Papers

On doubling metric measure spaces, we study nonlocal operators with non-standard $(p,q)$-Orlicz growth ($1<p\leq q<\infty$), where the interaction kernel is given by a general, non-translation-invariant measure. Under natural assumptions on the interaction measure - namely symmetry, a suitable nonlocal Poincaré inequality, and a tail bound - we prove that every weak solution to the corresponding homogeneous nonlocal equation admits a locally Hölder continuous representative. These regularity results are new even in the Euclidean setting.

The topological Halmos–von Neumann theorem is a fundamental result of topological dynamics that completely classifies continuous actions of abelian topological groups which are ergodic and have discrete spectrum. It rests on a correspondence between such actions (up to the choice of a distinguished point), compactifications of the acting group and subgroups of its dual group. We generalize both the classical statement and the underlying categorical equivalences to the framework of open group bundles over locally compact base spaces. As a byproduct, we establish a Pontryagin-type duality between étale Hausdorff abelian group bundles and open proper Hausdorff abelian group bundles, extending the classical duality between discrete and compact abelian groups.

We use tools and techniques from $p$-adic analysis and algebraic number theory to study the algebraicity of the hard square entropy constant and its high dimensional analogues. Specifically, we study arithmetic properties of $a_d(n)$, the number of independent sets in the $d$-dimensional discrete torus, and the associated entropy constants $\kappa_d=\lim_{n\to\infty}a_d(n)^{1/n^d}$. It is not known whether $\kappa_d$ is algebraic or transcendental for $d>1$. Using the fact that the sequence $a_d(p^k)$ converges $p$-adically for every prime $p$ and other arithmetic facts, we present a collection of criteria for the algebraicity of $\kappa_d$ and bound the number of possible values of prime powers $p^k$ for which $a_d(p^k)=\kappa_d^{p^{kd}}$.

We study which probability measures on $[0,\infty)$ can occur as the distribution of the distance between two independent points sampled from a complete separable metric space equipped with a Borel probability measure. We prove that an atomless Borel probability measure can be realised in this way if and only if its support contains the origin. This settles the conjecture of Aldous, Blanc, and Curien for absolutely continuous laws and also covers atomless measures that are singular with respect to Lebesgue measure. Moreover, for every $L>1$, we show that the realising metric may be chosen $L$-bi-Lipschitz equivalent to an ultrametric. The proof constructs the space from compact components whose mutual distances produce prescribed parts of the measure, while distances within the components are assigned to smaller scales.

We prove that the Fourier coefficients of the canonical critical Gaussian multiplicative chaos on the circle vanish almost surely at infinity. More precisely, let $M_\phi^{\mathrm{crit}}$ be the canonical critical chaos associated with the centered circle field $\phi$ of covariance $\mathbb{E}[\phi(\theta)\phi(\theta')] = \log\frac{1}{\lvert e^{i\theta}-e^{i\theta'}\rvert}$. Then, almost surely, $\widehat{M_\phi^{\mathrm{crit}}}(n)\longrightarrow0$ as $\lvert n\rvert\to\infty$. This resolves the almost-sure critical Rajchman problem for the canonical circle field. Since critical chaos has Fourier dimension zero almost surely, no positive polynomial Fourier-decay rate can hold; the theorem therefore exhibits qualitative Fourier cancellation beyond the regime of positive Fourier dimension. The proof addresses two coupled difficulties: the heavy, nonuniform cell masses of critical chaos and the need to control exponentially many frequencies in each dyadic annulus. For an auxiliary periodized compact-range star-scale field, a derivative-rooted Bessel regression yields weighted small-cell summability and moving-tail control of exceptional large cells. After conditioning at a coarse scale below the Fourier scale, finite-range independence and conditional Bernstein concentration reduce uniform control of the terminal Fourier coefficients over each dyadic annulus to a spatial-variation estimate for a coarse predictable measure. A smooth positive-definite covariance correction and critical-chaos uniqueness then transfer the Rajchman property to the canonical critical chaos of the exact circle field.

