Papers

It is well known that every compact oriented 3-manifold admits an ideal triangulation, and that any two such triangulations with at least two ideal tetrahedra are related by a sequence of Pachner $2$-$3$ moves. Motivated by constructions in quantum topology, we give a combinatorial description of closed $3$-manifolds in terms of ordered ideal triangulations and ordered Pachner $2$-$3$ and $0$-$2$ moves.

Ultra-reliable low-latency communication (uRLLC) is a pivotal enabler for B5G/6G networks, yet it faces severe challenges from rare but critical extreme events, which are characterized by heavy tails in the delay distribution. While the cell-free radio access network (CF-RAN) architecture offers essential spatial diversity to combat these uncertainties, conventional user-centric clustering designs typically focus on average metrics, thereby inadequately addressing such tail behaviors. We propose a novel, tail-risk-aware, user-centric clustering framework operating within the finite blocklength (FBL) regime. Our approach employs extreme value theory (EVT), specifically the peaks-over-threshold (POT) model, to accurately quantify the probability of queue latency violations. This framework is applied to formulate an energy efficiency (EE) maximization problem under strict tail latency constraints. The problem is solved via an efficient online algorithm that integrates Lyapunov optimization with successive convex approximation (SCA). Simulation results demonstrate that the proposed scheme, through its dynamic adaptation of cluster formation to mitigate tail risks, achieves a superior reliability-efficiency trade-off and leads to a significant suppression of extreme latency events.

The determination of the maximal length of maximum distance separable (MDS) codes arising from elliptic curves is a central problem in coding theory. For an elliptic curve $E$ over $\mathbb{F}_q$, let $\operatorname{MEC}(k,q)$ denote the maximal length of a $q$-ary MDS elliptic code of dimension $k$. It was recently shown that $\operatorname{MEC}(k,q)\le\frac{q+1}{2}+\sqrt{q}$ for $q\ge289$ and $3\le k\le(q+1-2\sqrt{q})/10$, with equality for odd $k$ when $q$ is an odd square. This paper investigates the remaining open cases, namely even dimension $k$, non-square $q$ and fields of characteristic $2$, and provides a complete resolution of the tightness question for the two natural parity regimes of $q+1+\lfloor 2\sqrt{q}\rfloor$. We prove that if the support of $G$ (used to define the code) consists of $\mathbb{F}_q$-rational points, the bound decreases to $\frac{q+1}{2}+\sqrt{q}-1$ for even $k$. Without this restriction, we construct MDS codes attaining $\frac{q+1}{2}+\sqrt{q}$ for even $k$. More generally, we establish $\operatorname{MEC}(k,q)=\frac{q+1+\lfloor2\sqrt{q}\rfloor}{2}$ when $q+1+\lfloor2\sqrt{q}\rfloor$ is even, and $\operatorname{MEC}(k,q)=\frac{q+\lfloor2\sqrt{q}\rfloor}{2}$ when it is odd.

A non-Abelian gauge field framework is proposed using the hypercomplex ring formalism. This extension generates non-compact hyperbolic symmetries, which, alongside the compact gauge symmetries, double the internal degrees of freedom. This will enable the description of bipartite gauge systems and demonstrate how field dissipation operates at the dynamical level. Working within a commutative ring allows for the decoupling of the algebraic structures and facilitates the construction of solutions to the equations of motion.

We construct a Poincaré sheaf on the compactified Prym variety associated with an étale double cover of integral curves with planar singularities, and prove that the associated Fourier-Mukai transform is an autoequivalence of its derived category. As an application, we prove the motivic decomposition conjecture of Corti-Hanamura for the Laza-Saccà-Voisin fibration, and construct a multiplicative motivic perverse filtration lifting the cohomological one.

