Papers

Random sums of independent random variables have been extensively studied in classical probability theory. We consider random sums of self-adjoint variables from a non-commutative probability space, and establish several $*$-convergence results. In particular, we show that the joint $*$-convergence of the standardized random sum of identically distributed self-adjoint variables and the standardized stopping random variable (rv) is equivalent to the convergence of all moments of the stopping rv together with the convergence of the ratio of its mean to its variance. We obtain central limit theorems for the random sums of free, independent and half independent self-adjoint variables with both deterministic and random scaling. Furthermore, we derive some scaling $*$-convergence limits for randomly indexed self-adjoint variables.

Ferrers diagram rank-metric codes are rank-metric codes with prescribed support, and their dimension is bounded from above by the Etzion–Silberstein bound. In this paper, we study this problem for block Ferrers diagrams, namely Ferrers diagrams whose dots are grouped into square blocks of a fixed size. Motivated by the diagonal construction for MDS-constructible Ferrers diagrams, we introduce the notion of MSRD-constructibility, where MDS codes on diagonals are replaced by maximum sum-rank distance (MSRD) codes on block diagonals. We show that MSRD-constructible pairs yield optimal Ferrers diagram rank-metric codes over sufficiently large finite fields. We then relate MSRD-constructibility of a block Ferrers diagram to MDS-constructibility of its contraction, proving an equivalence when the distance is compatible with the block size and giving lifting criteria in the general case. As a consequence, we obtain MSRD-constructibility for strictly block-monotone and initially block-convex diagrams. Finally, we prove a reduction to block triangular diagrams and use it to obtain new arbitrary-field cases of the Etzion–Silberstein conjecture for MSRD-constructible block Ferrers diagrams.

In this paper, we propose and analyze a Continuous Interior Penalty (CIP) stabilized Virtual Element Method (VEM) for three-dimensional advection-diffusion-reaction equations on general polyhedral meshes. While CIP-VEM schemes have been recently explored in a two-dimensional setting, their analysis heavily relies on global mesh quasi-uniformity and constant physical parameters. We overcome these limitations by introducing a novel three-dimensional variant of the Oswald-type quasi-interpolant. This allows us to establish robust, uniform error estimates in the hyperbolic limit under a realistic local quasi-uniformity assumption and variable coefficients. Finally, we provide a comprehensive set of three-dimensional numerical experiments to validate the theoretical convergence rates and demonstrate the absence of non-physical oscillations.

Circular coordinates obtained from persistent cohomology reveal loop structure in data, but they usually remain abstract: A detected circle does not tell us which measured angle, phase, torsion, or decoder explains it. We propose a method for selecting interpretable circle-valued coordinates from a user-supplied dictionary of scientifically meaningful candidates explaining the detected cohomology. In the continuous setting, each candidate is represented by the cohomology class of its pulled-back angular form, and selecting a minimum-energy set of candidates spanning the relevant $H^1$ subspace becomes a minimum-weight basis problem in a vector matroid. We then introduce CIRCOL, a method for discrete point clouds sampled from the manifold. We prove that the introduced cochain inner product is a consistent estimator of the $L^2$ inner product of fixed smooth 1-forms under non-uniform sampling. The resulting projection matrix both helps selecting a basis of low-energy dictionary coordinates and diagnoses topologically trivial candidates or unexplained persistent classes. Finally, we verify the effectiveness of our method on synthetic examples, on molecular simulations, and neural recordings of head-direction cells.

In this paper, we investigate a class of so-called sub-infinite horizon stochastic linear-quadratic optimal control problems, in which the initial time $t$ is arbitrarily taken from $[0,\infty)$ and the running cost is defined over $[t,t+T]$ for a given $T>0$. The optimal control of this type of problem can be obtained by standard methods; however, it is shown that the resulting optimal control is generally time-inconsistent. Thus, instead of seeking an optimal control, which is time-inconsistent, we aim to find a time-consistent, locally optimal, and time-invariant equilibrium strategy, by introducing a new and very interesting type of Riccati equation. Its main feature is that the generator depends on a delay term of the unknown. In other words, this Riccati equation is a backward ordinary differential equation (ODE) with delay, which is equivalent to a forward ODE with advanced terms. Such an equation is essentially a Fredholm integral equation, whose solvability is challenging. We overcome the difficulty by deriving a sharp a priori estimate and applying the Leray–Schauder fixed point theorem. To this end, we establish a comparison theorem between two matrix-valued nonlinear algebraic equations. The convergence behavior of the solution to the Riccati equation as $T\to\infty$ is also provided.

