Papers

Parameter estimation is a central task in data-driven learning of dynamical systems. It aims to recover the underlying physical parameters from observed time-series data, thereby providing interpretable insights into the physical mechanisms governing the system. Gradient/derivative matching methods based on Gaussian process provide an efficient way to perform parameter estimation. Those methods avoid repeated numerical integration and enforce local derivative consistency. However, such local matching may result in global inconsistency with the governing flow map, particularly under scarce and noisy observations. To address this limitation, we propose a framework based on Gaussian process learning with flow map refinement (GPL-FMR), a two-stage parameter estimation framework. The first stage is based on Gaussian process learning algorithm and the posterior obtained from which is transferred as an informative prior to the second stage based on flow-map refinement. The second stage further improves the parameter estimation via optimisation based on global dynamical constraints. We demonstrate and analyse its performance on multiple numerical examples, including the Van der Pol oscillator, the Lotka-Volterra model, and the Lorenz-63 system. The results show that the proposed framework consistently improves parameter estimation accuracy, particularly under scarce and noisy observations.

For $2 \leq a \leq d+1$, what is the largest integer $H_{a,d} (n)$ such that every set of $n$ points in $\mathbb{R}^d$ with no $a$ points on a common $(a-2)$-flat contains a subset of $H_{a,d} (n)$ points whose determined $(a-1)$-dimensional simplices have pairwise distinct $(a-1)$-dimensional volumes? We construct $n$-point sets that improve the best known upper bounds for $H_{a,d}(n)$ in several cases of $a$ and $d$. We also study a dual version of the problem. Let $D_d(n)$ the maximum number such that for any arrangement of $n$ hyperplanes in general position in $\mathbb{R}^d$, we can always find a subset of $D_d(n)$ hyperplanes for which all the $d$-dimensional simplices that they define have distinct $d$-dimensional volumes. We improve the current known upper bound for $D_d(n)$ and give the first nontrivial lower bound for $D_2(n)$ and $D_3 (n)$.

In this paper, for various pairs of topological spaces $X$ and $Y$, we introduce an analogue of the Bernšteĭn-Gel'fand-Ponomarev reflection functors between the categories of C*-algebras over $X$ and $Y$. We prove that these functors induce equivalences $\mathfrak{KK}(X)_{\mathrm{loc}} \simeq \mathfrak{KK}(Y)_{\mathrm{loc}}$ between Meyer and Nest's localizing subcategories in Kirchberg's ideal-related KK-theory, and $\mathfrak{E}(X) \simeq \mathfrak{E}(Y)$ between the entire categories in Dadarlat and Meyer's ideal-related E-theory. Both equivalences restrict to the respective bootstrap categories: $\mathcal{B}(X) \simeq \mathcal{B}(Y)$ and $\mathcal{B}_{\mathrm{E}}(X) \simeq \mathcal{B}_{\mathrm{E}}(Y)$. Combining these equivalences with a combinatorial argument in quiver representation theory, we obtain the same equivalences for any two finite $T_0$-spaces whose Hasse diagrams are orientations of the same tree. Moreover, we prove a rearrangement property of reflection functors, which yields an analogue of Coxeter functors. Finally, we apply reflection functors to give a conceptual alternative proof of Bentmann and Köhler's universal coefficient theorem for filtrated K-theory over finite $T_0$-spaces whose Hasse diagrams are orientations of Dynkin diagrams of type A.

