Papers

Let $q$ be a prime power, $F=\mathbb F_q(\!(t^{-1})\!)$, $G=\mathrm{SL}_2(F)$, $\Gamma=\mathrm{SL}_2(\mathbb F_q[t])$, and $K=\mathrm{SL}_2(\mathbb F_q[\![t^{-1}]\!])$, and let $U<G$ be the upper unipotent subgroup. We study right $K$-spherical averages along $U$ on $X=\Gamma\backslash G$. Expanding translates of compact $U$-orbits and compact-open F$\unicode{x00F8}$lner-ball averages become terminal layers of rooted descendant shadows in the Bruhat–Tits tree. In the even sector, we compute the Haar height law and signed finite-scale discrepancy exactly. This yields $K$-spherical equidistribution for compact-orbit translates and, for irrational boundary endpoints, for F$\unicode{x00F8}$lner-ball averages. For a depth-$N$ shadow rooted at height $k$, with cutoff $M=N-k\ge0$, bounded-profile errors are $O_q(q^{-M})$ in the backward state and $O_q(q^{-M/2})$ uniformly for moving roots, while the shadow law eventually agrees exactly with the Haar law on every fixed finite height window. For $|\Phi(2m)|\le Cq^{\alpha m}$, $\alpha<2$, three rate regimes arise, with a linear-in-scale factor at $\alpha=1$ and explicit moving-root dependence. Artin continued-fraction digits eventually encode the cutoff and these rates excursion by excursion through individual digit degrees.

We study the spectrum of the adjacency matrix $A_n$ of directed inhomogeneous random graphs on $n$ vertices. We assume that $A_n$ has independent entries and diverging average degree scale $s_n$. This framework includes, as special cases, the directed Chung–Lu random graph and directed stochastic block models. Assuming boundedness of the variance profile and that $s_n$ diverges faster than $\log(n)^2$, we show that the rank-one Chung–Lu model satisfies a non-homogeneous version of the circular law, which in some situations allows for an explicit expression. Moreover, under mild conditions, we identify the asymptotic singular value distribution using tools from free probability. Finally, for finite-rank directed models, we establish that eigenvalues outside the bulk have joint Gaussian fluctuations at the scale $\sqrt{s_n/n}$, with an explicit covariance matrix. These results extend the theory of spectral outliers and their fluctuations to directed inhomogeneous random graphs.

In this paper, we introduce nonstandard versions of multistep multistage methods. While proving the convergence of these schemes, we also define nonstandard general linear methods. We show that the nonstandard methods can attain the same order as their standard counterparts while preserving certain qualitative properties (e.g., boundedness) for all positive step sizes. These results are also demonstrated by some numerical experiments.

We give a self-contained short proof of optimal regularity for minimizers of the Alt-Phillips Free Boundary Problem for $\gamma \in (0, 1)$. We adopt a dichotomy argument that originates from [DS20].

We construct a measure that exhibits two aspects of a new type of universality and dramatically simplifies the integration of tensors $T_{a_1,a_2,\ldots,a_D} \in \mathbb{C}$ ($a_1,\ldots,a_D=1,\ldots,N$) at large $N$. In contrast to matrix integration, in which matrix traces canonically yield the integrand, tensors need additional information (equivalent to a $D$-coloured graph $B$) to contract their indices and form a tensor trace $B(T)$. We show that, whenever each $B_1,\ldots, B_n$ can be obtained by a recursive construction known as melonicity, then the leading order in $N$ of the integral of $ {B_1}(T) {B_2}(T) \cdots {B_n}(T) $ is independent of the – often intricate – combinatorics of the traces $B_i$, but also, to our surprise, independent of $D$ as far as $D\geq 3$. Instead, at large $N$, these integrals are some functions (indexed by $n$) of the number of vertices $2p_i$ of $B_i$ which we call melonic polynomials. Melonic traces cumulants with respect to any ('interacting') measure \[ \exp\Big\{-N^{D-1} \sum_{i=1}^m g_i {B_i}(T)\Big\} \mathrm{d}\mu_0(T) \quad (g_1,\ldots,g_m \in \mathbb{R}, \mathrm{d}\mu_0(T) =\text{the tensor Gaussian}) \] with each $B_i$ melonic, can be computed with our universal measure that replaces each $B_i$ by a canonical trace depending only on $p_i$. We prove that any two melonic tensor models are indistinguishable at large-$N$, independently of the number of tensor indices (first universality aspect), and of the fine-grainedness of their interactions (second universality), being a sufficient condition that the couplings (the parameters $g_i$ above) agree and their respective traces are monomials with the same degree in $T$.

