Papers

The goal of this paper is to understand the set $\mathrm{End}(W)$ of endomorphisms of an irreducible spherical reflection group $W$. We do this in two ways: numerically, by deriving an explicit formula for $|\mathrm{End}(W)|$; and probabilistically, by exploring the question \textit{what does a random endomorphism of $W$ look like?} For example, we show that as $n\to\infty$ the probability that a random endomorphism of $W_n$ is an automorphism tends to $\frac{1}{2}$ if $W_n=C_{2n}$ or $D_n$, to $\frac{1}{4}$ if $W_n=C_{2n+1}$, and to $1$ if $W_n=A_n.$

In this paper, we study discrete approximations of semi-Dirichlet forms obtained by adding non-symmetric drift terms, expressed in terms of mutual energy measures, to resistance forms whose associated resistance metric spaces are compact. The main novelty is that the drift terms need not to be absolutely continuous with respect to the underlying measure. Under suitable smallness assumptions, we prove generalized Mosco convergence of the approximating forms and derive weak convergence of the associated Feller processes. We also demonstrate the applicability of our main theorem to post-critically finite self-similar fractals and describe the resulting perturbation of the approximating jump chains in the case of the Sierpiński gasket.

This essay examines the relationship between artificial intelligence and the historical evolution of modern Mathematics. Rather than viewing AI as an external rupture, we argue that its effectiveness reveals a structural tendency already present in the autonomization of Mathematics itself. Modern Mathematics progressively developed formal environments that became increasingly autonomous, internally stable, and structurally navigable, reducing their dependence on concrete experience. In this context, the affinity between AI and contemporary mathematical practice appears less accidental than it may initially seem. The essay also discusses possible limits of formal navigability, particularly regarding the emergence of genuinely new conceptual regimes and forms of mathematical intelligibility. Husserl's reflections on mathematization and the distancing of science from the Lebenswelt provide a broader philosophical framework for understanding this process. We finally suggest that the contemporary debate on AI may concern less a threat to Mathematics itself than a challenge to the historical image of the mathematician as the privileged interpreter of mathematical structures.

We develop an ind-Banach framework for revisiting analytification in complex geometry, inspired by Bambozzi-Chiarellotto-Vanni's work on tempered cohomology. We define several ind-Banach rings of overconvergent and holomorphic power series to endow $\mathbb{C}$-algebras with analytic structures. As an application, we obtain an abstract GAGA-type comparison in this setting.

In this paper, we investigate the asymptotic behavior of continuous-state branching processes in a Brownian random environment (CBBRE) conditioned on non-extinction. For the subcritical case, we prove the existence of the Yaglom limit and derive an explicit representation of its Laplace transform using Kummer confluent hypergeometric functions. Notably, we demonstrate that the Yaglom limit is strictly independent of the initial state of the process across all three subcritical regimes: weakly, intermediately, and strongly subcritical.

For an integer $\ell\geq 2$, let ${\cal{H}}_{\ell}$ denote the family of graphs which have girth $2\ell$ and have no even hole of length greater than $2\ell$. Wu, Xu and Xu conjectured that every graph in $\bigcup_{\ell\geq 2} {\cal{H}}_{\ell}$ is $3$-colorable. Chen showed that every graph in $\bigcup_{\ell\geq 5} {\cal{H}}_{\ell}$ is $3$-colorable. In this paper, we prove that every graph in ${\cal{H}}_4$ is $3$-colorable.