We analyze coefficient design in a one-parameter family of explicit palindromic electric–magnetic splittings for the time-domain Maxwell equations. After fourth-order staggered spatial discretization, the Fourier amplification matrix depends on the single scalar $g_2=a(1-2a)/2$. We prove that $a=1/4$ is the unique real coefficient maximizing the spectral CFL interval, with threshold $s_*=12/(7\sqrt d)$. We then identify a real-coefficient obstruction to higher phase accuracy: cancellation of the leading temporal phase defect requires $g_2=1/12$, whereas every real member satisfies $g_2\le 1/16$. The resulting complex-conjugate coefficients give fourth-order temporal phase accuracy for each fixed semidiscrete Fourier mode and have threshold $6\sqrt3/(7\sqrt d)$, while the complete field update remains globally second order in time. For real Maxwell data, the physical output is the real projection of the complex trajectory; this projection is branch independent and preserves the second-order error bound. We further give an exactly equivalent doubled real-arithmetic realization, which clarifies the role of the auxiliary imaginary component without changing the numerical method. A semidiscrete convergence result and numerical experiments confirm the distinction between stability optimization and phase optimization.

Let $M^4\hookrightarrow\mathbb S^5(1)$ be a closed CMC hypersurface with constant scalar curvature and constant third power sum $f_3=\sum_{i,j,k}h_{ij}h_{jk}h_{ki}$. We prove that if $M^4$ has exactly two distinct principal curvatures at some point, then it is isoparametric. More precisely, it is a Clifford torus of the form $\mathbb S^1(r)\times\mathbb S^3(\sqrt{1-r^2})$ or $\mathbb S^2(r)\times\mathbb S^2(\sqrt{1-r^2})$, where $0<r<1$. Under the additional Willmore condition, we obtain a complete classification: every closed CMC Willmore hypersurface $M^4\hookrightarrow\mathbb S^5(1)$ with constant scalar curvature is isoparametric. Consequently, it is congruent to a totally umbilic geodesic sphere, the minimal Clifford torus $\mathbb S^2(1/\sqrt2)\times\mathbb S^2(1/\sqrt2)$, the nonminimal Clifford torus $\mathbb S^1(\sqrt3/2)\times\mathbb S^3(1/2)$, or a Cartan minimal hypersurface. The proofs combine trace-free local tensor identities, an algebraic analysis of the possible principal curvature multiplicities, and a weighted differential $3$-form together with a cut-off argument near the set where principal curvatures coalesce. No sign condition on the scalar curvature is imposed.

The Minimum Matching Problem consists of finding an independent edge set of minimum weight $M_{\star}(G)$ in a given edge-weighted graph $G$. When $G$ is bipartite, this reduces to the Assignment Problem. We consider a variant of this problem defined by taking the union of optimal matchings across various slightly modified versions of the base graph: $H_{\mathcal{J}}(G)=\bigcup_{U \in \mathcal{J}} M_{\star}(G_{U})$. We establish two families of results: (1) In two distinct settings for the Assignment Problem, we prove that the resulting graphs $H_{\mathcal{J}}$, as well as certain associated graphs $\bar{H}_{\mathcal{J}}$, are spanning trees on the relevant base graphs $G$ and $\bar{G}$. (2) In these same settings, assuming the edge weights are given by the $p$-th power of Euclidean distances for point configurations in the plane, we show that for $p=1$ the tree $H_{\mathcal{J}}$ is non-crossing (i.e., its planar embedding has no crossing edges), whereas, remarkably, for $p=2$ the associated tree $\bar{H}_{\mathcal{J}}$ is non-crossing. Finally, we introduce novel conjectures in Statistical Mechanics, to be explored in future work: in the Random Euclidean Assignment Problem (where points are i.i.d.\ on a planar domain), we conjecture that for $p=2$ the trees $\bar{H}_{\mathcal{J}}$ are asymptotically distributed as Uniform Spanning Trees with free and wired boundary conditions in the two respective settings. In particular, suitable paths on the tree in the second setting, and on its planar dual in the first setting, are asymptotically distributed as $\text{SLE}_{\kappa}$ with $\kappa=2$.

In the unit ball, we study a semilinear system that gives rise to a free boundary of Alt-Phillips type, where the equation is governed by a mixed variable-exponent operator of $(p,q)$-Laplacian type. Under certain structural conditions, we establish the existence of nonnegative radial solutions that are $C^1$, with their norms depending on the structural data.