In this note we explore numerically the finite Bernoulli convolutions. We show that with a suitable choice of parameter, it might serve as a toy model for intermittent energy cascade in fully developed turbulence. We then show how the crossings of $\beta$-expansions distribute in $\beta$, and suggest that it might highlight the parameters with enhanced overlap structure that are related to measures that are singular continuous. We later introduce a notion of order to the $\beta$-expansions based on the lexicographical order of the $N$-binary words, and observe that for most sampled adjacent pairs when $\beta=2$, the distance in their order increases exponentially when $\beta$ decreases from 2 to 1. This suggests 'chaotic' behavior, with the 'Lyapunov exponents' bunched into several clusters that depend on $N$. We end the note with some 'order plots' and an interesting connection between the finite $\beta$-compactum with $\beta=g$ (g being the golden ratio) and binary reflected Gray code.

The problem of recovering an (approximately) low-rank Hermitian matrix $\pmb{M}_0 \in \mathbb{C}^{n \times n}$ of rank $r$ from quadratic sampling matrices of the form $\{\pmb{a}_k \pmb{a}_k^*\}_{k=1}^m$ arises in a variety of applications, including phase retrieval. To obtain rigorous recovery guarantees, the sampling vectors $\{\pmb{a}_k\}_{k=1}^m$ are typically modeled probabilistically. However, most existing theoretical results rely on Gaussian or sub-Gaussian assumptions, which may not accurately capture practical data models. In many applications, sampling vectors exhibit heavier tails, while theoretical understanding in such regimes remains scarce. In this paper, we bridge this gap. We show that two widely used convex approaches, nuclear norm minimization and semidefinite-constrained empirical risk minimization, achieve uniform, stable, and robust recovery under the mild assumption that the entries of the sampling vectors have only finite $4+\delta$ moments, with the optimal sample complexity $m = \mathcal{O}(rn)$ up to moment-dependent constants. The two main ingredients of our analysis are moment estimates for quadratic forms established via decoupling, together with recent advances in covariance estimation in heavy-tailed settings. As byproducts, we also establish the optimal sample complexity for low-rank matrix recovery under complex projective $4$-design sampling, thereby improving upon previous results, and obtain stability guarantees for phase retrieval under similarly weak moment assumptions.

We study stochastic barycentric estimators for proximal points and metric projections obtained by exponentially reweighting Gaussian perturbations. Our main result is an abstract comparison theorem for probability measures with densities proportional to an exponential weight, under a radial dominance condition relative to a prescribed profile. This yields an explicit upper bound for the norm of the associated barycenter in terms of a one-dimensional comparison measure. We also provide tractable sufficient conditions for radial dominance, including strong convexity, addition of nonnegative convex terms, and star-shaped constraints. As a consequence, we obtain a refined convergence rate for stochastic proximal estimators of weakly convex functions, together with asymptotic sharpness of the constant. The same framework yields a corresponding rate for stochastic projection estimators onto closed convex sets. We further establish basic structural properties of the barycentric approximation operator, such as smoothness and cocoercivity. Numerical experiments illustrate the predicted rate, the dimensional scaling of the constant, and its asymptotic sharpness.

