Papers

We study best modular approximation in Musielak-Orlicz spaces. A geometric classification near the origin of Orlicz functions is introduced to isolate weighted $L^1$-like behavior. We establish existence results under minimal hypotheses and provide a variational characterization of minimizers via lateral derivatives. Finally, we establish uniqueness criteria, including a one-dimensional Haar-type theorem and characterizations in $0$- and $1$-spaces in higher dimensions.

We determine the order of the low moments of partial sums of random multiplicative functions associated with Euler products of bounded degree whose unitary local parameters satisfy a prime-square cancellation condition, thereby generalizing Harper's seminal work. The result applies to irreducible compact-group representations of unitary or symplectic type, including the higher-rank Sato–Tate model $SU(d)$ and the odd symmetric-power Sato–Tate model $\operatorname{Sym}^{m}SU(2)$.

We prove an $\varepsilon$-regularity theorem for $m$-varifolds with mean curvature in $L^p$, $p>m$, with respect to an anisotropic integrand satisfying a Michael-Simon inequality and the quadratic exposed condition: near sufficiently flat density-one points, such varifolds are representable as $C^{1,\alpha}$ graphs. Combined with the recent proof of the anisotropic Michael-Simon inequality, this establishes an anisotropic Allard regularity theorem in arbitrary codimension for a large class of anisotropic integrands, including those close to the area functional. We also exhibit the first examples of anisotropies satisfying both the uniform scalar atomic condition and the Michael-Simon inequality, that are not close to any ellipsoidal norm. These include the $\ell^q$ norms in every dimension and codimension, for explicit ranges of $q$, and a new class of axisymmetric anisotropies.

Let $F$ be a field of characteristic zero and consider $M_2(F),$ the algebra of $2\times 2$ matrices over $F,$ with canonical $\mathbb{Z}_2$-grading and endowed with transpose superinvolution. In this paper, we present the generators of the $T_2^*$-ideal of $*$-identities and of the $T_2^*$-subspace of central $*$-polynomials of $M_2(F).$ As a consequence, we determine the sequences of $*$-codimensions and $\langle n\rangle$-cocharacters of $M_2(F)$. In particular, we prove that the $n$-th $*$-codimension grows like $4^nn^{-1/2}$.

The dimension of a Galois hull is an important parameter in the construction of entanglement-assisted quantum error-correcting codes. In this paper, we study generalized Roth-Lempel (GRL) codes with prescribed Galois hull dimensions over finite fields of arbitrary characteristic. We develop a common construction method based on normalized Lagrange coefficients and suitable choices of column multipliers, and obtain six explicit constructions from multiplicative cosets, trace fibers, additive-subspace cosets, and mixed additive–multiplicative fibers. For extension sizes $s=2$ and $s=3$, we construct MDS and AMDS GRL codes with Galois hulls of specific dimensions. The resulting codes provide EAQECCs with explicit dimension, entanglement consumption, and relatively large minimum distance. Taking zero-dimensional hulls also gives Hermitian LCD GRL codes and the corresponding maximally entangled EAQECCs.

We show that there exists a set $S \subset \mathbb{Z}^2$ containing no four points on a circle or a line such that $|S \cap [n]^2| = \Omega(n)$ as $n \rightarrow \infty$. Since any no-four-on-a-circle set in $[n]^2$ has size $O(n)$, this resolves (up to a constant) a question raised by the current authors and Keevash concerning the density of extensible no-four-on-a-circle constructions. Our construction is based on weighted random sampling from the integer lattice followed by careful deletion.

We prove that for each $3\le k\le n-1$ a convex $C^2$ entire solution of $\sigma_k$ equation in $\mathbb{R}^n$ with positive constant right hand side is a quadratic polynomial, without a growth assumption on the solution itself.

