Papers

It is known that there can be at most 3 disjoint nontrivial minimal graphs in R^3 with 0 boundary values, and conjectured that there can be at most 2. We explore properties that the graphs would have if there were 3.

Let $F$ be a field of positive characteristic, and let $A$ be a finitely generated commutative associative $F$-algebra. We study Lie subalgebras of $Der \, A$ generated by locally finite derivations. We prove that if all commutators of the generators satisfy a natural local finiteness condition, then the associative subalgebra of $End_{F}\, A$ generated by these derivations is finite-dimensional.

We study adaptive confidence sets for mean vectors of multivariate Gaussians with identity covariance under Huber contamination when the contamination proportion is unknown. We show that an information-theoretically optimal adaptive confidence set can be constructed by simultaneously quantifying uncertainty across all one-dimensional projections. The resulting confidence set achieves the optimal diameter and reveals a substantial adaptation cost compared with the minimax estimation error. Although statistically optimal, this construction is computationally infeasible. We establish a Statistical Query (SQ) lower bound showing that achieving diameter $O(m^{-1/2})$ requires either exponentially many queries or query tolerance $d^{-\Omega(m)}$. Thus, in contrast to robust mean estimation, adaptation to the unknown contamination proportion incurs both a statistical cost and a computational barrier within the SQ framework. Finally, we give a low-degree moment-matching algorithm that constructs an adaptive confidence set using $d^{O(m)}$ samples and time. Its diameter bound contains an additional $O(m^{-1/2})$ term beyond the information-theoretic rate, matching the dependence on $m$ in the SQ lower bound. Together, these results characterize the statistical limits of adaptive robust confidence sets under Huber contamination and quantify a computational barrier within the SQ framework.

We prove mixing local limit theorems for dynamical systems in the superdiffusive regime where there is convergence to a multivariate stable law (including the nonstandard domain of the central limit theorem). We apply our results to d-dimensional observables for intermittent maps and solenoids, as well as intermittent semiflows and flows.

We construct a smooth autonomous divergence-free vector field on $\mathbb{T}^3$ whose flow is exponentially mixing with respect to Lebesgue measure. This answers an open question about smooth autonomous exponential mixing in three dimensions. Starting from an exponentially mixing Anosov suspension flow, we perform surgery near four periodic orbits and show that the resulting manifold is diffeomorphic to $\mathbb{T}^3$. The resulting vector field has exactly sixteen hyperbolic equilibria. We also prove a general transfer theorem that preserves exponential mixing under certain singular changes of coordinates.

We establish sharp weighted Rogers–Shephard inequalities for Rényi differential entropy. For independent copies $X,X'$ of a log-concave random vector in $\R^n$, we prove $h_p(aX-bX')\le h_p(X)+n\log(a+b)$ for all $a,b>0$ and $1\le p\le\infty$, where orders $1$ and $\infty$ correspond to Shannon entropy and min-entropy, respectively. The Shannon case settles the entropic Rogers–Shephard conjecture of Madiman and Kontoyiannis and its higher-dimensional formulation. For $1\le p<\infty$, equality holds precisely for invertible affine images of random vectors with independent exponential coordinates. At $p=\infty$, the larger equality class is characterized by possibly extended-valued convex potentials that are positively homogeneous of degree one. For every $0<p<1$, uniform laws on sufficiently high-dimensional simplices violate the unweighted bound. Thus $p=1$ is the critical order for this bound to hold in every dimension. The proof combines weighted convex intersections with an exponential lift to a log-affine probability density. At the Shannon endpoint, this lift yields an exact deficit decomposition with a conditional mutual information term for an additive channel with asymmetric Laplace noise.

