Papers

We prove uniform small-data polynomial decay for the real Korteweg--de Vries equation on an interval of length $14\pi$, with homogeneous Dirichlet boundary conditions and zero Neumann datum at the right endpoint. This critical length has three undamped linear modes, including …

An irreversible reaction--diffusion system can have both a positive equilibrium and a stationary state at which the catalyst is absent. For the three-species system studied by Nguyen and Tang, we prove that every strictly positive compatible initial triple converges uniformly …

We consider a thermal insulation energy with a quadratic Dirichlet term and the nonlinear boundary potential associated with radiative heat transfer. Both the heated set and the surrounding domain may vary, with heated volume normalized to that of the unit …

We consider a variational kinetic mean-field game with quadratic acceleration cost, local quadratic congestion, and quadratic terminal congestion on $\T^d\times\R^d$. For a smooth, compactly supported probability datum, the density minimizers at velocity diffusivity $\nu$ and at zero diffusivity are at …

We construct an extreme solitary wave for the bidirectional Whitham system. Its velocity profile is positive, even, continuous, smooth away from its crest and strictly decreasing to zero on the positive half-line. The crest reaches the first critical value of …

Fix an integer \(i\ge2\). We count the distinct Ehrhart functions of convex half-integral polygons with exactly \(i\) interior lattice points and at least two boundary lattice points. The number is \((9i^3+72i^2+175i+106)/2\), as proposed by Bohnert and Springer. We describe the …

We give a bijection from $[2]\times[3]$ onto itself whose graph, extended periodically in four coordinates, has distinct nonzero ordered ordinary difference vectors in every $2\times3\times2\times3$ window. Its order is $6$, whereas the proposed minimum-order restriction of Rubio and Torres would …

Let $G$ be a finite simple graph of girth at least $\ell\ge3$, maximal in its fixed crossing-free embedding on an orientable surface with $g\ge1$ handles or a nonorientable surface with $g\ge1$ crosscaps. We prove that every simple cycle forming the …

We count convex half-integral polygons with a prescribed number of collinear interior lattice points, up to affine unimodular equivalence. For every $i\geq3$, the count is the degree-seven polynomial conjectured by Bohnert and Springer. The proof is computer-assisted: their strip theorem …

Let $X_1,\ldots,X_n$ be independent uniform points in a fixed compact convex polytope $A\subset\R^d$ with nonempty interior, where $d\geq3$. Let $L_n$ be the largest nearest-neighbour distance and let $M_n$ be the connectivity radius of their Euclidean random geometric graph. We prove …

Let $G(n,p)$ be the loop-free random digraph in which every ordered edge is present independently with probability $p$, with reciprocal edges permitted. We prove that its first non-regular path homology over $\Q$ vanishes with probability tending to one if $n^2p^3/\log^4(2n)\to\infty$. …

For geodesic balls of radius \(R\) in the hyperbolic plane of curvature \(-1\), we prove that the least parameter of a self-concordant barrier is of order \(R^2\). Self-concordance is required for arbitrary triples of tangent vectors. The lower bound holds …

Let $A$ be a compact convex polyhedron in $\mathbb R^3$ with nonempty interior, and sample independent uniform points in $A$. We prove that the point with the largest nearest-neighbour distance converges in distribution to normalized surface area on $\partial A$ …

We prove a central limit theorem for the first non-regular path Betti number of a random digraph with independent ordered edges of probability $p_n$, in the full range $np_n\to\infty$ and $n^{2/3}p_n\to0$. This establishes Chaplin's Conjecture B.2. The variance is asymptotic …

We construct a finite three-ample simplicial complex on $32{,}768$ vertices which retracts simplicially onto a four-cycle. Thus the prescribed extension of every subcomplex on at most three vertices is compatible with a nontrivial fundamental group in a finite complex. The …

Does a low codimension-one degree force zero into the spectrum of the normalized upper Laplacian of an infinite simplicial complex? We construct a countable pure three-dimensional complex with maximum triangle degree three whose upper Laplacian has spectrum bounded away from …

Network coding allows intermediate nodes to combine messages, whereas routing only forwards them. The Li-Li conjecture asserts that network coding and fractional routing have the same capacity region in undirected multiple-unicast networks. We give a counterexample to this conjecture and …

