Papers

We determine how wide a randomly initialized ReLU layer must be to distinguish every pair of inputs. For independent Gaussian weights and zero bias, as the dimensions grow at a fixed output-to-input ratio, the probability of global injectivity tends to zero below a sharp threshold and to one above it. We characterize this threshold exactly through a variational formula for the limiting minimum number of active neurons per input coordinate. The formula accounts for correlations among input directions at arbitrarily many levels. We prove this formula rigorously, establishing a prediction from the statistical physics of the spherical perceptron. The proof compares systems of nearby dimensions and controls a Gaussian interpolation by calibrating responses to auxiliary fields. A uniform volume estimate connects the soft minimum to the worst input direction. An exact-arithmetic calculation places the threshold below seven output neurons per input coordinate. Numerical evaluations suggest a value near 6.698.

Let $m,n\geq2$ be integers. We study the least exponent $t_{m,n}$ such that the $m$-fold additive energy of any subset $A$ of the discrete cube $\{0,1,\cdots,n-1\}^d$ in any dimension $d$ satisfies $E_m(A)\leq |A|^{t_{m,n}}$. For every fixed $m$, we obtain the asymptotic formula $$ t_{m,n}=2m-1-(1+o(1))\log_n \frac{(2m-1)^{m-1/2}}{(2m)^{m-1}} \qquad(n\to\infty). $$ For $m=2$, this gives $$ t_{2,n} = 3 - (1+o(1)) \log_n\frac{3\sqrt{3}}{4}, $$ proving a conjecture of the first author.

A family of paths in a graph $G$ strongly separates the edges of $G$ if for every ordered pair of distinct edges $(e,f)$ some path in the family contains $e$ and avoids $f$; the minimum size of such a family is denoted by $\operatorname{ssp}(G)$. Bonamy, Botler, Dross, Naia, and Skokan proved, for every $n$-vertex graph $G$, that $\operatorname{ssp}(G)\le 19n$; Liu, Xu, and Yang recently improved this to $\operatorname{ssp}(G)\le 10n-o(n)$. We prove, for every $n$-vertex graph $G$, that $\operatorname{ssp}(G)\le 7.75n$.

We provide examples of smooth sextic double solids with potentially dense sets of rational points, over any field of characteristic zero.

Let $1,\alpha_1,\alpha_2$ be a basis for a cubic number field $K\subset\mathbb{R}$, and for $\tau_1,\tau_2\ge 1$ let $G(\mathbf{\alpha},\mathbf{\tau})$ denote the number of distinct gaps between the fractional parts of the numbers $m_1\alpha_1+m_2\alpha_2$, with $m_1,m_2\in\mathbb{Z}$, $0\le m_1<\tau_1$ and $0\le m_2<\tau_2$. We show that $G(\mathbf{\alpha},\mathbf{\tau})$ is unbounded as $\mathbf{\tau}$ varies. When $K$ is totally real this is deduced from a density theorem of Shapira for orbits of the diagonal group on the space of lattices. When $K$ has a complex embedding we give an explicit construction, which also shows that there are constants $c,\kappa>0$ for which $G(\mathbf{\alpha},\mathbf{\tau})\ge c(\log\max\{\tau_1,\tau_2\})^{\kappa}$ for infinitely many $\mathbf{\tau}\in\mathbb{N}^2$.

We formulate a version of Schur–Weyl duality for the current Lie algebra $\mathfrak{gl}_V[x,y]=\mathfrak{gl}_V\otimes\mathbb{C}[x,y]$. Under this duality, the Garsia–Haiman modules of the $N!$-conjecture become cyclic and cocyclic $\mathfrak{gl}_V[x,y]$-modules, and we describe them as iterated fusion and cofusion products of tautological $\mathfrak{gl}_V$-modules. We show that Haiman's $N!$-theorem for a diagram is equivalent to the coincidence of the fusion and the cofusion filtrations on the tensor product of the local Weyl modules attached to its rows, and that the fusion and cofusion products of Garsia–Haiman modules are associative. Let $\lambda$ and $\mu$ be Young diagrams obtained by removing two different corners from the same diagram. We construct an iterated fusion product of $S^2V$ with local Weyl modules, which is a quotient of the common quotient of the Garsia–Haiman modules of $\lambda$ and $\mu$, fits into short exact sequences with each of them, and has Butler's intersection polynomial as its character. This gives a representation-theoretic proof of Butler's conjecture. The same construction gives lower bounds for the dimensions in the $N!/k$-conjecture.

