Papers

The classical More Sums Than Differences (MSTD) problem studies finite sets $A\subset\{0,1,\ldots,n\}$ for which $|A+A|>|A-A|$, where $A+A=\{a_1+a_2:a_1,a_2\in A\}$ and $A-A=\{a_1-a_2:a_1,a_2\in A\}$. As addition is commutative and subtraction is not, it was conjectured that as $n\to\infty$ almost all subsets $A$ chosen uniformly from the power set of $\{0,1,\ldots,n\}$ are difference-dominated, and it was thus a surprise when Martin and O'Bryant proved a positive percentage of sets are sum-dominant. We greatly generalize this model by introducing a Markov-chain framework, where the classical MSTD model is now just a special case. Let $(X_i)_{i=0}^n$ be a stationary two-state Markov chain on $\{0,1\}$ with transition probabilities $P(0,0)=p$ and $P(1,1)=q$, where $p,q\in(0,1)$. We include $i$ in $A$ exactly when $X_i=1$, and define $A=\{i\in\{0,\ldots,n\}:X_i=1\}$. The usual independent Bernoulli model is recovered when consecutive inclusion decisions are independent, equivalently when $p=1-q$. In particular, the uniformly random subset model corresponds to $p=q=1/2$. Using the fringe-middle method from the MSTD literature, we show that the middle sums and differences are filled with high probability, so the comparison between $|A+A|$ and $|A-A|$ is again governed by endpoint fringes. By fringe manipulation, we prove that the probabilities of sum-dominant, difference-dominant, and balanced sets tend to strictly positive limits as $n\to\infty$. We also give numerical estimates of these three probabilities for finite $n$ over a range of values of $p$ and $q$. Through combinatorial methods, we find a closed-form expression for $\mathbb{E}[|A-A|-|A+A|]$ as $n\to\infty$.

We consider the following class of singular elliptic problems \[ \begin{cases} -\Delta u = -\dfrac{u}{|u|^{\beta+1}}\chi_{\{|u|>0\}} +\lambda |u|^{p-1}u, & \text{in }\Omega,\\[1mm] u=0, & \text{on }\partial\Omega, \end{cases} \] where $\Omega\subset\mathbb R^N$ is a bounded smooth domain, $N\geq3$, $0<\beta,p<1$, and $\lambda>0$. The energy functional is not of class $C^1$ in $H_0^1(\Omega)$, and the standard critical point theory does not apply. We prove that a nontrivial nonnegative weak solution exists if and only if $\lambda$ exceeds a critical value $\bar\lambda$, which we identify and locate strictly between two Rayleigh-type parameters $0<\lambda_*<\bar\lambda<\lambda^*<+\infty$. This gives a sharp existence threshold. We also prove that for every $\lambda$ in a neighborhood on the left of $\lambda^*$, there exist at least two distinct nontrivial nonnegative weak solutions. The first is a minimizer on the Nehari manifold, with an energy changing sign at $\lambda^*$. The second has a strictly positive energy. For $\lambda<\lambda^*$ sufficiently close to $\lambda^*$, both solutions have positive energy. This contrasts with previous multiplicity results for the same problem, in which the two solutions obtained have energies of opposite signs.

The inherent non-convex minimum-separation constraints introduced by movable antennas present a formidable challenge to the joint optimization of antenna positions and transmission strategies, rendering conventional methods computationally infeasible, particularly in large-scale cell-free massive multiple-input multiple-output (MIMO). In this work, we propose the graph-based learning individual intrinsic reward heterogeneous-agent proximal policy optimization (GLIIR-HAPPO) algorithm, a novel heterogeneous multi-agent reinforcement learning (MARL) framework that fundamentally overcomes this impasse by systematically decomposing the original coupled optimization into coordinated subproblems. To ensure tractability, we embed the non-convex geometric constraints into a penalty-augmented reward structure and develop a specialized geometric solver that enables the positioning agents to efficiently navigate the high-dimensional action space. Specifically, we propose an architecture featuring a dynamic-interaction graph critic for adaptive cross-role coordination, together with role-conditioned federated distillation that synchronizes same-role policies through compact actor-output statistics. Beyond architectural design, we establish a rigorous theoretical analysis that derives monotonic performance improvement bounds and establishes convergence guarantees for the proposed bi-level optimization. Numerical simulations demonstrate that our framework yields significant sum-rate improvements over state-of-the-art MARL schemes. Moreover, the performance of our advanced architecture closely approaches its fully centralized counterpart, while drastically reducing communication overhead.