We design and analyse a hybrid high-order (HHO) method for the linear elasticity problem with a linear slip interface, i.e.\ with interface conditions of spring type, in which the jump of the displacement across the interface is proportional to the traction through a compliancy tensor $\bK=\alpha\bI+(\beta-\alpha)\udl{n}\otimes\udl{n}$. The mesh is fitted to the interface, the discrete unknowns are polynomials of degree $k\ge 1$ on the mesh faces and of degree $k+1$ in the mesh cells, and general polytopal cells are allowed. The two novelties of the method lie in the local symmetric strain reconstruction, which incorporates the interface condition, and in the interface stabilisation, which is built from a regularised interface stiffness $\bS_h$ in the spirit of Hansbo and Hansbo \cite{HH04}. As a consequence, a single formulation covers the whole range of compliancies, from the perfectly bonded interface $\alpha=\beta=0$, where the interface stabilisation acts as a Nitsche-type penalty, to the traction-free interface obtained as $\alpha,\beta\to+\infty$, and no unknown is attached to the jump. We prove that the discrete bilinear form is coercive and that the errors converge as $h^{k+1}$ in the energy norm and as $h^{k+2}$ in the $L^2$ norm, with constants that are independent of the compliancy parameters and of the Lamé coefficient $\lambda$, so that the method is also locking free. Various numerical examples and comparisons are provided to confirm the theoretical results.

In 1878, Pafnuty Chebyshev presented to the Association fran\c caise pour l'avan\-cement des sciences [French Association for the Advancement of the Sciences] an article \cite{Chebyshev1878} dealing with garment cutting. According to Chebyshev himself, his interest was sparked by a lecture given by Édouard Lucas that he had attended in 1876 \cite{Lucas1876}. There is a second story on the origin of Chebyshev's interest in garment cutting according to which in the 1850s, being short of money, Chebyshev got himself a job as a consultant to a clothing factory. At the time of the Crimean War (1853-1856), there was a great demand for uniforms. Chebyshev was allegedly asked to optimize the use of fabric, and it was there that his interest in garment cutting was born. This second story appears to have its origin in a post by Clive J. Grant to MacTutor in 1996 \cite{Grant1996}. However, this contribution contains no references, and no other source of information that I have found offers any first-hand documentation to support this story. Our conclusion is that this second story is a fabrication, invented out of whole cloth.

A coloring of $\R^n$ is proper for the forbidden distance segment $[1,\ell]$ if no two points of the same color are at a distance from $[1,\ell]$; the minimum number of colors is $\chi(\R^n,[1,\ell])$, and $\ell=1$ gives the classical chromatic number $\chi(\R^n)$ of the Nelson–Hadwiger problem. We prove the new upper bounds $\chi(\R^4)\le43$, $\chi(\R^5)\le132$, $\chi(\R^7)\le1029$, $\chi(\R^9)\le7203$, $\chi(\R^{10})\le45619$, improving the previously known $49$, $140$, $1372$, $17253$ and $3^{10}$; in particular, this refutes the conjecture of Arman, Bondarenko, Prymak and Radchenko that $49$ and $140$ are optimal among all lattice colorings of $\R^4$ and $\R^5$. The first four bounds come from explicit rational lattices — an Eisenstein lattice in $\R^4$, a lattice in general position in $\R^5$, and laminations of the Eisenstein colorings $E_6^*/343$ and $E_8/2401$ in $\R^7$ and $\R^9$ — and each is reduced, by one verification protocol, to a finite list of inequalities between explicitly written rational numbers checked in exact arithmetic. The fifth bound is analytic: we prove that for every Eisenstein lattice $\Lambda$ the distance between same-colored cells of $(3+\omega)\Lambda$ equals $\sqrt{7/3}\,\lambda_1(\Lambda)$, which gives the exact widths of all known colorings with $7^{n/2}$ colors, and a product rule $\sum_i1/d_i^2\le1$ for the widths of orthogonal products; together they yield $45619=2401\cdot19$, the first bound in $\R^{10}$ below $3^n$, as well as $\chi(\R^{25})\le4\cdot7^{12}$ and $\chi(\R^{26})\le19\cdot7^{12}$. We also show that no sublattice of $E_8$ of index below $2401$ defines a proper coloring. All code, exact certificates and data are open.

Quantum $p$-local spin glass Hamiltonians are natural quantum analogues of the classical spin glass models. We provide an asymptotic characterization of the maximal energy achievable by the product states. We prove that for every $p \ge 2$, the limit of ground state energy exists and is given by a Parisi-type variational formula. This settles a question left open by Anschuetz et. al. (2025). The variational formula also recovers the known large-$p$ asymptotics as a consequence. Finally, we prove that the limiting product state energy is universal for a broad class of non-Gaussian interactions.

