Papers

In their 2003 article in the Journal of the American Mathematical Society, Eisenbud, Schreyer and Weyman asked whether every embedded projective variety carries an Ulrich sheaf. In 2017 Beauville proposed a numerical route toward a surface with no Ulrich bundles: in Picard rank one, existence forces $H^2\ge K_S^2-8\chi(\mathcal O_S)$, suggesting a search near the Bogomolov–Miyaoka–Yau boundary. We show that the Picard-rank-one hypothesis is not needed for the obstruction: a rank-independent Bogomolov–Hodge argument gives the same inequality on every smooth polarized surface. Using additional Neron–Severi directions on Hirzebruch–Kummer resolutions, the exponent-$3$ Hesse surface admits a very ample class $H=4A-E$ with $H^2=7\cdot3^9<16\cdot3^9=K_Y^2-8\chi(\mathcal O_Y)$, hence a smooth counterexample to the Eisenbud–Schreyer–Weyman problem in its standard formulation. Consequently its Chow form has no ESW-type linear determinantal representation arising from an Ulrich sheaf on the embedded surface. Moreover, for every $n\ge3$ the Hesse pair $(Y_n,4A_n-E_n)$ is a counterexample, the surfaces are pairwise non-isomorphic, and $H_n^2/\sigma(Y_n)=7/(3n^2-11)\to0$, while $K_{Y_n}^2/\chi(\mathcal O_{Y_n})\to60/7\approx8.5714$. A general arrangement-theoretic Rees-algebra/Segre mechanism yields further infinite families.

In homage to Hofstadter, we call a compact, citable specification of definitions, theorem, proof mechanism, and examples a mathematical koan when a full paper can be reconstructed and verified from it, including in reader-specific forms generated by AI. Every AI-solvable problem is itself a koan, although reconstruction need not be cheap; the present article is an expansion of one. We introduce Cartan convexity for self-adjoint free functions: locally, such a function agrees with a locally bounded matrix-convex free function. If the variable set has cardinality $\tau$ and $\lambda=\max\{\tau,\aleph_0\}$, every bounded real free set has a universal direct sum on a Hilbert space of dimension at most $2^\lambda$; every point of the set is a reducing summand of an amplification of this sum. Applying the noncommutative Kraus–butterfly theorem at that universal point yields an extension to an open matrix-convex neighborhood of the noncommutative convex hull. For normal affine pencils, the extension domain also contains the bounded strong closure of the hull. We prove an analogous theorem for a graph embedding $\Gamma$. A local convex lift through $\Gamma$ extends to a neighborhood of $\Gamma^{-1}(\operatorname{co}_{\mathrm{nc}}\Gamma(K))$ and admits a butterfly realization in the $\Gamma$-coordinates. We distinguish this lift condition from intrinsic $\Gamma$-convexity. For $\Gamma(x,y)=(x,y,y^2)$, the intrinsically $\Gamma$-affine polynomial $xy+yx$ has no convex lift germ at the origin, whereas a single quadratic coordinate suffices to lift every uniformly real analytic germ.

An explicit formula for the prime-counting function $\pi(x)$, usually attributed to Riemann and von Mangoldt, is prominently stated as the equation $\pi(x)=R(x)-\sum_\rho R(x^\rho)$, where the sum runs over all zeros $\rho$ of the Riemann $\zeta$-function, the non-trivial ones being ordered by increasing absolute value of their imaginary parts and counted with multiplicity. This particularly entails the claim that the partial sums over the non-trivial zeros, $\Sigma R_T(x):=\sum_{0<|\Im m(\rho)|\le T} R(x^\rho)$ converge as ${T\to\infty}$. Writing $\Theta:=\sup\{\Re e(\rho):\ \zeta(\rho)=0,\ 0<\Re e(\rho)<1\},$ for what has recently been called “Riemann's constant”, we prove that, for every fixed $x>1$ and every $\theta<\Theta$, the sums $\Sigma R_T(x)$ are not $O(T^\theta)$. As a consequence, $\limsup_{{T\to\infty}}|\Sigma R_T(x)|=\infty$ and $\sum_\rho R(x^\rho)$ diverges. We conclude the paper by showing that an adapted, but simpler strategy also gives the analogous result – and hence divergence – for the contribution of the trivial zeros to $\sum_\rho R(x^\rho)$. In both cases the exponent $\Theta$ is proved to be sharp.

