficiently small time steps to ensure positivity and accuracy, validate the analytical results and illustrate the impact of stochastic fluctuations on system dynamics. The simulations confirm that environmental noise can induce viral extinction even in parameter regimes where deterministic analysis predicts persistence. These findings enhance the mathematical understanding of HBV infection dynamics and underscore the significant role of stochastic effects in shaping long-term disease outcomes.

A classic three-jug puzzle asks, given three jugs $A$, $B$, and $C$ with fixed maximum capacities, with jug $A$ filled with wine to its maximum capacity, whether is it possible to divide the wine into two halves by pouring it from one jug to another without using any other measuring devices. However, we consider a generic version of the three-jug puzzle and present an independent graph-theoretic model to determine whether the puzzle has a solution at all. If it has a solution, then the same can be determined using this model. We also present the sketch of an algorithm to determine the solution of the puzzle.

It follows from the Grothendieck-Ogg-Shafarevich formula that the rank of an abelian variety (with trivial trace) defined over the function field of a curve is bounded by a quantity which depends on the genus of the base curve and on bad reduction data. Using a function field version of classical $\ell$-descent techniques, we derive an arithmetic refinement of this bound, extending previous work of the second and third authors from elliptic curves to abelian varieties, and improving on their result in the case of elliptic curves. When the abelian variety is the Jacobian of a hyperelliptic curve, we produce a more explicit $2$-descent map. Then we apply this machinery to studying points on the Jacobians of certain genus $2$ curves over $k(t)$, where $k$ is some perfect base field of characteristic not $2$.

In this paper, we establish new bounds for classical prime-counting functions. All of our bounds are explicit and assume the Riemann Hypothesis. First, we prove that $|\psi(x) - x|$ and $|\vartheta(x) - x|$ are bounded from above by $$\frac{\sqrt{x}\log{x}(\log{x} - \log\log{x})}{8\pi}$$ for all $x\geq 101$ and $x \geq 2\,657$ respectively, where $\psi(x)$ and $\vartheta(x)$ are the Chebyshev $\psi$ and $\vartheta$ functions. Using the extra precision offered by these results, we also prove new explicit descriptions for the error in each of Mertens' theorems which improve earlier bounds by Schoenfeld.

A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$, denoted by $r_F(H)$, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This generalizes the Ramsey number and size Ramsey number of a graph. Addressing a question of Spiro, we prove that \[r_{K_3}(K_t)=\binom{r(K_t)}{3}\] for all sufficiently large $t$. We do so through a result on graph coloring: there exists an absolute constant $K$ such that every $r$-chromatic graph where every edge is contained in at least $K$ triangles must contain at least $\binom{r}{3}$ triangles in total.

We study simple set-theoretic solutions of the Yang-Baxter equation that are finite and non-degenerate. Such retractable solutions are fully described and to investigate the irretracble solutions we give a new algebraic method. Our approach includes and extends the work of Joyce for quandles and Castelli for involutive solutions, demonstrating that the simplicity of a solution can be understood through its associated permutation skew left brace. In particular, we show that this skew left brace must have the smallest non-zero ideal, and the quotient by this ideal gives a trivial skew left brace of cyclic type; clearly all simple skew left braces satisfy these assumptions. As an application of our approach we construct and characterise new infinite families of simple solutions that are neither involutive nor quandles. Additionally, we show that our method can be applied to simple skew left braces to generate further families of simple solutions.

Wiener spaces are in many ways the decisive setting for fundamental results on Gaussian measures: large deviations (Schilder), quasi-invariance (Cameron--Martin), differential calculus (Malliavin), support description (Stroock--Varadhan), concentration of measure (Fernique), etc. Analogues of these classical results have been derived in the "enhanced" context of Gaussian rough paths and, more recently, regularity structures equipped with Gaussian models. The aim of this article is to propose a similar notion directly on this enhanced level - an abstract Wiener model space - that encompasses the aforementioned. More specifically, we focus here on enhanced Schilder type results, Cameron--Martin shifts and Fernique estimates, offering a somewhat unified view on results of Friz--Victoir and Hairer--Weber.

For any CDG-ring $B^\bullet=(B^*,d,h)$, we show that the homotopy category of graded-projective (left) CDG-modules over $B^\bullet$ is equivalent to the quotient category of the homotopy category of graded-flat CDG-modules by its full triangulated subcategory of flat CDG-modules. The contraderived category (in the sense of Becker) $\mathsf D^{\mathsf{bctr}}(B^\bullet{-}\mathbf{Mod})$ is the common name for these two triangulated categories. We also prove that the classes of cotorsion and graded-cotorsion CDG-modules coincide, and the contraderived category of CDG-modules is equivalent to the homotopy category of graded-flat graded-cotorsion CDG-modules. Assuming the graded ring $B^*$ to be graded right coherent, we show that the contraderived category $\mathsf D^{\mathsf{bctr}}(B^\bullet{-}\mathbf{Mod})$ is compactly generated and its full subcategory of compact objects is anti-equivalent to the full subcategory of compact objects in the coderived category of right CDG-modules $\mathsf D^{\mathsf{bco}}(\mathbf{Mod}{-}B^\bullet)$. Specifically, the latter triangulated category is the idempotent completion of the absolute derived category of finitely presented right CDG-modules $\mathsf D^{\mathsf{abs}}(\mathbf{mod}{-}B^\bullet)$.

We show that the extension of Presburger arithmetic by a quadratic generalised polynomial of a specific form is undecidable.

Most existing necessary conditions for optimal control based on adjoining methods require both state and costate information, yet the unobservability of costates for a given feasible trajectory impedes the determination of optimality in practice. This paper establishes a novel theoretical framework for time-optimal control of controllable linear systems with a single input, proposing the augmented switching law (ASL) that represents the input control and the feasibility in a compact form. Given a feasible trajectory, the perturbed trajectory under the constraints of ASL is guaranteed to be feasible, resulting in a novel state-centric necessary condition without dependence on costate information. A first-order necessary condition is proposed that the Jacobian matrix of the ASL is not of full row rank, which also results in a potential approach to optimizing a given feasible trajectory with the preservation of arc structures. The proposed necessary condition is applied to high-order chain-of-integrator systems with full box constraints, contributing to some theoretical results challenging to reason by costate-based conditions.

These are expanded notes from graduate courses about Lie algebras and Chevalley groups held at the University of Stuttgart. In the 1950s Chevalley showed how linear groups over arbitrary fields could be obtained~ -- ~by a uniform procedure~ -- ~from the simple Lie algebras over $\C$ occurring in the Cartan--Killing classification. Together with subsequent variations, Chevalley's work had a profound and long-lasting impact on group theory and Lie theory in general. Classical, and widely used references are the lectures notes by Steinberg (1967) and the monograph by Carter (1972). Our aim here is to present a self-contained introduction to the theory of Chevalley groups, based on recent simplifications arising from Lusztig's fundamental theory of ``canonical bases''. A further feature of our text is that we explicitly incorporate algorithmic methods in our treatment, both for the handling of substantial examples and regarding some aspects of the general theory. Eventually, this may turn into a book project.

In this paper, the asymptotic behaviors of the transition probability for two-level avoided crossings are studied under the limit where two parameters (adiabatic parameter and energy gap parameter) tend to zero. This is a continuation of our previous works where avoided crossings are generated by tangential intersections and obey a non-adiabatic regime. The main results elucidate not only the asymptotic expansion of transition probability but also a quantum interference caused by several avoided crossings and a coexistence of two-parameter regimes arising from different vanishing orders.

We prove that the quotient of the Brauer group of a product of varieties over k by the sum of the images of the Brauer groups of factors has finite exponent. The bulk of the proof concerns p-primary torsion in characteristic p. Our approach gives a more direct proof of the boundedness of the p-primary torsion of the Brauer group of an abelian variety, as recently proved by D'Addezio. We show that the transcendental Brauer group of a Kummer surface over k has finite exponent, but can be infinite when k is an infinite field of positive characteristic. This answers a question of Zarhin and the author.

We extend classical bootstrap percolation by introducing two concurrent, competing processes on an Erdős--Rényi random graph $G(n,p_n)$. Each node can assume one of three states: red, black, or white. The process begins with $a_R^{(n)}$ randomly selected active red seeds and $a_B^{(n)}$ randomly selected active black seeds, while all other nodes start as white and inactive. White nodes activate according to independent Poisson clocks with rate 1. Upon activation, a white node evaluates its neighborhood: if its red (black) active neighbors exceed its black (red) active neighbors by at least a fixed threshold $r \geq 2$, the node permanently becomes red (black) and active. Model's key parameters are $r$ (fixed), $n$ (tending to $\infty$), $a_R^{(n)}$, $a_B^{(n)}$, and $p_n$. We investigate the final sizes of the active red ($A^{*(n)}_R$) and black ($A^{*(n)}_B$) node sets across different parameter regimes. For each regime, we determine the relevant time scale and provide detailed characterization of asymptotic dynamics of the two concurrent activation processes.

This paper concerns the characteristic-$p$ fibers of $\mathsf{GU}(q-2,2)$ Shimura varieties, which classify abelian varieties with additional structure. These Shimura varieties admit two stratifications of interest: the Ekedahl-Oort stratification, based on the isomorphism class of the $p$-torsion subgroup scheme, and the Newton stratification, based on the isogeny class of the $p$-divisible group. In this paper, we present several novel techniques that give a better understanding of the Ekedahl-Oort stratification and of the interaction between the two stratifications for a general signature $(q-2,2)$.

Our focus is on the fast diffusion equation driven by the $p$-Laplacian operator, that is $\partial_t u=\Delta_p u$ with $1<p<2$, posed in the whole space $\mathbb{R}^N$, $N\geq 2$. The nonnegative solutions are expected to converge in time toward a stationary profile. While such convergence had been previously established for $p$ close to $2$, no quantitative rates were known, and the asymptotic behaviour remained poorly understood across the full fast diffusion range. In fact, the long time behaviour of solutions to the $p$-Laplace Cauchy problem drastically change in different subranges of the $p$. Some of them are analysed here for the first time. In this work, we provide the convergence rates for nonnegative, integrable solutions in the so-called good fast diffusion range, $p_c=\tfrac{2N}{N+1} <p<2$, where mass is conserved. We prove that solutions converge to a self-similar profile with matching mass, with explicit rates measured in relative error. Our constructive proof is based on a new entropy method that remains effective even when the entropy is not displacement convex -- where optimal transport techniques fail. In the very fast diffusion range $1<p<p_c$, we give the first asymptotic analysis near the extinction time. We uncover new critical exponents -- especially in high dimensions -- that give rise to markedly different qualitative behaviour depending on the value of $p$. We also establish convergence rates for the gradients of radial solutions in the good fast diffusion range, again measured in relative error. Finally, we analyze the structural properties required for the entropy method to apply, thereby opening a broader investigation into the basin of attraction of Barenblatt-type profiles, particularly in the singular case of $p$ close to $1$.

We study $\mathsf{RCD}$-spaces $(X,d,\mathfrak{m})$ with group actions by isometries preserving the reference measure $\mathfrak{m}$ and whose orbit space has dimension one, i.e. cohomogeneity one actions. To this end we prove a Slice Theorem asserting that when $X$ is non-collapsed the slices are homeomorphic to metric cones over homogeneous spaces with $\mathrm{Ric} \geq 0$. As a consequence we obtain complete topological structural results (also in the collapsed case) and a regular orbit representation theorem. Conversely, we show how to construct new $\mathsf{RCD}$-spaces from a cohomogeneity one group diagram, giving a complete description of $\mathsf{RCD}$-spaces of cohomogeneity one. As an application of these results we obtain the classification of cohomogeneity one, non-collapsed $\mathsf{RCD}$-spaces of essential dimension at most $4$.

Efficiency of routing on a regular digraph often involves finding opitmal properties of the graph. For example, the diameter of a digraph is the maximum distance between any two vertices. We show how we can study these problems algebraically in terms of quasigroups, 1-factors, and permutation groups. Our investigation originated from the study of graphs as the Cayley graphs of groupoids with $d$ generators, a left identity, and right cancellation; that is, a right quasigroup. This enables us to provide compact algebraic definitions for some important graphs that are either given as explicit edge lists or as the Cayley coset graphs of groups larger than the graph. One such example is a single expression for the Hoffman-Singleton graph. From there, we notice that the groupoids can be represented uniquely by a set of disjoint permutations and we explore the consequences of that observation.

We prove a simple identity relating the length spectrum of a Riemann surface to that of the same surface with an arbitrary number of additional cusps. Our proof uses the Brownian loop measure introduced by Lawler and Werner. In particular, we express the total mass of Brownian loops in a fixed free homotopy class on any Riemann surface in terms of the length of the geodesic representative for the complete constant curvature metric. This expression also allows us to write the electrical thickness of a compact set in $\mathbb C$ separating $0$ and $\infty$, or the Velling--Kirillov Kähler potential, in terms of the Brownian loop measure and the zeta-regularized determinant of Laplacian as a renormalization of the Brownian loop measure with respect to the length spectrum.