In 2024, Ceballos and Chenevière introduced alt $\nu$-Tamari lattices, parameterized by a lattice path $\nu$ and an increment vector $\delta$, as a common generalization of $\nu$-Tamari and $\nu$-Dyck lattices. We study rowmotion on two families: the alt hook-Tamari lattice $\mathsf{H}_{\delta}(a,b)$ (where $\nu=EN^{a-1}E^{b-1}N$) and the alt $2$-row-Tamari lattice $\mathsf{T}_{\delta}(a,b)$ (where $\nu=E^aNE^bN$). We explicitly determine the orbit structures of $\mathsf{H}_{\delta}(a,b)$ and $\mathsf{T}_{\delta}(a,b)$ under rowmotion, and prove that their orbit structures are independent of the increment vector $\delta$. As a consequence, we show that rowmotion on $\mathsf{H}_{\delta}(a,b)$ exhibits the cyclic sieving phenomenon. We also compute orbit sums for several natural statistics. In the hook case, we evaluate the down-degree, peak, valley, and area statistics; in the $2$-row case, we focus on the down-degree statistic. All of these – except for the area statistic – are homometric under rowmotion. Regarding the methodology of this paper, our results in the hook case are obtained by applying a simple local modification to their Hasse diagrams. In the $2$-row case, we introduce a switching property for semidistributive lattices, which allows us to compare the orbit structures arising from different increment vectors.

In this paper, we present a quantum implicit-explicit (IMEX) scheme for multiscale ordinary and partial differential equations whose discretization parameters are independent of the scaling parameter $\varepsilon$. A key ingredient of our approach is a continuous-time formulation of classical IMEX schemes, which decouples the evolution time of the quantum algorithm from the physical time of the differential equation and is therefore particularly useful in multiscale settings. Building on this idea, we employ the Schrödingerization framework [Phys. Rev. Lett. 133 (2024), 230602] to implement IMEX schemes on quantum computers. Compared to previous HHL type quantum AP scheme [J. Comput. Phys. 471 (2022), 111641], this new method requires narrower – an extra logarithmic factor – auxiliary register numerical examples on linear heat and multiscale telegraph equations demonstrate the independence in $\varepsilon$ of the method.

We show that the affine cactus group is a CAT$(0)$ group for all degrees. Furthermore, we show that the affine cactus group $AJ_3$ of degree three is a hyperbolic group.

In this paper, we give a description of the cohomology groups of the symmetric powers of the tautological bundle associated with a sufficiently positive line bundle on the Hilbert scheme of 2 or 3 points on a smooth projective complex variety. The dimensions of these cohomology groups are expressed in terms of the Hilbert polynomials of the first and second secant varieties of the embedding given by the sufficiently positive line bundle. We also compute these Hilbert polynomials completely.

Let $L$ be the Dunkl Laplacian on the Euclidean space $\mathbb{R}^N$ associated with a normalized root system $R$ and a multiplicity function $k(\nu)\geq 0$, $\nu\in R$. We establish a Leibniz-type rule for the fractional powers of $L$ on Besov and Triebel–Lizorkin spaces in the Dunkl setting. Our approach exploits the interplay between spectral multipliers and the Dunkl transform, together with the support properties of the distributions associated with Dunkl translations. These results extend the corresponding Leibniz-type estimates previously established on $L^p$ spaces to the broader setting of Besov and Triebel–Lizorkin spaces.

We give a simple example of a polynomial contraction automorphism of $\mathbb C^d$, $ d\ge 3 $, with unbounded degree growth. Combined with Poincaré-Dulac theorem it provides an algebraic automorphism of $\mathbb C^d$, $ d\ge 3 $, which is holomorphically but not algebraically linearizable.

Solving high-dimensional PDE-governed inverse problems is often challenging due to complex non-Gaussian posterior distributions, expensive forward model evaluations, and misspecified prior information. To address these issues, we propose a deep adaptive dimension-reduction Bayesian inference framework based on the Variational Flow (VF) model. Since standard normalizing flows are restricted by bijective mappings and cannot directly reduce dimensions, VF overcomes this limitation by integrating VAE-based nonlinear dimension reduction with dual normalizing flows for the latent prior and encoder. This design provides a strictly higher evidence lower bound than VAE and allows more flexible approximation of complex posterior distributions. We further introduce an iterative prior updating strategy that gradually moves the prior mean toward high-probability posterior regions, avoiding manual prior tuning. These components form a closed adaptive loop together with an adaptively fine-tuned Fourier Neural Operator (FNO) surrogate: VF generates posterior-concentrated samples to refine the surrogate, while the updated surrogate further improves posterior inference. Numerical experiments on a 100-dimensional Rosenbrock problem and three standard PDE-governed inverse problems show that our method delivers competitive or superior accuracy compared with MCMC, UKI, and SVGD baselines across all tested configurations, with the most pronounced advantages emerging in challenging scenarios such as high-noise observations and high-dimensional parameter spaces.