In the present paper, a study is made of first-order differential properties of certain multifunctions, which are defined as a solution map associated with a family of parameterized “split" feasibility problems. The latter are a class of convex feasibility problems, whose specific structure makes them suitable for well-established applications in several areas of engineering and systems biology. As a result, inner and outer approximations of the graphical (contingent) derivative of the solution map are provided, which are expressed in terms of derivatives of the problem data. As an application, a perspective is also discussed for employing some of these achievements in the formulation of optimality conditions for mathematical programs with “split" feasibility constraints.

We establish local well-posedness for a class of non-linear heat equations on the torus $\mathbb{T}^n$ whose highest order non-linearities have scaling-critical dimension $-1/k, \, k\in \mathbb{N}$, when the initial condition is a random field satisfying a spectral gap inequality involving a suitable Lebesgue norm. The corresponding subcriticality condition matches that of the Gaussian free field in dimension $d<2+2/k$. This extends previous subcritical well-posedness results beyond the Gaussian setting. Moreover, our proof is simpler and avoids diagrammatic arguments.

We prove explicitly the n-point K-identities we proposed previously in the calculation of one tensor hard string scattering amplitudes (HSSA). These K-identities were the key to obtain the stringy scaling behavior in the saddle point calculation of the n-point HSSA and, on the other hand, to consistently match with the calculation of decoupling of zero norm states. Moreover, we introduce a G function to generate an infinite set of generalized K-identities (GKI). The lowest order set of these GKI is the scattering equations (SE) used in the calculation of field theory amplitudes in the CHY formalism. The next to leading order set of these GKI is the K-identities used previously in the calculation of one tensor HSSA. We expect that the higher order sets of these GKI can find applications in calculating other HSSA.

A $\lambda$-self-expander $x: M^n\to \mathbb{R}^{n+1}$ is the solution of the isoperimetric problem of weight $e^{\frac{|x|^2}{4}}$. In this paper, we prove that the closed $\lambda$-self-expander is a round sphere centered at the origin under some curvature conditions. These curvature conditions mainly include the scalar curvature, the mean curvature, and the squared norm of the second fundamental form.

We show that length functions on hyperbolic groups whose induced metrics are hyperbolic and weakly geodesic naturally give rise to compact quantum metric spaces, thus generalizing the result of Ozawa and Rieffel. As a result we obtain several new examples of compact quantum metric spaces, in particular arising from Green metrics associated with random walks on hyperbolic groups, and from proper cocompact isometric actions on proper geodesic hyperbolic spaces admitting a point with trivial stabilizer.

We study the Cauchy problem $$ v_{tt}-v_{xx}=\abs{v_t+v_x}^{m}\abs{v_t}^{n}, \qquad v(x,0)=\eta\varphi(x),\quad v_t(x,0)=\eta\psi(x), $$ on $\R$, with compactly supported profiles $\varphi\in C_0^2(\R)$, $\psi\in C_0^1(\R)$, exponents $m,n>1$, and amplitude $\eta>0$. We show that the sign of $P_0=\psi+\varphi'$ decides the behaviour of small solutions, whatever the sign of $\psi-\varphi'$. If $P_0$ is positive at some point, the lifespan $T(\eta)$ obeys explicit two-sided bounds of order $\eta^{-(m+n-1)}$ for every $\eta>0$, and $$ \lim_{\eta\to0^+}\eta^{m+n-1}T(\eta)=\frac{2^{n}}{(m+n-1)\,(\max_{\R}P_0)^{m+n-1}} . $$ If $P_0\le0$, the solution is global for every amplitude below an explicit threshold. If moreover $P_0\not\equiv0$, the component $v_t+v_x$ decays at the universal rate $\bigl(2^{n}/((m+n-1)t)\bigr)^{1/(m+n-1)}$, and the gradient of the solution converges uniformly to that of a free wave travelling to the right; if $P_0\equiv0$, the solution is itself such a travelling wave. The analysis rests on a localization property: the zero set of $v_t+v_x$ is invariant along its own characteristics, so that the nonlinear source stays in a slab of fixed width moving with speed one, and each characteristic of the other family is forced only during a bounded time. This yields the global existence, the long-time behaviour and the exact value of the limit. The results extend to the endpoint exponents $m,n\ge1$ and to sources $f(v_t+v_x)g(v_t)$, for which $T(\eta)$ is asymptotic to an Osgood-type integral.

