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arXiv:2606.02250v1 Announce Type: new Abstract: This paper investigates a single species reaction-diffusion system incorporating a spatiotemporal delay memory kernel, which models the cognitive map of animals, under Neumann boundary conditions. The model can be used to describe the process in which individuals are influenced by historical information during spatial diffusion. An equivalent system construction method with auxiliary variable....
arXiv:2606.02254v1 Announce Type: new Abstract: Let $E$ be an elliptic curve over $\mathbb{Q}$ having split multiplicative reduction at a prime number $p$. We describe the tame part of the $\mathcal{L}$-invariant of $E$ at $p$ in terms of automorphic $p$-adic periods introduced in the work of Darmon. More precisely, we prove an equality of refined $\mathcal{L}$-invariants using twisted versions of refined exceptional zero conjectures. When..
arXiv:2606.02257v1 Announce Type: new Abstract: Ross and Yong conjectured a $K$-theoretic Kohnert rule for Grothendieck polynomials. Robichaux exhibited a counterexample to the Ross--Yong rule and proposed a revised ghost $K$-Kohnert rule, proving both rules hold for 321-avoiding permutations. We provide counterexamples to Robichaux's rule and give an explicit bijection showing that both the Ross--Yong and Robichaux rules hold for 1432-avo..
arXiv:2606.02262v1 Announce Type: new Abstract: In spite of the intensive study of cyclic codes and the recent construction of an infinite family of self-dual binary cyclic codes whose minimum distances have the square-root bound in IEEE Trans. IT, vol. 71, no. 4, 2025, it is still a 70-year-old open problem whether there is an infinite family of self-dual binary cyclic codes whose minimum distances have a lower bound better than the squar..
arXiv:2606.02269v1 Announce Type: new Abstract: We construct a cohesive $\infty$-topos $\mathbf{H}_{\mathbb{Q}}$ equipped with a \emph{quantum modality} -- an idempotent product-preserving comonad $Q^{\diamond}$ with right adjoint $Q_{\bullet}$ satisfying the Beck--Chevalley compatibility conditions with the cohesive structure $(\Pi,\flat,\sharp)$. The model is the functor $\infty$-topos $\operatorname{Fun}(\mathbf{C}^{*}\mathbf{Alg}_{\mat....
arXiv:2606.02271v1 Announce Type: new Abstract: Exact \(k\)-leaf powers are graphs whose edges are exactly the pairs of leaves at distance \(k\) in a tree. We prove explicit structure theorems for exact leaf powers on several representative graph families that test different exact-distance phenomena. Our most detailed root-classification theorem concerns chordless cycles: all exact \(5\)-leaf roots of \(C_\ell\), \(\ell\ge 8\), are describ....
arXiv:2606.02279v1 Announce Type: new Abstract: For graphs $G$ and $F$, we say that $G$ is $F$-free if $F$ does not occur as an induced subgraph of $G$. This paper is concerned with the following question: Given an $F$-free graph $G$ having two vertices between which there exists at least one Hamiltonian path, how many Hamiltonian paths between these endpoints must exist (in terms of the order of $G$)? Our main result shows that there exis....
arXiv:2606.02284v1 Announce Type: new Abstract: We show that virtually nilpotent groups are coarse median if and only if they are virtually abelian. The main idea is that the sub-Riemannian geometry of the asymptotic cone obstructs the existence of a locally convex Lipschitz median of finite rank. As an application, we deduce that non-compact lattices in the isometry group of a rank 1 symmetric space of non-compact type other than real hyp..
arXiv:2606.02286v1 Announce Type: new Abstract: The deterministic incompressible Navier-Stokes equations are physically incomplete: any viscous fluid at finite temperature must exhibit thermal fluctuations whose form is dictated by the fluctuation-dissipation relation. We formulate the stochastic Navier-Stokes equations with the kinematically selected deformation Laplacian on compact Riemannian manifolds with strictly negative Ricci curvat....
arXiv:2606.02299v1 Announce Type: new Abstract: We study sign uncertainty principles for trigonometric polynomials of prescribed degree $N$ with respect to a symmetric Borel measure $\mu$ on the unit circle $\mathbb{R}/\mathbb{Z}$. For each such measure, we determine the smallest radius of the last sign change for trigonometric polynomials with non-positive $\mu$-integral. We further extend these results to polar measures on higher-dimensi..
arXiv:2606.02308v1 Announce Type: new Abstract: In this paper, we construct families of unimodal maps with wild Cantor attractors and study, via symbolic dynamics, the interplay between recurrence properties of the critical orbit. Using kneading and co-kneading techniques, we provide a symbolic characterization of persistent recurrence in terms of cutting and co-cutting times, and relate it to other recurrence conditions, including the Col....
