Papers

A signed graph $\Sigma=(G,\sigma)$ is a graph $G$ with edges given signs $1$ or $-1$ defined by the function $\sigma$. The adjacency matrix of $\Sigma$ is defined as per these signs. The relation between the largest eigenvalue of $G$ and $G-v$ has been studied in recent years, where $G-v$ is the graph obtained from $G$ by deleting the vertex $v$. In 2020, Sun and Das proved that the difference of the squares of the largest eigenvalues of the graphs $G$ and $G-v$ is bounded above by $2d(v)-1$ where $d(v)$ is the degree of $v$. A similar result need not be true for the largest eigenvalue of signed graphs. In this paper, we prove that the result is valid for the spectral radius of signed graphs. On the other hand, the signed graph version of Hoffman's chromatic number bound was proved by Wang et al. in 2021. They also discussed the difficulty in proving the extended version encompassing all eigenvalues of $\Sigma$ as was done for unsigned graphs by Wocjan and Elphick. We note down a consequence of Wocjan and Elphick's lower bound for the chromatic number in terms of all the eigenvalues of $\Sigma_+$; all the eigenvalues of $\Sigma$ and $\Sigma_-$, where $\Sigma_+$ (resp. $\Sigma_-$) is the spanning subgraph induced by the positive (resp. negative) edges. We give examples where the result fails even under various restrictions on the signed graph. Finally, we improve an upper bound for the $k$-th power of the largest eigenvalue given by Stanić in terms of walks in signed graphs and give lower bounds for the least eigenvalue in terms of various parameters of $\Sigma_+$ and $\Sigma_-$.

A left-regular bipartite graph $G$ of degree $d$ is called a $(t,\alpha)$-small-set-expander if every subset $X$ of left vertices of size at most $t$ has at least $\alpha |X|$ neighbors. Such a graph is an optimal small-set expander if small subsets have as many neighbors as possible. We characterize optimal expanders combinatorially via girth and prove the existence of $s$-optimal expanders for every $s$. We also prove that $s$-optimality yields new "transfer" lower bounds on the number of neighbors of sets of size $h\geq s$. Finally, as an application, we discuss the use of optimal small-set expanders in building good codes for key exchange protocols in post-quantum cryptography.

Hyperparameter tuning almost always means search: fit the model at every value on a grid, score each by cross-validation, and keep the winner. For spline regression that search is unnecessary. The optimal resolution can be solved for in closed form, to the accuracy an exhaustive search reaches, at a fraction of the compute. Three ingredients make this possible: classical approximation theory pins the squared bias to a known power of the resolution G, exactly the Kolmogorov n-width of the smoothness class; the basis dimension is an explicit polynomial in G; and leave-one-out error follows from a single fit via the PRESS identity. Balancing the two known curves gives the minimizer analytically. We extend this calculus to many coordinates by replacing ambient input dimension with interaction order, the number of active low-order components in an ANOVA decomposition, yielding a scaling law in which the optimal resolution and error are power functions of the effective density (sample size per active component), with input dimension absent from the exponent. The law becomes an algorithm. KORE (Kolmogorov-optimal Order-aware Resolution Estimation) fits two pilot resolutions, solves a leverage-calibrated 2x2 system for the bias and noise scales, and evaluates the closed-form plug-in resolution with a tiny leave-one-out certificate: about a dozen fits instead of a full grid sweep, with a consistency guarantee as the sample grows. Across additive and sparse pairwise targets up to 80 input dimensions, KORE matches exhaustive 3-fold cross-validation and the full classical ladder (GCV, Mallows' Cp, AIC, BIC) while fitting roughly 8x fewer models; on 36 real tabular datasets it ranks first among 21 methods in accuracy per unit of compute, ahead of tuned boosters and kernel machines. When complexity lives in low interaction order, solving for the resolution beats searching for it.

We extend the Levenberg-Marquardt method on Riemannian manifolds to a robust variant that allows to tackle problems from applications where outliers are to be expected. We formally state the framework for a block-wise variant of the Robust Riemannian Levenberg-Marquardt algorithm and discuss how known convergence results can be applied here as well. We further discuss several alternatives for phrasing the sub problem that has to be solved. Finally we illustrate how the accompanying open source implementation in Manopt$.$jl can be used to efficiently solve problems such as geodesic regression, Procrustes analysis, subspace Procrustes analysis and bundle adjustment robustly and compare the Levenberg-Marquardt solver to other solvers for nonsmooth Riemannian optimization.