Let $k$ and $\ell$ be positive integers. The family $C(k,\ell)$ consists of all digraphs obtained from two internally vertex-disjoint directed paths of lengths at least $k$ and $\ell$, respectively, and identifying their initial vertices and their terminal vertices. Addario-Berry, Havet and Thomassé (JCT-B, 2007) asked whether, for any positive integers $k$ and $\ell$ with $k+\ell \ge 4$, the chromatic number $\chi(D)$ is at most $k+\ell-1$ for every $C(k,\ell)$-free strongly connected digraph $D$. Let $D$ be a $C(k,\ell)$-free Hamiltonian digraph. Kim, Kim, Ma and Park (JGT, 2018) showed that $\chi(D) \le k+\ell$ and the bound is attained when $k+\ell=5$. In this paper, we prove that $\chi(D) \le k+\ell-1$ for $k+\ell\ge 6$ and this bound is best possible for all $k+\ell\geq 6$, which resolves the problem posed by Addario-Berry, Havet and Thomassé for Hamiltonian digraphs.

For a crystalline representation of the absolute Galois group of Q_p, with given Hodge-Tate weights, we obtain new constraints on the inertial weights of its mod p reduction. This allows us to formulate an explicit Serre weight conjecture, in the generality of L-parameters for unramified connected reductive groups over Q_p, and to prove the elimination direction of this conjecture. The proof uses prismatic techniques to show that the reductions modulo p of the Breuil-Kisin modules attached to crystalline Galois representations acquire a natural {\mu}_p-equivariant structure. Combining this with results on the geometry of the {\mu}_p-fixed points of affine Grassmannians leads to our new constraint.

We prove that the Cantor measure with base $b$ does not admit a Fourier frame whenever $b > 1$ is an odd integer. In particular, this answers a question of Strichartz on the existence of a Fourier frame for the middle third Cantor measure. A formalization of our main result in Lean 4 is also provided.

Temperature is one of the central concepts of thermodynamics, yet its meaning far from equilibrium remains unclear. The problem is especially challenging in isolated quantum many-body systems, whose states evolve unitarily, may remain far from equilibrium, and retain energy coherence, a genuinely quantum feature with no direct classical counterpart. Specifically, energy fluctuations in a nonstationary quantum state have two distinct origins. Part of these fluctuations comes from uncertainty in the energy populations and has the usual thermodynamic meaning. The rest comes from quantum coherence between energy sectors and is responsible for the state's time dependence. We propose that, even away from equilibrium, temperature identifies the state within the family of regular states sharing the same energy-coherence structure. The resulting inverse temperature is obtained from a principle of minimum discrimination information, which selects the least distinguishable state compatible with the prescribed energy and coherence structure. This replaces the usual maximum-entropy principle, which, applied naively, can yield a biased estimator of the inverse temperature. We also lay the groundwork for an extension of the construction to subsystems and show that, although the inverse temperature of a subsystem is not determined by its reduced state alone, under an alignment assumption, its instantaneous rate of change is a local quantity for sufficiently large subsystems.

Economic policies rarely affect only their direct targets. To study these spillovers, researchers summarize who else was treated with a simple exposure measure, such as the share of treated neighbors within a radius. But for many settings, economic theory provides little guidance on choosing the functional form (e.g., ring) of that measure or its parameters (e.g., radius). We show that the data can inform both choices. Correctly specified exposure measures imply orthogonality conditions that can be used for both estimation and testing. We establish consistency and asymptotic normality of the resulting estimator under spatial and network dependence in a design-based framework, with all randomness arising from treatment assignment. We then characterize the efficient moment conditions. Applied to two large-scale anti-poverty programs, the framework supports some prior radius estimates but rejects others. In the latter case, the revised radius yields substantively different policy-effect estimates.

We investigate a special class of nonlinear infinite dimensional systems. These are obtained by subtracting a nonlinear maximal monotone (possibly multi-valued) operator M from the semigroup generator of a scattering passive linear system. While the linear system may have unbounded linear damping (for instance, boundary damping) which is only densely defined, the nonlinear damping operator M is assumed to be defined on the whole state space. We show that this new class of nonlinear infinite dimensional systems is well-posed and incrementally scattering passive. Our approach uses the theory of maximal monotone operators and the Crandall-Pazy theorem about nonlinear contraction semigroups, which we apply to a Lax-Phillips type nonlinear semigroup that represents the whole system.