We count the integral solutions of x_1^2+x_2^2-x_3^2=D weighted by Phi(x_3/M), for M smaller than sqrt(D). Such solutions correspond to binary quadratic forms of discriminant D, and the count becomes a sum, over the geodesics of discriminant D, of a function of their distance to the point i, which we evaluate by the spectral theory of automorphic forms. For square D=n^2 the geodesics run from cusp to cusp, and Hecke theory reduces them to a single geodesic. For every n and both parities of x_3 we obtain an asymptotic formula with error O(n^{1+theta+eps} M^{-1/2}), theta=7/64, using the Kim-Sarnak bound and the large sieve of Deshouillers and Iwaniec; it is non-trivial for M >= n^{71/96+eps}. This improves the range n^{5/6+eps} of the elementary method and confirms the corrected main term of the author's earlier work. For non-square D=f^2 D_0 the geodesics are closed, and a Hecke relation for their traces reduces everything to the fundamental discriminant D_0. For odd D_0 we obtain an error O(D^eps f^{1+theta} D_0^{2/3} M^{-1/2}), using the formulas of Duke-Imamoglu-Toth and Baruch-Mao together with Young's cubic moment. For D = 5 mod 8 this counts all solutions, and the range M >= D^{1/2-1/1332} of Friedlander and Iwaniec improves to M >= D^{4/9+eps}. This paper was written in collaboration with Claude (Opus 5.5, High), an AI model developed by Anthropic.

We introduce the problem of permutation burst error-correction, motivated by applications in molecular data storage with nanopore sequencing read platforms. A permutation burst of length $k$ is an error that affects a substring of length $k$ of a string of length $n$ by permuting its entries. Permutation burst errors can be corrected using classical burst error-correcting codes, such as Fire and Reed-Solomon codes. Here, we study how to reduce the redundancy of these classical codes by taking into account that the burst error preserves the weight of the substring. In addition, we further restrict our model to handle cyclic permutation bursts only, which are of independent practical and mathematical interest. Our main results include proving/establishing the following: a) for $k=o(\sqrt{n}),$ the redundancy of any permutation burst error correcting code is at least $\log n-k+\log \binom{2k-2}{k-2}+o(1).$ This result follows from McDiarmid's concentration bound; b) for $k \geq 9$ and $k=o\left(\sqrt{\frac{n}{\log n}}\right)$, the redundancy of any cyclic permutation burst error correcting code is at least $\log n +\log (k(k-1)-56)-\log\,80+o(1).$ c) the asymptotic redundancy upper bounds for the two types of substring errors that are twice the lower bound; d) constructions for both permutation burst and cyclic permutation burst errors that involve Varshamov-Tenegoltz type constraints or represent generalizations and extensions of Fire and other codes. In particular, we exhibit polynomially encodable and decodable codes that can correct cyclic bursts of length $\leq k$ using $\log n + k + 49\log k + O(\log \log k)$ redundant bits.

Let \(\alpha, \beta \in \mathbb{T} = \mathbb{R}/\mathbb{Z}\), and let \((\xi_n)_{n \ge 1}\) be independent random variables taking the values \(\alpha\) and \(\beta\) with equal probability. We study the random walk \[X_n = \xi_1 + \cdots + \xi_n \pmod 1\] and present a deterministic Diophantine condition on \((\alpha, \beta)\) that implies almost sure Poisson pair correlation. To the best of our knowledge, this is the first result providing explicit examples in this framework where the increment measure is purely discrete. The classical Fourier contraction method fails in this setting; a different approach is required. Our proof uses the parabolic geometry of returns to \(1\) and an exact covariance formula for overlapping blocks.

We construct closed symplectic 4-manifolds admitting the structure of hyperbolic $\pi$-orbifold, addressing LeBrun's conjecture. Our simplest example has Euler characteristic $304$, is tessellated by $64$ copies of the right-angled hyperbolic $120$-cell and admits a four-fold branched covering over a product of two surfaces of genus $4$.

