Papers

This paper investigates the topological properties of intersections of balls in finite-dimensional normed spaces - a problem that naturally arises when constructing covers for estimating the Gromov-Hausdorff distance. We study the topology of a set obtained by removing a large closed ball from a finite intersection of small open balls. It is proved that in an arbitrary normed plane, such a set is always contractible, provided that it is non-empty.

We introduce a new model for $(\infty,n)$-categories as Segal sheaves on lax grids, which are pasting diagrams of lax cubes. This model allows for a direct construction of the Gray tensor product via Day convolution. We show that this agrees with Campion's construction of the Gray tensor product. These results will be applied in future work to equip the higher categories of cobordisms with a Gray-algebra structure given by the cartesian product of manifolds.

We study maximum-drawdown laws conditioned on extremes for a spectrally negative Lévy process and observed up to an independent exponential time. The main contribution is a set of scale-function characterizations of the pre-infimum path arising from two decompositions of the process. The first is the decomposition at the infimum into pre-infimum and post-infimum components. The second, under the ordering in which the infimum is attained before the supremum, decomposes the path into pre-infimum, intermediate, and post-supremum components. We also identify the distribution of the supremum for the pre-infimum process in the first decomposition. The resulting conditional laws are expressed as Doob $h$-transforms of killed spectrally negative Lévy processes and they yield explicit formulas for the maximum drawdown on each independent path component. The results confirm the classical decompositions for Brownian motion.

A Tychonoff space $X$ has the two-disjoint-copies property (2DCP) if there exists a sequence $(K_n)_{n\in\omega}$ of non-empty compact subsets of $X$ such that each $K_n$ contains two disjoint subsets homeomorphic to $K_{n+1}$. Banakh, Kąkol and Śliwa showed that 2DCP yields an infinite-dimensional metrizable quotient of $C_p(X)$, while it is still a long-standing open question whether $C_p(X)$ has such a quotient for any infinite compact space $X$. The above concept as well as the last problem are closely related to Efimov's problem that has remained open for 40 years. We will discuss a number of conditions that imply 2DCP. For example, every locally homogeneous compact space, every space containing a copy of $\beta\omega$ or $2^\omega$ has 2DCP although compact $h$-homogeneous spaces with 2DCP without such copies exist in ZFC. We prove that no scattered compact space has 2DCP and there exist in ZFC compact perfect spaces without 2DCP. This implies that for compact metric spaces $X$ the 2DCP is equivalent to uncountability of $X$. There exist explicit uncountable separable compact spaces failing 2DCP, for example the Isbell-Mrówka compacta. We give positive classes among zero-dimensional compact spaces; for example, the Brech, as well as the Sobota-Zdomskyy compact spaces of Efimov type have 2DCP. Open questions are included.

Most approaches for accelerating Markov chain mixing either rely on incorporating expensive geometric information in the proposals, or reduce the per-step cost of sampling via surrogate densities. We propose a localisation principle that allows a surrogate-based Metropolis-Hastings proposal to exploit gradient-level geometric information of the target density, without evaluating either the target gradient or the surrogate gradient. The construction relies on regularisation and tempering of the proposal measure. We show that the expected proposal displacement coincides with the Langevin drift up to controlled error. The resulting framework, Delayed Acceptance with Regularisation and Tempering (DART), achieves an $O(\kappa \max\{\kappa, d\})$ mixing time from warm start for strongly log-concave targets with condition number $\kappa$ in $d$ dimensions. This matches the known $O(\kappa d)$ rate for MALA when $d \ge \kappa$, and scales as $O(\kappa^2)$, independent of dimension, otherwise. This is, to our knowledge, the first mixing time guarantee for a surrogate-transition-based MCMC method. We demonstrate DART on a hierarchical spatial generalised linear mixed model. In this setting, the Dirichlet-Neumann averaging parametrisation, originally introduced for the efficient simulation of Gaussian processes, is repurposed to supply the surrogate, and its linear memory and log-linear arithmetic scaling in the number of observation sites carry over to inference.

We study two proximal point type methods for finding equilibrium points of pseudomonotone and strongly quasiconvex bifunctions. Extending results by A. Iusem and F. Lara, we prove the strong convergence of these methods over general complete geodesic metric spaces of nonpositive curvature, so-called Hadamard spaces. Our arguments are quite elementary and in particular effective, yielding sublinear non-asymptotic guarantees for the distance of the iterates towards the solution. These quantitative results are novel even in the context of Euclidean spaces, the original setting of the work by Iusem and Lara, and the simplicity of our arguments allows us to either weaken or even fully discharge various assumptions featuring in this previous work. We also provide an existence result for solutions of equilibrium problems generated by suitably semicontinuous, pseudomonotone and strongly quasiconvex bifunctions over general Hadamard spaces, and derive from this that every lower-semicontinuous strongly quasiconvex function over a Hadamard space has a minimizer, answering a question of the second author.