We study insensitizing control problems for a class of coupled linear stochastic parabolic systems. We establish the existence of controls such that a sentinel functional, involving localized observations of the state variables and their spatial gradients, is insensitive to small perturbations of the null initial data. We first reformulate the insensitizing control problem as a null controllability problem for a coupled forward–backward stochastic parabolic system, in which the observation terms induce both zeroth- and second-order coupling terms. By duality, the analysis is reduced to an observability inequality for the corresponding adjoint system. The main analytical contribution is the derivation of new global Carleman estimates for coupled stochastic parabolic systems with zeroth- and second-order coupling terms, under suitable geometric assumptions on the control and observation regions. These estimates yield the required observability inequalities and, consequently, the existence of insensitizing controls. Furthermore, depending on the value of a weighting parameter $\beta\in[0,1]$, which determines the relative contributions of the two state components to the sentinel functional, we consider two cases. If $\beta\in\{0,1\}$, the sentinel functional depends on only one state component, and a single localized control acting in the drift of the first equation is sufficient. In contrast, if $\beta\in(0,1)$, both state components contribute to the sentinel functional, and two localized controls acting in the drift terms of the two equations are sufficient. Moreover, the control strategy in this paper involves two additional controls acting throughout the diffusion terms.

The study of evolution in temporally fluctuating environments often relies on the analysis of Lyapunov exponents, which quantify the exponential growth of populations. However, when model parameters depend on a stochastic process, calculating the selection gradient, a key tool for predicting the evolution of phenotypic traits, becomes a mathematical challenge. While the periodic case has been resolved, a general approach for random environments remains to be developed. This article proposes a rigorous method to: Establish the differentiability of the leading Lyapunov exponent with respect to a parameter, providing an explicit integral formula for its derivative. Develop a numerical algorithm to approximate the derivative by solving an extended differential equation. Apply these results to the analysis of mutant invasion in a resident population at equilibrium, identifying the selection gradient as the derivative of the top Lyapunov exponent.

Let $(M,\mathcal{D},g)$ be a smooth equiregular sub-Riemannian manifold equipped with a smooth positive measure $\mu$. We study the small-scale limit of the metric-ball mean-value operator \[ A_hf(x)=\frac{1}{h^{2}\mu(B(x,h)) }\int_{B(x,h)}(f(q)-f(x))\, d\mu(q). \] Exact homogeneity yields the pointwise limit on Carnot groups. On a general equiregular manifold, convergence for every smooth test function is equivalent to convergence of the rescaled horizontal first moments of metric balls in first-kind privileged coordinates. This criterion is independent of $\mu$; when it holds, the principal symbol is determined by the normalized second-moment tensor of the tangent unit ball, and the drift satisfies an explicit change-of-measure formula. In step at most two, we verify the criterion by combining a real-analytic finite-jet reduction with tame integration, which rules out oscillation of the normalized moments. We also compute the limit on Lie groups and prove unconditional distributional convergence of the volume-weighted operators on every equiregular manifold.

Using the newly developed $3$d quantum trace map, we compile more evidence for the length conjecture, a refinement of the all-order volume conjecture that incorporates insertions of additional links in the skein module of the knot complement. In particular, we prove that the length conjecture holds up to first order for an infinite family of twist knots. Along the way, we form a conjecture that the $3$d quantum trace map behaves naturally with respect to Dehn filling.

The Erdős–Dushnik–Miller theorem states that for every aleph $\kappa$, \[ \kappa\to(\kappa,\omega); \] that is, every coloring $c:[\kappa]^2\to2$ has either a $0$-homogeneous set of cardinality $\kappa$ or a $1$-homogeneous set of cardinality $\omega$. In this article, we present a purely combinatorial proof of this theorem in $\mathsf{ZF}$ (i.e., Zermelo–Fraenkel set theory without the axiom of choice), avoiding any metamathematical considerations.