arXiv:2606.02312v1 Announce Type: new Abstract: Since Green (2005), the Fourier-analytic transference principle has dominated the landscape of combinatorial theorems relative to sparse arithmetic sets. We demonstrate a different approach using arithmetic regularity. This is more versatile and has the potential to succeed when no obvious `dense model' is forthcoming. Moreover, we contend that, just as the traditional circle method disassemb..
arXiv:2606.02316v1 Announce Type: new Abstract: Let $\mathcal{R}$ be an expansion of the real field such that every subset of $\mathbb{R}$ definable in $\mathcal{R}$ either has interior or is a finite union of discrete sets. Answering a question by Chris Miller, we show that for every $n\in \mathbb{N}$ and every definable subset $A\subseteq \mathbb{R}^{n+1}$ there is $N\in \mathbb{N}$ such that for all $x\in \mathbb{R}^n$ either $A_x$ has ..
arXiv:2606.02323v1 Announce Type: new Abstract: Continuous-variable quantum key distribution (CV-QKD) requires highly efficient reconciliation techniques to operate at low signal-to-noise ratios and long distances. Multidimensional reconciliation addresses this challenge by transforming the physical Gaussian quantum channel into a virtual binary-input additive white Gaussian noise (BIAWGN) channel, enabling the use of modern errorcorrectin....
arXiv:2606.02324v1 Announce Type: new Abstract: Necklace polynomials arise in various fields of mathematics, including combinatorics, Lie theory, and Galois theory over finite fields. Their arithmetic properties have been extensively studied in recent years. In this article, we introduce a new class of rational necklace functions that unifies two well-studied families of polynomials: necklace polynomials and Fekete polynomials. We describe..
arXiv:2606.02340v1 Announce Type: new Abstract: Paired-comparison data are naturally represented by tournaments, where transitivity corresponds to the existence of a global ranking consistent with all pairwise outcomes. Accordingly, the classical Kendall-Smith coefficient of consistency measures deviations from transitivity in a tournament by counting the number of circular triads (directed $3$-cycles). In this paper, we characterize the f....
arXiv:2606.02356v1 Announce Type: new Abstract: Burr and Erd\H{o}s conjectured in 1976 that for all integers $k>\ell\geq 0$ such that $k\mathbb{Z}+\ell$ contains an even integer, every $n$-vertex graph without cycles of length $\ell$ modulo $k$ has at most a linear number of edges in $n$. Bollob\'{a}s confirmed the conjecture in 1977, and Erd\H{o}s further asked for the exact extremal number. To the best of our knowledge, this problem has ....
arXiv:2606.02364v1 Announce Type: new Abstract: Let $m,l\in\N$ be relatively prime and let \[ \Omega_{m/l}^{n+1}=\bigl\{(z,w)\in\C^n\times\C:\ \norm{z}^{m}<\abs{w}^{l}<1\bigr\} \] be the rational generalized Hartogs triangle of exponent $m/l$ in $\C^{n+1}$. In this paper, we study the Berezin transform $\BB_{m/l,n}$ associated with the Bergman kernel of $\Omega_{m/l}^{n+1}$, and prove that $\BB_{m/l,n}$ is bounded on $L^p(\Omega_{m/l}^{n+1..
arXiv:2606.02377v1 Announce Type: new Abstract: We establish a descent principle for $\otimes$-localizing subcategories along smooth presentations using a notion of descendability due to Balmer and Mathew. This allows us to classify $\otimes$-localizing subcategories of the derived category of complexes with quasi-coherent cohomology on suitable algebraic stacks in terms of subsets of its underlying topology.
arXiv:2606.02389v1 Announce Type: new Abstract: An old conjecture of Burr and Erd\H os states that the Ramsey number of any $n$-vertex tree $T$ is at most $2n-2$. In 2012, Schelp asked whether a degree version of the Burr--Erd\H{o}s conjecture holds. More precisely, Schelp asked if is it true that for any $\varepsilon>0$ and $\Delta\ge 2$, if $G$ is a graph on $N\ge (2+\varepsilon)n$ vertices and minimum degree $\delta(G)\ge \lfloor 3N/4\r..
arXiv:2606.02390v1 Announce Type: new Abstract: Every knot $K \subset S^3$ that admits a symmetric union presentation bounds an immersed ribbon disk in $S^3$, while the converse is an open problem due to Christoph Lamm. The symmetric ribbon number $r_s(K)$ of $K$ is the minimum number of ribbon singularities in any symmetric ribbon disk bounded by $K$. In this paper, we undertake a systematic investigation of symmetric ribbon numbers of kn..