Our first main result is a rigidity theorem for complete self-expanders in the Euclidean space with higher codimension, assuming an integral cur More precisely, any smooth complete self-expander $x:M\to \mathbb R^{n+p}$($n\geq3$) that satisfies both $(\int_{M}|A|^{n}d\mu)^{\frac{1}{n}}<K(n)$ and $\int_{M}|A|^{n}e^{\frac{|x|^{2}}{2}}d\mu<\infty$ for a positive constant $K(n)$ depending only o isometric to $\mathbb R^{n}$. Moreover, we show that the rigidity result persists when the pinching condition is expressed in terms of the trace-free second fundamental form.

We study inhomogeneous long-range first-passage percolation on $\mathbb{Z}^d$ with edge passage times $\lVert x-y\rVert^\alpha\omega_{xy}/(V_xV_y)$, where vertex weights have polynomial upper-tail exponent $\gamma$ and edge noises have polynomial lower-tail exponent $\theta$ at zero. The model interpolates between long-range first-passage percolation and scale-free percolation and exhibits competition between reusable heavy-vertex hubs and pair-specific small-noise bridges. We conjecture an eight-regime phase diagram, organized into five growth phases governed by $q_{\rm hub}=d/\gamma$ and $q_{\rm edge}=d/\theta$. For $T_n=T(0,\lceil nx\rceil)$, we prove upper bounds of the conjectured order in every regime and matching lower bounds in phases I and II. Specifically, $T_n=0$ a.s. when $\alpha<q_{\rm hub}\vee q_{\rm edge}$, while $T_n=\Theta_{\mathbb{P}}(1)$ when $q_{\rm hub}\vee q_{\rm edge}<\alpha<2q_{\rm hub}$. In the edge-dominated intermediate regime, $T_n=O_{\mathbb{P}}((\log n)^{\Delta_{\rm III}+\varepsilon})$, where $\Delta_{\rm III}=\log 2/\log(2q_{\rm edge}/\alpha)$. The two power-law regimes satisfy $T_n=O_{\mathbb{P}}(n^{\alpha-2q_{\rm hub}+\varepsilon})$ and $T_n=O_{\mathbb{P}}(n^{\alpha-2q_{\rm edge}+\varepsilon})$, and the linear regime satisfies $T_n=O_{\mathbb{P}}(n)$. All upper bounds are constructive, based on hub-chain and binary edge-bridge multiscale constructions.

Binary modular Laplacian dynamics is organized by repeated replication, growth, and collapse on dyadic scales. We study how isolated and periodically repeated non-binary updates modify this organization and whether they can sustain densely occupied geometric structures. Computational experiments across multiple finite seeds, neighborhood masks, inserted moduli, and periodic schedules show that constant prime moduli retain the expected \(p\)-adic replication hierarchy. A single non-binary insertion mainly acts as an effective-seed replacement: subsequent evolution remains binary-like, but may resume at an integer phase shift of the collapse–recovery cycle, with only modest finite-time densification. A qualitatively different response occurs for periodic schedules \([2,k,2^s]^\infty\), where \(2^s\) denotes \(s\) consecutive binary updates. Odd-modulus insertions can sustain long-lived, spatially coherent carpet-like regimes with comparatively high and stable occupation, whereas even-modulus insertions remain binary-like. Ternary insertion gives the broadest and most robust high-density response. For \(k=3\), density depends non-monotonically on the binary-tail length, with troughs near \(s=7,15,23\) that reveal phase-sensitive interaction with the binary epoch structure. Cross-modulus synchronization also occurs: for degree-four diagonal Neumann and von Neumann masks, \(k=5,7,9\) repeatedly collapse onto the same complete configuration at subsequent binary phases. A parity argument explains this loss of memory. Increasing seed extent generally raises occupation while reducing tail-length sensitivity. Overall, periodic modular insertions act as phase-selective, geometry-dependent mechanisms transforming \(p\)-adically organized replication into long-lived carpet-like dynamics.

The linking number and coefficients of the Conway polynomial are two examples of finite type invariants. Goussarov-Polyak-Viro proved that any finite type knot invariant can be computed in a two step process, where the second step is referred to as the GPV map for the invariant. This paper computes the GPV maps for the linking number and low-degree coefficients of the Conway Polynomial. Chmutov-Khoury-Rossi gave one description of the GPV maps for coefficients of the Conway polynomial in terms of arrow diagrams and state-sum calculations. We take a much more grounded approach using fundamental linear algebra to describe the GPV maps in terms of a basis for the vector space of Gauss diagrams.