An Euler circuit of a graph is a closed path that visits every edge of the graph exactly once. Euler circuit and circuit decomposition problems can also be formulated for higher dimensional simplicial complexes. An Euler k-circuit in K is a cyclic sequence of vertices v_1...v_n such that every k+1 adjacent terms { v_i,v_{i+1},...,v_{i+k} } (indexed modulo n) form a k-simplex, and every k-simplex of K appears exactly once in the sequence v_1v_2...v_n(v_1v_2...v_k). We investigate the 2-circuit decomposition problem for triangulated closed compact surfaces. For an orientable triangulated surface we use interior angles of paths to define an obstruction that lives in the first cohomology of the surface. It vanishes if and only if the surface has a 2-circuit decomposition. We also show that a non-orientable triangulated surface has a 2-circuit decomposition if and only if its orientable 2-fold cover does.

Motivated by the problem of finding obstructions to the existence of symplectic structures in dimensions higher than four, we introduce an elliptic system of equations on almost Hermitian manifolds that reduces to the Seiberg–Witten equations with Taubes' perturbation in dimension four. To define the equations, one needs to choose a spin$^{\mathbb C}$ structure on the manifold. We prove that the system has index zero for the canonical spin$^{\mathbb C}$ structure. Moreover, on almost Kähler (symplectic) manifolds, we construct a canonical solution and prove its transversality with large values of the perturbation parameter, while in dimension six we obtain uniqueness of the canonical solution under a certain assumption. We also show that on Kähler manifolds the equations reduce, under a natural ansatz, to the vortex equations. These results provide evidence toward a possible higher dimensional gauge theory capable of producing invariants of almost complex structures and ultimately, obstructions to compatible symplectic forms.

How small can the positivity loss be for an $N$-bandlimited spectral operator that exactly reproduces all modes up to degree $L$? For spherical polynomial approximation on $\mathbb S^d$, we prove that the smallest possible excess of the uniform operator norm above 1, equivalently the least positivity loss, is of sharp order $\left({L}/{(N+1)}\right)^2$ when $1\le L< N$. The lower bound follows from a Fejér peak test and a concentration estimate for bandlimited kernels, while a matching upper bound is obtained by correcting a positive Jackson operator with a smooth filter. We illustrate the result in three settings. On the circle, taking $N=sL-1$, this determines the sharp order of the generalized-projection constant above 1 and identifies the gap between the $s^{-1}$ excess of delayed de la Vallée–Poussin means and the optimal $s^{-2}$ order. For filtered hyperinterpolation, whose operator norm has long been known to be uniformly bounded, we give a quantitative lower bound on its separation from the positivity threshold 1. Finally, we identify an operator-level obstruction to maximum principles.

In this paper, we derive some new relative volume comparison theorems and Bishop-Gromov volume comparisons on Finsler metric measure manifolds, all of which are controlled by the integral weighted Ricci curvature. In particular, we establish a Bishop-Gromov volume comparison theorem for nonconcentric balls. Based on these, we prove a theorem of Bonnet-Myers type on Finsler metric measure manifolds with integral weighted Ricci curvature bounds.

In this paper, we study the large deviation principle (LDP) for a tensor-weighted functional of i.i.d. random variables, when the sequence of tensors converges under a variant of the "bad" cut norm. Using the LDP, we analyze a Gibbs measure with a tensor-valued Hamiltonian, and characterize the optimizers of the limiting variational problem in terms of a functional fixed point equation. As applications, we focus on several concrete examples, which include monochromatic subgraph counts in sparse random graphs, Erdős-Rényi hypergraphs, and a generalized Potts statistic of order $v\ge 2$. Studying the optimization problem, we give sufficient conditions for uniqueness of the optimizer, as well as for existence of constant optimizers (replica symmetry). Our results demonstrate universal weak laws for a large class of tensor Gibbs models with approximately regular tensors.