The Lonely Runner Conjecture states that if $k+1$ runners start at the same point on a unit-length circular track and run with distinct constant speeds, then each runner is at some time at least $1/(k+1)$-distant from every other runner. Equivalently, for every tuple of $k$ distinct positive integer speeds $s_1,\ldots,s_k$, there is a real number $t$ such that $\|s_i t\|\geq \frac{1}{k+1}$ for all $i$. We introduce and study a version of the conjecture in which the required distances may vary with $i$. For $\mathbf d=(d_1,\ldots,d_k)\in(0,1/2]^k$, let $\mathsf{MLPS}_k$ be the set of vectors such that, for every choice of distinct positive integer speeds $s_1,\ldots,s_k$, there is a real number $t$ with $\|s_i t\|\geq d_i$ for all $i$. We give an exact characterization of $\mathsf{MLPS}_2$. We also use Fourier series for distance-threshold indicator functions to obtain an arithmetic progression summation formula and an exact two-function integral formula for unequal thresholds.

Let $G$ be a finitely generated virtually abelian group and $[\sigma]\in H^2(G;\mathbb{T})$ such that $\sigma(x,y)$ is always a root of unity. We show that the nuclear dimension of the twisted group $C^*$-algebra $C^*(G,\sigma)$ is equal to the rank of a finite index abelian subgroup of $G$. We also show that $\mbox{dim}_{\text{nuc}}(C^*(\mathbb{Z}^r,\sigma))=r$ if and only if $\sigma$ is type I.

The notion of good integers, namely the divisors of the sequence $(a^s+b^s)_{s\ge 1}$ for nonzero coprime integers $a$ and $b$, together with their subfamilies such as oddly-good and evenly-good integers, has become an important arithmetic tool in the study of Euclidean and Hermitian dualities for abelian and cyclic codes. Building on this perspective, this paper introduces and studies another interesting subclass of good integers arising from the sequence $\bigl(a^{ks+T}+b^{ks+T}\bigr)_{s\ge 1}$ for some integers $0\leq T<k$, whose divisors are called $(T,k)$-good integers with respect to $(a,b)$. An arithmetic theory of these integers is developed, including a characterization at odd prime powers, a general characterization for odd integers in terms of $2$-adic valuations, and a treatment of even integers. An explicit algorithm is also given for deciding whether a given integer $d$ is $(T,k)$-good with respect to $(a,b)$ and, when it is, for computing an exponent $s$ such that $d\mid \bigl(a^{ks+T}+b^{ks+T}\bigr)$. Applications in coding theory are then obtained from the specialization $(a,b)=(q,1)$, where $q$ is a prime power. In particular, the $q^k$-cyclotomic classes of the cyclic group $\mathbb Z_n$ characterize the Galois self-reciprocal irreducible factors of $x^n-1$ over $\F_{q^k}$, give a description and enumeration of Galois LCD cyclic codes of length $n$ over $\F_{q^k}$, and lead to a characterization of Galois self-dual cyclic codes.

This paper investigates uplink multiple access for the coexistence of enhanced mobile broadband+ (eMBB+) and massive machine-type communications+ (mMTC+) in terminal-centric cell-free massive MIMO (CF-mMIMO) systems. We propose a non-orthogonal scheme in which low-rate mMTC+ transmissions are spread across the time-frequency grid shared with eMBB+ users, enabling efficient resource reuse. In the presence of imperfect channel state information, we derive closed-form expressions for the achievable rates of both services based solely on statistical channel knowledge. For mMTC+ devices, the analysis also incorporates finite blocklength (FBL) modeling to capture short-packet transmissions. To support heterogeneous service requirements, we formulate a power-control problem that maximizes the minimum energy efficiency of mMTC+ devices subject to quality-of-service constraints on eMBB+ users. The resulting nonconvex problem is solved via sequential fractional programming, accounting for both the Shannon and FBL regimes. To enable real-time operation, we further propose a graph neural network (GNN) with multi-head attention to approximate the model-based solution. Constraint satisfaction during training is enforced via an augmented Lagrangian loss. Numerical results demonstrate effective multiplexing of the two data services and show that the proposed GNN algorithm achieves near-optimal performance with a significantly lower computational complexity.

In this article, for every integer $g>1$, we construct the first example of an elliptic surface with infinitely many smooth $(-1)$-curves of genus $g$, settling an open question of Bauer et al. [Duke Math. J. 162 (10) (2013), 1877-1894].