In the first half of the paper we revisit the question regarding MaxEnt probability distributions on spheres. We derive families of MaxEnt distributions on spheres in real and complex vector spaces with fixed expected Busemann function (energy). As a particular case, we deduce sub-families on canonical energy levels where inverse temperature equals the dimension of the sphere. In the second part we focus on the information manifold of canonical MaxEnt distributions on the sphere in the complex vector space. This manifold is isomorphic to the Bergman ball. We introduce the reproducing kernel on this manifold and use the RKHS theory to elaborate the model of cognition. In particular, we state the principle of minimal cognitive effort in RKHS and demonstrate its dual relationship with the MaxEnt principle for probability distributions on the boundary sphere.

Two-timescale stochastic approximation (TTSA) is a fundamental tool for analyzing coupled iterative algorithms in reinforcement learning, optimization, and stochastic control. However, finite-time guarantees for nonlinear two-timescale schemes remain difficult to obtain, especially under constant step-sizes. In this paper, we study nonlinear TTSA with step-sizes $\alpha\gg\beta$. Under standard stability, regularity, and Markovian noise assumptions, we upper bound the mean-squared error and the bias of both iterates around their limiting equilibria. Our bounds scale as $O(\alpha+\beta^2/\alpha^2)$, which we prove to be tight when $\beta\le\alpha^{3/2}$. The analysis separates the contributions of initial conditions, fast-timescale tracking error, Markovian dependence, and timescale coupling, thereby clarifying the origin of the $\beta^2/\alpha^2$ term. Our results reveal qualitative differences from the linear TTSA setting previously studied, showing that nonlinear dynamics introduce additional finite-time effects that are absent in the linear case.

We classify the automorphism groups of smooth cubic threefolds and fourfolds. We also show that there are $156$ (respectively, $40$) connected families of smooth cubic fourfolds (respectively, threefolds) with specified automorphism group action. For each family, the group action and defining equations are also given. The classification combines both representation theory (based on GAP) and lattice theory (based on SageMath).

Let $q=3^m$, let $K=\mathbb F_{q^2}$, and let \[\mathcal T=\{x\in K^*:\operatorname{N}_{K/\mathbb F_q}(x)=1\}.\] For both cyclic and constacyclic Gashkov-Sidel'nikov codes, we show that the set of signed parity-check column labels is precisely $\mathcal T$. Consequently, the decoding problem separates into two stages: determining the minimum error weight associated with a syndrome $S$ and constructing an error vector attaining this minimum. We identify the former quantity with the minimum additive length of $S$ with respect to $\mathcal T$ and determine it exactly by the norm and the quadratic character of $\mathbb F_q$. We also determine the complete coset-weight distribution and recover the known covering radius $3$. For the constructive part, we use quadratic-character sums and Weil bounds to construct a coset leader for every syndrome of coset weight three. The resulting procedures give complete maximum-likelihood decoders.

We introduce a kernel–wreath construction for finite skew braces. Let $C$ be a finite skew brace, write $(C,+)=A$ and $(C,\circ)=R$, and let $\chi:R\twoheadrightarrow T$ be an epimorphism onto a non-trivial finite abelian group. We show that a natural index-$|T|$ kernel in the permutational wreath product $R\wr T$ acts regularly on $A^T$, and hence defines a new skew brace $\KT_T(C,\chi)$ with additive group $A^{|T|}$. The construction preserves every finite abelian quotient of the multiplicative group, as well as solvability, and contains the seed brace $C$ as a diagonal subbrace. It can therefore be iterated indefinitely. More precisely, if $Q$ is a non-trivial finite abelian quotient of $R$ and $p\in\pi(Q)$, then one obtains an infinite tower \[ C=C_0\hookrightarrow C_1\hookrightarrow C_2\hookrightarrow\cdots \] with $(C_m,+)\cong A^{p^m}$ for every $m\geq0$. Our main permanence result shows that if $C$ is simple and is not a trivial skew brace, then $\KT_T(C,\chi)$ is again simple. Consequently, a single simple seed whose multiplicative group has a non-trivial finite abelian quotient gives rise to infinite families of finite simple skew braces. In particular, starting from suitable solvable regular subgroups of the holomorph of a finite non-abelian simple group $S$, we obtain infinite families of simple skew braces with additive groups $S^{p^m}$ and solvable multiplicative groups. Further applications are given to the simple skew braces of order $12$ and to Byott's family.