Theory of Rota–Baxter operators on rings and algebras has been developed since 1960. In 2021, L. Guo, H. Lang, Y. Sheng have defined the notion of Rota—Baxter operator on a group. We classify all Rota—Baxter operators on the finite Heisenberg group and the extraspecial group of order $p^3$. We also investigate an analogue of the spectral property for Rota—Baxter operators on algebras.

The aim of this article is to discuss the proof that the norm principle holds for the group of similitudes ${\rm Go}(q)$ and the spinor group ${\rm Spin}(q)$ for a quadratic form $q$ defined over $F$.

We study regular unimodular triangulations, the integer decomposition property, and Ehrhart-theoretic properties of two families of Hermite normal form simplices. We first consider the one-row case associated with the vector $(N - 1, \dots ,N - 1 , N)\in \mathbb{N}^d$, and completely characterize when the corresponding simplices admit a regular unimodular triangulation. Our constructions are explicit and also yield closed formulas for the $h^\ast$-polynomial and the local $h^\ast$-polynomial. Moreover, we prove Ehrhart positivity and derive explicit dimension-dependent conditions under which the Ehrhart polynomial is not unimodal. Finally, we extend our approach to the two-row cases associated with $(1, \dots ,1 , N)\in\mathbb{N}^d$ and $(M-1, \dots ,M-1, M, 0)\in\mathbb{N}^d$. In these cases, we construct regular unimodular triangulations, derive closed formulas for the $h^\ast$-polynomial and the local $h^\ast$-polynomial, and prove Ehrhart positivity.

The reduction of mechanical presymplectic structures and its application to the reduction of time-dependent Hamiltonian systems were developed in a recent paper (Gutiérrez Sagredo et al, 2025). This approach overcomes some limitations of the original Albert reduction (Albert, 1989). In this paper we will describe the reduction of time-dependent Hamiltonian systems using the extended formalism. This process combines cotangent bundle reduction with reduction of presymplectic structures of corank 1 and 2. These results generalize previous work (Lacirasella et al, 2012), in the same direction. Several examples will be presented to illustrate the theory from this new perspective.

Let $n\ge2$ be even, let $\lambda=(\lambda_1,\ldots,\lambda_n)\in\mathbb{R}^n$ have pairwise distinct coordinates, and define the difference-power matrix \[ A_d(\lambda) := \bigl[(\lambda_r-\lambda_s)^d\bigr]_{r,s=1}^n, \qquad d\in\mathbb{N}. \] In 1928, Colombo proved that $\det A_{n-1}(\lambda)\ne0$—and hence $\det A_{n-1}(\lambda)>0$—and that $\operatorname{rank} A_d(\lambda)=d+1$ for $0\le d<n-1$. He conjectured that \[ \det A_d(\lambda)\ne0 \qquad\text{for every } d\ge n-1. \] For even $d$, the conjectured nonsingularity follows from previously published results on distance-power matrices. The remaining open cases were therefore the supercritical odd exponents $d\ge n+1$. We prove nonsingularity for all these odd exponents, thereby completing Colombo's conjecture. Consequently, \[ \operatorname{rank} A_d(\lambda)=\min\{n,d+1\} \qquad(d\in\mathbb{N}). \] Our proof converts a hypothetical kernel vector into a real binary form having more projective real linear factors, counted with multiplicity, than its real Waring length permits.