In multi-objective optimization, set-based quality indicators are a cornerstone of benchmarking and performance assessment. They capture the quality of a set of trade-off solutions by reducing it to a scalar number. One of the most commonly used set-based metrics is the R2 indicator, which describes the expected utility of a solution set to a decision-maker under a distribution of utility functions. Typically, this indicator is applied by discretizing the latter distribution, yielding a weakly Pareto-compliant indicator. In consequence, adding a nondominated or dominating solution to a solution set may -- but does not have to -- improve the indicator's value. In this paper, we reinvestigate the R2 indicator under the premise that we have a continuous, uniform distribution of (Tchebycheff) utility functions. We analyze its properties in detail, demonstrating that this continuous variant is indeed Pareto-compliant -- that is, any beneficial solution will improve the metric's value. Additionally, we provide efficient computational procedures that (a) compute this metric for bi-objective problems in $\mathcal O (N \log N)$, and (b) can perform incremental updates to the indicator whenever solutions are added to (or removed from) the current set of solutions, without needing to recompute the indicator for the entire set. As a result, this work contributes to the state-of-the-art Pareto-compliant unary performance metrics, such as the hypervolume indicator, offering an efficient and promising alternative.

Druzkowski s reduction of the Jacobian Conjecture gives rise to a class of highly complicated linear equations with polynomial coefficients. We find an unrecorded algebraic identity and apply it as a variant of the Laplace determinant expansion formula to solve such equation in a way analogous to(and no less elegant than) Cramer rule. Three consecutive cases are addressed where the situation upgrades from one case to the next.

We prove that the transcendental Brauer group of a K3 surface X over a finitely generated field k is finite, unless k has positive characteristic p and X is supersingular, in which case it is annihilated by p.

We generalize random-to-random shuffling from a Markov chain on the symmetric group to one on the Type A Iwahori Hecke algebra, and show that its eigenvalues are polynomials in q with non-negative integer coefficients. Setting q=1 recovers results of Dieker and Saliola, whose computation of the spectrum of random-to-random in the symmetric group resolved a nearly 20 year old conjecture by Uyemura-Reyes. Our methods simplify their proofs by drawing novel connections to the Jucys-Murphy elements of the Hecke algebra, Young seminormal forms, and the Okounkov-Vershik approach to representation theory.

Given a K3^[2]-type manifold X with a symplectic involution i, the quotient X/i admits a Nikulin orbifold Y as terminalization. We study the symplectic action of a group G of order 4 on X, such that i belongs to G, and the natural involution induced on Y (the two groups give two different results). We give a lattice-theoretic classification of X and Y in the projective case, and give some explicit examples of models of X. We also give lattice-theoretic criteria that a Nikulin-type orbifold N has to satisfy to admit a symplectic involution that deforms to an induced one.

We show small and large Carleson perturbation results for the parabolic Regularity boundary value problem with boundary data in $\dot{L}_{1,1/2}^p$ and small Carelson perturbation results for the Neumann problem with boundary data in $L^p$. The operator we consider is $L:=\partial_t -\mathrm{div}(A\nabla\cdot)$ and the domains are parabolic cylinders $\Omega=\mathcal{O}\times\mathbb{R}$, where $\mathcal{O}$ is a Lipschitz domain.

Recent studies have shown that many nonconvex machine learning problems satisfy a generalized-smooth condition that extends beyond traditional smooth nonconvex optimization. However, the existing algorithms are not fully adapted to such generalized-smooth nonconvex geometry and encounter significant technical limitations on their convergence analysis. In this work, we first analyze the convergence of adaptively normalized gradient descent under function geometries characterized by generalized-smoothness and generalized PŁ condition, revealing the advantage of adaptive gradient normalization. Our results provide theoretical insights into adaptive normalization across various this http URL stochastic generalized-smooth nonconvex optimization, we propose \textbf{I}ndependent-\textbf{A}daptively \textbf{N}ormalized \textbf{S}tochastic \textbf{G}radient \textbf{D}escent algorithm, which leverages adaptive gradient normalization, independent sampling, and gradient clipping to achieve an $\mathcal{O}(\epsilon^{-4})$ sample complexity under relaxed noise assumptions. Experiments on large-scale nonconvex generalized-smooth problems demonstrate the fast convergence of our algorithm.

We obtain some properties of a class $\mathcal{A}$ of $q$-hypergeometric orthogonal polynomials with $q=-1$, described by a uniform parametrization of the recurrence coefficients. We construct a class $\mathcal{C}$ of complementary $-1$ polynomials by means of the Darboux transformation with a shift. We show that our classes contain the Bannai-Ito polynomials and their complementary polynomials and other known $-1$ polynomials. We introduce some new examples of $-1$ polynomials and also obtain matrix realizations of the Bannai-Ito algebra.

For a closed negatively monotone symplectic manifold, we construct quasi-isometric embeddings from the Euclidean spaces to its Hamiltonian diffeomorphism group, assuming it contains an incompressible heavy Lagrangian. We also show the super-heaviness of its skeleton with respect to a Donaldson hypersurface.

This work presents a new three-operator splitting method to handle monotone inclusion and convex optimization problems. The proposed splitting serves as another natural extension of the Douglas-Rachford splitting technique to problems involving three operators. For solving a composite convex minimization of a sum of three functions, its formula resembles but is different from Davis-Yin splitting and the dual formulation of the classical three-block ADMM. Numerical tests suggest that such a splitting scheme is robust in the sense of allowing larger step sizes. When two functions have orthogonal domains, the splitting operator can be proven 1/2-averaged, which implies convergence of the iteration scheme using any positive step size.

In this paper, we introduce a comprehensive axiomatization of structure-preserving discretization through the framework of commutative diagrams. By establishing a formal language that captures the essential properties of discretization processes, we provide a rigorous foundation for analyzing how various structures (such as algebraic, geometric, and topological features) are maintained during the transition from continuous to discrete settings. Specifically, we establish that the transition from continuous to discrete differential settings invariably leads to noncommutative structures, reinforcing previous observation on the interplay between discretization and noncommutativity. We demonstrate the applicability of our axiomatization by applying it to the Berezin-Toeplitz quantization, showing that this quantization method adheres to our proposed criteria for structure-preserving discretization. We establish in this setting a precise limit theorem for the approximation of the Laplacian by a sequence of matrix approximations. This work not only enriches the theoretical understanding of the nature of discretization but also sets the stage for further exploration of its applications across various discretization methods.

Given a tied link $L$, the invariant $\langle\langle\cdot\rangle\rangle$ generalizes the Kauffman bracket of classical links. However, the analogues of Kauffman states and their relationship to this invariant are not immediately clear. We address this question by defining the Aicardi-Juyumaya states, and show that the contribution of each AJ-state to $\langle\langle\cdot\rangle\rangle$ does not depend on the chosen resolution tree. We also present an algorithm to compute the double bracket of a tied link diagram, and use it to find pairs of examples of (oriented) tied links sharing the same Homflypt polynomial but different tied Jones polynomial.

We consider three conditions on metric manifolds with finite volume: (1) the existence of a metric fundamental class, (2) local index bounds for Lipschitz maps, and (3) Gromov--Hausdorff approximation with volume control by bi-Lipschitz manifolds. Condition (1) is known for metric manifolds satisfying the LLC condition by work of Basso--Marti--Wenger, while (3) is known for metric surfaces by work of Ntalampekos--Romney. We prove that for metric manifolds with finite Nagata dimension, all three conditions are equivalent and that without assuming finite Nagata dimension, (1) implies (2) and (3) implies (1). As a corollary we obtain a generalization of the approximation result of Ntalampekos--Romney to metric manifolds of dimension $n\ge 2$, which have the LLC property and finite Nagata dimension.

Motivated by a recent work of Schippa (2022), we consider local smoothing estimates for Schrödinger equations in modulation spaces. By using the Córdoba-Fefferman type reverse square function inequality and the bilinear Strichartz estimate, we can refine the summability exponent of modulation spaces. Next, we will also discuss a new type of randomized Strichartz estimate in modulation spaces. Finally, we will show that the reverse function estimate implies the Strichartz estimates in modulation spaces. From this implication, we obtain the reverse square function estimate of critical order.

Commutative analogues of Clifford algebras are algebras defined in the same way as Clifford algebras except that their generators commute with each other, in contrast to Clifford algebras in which the generators anticommute. In this paper, we solve the problem of finding multiplicative inverses in commutative analogues of Clifford algebras by introducing a matrix representation for these algebras and the notion of determinant in them. We give a criteria for checking if an element has a multiplicative inverse or not and, for the first time, explicit formulas for multiplicative inverses in the case of arbitrary dimension. The new theorems involve only operations of conjugation and do not involve matrix operations. We also consider notions of trace and other characteristic polynomial coefficients and give explicit formulas for them without using matrix representations.

We make an application of ideas from partition theory to a problem in multiplicative number theory. We propose a deterministic model of prime number distribution, from first principles related to properties of integer partitions, that naturally predicts the prime number theorem as well as the twin prime conjecture. The model posits that, for $n\geq 2$, $$p_{n}\ =\ 1\ +\ 2\sum_{j=1}^{n-1}\left\lceil \frac{d(j)}{2}\right\rceil\ +\ \varepsilon(n),$$ where $p_k$ is the $k$th prime number, $d(k)$ is the divisor function, and $\varepsilon(k)$ is an explicit error term that is negligible asymptotically; both the main term and error term represent enumerative functions in our conceptual model. We refine the error term to give numerical estimates of $\pi(n)$ similar to those provided by the logarithmic integral, and much more accurate than $\operatorname{li}(n)$ up to $n=10{,}000$ where the estimates are {\it almost exact}. We then perform computational tests of unusual predictions of the model, finding limited evidence of predictable variations in prime gaps.

In [J. Wen, Y. Shi, Stat. Probab. Lett. 156 (2020) 108599] the authors first introduced a kind of anticipated backward stochastic Volterra integral equations (anticipated BSVIEs, for short). By virtue of the duality principle, it is found in this paper that the anticipated BSVIEs can be applied to the study of stochastic differential games. Naturally, in order to develop the relevant theories and applications of BSVIEs, in this paper we deeply investigate a more general class of anticipated BSVIEs whose generator includes both pointwise time-advanced functions and average time-advanced functions. In theory, the well-posedness and the comparison theorem of anticipated BSVIEs are established, and some regularity results of adapted M-solutions are proved by applying Malliavin calculus, which cover the previous results for BSVIEs. Further, using linear anticipated BSVIEs as the adjoint equation, we present the maximum principle for the nonzero-sum differential game system of stochastic delay Volterra integral equations (SDVIEs, for short) for the first time. As one of the applications of the principle, a Nash equilibrium point of the linear-quadratic differential game problem of SDVIEs is obtained.

Denote the symmetric group of degree $n$ by $S_n$. Let $\rho$ be an irreducible representation of $S_n$ over the field of complex numbers and $\sigma\in S_n$. In this paper, we describe the set of eigenvalues of $\rho(\sigma)$. Based on this result, we also obtain a description in the case of alternating groups.

Heating generated by high-intensity focused ultrasound waves is central to many emerging medical applications, including non-invasive cancer therapy and targeted drug delivery. In this study, we aim to gain a fundamental understanding of numerical simulations in this context by analyzing conforming finite element approximations of the underlying nonlinear models that describe ultrasound-heat interactions. These models are based on a coupling of a nonlinear Westervelt--Kuznetsov acoustic wave equation to the heat equation with a pressure-dependent source term. A particular challenging feature of the system is that the acoustic medium parameters may depend on the temperature. The core of our new arguments in the \emph{a prior} error analysis lies in devising energy estimates for the coupled semi-discrete system that can accommodate the nonlinearities present in the model. To derive them, we exploit the parabolic nature of the system thanks to the strong damping present in the acoustic component. Theoretically obtained optimal convergence rates in the energy norm are confirmed by the numerical experiments. In addition, we conduct a further numerical study of the problem, where we simulate the propagation of acoustic waves in liver tissue for an initially excited profile and under high-frequency sources.

The diffraction spectra of the Hat and Spectre monotile tilings, which are known to be pure point, are derived and computed explicitly. This is done via model set representatives of self-similar members in the topological conjugacy classes of the Hat and the Spectre tiling, which are the CAP and the CASPr tiling, respectively. This is followed by suitable reprojections of the model sets to represent the original Hat and Spectre tilings, which also allows to calculate their Fourier--Bohr coefficients explicitly. Since the windows of the underlying model sets have fractal boundaries, these coefficients need to be computed via an exact renormalisation cocycle in internal space.