We study the natural superconvergence points and asymptotic expansions of one-dimensional spline finite element approximations. For a spline space of degree $k$ and any smoothness $0\le\mu\le k-1$, we prove that the $s$-th derivative of the error exhibits enhanced convergence of order $O(h^{k+2-s})$ at points where $k-s$ is even, provided the mesh is symmetric within a region of size $Ch|\ln h|$ around the point. This condition is known to be optimal for the cases of low derivative order $s=0,1$; the present analysis shows that the same local condition is sufficient for all admissible $s$. Moreover, by expanding the error in Legendre polynomials, a closure theorem determines the leading-order Legendre coefficients (the asymptotic expansion of the error) by combining the Galerkin orthogonality with the superconvergence conditions. For $\mu=k-1$ (B-splines) and $\mu=k-2$, the Galerkin orthogonality conditions vanish and the coefficients are determined solely by the superconvergence conditions. The asymptotic expansion can be expressed through a simple antiderivative recurrence on Legendre polynomials. The resulting polynomial's zeros encode the complete set of superconvergence points for all derivative orders. Numerical experiments for selected $(k,\mu)$ pairs confirm the theoretical predictions.

This paper formulates and analyzes a delayed SVIRS epidemic model incorporating media coverage effects, vaccination, waning immunity, temporary post-recovery immunity, saturated treatment, and delayed behavioral responses induced by media coverage. The positivity and uniform boundedness of solutions are established, the basic reproduction number is derived, and the local and global asymptotic stability of the disease-free and endemic equilibria is investigated. Taking the media-induced behavioral delay as the Hopf bifurcation parameter, a critical delay threshold is obtained, beyond which the endemic equilibrium loses stability and periodic oscillations emerge. Center manifold and normal form theories are applied to determine the direction of the local Hopf bifurcation and the stability of the bifurcating periodic solutions, while a global Hopf bifurcation theorem is used to establish the unbounded continuation of the periodic solution branch. Numerical simulations confirm the theoretical results and indicate that stronger media intervention can suppress epidemic oscillations and enhance system stability. These findings reveal the coupled effects of multiple epidemiological mechanisms and delayed media responses, providing theoretical support for the design of effective infectious disease control strategies.

Determining the properties of a random polynomial has fuelled significant investigation over the past century. One driving force of this research is a 1936 paper of Van der Waerden. Fix $n \ge 3$ and let $E_n(B)$ be the number of monic, irreducible, non-$S_n$ polynomials $f = X^n + a_1 X^{n-1} + \cdots + a_n$ with $|a_j| \leq B$ for all $j$. A recent breakthrough of Bhargava bounds $E_n(B) \ll B^{n-1}$. This spectacularly resolves a conjecture of Van der Waerden, but leaves open its stronger form, namely that $E_n(B) = o(B^{n-1})$. Inspired by recent progress, we now prove this stronger form for all $n \ge 40$. Bhargava's result, together with work of Chow and Dietmann, essentially reduces the strong Van der Waerden conjecture to the claim that the number of polynomials $f$ with Galois group $A_n$ is $o(B^{n-1})$. We connect this claim to Chowla's conjecture, an area driving major mathematical progress. Indeed, we formulate a twisted function field version of Chowla's conjecture, and show how Van der Waerden's conjecture reduces to this conjecture for $n \ge 8$. Finally, we make significant progress towards this twisted Chowla conjecture, leading to our resolution of the strong form of Van der Waerden's conjecture for all $n \geq 40$.

