Papers

The positivity of solutions to subdiffusion equations has been widely studied, mainly in the context of homogeneous problems for single equations. In this article, we fill the missing strict positivity for inhomogeneous subdiffusion equations with nonnegative and nontrivial source terms by connecting Green's functions for fractional and classical diffusion equations via special functions. As a direct application, we further investigate the strict positivity property of coupled subdiffusion systems with nonnegative and partially nontrivial initial values or sources. Under suitable cooperativeness and connectivity conditions, the strict positivity turns out to propagate not only in time but also across different components of the system, reflecting the intrinsic interactions induced by the coupling structure. These results provide a unified framework for understanding positivity properties of both scalar and coupled subdiffusion equations, offering new insights beyond the classical maximum principle approach.

Let $G = (X, Y; E)$ be a bipartite graph with parts $X$ and $Y$ where $|X|=m$ and $|Y|=n$. We show that every bipartite graph with more than $mn - D(m,n)$ edges has positive Lin–Lu–Yau curvature, where $D(m,n)=m-2+\lceil{\frac {n}{2}\rceil} \text{ if $n\geq 2m$}, \mbox{and} \ n-1 \text{ if $m\leq n< 2m$}.$ We also show that every bipartite graph of order $m+n$ with $m \geq n$ and minimum degree at least $\min\{n, \lfloor{\frac{m+n}{3}\rfloor}+1\}$ has positive Lin–Lu–Yau curvature. Both bounds are sharp. Meanwhile probabilistically we can relax the edge density conditions in above results. It is shown that relatively dense random bipartite graph is positively curved. All of our proofs are based on a new formula for Lin–Lu–Yau curvature of bipartite graphs.

This paper studies a multi-player non-zero-sum optimal stopping game driven by a spider Brownian motion. We study a specific, tractable family of games and the focus is on the existence and uniqueness of subgame-perfect Nash equilibria (SPNE). We begin by analyzing the two-player setting, in which the underlying process is a skew Brownian motion, and establish sufficient conditions for the existence of an SPNE. We then derive sufficient conditions for the uniqueness of the SPNE in the symmetric case, (where the skew Brownian motion reduces to a standard Brownian motion and both players have the same payoff structures). We also give examples which show that there can be multiple SPNE or no SPNE. We then extend the framework to the multi-player setting and provide sufficient conditions for the existence of an SPNE in this more general setting with a spider Brownian motion.

We consider an irreducible, supercritical multi-type general branching process (Crump-Mode-Jagers process) with finite type space, counted with random characteristic. We prove that, under certain second moment assumptions, a central limit theorem holds for the process. Our result extends the single-type central limit theorem obtained recently in [Ann. Probab. 52 (2024), no. 4, 1538-1606] and unifies various limiting results for specific branching processes.

In 1976, George Spencer-Brown announced a proof of the four color theorem, using operations on Tait colorings for trivalent plane graphs. In subsequent work he formulated these operations in terms of an algorithm that he called a parity-pass and claimed that when the parity pass algorithm is performed on a non-polar pentagon region, it necessarily terminates in an edge coloring that is extendable to the entire graph. We provide here a counterexample to show that this claim is false. We then raise questions related to the existence of this sort of counterexample.

The biharmonic equation is a prototypical fourth-order partial differential equation whose high-dimensional discretization suffers from rapidly growing degrees of freedom and severe ill-conditioning. We develop QSVT–VTAA quantum linear-system algorithms by constructing explicit block-encodings tailored to periodic, simply supported, and Dirichlet–Neumann boundary conditions. For periodic and simply supported problems, Fourier and sine-transform diagonalizations yield augmented Poisson systems with the condition-number scaling of a second-order operator. For Dirichlet–Neumann problems, we introduce a second-order boundary-corrected finite-difference discretization, establish mesh-independent stability, and construct an explicit block-encoding of the resulting nonsymmetric matrix. We also formulate a coupled-Laplace system with additional boundary unknowns and characterize its complexity in terms of the condition number of the complete augmented matrix. The analysis covers discretization error, block-encoding normalization, gate complexity, and solution extraction under an amplitude-input and quantum-state-output model. Numerical experiments validate the proposed discretizations and the corresponding linear solves.