In this paper, we present a complete characterization of spherically symmetric Finsler metrics that satisfy the $\sigma T$-condition. We further investigate the subclass of such metrics within the Landsberg category and identify the precise conditions under which spherically symmetric Finsler metrics satisfying the $T$-condition must also be Berwaldian. In addition, we construct new non-regular solutions to the classical unicorn problem, providing explicit examples of Landsberg metrics that are not Berwaldian.

We introduce a categorical analogue of weak hypercube representations of finite posets by means of faithful embeddings into categories of subwords of finite words. For finite acyclic categories, we characterize those admitting such a weak subword representation: they are precisely the monic categories whose hom-sets carry a left-compatible local total order. The proof is constructive and gives an explicit word representation. We also introduce a query game for categories, generalizing a Boolean query game for posets, and show how winning sets produce explicit word representations and hence upper bounds for the weak word dimension.

The proximal operator is a fundamental tool in variational analysis and optimization. In the setting of a Hilbert space, given a proper, lower semicontinuous convex functional, its proximal operator is non-expansive, that is, 1-Lipschitz continuous. In the Wasserstein setting, the contraction properties of this operator have been investigated from different perspectives by Carlen and Craig and by Adve and Mészáros, among others, and are not completely understood. In this paper, we study the stability properties of proximal maps, with a particular focus on non-expansivity, under various notions of convexity of the functional that can be considered in the Wasserstein space.

While intuitive, naïve infinitesimal reasoning is classically inconsistent, and rigorous nonstandard analysis relies on non-constructive machinery. We resolve this tension by constructing an explicit, totally ordered field $\mathbb{R}^{\mathbb{Z}_{<}}$ using only real sequences and Cauchy convolution. We model the combined real and hyperreal axioms via the Chunk and Permeate strategy, a paraconsistent technique that isolates contradictions without global collapse. Equipping $\mathbb{R}^{\mathbb{Z}_{<}}$ with a two-tier topology, we develop a calculus where infinitesimal derivatives and integrals permeate cleanly to their classical counterparts. We further introduce a $(k,n)$-continuity hierarchy capturing infinitesimal smoothness invisible to standard or transfer-based models. Finally, $\mathbb{R}^{\mathbb{Z}_{<}}$ yields a direct algebraic consistency proof for Sergeyev's Grossone arithmetic, and we establish strict computability bounds on field operations. By guaranteeing infinitesimal contradictions never reach the classical chunk, this work bridges paraconsistent logic, constructive mathematics, and nonstandard analysis into a transparent, computationally tractable framework for infinitesimal reasoning.

The effect of a moving impurity in a dilute Bose-Einstein condensate is investigated by means of the one-dimensional Gross-Pitaevskii model (GP) with non-zero boundary conditions at infinity. The impurity is modeled as a localized external potential, that travels at constant speed $v \in \mathbf{R}$. In a co-moving reference frame, we study the existence and stability of time-independent solutions. The latter are of physical relevance, being associated with the superfluid behavior of the condensate. For every non-zero velocity $v$ in the subsonic regime, we show the existence of a family of time-independent solutions which bifurcates from a (displaced) gray soliton $\phi_{0,v}(x-s_0)$, with $s_0 \in \mathbf{R}$, of the GP equation. The position $s_0$ is determined as an extremal point of an effective potential explicitly defined. Moreover, we study the spectral stability of these states. For small values of the potential strength, we show that the families originating from the maxima of the effective potential are spectrally unstable. For this last result, we employ an Evans function approach. Finally, we formally apply the instability result to the case of a repulsive delta potential.

We develop the theory of probabilistic pseudo knots, providing a framework for modeling knot diagrams with unresolved crossing information. Pseudo knot diagrams generalize classical diagrams by allowing certain crossings to remain unspecified; in the probabilistic setting, each such pre-crossing, namely a crossing with undetermined over–under information, is assigned a probability describing the likelihood of resolving as a positive crossing, with complementary probability assigned to the negative resolution. This induces a probability distribution on complete classical resolutions and, by aggregation, a distribution on classical knot types, capturing uncertainty arising in physical, biological, and computational contexts. We introduce probabilistic equivalence, defined via total variation distance between resolution distributions, and extend classical numerical quantities such as writhe and linking number to this setting. We also develop new probabilistic constructions, including the probabilistic chirality index, minimal resolution genus, probabilistic Seifert surface distributions, and polynomial invariants extending the Kauffman bracket. We further discuss matrix-based constructions, including probabilistic Seifert and Goeritz-type matrices, as well as probabilistic surgery producing distributions over 3-manifolds. Finally, we discuss potential applications in molecular biology, materials science, and computational topology.