We develop an operator-theoretic framework for finite-dimensional, regime-dependent Volterra equations with completely monotone memory kernels, dissipative network coupling, and Hawkes-type self-excitation. For each fixed regime we construct the associated Volterra resolvent family and prove global well-posedness, continuity across regime switches, and explicit a priori bounds. The main stability result is sharp in the commuting case: after simultaneous diagonalization of the network Laplacian and the excitation operator, each mode obeys a scalar characteristic equation, and global asymptotic stability holds exactly when every modal branching ratio lies below the intensity damping threshold. We also give a norm-based sufficient condition for noncommuting operators and a Perron–Frobenius spectral criterion for nonnegative intensity blocks, showing when norm estimates are conservative. Beyond mean stability, we prove a pathwise finite-range power law for burst amplitudes generated by residence in a Hurwitz but nonnormal regime: under a cone-alignment event, the survival exponent is the ratio of the regime exit rate to a cone-corrected finite-time growth rate bounded above by the logarithmic norm of a fixed Markovian realization in the chosen Euclidean metric. A complementary idealized-feedback result shows how a logarithmic-norm contraction caps the amplification band. Finally, we derive the deterministic intensity block as a mean-field limit of a relaxing long-memory Hawkes system with regimes. Numerical experiments on modal equations, a small-world network, and a switched nonnormal ODE validate the sharp threshold and the finite-range amplification mechanism without using the closed-form tail formula as input.

We consider expectations of the type $E \exp \left\{\sum_{i=1}^m \phi_i \right\}$, where $\phi_i: {\Bbb R}^n \longrightarrow {\Bbb C}$ are functions, each depending on a few coordinates of a point in ${\Bbb R}^n$, and the expectation is taken with respect to the standard Gaussian or symmetric exponential probability measures. We prove sufficient conditions, in terms of the Lipschitz constants of $\phi_i$ and the combinatorics of their dependencies, for the integral to be non-zero, and, consequently, to be amenable to a computationally efficient approximation. We discuss applications to computing volumes of bodies and statistics on integer points in polyhedra in ${\Bbb R}^n$.

The idea of the exceptional series is that the exceptional simple Lie algebras should form a series. Since all four simple Lie algebras in the fourth row of the Freudenthal magic square are exceptional it is natural to ask if the remaining rows form a series. A stronger version of this question is that, for the first two rows (corresponding to the real and complex numbers), there is a category defined by a presentation which is a reasonable candidate for the series. Our main results show that neither of these candidates is a series but each consists of a finite set of points. In each case the series is defined by a parameter and we show that the relations imply that this parameter satisfies a polynomial. These two results were obtained by a computer calculation. Our calculation is supported by a website for inspection, and the calculations are certified by Lean 4.

A random walk with small positive drift $\mu$ sets new records at a rate $\lambda(\mu)$ that vanishes as $\mu \to 0$. For Gaussian and strictly stable centered steps whose stable law $Y$ has index $1 < \alpha \leq 2$ and positivity parameter $\rho = \mathbb{P}(Y>0)$, we find $\lambda(\mu) \sim K\mu^{(1-\rho)/\nu}$ as $\mu \to 0$, where $\nu=1-1/\alpha$ and $K$ is explicit. Throughout their domains of attraction, the exponent persists, with a slowly varying factor replacing the constant $K$. The exponent is set by the asymmetry only through $\rho$, sweeping the interval $[1,\,1/(\alpha-1)]$ as the skewness varies. For centered strictly stable steps, $\rho$ also governs the driftless record growth, $\langle R_{N}\rangle \sim N^{\rho}/\Gamma(1+\rho)$, which the small-drift law meets at the crossover where the drift takes over. The formula recovers the Gaussian linear law $\lambda(\mu) \sim \sqrt{2}\mu/\sigma$ and, for symmetric heavy tails, the power $\mu^{\alpha/2(\alpha-1)}$. It follows from one Mellin transform of the harmonic sum in the Spitzer–Baxter identity, which factorizes into a kernel transform carrying the step distribution and a Riemann zeta function carrying the harmonic weights. Its poles deliver the leading law, its prefactor, and a correction ladder reproducing the known Gaussian and stable series, unifying diffusive, heavy-tailed, and skewed walks. The power law ends at the Cauchy point $\alpha=1$, where $\mu$ is a pure location shift: for the strictly Cauchy family the limiting rate vanishes and records accumulate sublinearly with a shift-dependent exponent.