Estimating a fractal dimension from a finite stochastic trajectory is a finite-size scaling problem: the apparent box-counting exponent is shaped by an occupancy crossover between the resolved range of scales and the finite number of sampled points, and need not equal the dimension of the limiting process. We model this crossover with a balls-in-boxes occupancy law, which predicts the box-count curve, the finite-size saturation scale, and a scaling function for the normalized local slope. Across random-walk traces, fractional Brownian graphs, and Levy flights, the normalized local slope collapses onto a single crossover curve, while the windowed box-counting bias collapses when the regression window is positioned relative to the saturation scale. Inverting the occupancy model gives a finite-size bias correction that reduces error on controlled stochastic trajectories and transfers across held-out model classes. Comparisons with correlation dimension, detrended fluctuation analysis, the variogram, and Higuchi's method show that the dominant bias is specific to point-sampled box-counting over finite scale windows, and that local-slope stability alone is not a reliable diagnostic. A DNA-walk example illustrates the workflow on measured data, and all figures, tables, and in-text numbers are regenerated from released single-seed code.

In this paper, by considering two non-isospectral problems with matrices chosen on the color Lie algebra $\mathfrak{sp}_{1}(6)$, we construct (1+1)-dimensional and (2+1)-dimensional super integrable systems on $\mathfrak{sp}_{1}(6)$. Moreover, based on the supertrace identity, their super Hamiltonian structures are also constructed.

Quadratically regularized optimal transport (QOT) is a sparse alternative to entropic optimal transport. We develop a quantitative stability theory for QOT under perturbations of the marginals, the transport cost function, and the regularization parameter. The centerpiece is an $L^\infty$-stability result for the dual potentials. Starting from an error bound in an $L^2$-space that varies with the marginals, we use a self-bound for the potentials to derive a local $L^\infty$-Lipschitz bound that is uniform over marginals. This bound also yields stability of the optimal coupling and of its support. In particular, we show that for the quadratic transport cost, the support of the optimal coupling is locally Lipschitz in Hausdorff distance under perturbations of the marginals. To the best of our knowledge, this is the first stability result for the optimal support in regularized optimal transport.

In this paper, we give a new definition of the $q$-weight of zeros, which reduces to the multiplicity of zeros as $q\to 1$. Furthermore, we obtain a $q$-difference version of the Stothers-Mason theorem by means of the new definition of the $q$-difference radical, which covers the classical Stothers-Mason theorem as $q\to 1$. As applications, we study the polynomial solutions of $q$-difference Fermat type functional equations.

The octahedral decomposition of classical link complements has been considered and utilised by Weeks, Rubinstein, Sakuma etc. It is even more natural to consider the octahedral decomposition of virtual link complements. In this paper, we explore and generalise the octahedral decomposition used in snappy to links in thickened surface. Using the decomposition, we prove nonpositive curvature of the complement and essential-ness of edges in the decomposition.

We establish global well-posedness of the Benjamin–Ono equation for sufficiently small, zero-mean periodic initial data in the analytic Sobolev space $H^{\rho,1}_0$. The proof combines regularized commuting flows with an exponential spectral energy. A local uniform convexity property of this energy upgrades weak endpoint compactness to strong continuity, yielding a global flow in $C(\mathbb{R};H^{\rho,1}_0)$ and continuous dependence in the same analytic topology, with no dynamic loss of the prescribed analytic width.