In this paper, we study the asymptotic behaviours of a critical branching random walk in $\mathbb{R}^d$ under the assumption that the offspring distribution belongs to the domain of attraction of an $\alpha$-stable law with $\alpha\in(1,2]$, and that the jump distribution has a finite $\frac{2\alpha}{\alpha-1}$-th moment. First, we establish the precise decay rate for the tail probability of the all-time maximal displacement $M^d$. Next, we investigate the maximal displacement $M_n^d$ at generation $n$ and prove a conditional limit theorem for the distribution of $M_n^d$ given that the process survives up to generation $n$. These results extend the corresponding 1-dimensional results of Lalley and Shao (2015) to the case $d\ge2$. Finally, we study the asymptotic behaviour of the total progeny $\zeta$. In particular, we show that, conditioned on the event $\{M^d\ge x\}$, $\zeta$ converges in distribution under an appropriate normalization. This result reveals a quantitative relationship between the maximal displacement and the total progeny size.

This paper revisits the adaptive extended golden-ratio primal–dual algorithm (aEGRPDA) proposed by Soe et al. (2026) for structured convex optimisation problems involving a differentiable term that is only locally smooth. We prove that the artificial upper bound imposed on the primal step-size in aEGRPDA is redundant, since the adaptive rule itself keeps the step-sizes bounded above. As a consequence, the ergodic $\mathcal O(1/N)$ estimates for the objective residual and feasibility violation, where $N\ge1$ denotes the number of iterations, are independent of this hyperparameter. Consequently, the resulting adaptive golden-ratio primal–dual method, therefore, requires neither a step-size cap, nor a linesearch procedure, nor a known global Lipschitz constant. We establish linear convergence of the algorithm when both the primal and dual functions are strongly convex. Furthermore, we develop two accelerated variants, in addition to the local smoothness assumption: one for the case where the nonsmooth primal component is strongly convex, and another for the case where the differentiable term is globally strongly convex. For these accelerated methods, we prove an ergodic $\mathcal O(1/N^2)$ convergence rate. Preliminary numerical experiments on a Poisson imaging problem illustrate the efficiency and robustness of the proposed approaches.

The arithmetic intensity of algorithms for computing finite element operators increases with increasing polynomial degree. This has made high degree methods particularly attractive on modern CPU and GPU architectures, since performance at low degree is limited by memory bandwidth and only a small fraction of the floating point capacity of the processor is used. Higher degree methods can exploit a significantly greater fraction of the available compute power. While stable methods for computing high-degree finite element bases are well-established, there is no universal and automated algorithm for the efficient construction of the degree-of-freedom map for arbitrary degree elements. We introduce a new algorithm that can be used in computing degree-of-freedom maps for arbitrary Ciarlet-type finite elements using only the element's definition and properties of the reference cell, without requiring a specific implementation for each element. This method is implemented in the library Basix, a component of the FEniCSx libraries. As well as allowing vast simplifications of parts of a codebase, the algorithm has allowed us to support user-defined custom elements that a user can create at runtime without requiring the user to input any information about transformations required to construct a degree-of-freedom map.

Let $G$ be a finite group and let $\mathrm{P}_{\mathcal{U}}(G)$ denote the probability that two randomly chosen elements of $G$ generate a supersoluble subgroup. We prove that if $\mathrm{P}_{\mathcal{U}}(G) > 16/25$ then $G$ is supersoluble, and that the bound $16/25$ is sharp, being attained by the group $G = (C_5 \times C_5) \rtimes Q_8$, where $Q_8$ acts faithfully and irreducibly on $C_5 \times C_5$.

We prove local asymptotics for near-central complex $(p,q)$-binomial coefficient modulus ratios allowing an imaginary parameter perturbation of order $n^{-3/4}$ at a $\sqrt{n}$ length scale from the centre. Moreover, we obtain ratio asymptotics for a smaller imaginary perturbation of order $n^{-5/4}$ at the length scale $n^{3/4}$. These results were obtained by reducing the two-parameter coefficients to just one parameter, giving a branch-free logarithmic representation of the second-order ratio and, hence, uniform complex curvature asymptotics for near-central ratios.