We study the subclass of potential mean-field games in which the running interaction cost and the terminal target cost are both expressed through reproducing-kernel maximum mean discrepancy (MMD) penalties, and develop a computational framework that exploits this kernel structure. Both costs are estimated from finite-sample empirical distributions using a random Fourier U-statistic representation that is unbiased and has linear cost in the batch size. The drift of the controlled diffusion is parametrized by a neural network and trained via stochastic gradient descent. For population near-minimizers we prove convergence to the terminal-constrained problem as the penalty diverges, and show that the same limit is recovered almost surely when the learned controls are evaluated on independent finite samples under explicit coupling conditions on the penalty, random-feature count and sample size. The framework includes the kernel-MMD-penalty Schrödinger bridge problem as the special case of a vanishing interaction cost. Numerical experiments illustrate the method on the Schrödinger bridge problem in dimensions up to one hundred, and on an electric vehicle charging coordination problem with per-vehicle physical heterogeneity, where an aggregate-demand congestion cost represents price-feedback competition at the population level and the terminal MMD penalty shapes the state-of-charge distribution at the deadline.

We propose a high-order numerical methodology for computing the ground state and time evolution of the two-dimensional Gross-Pitaevskii equation with harmonic trapping potential. The ground state is obtained by combining normalized gradient flow with time-splitting schemes featuring strictly positive coefficients, which makes them suitable for both dissipative and Hamiltonian regimes and avoids the negative time steps required by classical symplectic methods. The computational cost of these methods grows quadratically with the order, in contrast to the exponential growth of symplectic alternatives. Numerical benchmarks assess ground state convergence under different initializations, long-time preservation of mass and Hamiltonian energy for varying mass constraints, and the cost-accuracy trade-off for orders q = 2, 4,..., 14.

We study minimal attention-only transformers under all-token corruption and show they admit a two-stage empirical Bayes interpretation. A single attention step computes a kernel-weighted posterior mean with respect to the empirical distribution defined by the context. Depth refines this distribution through particle dynamics (Stage 1), while a long-range skip-connection carries the noisy input as a query for posterior inference (Stage 2), revealing distinct statistical roles for depth and attention residuals. The framework isolates a minimal setting in which the context itself induces a depth-dependent energy landscape governing in-context inference. We show that effective denoising can emerge without an explicit noise schedule: a fixed kernel bandwidth and finite integration horizon suffice, yielding a principled depth-noise relationship. We further establish a posterior-mean recovery guarantee for Gaussian priors, showing the empirical estimator converges to the Bayes-optimal predictor under sequential asymptotic conditions. Connecting these dynamics to reverse-diffusion limits, our results provide a statistical interpretation of attention as in-context inference via sample-based posterior estimation, without explicit density modeling.

Alexandroff paratopological groups provide a natural setting in which order-theoretic and algebraic properties interact. In this short note we prove obstruction results when considering continuous homomorphisms from Alexandroff paratopological groups into topological groups. Particularly and, as a consequence of these results, we provide an alternative proof of the fact that the only connected Alexandroff topological group is the one that carries the trivial topology.

We construct a family of Whittaker functions for $SL(m,\mathbb{Z})$ induced directly from Whittaker functions for $SL(n,\mathbb{Z})$, for any $2 \leq m<n$. Given Jacquet's Whittaker function $W_{\alpha,N}^{(n)}$ on the generalized upper half-plane $\mathfrak{h}^n$, we show that the function $V_{\alpha,N}^{(m)}:\mathfrak{h}^m\to\mathbb{C}$ defined by restricting $W_{\alpha,N}^{(n)}$ to the block-diagonal embedding $\mathfrak{h}^m\hookrightarrow\mathfrak{h}^n$ is a Whittaker function for $SL(m,\mathbb{Z})$, provided the Langlands parameters $\alpha=(\alpha_i)_{1\leq i\leq n}$ satisfy $\sum_{i=1}^m\alpha_i = m(m-n)/2$. Under this condition, the induced function carries Langlands parameters $\bigl(\alpha_i+\frac{n-m}{2}\bigr)_{1\leq i\leq m}$ and inherits the first $m-1$ entries of the character tuple of $W_{\alpha,N}^{(n)}$. This result complements the propagation formulas of Ishii and Stade, which relate Whittaker functions on $GL(n,\mathbb{R})$ to those on $GL(n-1,\mathbb{R})$ and $GL(n-2,\mathbb{R})$. In contrast, our construction passes directly from $GL(n,\mathbb{R})$ to $GL(m,\mathbb{R})$ for any $m < n$ in a single step.