We study a family of preconditioned primal dual algorithms for convex-concave saddle point problems by the dynamics introduced in \cite{apidopoulos2026preconditioned}. The proposed framework exploits the possible smooth + nonsmooth structure of the saddle point formulation. It includes, but is not limited to, linearly constrained convex optimization problems. The proposed antisymmetric preconditioners allow us to establish non ergodic convergence rates, accounting for possible computational errors in the implementation of the method. Finally, we present numerical experiments to indicate our well performed preconditioned primal dual algorithms.

We construct an uncountable family of extremal Gibbs states of the low temperature Ising model on hyperbolic lattices embedded in the hyperbolic plane $\mathbb{H}_2$ whose interfaces are complete geodesics of $\mathbb{H}_2$. These states are extracted from the states constructed by D'Achille, Coquille and Le Ny in arXiv:2504.19553v2 by considering path in the dual lattice at close enough distance from geodesics of $\mathbb{H}_2$ thanks to the Morse–Mostow lemma.

We prove that $S^3$ endowed with an arbitrary Riemannian metric $g$ admits at least two embedded minimal spheres. The proof is based on an iterative scheme of relative min-max constructions.

The Balister–Győri–Schelp (BGS) conjecture asks whether every zero-sum list of $2^{s-1}$ nonzero vectors in $\mathbb{F}_2^s$ is the prescribed-difference profile of a perfect matching. The conjecture remains open in general, whereas the classical Hall hyperplane case is solved when all prescribed differences cross between two affine copies of a hyperplane. We isolate the smallest mixed case beyond Hall: exactly two prescribed differences are internal. Although only two requests have changed type, the complete Hall permutation is replaced by a prescribed-difference bijection between two punctured copies of the hyperplane, with two unknown deleted vertices on each side. We call this the two-hole problem. We develop a new combinatorial method for prescribed-difference matchings, based on counting and the character structure of the binary vector space. Unlike the known Hall-type methods, which construct a matching through a sequence of local algorithmic choices, our approach proves existence through a global noncancellation phenomenon. This loss of algorithmic structure is compensated by a different advantage: the method can retain global boundary information that local exchanges do not control. As a first application, it gives a new proof of the binary Hall theorem, and it then yields a complete solution of the two-hole problem with no multiplicity assumption. We also give direct constructive proofs for symmetric even-multiplicity two-hole and four-hole families. More broadly, the new technique provides a framework for studying further subfamilies of the BGS problem by measuring how far their matching structure departs from the Hall case.

We study the problem of matching two sets of noisy feature vectors when underlying true features are related by an unknown scalar affine transformation. Our method comprises two primary steps. First, we standardize the feature vectors to estimate the unknown scalar affine transformation. Subsequently, we estimate the permutation by minimizing the Least Sum of Logarithms (LSL) between two sets of observations using the estimated transformation. Our main result shows that the unknown permutation can be perfectly recovered given that the minimal separation distance of true feature vectors scales as $\sqrt{\rho_\sigma} \vee (d\log n)^{1/4} \vee \sqrt{\log n}$, where $d$ is the ambient dimension, $n$ is the sample size, and $\rho_\sigma$ is the maximal ratio of noise magnitudes. Interestingly, the obtained rate, under mild heteroscedasticity, coincides with that of the non-affine setting. We additionally demonstrate that there exist configurations requiring a larger minimal separation distance for perfect recovery. The latter makes the matching problem more challenging from minimax perspective compared to the non-affine setting. Consequently, we show that in the problem of feature matching, standardizing the data implicitly estimates the scalar affine parameters. As part of our analysis, we prove non-asymptotic concentration bounds for the affine parameter estimators in the presence of heterogeneous noise magnitudes.

We construct the first example of a higher rank, irreducible deformed Hermitian-Yang-Mills (dHYM) connection in the small radius regime. We also construct these in the large radius regime on infinitely many different bundles and make some contributions to the rank one equation as well. In particular, we investigate solutions away from the supercritical regime, showing the existence of solutions with any possible angle, and rule out some possible stability conditions.