We develop a new method for proving a weak functional inequality by first proving it for a sufficiently regular sequence of distributions approximating the stochastic localization (SL) process, and then transferring it to the desired distribution via regularity of the SL process and conductance arguments. We use this strategy to prove a weak Poincaré inequality (WPI) holds for the Gibbs measure of the Sherrington-Kirkpatrick model when $\beta < \frac{1}{2}$. A prior result of the authors [arXiv:2605.03718, 2026] proves the ASL process for the Sherrington-Kirkpatrick model satisfies the required regularity conditions. A consequence of the WPI is that a much simpler algorithm – Glauber dynamics with a warm-start – efficiently samples the Gibbs measure of the SK model at $\beta < \frac{1}{2}$. This is a significant structural step towards resolution of the conjecture that Glauber dynamics mixes fast in the replica-symmetric regime for the Sherrington-Kirkpatrick model [arXiv:2504.20539, Open-Problem 15, 2025].

In this paper, we study an age-dependent branching process. In the simplest setting, the population is divided into two age groups, namely juveniles and adults. Our objective is to estimate the model parameters using observations of the total population size only (i.e., juveniles plus adults). Focusing on the ergodic regime of the model, we introduce a method-of-moments estimator and establish its asymptotic normality. Several extensions are discussed, including models with more than two age groups.

The symmetry method is a powerful and systematic approach for solving difference equations. It leverages the concept of transformations that leave a difference equation invariant, simplifying its structure and often reducing the equation to a solvable form. In this paper, the symmetry method is employed to study some class of difference equations. Using analytical techniques and computational tools, we derive explicit solutions for these equations and establish conditions for the existence of periodic solutions. Stability analysis is performed to identify non-hyperbolic points. Furthermore, some asymptotic properties of the difference equations are explored, with results and graphs illustrating how initial conditions and parameter values influence the behavior of the solutions.

Through-the-wall radar (TWR) human activity recognition (HAR) is important for non-line-of-sight indoor sensing, security monitoring, and emergency rescue. However, structured distribution shifts caused by person variation, observation-view variation, and wall-condition variation severely degrade recognition generalization, while the origin of the target-domain error still lacks a rigorous theoretical explanation. To address this issue, a generalization-analysis framework for TWR HAR is proposed in this paper. First, models for indoor human kinematics, TWR echo generation, radar image formation, feature representation, and bounded-weight neural networks are established within a unified source-to-target learning formulation. Then, the source risk, target risk, empirical risk, and admissible physical domain descriptor are defined, and a unified target-domain generalization bound is derived. Next, the structured shift term is decomposed into cross-person, cross-view, and cross-wall components, and the bound-tightening effects of physical low-dimensional representations, multi-source training, and parameter-space coverage are analyzed. Simulated and measured experiments jointly support the resulting theoretical analysis and illustrate its application value.

We consider a nonlinear PDE describing a nonlinear electrostatic medium with nonlocal dielectricity. The existence proof for the corresponding equation is based on Schauder's theorem and a new compactness theorem for moving coefficients (“Helga's Theorem”). This technique uses insights from (operator-theoretic/topological) homogenisation theory. Surprisingly, even though monotonicity assumptions are neither used nor valid, the underlying domain is only required to be weak Lipschitz and no assumption on the derivatives of the nonlinearity is needed.

We introduce and study the notion of commutators and anti-commutator of higher degrees for ring elements, which generalize the concept of commutator and anti-commutator of ring elements. In particular, we study the invertibility of the degree $n$ commutators and anticommutator of $n$-potent elements. Under natural conditions on the ring, we relate the invertibility of degree $n$ commutators and anticommutator of $n$-potent elements. We also relate the invertibility of degree $n$ commutators and anticommutator of $n$-potent elements with the invertibililty of higher commutators and anticommutator. Finally, we study ring extensions in which the invertibility of degree $n$ commutators and anticommutator of $n$-potent elements is inherited from its base ring.

Special sets of the Hermitian surface $\mathsf{H}(3,q^2)$, $q$ odd, were introduced by Shult and Thas (1995) in order to construct new finite generalised quadrangles, yet only one example is known to exist and it gives rise to a classical generalised quadrangle. We show that there can be no other special sets of the Hermitian surface.

A central problem in quantum information theory is understanding how quantum resources can be used to communicate information more efficiently than classical resources. We introduce quantum dense network coding – a protocol that transmits the output of a non-Boolean function to a receiver using provably half as many qubits as bits for each sender by not transmitting the entirety of the function inputs. We show this advantage requires both shared entanglement and quantum communication, is robust to noise, and the gap in success probability between quantum and classical communication can be amplified exponentially in the number of senders. Finally, we show that dense network coding gives rise to a novel, information-theoretically secure, quantum cryptographic protocol, which we call measurement-device-independent quantum key growing.