Affine Coxeter groups are fundamental objects in mathematics and in crystallography. If two group elements are conjugate, then they have very similar algebraic and geometric properties. Using recent structural results of Milićević, Schwer and the second author, we develop an app to visualize conjugation in affine Coxeter groups in dimensions 2 and 3. We explain the mathematics underlying this visualization, and connect its visual features to the arts.

This work presents a weighted quadrature (WQ) method to fast assemble Galerkin matrices based on unstructured spline surfaces. The method is developed upon a particular variant of unstructured splines, namely the bicubic analysis-suitable unstructured T-splines (ASUTS). While existing WQ approaches have significant speedup for structured splines (e.g., B-splines), their extension to unstructured splines faces several challenges: (1) lack of a global parametric domain for defining quadrature points, (2) a varying number of basis functions across elements that complicates the determination of the optimal number of quadrature points, and (3) ill-conditioned underdetermined linear systems that must be solved to find the quadrature weights. To solve these issues, we first define the WQ rule directly in the physical domain. Second, we specify the number of quadrature points function-wise (rather than element-wise), which naturally satisfies the well-posedness condition, namely the number of unknown weights no less than that of exactness constraints. Third, we employ the truncated Singular Value Decomposition to improve the conditioning of the underdetermined systems by discarding extremely small singular values, which are caused by the splines around extraordinary points. Several different model problems are studied, such as Poisson's problem, the biharmonic problem, and the nonlinear heat transfer problem. In the end, a variety of numerical tests are performed to demonstrate the accuracy and efficiency of the proposed method.

We study the long-time behavior of the (critical) Kohn–Sham equation in two and three dimensions, i.e.,\[ \mathrm{i} \partial_t {\gamma} = \Big[-\frac{1}{2}\Delta + \lambda \, |\cdot|^{-1} \ast \rho_{{\gamma}} + \mu \, \rho_{{\gamma}}^{1/d}, {\gamma} \Big] \quad \text{for} \quad d=2,3. \] By introducing a suitable ”square root” of the density matrix and exploiting the pseudo-conformal transform, we establish global well-posedness for small initial data in an appropriate weighted Schatten norm. We also prove the optimal time decay of the particle density and establish modified scattering for small and localized solutions. In particular, our results provide a resolution to the open problems proposed by Pusateri and Sigal (2021) for the critical and subcritical regimes, rigorously proving their conjectures regarding modified scattering in the critical case and linear scattering in the subcritical cases. Our results place these scattering phenomena in the operator-valued setting of density matrices, thereby extending the classical scalar theory to a broader framework.

We establish a generalization of Anush Tserunyan and Jenna Zomback's 2024 Backward Ergodic Theorem. We remove the countable-to-one assumption and thus provide a backward ergodic theorem for arbitrary measure-preserving transformations. However, this new setting introduces measurability concerns as unlike the countable-to-one case, we no longer have a collection of Borel right inverses. Instead, we must rely on the Jankov, von Neumann uniformization theorem. Towards this, we use Borel and measured field structures introduced by Stefaan Vaes and Lise Wouters.

Nagano spaces are compact symmetric spaces that admit large transformation groups. They include for instance all the Grassmannians and the Einstein Universes. In this paper, we study a Kobayashi-type pseudometric on domains in real-type Nagano spaces. When the Nagano space is real projective space, this metric coincides with the classical Kobayashi pseudometric. For a dually convex domain of a general real-type Nagano space, we prove that this pseudometric is a genuine metric if and only if the domain does not contain a photon minus a point. We compute this metric on the proper symmetric domains and prove that it is obtained by integrating the $L^1$-norm along flats. We prove that in higher rank, the Kobayashi metric of a strongly $\mathcal{R}$-proper dually convex divisible domain is never Gromov hyperbolic. This contrasts with the rank-one case corresponding to real projective space, where a classical result of Benoist shows that this metric is Gromov hyperbolic if and only if the domain is strictly convex.