We consider a free energy functional defined on probability densities on the unit sphere $\mathbb{S}^d$, and investigate its global minimizers. The energy consists of two components: an entropy and a nonlocal interaction energy, which favour spreading and aggregation behaviour, respectively. We find a threshold value for the size of the attractive interactions, and establish the global energy minimizers in each case. The bifurcation at this threshold value is investigated. We also generalize the results to spaces consisting of an arbitrary number of spheres (e.g., the flat torus $\mathbb{S}^1 \times \mathbb{S}^1$).

We prove that the rescaled ``true'' self-avoiding walk $(n^{-2/3}X_{\lfloor nt \rfloor})_{t\in\mathbb{R}_+}$ converges weakly as $n$ goes to infinity to the ``true'' self-repelling motion constructed by Tóth and Werner. The proof features a joint generalized Ray-Knight theorem for the rescaled local times processes and their merge and absorption points as the main tool for showing both the tightness and convergence of the finite dimensional distributions. Thus, our result can be seen as an example of establishing a functional limit theorem for a family of processes by inverting the joint generalized Ray-Knight theorem.

We study the fundamental problem of calibrating a linear binary classifier of the form $\sigma(\hat{w}^\top x)$, where the feature vector $x$ is Gaussian, $\sigma$ is a link function, and $\hat{w}$ is an estimator of the true linear weight $w^\star$. By interpolating with a noninformative $\textit{chance classifier}$, we construct a well-calibrated predictor whose interpolation weight depends on the angle $\angle(\hat{w}, w_\star)$ between the estimator $\hat{w}$ and the true linear weight $w_\star$. We establish that this angular calibration approach is provably well-calibrated in a high-dimensional regime where the number of samples and features both diverge, at a comparable rate. The angle $\angle(\hat{w}, w_\star)$ can be consistently estimated. Furthermore, the resulting predictor is uniquely $\textit{Bregman-optimal}$, minimizing the Bregman divergence to the true label distribution within a suitable class of calibrated predictors. Our work is the first to provide a calibration strategy that satisfies both calibration and optimality properties provably in high dimensions. Additionally, we identify conditions under which a classical Platt-scaling predictor converges to our Bregman-optimal calibrated solution. Thus, Platt-scaling also inherits these desirable properties provably in high dimensions.

A generalization of the determinant appears in particle physics in effective Lagrangian interaction terms that model the chiral anomaly in Quantum Chromodynamics (PRD 97 (2018) 9, 091901 PRD 109 (2024) 7, L071502), in particular in connection to mesons. This \textit{polydeterminant function}, known in the mathematical literature as a mixed discriminant, associates $N$ distinct $N\times N$ complex matrices into a complex number and reduces to the usual determinant when all matrices are taken as equal. Here, we explore the main properties of the polydeterminant applied to (quantum) fields by using a formalism and a language close to high-energy physics approaches. We discuss its use as a tool to write down novel chiral anomalous Lagrangian terms and present an explicit illustrative model for mesons. Finally, the extension of the polydeterminant as a function of tensors is shown.

Let $I(\Delta)^{[k]}$ denote the $k^{\text{th}}$ square-free power of the facet ideal of a simplicial complex $\Delta$ in a polynomial ring $R$. Square-free powers are intimately related to the `Matching Theory' and `Ordinary Powers'. In this article, we show that if $\Delta$ is a Cohen-Macaulay simplicial forest, then $R/I(\Delta)^{[k]}$ is Cohen-Macaulay for all $k\ge 1$. This result is quite interesting since all ordinary powers of a graded radical ideal can never be Cohen-Macaulay unless it is a complete intersection. To prove the result, we introduce a new combinatorial notion called special leaf, and using this, we provide an explicit combinatorial formula of $\mathrm{depth}(R/I(\Delta)^{[k]})$ for all $k\ge 1$, where $\Delta$ is a Cohen-Macaulay simplicial forest. As an application, we show that the normalized depth function of a Cohen-Macaulay simplicial forest is nonincreasing.

We show that $\kgl$-linear cohomology theories over an affine Dedekind scheme $S$ admit a canonical weight filtration on resolvable motives without inverting residual characteristics. Combined with upcoming work of Annala--Hoyois--Iwasa, this endows essentially all known logarithmic cohomology theories with weight filtrations when evaluated on projective sncd pairs $(X,D)$ over $S$. Furthermore, the weight-filtered cohomology is an invariant of the open part $U = X-D$. On variants of de Rham cohomology, we show that our weight filtration recovers the décalaged pole-order filtration defined by Deligne. One interpretation of this is that the spectral sequence associated to the pole-order filtration is an invariant of $U$ from the $E_2$-page onwards, which generalizes a result of Deligne from characteristic 0 to positive and mixed characteristic, and suggests that ``mixed Hodge theory'' is a useful invariant of $S$-schemes. Finally, we compute explicit examples of weight filtered pieces of cohomology theories. One of the computations reproves a slight weakening of a result of Thuillier stating that the singular cohomology of the dual complex associated to the boundary divisor of a good projective compactification does not depend on the chosen compactification. In the appendix, we prove the folklore results that the Whitehead tower functor is fully faithful and that perfect bivariant pairings with respect to the twisted arrow category correspond to duality.

The critical value of an eikonal equation is the unique value of a parameter for which the equation admits solutions and is deeply related to the effective Hamiltonian of a corresponding homogenization problem. We study approximation strategies for the critical value of eikonal equations posed on networks. They are based on the large time behavior of corresponding time-dependent Hamilton-Jacobi equations. We provide error estimates and some numerical tests, showing the performance and the convergence properties of the proposed algorithms.

Hopf braces are the quantum analogues of skew braces and, as such, their cocommutative counterparts provide solutions to the quantum Yang-Baxter equation. We investigate various properties of categories related to Hopf braces. In particular, we prove that the category of Hopf braces is accessible while the category of cocommutative Hopf braces is even locally presentable. We also show that functors forgetting multiple antipodes and/or multiplications down to coalgebras are monadic. Colimits in the category of cocommutative Hopf braces are described explicitly and a free cocommutative Hopf brace on an arbitrary cocommutative Hopf algebra is constructed.

Switching is an operation on a graph that does not change the spectrum of the adjacency matrix, thus producing cospectral graphs. An important activity in the field of spectral graph theory is the characterization of graphs by their spectrum. Hence, switching provides a tool for disproving the existence of such a characterization. This paper presents a general framework for counting the number of graphs that have a non-isomorphic cospectral graph through a switching method, expanding on the work by Haemers and Spence [European Journal of Combinatorics, 2004]. Our framework is based on a different counting approach, which allows it to be used for all known switching methods for the adjacency matrix. From this, we derive asymptotic results, which we complement with computer enumeration results for graphs up to $10$ vertices.

Tropical caustic of a convex domain on the plane is a canonically associated tropical analytic curve inside the domain. In this note we give a graphical proof for the classification of its intermediate vertices, implying in particular that they are always trivalent. Apart from that we explain how various known examples of tropical caustics are constructed and discuss the possibility of relaxing the convexity condition for the domain.

This study investigates the asymptotic dynamics of solutions to the Fokker-Planck-Kolmogorov (FPK) equation, with a specific focus on ship roll stability in dynamic sea conditions. Utilizing a fourth-order filter, we conduct a thorough analysis of the time evolution of the probability distributions for roll angles, roll speeds, and roll excitations. Our theoretical framework provides new insights into the long-term behavior of these systems, emphasizing the role of stochastic perturbations. Key findings reveal that the probability of capsizing remains constant over time, offering significant contributions to the stability assessment of maritime vessels under uncertain environmental conditions. This work paves the way for more robust models in maritime engineering and dynamic stability analysis.

Many geometric machine learning and image analysis applications, require a left-invariant metric on the 5D homogeneous space of 3D positions and orientations SE(3)/SO(2). This is done in Equivariant Neural Networks (G-CNNs), or in PDE-Based Group Convolutional Neural Networks (PDE-G-CNNs), where the Riemannian metric enters in multilayer perceptrons, message passing, and max-pooling over Riemannian balls. In PDE-G-CNNs it is proposed to take the minimum left-invariant Riemannian distance over the fiber in SE(3)/SO(2), whereas in G-CNNs and in many geometric image processing methods an efficient SO(2)-conjugation invariant section is advocated. The conjecture rises whether that computationally much more efficient section indeed always selects distance minimizers over the fibers. We show that this conjecture does NOT hold in general, and in the logarithmic norm approximation setting used in practice we analyze the small (and sometimes vanishing) differences. We first prove that the minimal distance section is reached by minimal horizontal geodesics with constant momentum and zero acceleration along the fibers, and we generalize this result to (reductive) homogeneous spaces with legal metrics and commutative structure groups.

Polynomial convergence bounds are considered for left, right, and split preconditioned GMRES. They include the cases of Weighted and Deflated GMRES for a linear system Ax = b. In particular, the case of positive definite A is considered. The well-known polynomial bounds are generalized to the cases considered, and then reduced to solving a min-max problem on rectangles on the complex plane. Several approaches are considered and compared. The new bounds can be improved by using specific deflation spaces and preconditioners. This in turn accelerates the convergence of GMRES. Numerical examples illustrate the results obtained.

Future greenhouse gas neutral energy systems will be dominated by renewable energy technologies whose energy output and utilisation is subject to uncertain weather conditions. This work proposes an algorithm for capacity expansion planning if only uncertain data is available for a year's operative parameters. When faced with multiple possible operating years, the quality of a solution derived on a single operating year's data is evaluated for all years, and the optimisation problem is iteratively modified whenever supply gaps are detected. These modifications lead to solutions with sufficient back-up capacity to overcome periods of cold dark lulls, and sufficient total annual energy supply across all years. A computational study on an energy system model of Germany for 40 different operating years shows that the iterative algorithm finds solutions that guarantee security of supply for all considered years increasing the total annual cost by 1.6-2.9% compared to a lower bound. Results also underline the importance of assessing the feasibility of energy system models using atypical time-series, combining dark lull and cold period effects.

Substantially extending previous results of the authors for smooth solutions in the viscous case, we develop linear damping estimates for periodic roll-wave solutions of the inviscid Saint-Venant equations and related systems of hyperbolic balance laws. Such damping estimates, consisting of $H^s$ energy estimates yielding exponential slaving of high-derivative to low-derivative norms, have served as crucial ingredients in nonlinear stability analyses of traveling waves in hyperbolic or partially parabolic systems, both in obtaining high-frequency resolvent estimates and in closing a nonlinear iteration for which available linearized stability estimates apparently lose regularity. Here, we establish for systems of size $n\leq 6$ a Lyapunov-type theorem stating that such energy estimates are available whenever strict high-frequency spectral stability holds; for dimensions 7 and higher, there may be in general a gap between high-frequency spectral stability and existence of the type of energy estimate that we develop here. A key ingredient is a dimension-dependent linear algebraic lemma reminiscent of Lyapunov's Lemma for ODE that is to our knowledge new.

In decentralized optimization, the choice of stepsize plays a critical role in algorithm performance. A common approach is to use a shared stepsize across all agents to ensure convergence. However, selecting an optimal stepsize often requires careful tuning, which can be time-consuming and may lead to slow convergence, especially when there is significant variation in the smoothness (L-smoothness) of local objective functions across agents. Individually tuning stepsizes per agent is also impractical, particularly in large-scale networks. To address these limitations, we propose AdGT, an adaptive gradient tracking method that enables each agent to adjust its stepsize based on the smoothness of its local objective. We prove that AdGT achieves linear convergence to the global optimal solution. Through numerical experiments, we compare AdGT with fixed-stepsize gradient tracking methods and demonstrate its superior performance. Additionally, we compare AdGT with adaptive gradient descent (AdGD) in a centralized setting and observe that fully adaptive stepsizes offer greater benefits in decentralized networks than in centralized ones.

A bootstrap program is presented for algebraically solving the $R$-matrix of a generic integrable quantum spin chain from its Hamiltonian. The Yang-Baxter equation contains an infinite number of seemingly independent constraints on the operator valued coefficients in the expansion of the $R$-matrices with respect to their spectral parameters, with the lowest order one being the Reshetikhin condition. These coefficients can be solved iteratively using Kennedy's inversion formula, which reconstructs the $R$-matrix after an infinite number of steps. For a generic Hamiltonian, the procedure could fail at any step, making the conditions useful as an integrability test. However in all known examples they all turn out to be satisfied whenever the lowest order condition is. It remains to be understood whether they are all implied by the Reshetikhin condition.

In this paper, we present a sufficient condition on a pair of nonnegative weights $v$ and $w$ such that we have a general weighted $L^{p}$-Hardy type identity. The result, for a certain choice of weights, gives weighted $L^{p}$-Hardy type inequalities and identities with explicit remainder terms, thereby improving previously known results. Furthermore, we obtain the corresponding general weighted Caffarelli-Kohn-Nirenberg type inequality with remainder terms, which, as a result, imply Heisenberg-Pauli-Weyl type inequalities.