Let \(\Omega\) be a compact Hausdorff space, and let \(\cA\) be a unital \(C^*\)-algebra. In this study, we continue our examination of the comparison between \(C^*\)-extreme points and linear extremal structures of the unit ball, in the vector-valued \(C^*\)-algebra \(C(\Omega, \cA)\), building upon the work initiated in \cite{HR}. We first enlarge the class of $C^\ast$-algebras in which a $ C^\ast$-extreme point is an extreme point. We demonstrate that if \(\cA\) has a faithful tracial state, then any \(C^*\)-extreme point of the unit ball \(C(\Omega, \cA)_1\) is a unitary. Additionally, we identify a classes of \(C^*\)-algebras where the concepts of \(C^*\)-extreme and pointwise \(C^*\)-extreme points in \(C(\Omega, \cA)_1\) coincide. We show this holds if a von Neumann algebra has a separable predual with the Radon-Nikodým property.

We investigate the power-law decay of the survival probability of a Brownian particle diffusing in an infinite planar wedge whose two sides are partially reactive and described by Robin boundary conditions. We employ matched asymptotic analysis to relate the long-time asymptotics to a stationary harmonic Robin problem near the apex. This approach determines the persistence exponent for arbitrary opening angles and yields the prefactor explicitly for the family of wedges with $\alpha=\pi/n$ ($n=1,2,\ldots$). A simple extension of the apex prefactor to arbitrary angles is conjectured and supported numerically. This asymptotic behavior describes the crossover to the well-known result for perfectly absorbing wedges. As immediate applications, we also deduce the long-time behavior of the probability density function of the boundary local time on wedge sides, as well as the probability density function of the associated first-crossing time.

The pairwise independent correlation gap is the ratio of the maximum expected value of a set function under arbitrary dependence to that under pairwise independence, measuring the loss from this independence restriction. Under mutual independence, this gap is universally bounded by $e/(e-1)$ for monotone submodular functions. With pairwise independence, a tighter $4/3$ upper bound was established for several special cases, including $n=3$, and conjectured to hold universally. A recent AI-assisted counterexample disproved this conjecture for $n=5$, leaving the validity of the $n=4$ bound and the tight worst case bound open. We resolve both questions. First, for $n=4$, we establish that the $4/3$ bound holds universally and is tight using an AI-assisted proof combining theoretical analysis and computational verification. The proof combines a structural characterization of optimal numerator vertices, permutation symmetry, cone certificate systems, Bernstein polynomial representations, recursive simplex subdivision, and verification of $2,745$ Bernstein coefficient systems. Second, we show that the worst case pairwise independent correlation gap attains $e/(e-1)$ asymptotically by constructing an instance with identical marginal probabilities and a monotone submodular union coverage function on a ground set partitioned into $m$ blocks. The number of blocks grows sublinearly with the ground set size. The result follows by constructing a feasible solution to a scaled asymptotic reduced dual of the pairwise independent linear program and immediately extends to $t$-wise independent random elements ($t\ge2$), since $t$-wise independence implies pairwise independence. Thus, pairwise independence, despite being the least restrictive form of independence in the $t$-wise independence hierarchy, can be as restrictive as mutual independence in the worst case.

The equivalence classes of boundary conditions of a gauge theory on a two-dimensional orbifold are the fibres of a marginal map, indexed by an affine semigroup: one generator per alphabet label, graded by weight, embedded by its local data at the fixed points. This note identifies that semigroup. Without weights the configuration has a name and a literature, whose results about our cases are attributed here: over $\mathbb{Z}_2$ it is the cut configuration of an explicit graph in the sense of Sturmfels-Sullivant — the four-cycle for $T^2/\mathbb{Z}_2$, the wheel $W_4$ for $S^1/\mathbb{Z}_2\times S^1/\mathbb{Z}_2$ — verified as an equality of configurations; over $\mathbb{Z}_m$ with equal cone orders, the group-based phylogenetic model on a claw tree; with unequal orders, a mixed-order variant we do not find in the literature; for higher products, the binary hierarchical model of a cross-polytope boundary complex. The product orbifold's ring is a row of a 2008 table — codimension, degree, minimal generators, normality — every invariant of which our machinery reproduced without knowing of it. What none of the three covers is the alphabet with weights, which arise from induction to higher-dimensional irreducibles of a non-abelian space group and from conjugate-pair recombination over real or quaternionic ground. That sector is adjacent to, but not identified with, the non-abelian direction Sturmfels and Sullivant raised in 2005, and is where our contributions sit: gluing trees for the weighted alphabets and the orthogonal and symplectic columns, and the group-based model on the tripod, a complete intersection exactly when the finite abelian group has order at most three. The first group beyond $\mathbb{Z}_3$ separates local from global: the $\mathbb{Z}_4$ tripod is a complete intersection on the Zariski-open set the phylogenetics literature works in, and not globally.