Our earlier results on effective codescent morphisms of Hausdorff topological spaces are strengthened and complemented. In particular, it is proved that any embedding $p:B\rightarrowtail E$ with compact $B$ and normal $E$ is an effective codescent morphism in the category of Hausdorff topological spaces.

We study non-compact Ricci solitons of finite volume whose potential vector field has constant length. Under the assumptions that the scalar curvature is constant along the integral curves of the potential field and that a natural divergence term is integrable on the unit tangent bundle, we prove that such Ricci solitons are necessarily trivial. As applications, we characterise $3$-dimensional Ricci solitons of constant length and of finite volume on almost $\alpha$-cosymplectic manifolds. In fact, we show that these spaces are indeed compact quotients of Euclidean spaces. We also study complete, non-compact and finite volume contact metric manifolds in dimension $3$ admitting a Reeb vector field as a Ricci soliton. We show that it is a compact spherical space form. We include some examples to illustrate the necessity of the finite-volume and constant-length assumptions.

We give a distributional definition of the $\frac{1}{2}$-fractional Laplacian for smooth functions with power-type singularities at the origin, including negative monomials and certain holomorphic functions with isolated singularities. The construction is based on a suitable space of test functions related to the Lizorkin space of test functions, for which both the test functions and their fractional Laplacians belong to the Schwarz class and vanish to infinite order at the origin, and thus compensate for any singularity of power type at the origin. This allows the $\frac{1}{2}$-fractional Laplacian to be defined by duality without requiring the underlying function to satisfy the usual integrability assumptions. The considered distributional framework is shown to be invariant under the $\frac{1}{2}$-fractional Laplacian, yielding a natural semigroup property. By applying the construction term by term to Laurent series we obtain new series that, remarkably, involve Jacobi polynomials. With this procedure we define the fractional Laplacian for a class of holomorphic functions and derive several applications, with particular emphasis on the recently developed theory of the quaternionic fine structures of the spectral theory on the $S$-spectrum, describing the functional calculi extending the classical holomorphic functional calculus in the various classes of holomorphic-type functions arising from the Fueter-Sce-Qian extension theorem.

We prove that there exist positive constants $A$ and $V$ such that the high-dimensional critical percolation $k$-point function is given by \[ T_{p_c}(x_1,x_2,\ldots,x_{k}) \sim V^{k-2} A^{2k-3} \sum_{T\in \mathsf{Tr}(k)} \sum_{\substack{\Phi:V(T)\to \mathbb{Z}^d \\ \Phi(i)=x_i \forall 1\leq i \leq k}} \prod_{\substack{u,v\in V(T)\\u\sim v}}G(\Phi(u),\Phi(v)) \] as $\min_{i\neq j}\|x_i-x_j\|\to \infty$, where $\mathsf{Tr}(k)$ is a set of isomorphism class representatives of trees with $k$ labelled leaves $\{1,\ldots,k\}$ and unlabelled internal vertices all of which have degree $3$ and $G$ is the lattice Green's function. This verifies a conjecture of Aizenman and Newman (1984) subject to the usual perturbative conditions needed for convergence of the lace expansion. It follows from this theorem that the law of the cluster of the origin, considered as the counting measure on its range, converges under rescaling to the canonical measure of the integrated super-Brownian excursion. By computing the asymptotics of various more complicated variations on the $k$-point function, we also prove the stronger result that the cluster converges as an embedded metric-measure space to the continuum random tree equipped with its Brownian embedding into $\mathbb{R}^d$. This convergence holds simultaneously with respect to the chemical distance, pivotal distance, and resistance distance on the cluster, which we prove are asymptotic to constant multiples of each other. This resolves conjectures of Hara and Slade (1998) and van der Hofstad and Slade (2003). As a corollary of our results we prove that there exists a positive constant $C$ such that $\mathbb{P}_{p_c}(0\leftrightarrow \mathbb{Z}^d \setminus [-r,r]^d)\sim C r^{-2}$, answering a question of Heydenreich and van der Hofstad (2017).