Given a cycle type, the corresponding conjugacy class of $S_n$ generates either $S_n$ or $A_n$ for sufficiently large $n$. We prove that the sequence of diameters of Cayley graphs is eventually polynomial on residue classes for any fixed cycle type. This result is also extended to finite unions of conjugacy classes.

We study a random walk in $\mathbb{Z}^d$ in a dynamical random environment constituted of independent null recurrent birth-and-death chains, and establish a scaling limit result for the position of the walker in terms of a sublinear time change of Brownian motion, thus establishing subdiffusivity, in contrast to a diffusivity result established previously in an ergodic version of the model.

We introduce new techniques to study mixed-characteristic special loci in moduli spaces and give several applications, focusing in particular on the moduli space $\mathsf{M}_g$ of curves. These applications include Zariski density results for split Jacobians, non-density results for supersingular loci, and lifting results for algebraic cycles, all under appropriate numerical conditions. Our methods generalize to more-or-less arbitrary moduli spaces mapping to Hodge-type Shimura varieties (and even some settings beyond the Shimura case), and relate to questions raised by Oort and de Jong.

Let $L$ be a number field containing a CM field $K$. We compute an explicit current representing the $\mathrm{GL}_{\mathcal{O}_K}(\mathcal{O}_L)$-equivariant algebraic Eisenstein-Kronecker classes constructed by Kings-Sprang. In the Bianchi case, where $K$ is imaginary quadratic and $[L:K]=2$, this yields a purely algebraic construction of Eisenstein cohomology classes, even though the associated locally symmetric space is not an algebraic variety. We express their constant terms at the cusp in terms of partial Hecke $L$-values of $K$.

We develop a spatially explicit nonlinear biomass model for exploited fish populations incorporating density-dependent Beverton-Holt production, natural mortality, diffusion driven spatial movement, and harvesting. Starting from a reduced biomass model motivated by spawning stock biomass data, we derive a reaction-diffusion model with Neumann boundary conditions and spatially distributed harvesting. We establish positivity, boundedness, and well-posedness of the biomass dynamics, and identify a critical harvesting threshold separating persistence and extinction regimes of spatially homogeneous equilibria. A distributed optimal harvesting problem is then formulated, and the corresponding state-adjoint optimality system and pointwise projection characterization of the harvesting control are derived. The adjoint variable provides a measure of the marginal future value of biomass within the harvesting objective and offers a diagnostic that complements biomass abundance alone. Numerical simulations examine the coupled evolution of biomass, the adjoint variable, and optimal harvesting effort. The results show that diffusion progressively reduces the initial spatial heterogeneity in biomass, while the computed adjoint and optimal harvesting fields exhibit limited spatial variation under the parameter regime considered. These results illustrate how the coupled state-adjoint-control framework links spatial biomass dynamics with future management value and harvesting decisions.

We study the Dunkl–Schrödinger operator $L=-\Delta_k+V$ on $\mathbb R^N$, $N>2$, where $V$ is a nonnegative potential in the reverse Hölder class ${\rm RH}^q(dw)$, $q>\mathbf N/2$, and $\mathbf N$ is the homogeneous dimension. We establish decay estimates for the resolvent kernel in terms of the critical radius and the orbit distance. For reflection-invariant potentials, we obtain $L^p(dw)$ bounds for $VL^{-1}$, second-order Dunkl derivatives of $L^{-1}$, and the Riesz transform $\nabla_kL^{-1/2}$, together with fractional potential and mixed estimates. The proofs combine local estimates for weak solutions, the Dunkl Fefferman–Phong inequality, and kernel estimates adapted to the Euclidean and orbit distances.