The aim of this paper is to introduce pairing theory for partitions. We begin with two statistics on integer partitions, the pairing index and the pairing rank. The pairing index is equidistributed with the number of parts, while a joint refinement identifies its two constituents with the numbers of even and odd parts. We further introduce the pairing width and prove that pairing index and pairing width are jointly equidistributed with the number of parts and the largest part. The resulting finite Gaussian generating function has a cyclotomic factorization from which Kummer's famous carry theorem for binomial coefficients follows. We also prove a mod-$5$ congruence for the excess of unpaired parts congruent to $1$ modulo $4$ over those congruent to $3$ modulo $4$ in the partitions of $5n+4$. A signed specialization exhibits that the parity of the pairing rank is governed by self-conjugate partitions. Motivated by this, we go on to introduce a second, diagrammatic pairing: after the two wings of the Durfee square are folded together, the unpaired cells break into connected diagonal blocks. These blocks may be reflected independently, giving a Boolean decomposition of the set of partitions with a unique representative having all successive ranks nonnegative. We then relate our theory to overpartitions and Frobenius representations, obtaining as a corollary a geometric realization of overpartitions in terms of partitions whose principal hooks are all even. Finally, we study simply paired partitions of negative pairing rank, obtaining identities involving odd divisors and overpartitions, a parity theorem for pairing rank $-2$, and a Toeplitz determinant whose coefficientwise limit is an explicit infinite product related to MacMahon's product for plane partitions.

We show that approximately counting the vertices of a bounded 0/1 polytope, presented as a system of rational linear inequalities, is, informally speaking, NP-hard. In particular, there is no FPRAS for this problem unless RP=NP. The proof is by a reduction from approximately counting homomorphisms from a given graph to a particular four-vertex graph. The main proof ideas were found using GPT-5.6 Sol Ultra.

Let $\Omega\subset\mathbb R^3$ be a smooth bounded domain and let $g:\partial\Omega\to\mathbb S^2$ be smooth. We prove that any global minimizer of the Oseen–Frank energy subject to the strong anchoring condition $n=g$ is smooth in a full neighborhood of $\partial\Omega$. The result holds for arbitrary positive splay, twist, and bend constants, without any small-anisotropy assumption. It resolves the boundary-regularity part of Lin and Liu's Problem~(c) for the pure Oseen–Frank problem and for its prescribed smooth magnetic-field perturbation.

In this paper, we propose a method to study the Uniform Boundedness Conjecture and the Lang-Silverman Conjecture for abelian varieties $A$ defined over a global field $K$; the latter is a uniform lower bound on the heights of non-torsion rational points. Our method is inspired by Vojta's proof of the Mordell Conjecture (Faltings's Theorem). Over function fields of characteristic $0$, a recent breakthrough of Looper-Yap (arXiv:2603.23396) proves both conjectures with inexplicit bounds. In our paper, we give a new proof of both conjectures with explicit bounds, which also depend polynomially on the field $K$ unless $A/K$ admits a factor of good reduction everywhere. We also prove explicit bounds for elliptic curves over function fields of characteristic $p>0$. Over number fields, we prove both conjectures under a suitable high-dimensional Szpiro conjecture (weaker than Hindry's version, Conjecture 3.4 of https://webusers.imj-prg.fr/~marc.hindry/MW-size.pdf) that we propose.

We prove several results concerning the arithmetic nature of multiple zeta values related to Zlobin's 2006 note. Our proofs are based on Zagier's theorem and Lindemann's theorem.

The exact nondeterministic automatic complexity $A_{Ne}(x)$ of a word $x$ is the minimum number of states of a nondeterministic finite automaton that accepts $x$ and no other word of length $|x|$; the unique nondeterministic automatic complexity $A_N(x)$ additionally requires the accepting computation to be unique. Chen, Kjos-Hanssen, Koswara, Richter, and Stephan (FSTTCS 2025) asked whether the two measures can differ. We answer this affirmatively: the binary word $w=1101000000100$ satisfies $A_{Ne}(w)=6<7=A_N(w)$. The separation is witnessed by a six-state automaton whose sole accepted word of length $13$ is $w$, accepted along exactly three computation paths. The matching lower bounds were established by exhaustive search and by SAT solvers, with proofs of unsatisfiability certified in the DRAT format. An exhaustive census shows that $13$ is the least length of a separating word over any alphabet, binary or not, and that up to renaming of letters the separating words of length $13$ are precisely $365$ letter-mergings of the single seven-letter word $0123444445126$, all with $A_{Ne}=6<7=A_N$; exactly eight of them are binary. The separating word and the core of the verification were found by an artificial intelligence agent.