In this paper, we show that the height function of the six-vertex model in simply connected domains with piecewise constant boundary conditions converges to the Gaussian Free Field (GFF) with Dirichlet boundary conditions. This implies the conformal invariance of the scaling limit of the model. The argument relies on the full plane GFF convergence combined with Reflexion Positivity and qualitative estimates. The proof works for a wide range of spectral parameters $\Delta$ (among which $\Delta$ close to $0$, and between $-1$ and $-1/2$) making this paper the first instance of a conformal invariance result for a continuum of models. The paper has consequences for other models, including the FK percolation with cluster-weight $q=4$ and the Ashkin-Teller model, which we will pursue in future work.

We prove that the set of real $\mathrm{SL}_n$-opers forms a discrete subset of the moduli space of irreducible flat connections.

The hypersimplex $\Delta_{k,n}$ is the convex hull of all 0/1-vectors of length $ n $ with coordinate sum $ k $. Early conjectured, and Kim proved, a combinatorial formula for the $ h^* $-polynomial of $\Delta_{k,n}$ in terms of hypersimplicial decorated ordered set partitions. In this paper, we refine the Early–Kim formula to the flag $h$-numbers of the alcove triangulation of $\hyp$. We also give a representation-theoretic interpretation of its flag $f$-numbers in terms of Young permutation modules.

A classical theorem of Figiel asserts that every isometric embedding $\varphi\colon X\to Y$ between real Banach spaces, with $\varphi(0)=0$, admits a linear contractive left inverse on the closed linear span of $\varphi(X)$. We show that this phenomenon breaks down completely once local convexity is lost. More precisely, for every $0<p<1$ and every nonzero real Banach space $X$, we construct a $p$-Banach space $Y$ and an isometric embedding $\varphi\colon X\to Y$ such that $\varphi(0)=0$ and the closed linear span of $\varphi(X)$ is the entire space $Y$, but there is no linear map, bounded or not, on the linear span of $\varphi(X)$ which is a left inverse of $\varphi$. The construction is carried out using the $p$-Banach counterpart of Lipschitz-free spaces. In fact, $Y$ may be chosen isomorphic to the Lipschitz-free $p$-space over $X$. The main ingredient is a quantitative estimate controlling the distance to elementary molecules in Lipschitz-free $p$-spaces, a phenomenon specific to the range $0<p<1$. As a further consequence, we show that a real Banach space has the isometric $p$-lifting property if and only if its dimension is at most one.

Mod $p$ Hecke eigenforms for congruence subgroups of $\operatorname{SL}_2(\mathbb{Z})$ which do not lift to characteristic $0$ have received much attention, and were named ethereal modular forms by G. Schaeffer. In this paper, we study liftability for mod $p$ Bianchi modular forms: computations suggest that a lift to characteristic zero may always be possible by increasing the level, as in the case of classical modular forms. We develop efficient algorithms for computing degeneracy maps in cohomology, which allow us to rigorously verify eigenvalue congruences. For small primes and certain fields, we also prove that lifts always exist as a consequence of the recent modularity theorem due to Caraiani-Newton. A Magma implementation, complemented by LMFDB data due to Cremona, shows that in favourable parameter ranges, most mod $p$ Bianchi modular forms lift.

We introduce extendable pattern avoidance for alternating sign rectangles (ASRs), the natural rectangular generalization of alternating sign matrices (ASMs). An ASR extendably avoids a pattern $\pi$ if it is the upper left corner of an ASM avoiding $\pi$. For each of the four length-three patterns in the equivalence class $\{312, 132, 213, 231\}$ we establish a complete system of recurrence relations enumerating extendably $\pi$-avoiding ASRs of size $r \times k$ with a prescribed number $d$ of nonempty rows. For $\pi = 312$ we further conjecture a closed-form expression and prove it on several diagonal slices via bijections involving Schröder ballot numbers, refined Schröder numbers and Delannoy paths that do not cross the main diagonal vertically. For ASRs of size $(r-1) \times (r+1)$ extendably avoiding $213$ we give a bijection to little Schröder paths of length $r$. The remaining patterns of length three, $123$ and $321$, are more elusive, mirroring the situation of ASMs.