We introduce a two-dimensional non-hydrostatic model for shallow water wave dispersion. The model is based on a locally adapted application of a non-hydrostatic correction to the hydrostatic shallow water equations (SWE) in a predictor-corrector scheme. Applying the non-hydrostatic correction uniformly to the entire domain demands a high computational cost, since an elliptic system of equations needs to be solved for the correction terms. We demonstrate that by determining the area where the non-hydrostatic effects are significant, and applying the correction only locally, the computational effort can be reduced by approximately 40% without sacrificing accuracy in tsunami-like scenarios. As indicators for the non-hydrostatic effect, we use the ratio between total water depth and surface elevation, as well as horizontal velocity norms. Results are shown for several well-known test cases, including wave trains over a semi-circular shoal, static, and moving bottom tsunami-like wave propagation.

We prove a finite-scale estimate for vortex stretching in spatially filtered three-dimensional Navier–Stokes flow. The positive near-field part of the filtered stretching is bounded by a pairwise defect of filtered vorticity directions. A magnitude-weighted direction inequality converts this angular defect into a first-order difference quotient of filtered vorticity, and the resulting term is absorbed by filtered diffusion up to a lower-order enstrophy reservoir. In the localized filtered enstrophy balance, the remaining positive surplus is assigned to far-field strain, commutator forcing, and localization residuals. The far-field term is reduced to weighted packing and conditional annular Carleson embedding. The differentiated commutator stress is controlled by a scale-invariant increment defect adapted to the filter and its derivative. At the critical exponent, bounded increment defects generate cylindrical generalized Young-measure profiles.

In this paper we review the connection among continuity of the diffraction spectrum, the (uniform) vanishing of the Fourier–Bohr coefficients and the so called consistent phase frequency.

Positivstellensätze provide certificates of positivity for polynomials. Extending these certificates to symmetric functions, uniformly across all dimensions, presents structural challenges. For instance, the underlying domain is not semialgebraic. In this paper, we prove two Positivstellensätze for symmetric functions that are uniformly bounded below by some $\varepsilon > 0$. These are infinite-dimensional analogs of theorems of Pólya and Reznick. The proof relates evaluations of the (truncated) power sum map $(p_2,p_3,\dots)$ to moments of discrete probability measures on the compact interval $[-1,1]$. This yields a characterization of the closure of the orbit space of the infinite symmetric group on the sphere. Finally, we provide an alternative proof of existing Positivstellensätze for normalized symmetric functions.

We investigate a class of drift transformations of multidimensional diffusion processes generated through Doob $h$-transforms. These transformations provide a systematic approach for constructing analytically tractable stochastic models with prescribed probabilistic and statistical properties. We derive sufficient conditions under which the transformed diffusion admits an explicit transition density expressed through a product form involving a strictly positive harmonic function. Particular choices of this function lead to mixture representations of the transition density and to bimodality. We further analyze the effects of the transformation on stochastic ordering, diffusions in potential landscapes, and Poissonian resetting dynamics. In particular, we show that the product-form relation is preserved under resetting, enabling explicit characterization of the corresponding stationary distributions. Two multidimensional examples based on Wiener and Ornstein–Uhlenbeck processes illustrate the theory, providing closed-form expressions for transition densities, weight functions, and effective potentials. The two-dimensional setting is explored in detail, including symmetry effects and absorbing boundaries.

A recent breakthrough of Chen, Chen, Chen, Yin, and Zhang shows rapid mixing for Glauber dynamics for the hard-core model on random regular graphs beyond the tree uniqueness threshold. Their approach builds upon the literature of various local-to-global techniques and applies to a more general setting of discrete distributions supported on downward-closed set families. We give a short and self-contained proof via a Bochner–Bakry–Émery approach and directly show a Poincaré inequality by expanding the Dirichlet form in terms of the $L^2$-norm of the generator applied to a test function and eliminating a sum of squares term. Our proof is a streamlined version of an argument of Kondratiev, Kuna, and Ohlerich used to study spatial birth-and-death dynamics for Gibbs point processes in the continuum, which we adapt to the discrete setting.