The principle of maximum entropy provides a fundamental framework for characterizing typical structures of large random networks subject to observable constraints. In their pioneering numerical experiments \cite{radin2014asymptotics}, Radin, Ren, and Sadun conjectured that entropy-maximizing graphons satisfying subgraph density constraints are stochastic block models a conjecture we term the RRS conjecture. While several special cases have been proven for single-relation graphs with specific constraint families, the general problem has remained open, particularly for multi-relational networks. We resolve the RRS conjecture for constrained multi-relational graphons in the non-extremal regime, proving that entropy-maximizing solutions are step functions with finitely many blocks under the condition the subgraph density constraints are analytically independent and for almost all feasible combinations of sufficient statistics. Our proof employs a differential geometric technique to study solutions of constrained optimization problems in function space via functions with a finite parametrization (step functions). The two cornerstones of this work are: the generalization of subgraph density notion to $h$-subgraph density and the proof that manifolds that define the constrained region for the solutions maintain topological stability without developing new connected components under refinement. Together, these enable proving that no new global optima emerge in higher-dimensional spaces.

We present results around an isoperimetric problem built on polyforms: What is the biggest enclosed area one can build using polyforms in each of the three plane tessellations? We give challenges to the readers and present Shimauchi's proof of the biggest area a fence made of pentominoes can enclose. A translation of the instance using integer linear programming is also given. A companion web app is available to test some of the challenges we propose and for activities, and we included extra pages with a cutout handout of the board and pieces of the puzzles, so that you can print and cut them to read this paper hands on.

Multiphoton light-matter interactions, in which a bosonic mode exchanges $k$ excitations at a time with a quantum system, are a source of genuine nonlinearity in quantum optics and are increasingly accessible experimentally. Here we study the class of operators $H = H_{\rm mat}\otimes I + I\otimes\omega a^\ast a + \Sigma\otimes(a^\ast)^k + \Sigma^\ast\otimes a^k$ on $\mathcal{H}\otimes L^2(\mathbb{R})$, coupling a single bosonic mode to an arbitrary matter system through a bounded operator $\Sigma$. When $\Sigma$ is normal and nonzero, we prove that $H$ is self-adjoint if and only if $k\leq2$; for $k\geq3$ we compute the deficiency indices, parametrise all self-adjoint extensions, and show that every extension has purely discrete spectrum whenever the matter system is finite-dimensional. Our analysis rests on a block Jacobi decomposition paired with a suitable unitary transformation depending on the polar decomposition of $\Sigma$. The normality of $\Sigma$ is optimal: a $k$-photon Jaynes-Cummings model, with non-normal coupling, remains self-adjoint for every $k$. We illustrate our results on the $k$-photon Rabi and Dicke models.

For the heat equations driven by a stochastic measure on $[-L, L]$ with the Dirichlet boundary condition, we prove that solutions tend to the solution of the heat equations defined on ${\mathbb R}$ as $L\to \infty$. The estimate of the convergence rate is obtained. For a stochastic measure, we assume the $\sigma$-additivity in probability only.

We study the clique complex $X_G$ of a finite group: a simplicial complex whose faces are in one-to-one correspondence with the elements of the group, and arise via a decomposition of group elements into $p$-elements for primes $p$. The $1$-skeleton of this complex then has the classical Gruenberg-Kegel graph $\Gamma_G$ as a quotient and a finite simple group is determined up to isomorphism by its clique complex. The clique complex can also be endowed with a combinatorial sheaf — called the order sheaf — and the resulting homology can be computed for an arbitrary simplicial complex in terms of the homology of the links of vertices. The homology of the clique complex with coefficients in this order sheaf is then computed for some special classes of groups, in particular those for which the centralizers of elements are nilpotent.

Let $f:\mathbb{Z}\to\mathbb{C}$ be a multiplicative function. Assume there exist integers $a>1$ and $d>1$ such that $(a,d)=1$ and let $\mathcal{P}_{a,d}=\{a+kd:k\in\mathbb{Z}\}$. Under mild extra conditions on $d$ and $f(a)$, we prove that $f(n)=n\chi(n)$ for all $n$ outside an explicit exceptional set depending on $d$, and some Dirichlet character $\chi$.

We study small-noise asymptotics for controlled functional differential equations driven by additive Gaussian noise. The Gaussian noise is modeled on an abstract Wiener space, covering both classical Brownian perturbations and non-Markovian perturbations such as fractional Brownian motion. The drift coefficient is assumed to be non-anticipative, Lipschitz continuous in the state path, and of linear growth. For bounded uniformly continuous cost functionals, we prove game-theoretic lower and upper bounds for the small-noise logarithmic value functions and identify their limit whenever the associated deterministic zero-sum game has a value. In the limiting game, one player chooses the drift control, while the other selects a Cameron–Martin shift of the Gaussian noise, penalized by the corresponding quadratic energy cost. We further provide sufficient Fan-type convexity and concavity conditions under which the game has a value, thereby obtaining a full small-noise Laplace principle. The proof combines the Boué–Dupuis variational representation on abstract Wiener spaces with pathwise stability of the controlled solution map and adapted finite-dimensional approximations of Cameron–Martin shifts.