We introduce linear occupancy homomorphisms (LOHs), a class of Markov decision process (MDP) transformations that defines the MDP homomorphism over the feasible occupancy measure space of MDPs. In contrast to classic MDP homomorphism frameworks, which are defined through mappings over the state and state-action spaces, LOHs map between occupancy measure spaces and are representable as linear transformation matrices that expose important structural properties of the homomorphism itself. For this class of MDP transformations, we derive finite dimensional linear inequality constraints that certify when a candidate transformation matrix is a valid LOH and derive the corresponding transformed MDP. We further establish a connection between LOHs, soft MDP homomorphisms, and factored matrix robust MDPs. When the linear transformation matrix is invertible, we prove that the original MDP is equivalent to a constrained MDP after the homomorphic transformation, and derive necessary and sufficient conditions under which the LOH preserves the optimal value. We illustrate the benefits of LOHs in two settings: improving robust MDP solutions by transforming nonrectangular parameter uncertainty sets into homomorphic rectangular MDPs, and decomposing a large MDP into smaller independent MDPs.

We study the extent to which periodic orbits determine a one-dimensional dynamical system. In particular, we show that they determine every topologically mixing interval map. More generally, for a non-mixing interval map $f$ with dense periodic points, the periodic orbits need not determine $f$, but they always do so for $f^2$. Finally, for maps on compact intervals, we obtain a local version of this rigidity: for each basic set $B$ arising in the spectral decomposition of an interval map $f$, there exists an iterate $f^q$ whose restriction to $B$ is determined by the periodic orbits of $f$.

We construct a Banach lattice $d$-algebra with a strong order unit whose product has no positive associative bilinear extension to its Dedekind completion. This gives a negative answer to a question of Wickstead and to the $d$-algebra case of a question of Huijsmans. It also disproves an extension theorem of Chil. The example shows that norm completeness and a strong order unit do not ensure that multiplication extends to the Dedekind completion.

High-subsonic turbulent jets trap acoustic modes in their potential core. The modes lie in narrow frequency bands that are $O(0.1)$-wide in Strouhal number, $St$. By studying a Mach 0.9, Reynolds number $10^6$ jet, this work confirms an earlier hypothesis: each band is made up of $O(10)$ discrete frequencies, $\Delta St \sim O(0.01)$ apart. To resolve them, we collect $O(100\,000)$ snapshots from dedicated schlieren experiments and large-eddy simulations (LES), and apply spectral proper orthogonal decomposition (SPOD). Our SPOD algorithm uses sine tapers and enforces the jet's rotational and reflectional symmetries. For both the schlieren and the LES, the trapped SPOD modes are bounded in the $St$-$x$ and $St$-$k$ planes, where $x$ and $k$ are respectively the axial coordinate and wavenumber. The bounds quantise the modes in $St$, $x$, and $k$, such that the $n_x$th mode in $St$ has $n_x$ antinodes along $x$ and $k$. We then predict trapped modes with local, global, modal, and non-modal stability theories. We find evidence of a competition between modal and non-modal stability. From only the theoretical bounds, we build a simple in-phase standing-wave resonance model that predicts the quantised global-mode frequencies.

For a finite poset, we partition the edges of the order polytope and of the chain polytope into classes indexed by the connected convex subsets of the poset, and we establish that corresponding classes have the same cardinality. This yields a closed formula for the common number of edges, and it identifies the bijection of Hibi, Li, Sahara and Shikama, given by them through an explicit formula, as a disjoint union of simple bijections between corresponding classes, which explains why it is a bijection and how it acts on edge directions and lengths. As consequences, the two polytopes have equally many edge directions, with matching multiplicities; the chain polytope has at least as many edges parallel to each coordinate subspace as the order polytope; and the edge lengths of the chain polytope are dominated by those of the order polytope. Strict inequality occurs in the last two comparisons exactly when the poset contains a three-element chain. We discuss implications for linear and convex combinatorial optimization over ideals and antichains. We also express the number of edges in terms of the comparability graph, in a form that extends to stable-set polytopes of arbitrary graphs, give a recursion for series-parallel posets and closed formulas for layered, zigzag and crown posets, characterize the distributive lattices for which the two polytopes are unimodularly equivalent, and demonstrate that, although the two polytopes have the same number of edges, either one can have the larger diameter, by an arbitrary amount; on the other hand, both diameters are bounded by the width of the poset, and they coincide for ordinal sums, series-parallel posets, and zigzag and crown posets.