Let $\{\eta_i\}_{i\ge 1}$ be a sequence of dependent Bernoulli random variables. While the Poisson approximation for the distribution of $\sum_{i=1}^n\eta_i$ has been extensively studied in the literature, this paper establishes new convergence regimes characterized by non-Poisson limits. Specifically, under a Markovian dependence structure, we show that $\sum_{i=1}^n\eta_i,$ under suitable scaling, converges almost surely or in distribution as $n\to\infty$ to a geometric or Gamma random variable. These results provide a new tool for analyzing the limit distributions of sums of Markovian dependent Bernoulli random variables. We demonstrate these results in several applications: determining the limiting distribution of the number of weak cutspheres for a $d(\ge3)$-dimensional standard Brownian motion; deriving the limit law for weak cutpoints of geometric Brownian motion; and analyzing how often the population size reaches a given threshold in certain branching processes, both with and without immigration.

In this paper we construct a continuum family of non-isomorphic 3-generator groups in which the identity $x^n = 1$ holds with probability 1, while failing to hold universally in each group. This resolves a recent question about the relationship between probabilistic and universal satisfaction of group identities. Our construction uses $n$-periodic products of cyclic groups of order $n$ and two-generator relatively free groups satisfying identities of the form $[x^{pn}, y^{pn}]^n = 1$. We prove that in each of these products, the probability of satisfying $x^n = 1$ is equal to 1, despite the fact that the identity does not hold throughout any of these groups.

We show that the vertex-reinforced jump processes on a \(d\)-dimensional hierarchical lattice are recurrent for \(d < 2\) and transient for \(d > 2\). We also explore certain regimes when \(d = 2\). The proof of recurrence relies on an exponential decay estimate of the fractional moment of the Green's function, which, unlike the classical approach used for \(\mathbb{Z}^d\), requires additional entropy estimates via stability of the model distribution under coarse grain operation, which leverages its linear reinforcement.

This paper highlights a formal connection between two families of widely used matrix factorization algorithms in numerical linear algebra. One family consists of the Jacobi eigenvalue algorithm and its variants for computing the Hermitian eigendecomposition and singular value decomposition. The other consists of Gaussian elimination and the Gram-Schmidt procedure with various pivoting rules for computing the Cholesky decomposition and QR decomposition respectively. Both families are cast as special cases of a more general class of factorization algorithms. We provide a randomized pivoting rule that applies to this general class (which differs substantially from the usual pivoting rules for Gaussian elimination / Gram-Schmidt) which admits a unified analysis of the entire class of algorithms. The result is the same linear rate of convergence for each algorithm, irrespective of which factorization it computes. One important consequence of this randomized pivoting rule is a provable, effective bound on the numerical stability of the Jacobi eigenvalue algorithm, which addresses a longstanding open problem of Demmel and Veselić `92.

We review the main features of an unfitted finite element method for interface and fluid-structure interaction problems based on a distributed Lagrange multiplier in the spirit of the fictitious domain approach. We recall our theoretical findings concerning well-posedness, stability, and convergence of the numerical schemes, and discuss the related computational challenges. In the case of elliptic interface problems, we also present a posteriori error estimates.

The organisers of major sports competitions use different policies with respect to constraints in the group draw. Our paper aims to rationalise these choices by analysing the trade-off between attractiveness (the number of games played by teams from the same geographic zone) and fairness (the departure of the draw mechanism from a uniform distribution). A parametric optimisation model is formulated and applied to the 2018 and 2022 FIFA World Cup draws. A flaw of the draw procedure is identified: the pre-assignment of the host to a group unnecessarily increases the distortions. All Pareto efficient sets of draw constraints are determined via simulations. The proposed framework can be used to find the optimal draw rules and justify the non-uniformity of the draw procedure for the stakeholders.

In this paper, we consider four types of subvarieties of the variety of associative algebras. We study these subvarieties from the point of view of operads and show their connections with well-known classes of algebras, such as dendriform algebras and noncommutative Novikov algebras. Finally, we define the commutator and anti-commutator operations on these algebras and derive several identities satisfied by these operations.

Let $M$ be a compact smooth manifold with corners and $N$ be a finite dimensional smooth manifold without boundary which admits local addition. We define a smooth manifold structure to general sets of continuous mapings $\mathcal{F}(M,N)$ whenever functions spaces $\mathcal{F}(U,\mathbb{R})$ on open subsets $U\subseteq [0,\infty)^n$ are given, subject to simple axioms. Construction and properties of spaces of sections and smoothness of natural mappings between spaces $\mathcal{F}(M,N)$ are discussed, like superposition operators $\mathcal{F}(M,f):\mathcal{F}(M,N_1)\to \mathcal{F}(M,N_2)$, $\eta \mapsto f\circ \eta$ for smooth maps $f:N_1\to N_2$.

We establish the scaling limit of a class of boundary random walks to the full spectrum of Brownian-type processes on the half-line. By solving the associated martingale problem and employing weak convergence techniques, we prove that under appropriate scaling, the process converges to the general Brownian motion in the $J_1$-Skorokhod topology. The main novelty of our approach lies in a result on the asymptotic behavior of the local time of the boundary random walks, allowing us to derive a CLT result for several Brownian-type limit processes on the half-line.

We study when a Riemann difference of order $ n $ possesses the Marcinkiewicz-Zygmund (MZ) property: that is, whether the conditions $ f(h) = o(h^{n-1}) $ and $ Df(h) = o(h^n) $ imply $ f(h) = o(h^n) $. This implication is known to hold for some classical examples with geometric nodes, such as $ \{0, 1, q, \dots, q^{n-1}\} $ and $ \{1, q, \dots, q^n\} $, leading to a conjecture that these are the only such Riemann differences with the MZ property. However, this conjecture was disproved by the third-order example with nodes $ \{-1, 0, 1, 2\} $, and we provide further counterexamples and a general classification here. We establish a complete analytic criterion for the MZ property by developing a recurrence framework: we analyze when a function $ R(h) $ satisfying $ D(h) = R(qh) - A R(h) $, together with $ D(h) = o(h^n) $ and $ R(h) = o(h^{n-1}) $, forces $ R(h) = o(h^n) $. We prove that this holds if and only if $ A $ lies outside a critical modulus annulus determined by $ q $ and $ n $, covering both $ |q| > 1 $ and $ |q| < 1 $ cases. This leads to a complete characterization of all Riemann differences with geometric nodes that possess the MZ property, and provides a flexible analytic framework applicable to broader classes of generalized differences.

In this paper, we present a typed lambda calculus ${\bf SILL}(\lambda)_{\Sigma}$, a type-theoretic version of intuitionistic linear logic with subexponentials, that is, we have many resource comonadic modalities with some interconnections between them given by a subexponential signature. We also give proof normalisation rules and prove the strong normalisation and Church-Rosser properties for $\beta$-reduction by adapting the Tait-Girard method to subexponential modalities. Further, we analyse subexponentials from the point of view of categorical logic. We introduce the concepts of a Cocteau category and a $\Sigma$-assemblage to characterise models of linear type theories with a single exponential and affine and relevant subexponentials and a more general case respectively. We also generalise several known results from linear logic and show that every Cocteau category and a $\Sigma$-assemblage can be viewed as a symmetric monoidal closed category equipped with a family of monoidal adjunctions of a particular kind. In the final section, we give a stronger 2-categorical characterisation of Cocteau categories.

In this paper, we study the vertex functions of finite type A bow varieties. Vertex functions are K-theoretic analogs of I-functions, and 3d mirror symmetry predicts that the q-difference equations satisfied by the vertex functions of a variety and its 3d mirror dual are the same after a change of variable swapping the roles of the various parameters. Thus the vertex functions are related by a matrix of elliptic functions, which is expected to be the elliptic stable envelope of M. Aganagic and A. Okounkov. We prove all of these statements. The strategy of our proof is to reduce to the case of cotangent bundles of complete flag varieties, for which the q-difference equations can be explicitly identified with Macdonald difference equations. A key ingredient in this reduction, of independent interest, involves relating vertex functions of the cotangent bundle of a partial flag variety with those of a ``finer" flag variety. Our formula involves specializing certain Kähler parameters (also called Novikov parameters) to singularities of the vertex functions. In the $\hbar\to \infty$ limit, this statement is expected to degenerate to an analogous result about I-functions of flag varieties.

This paper studies a nonlinear diffusion equation with memory: $$u_t=\nabla\cdot \big( D(x)\cdot\int_0^t K(t-s) \nabla\cdot\Phi(u(x,s))ds \big)+f(x,t)$$ Where $K$ is memory Kernel and $D(x)$ is bounded. Under monotonicity and growth conditions on $\Phi$, the existence and uniqueness of weak solution is established. The analysis employs Orthogonal approximation, energy estimates, and monotone operator theory. The convolution structure is handled within variational frameworks. The result provides a basis for studying memory-type diffusion.

This short note modifies a reconstruction method by the author (Comm.~PDE, 45(9):1118--1133, 2020), for reconstructing piecewise constant conductivities in the Calderón problem (electrical impedance tomography). In the former paper, a layering assumption and the local Neumann-to-Dirichlet map were needed since the piecewise constant partition also was assumed unknown. Here I show how to modify the method in case the partition is known, for general piecewise constant conductivities and only a finite number of partial boundary measurements. Moreover, no lower/upper bounds on the unknown conductivity are needed.

In this note, we propose some open problems and questions about bounded convex domains in $\mathbb C^N$, specifically about visibility and iteration theory.

A pair of transverse contact distributions on a 3-manifold will in general admit no 1-parameter families of symmetries: a flow preserving both contact distributions. Here, we will determine local normal forms for such pairs admitting symmetries. In particular, we observe that orientable Anosov flows may be globally given by the intersection of a pair of oppositely oriented contact distributions admitting, around any point, maximal local symmetries.

This paper studies the family of sliced Cramér metrics, quantifying their stability under distortions of the input functions. Our results bound the growth of the sliced Cramér distance between a function and its geometric deformation by the product of the deformation's displacement size and the function's mean mixed norm. These results extend to sliced Cramér distances between tomographic projections. In addition, we remark on the effect of convolution on the sliced Cramér metrics. We also analyze efficient Fourier-based discretizations in 1D and 2D, and prove that they are robust to heteroscedastic noise. The results are illustrated by numerical experiments.

Computing with discrete representations of high-dimensional probability distributions is fundamental to uncertainty quantification, Bayesian inference, and stochastic modeling. However, storing and manipulating such distributions suffers from the curse of dimensionality, as memory and computational costs grow exponentially with dimension. Monte Carlo methods require thousands to billions of samples, incurring high computational costs and producing inconsistent results due to stochasticity. We present an efficient tensor train method for performing exact arithmetic operations on discretizations of continuous probability distributions while avoiding exponential growth. Our approach leverages low-rank tensor train decomposition to represent latent random variables compactly using Dirac deltas, enabling deterministic addition, subtraction and multiplication operations directly in the compressed format. We develop an efficient implementation using sparse matrices and specialized data structures that further enhances performance. Theoretical analysis demonstrates polynomial scaling of memory and computational complexity under rank assumptions, and shows how statistics of latent variables can be computed with polynomial complexity. Numerical experiments spanning randomized linear algebra to stochastic differential equations demonstrate orders-of-magnitude improvements in memory usage and computational time compared to conventional approaches, enabling tractable deterministic computations on discretized random variables in previously intractable dimensions.

We complete a uniform construction of canonical absolute parallelism for bracket generating rank $2$ distributions with $5$-dimensional cube on $n$-dimensional manifold with $n\geq 5$ by showing that the condition of maximality of class that was assumed previously by Doubrov-Zelenko for such a construction holds automatically at generic points. This also gives analogous constructions in the case when the cube is not $5$-dimensional but the distribution is not Goursat through the procedure of iterative Cartan deprolongation. This together with the classical theory of Goursat distributions covers in principle the local geometry of all bracket generating rank 2 distributions in a neighborhood of generic points. As a byproduct, for any $n\geq 5$ we describe the maximally symmetric germs among bracket generating rank $2$ distributions with $5$-dimensional cube, as well as among those which reduce to such a distribution under a fixed number of Cartan deprolongations. Another consequence of our results on maximality of class is for optimal control problems with constraint given by a rank $2$ distribution with $5$-dimensional cube: it implies that for a generic point $q_0$ of $M$, there are plenty abnormal extremal trajectories of corank $1$ (which is the minimal possible corank) starting at $q_0$. The set of such points contains all points where the distribution is equiregular.