Let $E/F$ be a CM extension and let $1\le r<n$. We determine primitive Fourier coefficients of Hermitian Klingen-Eisenstein series on $\mathrm{U}_{n,n}$ induced from cusp forms on $\mathrm{U}_{r,r}$, at principal congruence level. Using the pullback formula for Hermitian Eisenstein series, we express a normalized primitive Fourier coefficient of a Klingen-Eisenstein series as the corresponding primitive Fourier coefficient of a Siegel Eisenstein series multiplied by a Petersson inner product with an explicitly constructed holomorphic automorphic form. The automorphic form in the Petersson inner product is a finite sum of products of Hermitian theta series and Siegel Eisenstein series of degree $r$. Under some hypotheses, we also prove the $p$-integrality of these normalized primitive Fourier coefficients.

We give an introduction to the time-fractional stochastic heat equation driven by 1+d-parameter fractional time-space white noise, in the following two cases: (i) With additive fractional time-space white noise (ii) With multiplicative time-space Brownian noise The fractional time derivative is interpreted as the Caputo derivative of order $\alpha \in (0,2]$ and we assume that the Hurst coefficient $H=(H_0,H_1,H_2, ...,H_d)$ of the time-space fractional white noise is in $(\tfrac{1}{2},1)^{1+d}$. We find an explicit expression for the unique solution in the sense of distribution of the equation in the additive noise case (i). In the multiplicative case (ii) we derive the correct space–time convolution equation and its Wiener chaos expansion. For $1<\alpha\leq2$ a second initial condition is required. The endpoint $\alpha=2$ is treated separately and yields a stochastic wave equation with finite propagation speed. A solution $Y(t,x)$ is called mild if $E[Y^2(t,x)] < \infty$ for all $t,x$. In the multiplicative case the random-field condition is expressed by the square integrability of the forcing kernel. For the classical Laplacian and space–time white noise, the endpoint $\alpha=2$ admits a random-field solution only in space dimension $d=1$. This paper is partly a survey paper, explaining the concepts and methods behind the results. It is also partly a research paper, in the sense that some results are new, to the best of our knowledge.

We prove that the number of labelled embeddings of any $n$-vertex tree $T$ into an $n$-vertex graph $G$ of maximum degree $d$ satisfies $$ \mathrm{inj}(T,G) \leq (d/e)^n \exp(o_d(1) n). $$ The bound is sharp up to determining $o_d(1)$, even for paths, and the dependence of the error $\exp(o_d(1)n)$ on $d$ is necessary. As an immediate corollary, we obtain an optimal anticoncentration bound for the isomorphism class of a uniformly random spanning tree in a connected $d$-regular graph, answering a conjecture of H. Lee. The proof combines Brégman's inequality with entropy methods.

In this paper, we show that for any coquasi-Hopf algebra $H$ and any $H$-Majid bimodule $M$, the cotensor coalgebra $\operatorname{CoT}_H(M)$ admits a graded coquasi-Hopf algebra structure. We then define the standard filtration of a coquasi-Hopf algebra and show that the associated graded coquasi-Hopf algebra admits an embedding into a cotensor coquasi-Hopf algebra that preserves degrees zero and one. If, in addition, this coquasi-Hopf algebra is coquasitriangular (respectively, coribbon), then the cotensor coquasi-Hopf algebra is also coquasitriangular (respectively, coribbon), and the above embedding is a morphism of coquasitriangular (respectively, coribbon) coquasi-Hopf algebras. As a byproduct, we prove that the coquasi-antipode of a coquasi-Hopf algebra with the dual Chevalley property is bijective. Moreover, we establish graded quasi-coquasi-Hopf pairings between tensor quasi-Hopf algebras and cotensor coquasi-Hopf algebras.

We construct an example of a smooth embedding of $\mathbb{S}^{p+q-1}$ in $\mathbb{R}^{p+q}$ whose evolution under the mean curvature flow forms an isolated singularity at one point, after which it is a smoothly embedded copy of $\mathbb{S}^{p-1}\times \mathbb{S}^q$. In doing so, we prove some general results about $SO(p)\times SO(q)$-invariant mean curvature flow whose initial data is generated by the rotation of a graphical profile curve, including an analysis of bubblesheet singularities. The most involved part of our analysis is an instant smoothness result, which requires a delicate approximation and pseudolocality.

In this paper, we study the regularity of free boundaries for Monge-Ampère obstacle problem up to the boundary. We mainly focus on how the growth order of boundary value near the contact point impact the regularity. We also find an interesting phenomena that under quadratic growth boundary value, the free boundary enjoys better regularity in high dimensions than in two dimension.