Let $ X $ be a closed, oriented manifold with $ \dim X \geqslant 5 $. In this article, we show that 2006 Rosenberg's $ \mathbb{S}^{1} $-stability holds when $ X $ has zero Euler characteristic. The 2006 Rosenberg-Stolz Conjecture for $ X \times \mathbb{R} $ also follows under the same assumption, provided that the Riemannian metric $ g $ on $ X \times \mathbb{R} $ is complete, is of bounded curvature, and whose smallest eigenvalue is uniformly bounded below by some positive constant. We then show a $ \mathbb{T}^{n} $-stability theorem with the same hypothesis of $ X $.

Two transverse segments have local Hausdorff and upper box dimension one at their crossing, although they carry two independent tangent directions. We define the Point-Cross Dimension by assigning each tangent line the largest local upper box dimension captured by intersection with a regular \(C^1\) curve in that direction, then taking the supremum of their sums over finite independent families of tangent lines. Every Bouligand direction admits such a recurrent curve. The resulting profile is invariant under closure and covariant under local \(C^1\)-diffeomorphisms. Each directional weight is attained, and the profile is upper semicontinuous. An integral representation by ranks of weight superlevel sets gives submodularity and, together with projection bounds, exact Cartesian-product additivity. Germs with the same projective tangent support, local upper box dimension, and rank of segment directions can have different Point-Cross Dimensions. Further computations give exact directional profiles for chirps and the Point-Cross Dimension of a self-affine carpet. Finally, approximation of a ray with error \(O(t^d)\), for \(d>1\), forces directional weight at least \(1-1/d\), and this bound is sharp.

Nondegenerate cycle sets were introduced by Rump as an algebraic framework for nondegenerate, involutive solutions to the Yang–Baxter equation. Nondegenerate cycle set structures on abelian groups, such as translation-invariant and quasilinear cycle sets, are of particular interest when studying the retraction problem in the theory of the Yang–Baxter equation. In this article, we solve the retraction problem for finite quasilinear cycle sets by showing that each nontrivial quasilinear cycle set is retractable, thus proving a conjecture of Rump.

Recently, there has been significant interest in applying the method of moments developed by Wood and others to study distributions of finite abelian groups that arise in number theory and combinatorics. When the moments do not grow too fast, they determine a unique distribution. We construct large families of distributions that have the same moments. These families include several distributions that arise naturally in the study of sandpile groups of families of random graphs. Wood determined the distribution of Sylow $p$-subgroups of sandpile groups of Erdős–Rényi random graphs. This was extended by Mészáros to sandpile groups of random $d$-regular graphs, who observed an interesting special case when $d$ is even and $p = 2$. We study Sylow $p$-subgroups of sandpile groups of random bipartite graphs and similarly find a special case for $p =2$. Although this distribution differs from that of Mészáros, we show that they have the same moments and fit into our broader construction. To compute the moments of the distributions we study, we apply combinatorial tools from the theory of Hall–Littlewood functions.

For $r\geq 3$ we denote by $\mathcal{C}(D_r)$ the Cartan matrix of type $D_r$. Recently, Sun and Wang conjectured a Rogers–Ramanujan type identity for the Nahm sum associated with $\mathcal{C}(D_r)^{-1}$ and the zero vector. They further conjecture that there exist $r-1$ companion modular Nahm sums associated with nonzero vectors. We partially prove this conjecture by constructing $\lfloor (r+4)/2\rfloor$ modular Nahm sums for $\mathcal{C}(D_r)^{-1}$. To prove their modularity, we utilize the method of Bailey pairs to establish various Rogers–Ramanujan type identities. In particular, we confirm their conjectural identity.

If $\mathfrak{p}$ is a prime ideal of a number ring $\mathcal{O}$, then the top-degree cohomology of the principal congruence subgroup of level $\mathfrak{p}$ is naturally a representation of $\text{SL}_n(\mathcal{O}/\mathfrak{p}).$ We prove that the multiplicity of the Steinberg representation in this cohomology space is one. When $\mathcal{O}$ is Euclidean and $\mathfrak{p}$ is suitably small – for example a universal side divisor – then we prove that the multiplicity of the Steinberg representation in the next-highest-degree cohomology space is zero. Our proof relies on a computation of the cohomology of an $S$-arithmetic group ouside of a linear range of degrees, derived from work of Blasius–Franke–Grunewald.