The mixed Christoffel-Minkowski problem asks for necessary and sufficient conditions for a Borel measure on the Euclidean unit sphere to be the mixed area measure of some convex bodies, one of which, appearing multiple times, is free and the rest are fixed. In the case where all bodies involved are symmetric around a common axis, we provide a complete solution to this problem, without assuming any regularity. In particular, we refine Firey's classification of area measures of figures of revolution. In our argument, we introduce an easy way to transform mixed area measures and mixed volumes involving axially symmetric bodies, and we significantly improve Firey's estimate on the local behavior of area measures. As a secondary result, we obtain a family of Hadwiger type theorems for convex valuations that are invariant under rotations around an axis.

A pair of closed, smooth $4$-manifolds $M$ and $M'$ are stably exotic if they are stably homeomorphic but not stably diffeomorphic, where stabilisation refers to connected sum with copies of $S^2 \times S^2$. Orientable stable exotica do not exist by a result of Gompf, but Kreck showed that nonorientable examples are plentiful. We investigate which values of the fundamental group $\pi$ and the first and second Stiefel-Whitney classes $w_1$ and $w_2$ admit stably exotic pairs, providing a complete description if $H_5(\pi;\mathbb{Z})=0$. In particular we produce new stable exotica, and new settings in which they do not arise.

The virtual element method (VEM) is a stabilized Galerkin method that is robust and accurate on general polygonal meshes. This feature makes it an appealing candidate for simulations involving meshes with embedded interfaces and evolving geometries. However, similar to the finite element method, in such scenarios the VEM can also yield poorly conditioned stiffness matrices due to meshes having cut cells. With the objective of developing an embedded domain method, we propose a novel element agglomeration algorithm for the VEM to address this issue. The agglomeration algorithm renders the VEM robust over planar polygonal meshes, particularly on finite element meshes cut by immersed geometries. The algorithm relies on the element stability ratio, which we define using the extreme eigenvalues of the element stiffness matrix. The resulting element agglomeration criterion is free from nebulous polygon quality metrics and is defined independently of polygon shapes. The algorithm proceeds iteratively and element-wise to maximize the minimum element stability ratio, even at the expense of degrading elements with better ratios. The resulting method, which we label as CutVEM, retains node locations of cut elements unchanged, and yields discretizations that conform to embedded interfaces. This, in turn, facilitates straightforward imposition of boundary conditions and interfacial constraints. Through detailed numerical experiments that sample varied element-interface intersections, we demonstrate that CutVEM enjoys dramatically improved condition numbers of global stiffness matrices over the VEM. Furthermore, simulations of prototypical heat conduction problems with Dirichlet and Neumann boundary conditions on domains with immersed geometries show that element agglomeration does not noticeably degrade solution accuracy and that CutVEM retains the VEM's optimal convergence rate.

Recently, Andrews introduced separable integer partition classes and studied some well-known theorems. In this article, we will consider the types of partitions with restrictions on consecutive parts. We will show that such partitions are separable integer partition classes and then give the generating functions for such partitions.

We investigate a variation of Nicomachus's identity in which one term in the cubic sum is replaced by a different cube. Specifically, we study the Diophantine identity \[ \sum_{j=1}^{n} j^3 + x^3 - k^3 = \left( \sum_{j=1}^{n} j + x - k \right)^2 \] and classify all integer solutions $(k,x,n)$. A full parametric family of nontrivial solutions was introduced in a 2005 paper, along with a conjectural condition for when such solutions exist. We provide a complete proof of this characterization and show it is equivalent to a structural condition on the prime factorization of $ n^2 + n + 1 $. Our argument connects this identity to classical results in the theory of binary quadratic forms. In particular, we analyze the equation $a^2 + ab + b^2 = n^2 + n + 1$, interpreting it as a norm in the ring of Eisenstein integers $\mathbb{Z}[\omega]$, where $\omega = \frac{1 + \sqrt{-3}}{2}$. This yields a surprising connection between a modified combinatorial identity and the arithmetic of algebraic number fields.

In this paper, we obtain sharp remainder terms for the Hardy-Poincaré inequalities with general non-radial weights in the setting of Baouendi-Grushin vector fields (see Theorem 2.5). It is worth emphasizing that all of our results are new both in the Baouendi-Grushin and standard Euclidean settings. The method employed allows us to not only unify, but also improve the results of Kombe and Yener [KY18] for any $1<p<\infty$ while holding true for complex-valued functions and providing explicit constants (Corollary 2.7). As a result, we are able to obtain sharp remainder terms to many known weighted Hardy-type inequalities (see Section 3.1). Aside from weighted Hardy-type inequalities, we also recover a sharp remainder formula for the $L^{p}$-Poincaré inequality (Corollary 3.5). In the special case of radial weights, we are naturally able to introduce the notion of Baouendi-Grushin $p$-Bessel pairs (see Definition 2.9). Finally, we apply the technique to establish the sharp remainder term of the Heisenberg-Pauli-Weyl inequality in $L^{p}$ for $1<p<\infty$ (Corollary 3.13), which includes the sharp constant. This makes it possible to obtain the $L^{p}$-analogue for $2\leq p < n$ (Theorem 3.17) of a stability result by Cazacu, Flynn, Lam and Lu [CFLL24].

The Kromatic symmetric function (KSF) $\overline{X}_G$ of a graph $G$ is a $K$-analogue introduced by Crew, Pechenik, and Spirkl in arXiv:2301.02177 of Stanley's chromatic symmetric function (CSF) $X_G$. The KSF is known to distinguish some pairs of graphs with the same CSF. The first author showed in arXiv:2403.15929 and arXiv:2502.21285 that the number of copies in $G$ of certain induced subgraphs can be determined given $\overline{X}_G$, and conjectured that $\overline{X}_G$ distinguishes all graphs. We disprove that conjecture by finding four pairs of 8-vertex graphs with equal KSF, as well as giving several ways to use existing graph pairs with equal KSF to construct larger graph pairs that also have equal KSF. On the other hand, we show that many of the graph pairs from the constructions of Orellana and Scott in arXiv:1308.6005 and of Aliste-Prieto, Crew, Spirkl, and Zamora in arXiv:2007.11042 of graphs with the same CSF are distinguished by the KSF, thus also giving some new examples of cases where the KSF is a stronger invariant than the CSF.

While single-purchase choice models have been widely studied in assortment optimization, customers in modern retail and e-commerce environments often purchase multiple items across distinct product categories, exhibiting both substitution and complementarity. We consider the cross-category assortment optimization problem where retailers jointly determine assortments across categories to maximize expected revenue. Most prior work on the topic either overlooks complementarity or proposes models that lead to intractable optimization problems, despite being based on the multinomial logit (MNL) choice model. We propose a sequential multi-purchase choice model for cross-category choice that incorporates complementarity through a Markovian transition structure across categories, while allowing general Random Utility Maximization (RUM)-based choice models to capture the within-category substitution. We develop an Expectation-Maximization algorithm for estimation, and a polynomial-time algorithm for unconstrained assortment optimization that yields the optimal solution when the within-category substitution follows a Markov chain choice model. Furthermore, we introduce an empirical metric to quantify complementarity strength across product categories and conduct extensive numerical experiments on both synthetic data and a large-scale transaction-level dataset from a major US grocery store. Our model yields improvements in predictive accuracy, model fit, and expected revenue in setting with complementarity, and it reveals intuitive market structures such as brand-loyal cross-category purchasing. Overall, we believe that our model provides a theoretically-grounded and practical framework for modeling complementarity and making better cross-category assortment decisions.

The Macdonald operator is known to coincide with a certain element of the quantum toroidal $\mathfrak{gl}(1)$ algebra in the Fock representation of levels $(1,0)$. A generalization of this operator to higher levels $(r,0)$ can be built using the coproduct structure, it is diagonalized by the generalized Macdonald symmetric functions, indexed by $r$-tuple partitions and depending on $r$ alphabets. In this paper, we extend to the generalized case some of the known formulas obeyed by ordinary Macdonald symmetric functions, such as the $e_1$-Pieri rule or the identity relating them to Whittaker vectors obtained by Garsia, Haiman, and Tesler. We also propose a generalization of the five-term relation, and the Fourier/Hopf pairing. In addition, we prove the factorized expression of the generalized Macdonald kernel conjectured previously by Zenkevich.

Energy decay rates of damped waves on the torus depend on the behavior of the damping near the undamped region and on the geometry of the damped set. In this paper we refine these geometric considerations, by introducing the concept of order of a glancing undamped point, and estimating decay rates in terms of this order. The proof is based on generalizing an averaging argument due to Sun. We also show that damping sets which attain these improvements are generic among polygons and smooth curves.

In a previous paper, we recast Morgado hyperlattices and Sette implicative hyperlattices in lattice-theoretic terms. By utilizing swap structures induced by implicative lattices, we obtained a direct proof of soundness and completeness for da Costa's paraconsistent logic $C_\omega$ with respect to Sette's hyperalgebraic semantics. Inspired by Kalman functors in the context of twist structures, we introduce the notion of hyper swap structures, a novel class of hyperalgebras that naturally generalize swap structure semantics. We prove that these hyperalgebras, besides providing another class of hyperalgebraic models for $C_\omega$, induce a Kalman-style functor between the category of Sette implicative hyperlattices and the category of enriched hyperalgebras for $C_\omega$. Specifically, we exhibit an equivalence of categories between Sette implicative hyperlattices and their enriched hyperalgebraic counterparts using Kalman and forgetful functors. Similar results are extended to two axiomatic extensions of $C_\omega$.

Let $G$ be a complex reductive group. A folklore result asserts the existence of an $\mathbb{E}_2$-algebra structure on the Ran Grassmannian of $G$ over $\mathbb{A}^1_{\mathbb{C}}$, seen as a topological space with the complex-analytic topology. The aim of this paper is to prove this theorem, by establishing a homotopy invariance result: namely, an inclusion of open balls $D' \subset D$ in $\mathbb{C}$ induces a homotopy equivalence between the respective Beilinson--Drinfeld Grassmannians $\mathrm{Gr}_{G, {D'}^n} \hookrightarrow \mathrm{Gr}_{G, D^n}$, for any positive integer $n$. We use a purely algebraic approach, showing that automorphisms of a complex smooth algebraic curve $X$ can be lifted to automorphisms of the associated Beilinson--Drinfeld Grassmannian. As a consequence, we obtain a stronger version of the usual homotopy invariance result: namely, the homotopies can be promoted to equivariant stratified isotopies, where "equivariant" refers to the action of the arc group $\mathrm{L}^+G$ and "stratified" refers to the stratification induced by the Schubert stratification of $\mathrm{Gr}_G$ and the incidence stratification of $\mathbb{C}^n$.

The set of infinite upper-triangular totally positive Toeplitz matrices has a classical parametrisation proved by Edrei et al and originally conjectured by Schoenberg, that involves pairs of sequences of positive real parameters. These matrices (and their parameters) are central for understanding characters of the infinite symmetric group by work of Thoma. On the other hand there is a very different parametrisation theorem that applies to the finite analogue of this set. These finite Toeplitz matrices and their parameters relate to quantum cohomology of flag varieties and mirror symmetry. In this paper we replace the positive reals by a semifield with valuation to then construct tropical analogues for both parametrisation theorems. In the finite case we tropicalise using positive generalised Puiseaux series. This builds on work of Judd and Lüdenbach. In the infinite case we use a new valued semifield of continuous functions. We arrive at different natural infinite analogues of totally positive Toeplitz matrices, depending on a choice of topology on our valued semifield. We then prove an asymptotic result relating the tropical parameters from the finite case to the tropicalisations of the Schoenberg parameters. Moreover, we show that our finite type tropical parametrisation map is given by Lusztig's weight map from the theory of canonical bases. This results in a surprising connection between the classical Edrei theorem with its Schoenberg parameters and Lusztig's canonical basis parametrisation.

We show that if a graph $G$ admits a quasi-isometry $\phi$ to a graph $H$ of bounded path-width, then we can assign a non-negative integer length to each edge of $H$, such that the same function $\phi$ is a quasi-isometry to this weighted version of $H$, with error only an additive constant.

We suggest a dependence coefficient between a categorical variable and some general variable taking values in a metric space. We derive important theoretical properties and study the large sample behaviour of our suggested estimator. Moreover, we develop an independence test which has an asymptotic $\chi^2$-distribution if the variables are independent and prove that this test is consistent against any violation of independence. The test is also applicable to the classical~$K$-sample problem with possibly high- or infinite-dimensional distributions. We discuss some extensions, including a variant of the coefficient for measuring conditional dependence.

We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that solvability of the Regularity problem in $\dot{W}^{1,p}$ implies solvability of the adjoint Dirichlet problem in $L^{p'}$. Previously, Shen (2007) established a partial reverse result. In our work, we show that if we assume an $L^1$-Carleson condition on only $|\partial_t A|$ the full reverse direction holds. As a result, we obtain equivalence between solvability of the Dirichlet problem $(D)^*_{p'}$ and the Regularity problem $(R)_p$ under this condition. As a further consequence, we can extend the class of operators for which the $L^p$ Regularity problem is solvable by operators satisfying the mixed $L^1-L^\infty$ condition. Additionally in the case of the upper half plane, this class includes operators satisfying this $L^1$-Carleson condition on $|\partial_t A|$.