arXiv:2606.02395v1 Announce Type: new Abstract: Permuted-basement Macdonald polynomials $E_\alpha^\sigma(\mathbf{x};q,t)$ are nonsymmetric generalizations of symmetric Macdonald polynomials that form a basis for the polynomial ring $\mathbb{Q}(q,t)[\mathbf{x}]$ for each fixed $\sigma$. There are combinatorial formulas for them as generating functions over composition-shaped non-attacking fillings. In this extended abstract, we bijectively ..
arXiv:2606.02396v1 Announce Type: new Abstract: We show that any three-ball with mean convex boundary contains an embedded free boundary minimal disk. Moreover, when the three-ball is a strictly convex domain with nonnegative Ricci curvature (for instance, a compact convex domain in Euclidean three-space), we prove the existence of at least three embedded free boundary minimal disks. Our approach is based on a multiplicity-one theorem for ..
arXiv:2606.02397v1 Announce Type: new Abstract: We investigate the stability of discrete structures comprising of woven elastic beams in a plane, exposing a connection between admissible over / under patterns and the first-order and static rigidity of an associated tensegrity that is related through a natural polarity transformation. The relationship between the Airy and Whiteley stress functions for the tensegrity and weaving structures i..
arXiv:2606.02401v1 Announce Type: new Abstract: We prove that stable degeneration, the canonical degeneration associated to the normalized volume minimizer of a Kawamata log terminal (klt) singularity, preserves non-degenerate reflexive differential forms. In particular, the stable degeneration of a symplectic singularity is again symplectic. Combining this with a deformation-theoretic rigidity result for symplectic degenerations, we confi..
arXiv:2606.02405v1 Announce Type: new Abstract: We prove that the bilinear generating function for Wronskians of Hermite polynomials can be expressed as the classical Mehler kernel multiplied by a polynomial, thereby extending the result of Pupasov-Maksimov for exceptional Hermite polynomials. We establish several properties of the polynomials appearing in this extended version of the Mehler formula and present four conjectures about them.
arXiv:2606.02407v1 Announce Type: new Abstract: For the Lindel\"of Hypothesis concerning the Riemann zeta function $\zeta(s)$, upper bounds as $\Im(s)\to\infty$ have been extensively studied for many years. In particular, the Lindel\"of Hypothesis is one of the most important open problems in analytic number theory. It is also known to be equivalent to certain mean value estimates, which provide a fundamental connection between pointwise u..
arXiv:2606.02409v1 Announce Type: new Abstract: This note combines a result of B\"okstedt and Waldhausen concerning the so-called derivative map on tubes with the existence theorem for generating functions of tube type for nearby Lagrangian homotopy spheres due to Abouzaid, Courte, Guillermou and Kragh to obtain a restriction on the smooth structure of nearby Lagrangian homotopy spheres. Concretely, if a homotopy $n$-sphere $L$ admits a La....
arXiv:2606.02415v1 Announce Type: new Abstract: In this article, we study rational representations of $G=\operatorname{GL}_2(q)$, where $q$ is a prime power. Let $\rho$ be an irreducible representation of $G$ over $\mathbb{Q}$. Then $\rho$ affords the character \[ \Omega(\chi)=m_{\mathbb{Q}}(\chi)\sum_{\sigma\in\operatorname{Gal}(\mathbb{Q}(\chi)/\mathbb{Q})}\chi^{\sigma}, \] for some irreducible complex character $\chi$ of $G$, where $m_{....
arXiv:2606.02426v1 Announce Type: new Abstract: For the family of Dirichlet stadia $S_t$ whose rectangular part has height $\pi$ and half-length $\pi t/2$, $t \in [1,2]$, we show that for Lebesgue almost every $t$ there exist real eigenfunctions $u_j$ and a smooth mean-zero physical observable $Q$ for which $\langle Q u_j,u_j\rangle$ has a non-zero subsequential limit. Consequently, along the same subsequence, the eigenfunction mass fails ..
arXiv:2606.02427v1 Announce Type: new Abstract: Existing evaluations of neural operators and in-context operator learning rely primarily on prediction error, but accurate output prediction does not guarantee the correct local dynamical structure. A model may match solutions while exhibiting incorrect sensitivities, distorted frequency response, spurious mode coupling, or unstable tangent behavior. We introduce a Jacobian-based spectral aud....
arXiv:2606.02428v1 Announce Type: new Abstract: We determine the logarithmic growth exponents of the $L^p$ norms, $1\le p\le\infty$, of $L^2$-normalized Laplace eigenfunctions on the unit disk, for both Dirichlet and Neumann boundary conditions. We also prove sharp uniform $L^p$ upper and lower bounds for every $L^2$-normalized Dirichlet eigenfunction and every non-constant Neumann eigenfunction $u_{\lambda}$ on the disk. The proof uses st..