Erasure codes are widely used in distributed storage systems to provide fault tolerance. An $[n,k]$ erasure code encodes $k$ data symbols into $n$ coded symbols and distributes them across $n$ storage nodes. Once the code parameters are fixed, the achievable fault tolerance is also fixed. However, the failure rates of storage nodes may vary over time, and dynamically adapting the code parameters to these variations can substantially reduce storage overhead. Motivated by this observation, Maturana and Rashmi introduced convertible codes, which allow an $[n^I,k^I]$ initial code with redundancy $r^I=n^I-k^I$ to be transformed into an $[n^F,k^F]$ final code with redundancy $r^F=n^F-k^F$, while preserving the required code properties. Convertible codes whose initial and final codes are both MDS codes are called MDS convertible codes. They are of particular interest because MDS codes provide the maximum erasure tolerance for a given amount of storage overhead. Several works have established lower bounds and constructions for the bandwidth cost of MDS convertible codes in the split regime. However, the tight bound in the parameter range $r^F<k^F<r^I$ remains unknown. In this work, we establish a family of new rank inequalities for stable MDS convertible codes with linear conversion procedures and derive an improved lower bound on the bandwidth cost consisting of three cases for this remaining range. We prove that the bound is tight by presenting three explicit constructions, one for each case.

We construct distinct high-dimensional mean-zero finite range lattice random walks having pairwise-equal return probabilities for all step counts. The same examples provide pairwise-distinct shapes of discretizations of the standard Laplacian with pairwise-equal density state functions. The main contribution is a reconstruction theorem: to a colored trivalent graph one associates a quantum Clebsch–Gordan polytope, and this association is a full functor, in particular from the polytope one can uniquely recover the original graph. These polytopes appear as moment polytopes of toric degenerations of character varieties (moduli spaces of rank-2 bundles on curves), yielding a combinatorial non-abelian Torelli theorem. In symplectic geometry, the reconstruction theorem implies that monotone Lagrangian tori on odd character varieties associated with these degenerations are pairwise non-Hamiltonian isotopic. These results, and some of the applications, arose from the study of mirror symmetry for moduli spaces of vector bundles, and of the related Laurent phenomenon for mutations of graph potentials.

A planner in a network of strategic agents faces three entangled challenges: the optimum depends on agents' private information, queried agents may misreport to steer the outcome, and exact computation does not scale. We study these challenges in multi-activity network games with heterogeneous private technologies, in which the planner sets non-discriminatory prices. We show that the optimal prices admit a centrality-based decomposition of the welfare kernel: each agent's contribution scales with its squared centrality in a network reweighted by agents' preferences across activities. This decomposition motivates Poll, a polling algorithm in which the planner samples one agent per round, walks briefly through the agent's neighborhood, and updates the price from a local report. From the same decomposition flow three forms of efficiency: computationally, Poll uses significantly fewer operations than exact computation and other distributed methods, requiring up to three orders of magnitude less communication on a real-world network with over 300,000 agents; statistically, its query complexity scales with topology and preference heterogeneity rather than explicitly with population size; and economically, it converges to welfare-maximizing prices while admitting behavior-specific implementations that induce truthful reports and detect adversarial deviations.

Let $H>0$ and $\kappa,\tau\in\mathbb{R}$ satisfy $4H^2+\kappa>0$ and $\kappa\leq 4\tau^2$. We prove that an immersed stable disk of constant mean curvature $H$ and circular boundary in the space $\mathbb{E}(\kappa,\tau)$ is a spherical cap of the canonical rotational $H$-sphere.

Let $I$ be the defining ideal of a general set of $s$ points in projective $N$-space over an algebraically closed field. Harbourne and Huneke conjectured the strengthened symbolic containment $$ I^{(Nr)}\subseteq \mathfrak{m}^{r(N-1)}I^r $$ for every $r\ge 1$. For general points this was known in low dimension and for small numbers of points, while in arbitrary dimension and arbitrary cardinality the best general result was the stable version, valid on one dense Zariski-open set for all sufficiently large $r$. We show that the recent proof of Demailly's conjecture for arbitrary finite point sets by Hà and Sivakumar removes the stability threshold. The key observation is that the strict generic Waldschmidt estimate of Bisui and Nguyẽn allows one to select a single finite symbolic level $m_0$. Equality of the corresponding interpolation number is a Zariski-open condition. Demailly's inequality then propagates this one finite condition to lower bounds for every symbolic power, and a standard regularity criterion converts those bounds into the Harbourne–Huneke containment for every $r$. Consequently, for every $N\ge2$ and every $s\ge1$, one dense Zariski-open family of $s$-point configurations in $\mathbb{P}^N$ satisfies the Harbourne–Huneke containment simultaneously for all $r\ge 1$. The argument also clarifies a quantifier issue in the earlier literature: before the full Demailly theorem, estimates on the generic fibre naturally yielded either all $r$ on a very general set, or one open set only for $r\gg 0$. The new theorem replaces infinitely many symbolic conditions by a single finite interpolation condition.