We investigate the stabilizability of linear discrete-time switched systems with singular matrices, focusing on the spectral radius in this context. A new lower bound of the stabilizability radius is proposed, which is applicable to any matrix set. Based on this lower bound, more relationships between the stabilizability radius and joint spectral subradius are established. Detailed analysis of the stabilizability radius of a special kind of two-dimensional switched system, consisting of a singular matrix and a rotation matrix, is presented. The Hausdorff dimensions of the parameter sets such that the stabilizability radius of these systems equals a constant are also presented. Other properties of switched systems with singular matrices are also discussed along with examples.

Huh-Stevens and Ferroni-Schröter independently conjectured that Hilbert-Poincaré series of Chow rings of geometric lattices have only real zeros. Ferroni, Matherne and the second author extended this conjecture to Chow polynomials of Cohen-Macaulay poset. In this paper we address the above conjectures by providing new defining relations and properties of Chow functions of posets and matrices. These are used, in conjunction with new techniques on interlacing sequences of polynomials, to prove that Chow polynomials of totally nonnegative matrices have only real zeros, which, in turn, proves the above conjectures for a class of posets that contains projective and affine geometries, face lattices of cubical polytopes, dual partition and Dowling lattices, perfect matroid designs, and lattices of flats of paving matroids. We also study Chow polynomials of Toeplitz matrices in greater detail, and show how these are related the combinatorics of binomial and Sheffer posets, as well as to a family of generalized Eulerian polynomials with coefficients in the ring of symmetric polynomials that have been studied by e.g. Stanley, Brenti, Stembridge and Shareshian-Wachs.

Let $X_p, p\in\cP$ be a sequence of independent random variables s.t. $\bbP(X_p=\pm 1)=1/2$. Let $\te_j=\prod_{p|j}X_p$ if $j$ is square free and $\te_j=0$ otherwise. Denote $S_n=\sum_{\ell=1}^n\te_\ell$. The from this point of view proving limit theorems for $S_n$ is natural problem, since $S_n$ mimics the behavior of $e^{\sqrt{\ln(\beta)}}$. It is a natural guiding conjecture that $S_n/\sqrt n$ obeys the central limit theorem (CLT). However, S. Chatterjee conjectured (as expressed in \cite{[25]}) that the CLT should not hold. Chatterjee's conjecture was proved by Harper \cite{[17]}, and by now it is a direct consequence of a more recent breakthrough by Harper \cite{Har20} that $\frac{S_n}{b_n}\to 0$ in $L^1$, where $b_n=(n^{1/2}(\ln(\ln(n)))^{-1/4})u_n, u_n\to\infty$. In particular $S_n/\sqrt n\to 0$. Nevertheless, the question whether there exists a sequence $a_n=o(b_n)$ such that $S_n/a_n$ converges to some limit remains a mystery. Note that the corresponding problem in the Steinhaus Setting was recently resolved by \cite{Gor1}. In this paper make an attempt to shed some light on the convergence of $S_n/a_n$. Additionally, we obtain explicit estimates on hight moments of $S_n$ without restrictions on the size of the moment compared to $n$ like in \cite[Theorem 1.2]{Har19}, which is of independent interest. This is achieved by a martingale argument together with the Burkholder inequality, and it has applications in a natural number theoretic combinatorial problem. Using martingale techniques we will also obtain exponential concentration inequalities for $S_n$ (in the large deviations regime)

We provide a comprehensive overview of meshfree collocation methods for numerically approximating differential operators on continuously labeled unstructured point clouds. Meshfree collocation methods do not require a computational grid or mesh. Instead, they approximate smooth functions and their derivatives at potentially irregularly distributed collocation points, often called particles, to a desired order of consistency. We review several meshfree collocation methods from the literature, trace the historical development of key concepts, and propose a classification of methods according to their principle of derivation. Although some of the methods reviewed are similar or identical, there are subtle yet important differences between many, which we highlight and discuss. We present a unifying formulation of meshfree collocation methods that renders these differences apparent and show how each method can be derived from this formulation. Finally, we propose a generalized derivation for meshfree collocation methods going forward.

A graph on $n \ge 3$ vertices drawn in the plane such that each edge is crossed at most four times has at most $6(n-2)$ edges -- this result proven by Ackerman is outstanding in the literature of beyond-planar graphs with regard to its tightness and the structural complexity of the graph class. We provide a much shorter proof while at the same time relaxing the conditions on the graph and its embedding, i.e., allowing multi-edges and non-simple drawings.

In this article, we present a combinatorial formula for the Wedderburn decomposition of rational group algebras of nested GVZ $p$-groups, where $p$ is an odd prime. Using this formula, we derive an explicit combinatorial expression for the Wedderburn decomposition of rational group algebras of all two-generator $p$-groups of class $2$. Additionally, we provide explicit combinatorial formulas for the Wedderburn decomposition of rational group algebras of certain families of nested GVZ $p$-groups with arbitrarily large nilpotency class. We also classify all nested GVZ $p$-groups of order at most $p^5$ and compute the Wedderburn decomposition of their rational group algebras. Finally, we determine a complete set of primitive central idempotents for the rational group algebras of nested GVZ $p$-groups.

We prove that if a tile in $\mathbb Z^d$ has prime size $p$, then it must be spectral. The proof is by contradiction, it is simply shown that the tiling complement of such a tile can not annihilate all $p$-subgroups. In addition, with a simple transformation we prove that any $p$ points in general linear positions in $\mathbb Z^d$ must be both tiling and spectral if $d\ge p-1$.

In this paper, we present the Stochastic Origin Frank-Wolfe (SOFW) method, which is a special case of the block-coordinate Frank-Wolfe algorithm, applied to the problem of finding equilibrium flow distributions. By significantly reducing the computational complexity of the minimization oracle, the method improves overall efficiency at the cost of increased memory consumption. Its key advantage lies in minimizing the number of shortest path computations. We refer to existing theoretical convergence guarantees for generalized coordinate Frank-Wolfe methods and, in addition, extend the analysis by providing a convergence proof for a batched version of the Block-Coordinate Frank-Wolfe algorithm, which was not covered in the original work. We also demonstrate the practical effectiveness of our approach through experimental results. In particular, our findings show that the proposed method significantly outperforms the classical Frank-Wolfe algorithm and its variants on large-scale datasets. On smaller datasets, SOFW also remains effective, though the performance gap relative to classical methods becomes less pronounced. In such cases, there is a trade-off between solution quality, iteration time complexity, and memory usage.

We introduce a tree-based formulation for the minimum-cost multi-commodity flow problem that addresses large-scale instances. The method decomposes the source-based model by representing flows as convex combinations of trees rooted at source nodes, and solves the resulting formulation with column generation. The number of demand constraints now depends on the number of sources $|S|$, not commodities $|K|$, yielding a compact master problem when $|S| \ll |K|$. We conduct a computational study comparing tree-based decomposition against path-based column generation and direct LP solving. The results show speed-ups of up to one order of magnitude over direct LP solving, and improved scalability compared to path-based formulations. Tree-based decomposition enables solving instances with millions of commodities and hundreds of thousands of nodes. This makes it well-suited for applications in transportation and logistics networks where multiple demands often share common origins.

Given infinite cardinals $\theta\leq \kappa$, we ask for the minimal VC-dimension of a cofinal family $\mathcal{F}\subseteq[\kappa]^{<\theta}$. We show that for $\theta=\omega$ and $\kappa=\aleph_n$ it is consistent with ZFC that there exists such a family of VC-dimension $n+1$, which is known to be the lower bound. For $\theta>\omega$ we answer this question completely, demonstrating a strong dichotomy between the case of singular and regular $\theta$. We furthermore answer some relative and generalized versions of the above question for singular $\theta$, and answer a related question which appears in \cite{BBNKS}.

On one hand, exact structures were introduced by D. Quillen in the '70s. They can be defined as collections of short exact sequences in a fixed abelian category satisfying additional properties. On the other hand, in a recent work, A. Garver, R. Patrias, and H. Thomas introduced Jordan recoverability. Given a bounded quiver $(Q,R)$, a full additive subcategory of $\operatorname{rep}(Q,R)$ is said to be Jordan recoverable if any $X \in \mathscr{C}$ can be recovered, up to isomorphism, from the Jordan form of its generic nilpotent endomorphisms. Such a subcategory $\mathscr{C}$ is said to be canonically Jordan recoverable if, moreover, there exists a precise algebraic procedure that allows one to get back $X \in \mathscr{C}$ from that same Jordan form data. We introduce a new family of operators, called Gen-Sub operators $\operatorname{GS}_\mathcal{E}$, parametrized by the exact structures $\mathcal{E}$ of abelian categories. After showing some properties of those operators in hereditary abelian categories, by focusing on the setting of modules over type $A$ quivers endowed with the diamond exact structure $\mathcal{E}_\diamond$, we establish that the maximal canonically Jordan recoverable subcategories are precisely of the form $\operatorname{GS}_{\mathcal{E}_\diamond}(T)$ for some tilting object $T$.

This paper is dedicated to the analysis of forward backward stochastic differential equations driven by a L{é}vy process. We assume that the generator and the terminal condition are path-dependent and satisfy a local Lipschitz condition. We study solvability and Malliavin differentiability of such BSDEs. The proof of the existence and uniqueness is done in three steps. First of all, we truncate and localize the terminal condition and the generator. Then we use an iteration argument to get bounds for the solutions of the truncated BSDE (independent from the level of truncation). Finally, we let the level of truncation tend to infinity. A stability result ends the proof. The Malliavin differentiability result is based on a recent characterisation for the Malliavin Sobolev space D 1,2 by S. Geiss and Zhou.

We investigate the toric geometry of two families of generalised determinantal varieties arising from permutations: Matrix Schubert varieties ($\overline{X_w}$) and Kazhdan-Lusztig varieties ($\mathcal{N}_{v,w}$). Matrix Schubert varieties can be written as $\overline{X_w} = Y_w \times \mathbb C^d$, where $d$ is maximal. We are especially interested in the structure and complexity of these varieties $Y_w$ and $\mathcal{N}_{v,w}$ under the so-called usual torus actions. In the case when $Y_w$ is toric, we provide a full characterisation of the simple reflections $s_i$ that render ${Y_{w \cdot s_i}}$ toric, as well as the corresponding changes to the weight cone. For Kazhdan-Lusztig varieties, we consider how moving one of the two permutations $v,w$ along a chain in the Bruhat poset affects their complexity. Additionally, we study the complexity of these varieties, for permutations $v$ and $w$ of a specific structure. Finally, we consider the links between these determinantal varieties and two classes of statistical models; namely conditional independence and quasi-independence models.

We study limit sets of Teichmüller disks in the Thurston boundary of Teichmüller space of a closed surface S of genus at least 2. It is well known that almost every Teichmüller geodesic ray converges to a point on the boundary. We show that unlike rays, Teichmüller disks with smallest possible limit sets are extremely rare.

We undertake a study of extensions of unirational algebraic groups. We prove that extensions of unirational groups are also unirational over fields of degree of imperfection $1$, but that this fails over every field of higher degree of imperfection, answering a question of Achet. We also initiate a study of those groups which admit filtrations with unirational graded pieces, and show that one may deduce unirationality of unipotent groups from unirationality of certain quotients.

In this survey, we describe recent progress on asymptotic properties of various automorphic orbits in free groups. In particular, we address the problem of counting potentially positive elements of a given length. We also discuss complexity (worst-case, average-case, and generic-case) of Whitehead's automorphism problem and relevant properties of automorphic orbits, including orbit-blocking words.

When parameters are weakly identified, bounds on the parameters may provide a valuable source of information. Existing weak identification estimation and inference results are unable to combine weak identification with bounds. Within a class of minimum distance models, this paper proposes identification-robust inference that incorporates information from bounds when parameters are weakly identified. This paper demonstrates the value of the bounds and identification-robust inference in a simple latent factor model and a simple GARCH model. This paper also demonstrates the identification-robust inference in an empirical application, a factor model for parental investments in children.

Directed acyclic graphs (DAGs) are directed graphs in which there is no path from a vertex to itself. DAGs are an omnipresent data structure in computer science and the problem of counting the DAGs of given number of vertices and to sample them uniformly at random has been solved respectively in the 70's and the 00's. In this paper, we propose to explore a new variation of this model where DAGs are endowed with an independent ordering of the out-edges of each vertex, thus allowing to model a wide range of existing data structures. We provide efficient algorithms for sampling objects of this new class, both with or without control on the number of edges, and obtain an asymptotic equivalent of their number. We also show the applicability of our method by providing an effective algorithm for the random generation of classical labelled DAGs with a prescribed number of vertices and edges, based on a similar approach. This is the first known algorithm for sampling labelled DAGs with full control on the number of edges, and it meets a need in terms of applications, that had already been acknowledged in the literature.