Let $p$ be a prime. Suppose that integers $r$, $e$, $d$ such that $r \ge 2$, $e \ge 0$, $0 \le d \le p$ are given. Let $f(x)=s_0 x^r + s_1 x^{r-1} + \cdots + s_r$ be a generic polynomial of degree $r$ in characteristic $p$. We put $f(x)^e=\sum_{i \ge 0} c_i x^i$. We define a $d\times d$ matrix $M_d(f(x)^e)$ by $M_d(f(x)^e) = ( c_{i p + j - d -1})_{1 \le i,\, j \le d}$. In this paper, we shall be concerned with the divisibility of $\det M_d(f(x)^e)$ by powers of the discriminant $\Delta(f(x))$ of $f(x)$. First, assuming $s_0=1$, we study the condition under which $\det M_d(f(x)^e)$ is a positive power of $\Delta(f(x))$ multiplied by a non-zero constant in ${\mathbb F}_p$. Second, for such matrices when $d=r-1$, we present a formula for $M_d(f(x)^e)^{-1} M_d(f(x)^{e+1})$ involving the Bézout matrix of $f'(x)$ and $f(x)-\frac{1}{r} x f'(x)$. Finally, we present two similar experimental equalities, the first of which involves the determinant $\det M_d(f(x)^e)$.

A linearized function field $F$ can be viewed as a Galois extension of a rational function field $K(x)$. For a totally ramified place $Q$ of degree one in $F/K(x)$, we give a unified description of the set $G(Q)$ of gaps at $Q$. As a consequence, we explicitly provide a system of generators, the multiplicity, and the Frobenius number of the Weierstrass semigroup $H(Q)$. Moreover, we give a necessary and sufficient condition for $H(Q)$ to be symmetric. Then we investigate the minimal generating set of the Weierstrass semigroups at several totally ramified places of degree one. We not only explicitly describe the minimal generating set, but also provide functions whose coefficients of pole divisors lie in the minimal generating set. Finally, we investigate the linearized function field associated with the denominator of a separable polynomial and apply our results to present several examples.

In this paper, we further investigate and refine the subspace-constrained preconditioning technique to enhance the theoretical and numerical convergence properties of randomized iterative methods for solving linear systems. In particular, we design a QR-like factorization that transforms the original linear system into an equivalent block-orthogonal form, thus avoiding the full-rank assumptions adopted in existing work. Moreover, this reformulation reduces the problem to solving a smaller linear system with a favorable singular value distribution, provided an appropriate initial point is employed. The proposed framework can be implemented implicitly within the iteration and does not require explicitly constructing either a preconditioner matrix or a preconditioned linear system, which eliminates the prohibitive cost of forming a fully preconditioned system. Furthermore, we construct orthogonalized search directions from stochastic gradients and develop accelerated variants of the framework. We prove that the proposed algorithmic framework converges linearly in expectation. Numerical experiments demonstrate the benefits of the proposed preconditioning strategy.

This paper studies the nonconvex quadratic root-difference minimization under elliptic annulus constraints (QR). We first establish the Annulus Brickman theorem and equivalently reformulate (QR) as a 2-dimensional convex problem (HP) with hidden variables. We employ the Frank-Wolfe algorithm to globally solve (HP). A key finding is that the solutions of the Frank-Wolfe subproblems, which are traditionally viewed as mere auxiliary updates, are proven to be $O(1/\sqrt{k})$-approximate solutions of the original problem (QR). This transforms an algorithmic by-product into the primary output and completely bypasses the need to solve the computationally expensive quadratic system required for solution recovery. Leveraging this recovery-free property, we develop the efficient Iterative Minimum Generalized Eigenpair (IMGE) algorithm for globally solving (QR). Numerical experiments confirm that IMGE converges rapidly and significantly outperforms conventional methods, especially for large-scale problems.