In this paper, we study the well-posedness and exponential stability of a class of abstract boundary-coupled positive systems. Within a general semigroup framework, we establish well-posedness in terms of existence, uniqueness, and regularity of solutions. More concretely, we derive some simple and readily verifiable tests on the boundary coupling operators for the verification of the well-posedness of the coupled system and, at the same time, provide spectral criteria that guarantee exponential stability of the solutions. The analysis is based on two complementary approaches: the feedback-theoretic method and perturbation techniques for resolvent positive operators on Banach lattices (i.e., Banach spaces equipped with a compatible order structure). This unified framework allows for a systematic treatment of a broad class of boundary couplings in abstract dynamical systems. Several examples are presented to illustrate the applicability of the theoretical results.

We study the location of the largest spacing generated by the order statistics of a sample from the standard exponential distribution. Although the asymptotic behaviour of the largest spacing itself is well understood, considerably less is known about the index at which it is attained. Using the independence and unequal rates of exponential spacings, we derive exact finite-sample formulas and show that, when measured relative to the right endpoint, the location of the largest spacing converges in distribution to a non-degenerate probability distribution on the positive integers. We obtain explicit integral representations for the limiting probabilities and, using Euler's pentagonal number theorem, derive a series representation involving the generalized pentagonal numbers. This reveals that the Euler product appearing in the limiting distribution of the largest exponential spacing also governs the distribution of its location. The convergence result is also extended to the location of the largest $m$-spacing for every fixed $m\geq 1$, despite the dependence among overlapping $m$-spacings. Numerical values illustrate the concentration of the limiting distribution near the right endpoint and the rapid convergence of the finite-sample probabilities.

We prove long-time existence of the pluriclosed flow on certain compact quotients of Lie groups for non-invariant initial data, as well as on some holomorphic principal torus bundles over nonpositively curved Kähler manifolds. In particular, our results cover the cases of nilmanifolds and almost-abelian solvmanifolds, and provide a new proof of the long-time existence of the pluriclosed flow on certain complex surfaces, originally established by Garcia-Fernandez, Jordan, and Streets. These results follow from a general theorem on holomorphic submersions, which is of independent interest and, in particular, also implies the long-time existence of the pluriclosed flow on Oeljeklaus-Toma manifolds, as proved by Streets and Wang.

It has been observed that the mutual scalar products of the vectors in a SIC-POVM are given by algebraic units, and at least in some cases by square roots of Stark units. The full picture is somewhat more complicated, especially if non-minimal SIC-POVMs are considered. We present a mixture of exact and numerical evidence suggesting that the overlap units are always products of integral powers of square roots of Stark units from ray class fields all of which are attached to the maximal ring of integers in the base field. In the non-minimal case a lattice of such ray class fields is involved. In every second dimension (counted in a certain way) some of the overlap units equal $\pm 1$, and we show that this follows from a special property of the ray class fields. Our observations are complementary to but consistent with the claim that the overlap units can be calculated directly from the Shintani–Faddeev modular cocycle.

We prove the Riemannian positive mass theorem in all dimensions for asymptotically flat $L^\infty$-metrics with subcritical singular sets. More precisely, we consider complete asymptotically flat manifolds whose metrics are smooth away from a compact singular set of Minkowski dimension less than $n-3+\frac{2}{n}$, and whose scalar curvature is nonnegative on the regular set. We show that the ADM mass of each asymptotically flat end is nonnegative, and that the mass vanishes in some end only in the Euclidean case. For the rigidity statement, we require additionally that the Minkowski dimension of the singular set is not larger than $n-3+\frac{1}{n-1}$. This gives an asymptotically flat analogue of Schoen's codimension-three conjecture for positive scalar curvature. The proof combines a density theorem for singular asymptotically flat metrics, capacity estimates across the singular set, conformal blow-up inspired by Bi-Hao-He-Shi-Zhu [3], and a $\mu$-bubble dimension-descent argument adapted from Brendle-Wang [6].