We establish sharp convergence rates for quadratically regularized optimal transport with quadratic cost in the regime of small regularization $\varepsilon$. In particular, we quantify the sparsity of the support of the regularized optimizer. For smooth marginal densities in $\mathbb{R}^d$, this support lies within a distance of order $\varepsilon^{1/(d+2)}$ from the Brenier graph, and every section of the support is sandwiched between balls with radii of that order. The geometry of the support is closely linked to the dual potentials. We show that the potentials satisfy uniform two-sided Hessian bounds and converge uniformly at rate $\varepsilon^{2/(d+2)}$, while their gradients converge at rate $\varepsilon^{1/(d+2)}$. The rates are sharp, and we further identify the regularity threshold where convergence breaks down.

We propose sparse identification of port-Hamiltonian systems (SIPHy), enabling structure-preserving symbolic regression from noisy trajectory observations. The method applies to port-Hamiltonian systems, which provide a general framework for describing dynamical systems in terms of energy exchange, dissipation and control. Our algorithm can jointly identify the Hamiltonian as well as the dissipation and input matrices. Furthermore, we introduce Hamiltonian flow splines to better approximate derivatives of trajectory data corrupted by noise or with missing time points, a major challenge of system identification for differential equations. This method assembles flows of piecewise polynomial Hamiltonians to produce a smooth, differentiable trajectory necessary for sparse regression. Combining flow splines with SIPHy yields interpretable, physically grounded models from noisy and incomplete trajectory data. Because the Hamiltonian, dissipation, and input matrices are identified as separate components, the resulting models can be simulated under control inputs and dissipation regimes never observed during training, thus improving the robustness and generalization capabilities of model discovery methods.

Self-similar solutions to the porous medium equation with dominating spatially inhomogeneous absorption $$ \partial_tu=\Delta u^m-|x|^{\sigma}u^p, \quad (x,t)\in\real^N\times(0,\infty), \quad N\geq1, $$ with exponents $1<p<m$ and $\sigma>0$, are classified. Looking for solutions in the form $$ u(x,t)=t^{-\alpha}f(|x|t^{\beta}), \quad \alpha=\frac{\sigma+2}{\sigma(m-1)+2(p-1)}, \quad \beta=\frac{m-p}{\sigma(m-1)+2(p-1)}, $$ it is shown that all their profiles satisfy the behavior at infinity given by $$ \lim\limits_{\xi\to\infty}\xi^{\sigma/(p-1)}f(\xi)=\left(\frac{1}{p-1}\right)^{1/(p-1)}, $$ but the solutions strongly differ with respect to their behavior near the origin: there exist a unique solution with $f(0)>0$, $f'(0)=0$, another unique solution such that $f$ presents a dead-core; that is, $f\equiv0$ for $\xi\in[0,\xi_0]$ for some $\xi_0>0$, and, finally, there exists $K^*\in(0,\infty)$ such that, for any $K\in(0,K^*)$, there is at least a solution such that $$ \lim\limits_{\xi\to0}\xi^{-(\sigma+2)/(m-p)}f(\xi)=K. $$ The large time behavior of general solutions, making strong use of these three types of self-similar solutions, will be addressed in a companion work.

This work studies adaptive sparse-grid discontinuous Galerkin (DG) discretizations for the Bhatnagar–Gross–Krook (BGK) model, a kinetic equation posed in four- and six-dimensional phase-space. Standard DG methods are rendered impractical for the BGK model by the curse of dimensionality, motivating compressed representations that adapt to the solution in time. Using the adaptive sparse-grid DG method, we quantify accuracy and compression by comparing the adaptive degrees of freedom to full-grid DG methods and by assessing the resulting kinetic and fluid quantities in both fluid and rarefied regimes. Test cases include a relaxation problem, a multidimensional Sod shock tube, and shear/expansion flows used in prior low-rank BGK studies. To build an efficient Maxwellian evaluation without violating conservation, a central obstacle for structure-perserving BGK simulations, we introduce a hybrid interpolation strategy that exploits velocity separability to recover the correct discrete collision invariants and prove conservation of the resulting discrete collision operator on adaptive sparse grids. Our results show that the adaptive sparse-grid strategy can recover accurate and physically relevant solutions with sharp gradients, and the method reduces the active degrees of freedom by factors ranging from several-fold to several orders of magnitude, with the largest reductions occurring in the six-dimensional examples. All computations are performed with the open-source ASGarD adaptive sparse-grid DG library.