Process-induced warpage, ceramic stress and solder strain limit the reliability of insulated-gate bipolar transistor (IGBT) modules on direct-bonded copper (DBC) substrates. Surrogate-assisted design studies train regression models on finite-element analysis (FEA) databases, but rarely check the optimized designs against new FEA or report how surrogate error interacts with the optimizer. This paper builds and evaluates an open pipeline: an open-source process finite-element model, surrogates tuned by Simplified Swarm Optimization (SSO), multi-objective design search, FEA confirmation of selected designs and confirmation-driven infill. The model starts at the second reflow and reproduces measured warpage within 18.2%, 33.6% and 15.3% at the reflow, housing and molding stages without fitted parameters. On a balanced 60-design database, all stochastic tuners reach the same test accuracy, outperform the published grid on cross-validated performance for every output but generalize better only for warpage; the cross-validated ranking of tuners does not transfer to the test set. With equal result reporting, multi-objective SSO and the non-dominated sorting genetic algorithm II give comparable Pareto fronts; a corrected multi-objective particle swarm optimizer trails both. FEA confirmation shows that warpage predictions hold (mean absolute error 0.5%), whereas at the design-space bounds reached by the optimizers the ceramic-stress surrogate is optimistic by up to 26%. Two confirmation-driven infill rounds reduce this error to 1-10% and halve the out-of-sample error; no confirmed design improves on the database in ceramic stress. Ceramic-stress results are indicative, as the metric is mesh-sensitive at production resolution. The model, database, scripts and pre-registered and post-registration results are released.

We study a class of FitzHugh-Nagumo type systems with critical exponential growth in a smooth bounded domain of $\mathbb{R}^{2}$. The classical reduction method turns the strongly indefinite system into a scalar nonlocal problem whose quadratic part induces a norm equivalent to, but different from the standard Sobolev norm, so that the Trudinger-Moser threshold must be handled in this framework. We prove the existence of a ground state solution via a mountain pass argument combined with a Nehari level comparison. We also prove the existence, without imposing the monotonicity condition required in the ground state argument, of a nontrivial solution obtained through a saddle-point linking argument based on the spectral decomposition of $\operatorname{T}=-\Delta+B$.

In actuarial science and quantitative finance, convex order provides a natural way to compare risks with the same mean. In dimension one, convex order is well understood through several characterisations. In higher dimensions, a natural approach is to compare all one-dimensional projections, but, although necessary, the resulting condition is in general not sufficient for multivariate convex order. A simple counterexample due to Pinelis exploits Popoviciu's inequality. We revisit this example using gauge functions, and show that gauge functions themselves do not suffice to characterise multivariate convex order for symmetric distributions, even in dimension two. We then show that the radial case makes the source of the failure transparent and construct radial counterexamples in every dimension at least two. Nevertheless, projection tests are sufficient for some important families of distributions. In particular, this is the case for linear transformations of a common radially distributed random vector. This framework includes elliptical distributions generated from a common radial distribution and, in particular, centred Gaussian distributions. Beyond comparisons between Gaussian distributions, Jourdain and Pagès recently studied the comparison of a Gaussian distribution with a Gaussian mixture, obtaining necessary conditions and sufficient conditions whose equivalence was left open in general. In this paper, we settle the comparison of a centred Gaussian distribution with a mixture of two centred Gaussian distributions: in every finite dimension, the Gaussian is dominated by the mixture in multivariate convex order if and only if all one-dimensional projections are ordered in convex order.