We study admissible entire subsolutions and radial boundary blow-up solutions of $S_k^{\frac1k}[\Delta uI-D^2u]=f(u)+g(u)|Du|^q$ for $1\leq k<n$, continuing our study of the determinant case $k=n$ in Part I. Under structural assumptions on the continuous, non-decreasing coefficients $f$ and $g$, we establish necessary and sufficient conditions for the existence of entire subsolutions when $q<2$. Our results cover the non-Lipschitz case $0<q<1$ with a possibly vanishing zero-order term, the gradient-singular case $q<0$, and sign-changing gradient coefficients. We also characterize finite-radius blow-up of radial solutions and derive two-sided estimates for the blow-up radius, as well as pointwise upper bounds for their profiles in terms of inverse Keller–Osserman functions.

Let $q=2^n$ with $n=2m+1$ for an integer $m\ge1$, and let ${\mathbb F}_q$ be the finite field with $q$ elements. Let $W_n(x)=x^{2^m+3}$ be the Welch function on ${\mathbb F}_q$. For $1\le k\le n$ and $b\in{\mathbb F}_q$, denote by $N_{n,k}(b)$ the number of $k$-dimensional ${\mathbb F}_2$-subspaces $E$ of $\mathbb F}_q$ such that \[ \sum_{x\in E}W_n(x)=b. \] In this paper, we determine $N_{n,k}(b)$ for all $k$. The only nontrivial case is $k=3$, for which the answer is given in terms of a binary Kloosterman sum. As a consequence, we prove a recent conjecture of Hou and Zhao on the third-order sum-freedom of the Welch function.

We study positive $(n-1)$-convex entire subsolutions of the normalized real $(n-1)$ Monge–Ampère equation $\operatorname{det}^{\frac1n}\bigl(\Delta uI-D^2u\bigr)=f(u)+g(u)|Du|^q$ in $\mathbb R^n$. Under two sets of structural assumptions on the continuous, non-decreasing coefficients $f$ and $g$, we establish necessary and sufficient conditions for the existence of entire subsolutions when $q<2-\frac1n$. Our results cover the pure-gradient degenerate case $f\equiv0$ with $0<q<1$, the gradient-singular case $q<0$, and the case where $g$ changes sign. For $q=2-\frac1n$, we recover the Keller–Osserman condition for the gradient-independent equation, prove nonexistence when $g$ is eventually positive, and obtain an exponentially weighted condition when $g$ has a nonpositive limit at infinity. This is the first of two companion papers and treats the determinant case.

We establish uniform score bounds and Lipschitz continuity for truncated Gaussian, Cauchy, Boltzmann, and softmax policies under explicitly stated parameterizations.

Flow matching can learn structure directly from data, raising the question of when additional constraints improve generation. We study penalties that encode such constraints, with physics-informed flow matching as a motivating example. Satisfying a constraint more closely need not make the generated distribution more accurate. We therefore compare penalised and unpenalised generative modelling through their expected squared $2$-Wasserstein error. Our analysis covers both vector-valued and functional data in a Hilbert-space setting. We first derive geometric criteria that account for constraint violations and the distribution of samples after projection onto the constraint set. We then solve the statistical comparison explicitly for specific velocity-field classes and constraints via flow matching. In this model, penalisation improves accuracy exactly when the reduction in estimation variance outweighs the increase in squared bias. The resulting formula shows how this trade-off depends on sample size, the model class, the noise schedule, and the penalty weight.

We generalize Nakajima's theory of Hilbert schemes of points on surfaces to Hilbert schemes of points on orbisurfaces, and obtain initial factorization based on stratification by stabilizers. While the usual formula is retained for the trivial stabilizer stratum and for those strata of codimension $1$, genuine difficulties arise with stacky points strata. In the local picture of an isolated stacky point, the theory reduce to the study of $\Quot^\bfv[\CC^2/G]$ for a small finite subgroup of $GL_2(\CC)$. The generating series for their total cohomology is a geometric correction of the generating series for total cohomology of ordinary smooth surface schemes. In addition, we obtain an action of the Borel-Moore homology of the punctual fibre stacks on the total Hilbert cohomology by virtual correspondences, in analogy with the Fock space description in Nakajima's theory. The punctual fibre stacks are genuinely different from the symmetric stack, and has natural realizations via motivic integration.