We study Langevin-based methods for non-convex optimization under smoothness and dissipativity assumptions, focusing on non-asymptotic expected excess-risk bounds rather than sampling guarantees for the full target distribution. A main methodological message is that relative-entropy sampling guarantees can be converted directly into expected objective-value guarantees, without passing through Wasserstein distance. This direct KL-to-objective route yields sharper dependence on the Log-Sobolev constant, which may scale exponentially with inverse temperature and dimension in non-convex problems. We first analyze the Unadjusted Langevin Algorithm with exact gradients and derive explicit bounds in terms of inverse temperature, dimension, stepsize, smoothness and dissipativity parameters, and the Log-Sobolev constant. We then extend the result to inexact ULA, allowing biased and stochastic gradient surrogates whose mean-square error grows at most quadratically in the state. This framework covers stochastic gradients and zeroth-order estimators based on function evaluations. We show that Gaussian and spherical finite-difference estimators fit the theory and obtain explicit function-evaluation complexity bounds. To the best of our knowledge, these are the first non-asymptotic global non-convex optimization complexity bounds for zeroth-order ULA. We also provide numerical experiments illustrating the spherical zeroth-order scheme.

In 2011/12, Ruskey, Sawada, Stevens and Williams showed that a binary universal sequence can be constructed containing all $n$-tuples of Hamming weight either $r$ or $r+1$ for every possible $r$, and no others. In this paper we examine extensions of this result that require choosing a weight-like function applying to $n$-tuples with symbols taken from an alphabet of arbitrary (finite) size. We consider three possible such weight functions, and in each case establish an analogous result to that of Ruskey, Sawada, Stevens and Williams.

In this paper, we study the congruences for the coefficients of MacMahon-like series introduced by Bringmann–Craig–van Ittersum–Pandey. We prove all the congruences conjectured in their paper. In addition, we establish several new congruence relations of a similar type.

Starting from a positive integer $n$ and no a priori information about the primes above it, we construct a polynomial quotient ring that recovers exactly the primes in $(n,2n]$ from a single modular exponentiation. The primes occur simultaneously as the nonzero monomial degrees of the resulting polynomial remainder, and each coefficient independently certifies its corresponding prime through its additive order. When $n=p_k$ is prime, the least nonzero degree is $p_{k+1}$. Thus the next prime is recovered from the preceding prime alone, without using the index $k$, the prime-counting function, a prime table, nor any primality tests. We develop the underlying ring structure, give equivalent annihilator and quotient formulations, extend the result to shorter intervals, and provide a SageMath implementation.

We introduce the so-called fractional analog of the Kontorovich-Lebedev transform, modifying its kernel, which is the Macdonald function $K_\nu(z)$. Properties of this kernel as well as mapping properties of the new index transform are investigated. The inversion formula in suitable spaces of functions is established.

We study generalizations of the Scorza correspondence associated with a general $(C,\eta)$ in the moduli space of even spin curves $S_g^+$. We first give a new proof of the smoothness of the classical Scorza curve, originally established by Farkas-Verra. We then introduce higher order analogues of the Scorza correspondence and investigate their geometry. In particular, we compute their classes in the Néron-Severi group and study their singularities and genera. Finally, we consider a higher-dimensional version of the construction and focus on the threefold case, where we describe its class and prove a smoothness result outside a naturally defined degeneracy locus.

We develop criteria for collision and non-collision of reversible infinitely many interacting diffusion processes in the real line. The approach is potential-theoretic and is based on capacity estimates for symmetric Dirichlet forms on the configuration space. Our main results are model-independent in the sense that no determinantal or Pfaffian structure, prescribed interaction potential, or explicit labelled stochastic differential equation is required. The non-collision criterion involves only the second and third correlation functions of the reversible measure, whereas the collision criterion is based on a local lower bound for a finite-volume conditional density of the reversible measure. As an application, we identify the sharp collision threshold for the diffusion associated with the $\mathsf{Sine}_\beta$-symmetric Dirichlet form: the collision set is polar if and only if $\beta\ge 1$. This provides an infinite-particle counterpart of the classical collision threshold for the finite-particle Dyson Brownian motion with inverse temperature $\beta$.