Repeated sample splitting, cross-fitting, conformal ensembling, and related randomized workflows often produce multiple dependent valid p-value functions for the same target. Existing p-merging theory guarantees pointwise validity under arbitrary dependence, but turning the merged output into a confidence region typically requires repeated evaluation on a grid over the parameter space. This inversion step can be computationally costly, approximation-dependent, and increasingly difficult to scale. We study when confidence regions can instead be computed exactly from split-wise regions without gridding the parameter space. Our main structural result shows that mergers induced by step calibrators form a broad exactly executable class at the region level, and a partial converse within the calibrator-induced family indicates that exact executability is closely tied to step structure. Within this class, we develop exact voting algorithms, including an adaptive multi-quantile contour aggregator that avoids pre-specifying a single order-statistic threshold while preserving finite-sample validity under arbitrary dependence. Simulation studies on repeated-split regression and conformal prediction, together with real-data regression examples, show that the proposed methods provide stable, robustness-oriented exact inference with substantial runtime gains over grid-inversion comparators. A grid-resolution benchmark shows that fixed-k voting and adaptive multi-quantile voting are essentially insensitive to inversion-grid refinement, while a multidimensional single-threshold stress test illustrates the dimensional blow-up faced by grid-inversion baselines in a simple box-geometry setting. Taken together, the results show that arbitrary-dependence contour merging can be turned into an exactly executable region-computation framework rather than merely a pointwise validity device.

Let $M^3\to X^4$ be a complete, connected, two-sided stable minimal immersion. We prove that if the ambient sectional curvature is nonnegative and the ambient scalar curvature has a positive uniform lower bound, then $M$ is totally geodesic and its normal Ricci curvature vanishes. No weak bounded geometry assumption and no upper curvature bound are imposed. We also construct a complete metric of strictly positive sectional curvature on $\mathbb{R}^4$ admitting a complete, embedded, one-ended, nonparabolic, two-sided stable minimal hypersurface diffeomorphic to $\mathbb{R}^3$ which is not totally geodesic. The rigidity proof combines spectral splitting theory, a warped $\mu$-bubble construction, and a harmonic function level set argument. The example is obtained by a compactly supported deformation of an example in \cite{CLS}.

In this paper, we consider the convergence of some popular Kaczmarz-type methods, including the randomized Kaczmarz (RK) method, the randomized extended Kaczmarz (REK) method, the randomized block Kaczmarz (RBK) method, and the randomized double block Kaczmarz (RDBK) method for doubly noisy linear system. To the best of our knowledge, this is the first work that considers and compares the convergence of the Kaczmarz-type methods for consistent linear systems, as well as that for inconsistent linear systems to solve double noisy linear system. We prove that these methods converge to a neighborhood of the smallest norm least squares solution of the underlying noiseless system. Specifically, RK and RBK that are designed for consistent linear systems, often outperform REK and RDBK that are designed for inconsistent linear systems, respectively, in terms of the convergence rate or the convergence horizon. Moreover, we establish the convergence of the more general sketch-and-project framework for the doubly noisy linear system. The analysis provides more comprehensive results without imposing any additional restrictions. Numerical experiments are performed to validate our theoretical results.