Fix a bound on the number of meetings between any two edges of a complete graph drawn in the unit square. We prove that, as the number of vertices tends to infinity, some nonzero mod-two cycle has area tending to …

In the lower multiparameter model of random simplicial complexes, a simplex is retained when all its nonempty subfaces pass independent Bernoulli trials. We prove that, when every trial has the same fixed probability in $(0,1)$, the resulting complex is path-connected …

The proposed coboundary expansion of random balanced Cayley complexes with binary coefficients and logarithmically many group-product layers fails in every fixed dimension $k\ge2$. For $G=(\F)^d$ and every $A\subseteq G$, we prove \[ _{}H^{k-1}(Y_{A,k};) _{j=0}^{(k,d)} dj-|A|. \] The proof maps a …

Starting from $0$ and $1$, repeatedly apply the operation $(a,b)\mapsto(1-\lambda)a+\lambda b$, with a fixed complex parameter $\lambda$. We prove that the resulting set is discrete exactly when $\lambda$ is an algebraic integer whose other conjugates, apart from $\lambda$ and $\overline\lambda$, …

We construct a closed connected PL four-manifold with odd Euler characteristic that admits a finite flag-no-square triangulation. Here flag means that every clique is a simplex, and no-square excludes induced four-cycles in the graph. The construction identifies antipodal points on …

Starting from $0$ and $1$, repeatedly apply the operation $(a,b)\mapsto(1-\lambda)a+\lambda b$, with a fixed complex parameter $\lambda$. We prove that the generated set has bounded holes in the real line for every real $\lambda\notin[0,1]$, and in the complex plane for …

We introduce the Enhanced Neumann Pólya inequality, which is stronger than Pólya's conjecture, and verify it for many classes of domains, including strip-tiling domains, Euclidean balls, thin product domains, and certain higher-dimensional tiling domains. We develop a geometric removal approach …

Fix \(d\geq3\), \(1\leq k\leq d-2\), \(p\geq k+2\), and a fatness parameter \(\rho\geq1\). We prove that a finite family of \(\rho\)-fat compact convex bodies in \(\R^d\) can be intersected by a bounded number of affine \(k\)-flats whenever every \(p\) members contain …

A lattice polytope is reflexive if it contains the origin in its interior and its polar is also a lattice polytope. We construct a $45$-dimensional reflexive lattice polytope with $2^{32}\binom{40}{8}+5$ facets, exceeding $6^{45/2}$. This disproves the unrestricted facet bound conjectured …

For every integer \(k\geq1\), an arrangement of \(3k\) disjoint red straight-line segments and \(3k\) disjoint blue straight-line segments, with every red--blue pair crossing in both relative interiors, contains \(k\) segments of each color forming an embedded rectangular grid. This answers …

The center of a largest centrally symmetric subset of a convex body need not satisfy the usual dimension bound on asymmetry. We compute this center for a square pyramid in $\mathbb R^3$ and show that reflection about it, followed by …

The persistent homology of a finite simplicial complex is determined by a canonical grid whose coordinate values occur in minimal generators and relations. We prove that the number of values in each coordinate is bounded by the homological Morse lower …

We prove that the space of all finite persistence diagrams, with no common cardinality or coordinate bound, does not coarsely embed into Hilbert space for any Wasserstein exponent \(1\le p\le2\). The metric uses the usual \(\ell^\infty\) ground distance and matching …

For every integer \(k\geq3\), we determine the homotopy type of the closed Vietoris--Rips complex of the \(2k\) by \(2k\) periodic Manhattan grid at radius \(k\). It is \(S^3\vee\bigvee^{4k}S^{2k-1}\), as conjectured by Adams, Adetowubo, Barriga-Acosta, Feng and Sterling. Adjacent pairs of …

For a finite connected simplicial complex, consider all simplex-wise filters whose ordinary persistent homology has one infinite degree-zero bar born at time 1 and all positive-degree bars born at time 2, with no finite bars. We prove that this fiber …

We study the finite clique complexes of sequential affine preferential-attachment graphs. For every $q\ge2$, $m\ge2q$ and $\delta>-m$ with $x=(m+\delta)/(2m+\delta)<1/(2q)$, the $q$th rational Betti number, divided by its expectation, converges to a nondegenerate random variable. We identify its law as a …