We propose restarted accelerated primal-dual algorithms with (non-monotone) backtracking (rAPDB) for convex nonlinear conic programs, with quadratically constrained quadratic programs (QCQPs) as a special case. Unlike linear and quadratic programs, these problems give rise to convex-concave minimax reformulations with non-bilinear coupling terms; therefore, the existing primal-dual methods for bilinear couplings are not applicable. To address this challenge, we build on the accelerated primal-dual method with adaptive stepsize search – as it adapts to the local curvature – and develop both fixed-frequency and adaptive restart schemes, incorporating both monotone and non-monotone adaptive step-size search strategies. The resulting algorithms require only first-order information and matrix-vector products, making them suitable for large-scale and GPU-accelerated implementation. Under metric subregularity of the KKT mapping, we prove a quadratic growth property for a self-centered smoothed duality gap and establish global linear convergence of the proposed restarted methods. We also establish sufficient conditions under which the metric subregularity holds even for general nonconvex problems over convex polyhedral cones. These results are new and may be of independent interest. Numerical experiments on random QCQPs and kernel matrix learning instances show that the proposed methods, especially with non-monotone adaptive stepsizes and GPU acceleration, achieve strong practical performance.

We study the anisotropic Finsler $p$-Laplacian equation \begin{equation*} \left\{ \begin{aligned} &-\Delta ^{H}_{p}u=f(u) \quad\,\,\, &{\rm{in}} \,\, \mathcal{C}, &{\bf{a}}(\nabla u)\cdot \nu =0 \quad\,\,\, &{\rm{on}} \,\, \partial\mathcal{C}, \end{aligned} \right. \end{equation*} where $N\geq3$, $1<p<N$, $\mathcal{C}\subseteq\mathbb{R}^{N}$ is an open convex cone and $\Delta ^{H}_{p}$ is the anisotropic Finsler $p$-Laplacian operator. If $f(u)$ is nonnegative and subcritical, we prove that every bounded nonnegative solution in $\mathcal{C}$ is identically zero. In particular, for $f(u)=u^{q}$ with $0<q<p^*-1$, we establish a pointwise decay estimate in $\mathcal{C}$ via the doubling argument and blowing-up method and prove that all nonnegative solutions must be zero without the boundedness assumption. Our results are the subcritical counterpart of the classification result for the critical case in \cite{CFR}, and extend the Liouville type theorems in $\mathbb{R}^{N}$ for the standard $p$-Laplacian in \cite{SZ} and for the anisotropic $p$-Laplacian in \cite{CFV, CHN} to general convex cones $\mathcal{C}$. In the critical case $f(u)=u^{p^*-1} $ and typical case $H(\xi)=|\xi|$, for $\frac{N+1}{3}<p<N$, we classified the positive solutions of the critical $p$-Laplacian equation in convex cones $\mathcal{C}$ without finite-energy assumption. This extends the classification result of \cite{Ou} in $\mathbb{R}^{N}$ to general convex cones $\mathcal{C}$, and removes the finite-energy assumption in \cite{CFR} in the typical case $H(\xi)=|\xi|$.

A crucial component of machine learning algorithms is minimizing loss functions with less computational cost and less oscillations. While adaptive learning rate-based optimizers have been widely used for real-world tasks, they do not guarantee convergence, which is why AMSGrad was later introduced to investigate the non-convergence behaviour of Adam. In this paper, popular adaptive optimization methods like Adam and AMSGrad are critically reviewed with an emphasis on their fundamental design concepts. To address limitations of the above mentioned optimizers, a new optimizer variant, C-Adam, is proposed based on the line of sight approach. A theoretical proof for convergence is also provided and the optimizer is validated through a number of real-life based numerical experiments.

Motivated by the introduction of the Zakharov-Kuznetsov equation as a higher dimensional generalization of the Korteweg-de Vries equation, in this paper we introduce the modified Zakharov-Kuznetsov (mZK) equation as a 2-dimensional generalization of the complex-valued modified Korteweg-de Vries equation. We initiate the mathematical analysis of the mZK equation on $\mathbb{T}^2$ by proving a local well-posedness result in Sobolev spaces and by establishing a failure of uniform continuity. We also note that the real-valued version of the mZK equation has physical significance.