Let $\Bbbk$ be an algebraically closed field of characteristic zero. We classify the locally finite derivations of arbitrary Ore extensions of $\Bbbk[x]$, thus extending van den Essen's \cite{V92} classification for the polynomial algebra $\Bbbk[x,y]$ to this noncommutative setting. More precisely, we consider the three families arising in the classification of Ore extensions of $\Bbbk[x]$: the quantum plane, the first quantum Weyl algebra, and the differential Ore extensions \[ A_h=\Bbbk[x][t;h(x)\partial_x]. \] For both the quantum plane and the first quantum Weyl algebra, we determine the locally finite derivations and explain how the resulting classifications are related to the work of Suárez-Alvarez and Vivas \cite{SuarezVivas} on generalized Weyl algebras. For the algebras $A_h$, with $h$ nonconstant, we obtain a complete classification in both the square-free and non-square-free cases. As a consequence, we show that $\LFD(A_h)$ is a solvable and weakly locally finite Lie subalgebra of $\Der(A_h)$, although it is not locally finite as a set of derivations.

The basic issue concerns sets of values of rational functions at rational points of an algebraic variety, namely image f(X(k)), where X is an algebbaric variety and f is a rational function on X, defined over the number field k. For instance, we shall prove that for X an abelian variety, the map between rational points is never surjective. This is reminiscent of the Hilbert Property, but here the fibers may have arbitrary dimension. One of our examples concerns the classical Hilbert Property: we produce a simply connected affine surface whose set of integral points is Zariski-dense and thin, disproving a plausible expectation. We shall also discuss hieghts and integrality issues; in this context, a role will be played by 'gcd estimates'. In the first Appendix, written by D. Masser, an effective estimate of some relevant gcd is provided.

In this paper we study irreducible simplicial arrangements of projective planes in $\mathbb{P}^{3}(\mathbb{R})$ from combinatorial and projective geometry viewpoints. We first formulate a simpliciality criterion in terms of incidences between rank-two and rank-three flats, together with equivalent formulations using face numbers, reduced restrictions, and characteristic polynomials. We also relate the classical planar restriction data of Grünbaum-Shephard to Ziegler multirestrictions. Our principal result concerns the rank-four special-vertex property: among the irreducible crystallographic Coxeter arrangements of rank four, the arrangements of types $A_4$ and $B_4$ admit a special vertex, whereas those of types $D_4$ and $F_4$ do not. A simplicial deletion chain inside $B_4$ supplies further irreducible examples with a special vertex. Finally, we compare rank-flat, Purdy-type, and Grünbaum-Shephard defects for these arrangements.

We prove a dimension-free count of lattice points in high-dimensional $\ell^q$ balls and spheres with small radii. Our result complements those already established for $q=1,2$ by covering the whole range of integers $q\ge 2$.

In this paper, we investigate the behavior of associated primes of powers of monomial ideals, with particular emphasis on the presence of the maximal ideal and the phenomenon of fluctuation. We establish criteria for determining when the maximal ideal belongs to the set of associated primes of powers of monomial ideals in $K[x,y,z]$, and provide examples illustrating its appearance and disappearance among the associated primes of successive powers. Furthermore, we prove that for every $n\geq 3$, there exist infinitely many monomial ideals in $K[x_1,\ldots,x_n]$ whose powers exhibit fluctuations in their sets of associated primes. Finally, we construct infinitely many examples of nearly normally torsion-free (respectively, co-nearly normally torsion-free) monomial ideals that fail to satisfy the persistence (respectively, copersistence) property.

Suppose $(M, g)$ is a smooth, closed, $5$-dimensional Riemannian manifold with positive Yamabe invariant $Y(M, [g]) > 0$ and positive Yamabe-type $Q$-curvature invariant $Y_{4}^{\ast}(M, [g]) > 0$. Using the continuity method, we prove the existence of a metric in the conformal class $[g]$ with positive scalar curvature and positive $Q$-curvature, assuming an additional condition on an “initial metric" in the class $[g]$.

We solve the matricial univariate rational truncated moment problem on the real line and on a half-line. We give explicit necessary and sufficient conditions for the existence of a positive matrix-valued representing measure and show that every solvable problem admits a finitely atomic representing measure whose total atomic multiplicity equals the rank of the associated moment matrix. The rational problem is reduced to an ordinary matricial moment problem with the additional requirement that the representing measure avoid the real poles of the rational data. The main new ingredient is a simultaneous prescribed-node result: the smallest attainable multiplicities at finitely many prescribed points can be realized by a single minimal representing measure. Our proofs are constructive and use flat extensions, block-column relations, and localizing conditions.