For $n\geq 1$ and $p$ a prime, the Hecke congruence subgroup $\Gamma_{0,n}(p)\leq \mathrm{SL}_n(\mathbb{Z})$ is the subgroup of matrices whose first column is of the form $(*,0,\dots,0)^t\bmod p$. Borel–Serre showed that $\Gamma_{0,n}(p)$ has virtual cohomological dimension $\binom{n}{2}$. The first author proved that the rational cohomology in this top degree $\binom{n}{2}$ vanishes for $n$ sufficiently large compared to $p$. We prove analogous results in codimension $1$ and $2$.

Baker asked, as recorded by Poonen, whether a fixed smooth quasiprojective variety over a finite field must have a smooth rational hyperplane section after every sufficiently high-dimensional linearly nondegenerate embedding. Poonen predicted a negative answer for every positive-dimensional variety. We prove this predicted negative answer for each prescribed variety: if $X$ is nonempty, smooth, quasiprojective, and of pure positive dimension over $\F_q$, then for every sufficiently large $N$ there is a locally closed embedding $X\hookrightarrow\PP^N_{\F_q}$ whose components remain linearly nondegenerate after arbitrary scalar extension, but whose every $\F_q$-rational hyperplane section is singular. The construction assigns one closed point of $X$ to each rational hyperplane and forces the pulled-back linear form to have zero first-order jet at that point.

For any infinite one-relator group $\Gamma=\langle S \mid w^m\rangle$, we prove that $\mathrm{cost}(\Gamma)=|S|-\frac{1}{m}$. For such groups, this gives $\beta^{(2)}_1(\Gamma)=\mathrm{cost}(\Gamma) - 1$, answering a special case of Gaboriau's question on the relationship between cost and first $\ell^2$-Betti number.

High-order discontinuous Galerkin spectral element methods (DGSEM) based on Legendre-Gauss-Lobatto (LGL) nodes provide accurate and efficient discretizations for conservation laws. However, their cost, memory footprint, and time-step restrictions increase rapidly when the degree of the polynomial increases. This paper develops a corrected $\mathbb{P}_n\mathbb{P}_m$ ($c\mathbb{P}_n\mathbb{P}_m$) approach for DGSEM-LGL discretizations that aims to recover the accuracy of an $m^{th}$-order approximation while evolving only the degrees of freedom associated with an $n^{th}$-order representation, with $n<m$. The projected evolution of the high-order components is derived first at the continuous level and then in the fully discrete DGSEM-LGL setting. The discrete analysis shows that because LGL quadrature is not exact for the highest Legendre mode, a correction term for that mode is required to preserve the order of convergence. A compact projection-based reconstruction operator is then introduced to recover high-order components without solving the enlarged constrained least-squares systems used in standard reconstruction procedures. For sufficiently smooth solutions, the resulting $c\mathbb{P}_n\mathbb{P}_m$ scheme is shown to achieve the expected $m+1^{th}$ convergence order. Numerical experiments for one- and two-dimensional conservation laws, including Euler, viscous Burgers, and 2D decaying homogeneous isotropic turbulence, confirm theoretical convergence behavior and demonstrate competitive accuracy relative to computational cost, with particularly clear efficiency gains for viscous flows.

In this paper we show that bounded entropy solutions of genuinely nonlinear $2\times 2$ systems of hyperbolic conservation laws admit a strong trace on Lipschitz curves. The main argument relies on a new half-space Liouville-type theorem for isentropic solutions with constant normal traces, providing the first extension to general genuinely nonlinear systems of the strong trace properties available for scalar conservation laws.