We consider the problem of approximating the regression function $f_\mu:\, \Omega \to Y$ from noisy $\mu$-distributed vector-valued data $(\omega_m,y_m)\in\Omega\times Y$ by an online learning algorithm using a reproducing kernel Hilbert space $H$ (RKHS) as prior. In an online algorithm, i.i.d. samples become available one by one via a random process and are successively processed to build approximations to the regression function. Assuming that the regression function essentially belongs to $H$ (soft learning scenario), we provide estimates for the expected squared error in the RKHS norm of the approximations $f^{(m)}\in H$ obtained by a standard regularized online approximation algorithm. In particular, we show an order-optimal estimate $$ \mathbb{E}(\|\epsilon^{(m)}\|_H^2)\le C (m+1)^{-s/(2+s)},\qquad m=1,2,\ldots, $$ where $\epsilon^{(m)}$ denotes the error term after $m$ processed data, the parameter $0<s\leq 1$ expresses an additional smoothness assumption on the regression function, and the constant $C$ depends on the variance of the input noise, the smoothness of the regression function, and other parameters of the algorithm. The proof, which is inspired by results on Schwarz iterative methods in the noiseless case, uses only elementary Hilbert space techniques and minimal assumptions on the noise, the feature map that defines $H$ and the associated covariance operator.

Finding the parameters of a latent variable causal model is central to causal inference and causal identification. In this article, we show that existing graphical structures that are used in causal inference are not stable under marginalization of Gaussian Bayesian networks, and present a graphical structure that faithfully represent margins of Gaussian Bayesian networks. We present the first duality between parameter optimization of a latent variable model and training a feed-forward neural network in the parameter space of the assumed family of distributions. Based on this observation, we develop an algorithm for parameter optimization of these graphical structures based on a given observational distribution. Then, we provide conditions for causal effect identifiability in the Gaussian setting. We propose an meta-algorithm that checks whether a causal effect is identifiable or not. Moreover, we lay a grounding for generalizing the duality between a neural network and a causal model from the Gaussian to other distributions.

In a continuous-time economy, this paper formulates the Epstein-Zin preference for discounted dividends received by an investor as an Epstein-Zin singular control utility. We introduce a backward stochastic differential equation with an aggregator integrated with respect to a singular control, prove its well-posedness, and show that it coincides with the Epstein-Zin singular control utility. We then establish that this formulation is equivalent to a robust dividend policy chosen by the firm's executive under the Maenhout's ambiguity-averse preference. In particular, the robust dividend policy takes the form of a threshold strategy on the firm's surplus process, where the threshold level is characterized as the free boundary of a Hamilton-Jacobi-Bellman variational inequality. Therefore, dividend-caring investors can choose firms that match their preferences by examining stock's dividend policies and financial statements, whereas executives can make use of dividend to signal their confidence, in the form of ambiguity aversion, on realizing the earnings implied by their financial statements.

Spectral properties of bounded linear operators play a crucial role in several areas of mathematics and physics. For each self-adjoint, trace-class operator $O$ we define a set $\Lambda_n\subset \mathbb{R}$, and we show that it converges to the spectrum of $O$ in the Hausdorff metric under mild conditions. Our set $\Lambda_n$ only depends on the first $n$ moments of $O$. We show that it can be effectively calculated for physically relevant operators, and it approximates the spectrum well without diagonalization. We prove that using the above method we can converge to the minimal and maximal eigenvalues with super-exponential speed. We also construct monotone increasing lower bounds $q_n$ for the minimal eigenvalue (or decreasing upper bounds for the maximal eigenvalue). This sequence only depends on the moments of $O$ and a concrete upper estimate of its $1$-norm; we also demonstrate that $q_n$ can be effectively calculated for a large class of physically relevant operators. This rigorous lower bound $q_n$ tends to the minimal eigenvalue with super-exponential speed provided that $O$ is not positive semidefinite. As a by-product, we obtain computable upper bounds for the $1$-norm of $O$, too. Numerical examples demonstrate the relevance of our approximation in estimating entropy and negativity, which is useful, among others, in quantum optical and in open quantum system models. The results can be directly applicable to problems in quantum information, statistical mechanics, and quantum thermodynamics, where using traditional techniques based on diagonalization is impractical.

The Pandora's Box problem models the search for the best alternative when evaluation is costly. In the simplest variant, a decision maker is presented with $n$ boxes, each associated with a cost of inspection and a hidden random reward. The decision maker inspects a subset of these boxes one after the other, in a possibly adaptive order, and gains the difference between the largest revealed reward and the sum of the inspection costs. Although this classic version is well understood (Weitzman 1979), there is a flourishing recent literature on variants of the problem. Here we introduce a general framework -- the Pandora's Box Over Time problem -- that captures a wide range of variants where time plays a role, e.g., by constraining the schedules of exploration and influencing costs and rewards. In our framework, boxes have time-dependent rewards and costs, whereas inspection may require a box-specific processing time. Moreover, once a box is inspected, its reward may deteriorate over time. Our main result is an efficient constant-factor approximation to the optimal strategy for the Pandora's Box Over Time problem, which is generally NP-hard to compute. We further obtain improved results for the natural special cases where boxes have no processing time, boxes are available only in specific time slots, or when costs and reward distributions are time-independent (but rewards may still deteriorate after inspection).

We study relaxations of entanglement-assisted quantum channel coding and establish that non-signaling assistance and a natural semi-definite programming relaxation\, -- \,termed meta-converse\, -- \,are equivalent in terms of success probabilities. We then present a rounding procedure that transforms any non-signaling-assisted strategy into an entanglement-assisted one and prove an approximation ratio of $(1 - e^{-1})$ in success probabilities for the special case of measurement channels. For fully quantum channels, we give a weaker (dimension dependent) approximation ratio, that is nevertheless still tight to characterize the strong converse exponent of entanglement-assisted channel coding [Li and Yao, IEEE Tran.~Inf.~Theory (2024)]. Our derivations leverage ideas from position-based coding, quantum decoupling theorems, the matrix Chernoff inequality, and input flattening techniques.

Pixel antennas, based on discretizing a continuous radiation surface into small elements called pixels, are a flexible reconfigurable antenna technology. By controlling the connections between pixels via switches, the characteristics of pixel antennas can be adjusted to enhance the wireless channel. Inspired by this, we propose a novel technique denoted antenna coding empowered by pixel antennas. We first derive a physical and electromagnetic based communication model for pixel antennas using microwave multiport network theory and beamspace channel representation. With the model, we optimize the antenna coding to maximize the channel gain in a single-input single-output (SISO) pixel antenna system and develop a codebook design for antenna coding to reduce the computational complexity. We analyze the average channel gain of SISO pixel antenna system and derive the corresponding upper bound. In addition, we jointly optimize the antenna coding and transmit signal covariance matrix to maximize the channel capacity in a multiple-input multiple-output (MIMO) pixel antenna system. Simulation results show that using pixel antennas can enhance the average channel gain by up to 5.4 times and channel capacity by up to 3.1 times, demonstrating the significant potential of pixel antennas as a new dimension to design and optimize wireless communication systems.

In a gauge theory, a collection of kinematical degrees of freedom is used to redundantly describe a smaller amount of gauge-invariant information. In a quantum error correcting code (QECC), a collection of computational degrees of freedom that make up a device's physical layer is used to redundantly encode a smaller amount of logical information. We elaborate this parallel in terms of quantum reference frames (QRFs), which are a universal toolkit for dealing with symmetries in quantum systems and which define the gauge theory analog of encodings. The result is a precise dictionary between QECCs and QRF setups within the perspective-neutral framework for gauge systems. Concepts from QECCs like error sets and correctability translate to novel insights into the informational architecture of gauge theories. Conversely, the dictionary provides a systematic procedure for constructing symmetry-based QECCs and characterizing their error correcting properties. In this initial work, we scrutinize the dictionary between Pauli stabilizer codes and their corresponding QRF setups. We show that there is a one-to-one correspondence between maximal correctable error sets and tensor factorizations splitting system from error-generated QRF degrees of freedom. Relative to this split, errors corrupt only redundant frame data, leading to a novel characterization of correctability. When passed through the dictionary, standard Pauli errors behave as electric excitations that are dual, via Pontryagin duality, to magnetic excitations related to gauge-fixing. This gives rise to a new class of correctable errors and a systematic error duality. We illustrate our findings in surface codes, which themselves connect quantum error correction with gauge systems. Our exploratory investigations pave the way for foundational applications to gauge theories and for eventual practical applications to quantum simulation.

Matrix Product Unitaries (MPUs) have emerged as essential tools for representing locality-preserving 1D unitary operators, with direct applications to quantum cellular automata and quantum phases of matter. A key challenge in the study of MPUs is determining when a given local tensor generates an MPU, a task previously addressed through fixed-point conditions and canonical forms, which can be cumbersome to evaluate for an arbitrary tensor. In this work, we establish a simple and efficient necessary and sufficient condition for a tensor $M$ to generate an MPU of size $N$, given by $\operatorname{Tr}(\mathbb{E}_M^N) = \operatorname{Tr}(\mathbb{E}_T^N) = 1$, where $\mathbb{E}_M$ and $\mathbb{E}_T$ are the transfer matrices of $M$ and $T = MM^\dagger$. This condition provides a unified framework for characterizing all uniform MPUs and significantly simplifies their evaluation. Furthermore, we show that locality preservation naturally arises when the MPU is generated for all system sizes. Our results offer new insights into the structure of MPUs, highlighting connections between unitary evolution, transfer matrices, and locality-preserving behavior, with potential extensions to higher-dimensions.

We present Riemannian Gaussian Variational Flow Matching (RG-VFM), a geometric extension of Variational Flow Matching (VFM) for generative modeling on manifolds. In Euclidean space, predicting endpoints (VFM), velocities (FM), or noise (diffusion) are largely equivalent due to affine interpolations. On curved manifolds this equivalence breaks down, and we hypothesize that endpoint prediction provides a stronger learning signal by directly minimizing geodesic distances. Building on this insight, we derive a variational flow matching objective based on Riemannian Gaussian distributions, applicable to manifolds with closed-form geodesics. We formally analyze its relationship to Riemannian Flow Matching (RFM), exposing that the RFM objective lacks a curvature-dependent penalty - encoded via Jacobi fields - that is naturally present in RG-VFM. Experiments on synthetic spherical and hyperbolic benchmarks, as well as real-world tasks in material and protein generation, demonstrate that RG-VFM more effectively captures manifold structure and improves downstream performance over Euclidean and velocity-based baselines.

Let $G=(V,E)$ be an undirected unweighted multi-graph and $S\subseteq V$ be a subset of vertices. A set of edges with the least cardinality whose removal disconnects $S$, that is, there is no path between at least one pair of vertices from $S$, is called a Steiner mincut for $S$ or simply an $S$-mincut. Connectivity Carcass is a compact data structure storing all $S$-mincuts in $G$ announced by Dinitz and Vainshtein in an extended abstract by Dinitz and Vainshtein in 1994. The complete proof of various results of this data structure for the simpler case when the capacity of $S$-mincut is odd appeared in the year 2000 in SICOMP. Over the last couple of decades, there have been attempts towards the proof for the case when the capacity of $S$-mincut is even, but none of them met a logical end. We present the following results. - We present the first complete, self-contained exposition of the connectivity carcass which covers both even and odd cases of the capacity of $S$-mincut. - We derive the results using an alternate and much simpler approach. In particular, we derive the results using submodularity of cuts -- a well-known property of graphs expressed using a simple inequality. - We also show how the connectivity carcass can be helpful in efficiently answering some basic queries related to $S$-mincuts using some additional insights.

Stochastic policies (also known as relaxed controls) are widely used in continuous-time reinforcement learning algorithms. However, executing a stochastic policy and evaluating its performance in a continuous-time environment remain open challenges. This work introduces and rigorously analyzes a policy execution framework that samples actions from a stochastic policy at discrete time points and implements them as piecewise constant controls. We prove that as the sampling mesh size tends to zero, the controlled state process converges weakly to the dynamics with coefficients aggregated according to the stochastic policy. We explicitly quantify the convergence rate based on the regularity of the coefficients and establish an optimal first-order convergence rate for sufficiently regular coefficients. Additionally, we prove a $1/2$-order weak convergence rate that holds uniformly over the sampling noise with high probability, and establish a $1/2$-order pathwise convergence for each realization of the system noise in the absence of volatility control. Building on these results, we analyze the bias and variance of various policy evaluation and policy gradient estimators based on discrete-time observations. Our results provide theoretical justification for the exploratory stochastic control framework in [H. Wang, T. Zariphopoulou, and X.Y. Zhou, J. Mach. Learn. Res., 21 (2020), pp. 1-34].