In this work, we show how heteroclinic networks can arise in a simple class of network dynamical systems through the application of the Invariant Sphere Theorem. Ring-coupled systems are ODE networks in $\mathbb{R}^n$ in which each variable $x_i$ interacts only with its predecessor $x_{i-1}$. We derive conditions on the coefficients of a cubic polynomial that guarantee the existence of a globally attracting invariant sphere via the Invariant Sphere Theorem. Moreover, setting one of these coefficients to zero, we identify conditions on the remaining coefficients that guarantee the existence of a heteroclinic network on the invariant sphere. Focusing on the case $n=3$, we investigate perturbations of the vanishing coefficient in a neighbourhood of the heteroclinic network. Under such perturbations, the heteroclinic network is destroyed and periodic orbits emerge. When the original heteroclinic network is asymptotically stable, the resulting periodic orbits shadow the network. In one parameter regime, a unique attracting periodic orbit appears and shadows the entire heteroclinic network. By contrast, when the heteroclinic network is not stable, repelling periodic orbits arise that shadow only part of the heteroclinic structure, namely half of the heteroclinic connections. Symmetry plays a fundamental role throughout the analysis. Exploiting the symmetries of the system, we reduce the heteroclinic network to two homoclinic orbits. Furthermore, the local dynamics near the heteroclinic network can be studied on a quotient space consisting of an annulus attached to a Möbius band, with each homoclinic orbit lying on one of these components. This reduction provides a geometric framework for understanding the bifurcations and the emergence of the periodic dynamics.

This paper determines the exact lengths of shortest self-orthogonal and LCD embeddings of linear codes over $\mathbb{F}_q+u\mathbb{F}_q$. By decomposing Gram matrices over $\mathbb{F}_q+u\mathbb{F}_q$ into pairs of symmetric matrices over the finite field $\mathbb{F}_q$, the embedding problems are reduced to the congruence classification of symmetric and alternate matrices over finite fields. Complete formulas for the shortest self-orthogonal embedding length are obtained, with two distinct cases arising in both even and odd characteristic. We also show that every self-orthogonal code over $\mathbb{F}_q+u\mathbb{F}_q$ with nonzero free rank can be viewed as a shortest self-orthogonal embedding of another code. We use Witt theory to construct all shortest self-orthogonal embeddings. A complete characterization of shortest LCD embeddings is also established in terms of invertible and arbitrary matrices of prescribed sizes appended to a generator matrix. Examples of self-orthogonal and LCD embeddings with the largest minimum distance for the code considered are also presented, some of whose Gray images are optimal codes over $\mathbb{F}_q$.

We study the non-uniqueness of exact two-sided $p$-values in continuous one-parameter natural exponential families (NEFs). For directed one-sided problems, the tail $p$-value agrees with the $p$-values using UMP, UMPU, and likelihood-ratio (LR) tests. For a two-sided simple null, we distinguish four constructions: equal-tail, density-ordered, UMPU, and LR $p$-values. At a fixed null parameter, UMPU and equal-tail $p$-values coincide if and only if the null law is symmetric about its mean; under a regular two-branch density-level condition, the same fixed-null symmetry characterization holds for UMPU versus density ordering and equal-tail versus density ordering. Requiring any of these coincidences throughout the NEF characterizes the Gaussian family. We combine these results with the theorem of Bar-Lev, Bshouty and Letac that UMPU and LR $p$-values coincide throughout a continuous NEF precisely for the normal, gamma and inverse-Gaussian families. We also investigate the two LR pairings not covered by those results. If equal-tail and LR $p$-values coincide throughout a NEF satisfying our standing regularity assumptions, then $(V^{2/3})^{\prime \prime \prime }=0$ on the mean domain. The same coincidence also forces an explicit density-at-the-mean identity. For an i.i.d.\ sample with canonical sufficient statistic $T_n=\sum_{i=1}^nX_i$, persistence of equal-tail-LR coincidence throughout the family along an unbounded sequence of sample sizes forces Gaussianity. A corresponding density-LR statement is given conditionally on an explicitly stated differentiated local Edgeworth expansion. Finally, inverse-Gaussian and hyperbolic-secant examples quantify numerical $p$-value differences, disagreement of rejection decisions, and differences in power.