In this paper, we prove that the combination of dense klt type and $\mathcal{D}$-simplicity property establishes the klt type property.

For perfect state transfer (PST) on unweighted mixed graphs, we classify the normalized transfer times of complete PST classes. A finite set $\Lambda\subset\mathbb R/\mathbb Z$ containing zero occurs at a nonstationary periodic vertex if and only if $\cos(2\pi(x-y))\in\mathbb Q$ for all $x,y\in\Lambda$. Every admissible set has a connected oriented realization. We also classify the possible return phases of oriented realizations at the minimum vertex period. Transfers at rational multiples of the common minimum vertex period partition a complete class into sets of size at most six, or at most three in an oriented graph with return phase $-1$; both bounds are sharp. We construct complete classes of every finite size, including classes in which all transfers between distinct vertices occur at irrational multiples of the period and no switching automorphism maps a class vertex to a distinct class vertex. We also characterize simultaneous realization in connected oriented graphs with prescribed relative minimum vertex periods and return phases. The proof combines an imaginary quadratic field restriction with an unweighted construction that selects the complete target set.

We present an AI-assisted Lean 4 formalization of the Poincaré conjecture. The project began with limited reusable formal infrastructure for the geometric analysis behind the proof. To organize this work, we combined a proof blueprint prepared by mathematicians with explicit milestone statements. These milestones enabled parallel agent work and gave mathematicians clear points to locate blockers and provide effective mathematical guidance. Our analysis identifies the human interventions and organizational choices behind this workflow. The project provides a starting point toward reusable infrastructure for future formalization projects; such infrastructure, once developed, could eventually reduce the cost of verifying mathematical results in geometric analysis.

This paper investigates many-valued generalisations of the classical essence and accident modalities. In two-valued logic, a proposition is essentially true (resp. false) if, whenever it is true (resp. false), it is necessarily true (resp. false); it is accidentally true (resp. false) if it is true (resp. false) but not necessarily so. Many-valued logics provide a natural setting for introducing further modalities of this kind. We focus on Belnap-Dunn's First-Degree Entailment (FDE), a four-valued system that generalises the classical truth values. More precisely, we consider an extension of FDE with Boolean negation and implication. In addition to modalities of essential and accidental truth and falsity, we define modalities of essential and accidental inconsistency and indeterminacy. We present a four-valued S5-based Kripke semantics and cut-free hypersequent calculi for the resulting logics. We then prove semantic and syntactic embedding theorems for these logics into a four-valued version of S5 with necessity and possibility modalities. These embeddings clarify the intended interpretation of the Belnapian essence and accident modalities and yield soundness, completeness, and cut-admissibility results.

Stability of matter is the property that the ground state energy of $N$ charged quantum particles is bounded below by $-CN$, with $C$ independent of $N$. We ask whether this holds for colour-charged fermions interacting with a classical non-abelian gauge field. The field is a dynamical variable on the same footing as the matter, and the energy is minimized over both. The charges are matrices, and every pair of particles has an attractive colour channel. Non-relativistic spinless fermions are stable for every value of the coupling constant $\alpha$. Relativistic spinless fermions and non-relativistic spin-$\frac12$ fermions are stable for small $\alpha$ and unstable for large $\alpha$. For Dirac fermions the choice of positive-energy subspace decides: with that of the full Dirac operator they are stable for small $\alpha$, with the free one unstable for every $\alpha$. Two species of bosons are unstable. In every case the outcome is the same as for electric charges, although essential steps of the abelian proofs fail. The key tool is an electrostatic inequality for matrix-valued charges, uniform in the field, which replaces the uniformly charged sphere of Newton's theorem by the surface charge of a covariant harmonic extension. The stability results hold for every compact gauge group, and most of them for non-smooth fields of finite energy.