We prove that two completely different oscillators synchronise through arbitrarily weak reciprocal impulses. Any two planar oscillators with hyperbolic attracting limit cycles and equal natural periods lock stably one-to-one under translation impulses with generic directions and a suitable range of positive strength ratios, whatever their vector fields, limit cycles and transverse contraction rates. The mechanism is topological: the winding of the asymptotic phase gradient forces phase advances and delays, and the relative winding number of an effective impulse field gives lower bounds on attracting and unstable locked states. Equal periods are not essential: locking persists under small nonzero mismatches, with persistence and loss governed by a shifted phase balance. The analysis covers the full state-dependent impulsive dynamics, including transverse stability. Nearly simultaneous emissions may make the return discontinuous; under an explicit regularity condition the global return has a normally attracting Lipschitz invariant circle with topologically Morse–Smale dynamics.

We present a faithful translation from Aloni's logic BSL into a classical modal logic over semilattices combining ideas from earlier modal translations of truthmaker semantics with algebraic representation theory. We show how our method can also be made to work for other current hyperintensional logics. Finally, we place our results in a broader setting of translating hyperintensional logics into classical modal ones, and discuss applications and new challenges that arise for both areas of philosophical logic connected in this way.

The bipartite edge frustration of a graph $G$, $\varphi(G)$, is the minimum number of edges that must be removed from $G$ to obtain a graph that is bipartite. Core graphs are singular graphs on $2$ or more vertices, for which the kernel of the adjacency matrix contains a full eigenvector. A nut graph is a core graph of nullity one. Nut graphs $G$ are non-bipartite. It is proved here that $\varphi(G) \geq 2$ if $G$ is a nut graph, i.e. every nut graph requires deletion of at least two edges to produce a bipartite graph. In addition, we give a constructive proof that for every integer $k \geq 2$ there exist infinitely many chemical nut graphs $G$ with $\varphi(G) = k$. Furthermore, for $3 \leq d \leq 8$, the values $k$, for which a $d$-regular nut graph $G$ with $\varphi(G) = k$ exists, are characterised. It is shown here that for every integer $k \geq 2$ there exist infinitely many cubic polyhedral nut graphs $G$ with $\varphi(G) = k$. Two ways of producing larger nut graphs from smaller are described: the cut and frustration constructions. One straightforward consequence is that total double subdivision of a nut graph (i.e. simultaneous insertion of two degree-$2$ vertices into every edge) yields a nut graph.

In this paper, we develop a greedy double-block nonlinear Bregman-Kaczmarz method to efficiently find sparse solutions of nonlinear equations. The theoretical convergence rate of the proposed method is established under certain assumptions. Numerical experiments illustrate the competitive performance and superiority of the proposed method.

Let $h(x)$ be the number of pairs $1\le a<b<x$ with $\gcd(a,b)=1$ and $\sigma(a)=\sigma(b)$. Erdős Problem #824 asks whether $h(x)>x^{2-o(1)}$. By an argument of Pollack and Pomerance, this follows if for every fixed $\delta>0$ there are $y^{1-o(1)}$ primes $p\le5y$ with $p+1$ free of prime factors exceeding $y^\delta$. We write this implication out with explicit parameters and formalize it in Lean 4. We then check that the argument of the recent OpenAI preprint "Weighted dilation graphs, smooth shifted primes and totient fibers", which claims the corresponding statement for $p-1$, applies to $p+1$ after a change of sign in two of its sections. Consequently $h(x)=x^{2-o(1)}$, conditionally on the correctness of that preprint, which has not been refereed and whose main new estimates we have not independently verified.