We investigate the entanglement dynamics of quantum states defined on graphs with non-local Ising interactions governed by the Katz kernel of the underlying network. The interaction pattern is physically motivated by a gapped fermionic mediator propagating on the same graph, whose perturbative elimination yields an effective Katz-weighted Ising Hamiltonian. Using the Entanglement Distance, we derive an exact analytical expression for the entanglement generated from an initially separable state and apply it to representative deterministic graph families. We then characterize the dynamics in different propagation regimes. In the weak-Katz regime, the dynamics admits a systematic motif expansion with triangles entering at first order order and four-cycles, local degree structure, and overlappin triangles appearing at second order. In the strong-propagation regime, the interaction is instead dominated by the principal adjacency mode and by the localization properties of its eigenvector. For Erdős–Rényi graphs, the weak-propagation expansion can be averaged analytically, revealing a locally tree-like contribution in the sparse regime and saturation of the Entanglement Distance density in the dense regime. Our results connect entanglement dynamics with both the walk-based and spectral structure of complex networks.

Let $\mathbb N$ denote the set of nonnegative integers. For an integer $k>1$ and a set $A\subseteq\mathbb N$, let $R_{1,k}(A,n)$ denote the number of solutions of $n=a_1+ka_2$ with $a_1,a_2\in A$. For integers $k>1$ and $t\ge1$, Yang and Chen defined $f_k(t)$ to be the number of sets $A\subseteq\mathbb N$ for which \[ R_{1,k}(A,n)=R_{1,k}(\mathbb N\setminus A,n) \] for all integers $n\ge t$, and asked whether $f_k(t)$ and $f_l(t)$ are eventually equal for any integers $k,l>1$. We prove that \[ f_k(t)\asymp_k \frac{2^t}{t^{k/2}}, \] which gives a negative answer to their problem.

We prove that the Bézout inequality for mixed volumes characterizes simplices among full dimensional convex bodies in every dimension, resolving a conjecture of Soprunov and Zvavitch. We give two separate proofs of the conjecture. Along the way, we also prove characterizations of simplices in terms of longest chords or relative inradii.

The Solvability Complexity Index (SCI) provides an extensional limit-height formalism for recovering a target map $\Xi$ from finite samples of an evaluation interface $\Lambda$ by finite-height towers of pointwise limits. At first sight this sounds like a typical Type-2 question, so it seems natural to ask how deep and rich the connection of SCI approximation questions and Type-2 computability questions is. This note aims to explain this connection further with explicit examples mostly from the literature.

We study a variable-exponent double-phase Neumann problem on a domain in a metric measure space, where the boundary measure satisfies an upper codimension-\(\theta\) bound for \(0\le\theta<1\). Under a local comparability condition on the double-phase growth function, we establish a modular estimate for a fractional maximal operator and use it to prove the existence and boundedness of the associated trace operator. We then establish coercivity of the associated energy functional on the normalized double-phase Newtonian space and, under reflexivity, prove the existence of minimizers. We further show that the set of minimizers is closed and convex, and that any two minimizers have the same minimal weak upper gradient almost everywhere. Finally, we prove the stability of minimizers under perturbations of the Neumann data.

Mathematical Programs with Blocks of Vanishing Constraints (MP-BVCs) are a challenging class of optimization problems due to the loss of standard constraint qualifications. Based on existing theory for classical MPVCs, i.e., MPBVC with block size one, we extend the constraint qualification theory for MPBVCs and show that the known LICQ-type condition for MPBVCs implies the Guignard Constraint Qualification (GCQ). A novel piecewise gradient flow approach converges under GCQ. It directly computes strongly stationary points and is an extension of the piecewise gradient flow approach for classic MPVC problems. Numerical experiments on several truss topology design instances with two and even three applied load forces demonstrate the robustness and effectiveness of the method.