A topological space is $\varkappa$-Fréchet–Urysohn if every point in the closure of an open set $U$ is the limit of a sequence of points of $U$. We prove that every $\varkappa$-Fréchet–Urysohn subgroup and every Baire subgroup of the free topological group on a Tychonoff space $X$ in any full variety of topological groups (in particular, of the free topological group $F(X)$, the free Abelian topological group $A(X)$, and the free Boolean topological group $B(X)$) is discrete. In particular, each of these groups is $\varkappa$-Fréchet–Urysohn (or Baire) only if $X$ is discrete, and all their metrizable and even Fréchet–Urysohn subgroups are discrete. Along the way we characterize strongly $\varkappa$-Fréchet–Urysohn spaces in the spirit of Michael's theorem and study countable families of closed sets absorbing convergent sequences in $\varkappa$-Fréchet–Urysohn spaces and groups.

We formally initiate the study of $\lambda$-fold $(s,t)$ near-factorizations in association schemes. Namely, given an association $(X, \mathcal{A})$, we consider the existence of a factorization of $\lambda(J-I)$ into 01-matrices $S$ and $T$ with the constraint that $S$ and $T$ must belong to the adjacency algebra of $(X, \mathcal{A})$. We establish basic properties of $\lambda$-fold $(s,t)$ near-factorizations in association schemes and calculate bounds for $\lambda$, $s$, and $t$ relative to the order of the association scheme. We completely determine all near-factorizations in symmetric and asymmetric 2-class association schemes. Furthermore, we construct near-factorizations in certain Hamming schemes, cyclotomic association schemes, Schurian schemes on small primitive groups, and the folded cube and halved cube association schemes. Finally, we establish that certain Hamming schemes, Johnson schemes, and Grassmann schemes do not admit a $\lambda$-fold near-factorization.

Let $A$ be a positive continuous additive functional of a strong Markov process and $\alpha$ an independent finite-state Markov chain. For bounded $h$, we study \[ J_\varepsilon(t)=\int_0^t\bigl(h(\alpha_{s/\varepsilon})-\pi(h)\bigr)\,dA_s . \] If $\sup_y \mathrm E_y A_t\le C_T t^\vartheta$, then for every bounded random time determined by the base process, not necessarily a stopping time, \[ \mathrm E|J_\varepsilon(\tau)|^2=O(\varepsilon^\vartheta),\qquad \mathrm E\sup_{t\le T}|J_\varepsilon(t)|^2 =O\!\left(\varepsilon^\vartheta(1+|\log\varepsilon|)^2\right). \] For reflected Brownian boundary local time, the exponent $1/2$ is sharp at fixed time. We apply this principle to reflected Brownian motion with rapidly switching Wentzell boundary dynamics. The homogenized boundary condition is again of Wentzell type, with stickiness and killing coefficients given by invariant averages. For bounded Lipschitz observables, the quenched Feynman–Kac error has every polynomial rate below $1/8$, uniformly over deterministic times and in expected uniform norm. For smooth Neumann-compatible observables with proportional killing and stickiness, the rate improves to $O(\varepsilon^{1/4})$, up to a logarithmic factor in the uniform-in-time estimate. We also construct the switching process, derive quenched and annealed backward evolutions, and transfer the estimates by duality to the forward measure evolution. For the forward evolution, we obtain quantitative convergence in bounded-Lipschitz distance and the sharper $1/4$ rate for smooth observables, including surviving mass. Numerical experiments illustrate the finite-scale convergence.

We show that, in random hypergraphs with several permitted edge sizes, non-uniformity affects existential and algorithmic bounds for 2-colorability (Property B) in fundamentally different ways. Assigning weight $2^{-k}$ to each $k$-edge, we obtain upper and lower bounds in terms of the total edge weight that asymptotically match the uniform bounds as the minimum permitted edge size grows, regardless of how edges are distributed among sizes. We also extend the best-known algorithm for 2-coloring random uniform hypergraphs to the non-uniform setting. With weight $\frac{k}{2^k}$ assigned to each $k$-edge, we construct non-uniform instances whose expected total edge weight per vertex is arbitrarily large, yet the algorithm finds a proper coloring asymptotically almost surely. In the uniform setting, the algorithm fails with high probability once this quantity exceeds a constant. Our construction uses sufficiently separated edge sizes, so that edges of different sizes become relevant at well-separated stages of the execution and their effects are essentially independent.