We use the Pfaffian minor summation formula together with Jacobi–Trudi Toeplitz matrices to derive Pfaffian expansions in skew Schur functions. This yields explicit Schur expansions for several generating functions involving products of skew Schur functions. In particular, using sparse skew-symmetric matrices, we provide a Pfaffian proof of a Schur-positive identity arising in the study of the $q$-log-convexity of the Narayana polynomials. As the main application, we prove that the Baxter polynomials form a $q$-log-convex sequence. We further show that the Baxter transformation defined by the refined Baxter numbers preserves log-convexity. Finally, by realizing the $q$-refined Baxter numbers as principal specializations of rectangular Schur functions, we prove that the array of $q$-refined Baxter numbers is $q$-log-concave both along each row and along each column. Both $q$-log-concavity results extend naturally to the $q$-analogues of the $d$-Hoggatt numbers.

Let ${T}_n$ be a random $2$-dimensional determinantal hypertree on $n$ vertices. Given any prime $p$, we answer the following question: If a cocycle in $Z^1({T}_n,\mathbb{F}_p)$ has small support, what does the support typically look like? More precisely, we characterize all the finite connected graphs $G$ for which there is a constant $c_G>0$ with the following property: For all large enough $n$, with probability at least $c_G$, we have a cocycle $f\in Z^1({T}_n,\mathbb{F}_p)$ such that after removing all the isolated vertices, the support of $f$ is isomorphic to $G$. We prove that for $p>2$, we do not have any such graph. For $p=2$, a connected graph has the property above if and only if it has a unique cycle such that this unique cycle has odd length, moreover, if the unique cycle is a triangle, then we also need to require that all the vertices of the triangle have degree at least $3$.

We establish sharp Barron regularity for Coulombic many-electron wave functions after extraction of the universal cut-off Jastrow factors. Following the factorization of Fournais et al.~\cite[Definition~1.4]{FournaisEtAl2005}, for a Coulombic eigenfunction $\psi$ we define the successive quotients by \[ \phi=e^{-F_{2,\mathrm{cut}}}\psi \quad\text{and}\quad \phi_3=e^{-F_{3,\mathrm{cut}}}\phi=e^{-(F_{2,\mathrm{cut}}+F_{3,\mathrm{cut}})}\psi. \] Then \[ \phi,\phi_3\in\mathcal{B}^s(\mathbb{R}^{3N}) \qquad\text{for every }s<2. \] This range is optimal among universal factorizations. No factor depending only on the particle number and the nuclear data, but not on the eigenfunction or its eigenvalue, can make every corresponding quotient belong to $\mathcal{B}^2$. We also determine the exact endpoint growth. Writing $\varepsilon=2-s$, we prove that, for either $u=\phi$ or $u=\phi_3$, there is a computable constant $M$ independent of $\varepsilon$ such that \[ \left\|u\right\|_{\mathcal{B}^{2-\varepsilon}}\leq\frac{M}{\varepsilon^2}\left\|u\right\|_{\mathcal{B}^1}. \] For the unperturbed two-electron atom we prove, with a constant independent of $\varepsilon$, \[ \left|\left\|\phi_3\right\|_{\mathcal{B}^{2-\varepsilon}}-\frac{32\pi Z\lvert\phi_3(0,0)\rvert}{\varepsilon^2}\right|\leq\frac{C}{\varepsilon}. \] Hence the quadratic rate in the upper bound is sharp whenever $\lvert\phi_3(0,0)\rvert\neq0$, as is the case for the ground state.