We show that under a suitable additional hypothesis the restricted Zassenhaus $\F_p$-Lie algebra or the rational Magnus Lie algebra of a free amalgamated product is the free amalgamated product of the corresponding Lie algebras of the factors. This generalises a Theorem of A.I.\,Lichtman, who proved the analoguous statement for free products. Our conditions include the case when the amalgamated product is a retract in both factors. As a by-product, we show that a free product of residually torsion free nilpotent groups amalgamated along retracts is also residually torsion free nilpotent and obtain also some results on cohomological completeness. In the final sections we apply our main results to two recently raised open questions.

For an acyclic cluster algebra, the $c$-vectors are, up to sign, the real Schur roots of the associated root system. We study the two-coordinate projections $(c_v, c_w)$ of this configuration: when the difference $c_v - c_w$ is bounded, the image lies in a finite band of lattice lines, and we ask when the projection fills every lattice point of that band. In affine type, boundedness is equivalent to $\delta_v=\delta_w$ for the null root $\delta$. Writing this common coordinate as $r$, we prove that a band line fills exactly when its transjective root classes cover every residue modulo $r$. This yields a complete classification for every acyclic orientation of every simply-laced affine diagram. For a source-sink $\widetilde A_n$ quiver, every projection fills except the source-sink diagonal. In tree type, every width-one band fills, while every width-two band fails on its two boundary lines because of a congruence gap. The Auslander–Reiten defect further determines whether such a boundary line is finite or contains an infinite arithmetic progression; finiteness is governed by a balanced-geodesic criterion. On the annulus, the absolute defect is the crossing number with the core curve, and the $\delta$-shift is the Dehn twist along it.

We consider clusters formed by a Poisson ensemble of random walk loops on the $d$-regular tree with an intensity parameter $\alpha>0$ and a killing parameter $\kappa>-1$; the latter penalizes ($\kappa > 0$) or favors ($\kappa <0$) the appearance of large loops. We obtain an implicit formula for the critical curve $\kappa\mapsto \alpha_c(\kappa)$ for the percolation phase transition; the curve is positive if and only if $\kappa>\kappa_c = \frac{2\sqrt{d-1}}{d}-1$, differentiable away from $\kappa_c$, and has order $\sqrt{\kappa-\kappa_c}$ as $\kappa\downarrow\kappa_c$ and order $(1+\kappa)^2$ as $\kappa\to\infty$. We show that for each $\kappa>-1$, an infinite cluster exists exactly when $\alpha>\alpha_c(\kappa)$. Finally, we identify the near-critical behavior of the susceptibility and the percolation probability: for $\kappa>\kappa_c$, the critical exponents take the mean-field values, while for $\kappa=\kappa_c$, the phase transition is of a higher order with the percolation probability decaying quadratically in $\alpha-\alpha_c$.

A Class-Uniformly Resolvable Design (CURD) is a resolvable design in which each parallel class has the same block structure. We study CURDS in which each parallel class contains one block of size $m$ and the remaining blocks have size $2$, for $m \ge 3$. In addition to establishing necessary conditions for such a CURD to exist, we present two general constructions. The first transforms a particular type of cyclic design with block size $k$ into a CURD with partition $m^12^{\frac{n-m}{2}}$ where $m = 2k$. This construction is used to generate CURDS with 26 varieties (where $m=6$) and with 82 varieties (where $m=10$). The second constructs a CURD with partition $m^12^{\frac{n-m}{2}}$ for every value of $m$ that is the power of an odd prime.

We establish two Korovkin-type approximation theorems for sequences of positive linear operators acting on continuous functions on $[0,\infty)$. The first result shows that pointwise convergence on all polynomial test functions implies pointwise convergence for every function in the domain that is dominated by a polynomial. The second result provides an analogous criterion for functions of exponential growth, where the test functions are the exponentials $e^{\lambda x}$, $\lambda\in\mathbb{R}$. The results are applied to several classical and modified positive linear operators, including Baskakov, Szász–Mirakjan, Phillips, Post–Widder, Gamma, and Durrmeyer-type operators. Further applications are given to operators associated with modified Bessel functions and hyperbolic functions. The approach provides a unified framework for deriving pointwise convergence on unbounded intervals from convergence on suitable test functions.

Assuming the Generalised Riemann Hypothesis, we establish conjecturally sharp upper bounds for shifted moments of products of Dedekind zeta functions of arbitrary number fields. This improves results of Milinovich and Turnage-Butterbaugh and extends a recent result of Hagen. As applications, we obtain mean-square bounds for short-interval sums of the coefficients of Dedekind zeta functions and upper bounds for the large deviations of Dedekind zeta functions. Our results apply to both Galois and non-Galois extensions.