This paper establishes a new comparison principle framework for the subquadratic fractional $p$-Laplacian, i.e.~$1 < p < 2$ under minimal regularity assumptions, that has remained a significant challenging issue due to the singularity of the operator. Our results provide the essential analytical tools required for the moving plane method in this setting. We prove first a weak comparison principle for $(-\Delta)_p^s u = f(u)$ in bounded domains with sufficiently small measure, where only the boundedness of the weak solution is required. More importantly, we establish a strong comparison principle for continuous weak solutions in the parameter range $s \in (0, \frac{1}{2})$ and $\frac{1}{1-s} < p < 2$. Our proof introduces a localized barrier function and does not require any Hölder regularity of the weak solution, nor any smoothness of the domain. This presents a substantial contrast over previous study, which relied heavily on Hölder or even $C^{1,1}$ regularity. As a direct application, we employ these comparison principles to prove the symmetry of weak solutions to $(-\Delta)_p^s u = f(u)$ under mild assumptions, which significantly extend existing symmetry theories for nonlocal quasilinear equations.

For a bipartite graph $G$ and an integer $t\geq2$, let $\NM_t(G)$ be its $t$-non-matching complex. We prove that adding an edge while preserving bipartiteness induces an injection on reduced homology in degree $2t-3$. Combined with the cyclic-polytope model for non-matching complexes of cycles, this shows that $\NM_t(G)$ has Leray number $2t-2$ whenever $G$ contains a cycle of length at least $2t$. Under the same hypothesis, Hochster's formula yields regularity $2t-1$ for the Stanley-Reisner ideal $I_{\NM_t(G)}$, together with explicit lower bounds for the Betti numbers on its top regularity strand and for its projective dimension. If $G$ contains a $2t$-cycle, we also determine the maximal shifts of $I_{\NM_t(G)}$ through homological degree $|E(G)|-2t+1$. For $G=K_{r,s}$ with $2\leq t\leq r\leq s$, we determine the depth, projective dimension, all maximal shifts, and the unique extremal Betti number of $I_{\NM_t(K_{r,s})}$, thereby settling a conjecture on facet ideals of chessboard complexes.

We study the classical $XY$-model on random geometric graphs $\mathcal{G}_{n, \varepsilon}$, which are obtained by sampling $n \in \mathbb{N}$ independent points in a finite domain $\Omega \subset \mathbb{R}^d$, $d \geq 2$, and connecting two points by and edge if their distance is of order $\varepsilon > 0$. We refer to $\mathcal{G}_{n, \varepsilon}$ as the random environment. Letting $\varepsilon \to 0$ as $n \to \infty$ at a sufficiently slow rate, these graphs capture the geometry of $\Omega$. Denoting the inverse temperature by $\beta$, we show that in the limit $\beta \to \infty$ at a rate depending on $n$ and $\varepsilon$, the $XY$-model on $\mathcal{G}_{n, \varepsilon}$ exhibits long range order in the sense that we prove a lower bound away from zero on the two-point function. Our result is quenched in the random environment: long-range order holds with large probability, converging to one as $n \to \infty$. To prove the statement, we show that with high probability the environment is sufficiently regular to apply a convexity argument and the Brascamp–Lieb inequality.

We give a complete classification of the automorphisms of twisted Chevalley groups of types ${}^2A_3$ and ${}^2A_4$, and of their elementary subgroups, over commutative local rings $R$ containing $1/2$ and equipped with a non-trivial involution $\theta$. The classification covers all isogeny types and, together with our higher-rank result~\cite{EB&DM1}, completes the description for type ${}^2A_{\ell}$ with $\ell\geq3$.