arXiv:2606.02429v1 Announce Type: new Abstract: This note gives a simple algorithm for the following effectivity problem: given a genus $2$ curve $X$ together with a nonconstant map $\pi:X\to E$ to an elliptic curve, determine an elliptic curve $E'$ and a map $\pi':X\to E'$ independent of $\pi$. Equivalently, we compute the complementary elliptic factor in the decomposition of $\operatorname{Jac}(X)$ up to isogeny. While the problem has be..
arXiv:2606.02452v1 Announce Type: new Abstract: In this paper, we prove that all normalized symplectic capacities agree for dynamically convex domains in $\mathbb{C}^2$ that are invariant under any Hamiltonian $S^1$-action isotopic to the Hopf diagonal action. We also give necessary and sufficient conditions for $S^1$-invariant domains to be dynamically convex.
arXiv:2606.02464v1 Announce Type: new Abstract: We study admission control to an unobservable M/M/1 queue. A memoryless controller can only randomly thin arrivals (random routing, RR). We show that a gated admission (GA) policy, blocking arrivals for a fixed period after each admission, stochastically dominates RR at equal throughput, improving social welfare under any sojourn-based cost. We characterize the welfare-maximizing threshold an..
arXiv:2606.02468v1 Announce Type: new Abstract: In this thesis we present the combinatorial game theory of the game Rex+. Rex+ is a variant of the game Hex, played on a four sided board made out of hexagons. Both players take turns placing as many stones of their colour as they would like on the board, with the objective being to force the other player to connect their two sides. We describe a new ordering, and present some preliminary res..
arXiv:2606.02469v1 Announce Type: new Abstract: This paper introduces the L-shaped refinement chain cuts method, a novel approach for solving two-stage stochastic programs. The proposed method integrates the refinement chain of scenarios within the classical L-shaped decomposition framework. In the proposed approach, the full scenario set is partitioned into subgroups at each level of the refinement chain, and one subproblem is solved for ....
arXiv:2606.02471v1 Announce Type: new Abstract: Using our new notion of folding shuffle algebras, we prove a conjecture of Hernandez on the equality between certain $q$-characters of quantum untwisted affine algebra modules and their twisted counterparts. We generalize this result to the setting of arbitrary quivers with automorphisms, in particular by defining and describing twisted quantum toroidal algebras.
arXiv:2606.02472v1 Announce Type: new Abstract: We introduce and study a new model of correlated uniform attachment (UA) trees, where correlation is sprinkled throughout the time evolution of the process. In this model, two UA trees are grown in parallel, and at each time step a new node is added to each tree, with an edge between it and a uniformly chosen existing vertex in the respective tree. The two choices of attachment are correlated....
arXiv:2606.02475v1 Announce Type: new Abstract: Classical finite-difference solvers remain reliable tools for partial differential equations, but their efficiency depends on where mesh resolution is placed. Uniform refinement can waste degrees of freedom when solution difficulty is localised near sharp gradients, fronts, oscillations, or constraint-sensitive regions. This paper studies a hybrid strategy in which a physics-informed neural n....
arXiv:2606.02477v1 Announce Type: new Abstract: In the representation theory of Lorentzian orthogonal groups there are well known arguments as to why the parity operator $\mathcal{P}$ and the time reversal operator $\mathcal{T}$ should be realized as linear and anti-linear operators respectively (Wigner 1932). In this paper it is shown that the only operators satisfying the requisite properties for the spinor representation of the de Sitte..
arXiv:2606.02478v1 Announce Type: new Abstract: A regular clock map is a regular map of directed spaces from a saturated directed space to the directed circle. We prove that the category of regular clock maps is a small-orthogonality class of the category of clock maps. Hence it is locally presentable. Any geometric realization of precubical sets and of transverse sets gives rise to a regular clock map. Finally, we prove that for the under..
arXiv:2606.02495v1 Announce Type: new Abstract: Switched systems are a family of dynamical systems where a switching rule indicates which system is "switched on". This rule can be dependent on time and/or position in the state space. Stability of switched systems is a property that is often investigated using the existence of one (or multiple) Lyapunov function(s). We develop an algorithm using a scattered approximation method to construct..
arXiv:2606.02496v1 Announce Type: new Abstract: We introduce the notion of timelike ideal boundary of a Lorentzian length space as the set of asymptotic classes of future or past-directed timelike geodesic rays, a construction complementary to the causal boundary in the sense of Geroch-Kronheimer-Penrose and akin to the concept of ideal boundary of a metric space. We endow such a timelike ideal boundary with a natural cone topology and an ..
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