For $k\geq 1$, we prove that \[ [q^n z^s]J_k(z,q)\geq 0, \qquad (n\geq 0,\ s\in\mathbb Z) \] for the normalized Jacobi triple product tails \[ J_k(z,q) = \frac{ \sum_{j=k}^{\infty}(-1)^{j-k} q^{\binom{j+1}{2}}(z^{-j}+\cdots+z^j)} {(zq,q/z;q)_\infty}. \] This result not only implies Merca's stronger nonnegativity conjecture on truncated Jacobi triple product series in full generality, but also yields infinite families of linear inequalities for two-colored partitions and partitions with parts in the residue classes $\pm S \pmod{R}$. We present a combinatorial proof wherein a sign-reversing involution reduces the normalized Jacobi triple product tails to the invariant subsets according to the generalized minimal-excludant of partitions. Furthermore, by combing an invertible lift operator on Frobenius arms with Konan's size- and length-preserving bijection, an injection is constructed between the consecutive invariant subsets, which implies the coefficientwise positivity of the normalized Jacobi triple product tails.

The term tropical pseudonorm refers to a family of (not necessarily symmetric) gauge functions that arise in tropical or idempotent geometry. An important characteristic of these gauges is their invariance under translation by a constant vector, allowing them to descent naturally to tropical projective spaces. In this work, we explore the tropical one-infinity pseudonorm, a polyhedral hybrid gauge that allows for tunable asymmetry, in the context of a Fermat–Weber location problem. We extend previous formulations in considering non-finite data, and we investigate several variants of the inverse problem, providing linear programming formulations for their solution.

In a recent paper, Antić et al.~studied collections of planar orthogonal drawings of a graph where every edge is unbent in at least one drawing. This paper generalizes this concept to non-planar drawings, and shows that then two drawings always suffice (for planar drawings three drawings are sometimes needed). The results can also be generalized to $s$-grid drawings for $s\geq 3$.

We study intersections of conjugacy classes of square matrices over a finite field with affine coordinate subspaces, or equivalently matrices in a fixed adjoint orbit with prescribed entries. Our main result treats the case of prescribed columns: for a partially defined linear map we give a Hall scalar product formula for the number of extensions to an endomorphism with prescribed similarity invariants. This formula is expressed in terms of skew modified Hall–Littlewood functions and $q$-Whittaker functions. As applications, we count monic matrix polynomials over $\mathbb{F}_q$ with prescribed Smith normal form and with prescribed determinant, and recover the Gerstenhaber–Reiner formula for the number of square matrices with a fixed characteristic polynomial. We also note that known point-count formulas for Hessenberg varieties imply related formulas for Hessenberg supports involving chromatic quasisymmetric functions, motivating polynomiality questions for more general supports and prescribed affine slices.

This paper introduces and studies $(2,3)$-palintropic algebras, a class of commutative algebras defined by the identity $(x^{2})^3 - (x^{3})^2 = 0$. This specific relation is the simplest generator of the $2$-dimensional space of minimal-degree evanescent identities in degree $6$, and encompasses several well-studied structures, including Jordan and medial algebras. The primary motivation for investigating these algebras lies in their trivial Peirce polynomials, which removes a priori restrictions on the spectrum of the multiplication operator associated with an idempotent. This paper reviews and further develops the theory of Peirce operators, Peirce polynomials, and second-order linearizations. It shows that despite the triviality of the Peirce polynomial, any idempotent $c$ admits well-behaved, explicit fusion rules for multiplication between its $\lambda$-Peirce spaces for $\lambda \neq \tfrac{1}{2}$. Furthermore, multiplication by a nonsingular idempotent is an algebra homomorphism. Finally, there are given concrete examples of $(2,3)$-palintropic algebras, together with applications of these algebraic structures to commutative polynomial maps.

We establish a correspondence between the free-field construction and the minimal-model construction of the Calabi–Yau sector of the four-dimensional heterotic string compactified on Berglund–Hübsch type Calabi–Yau manifolds and their orbifolds. For Fermat-type polynomials the Calabi–Yau vertex operators expressed in terms of free fields are shown to correspond to products of primary fields of $\mathcal{N}{=}2$ minimal models. Using this correspondence we verify modular invariance of the free-field construction and extend it to Berglund–Hübsch Calabi–Yau orbifolds, deriving the conditions on complete vertex operators that parallel those of the minimal-model construction.