We prove two new results about the number of classes of positive definite eutactic forms in n variables.

Particle swarm optimization (PSO) is a popular class of gradient-free methods for global optimization. We study a continuous-time model for momentum-free PSO with personal-best memory and global-best interaction. Unlike previous models which rely on smooth approximations, we keep the original discontinuous personal- and global-best processes. We study the small-step limit and show that weak limit points satisfy a stochastic differential inclusion: personal-best variables obey a relaxed record condition, while fast switching of the global-best selector leads to a convexified interaction. We then give conditions under which the true original selectors are recovered. Under suitable regularity and nondegeneracy assumptions, Brownian noise allows us to identify the personal-best processes with the latest record positions of the limiting trajectories and to prove that the particle attaining the global best is unique at almost every time. The resulting continuous-time dynamics retain both selection rules without smoothing them.

DNA storage systems retrieve information by randomly sampling synthesized DNA strands, making the number of reads required for successful recovery a fundamental performance measure. This motivates the coverage depth problem: for given code parameters, determine a linear code that minimizes the expected number of randomly sampled columns required to span the entire information space. While MDS codes are known to be optimal whenever they exist, identifying optimal codes in parameter regimes where MDS codes do not exist remains largely open. In this work, we derive a general formula for the expectation in the coverage depth setting, and apply this result to establish the optimality of the $q$-ary simplex code for the parameters that allow its existence. Moreover, we show the optimality of the $q$-ary Hamming code by relating coverage depth to independent sets in the dual code and exploiting log-concavity properties of matroid basis-generating polynomials. We further analyze random linear codes, derive an exact expression for their expected coverage depth, and show that, in the constant-rate regime, their additive gap from the MDS benchmark remains bounded independently of the code length.

The goal of the partial information decomposition (PID) is to quantify the redundant and synergistic information that multiple sources provide about a target. PID has many applications in machine learning, neuroscience, and other fields, but defining and computing it for high-dimensional continuous systems remains challenging. Here, we define a PID for multivariate Gaussian systems based on the Blackwell order, which formalizes when one channel is more informative than another. We prove that Gaussian channels are optimal for extracting both redundant and union information, yielding an intuitive geometric interpretation and an efficient numerical algorithm for the PID. Our union information and synergy coincide with the well-known BROJA measures, and we derive closed-form expressions for both in the case of two sources. We also argue that Blackwell redundancy (which differs from BROJA) is the only existing redundancy measure that satisfies a set of natural desiderata. We demonstrate the scalability of our method on systems with up to a thousand dimensions or sources. Our approach is illustrated on an optimal control problem, where it identifies redundant and synergistic interactions between sensor and memory.

We prove that the Boca–Cobeli–Zaharescu (BCZ) map is strongly mixing with respect to normalized Lebesgue measure on the BCZ triangle.

A single spatial snapshot of a system with multiple interacting populations captures positions but not the underlying dynamics that have led to the observation. This work asks what the stationary second-order spatial statistics of a snapshot can reveal about the structure of the linear reaction-diffusion dynamical model, and in particular about the pairwise directed entries of its interaction operator. The observation is the matrix-valued structure factor $S(k)$ as a one-parameter family in the diffusion-inherited dependence on the wavenumber $\kappa=|k|^2$. The main result is that the directed interaction operator $A$ and the diffusivity matrix $D$ are generically identifiable for any number of interacting populations from $\{S(\kappa)\}$ up to a single global time rescaling. The findings do not contradict known bounds and the recent result that spatial observational data only determine the Markov equivalence class of the underlying graph. In this work, we identify physical assumptions that lift this barrier. The absolute time scale is not identifiable from a single snapshot of any sample size and requires an external clock. For tumor-immune tissue, the results imply that directed cell-cell interactions are in principle recoverable from one section, that the direction cannot be inferred from the mean densities or a covariance matrix, but from the wavenumber dependence of the cross and autospectra, and that a baseline section determines the interaction structure but not the absolute rates.