Discrete-time affine processes are widely used in finance and economics and encompass count, positive, and nonnegative-valued processes. This paper develops near-unit-root asymptotic theory for this class of models. Unlike linear AR(1) processes, affine processes exhibit time-varying conditional variance that remains asymptotically non-negligible near unity, leading to qualitatively different scaling limits and estimator behavior. We show that the local-to-unity regime suffers from the usual nuisance-parameter problem, whereas the mildly explosive regime, while free of it, still does not allow consistent estimation of the intercept. By contrast, the mildly stationary framework is more tractable: the OLS estimator is asymptotically normal, the resulting trajectories are more realistic than those of linear AR(1) models, and inference is possible through both a plug-in method or bootstrap. The theoretical results are supported by simulation evidence and illustrated through applications to insurance and financial data.

This paper introduces a novel, robust, and computationally efficient framework for high-quality quadrilateral mesh generation on general two-dimensional domains. The core of the proposed approach is a novel method for computing cross fields by minimizing a modified and relaxed Ginzburg–Landau-type energy functional. A key innovation is the extension of the problem from the original, potentially complex domain to a larger regular computational domain. This extension transforms the central computational procedure into an iterative scheme that requires only two straightforward and efficient operations: linear diffusion solved globally via the Fast Fourier Transform (FFT) and point-wise normalization. Notably, our method eliminates the conventional need for generating an intermediate triangular mesh or solving complex nonlinear optimization problems on the irregular domain. We provide a rigorous theoretical analysis, proving that the proposed iterative algorithm guarantees unconditional monotonic decay of the objective functional. Comprehensive numerical experiments demonstrate the method's robustness across a wide range of complex geometries, its significant computational efficiency afforded by the FFT-based diffusion, and its consistent generation of high-quality quadrilateral meshes. This work presents a reliable and theoretically sound alternative to existing mesh generation techniques, with strong potential for practical applications in scientific computing.

In this paper, we propose a spectral framework that embeds 1D and 2D quasiperiodic Helmholtz eigenvalue problems into higher-dimensional (2D and 4D) periodic spaces via the projection method \cite{jiang2014numerical, jiang2024numerical}. To effectively map the elevated high-dimensional states back to the original physical space, we establish a novel validation framework based on the weighted expectation of pointwise Rayleigh quotients. Supported by comprehensive error and spectral analysis, we demonstrate that the eigenvalues derived from this expectation align more authentically with the original quasiperiodic model, ultimately yielding a more appropriate and reliable eigenpair solution. Numerical experiments on continuous media demonstrate that our approach offers an accurate, robust, and scalable tool for solving quasiperiodic Helmholtz eigenvalue problems.

We study a class of spectral design problems in which a prior positive semidefinite information matrix is updated by a sum of rank-one matrices constructed from chosen design vectors subject to a bound on their Euclidean norm. The objective of a spectral design problem is any symmetric convex function of the eigenvalues of the updated information matrix. This framework unifies classical optimal experimental design criteria, including A-, D-, and E-optimality. It also arises in model-based derivative-free optimization, where sampling directions determine the conditioning and accuracy of regression models. Although the objective is symmetric and convex in the eigenvalues, the optimization problem with design vectors/matrix as decision variables is nonconvex, and optimal solutions of their convex relaxations may not be feasible for the spectral design problem. We use tight eigenvalue relaxations to obtain a convex reformulation, and we apply the Schur–Horn theorem to construct a simple polynomial-time algorithm for solving the spectral design problem. We illustrate the optimal spectral designs computed by our algorithm. Moreover, a small set of numerical experiments shows the potential of spectral designs for derivative-free optimization.

We study a multiplicative function analogue of Linnik's problem on the least prime in an arithmetic progression. Let $h\colon \mathbb{N}\to\mathbb{R}\setminus\{0\}$ be a multiplicative function, and let $a \pmod q$ be a reduced residue class. We ask how far one must go before finding square-free integers $n_1,n_2\equiv a \pmod q$ with $h(n_1)<0<h(n_2)$. We show that one can always find such integers with $n_1,n_2\le q^{2+o(1)}$, unless the sign of $h$ strongly pretends to be a real Dirichlet character modulo $q$. Thus, apart from this natural character obstruction, sign changes of a multiplicative function occur in every reduced residue class at a scale corresponding essentially to the square root barrier. In the special case of the Liouville function $\lambda$ this improves on a recent result of Ford and Radziwiłł and matches, up to $q^{o(1)}$ factors, what was previously known conditionally under the generalized Riemann hypothesis.