We resolve the long-standing non-extinction problem for Brownian Fleming-Viot systems in bounded domains. For every fixed N >= 3, we construct a bounded connected domain in R3 in which branching explosion occurs with positive probability before every prescribed positive time, …

We compute the fourth integral Betti number of the closed radius-five Vietoris--Rips complex on the thirteen-by-thirteen torus grid with its periodic Manhattan metric. It is two, giving a counterexample to the integer-rank formula in AIM Problem 531, which predicts twenty-six …

Let $S$ be the closed oriented surface of genus two. The aspherical four-manifold $M=S\times S$ has a rational fundamental class of positive homological $\ell^1$ seminorm, but every nonzero $u\in H^4(M;\Q)$ satisfies $\wt(1\times u-u\times1)\le2$. This gives a negative answer to Neofytidis's …

We study optimal transport with a non-convex polyhedral target in $\mathbb{R}^n$. The closure of the singular set is locally a smooth hypersurface outside a relatively closed set of Hausdorff dimension at most $n-2$. For partial transport between domains separated by …

Let \(M\) and \(N\) be closed, connected, oriented aspherical manifolds of the same dimension \(n\geq 3\). We prove that their connected sum has ordinary unreduced topological complexity \(2n+1\). Thus joining two aspherical summands always attains the dimensional upper bound, although …

We construct a closed connected oriented flat five-manifold whose fundamental group has centre of rank two and whose unreduced topological complexity is at least ten. Since its universal cover is Euclidean space, this gives a negative answer to the proposed …

For every \(n\geq3\), we identify the homotopy type of the strict Vietoris--Rips complex of the unit round sphere on an interval immediately above the regular-simplex threshold \(r_n=\arccos(-1/(n+1))\). It is the join of \(S^n\) with \(\SO(n+1)/A_{n+2}\), where \(A_{n+2}\) acts by the …

We consider the topology-preserving cube dynamics in the higher-dimensional extension of the Ising-disk model. The states are all finite regular contractible unions of lattice cubes containing a fixed cube. A removal must admit a strong deformation retraction that keeps the …

The affine projection from a Euclidean Vietoris--Rips complex to its shadow need not be injective on fundamental groups. We give six rational points in three-dimensional space whose Rips complex retracts onto a four-cycle, while its shadow is the union of …

We determine the exponential rate at which a topology-preserving planar Ising disk reaches a large area during one excursion from a single square. For addition rate $\beta$, deletion rate $1$, and every $0<\beta<1/\kappa$, the excursion maximum $\xi$ satisfies $\lim_{T\to\infty}\Prob(\xi\ge T)^{1/T}=\kappa\beta$, …

The real flat norm decomposes a current into a residual and the boundary of a filling, minimizing their total mass. We prove that every compactly supported integral $d$-current in $\R^{d+1}$ has an integral optimal decomposition, including when its boundary is …

Let $X_N$ be a normalized Gaussian unitary ensemble matrix of size $N^2$. We place the same realization on two adjacent pairs of tensor factors, as $X_N\otimes I_N$ and $I_N\otimes X_N$ on $(\C^N)^{\otimes3}$. For every fixed complex noncommutative polynomial, its operator …

We prove that two Dirichlet boundary controls can drive a one-dimensional two-phase Stefan system to zero temperature and any prescribed interior interface position in every positive time. The initial temperatures may be arbitrary real smooth functions compactly supported in their …

For three-player fair gambler's ruin, we prove that the probability of each elimination order differs from its Brownian scaling limit by $O((nN)^{-4})$ when fixed positive capitals with total $N$ are multiplied by $n$. This is Conjecture 4.2(b) of Diaconis and …

We prove that the covariance of finitely many Ising replicas conditioned on fixed magnetizations has a bound independent of the number of replicas, the conserved totals and all replica-dependent external fields, for every fixed interaction matrix. The constraints may conserve …

We prove a terminal strong convergence bound for the GfBm\_EM Euler integrator of a scalar stochastic differential equation driven by multiplicative fractional Brownian noise in the Wick--Itô--Skorohod sense. For every Hurst parameter \(H\in(0,1)\) and every autonomous globally Lipschitz drift, the …

Independently delete each vertex of the nearest-neighbor lattice $\Z^d$ with probability $p$. Two players, knowing the entire resulting board, move a token alternately to an undeleted neighbor that has never been visited; a player with no legal move loses. A …