Several new and sharp informational inequalities are derived as a byproduct of Stam-like and moment-entropy-like inequalities in the relative framework and a recently established inequality mixing the Rényi entropy, the Rényi divergence and the Rényi cross entropy of suitable probability density functions. More precisely, we obtain a Stam-like inequality connecting the Rényi entropy power, the recently introduced scaling-invariant relative Fisher information and the Rényi cross entropy. Furthermore, we derive an inequality involving only Fisher-like informational measures and another inequality involving only moment-like functionals of non-relative, relative and cross types, respectively. All the inequalities are sharp. The minimizers of the Stam-like inequality are, in certain cases, pairs of Gaussian or stretched Gaussian probability densities; in contrast, each minimizer of the moment-like inequality is the probability density of the generalized Beta distribution.

This work investigates time-harmonic electromagnetic scattering governed by the Maxwell system, where bounded anisotropic scatterers are embedded in a homogeneous electromagnetic background. We focus on the localized enhancement of the gradients of the total electric and magnetic fields in small boundary-attached neighborhoods of finitely many prescribed points near boundaries of anisotropic electromagnetic scatterers. We show that, through a suitable construction of incident electromagnetic waves, the gradients of both the total electric field and the total magnetic field can be made arbitrarily large in these neighborhoods. The main strategy is based on the introduction of auxiliary boundary-attached electromagnetic neighborhoods and the associated electric and magnetic fields, which exhibit strong gradient variation near the prescribed points. Using the approximation property of Maxwell Herglotz wave functions, these auxiliary fields are then approximated by physically admissible incident waves in the neighborhood of the scatterers. Together with the well-posedness and continuous dependence of the anisotropic scattering problem, this implies that the corresponding scattered field can be controlled to be sufficiently weak in the relevant region. Consequently, the total field is dominated by the incident field near the prescribed points and inherits its large-gradient behavior. The result provides a theoretical mechanism for localized gradient enhancement in anisotropic electromagnetic scattering and may have implications for field concentration, high-resolution probing, and sensitivity analysis of electromagnetic responses in complex media.

We study a class of random inner functions $\varphi$ whose Clark measure at $1$ is the weighted sum of point masses supported on independent uniformly distributed points of $\mathbb T$. Our first result shows that $\varphi$ is almost surely a Blaschke product. We then investigate when $\varphi$ admits angular derivative almost surely and we provide a $0 - 1$ law. These conditions have a direct interpretation in terms of the other Clark measures associated with $\varphi$. Finally, we obtain quantitative estimates for the zeros of $\varphi$, proving that, in suitable regimes, their distribution satisfies summability conditions stronger than the classical Blaschke condition.

We study an equation-free screen for finite-window Mittag-Leffler compatibility, in which multihorizon relaxation profiles are constructed out of sample by k-nearest-neighbor prediction from ensembles of scalar trajectories and compared with candidate temporal laws. The screen assesses compatibility, not the generating mechanism. In a matched, noise-free calibration, it reproduces the generating Mittag-Leffler behavior to numerical precision, while relaxation-rate heterogeneity shifts the fitted Mittag-Leffler order toward lower values. For noise-free classical distributed-rate ensembles, a Mittag-Leffler preference over a single exponential disappears when a biexponential alternative is included. Under observational noise, this recovery is incomplete: with three candidate families, the Mittag-Leffler law is still selected in 60% of runs for a classical ensemble, against 72% for a fractional reference. In the tested noise-free settings, simpler profile constructions reproduce the screening decisions, whereas kNN predicts individual trajectories far more accurately than a constant predictor. The screening interpretation of a single-record example, initially investigated as a chaotic test case, is withdrawn as evidence about the continuous system after numerical verification revealed a nonrobust simulated regime. Mittag-Leffler compatibility is therefore a comparative, condition-dependent screening outcome, not identification of fractional origin.