This paper proposes an analytical and structural reformulation of the classical Brocard Problem. Through modular reduction using primorial cycles, the study establishes that the global solution space is governed by a structural invariant tightly restricted to pairs of consecutive $n$-smooth integers. The work mathematically proves the Triangular Equivalence, demonstrating that the existence of a Brown number is strictly equivalent to the condition that one-eighth of the factorial of $n$ is a perfect triangular number. Additionally, based on the profound structural tension and $p$-adic density imbalances, the manuscript introduces a generative $p$-Adic Sieve algorithm. This computational innovation shifts the traditional search from continuous floating-point extractions to discrete combinatorial partitions, identifying the potential for polynomial-time approximations via Lenstra–Lenstra–Lovász (LLL) lattice reduction. Finally, utilizing the Dickman-de Bruijn function and structural mapping to the Erdős-Selfridge Theorem, the research asymptotically justifies the extreme geometric rarity of the solutions and demonstrates the universality of the methodological framework by establishing recursive reductions for even shifts and immediately invalidating the generalized equation $n! + 3 = m^2$ for any $n \ge 4$.

Over the past decade, deep neural networks (DNNs) have achieved remarkable success on complex machine-learning tasks, yet the theoretical foundations of their performance remain incomplete. From a statistical viewpoint, a natural question is: can DNNs attain feature-learning and prediction consistency comparable to that of classical models? While a full characterization is open, we provide positive results for a broad subclass. We establish feature-learning consistency guarantees for sublinearly structured DNNs-architectures whose input/output dimensions and number of hidden neurons grow sublinearly with the sample size-when learning hierarchically compositional target functions. Importantly, this consistency still holds even in the conventional "over-parameterized" regime where the total number of parameters exceeds the number of training samples. Empirically, sublinearly structured DNNs match or surpass wide DNNs in prediction. A structural audit further indicates that widely used convolutional neural networks (CNNs), including AlexNet, VGGNet, ResNet, GoogLeNet, are sublinearly structured on their image classification benchmarks. We further prove that the sublinearly structured DNNs achieve universal approximation for hierarchically compositional functions in the large-sample limit. Moreover, images exhibit an inherent hierarchical, compositional structure. Taken together, these results explain, through a statistical lens, why many large-scale deep learning models succeed after adequate training on massive image datasets.

Continuing our previous work, this paper closely studies the relationship between the cluster algebra structures on the coordinate ring of Schubert cells and those on the coordinate ring of partial flag varieties. We give a finite-type classification for these cluster structures and point out several results that were left open in our previous work.

The Block Decomposition Method (BDM) was introduced as an alternative to popular lossless compression methods such as LZW for estimating algorithmic complexity from the principles of algorithmic probability and classical information theory. It extends the Coding Theorem Method (CTM) from small objects to larger ones by combining local estimates of algorithmic complexity with a global account of repetition based on Shannon entropy. Here, we introduce a version of BDM in which dependencies between blocks are utilized to reduce the length of the description based on reusable program code in the decomposition of an object, and on conditional descriptions capable of accounting for shared structure between observations. We formalize this allocation of descriptive resources as algorithmic attention. Repeated or related components need not be described independently, and the resulting reduction in description length is governed by the amount of shared algorithmic information. We formulate this extension as a reuse optimization problem, show that exact optimization is NP-hard, derive conditions under which it improves upon independent descriptions, relate the achievable gains to algorithmic mutual information, prove the relationship with the previous BDM version, and provide a roadmap for its implementation using CTM-derived complexity and conditional complexity estimates.

In the 1980s, Erdős and, independently, Frankl and Pach conjectured that, for sufficiently large $n$, every $(d+1)$-uniform family on $\{1,\ldots,n\}$ with VC-dimension $d$ has size at most $\binom{n-1}{d}$, the size of a star. Ahlswede and Khachatrian disproved this conjecture in 1997 by giving a family of size $\binom{n-1}{d}+\binom{n-4}{d-2}$. This value has since been widely believed to be best possible, and Mubayi and Zhao explicitly conjectured its optimality in 2007. Very recently, Wang, Xu and Zhang proved their conjecture for $d=2$ and $n\ge 7$, providing further support for this belief. Surprisingly, we show that the Mubayi-Zhao conjecture is false for every $d\ge 3$ by constructing families larger than the Ahlswede–Khachatrian bound. Our constructions suggest that the answer to the Erdős–Frankl–Pach problem depends delicately on both $n$ and $d$.

Ferdinand Ihringer proved that Boolean degree one functions on the Grassmann scheme $J_q(n,k)$ are trivial when $\min(k,n-k) \ge 2$ and $n$ is large enough. We provide a mostly self-contained exposition of this result. All results were formalized by ChatGPT 5.6 Sol, and are available on GitHub at YuvalFilmus/grassmann-degree-one.