For the degenerate parabolic equation $u_t=\nabla\cdot (u^{m-1}\nabla u)$, the homogeneous Neumann problem in smoothly bounded domains $\Omega\subset {\mathbb{R}}^n$ is considered in the (super-fast) diffusion range when $m\in (-\infty,1)$. Upper and lower estimates for solutions are inferred by means of two Moser-type iterations, and optimality properties of these bounds are derived using comparison arguments. On the basis of the accordingly obtained information on regularization into spaces of bounded and uniformly positive functions, a result on global solvability in an associated initial-boundary value problem is achieved for all nonnegative and nontrivial initial data belonging to $L^1(\Omega) \cap \bigcup_{s>n(1-m)/2} L^s(\Omega)$.

We study multidimensional entry-exit phenomena in slow-fast systems whose fast linearization contains a nilpotent component. We show that the classical spectral balance still determines the exit point as $\varepsilon \to 0$, however the nilpotent component induces an algebraic amplification invisible to the spectrum, producing a $\varepsilon|\log\varepsilon|$-shift of the exit point for $0 <\varepsilon \ll 1$. We further extend the analysis to systems with several spectral branches and show that mode dominance follows a hierarchical, lexicographic selection principle: accumulated spectral growth determines the leading order, while the nilpotent component resolves degeneracies between spectrally equivalent modes.

A topological group, defined in terms of “composable minimal geodesics” is constructed and shown to be homotopy equivalent to the pointed loop space of a path-connected, simply connected Riemannian manifold $M$ of positive convexity radius. This construction is analogous to John Milnor's free group model for the pointed loop space of a connected simplicial complex. Related models built from composable minimal geodesics are also obtained for the free loop space of $M$ and for the space of maps from any finite two-dimensional CW-complex into $M$.

We study the communication cost of exact sampling under Campbell's exponential coding criterion. Given a target distribution $P$ and a proposal distribution $Q$, the sender and receiver share i.i.d. $Q$-distributed samples, and the sender communicates an index so that the selected sample has distribution exactly $P$. We introduce a new information measure Rényi channel simulation divergence $R_\alpha(P\Vert Q)$, a Rényi generalization of the channel simulation divergence. We then show that it lower bounds the Campbell cost of every exact sampler and is achievable, via a new layered sampling scheme, within an additive constant depending only on the Campbell parameter. This yields a tighter one-shot characterization than previous bounds based on Rényi divergences of different orders. For i.i.d. product distributions, under suitable non-degeneracy and moment conditions, we further show that the gap beyond the leading Rényi-divergence term is $\frac{1}{2}\log n+O(1)$.

Stationary scattering admits a dynamical formulation in which the scattering data are extracted from the evolution operator for an effective non-stationary and non-unitary quantum system. This provides a mathematical method of solving scattering problems. We consider the possibility of realizing such a quantum system and relating the scattering information to outcomes of quantum measurements performed on it. To this end, we examine stationary scattering problems in one dimension where the scattering data consist of reflection and transmission amplitudes and the corresponding effective quantum system is a non-unitary two-level system. The states of this system identify points in a Bloch sphere $S^2$ and their evolution yields trajectories in $S^2$. Quantum measurements turn out to be capable of providing information about the reflection amplitudes. We identify configurations displaying reflectionlessness with situations where one or both of the poles of $S^2$ undergo cyclic evolutions. Spectral singularities and coherent perfect absorption correspond to dynamical spin flips which map one of the poles of $S^2$ to the other.

For any $n\geq 1$, we construct a continuum of non-conjugate actions by homeomorphisms of $\operatorname{SL}_{n+1}(\mathbb{Q})$ on the $n$-sphere $S^{n}$ that do not extend to a continuous action of $\operatorname{SL}_{n+1}(\mathbb{R})$. We do this by adapting a blow-up construction of Bowden and Mann. We motivate studying these actions and note down some related observations.