We study decentralized online convex optimization when agents communicate over a graph and messages may be compressed. Classical decentralized online methods typically require learning-rate choices that depend on the horizon, comparator scale, or other problem parameters, while compressed communication introduces additional disagreement that must be controlled. We propose DECO-EF (DEcentralized COin-betting with Error Feedback), a decentralized parameter-free online learning algorithm that combines coin-betting predictions with compressed difference-based gossip. Each agent maintains a clean accumulated state and a compressed tracker, and communicates only compressed state differences during gossip steps. The method is parameter-free in the online-learning sense: it does not tune to the horizon, the comparator norm, or the learning rate. We prove expected comparator-adaptive network-regret bounds for DECO-EF under compressed communication. To the best of our knowledge, this gives the first expected sublinear network-regret guarantees for parameter-free decentralized online learning under compressed communication.

A sprout sequence is a sequence $\frakr=(R_0=1,R_1,R_2,\dots)$ of symmetric functions in the variables $\bmx=(x_1,x_2,\dots)$ over a field $K$ generated from a power series $F(t)=1+a_1t+a_2t^2+\cdots$ by the rule $\sum_{n\geq 0}R_nt^n = \prod_{i\geq 1} F(x_it)$. The power series $F(t)$ is called the seed of $\frakr$. This concept originated in the work of Littlewood and Richardson (though not with the name “sprout sequence”), and numerous examples of sprout sequences have appeared in the literature. They are related to chromatic Tutte polynomials of complete graphs and complete hypergraphs, binomial posets, upper homogeneous (upho) posets, topological genera, etc. We first develop the basic theory of sprout sequences and then look at the special case $F(t)=\sec(\sqrt{t})$. We give five characterizations of sprout sequences and consider the expansion of sprout symmetric functions in terms of well-known symmetric function bases. The Schur positivity, elementary symmetric function positivity, and complete homogeneous symmetric function positivity of $R_n$ for all $n$ are completely characterized using the Edrei-Thoma theorem from the theory of total positivity. The seed $F(t)=\sec(\sqrt{t})$ is especially interesting. The expansion of $R_n$ in the power sum or monomial basis is related to alternating permutations. The Schur function expansion is related to standard Young skew tableaux. The expansion in terms of the complete symmetric functions has nonnegative integer coefficients, but we don't know a combinatorial interpretation. Finally we give a formula for $R_n$ as a sum of chromatic symmetric functions of interval orders.

We study a system of $N$ inertial particles on a two-dimensional torus $\T^2$, evolving under a second-order stochastic dynamics with position-dependent friction $\lambda$ and noise amplitude $\sigma$, and undergoing coalescence at rate $R_0$ when their distance falls below a threshold $\delta$. In the joint small-mass / small-correlation limit $\mu(\eps)\to 0$, $\mu(\eps)/\eps\to\al\in(0,\infty)$, the empirical measure of the surviving particles converges to a stochastic continuity equation with inertial drift~$g_\al$. Assuming that $\sigma$ is tangent to the level sets of a Hamiltonian $H=h_1(x_1)\,h_2(x_2)$ satisfying mild non-degeneracy and convexity-type conditions, and that $\lambda$ and the amplitude $\rho$ of $\sigma$ along $\xi=\nabla^\perp H$ are aligned with $H$, we prove that the expected total mass decays exponentially in time, with an explicit rate depending on $\al$ and on the values of $\lambda$ and $\rho$ on the separatrix $\{H=0\}$. The proof rests on a cell-by-cell analysis of the sign of $\div\,g_\al$ on the level sets of $H$, showing that the inertial drift pushes trajectories toward the separatrix at a quantitative rate.