Let $A_n$ be a subset of $\{1,2,\ldots,n\}$ obtained by retaining each integer independently with fixed probability $\theta\in(0,1)$, and let $L_n$ be the least common multiple of the integers in $A_n$. We prove a functional large deviation principle, a functional moderate deviation principle, and a Strassen-type functional law of the iterated logarithm for the process $(\log L_{\lfloor{nt}\rfloor})_{0\le t\le1}$. The large deviation rate function is given by an entropy contraction for geometric marks, while the moderate deviation rate function and LIL cluster set are described by the reproducing kernel Hilbert space associated with the Gaussian limit process.

The classical notion of metric dimension has led to a wide range of extensions, such as the local, strong, fractional, and k-metric dimensions. This naturally raises the question of whether analogous variants can be formulated and studied within the multiset framework. While some progress has been made, particularly on the local multiset dimension, outer multiset dimension, local outer multiset dimension, and edge multiset dimension, however, the area remains far from fully explored. In this paper, we survey the existing variants and consolidate the results currently available in the literature. Furthermore, we identify several directions for future work.

Inspired by the non-meager independence $\downarrow^{nm}$ introduced by Krupiński [3], we study further possible independence relations induced by ideals and provide a general framework for this topic. We show that the independence relation $\downarrow^\mathrm{Haar}$ induced by Haar null ideals is a good example for locally compact groups and provide an example showing $\downarrow^\mathrm{Haar}\neq \downarrow^{nm}$. Moreover, we introduce an order on the collection of all well-behaved independence relations and conjecture that $\downarrow^{nm}$ is the maximum one for Polish groups. We prove the conjecture under the extra hypothesis of $\sigma$-compactness. For Lie groups, we discuss $\mathrm{SO}(3)$ and $\mathrm{SE}(2)$ as examples, and prove that the independence relation is unique for nilpotent Lie groups.

Using the methods of holomorphic dynamics we investigate planar birational mappings that arise from the theory of cluster algebras and integrable systems. Computing dynamical degrees of these mappings, many of which are greater than one, allows us to show that many of the mappings do not have a conserved quantity (nor an invariant fibration). In most of the examples, invariant fibrations can also be ruled out by finding superattracting periodic points. This answers a question posted by Machacek and Ovenhouse 2024 and by Chen and Li 2024. Moreover, having found a good algebraically stable model for the mappings and having computed the dynamical degree, we can then apply results from the ergodic theory of birational maps to produce invariant measures with positive entropy and positive Lyapunov exponents.

In this article, we establish explicit and uniform $L^{\infty}$-norm bounds for $L^{2}$-normalized Siegel–Jacobi cusp forms of integral weight $k$ and index $m$ for the Siegel modular group $\Gamma_{0}=\mathrm{Sp}_{2g}(\mathbb{Z})$ for arbitrary genus $g\geq 1$. Using the generalization of the classical Eichler–Zagier theta decomposition to higher genus, any such Siegel–Jacobi cusp form can be written as a finite linear combination of Siegel cusp forms of half-integral weight $k-1/2$ multiplied by the higher-dimensional analogues of the classical Jacobi theta functions. By building upon the uniform $L^{\infty} $-norm bounds on average for Siegel cusp forms established by J.~Kramer and A.~Mandal~\cite{k1} via the associated Bergman kernels, we prove that for $k\in\mathbb{Z}_{\geq g+1}$, $m\in\mathbb{Z}_{\geq 1}$, and a given $\epsilon>0$, the $L^{\infty}$-norm bound \begin{equation*} \Vert\phi\Vert_{L^{\infty}}=\sup_{(\tau,z)\in\mathbb{H}_{g}\times\mathbb{C}^{g}}\Vert\phi(\tau,z)\Vert_{\mathrm{Pet}}=O_ {\Gamma_{0},\epsilon}\big(k^{(3g^{2}+5g)/8}\,m^{g^{2}+5g/4+\epsilon}\big) \end{equation*} holds for any Siegel–Jacobi cusp form $\phi$ that is $L^{2}$-normalized with respect to the Petersson inner product. These estimates provide the first explicit upper bounds in terms of both parameters $k$ and $m$ for arbitrary genus $g$.

We show that a fully inhomogeneous uniform Littlewood type problem has a negative answer and the counterexamples form a dense $G_{\delta}$ set. This extends the author's recent analogous result for the homogeneous case. The main difficulty in the general setting is the semi-homogeneous case where one linear form is homogeneous and the other inhomogeneous with irrational shift. We further address the higher dimensional analogue.