Tipping points—abrupt, potentially irreversible reorganizations of a system's statistical state—are commonly anticipated through critical slowing down: recovery slows, autocorrelation and variance rise, and spectra redden. This paradigm is powerful near simple equilibrium bifurcations but is not a general theory for stochastic, multistable, or metastable systems. We develop such a theory from the Ruelle–Pollicott (RP) spectrum of Kolmogorov generators of hypoelliptic Itô diffusions. Resolving the sensitivity of invariant statistics on RP spectral blocks shows that each contribution factorizes into a spectral denominator and a residue coupling the block to both the observable and perturbation direction. Small denominators permit large responses; residues produce them. Thus a closing RP gap is not sufficient for an early warning, while growing residues can generate the classical signature with no gap closure. An early-warning signal is therefore a property of a triple: RP block, observable, and perturbation direction. We demonstrate this on a stochastic non-normal system whose RP spectrum is exactly frozen, yet classical indicators become more alarming than during genuine gap closure. The RP decomposition attributes this to residue growth and yields an index $\mathcal{N}(f)$ satisfying $\mathcal{N}(f)\leq 1$ for reversible dynamics; hence $\mathcal{N}(f)>1$ certifies residue-driven amplification. For metastable systems, one killed problem yields two complementary spectral objects: the Doob $Q$-process isolates in-well recovery, while escape clocks and committor-weighted destination probabilities govern interwell transitions. In a one-dimensional fold, these scale as $(\epsilon_c-\epsilon)^{1/2}$ and $(\epsilon_c-\epsilon)^{3/2}$, separating bifurcation-induced from noise-induced tipping. Doob drift also connects to optimal Girsanov sampling.

In this work, we study nonnegative viscosity solutions to Hénon-type equations driven by the infinity-Laplacian with a degenerate weight, strong absorption and an additional source term. We focus on free boundary points lying in the degeneracy set of the weight. We prove the degeneracy set comparison principle, which yields uniqueness within each slice determined by the value on the degeneracy set of the weight, although global uniqueness is not available in general. We establish sharp improved regularity estimates near free boundary points, with the intrinsic growth rate $r^{\frac{4+\alpha}{3-m}}$. Finally, we obtain a matching non-degeneracy estimate at free boundary points; the property holds in the inhomogeneous case with suitable condition. Consequently, solutions detach from their zero phase at this rate.

We determine the frame set of the first Hermite function, thereby proving a conjecture of Lyubarskii and Nes. A characterization of semi-regular Gabor frames due to Gröchenig, Romero, and Stöckler reduces the problem to a uniqueness question for entire functions in a Gaussian shift-invariant space. We address this question by forming Wronskians of finitely many translates of a Gaussian shift-invariant function. Their common critical points become zeros of increasing multiplicity, while automorphy shows that the Wronskians remain within a Gaussian shift-invariant class. A sharp zero-density theorem then rules out all possible failures of the frame property except at the known rational obstructions. This yields the complete rectangular frame set of the first Hermite function and introduces Wronskian amplification as a new method in Gabor frame theory.

This paper investigates the global dynamics of a doubly degenerate nutrient-taxis system with indirect consumption: \begin{equation*} \left\{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-\nabla \cdot (u^{2}v\nabla v)+\ell vw,&x\in \Omega,\, t>0,\\ & v_{t}=\Delta v-vw,&x\in \Omega,\, t>0,\\ &w_t=\Delta w-w+u,&x\in\Omega,t>0 \end{aligned} \right. \end{equation*} posed on a smooth bounded domain $\Omega\subset\mathbb{R}^{3}$ with no-flux boundary conditions. It is shown that for suitably regular initial data $(u_0,v_0,w_0)$, the associated initial-boundary value problem admits a global weak solution. Furthermore, in an appropriate topological setting, this solution converges to an equilibrium $(u_\infty, 0,w_\infty)$ as $t\rightarrow \infty$. Notably, when $u_0$ is nonconstant and the mass of $v_0$ is sufficiently small, the limiting profiles $u_{\infty}$ and $w_{\infty}$ are are spatially nonhomogeneous, capturing emergent patterning in nutrient-depleted environments. A cornerstone of our analysis is the introduction of novel functional inequalities, which provide estimates from below for the integral $\int_{\Omega}u^{k}v|\nabla u|^2$ with some $k>-1$.