Recent numerical computations and stochastic modeling by Brevitt and Klages suggest that introducing a hole in a Pomeau–Manneville map can suppress survivor-conditioned Lyapunov stretching. We prove a deterministic renewal theorem which explains this phenomenon and its observable-level generalizations. For an open intermittent map induced on a base away from the neutral fixed point, we describe the asymptotic distribution of the number of completed survivor returns to the base, conditioned on survival up to time $t$. The limiting law is expressed in terms of the killed induced transfer operator; for the conditionally invariant density of the killed induced system it is geometric. We then prove two reward results for additive observables. A reward domination theorem gives bounded survivor-conditioned Birkhoff sums, while a stronger final-tail asymptotic gives convergence to a finite limit. For generalized Pomeau–Manneville maps, bounded observables satisfying $\lvert \psi(x) \rvert \leq C x^{\gamma}$ near the neutral fixed point and a mild variation condition satisfy the domination hypotheses. When the neutral branch and final tails satisfy the corresponding regularity assumptions, asymptotically regular observables satisfy the convergence hypotheses. In particular, $\psi=\log\lvert f' \rvert$ gives bounded survivor-conditioned Lyapunov stretching for the generalized class; under these additional regularity assumptions, it converges. Under an additional entropy-domination assumption, we also derive a zero entropy-rate consequence for survivor return-length names and record the complementary linear growth of stretching when the hole contains a neighborhood of the neutral fixed point.

We extend the study of contact cosmetic surgeries to Legendrian knots in integer homology sphere L-spaces . We prove that the contact cosmetic surgery conjecture holds for all non-trivial Legendrian knots in this setting, with the possible exception of Lagrangian slice knots. Our argument adapts and refines techniques from the S3 case to the broader context of L-spaces, incorporating constraints arising from Heegaard Floer theory

Generalizing a construction due to Argyros and Motakis \cite{AM}, we define a nonseparable abstract interpolation space associated to any given reflexive space with an unconditional basis together with the Schreier spaces associated to an increasing sequence of compact families of finite subsets of an uncountable set. Under a mild complexity assumption on the families, we prove that the interpolation space admits a $2$-rotund norm with an uncountable $1$-unconditional basis.

We consider the cubic nonlinear Schrödinger (NLS) equation on two-dimensional flat tori with varying aspect ratios. In this formulation, the choice of aspect ratio governs the Fourier resonance structure, so rational and irrational geometries can exhibit different high-frequency cascade behaviors. We present a geometry-conditioned Fourier neural operator (FNO) for the cubic defocusing NLS equation, where the input consists of the real and imaginary parts of the solution together with the aspect-ratio parameter \(\omega^2\). The model is trained to approximate the one-step solution operator and is evaluated on unseen trajectories generated from random-phase initial data using Fourier pseudospectral method. Our numerical experiments show that the learned operator captures the main solution dynamics on both tori and reproduces the distinct Sobolev norm behavior of the two geometries, with stronger \(H^2\)-growth on the rational torus and more constrained behavior on the irrational torus, consistent with the findings of \cite{hrabski2021energy}. We perform ablation studies to examine the roles of retained Fourier modes, activation functions, Fourier-layer depth, and explicit geometry conditioning. The results indicate that including $\omega^2$ improves long-time predictive accuracy, especially for the rational geometry, and supports the use of geometry-aware neural operators for learning spectral-transfer phenomena in nonlinear dispersive partial differential equations.

In this paper, we study totally disjoint diametral paths in simple connected graphs. A diametral path in a graph is a shortest path that connects two vertices whose mutual distance is equal to the diameter of the graph. Totally disjoint paths are paths that have no vertices in common, including their end vertices. We show that the problem of deciding whether a graph $G$ has $k$ totally disjoint diametral paths is NP-complete. We consider restricted classes of graphs for which the problem of determining the maximum size of a set of totally disjoint diametral paths is readily solved. We then give a linear-time algorithm for a subclass of maximal outerplanar graphs called 2-paths, define a polynomial-time algorithm for threshold graphs, and establish a structural bound for proper interval graphs. Finally, we define classes of extremal graphs with $k$ totally disjoint diametral paths of length $d$ having the fewest possible number of edges.

We present a study on the time-domain propagation of electromagnetic waves in dielectric materials modeled by a nonlinear Debye medium with random perturbations. Polynomial Chaos Expansions are employed to transform the random nonlinear Debye polarization model into a deterministic framework. We extend the Yee discretization to the resulting coupled system, establish second order accuracy, and verify convergence numerically. We investigate the sensitivity of nonlinear properties to uncertainty, particularly when the amplitude of the input signal is large. Given the challenges in manufacturing where uncertainties can cause optimal parameters to vary and potentially disrupt nonlinear effects, our approach incorporates these uncertainties within the simulation. This can enable the model-based design identification of realizable materials that maintain their desired effects despite variations. The findings from this study contribute to a deeper understanding of wave propagation in complex media, with potential implications for applications in optical communications, material science, and electromagnetic wave control.