High-order direct arbitrary Lagrangian–Eulerian discontinuous Galerkin (ALE-DG) methods can suffer severe order reduction under nonsmooth mesh motion despite satisfying the geometric conservation law (GCL). To address this limitation, we propose a new ALE-DG framework applicable to moving triangular and quadrilateral meshes. This framework successfully overcomes the uniform Lipschitz constraint inherent in existing direct ALE-DG methods, retaining high-order accuracy even when the mesh-velocity gradients are unbounded. This capability is driven by transport polynomial exactness (TPE($k$)), which is a structural property stronger than the GCL. TPE($k$) ensures the exact transport of polynomials of degree at most $k$ under arbitrary mesh motion. To incorporate TPE($k$) into a direct ALE-DG method, we introduce a coefficient-evolution (CE) formulation. Reconciling strict local conservation, an indispensable requirement for accurate shock capturing, with the preservation of the TPE(\(k\)) property constitutes the principal technical challenge. We resolve this by evolving the coefficients using the third-order strong-stability-preserving Runge–Kutta method, paired with a conservative correction executed via Simpson's rule integration. Crucially, this correction strictly vanishes for exactly transported quadratic polynomials, thereby ensuring that our third-order scheme rigorously satisfies the TPE($2$) property. Numerical experiments demonstrate that the proposed scheme retains third-order accuracy when the discrete mesh-velocity Lipschitz estimate grows as $\mathcal O(h^{-1})$, whereas a representative existing direct ALE-DG scheme exhibits order reduction under the same mesh motion. Tests with discontinuous solutions further demonstrate robust and high-resolution performance under random and Lagrangian mesh motions.

It follows from the compactness of the connectedness locus that the locus of PCF polynomials in the moduli space of polynomials of degree d (at least 2) over the complex number is a countable union of zero-dimensional subvarieties. Over fields of characteristic p the analogous claim is certainly false, so it is natural to ask how large the components of the PCF locus can be. We show that the components of the PCF locus in the moduli space of polynomials of degree d over an algebraically closed field of characteristic p>0 have dimension at most d/p and that this bound is sharp, in the strong sense that for every p and d there are infinitely many distinct PCF components whose dimension is the greatest integer no greater than this bound (except in the case $p=d=2$).

Let $\mathbf{T}=(T_1,\ldots,T_n)$ be a commuting tuple of bounded linear operators on a separable Hilbert space $\mathcal{K}$, and let $\{a_\alpha\}_{\alpha\in\mathbb{Z}_+^n}$ be strictly positive weights. Under mild assumptions on the weights, we characterize the weighted frame vectors and weighted Parseval frame vectors, that is, those vectors $\xi$ for which $\{a_\alpha \mathbf{T}^\alpha\xi\}_{\alpha\in\mathbb{Z}_+^n}$ is a frame or a Parseval frame. We reduce the problem to characterizing surjective module maps from the reproducing analytic Hilbert module associated with the weights onto the module $\mathcal{K}$, equipped with the module structure defined by $\mathbf{T}$. It is shown that weighted Parseval frame vectors differ by a unimodular scalar whenever they exist. If the kernel defined by the weights has the complete Nevanlinna–Pick property, then any two associated weighted frame vectors are similar whenever they exist. We also focus on the case of the Hardy kernel on the polydisc, that is, the case $a_\alpha=1$ for all $\alpha\in\mathbb{Z}^n_+$. This is related to the famous Douglas–Foias problem: whether two similar quotient modules of the Hardy spaces $H^2(\mathbb{D}^n)~(n>1)$ are equal. Our result shows that the answer is negative. This leads to the conclusion that frame vectors for representations of $\mathbb{Z}_+^n$ $(n>1)$ may fail to be similar. It means that, for the case $a_\alpha=1$, there exists a tuple $\mathbf{T}=(T_1,\dots,T_n)$ with $n>1$ on $\mathcal{K}$ and two distinct frame vectors for $\mathbf{T}$ which are not similar.