We introduce and analyze a fluctuating homogeneous Landau equation with regularised thermal noise. The model is motivated by the nonlocal gradient flow structure of the deterministic Landau equation, the fluctuation–dissipation principle, and the covariance of the martingale fluctuations of a Kac-like conservative Landau particle system. The noise is written in Landau-divergence form, is antisymmetric in the pair of velocities, and is interpreted in the Stratonovich sense after introducing a velocity correlation. To handle the vacuum singularity of the square-root mobility and the nonlocal Stratonovich-to-Itô correction, we replace the mobility by a regular coefficient. For moderately soft potentials, we prove the existence of probabilistic weak solutions to the regularised fluctuating homogeneous Landau equation. The proof is based on a three-level approximation scheme combining Galerkin approximations, coefficient regularisations, artificial diffusion, and compactness in both $L^2$ and $L^1$ frameworks. The solutions satisfy mass conservation, an energy inequality, and the entropy dissipation estimate. Finally, for a special class of admissible noise bases satisfying a tangential divergence-free condition, we obtain a refined entropy inequality in which the expected entropy is non-increasing relative to the initial entropy.

We introduce a stochastic model for the papal conclave in which $n$ cardinals vote repeatedly among themselves until one cardinal receives all the votes. In each round, the probability that a cardinal votes for a given candidate is proportional to the $\alpha$-th power of that candidate's vote count in the preceding round. For $\alpha=1$, the model reduces to the Wright-Fisher model and is dual to Kingman's n-coalescent. We reveal a sharp transition in the absorption time $\mathcal{T}$ at $\alpha=1$. It was known that when $\alpha=1$, $\mathcal{T}$ is typically of order $n$. We prove that for $\alpha>1$, it drops to order $\textit{loglog n.}$ In contrast, for $\alpha<1$, $\mathcal{T}$ is typically at least $\exp(\Omega(n))$. We also prove a sharp phase transition in the identity of the winner when $\alpha>1$. For every positive integer $k$, if $2^{1/k}<\alpha<2^{1/(k-1)}$ (where we write $2^{1/0} = +\infty$), with probability tending to 1 as $n\to\infty$, the eventual winner is the unique leader after round $k$. These results show that reinforced voting processes reach consensus remarkably quickly even for large electorates.

We introduce a novel coupling of Rulkov neural maps, proposing a heuristic biological interpretation for the transition to non-small values of the perturbations acting on the slow variables. We analytically prove that the coupling of two identical neurons preserves the emergence of Devaney chaos through the existence of a generalized snap-back repeller, provided that a snap back repeller exists for the original system. We present numerical simulations for the coupling of two different neurons showing the arising of a potential global strange attractor, whose fractal structure is strongly suggested by the computation of a non-integer Kaplan-Yorke dimension. Furthermore, we perform standard numerical studies concerning time series, Lyapunov exponents spectra, bifurcation diagrams and basins of attraction. Finally, we briefly propose a generalization of the coupling to an arbitrary number of neurons.

We prove uniqueness of $q$-rational numbers up to conjugacy, using character varieties. Despite the uniqueness, we exhibit a two-parameter family of deformations of rationals with a modular symmetry. We prove that there are exactly two deformations which deliver the usual $q$-integers: the original $q$-rationals defined by Morier-Genoud and Ovsienko, and another new one. Although the new family can be obtained by conjugacy from the old one, new positivity properties appear. In addition, this new family provides a direct computation of the Jones polynomial of rational knots.

We consider a model of thermal explosion in reactive turbulent jets introduced in [GHH], which falls into the general class of Gelfand-type problems. Our main result closes the gap in [GMN], where the question of finding an explicit sharp bound for the $L^{\infty}$ norm of the extremal solution in the asymptotic limit of strong flows was left partially unresolved. Namely, we obtain the matching upper and lower bounds under no additional assumptions on the nonlinear reaction rate.

We solve the isomorphism problem for graph products of groups. We give a generating set for the automorphism group of a graph product of groups, generalizing the one given by Laurence and Servatius for right-angled Artin groups.