In this paper, we introduce a generalized framework for star products that preserve associativity while not necessarily obeying the canonical commutation relations. Within this framework, we formulate the augmented star product and investigate the associated augmented star exponentials. We further demonstrate several applications of these constructions to special functions and to problems arising in quantum physics.

We show that the homeomorphism groups of the following spaces have precisely $2^{2^{\aleph_0}}$ maximal subgroups: the rational numbers $\mathbb{Q}$, the Baire space $\mathbb{N}^{\mathbb{N}}$, the space $\mathbb{N}\times 2^{\mathbb{N}}$ where $2^{\mathbb{N}}$ is the Cantor set, the ordinal $\omega^2$ under its order topology, and the Sorgenfrey line $\mathbb{S}$. More generally, we find sufficient conditions on a group $G$ acting on a topological space which imply that $G$ has at least $2^{2^{\aleph_0}}$ maximal subgroups. Moreover, if the groups $\operatorname{Homeo}(\mathbb{Q})$ and $\operatorname{Homeo}(\mathbb{N}^\mathbb{N})$ are equipped with the pointwise topology, then it is shown that $\operatorname{Homeo}(\mathbb{N}^\mathbb{N})$ has precisely $2^{\aleph_0}$ open maximal subgroups, and $\operatorname{Homeo}(\mathbb{Q})$ has precisely $\aleph_0$ open maximal subgroups and $2^{\aleph_0}$ closed maximal subgroups.

We consider an inverse scattering problem for the scalar wave equation in which the point emitter, a point-like scatterer, and multiple receivers are all moving. The goal is to reconstruct the trajectory of a moving point-like scatterer from time-dependent measurements of the scattered field. To this end, we establish a rigorous point-interaction model for the moving scatterer in the time domain, proving well-posedness of the forward scattering problem, including existence, uniqueness and continuous dependence of the scattered field on the scatterer trajectory and scattering parameter. By exploiting the retarded-time structure, we introduce distance functions that connect emission, scattering, and observation processes, and show that they satisfy a coupled system of nonlinear ordinary differential equations determined by the measurement data. Based on this formulation, we propose a reconstruction algorithm that combines initial localization with the solution of the derived ODE system. Numerical experiments demonstrate that the method is accurate and robust with respect to noise, with reconstruction errors remaining stable under moderate perturbations.

Spin-boson models are paradigmatic examples of open quantum systems and serve as the theoretical paradigm for current quantum computers based on single-ion trap technology. Despite their ubiquity, a complete spectral diagonalization of these models remains an open problem, except in highly singular regimes. This paper establishes a rigorous framework for the approximate diagonalization of generalized spin-boson systems. Our approach is inspired by the Brockett-Wegner double-commutator flow, which is a non-linear differential equation governing the evolution of (here unbounded) operators. Unlike relatively recent applications of this flow to quadratic Hamiltonians in quantum field theory, it does not close in the spin-boson context. We overcome this fundamental obstruction by performing a detailed analysis of the resulting non-closed algebraic structure, allowing us to explicitly bound the higher-order error term with respect to the spin-boson coupling strength. Consequently, this work provides the first mathematically rigorous justification for several heuristic diagonalization techniques widely employed in the theoretical physics literature for small-coupling regimes, in the simplest non-trivial cases. More broadly, our framework renders a flow-based algorithm feasible for systematic higher-order diagonalization and self-energy renormalization. This strategy is conceptually akin to multi-scale analysis, or, much more recently, to the iterative, local Lie-Schwinger block-diagonalization method by Fröhlich and Pizzo.