We prove the Lonely Runner Conjecture for fourteen and fifteen runners. Our proof combines a stronger bound on the speed product in a primitive counterexample with exhaustive computations modulo primes. To obtain the bound, we work with a projected lattice basis and use cases of the conjecture with fewer runners to bound partial sums of its squared Gram-Schmidt lengths. For fifteen runners, the product bound derived from the work of Malikiosis, Santos, and Schymura gives a logarithmic threshold of about $810$; ours reduces this to about $415$, making the computation feasible. For fourteen runners, the new bound reduces the required number of primes from $111$ to $61$. The computations start with a complete two-branch covering search, followed by binary lifting. Since $14$ and $15$ are composite, the polynomial argument used in earlier work does not apply directly. We finish the fourteen-runner case by a direct search. For fifteen runners, we use the factorisation $15=3\cdot5$ and a shifting argument when all but a few speeds share a common divisor. The code and certificates are publicly archived.

The Calderón-Zygmund theory establishes the boundedness of singular integral operators on $L^p$ spaces for $1 < p < \infty$, yet it encounters a failure at the endpoint $p = 1$. While radial counterexamples in $\mathbb{R}^n$ are well-documented, Pan-Shao-Wang-Wu \cite{psww2026} has showed that every nonconstant holomorphic function provides a counterexample to the Poisson equation within the Calderón-Zygmund framework, with the singular locus being a complex subvariety of codimension one. In this paper, we focus on the complex one-dimensional case and establish stronger results. We prove asymptotic formulas with explicit constants for both the level-set integral and the sublevel-set energy. Then we give simplified proofs of the universal counterexamples to Calderón-Zygmund theory at $p = 1$ in $\mathbb{C}$. Additionally, we construct a new family of counterexamples at the endpoint $p = \infty$, showing that the failure of $W^{2,\infty}$-regularity is also a universal phenomenon in complex one dimension.

We study the relationship between quasi-uniform spaces, topological spaces, and quantale-valued metric spaces. Our main result is a metrization theorem establishing an equivalence between the category of quasi-uniform spaces and a category of quantale-valued metric spaces. We also obtain a quantale-based metrization theorem for arbitrary topological spaces that refines existing constructions. These results identify quasi-uniformities as the appropriate qualitative counterpart of quantale-valued metrics. Building on this correspondence, we show that the Hausdorff-Smyth monad on quantale-valued metric spaces, which is used in quantitative robustness analysis, arises as a lifting of a corresponding monad on quasi-uniform spaces along the equivalence. This provides a unified categorical framework connecting topology, quasi-uniformity, and quantitative robustness analysis.

In this paper, we establish a finite–tame–wild trichotomy for the representation type of $(n+1)$-preprojective algebras of type $A_m$. We also provide a complete classification of such algebras with respect to $\tau$-tilting finiteness. Moreover, for $n\ge 2$, we give a precise characterization of when each indecomposable module is $\tau$-rigid.

Let X be a centered infinitely divisible random vector with finite second moment and covariance matrix Sigma. We define the radial pole C_X(t) as the smallest scale in a right sub-gamma bound for <t,X> whose quadratic proxy is fixed at the true variance t^T Sigma t. A canonical directional measure and an independent Beta(1,2) multiplier give an exact variational formula for C_X. The resulting extended-valued function is positive homogeneous and is pointwise least among all homogeneous denominators compatible with the covariance quadratic form. We prove linear-map, convolution, and Levy-time rules, and show that the full family of directional remainders determines the law of X. Geometrically, C_X lies between the Minkowski functional of the moment-generating-function domain and one third of the positive support function of the Levy measure; the upper constant is sharp, and the zero set is a polar cone. The pole need not be subadditive. It is continuous on the sphere under global exponential moments and positive-definite covariance, whereas finite variance alone permits a jump from zero to infinity in nearby directions. For additive gamma-ray models, C_X equals the domain gauge and has a finite-polytope formula.