We study future-pointing timelike solutions of the Lorentz force equation in the sub-extremal Kerr–Newman spacetime, with special attention to the time-machine region $\mathfrak T$, where the axial Killing field $\partial_\phi$ is timelike. We first construct smooth closed electromagnetic orbits tangent to $\partial_\phi$ in the positive equatorial part of $\mathfrak T$: the radius of such a circle determines, and is determined by, the charge-to-mass ratio of the particle which must have opposite sign to that of the black hole charge. We then prove the existence of spherical electromagnetic orbits contained in the equatorial time-machine region and derive explicit relations between their radius, charge-to-mass ratio, energy and angular momentum. Next we give sufficient conditions ensuring that an equatorial electromagnetic orbit is a flyby orbit with radial turning point in $\mathfrak T$. Finally, to describe charged-particle decay processes whose fragments have different charge-to-mass ratios, we introduce the notion of a broken electromagnetic orbit: a continuous, piecewise smooth, future-pointing worldline whose smooth pieces solve the Lorentz force equation. Imposing conservation of kinetic four-momentum and electric charge at the decay vertices, we exhibit an energy extraction process followed by a causality-violating one. The latter is realized by a closed broken electromagnetic orbit, which we construct in both the $r$-positive and the $r$-negative regions inside the inner horizon, by concatenating two flyby branches sharing a common radial turning value $\bar r$, and having charge-to-mass ratios with opposite sign to that of the black hole charge, with a spherical electromagnetic orbit of radius $\bar r$.

This paper studies probabilistic properties of pairwise stability for a network model where individual utilities are random variables. We study the probability that a given network is pairwise stable and the expected number of pairwise stable networks. We provide a closed-form solution for the latter number. As the evaluation of the exact expression is computationally challenging for large populations, we provide tractable lower and upper bounds for this expression which allow us to pin down the asymptotic behavior of the expected number of pairwise stable networks up to a multiplicative constant. This asymptotic behavior is described by the number of networks $ 2^{n(n-1)/2} $ times $ (2/n+1)^{n} $, a sequence that tends to infinity fast. We normalize the number of pairwise stable networks by this sequence and show that the variance of the normalized number of pairwise stable networks converges to zero as $ n $ tends to infinity. We conclude that almost surely the number of pairwise stable networks tends to infinity, while the fraction of pairwise stable networks tends to $ 0 $ as $ n $ goes to infinity.

We present an algorithm for exact-size sampling of uniformly random Pólya trees with linear expected runtime.

A convex splinter $K$ is a union of convex sets in $\mathbb{R}^d$ such that every minimal affine dependent set of $\mathbb{R}^d$ contained in $K$ is contained in one of the sets. The study of convex splinters was motivated by the study of flat transversals to convex sets. We extend several variations of Helly's theorem from convex geometry to convex splinters. These include fractional and colorful variations of Helly's theorem. We also extend Tverberg's theorem to convex splinters.

We study a dynamical invariance principle for interacting particle systems with mean-field interactions and common noise emerging through a collective stochastic perturbation. The particle dynamics combine autonomous evolution with a weakly scaled random bombardment whose cumulative effect generates a Brownian common noise in the large population limit. Working in a general abstract framework based on stochastic flows and the stochastic sewing lemma, we establish quantitative conditional propagation of chaos estimates for both discrete Euler schemes and their continuous-time flow limits. Our approach yields explicit Wasserstein convergence rates and applies in particular to jump-diffusion models motivated by interacting spiking neuron systems. The analysis relies on quantitative central limit theorems of Rio and Bonis together with a stochastic sewing argument adapted to $L^2$-valued flows.

A well-known procedure for the asymptotic evaluation of Laplace transforms is Laplace's method. Despite its wide applicability, however, it is easy to find relevant examples where the technique is infeasible, because the integrand admits no power series expansion near the critical point, e.g., due to exponential behaviour there. We circumvent this issue through an extension of the method of Mellin transforms, based on an integral representation for the kernel and a generalization of the Taylor expansion. Our main result then extends the Laplacian method in the sense that it provides asymptotic expansions for a wider scope of Laplace transforms in terms of known special functions, rather than only in powers of the asymptotic parameter. The results are illustrated in two examples.