In this paper, we study two questions concerning perfect powers in $S$-unit equations and sparse representations of perfect powers. Corvaja-Zannier \cite{corvaja2013finiteness} proved that there are only finitely many odd perfect powers in $\N$ having precisely four non-zero digits in their binary expansion. At first, we prove the finiteness of the set of solutions to the equation \begin{equation*} y^d=1+c_1g^{m_1}+c_2g^{m_2}+c_3g^{m_3}, \quad 0<m_1<m_2<m_3, \end{equation*} where $c_i$ are $S$-units for a suitable set $S$. The proof combines consequences of the Subspace Theorem concerning integer values of analytic series at $S$-unit points with Roth's theorem for number fields. As a second result, we extend the work of Bennett and Bugeaud \cite{bennettbugead2013perfect} by proving that, for every fixed integer base $g\geq 2$, there exists a constant $q_0(g)$ such that, for every prime $q>q_0(g)$, no perfect $q$-th power admits a representation with five non-zero digits in base $g$. The proof relies on explicit lower bounds for linear forms in both Archimedean and non-Archimedean logarithms.

We study the quantum Cayley graphs associated with the Kac-Paljutkin quantum group. We determine which projections give rise to undirected, irreflexive and connected quantum graphs and classify them up to isomorphism. Their spectra and chromatic numbers are computed. For the special case of the projection $p = I_2$ we also compute the associated quantum Cuntz-Krieger algebra.

In this paper, we investigate connections between dependence logics, viewed as a subclass of relating logics, and three-valued logics. More specifically, we identify common features of Epstein's subject-matter semantics and the variable-inclusion conditions characteristic of some infectious many-valued logics. Inspired by Del Cerro and Lugardon's sequent calculi for dependence logics, in which classical connectives are combined with connectives satisfying Epstein-style conditions, we introduce two three-valued dependence-like logics that combine classical conjunction and disjunction with infectious negation and implication. We also provide sound, complete, and cut-free bisequent calculi for these logics.

Many two-sample tests for populations of persistence diagrams assess global differences without identifying the regions of the birth-death plane that contribute to them. We study simultaneous inference for local mean contrasts when the number of available diagrams is fixed. They are differences in expected weighted feature mass within $\ell_\infty$ neighborhoods at several centers and radii. We estimate these contrasts using additive landmark responses. A Gaussian multiplier bootstrap calibrates simultaneous confidence intervals while allowing unequal group covariances. The neighborhoods whose intervals exclude zero form a map with approximate family-wise error control, and selecting a subset of original intervals for display preserves their joint coverage guarantee. On the simultaneous coverage event, every reported neighborhood lies within twice its radius of the support of the mean-measure difference. A geometric result gives sufficient radius conditions for a displaced feature to produce a nonzero contrast. A comparison of sufficient detection thresholds quantifies the tradeoff between reducing the number of tested coordinates and reserving observations for an independent pilot. In simulations with 40 to 120 diagrams per class, the bands achieved 94%-98% simultaneous coverage under both the strict null and equal means with unequal covariances. In the latter setting, a permutation maximum and the pooled-t implementation of the two-stage persistence-image test of Moon and Lazar rejected in up to 32% and 26% of runs, respectively. In the fixed-budget simulations, spending a third of the observations on a pilot to choose landmarks or radii located changes less often than a prespecified grid at a single radius. On the MUTAG benchmark, the localized region concentrates on rings of fused-ring systems, an exploratory reading.

We study Hodge–Tjurina numbers and the Tjurina subspectrum of isolated hypersurface singularities. We relate these invariants to Hodge ideals and microlocal $V$-filtrations and investigate their behavior under Thom–Sebastiani sums. For separated sums, we obtain a filtered defect formula and, in particular, give an affirmative answer to the question raised by Almirón concerning the Thom–Sebastiani property of the Tjurina subspectrum: it holds if and only if the Tjurina number is multiplicative. We also discuss rational sampling, spectral reconstruction, symmetry bounds, and examples illustrating the information captured and lost by integer Hodge–Tjurina data.