We explore the potential for quantum speedups in convex optimization using discrete simulations of the Quantum Hamiltonian Descent (QHD) framework, as proposed by Leng et al., and establish the first rigorous query complexity bounds. We develop enhanced analyses for quantum simulation of Schrödinger operators with black-box potential via the pseudo-spectral method, providing explicit resource estimates independent of wavefunction assumptions. These bounds are applied to assess the complexity of optimization through QHD. Our findings pertain to unconstrained convex optimization in $d$ dimensions. In continuous time, we demonstrate that QHD, with suitable parameters, can achieve arbitrarily fast convergence rates. The optimization speed limit arises solely from the discretization of the dynamics, mirroring a property of the classical dynamics underlying QHD. Considering this cost, we show that a $G$-Lipschitz convex function can be optimized to an error of $\epsilon$ with $\widetilde{\mathcal{O}}(d^{1.5}G^2 R^2/\epsilon^2)$ queries. Moreover, under reasonable assumptions on the complexity of Hamiltonian simulation, $\widetilde{\Omega}(d/\epsilon^2)$ queries are necessary. Thus, QHD does not offer a speedup over classical zeroth order methods with exact oracles. However, we demonstrate that the QHD algorithm tolerates $\widetilde{\mathcal{O}}(\epsilon^3/d^{1.5}G^2 R^2)$ noise in function evaluation. We show that QHD offers a super-quadratic query advantage over all known classical algorithms tolerating this level of evaluation noise in the high-dimension regime. Additionally, we design a quantum algorithm for stochastic convex optimization that provides a super-quadratic speedup over all known classical algorithms in the high-dimension regime. To our knowledge, these results represent the first rigorous quantum speedups for convex optimization achieved through a dynamical algorithm.

We introduce a measure for evaluating the efficiency of finite universal quantum gate sets $\mathcal{S}$, called the Quantum Circuit Overhead (QCO), and the related notion of $T$-Quantum Circuit Overhead ($T$-QCO). The overhead is based on the comparison between the efficiency of $\mathcal{S}$ versus the optimal efficiency among all gate sets with the same number of gates. We demonstrate the usefulness of the ($T$-)QCO by extensive numerical calculations of its upper bounds, providing insight into the efficiency of various choices of single-qubit $\mathcal{S}$, including Haar-random gate sets and the gate sets derived from finite subgroups, such as Clifford and Hurwitz groups. In particular, our results suggest that, in terms of the upper bounds on the $T$-QCO, the famous T gate is a highly non-optimal choice for the completion of the Clifford gate set, even among the gates of order 8. We identify the optimal choices of such completions for both finite subgroups.

In this paper, we propose a method for transferring feature representation to lightweight student models from larger teacher models. We mathematically define a new notion called \textit{perception coherence}. Based on this notion, we propose a loss function, which takes into account the dissimilarities between data points in feature space through their ranking. At a high level, by minimizing this loss function, the student model learns to mimic how the teacher model \textit{perceives} inputs. More precisely, our method is motivated by the fact that the representational capacity of the student model is weaker than the teacher model. Hence, we aim to develop a new method allowing for a better relaxation. This means that, the student model does not need to preserve the absolute geometry of the teacher one, while preserving global coherence through dissimilarity ranking. Importantly, while rankings are defined only on finite sets, our notion of \textit{perception coherence} extends them into a probabilistic form. This formulation depends on the input distribution and applies to general dissimilarity metrics. Our theoretical insights provide a probabilistic perspective on the process of feature representation transfer. Our experiments results show that our method outperforms or achieves on-par performance compared to strong baseline methods for representation transferring.

We present a rigorous framework for determining equilibrium configurations of uniformly rotating self-gravitating fluid bodies. This work addresses the longstanding challenge of modeling rotational deformation in celestial objects such as stars and planets. By integrating classical Newtonian potential theory with modern mathematical tools, we develop a unified formalism that improves both the precision and generality of shape modeling in astrophysical contexts. Our method employs Lie group theory and exponential mapping to characterize vector flows associated with rotational deformations. We derive functional equations for perturbations in density and gravitational potential, resolved analytically using the shift operator and Neumann series. This extends Clairaut's classical linear theory into the nonlinear regime. The resulting formulation yields an exact nonlinear differential equation for the shape function, describing hydrostatic equilibrium under rotation without assuming slow rotation. This generalized Clairaut equation incorporates nonlinear effects and accommodates large rotational speeds. We validate the theory by deriving exact solutions, including the Maclaurin spheroid, Jacobi ellipsoid, and the unit-index polytrope. We also introduce spectral decomposition techniques to analyze radial harmonics of the shape function and gravitational perturbations. Using Wigner's formalism for angular momentum addition, we compute higher-order spectral corrections and derive boundary conditions for radial harmonics. This enables accurate computation of Love numbers and gravitational multipole moments, offering a comprehensive, non-perturbative approach to modeling rotational deformations in astrophysical systems.

Machine learning techniques offer an effective approach to modeling dynamical systems solely from observed data. However, without explicit structural priors -- built-in assumptions about the underlying dynamics -- these techniques typically struggle to generalize to aspects of the dynamics that are poorly represented in the training data. Here, we demonstrate that reservoir computing -- a simple, efficient, and versatile machine learning framework often used for data-driven modeling of dynamical systems -- can generalize to unexplored regions of state space without explicit structural priors. First, we describe a multiple-trajectory training scheme for reservoir computers that supports training across a collection of disjoint time series, enabling effective use of available training data. Then, applying this training scheme to multistable dynamical systems, we show that RCs trained on trajectories from a single basin of attraction can achieve out-of-domain generalization by capturing system behavior in entirely unobserved basins.

We investigate the accretion of a collisionless, relativistic kinetic gas by a rotating Kerr black hole, assuming that at infinity the state of the gas is described by a distribution function depending only on the energy of the particles. Neglecting the self-gravity of the gas, we show that relevant physical observables, including the particle current density and the accretion rates associated with the mass, the energy, and the angular momentum, can be expressed in the form of closed integrals that can be evaluated numerically or approximated analytically in the slow-rotation limit. The accretion rates are computed in this manner for both monoenergetic particles and the Maxwell-Jüttner distribution and compared with the corresponding results in the non-rotating case. We show that the angular momentum accretion rate decreases the absolute value of the black hole spin parameter. It is also found that the rotation of the black hole has a small but non-vanishing effect on the mass and the energy accretion rates, which is remarkably well described by an analytic calculation in the slow-rotation approximation to quadratic order in the rotation parameter. The effects of rotation on the morphology of the accretion flow are also analyzed.

Privacy regulations require the erasure of data from deep learning models. This is a significant challenge that is amplified in Federated Learning, where data remains on clients, making full retraining or coordinated updates often infeasible. This work introduces an efficient Federated Unlearning framework based on information theory, modeling leakage as a parameter estimation problem. Our method uses second-order Hessian information to identify and selectively reset only the parameters most sensitive to the data being forgotten, followed by minimal federated retraining. This model-agnostic approach supports categorical and client unlearning without requiring server access to raw client data after initial information aggregation. Evaluations on benchmark datasets demonstrate strong privacy (MIA success near random, categorical knowledge erased) and high performance (Normalized Accuracy against re-trained benchmarks of $\approx$ 0.9), while aiming for increased efficiency over complete retraining. Furthermore, in a targeted backdoor attack scenario, our framework effectively neutralizes the malicious trigger, restoring model integrity. This offers a practical solution for data forgetting in FL.

Polarimetry, which is the ability to measure the scattering response of the environment across orthogonal polarizations, is fundamental to enhancing wireless communication and radar system performance. In this paper, we utilize the Zak-OTFS modulation to enable instantaneous polarimetry within a single transmission frame. We transmit a Zak-OTFS carrier waveform and a spread carrier waveform mutually unbiased to it simultaneously over orthogonal polarizations. The mutual unbiasedness of the two waveforms enables the receiver to estimate the full polarimetric response of the scattering environment from a single received frame. Unlike existing methods for instantaneous polarimetry with computational complexity quadratic in the time-bandwidth product, the proposed method enables instantaneous polarimetry at near-linear complexity in the time-bandwidth product. Via numerical simulations, we show ideal polarimetric target detection and parameter estimation results with the proposed method, with improvements in performance and computational complexity over comparable baselines.

We provide new algorithms and provide example constructions of stabilizer models for the gapped boundaries, domain walls, and $0D$ defects of Abelian composite dimensional twisted quantum doubles. Using the physically intuitive concept of condensation, our algorithm explicitly describes how to construct the boundary and domain-wall stabilizers starting from the bulk model. This extends the utility of Pauli stabilizer models in describing non-translationally invariant topological orders with gapped boundaries. To highlight this utility, we provide a series of examples including a new family of quantum error-correcting codes where the double of $\mathbb{Z}_4$ is coupled to instances of the double semion (DS) phase. We discuss the codes' utility in the burgeoning area of quantum error correction with an emphasis on the interplay between deconfined anyons, logical operators, error rates and decoding. We also augment our construction, built using algorithmic tools to describe the properties of explicit stabilizer layouts at the microscopic lattice-level, with dimensional counting arguments and macroscopic-level constructions building on pants decompositions. The latter outlines how such codes' representation and design can be automated. Going beyond our worked out examples, we expect our explicit step-by-step algorithms to pave the path for new higher-algebraic-dimensional codes to be discovered and implemented in near-term architectures that take advantage of various hardware's distinct strengths.

This paper investigates the Teleparallel Robertson-Walker (TRW) $F(T)$ gravity solutions for a Chaplygin gas, and then for any polytropic gas cosmological source. We use the TRW $F(T)$ gravity field equations (FEs) for each $k$-parameter value case and the relevant gas equation of state (EoS) to find the new teleparallel $F(T)$ solutions. For flat $k=0$ cosmological case, we find analytical solutions valid for any cosmological scale factor. For curved $k=\pm 1$ cosmological cases, we find new approximated teleparallel $F(T)$ solutions for slow, linear, fast and very fast universe expansion cases summarizing by a double power-law function. All the new solutions will be relevant for future cosmological applications on dark matter, dark energy (DE) quintessence, phantom energy, Anti-deSitter (AdS) spacetimes and several other cosmological processes.

We derive an exact solution representing a Bondi-type stationary accretion of a kinetic (Vlasov) gas onto the Kerr black hole. The solution is exact in the sense that relevant physical quantities, such as the particle current density or the accretion rates, are expressed as explicit integrals, which can be evaluated numerically. We provide an analytic approximation which allows us to obtain simple formulas for the mass, energy, and angular momentum accretion rates. These formulas are used to derive characteristic time scales of the black hole mass growth and the associated spin-down in two different scenarios: assuming that the ambient energy density is either constant or decreases on a cosmological scale.

Simulation-Grounded Neural Networks (SGNNs) are predictive models trained entirely on synthetic data from mechanistic simulations. They have achieved state-of-the-art performance in domains where real-world labels are limited or unobserved, but lack a formal underpinning. We place SGNNs in a unified statistical framework. Under standard loss functions, they can be interpreted as amortized Bayesian predictors trained under a simulator-induced prior. Empirical risk minimization then yields convergence to the Bayes-optimal predictor under the synthetic distribution. We employ classical results on distribution shift to characterize how performance degrades when the simulator diverges from reality. Beyond these consequences, we develop SGNN-specific results: (i) conditions under which unobserved scientific parameters are learnable via simulation, and (ii) a back-to-simulation attribution method that provides mechanistic explanations of predictions by linking them to the simulations the model deems similar, with guarantees of posterior consistency. We provide numerical experiments to validate theoretical predictions. SGNNs recover latent parameters, remain robust under mismatch, and outperform classical tools: in a model selection task, SGNNs achieve half the error of AIC in distinguishing mechanistic dynamics. These results establish SGNNs as a principled and practical framework for scientific prediction in data-limited regimes.

Many relevant machine learning and scientific computing tasks involve high-dimensional linear operators accessible only via costly matrix-vector products. In this context, recent advances in sketched methods have enabled the construction of *either* low-rank *or* diagonal approximations from few matrix-vector products. This provides great speedup and scalability, but approximation errors arise due to the assumed simpler structure. This work introduces SKETCHLORD, a method that simultaneously estimates both low-rank *and* diagonal components, targeting the broader class of Low-Rank *plus* Diagonal (LoRD) linear operators. We demonstrate theoretically and empirically that this joint estimation is superior also to any sequential variant (diagonal-then-low-rank or low-rank-then-diagonal). Then, we cast SKETCHLORD as a convex optimization problem, leading to a scalable algorithm. Comprehensive experiments on synthetic (approximate) LoRD matrices confirm SKETCHLORD's performance in accurately recovering these structures. This positions it as a valuable addition to the structured approximation toolkit, particularly when high-fidelity approximations are desired for large-scale operators, such as the deep learning Hessian.