Direct search is one of the most popular derivative-free optimization paradigms, that relies on exploring the variable space using polling directions. To analyze and implement direct search, one typically relies on positive spanning sets. This concept is somewhat decorrelated from interpolation-based sets used in model-based algorithms, another class of derivative-free optimization methods. This discrepancy is even more pronounced for constrained problems, where recent advances in the interpolation-based setting have produced a unified picture that is still lacking in the direct-search case. In this paper, we introduce a new theoretical underpinning for direct-search methods, that can be defined for general convex constraints and lead to complexity guarantees, as in the model-based setting. By focusing on polyhedral convex constraints, we are able to construct polling sets that meet our new theoretical requirements. In particular, our polling sets necessarily include directions outside of the approximate tangent cone, giving theoretical justification to existing practical heuristics which incorporate this idea. Our numerical results confirm that adding these extra directions significantly improves practical performance.

In this work, we formulate a two-species Lotka–Volterra competition model on time scales. To derive the global dynamics of solutions of this planar model, we introduce a dynamic augmented phase plane analysis that extends the traditional phase plane analysis known to be a powerful tool in the analysis of planar differential equations. In the case of a constant graininess, the method is consistent with the augmented phase portrait introduced for the discrete space. However, for non-constant graininess, this augmented phase plane is time-dependent and therefore dynamic. Despite the dynamic character, we are able to identify time-independent information that allows the conclusion of the global dynamics of the introduced Lotka–Volterra competition model on time scales. The dynamic phase plane method, albeit only introduced in the context of this particular two-species competition model, it can be extended to other planar systems on time scales, providing a novel technique to study the global dynamics of planar systems on time scales.

The Lenard-Balescu equation is a collisional kinetic model widely used in plasma physics as a Bogoliubov correction to the meanfield Vlasov theory. Unlike the classical Landau and Boltzmann collision operators, the Lenard-Balescu collisional kernel not only accounts for the binary interaction between particles, but also includes the collective meanfield effects. In this paper, we construct global smooth solutions to the regular Lenard-Balescu equation near global Maxwellians in a periodic box, thus extending the previous work by Duerinckx-Winter that treats the spatially homogenous case to the inhomogenous setting.

We define an indicial polynomial of a $D$-module along an arbitrary subvariety as a generalization of both the classical indicial polynomial for a single linear differential equation and the Bernstein-Sato polynomial of a variety defined by Budur-Mustata-Saito. An indicial polynomial is also a generalization of the $b$-function of a $D$-module along a submanifold and can be used in the computation of the $D$-module theoretic inverse image by the embedding instead of the $b$-function. We consider properties of indicial polynomials and relations with $b$-functions. An indicial polynomial may exist even if the $b$-function does not, and gives the set of the roots of the $b$-function if it exists. Computation of an indicial polynomial is easier than the $b$-function and naturally includes the case with parameters.

The Variational Quantum Eigensolver (VQE) is a fundamental algorithm in quantum computing, yet a coherent geometric characterization of VQE remains missing due to fragmented analyses across fixed-ansatz and adaptive-circuit formulations. In this paper, we establish a geometric analysis of VQE in terms of optimization landscape, initialization guarantee, and noise robustness. First, we study the optimization landscape via an ansatz-free product-unitary formulation over the unitary group, unifying both paradigms. For the single-unitary case, we establish linear convergence of Riemannian gradient descent (RGD) and prove the strict saddle property. For the product-unitary case, we show the convergence rate deteriorates polynomially with circuit depth, providing a geometric explanation of the barren plateau phenomenon. Second, we prove that small-angle random Pauli-rotation circuits satisfy the required initialization conditions with high probability. Third, we show that RGD retains linear convergence under finite-shot measurements, and that coefficient-adaptive allocation achieves strictly lower statistical error than uniform sampling under a fixed measurement budget.