Let $k_{1,\varepsilon}(n)$ be the smallest number of real linear measurements needed by a randomized oblivious sketch that estimates the nuclear norm of every fixed real $n\times n$ matrix within a factor $1\pm\varepsilon$, with probability at least $2/3$. For every fixed $0<\varepsilon<1$, we prove \[ \frac{n^2}{(\log n)^{A_\varepsilon}} \le k_{1,\varepsilon}(n) \le C_\varepsilon \frac{n^2\{\log\log(e^e n)\}^2}{\log(e n)}. \] Previously, the best unrestricted bounds for general linear sketches of the Schatten–1 norm were $\Omega(n)$ and the trivial $O(n^2)$ upper bound (Li, Nguyen, Woodruff'19), leaving a polynomial gap. Our bounds close that gap up to polylogarithmic factors and give a nontrivial logarithmic saving below the $n^2$-measurement storage bound. The result extends much further. Write $k_{p,\varepsilon}(n)$ for the analogous sketch dimension for the Schatten–$p$ norm. For every fixed finite $p>0$ that is not a positive even integer, there are positive constants $A_{p,\varepsilon},C_{p,\varepsilon},c_p$ such that \[ \frac{n^2}{(\log n)^{A_{p,\varepsilon}}} \le k_{p,\varepsilon}(n) \le C_{p,\varepsilon}\frac{n^2}{(\log n)^{c_p}}, \] so $k_{p,\varepsilon}(n)=n^{2-o(1)}$ throughout the non-even regime. Together with the known tight bounds $\Theta_{p,\varepsilon}(n^{2-4/p})$ for positive even $p$ and $\Theta_\varepsilon(n^2)$ for $p=\infty$ (Li, Woodruff'16), our results close the remaining polynomial gap across the Schatten family and complete, up to polylogarithmic factors, the polynomial-order classification of general linear sketches for all Schatten-$p$ norms.

We develop a componentwise geometric and monodromy theory for the one-support generalized Lamé equation on an elliptic curve, with singularities at \(0\) and \(\pm p\). Its log-free curve decomposes canonically into irreducible even and non-even components, the former being governed by elliptic Painlevé~VI. For the non-even component, we construct the hyperelliptic spectral curve, Baker–Akhiezer functions, and addition map, and prove that \[ \deg \sigma_{n,p}^{(1)}=n(n+1). \] After quotienting by the involution \(T\mapsto -T\), we identify the non-even spectral curve with the classical Lamé spectral curve of weight \(n\), compatibly with the addition map and the rational function \(\kappa\). This identification is realized by \[ \widetilde B=T^2-n(n+1)\wp(p), \] and associates every non-even generalized Lamé equation with a unique classical Lamé equation on the same elliptic curve having equivalent period monodromy. Fixing \(\widetilde B\) yields an isomonodromic deformation with \(\tau\) fixed. Together with the Painlevé-VI deformation on the even component, it gives a componentwise interpretation of the collision \(p\to0\), and yields a finite descent on the admissible completely reducible locus. The classical spectral, finite-gap, finite-monodromy, and curvature theories consequently transfer to the non-even component. Finally, within the symmetric family $\left(n_0,n_1,n_2,n_3,\frac12,\frac12\right),$ the one-support case forms an affine genus-zero hierarchy, whereas for $\left(1,1,0,0,\frac12,\frac12\right)$, the non-even normalization is generically elliptic and becomes rational on the discriminant locus, while the full compactified log-free curve retains arithmetic genus two. This first genus jump marks the boundary of the affine theory and motivates a genus-dependent componentwise geometry.

Quasiperiodic parabolic equations (QPEs) describe various physical processes in systems with long-range order without decay, such as heat conduction, diffusion, and transport, and are naturally posed on the whole space. However, practical computations could be performed on finite domains, where preserving the global quasiperiodic structure through appropriate boundary conditions remains a key challenge. Traditional periodic boundary conditions (PBCs) cannot capture quasiperiodic information beyond the computational domain and suffer from Diophantine errors caused by rational approximations of irrational numbers. To address this issue, we propose a class of quasiperiodic boundary conditions (QBCs) based on a homomorphism between the physical domain and a high-dimensional torus, enabling the long-range quasiperiodic structure to be captured at the boundaries. We further establish rigorous convergence results for numerical discretizations of finite-size QPEs with QBCs. Numerical experiments demonstrate that QBCs avoid the influence of Diophantine errors and enable accurate and efficient computations for both high- and low-regularity cases, with improved convergence over PBCs.

Reinforcement Learning (RL) has achieved tremendous success in recent years. However, the classical foundations of RL do not account for the risk sensitivity of the objective function, which is critical in various fields, including healthcare, finance, etc. A popular approach to incorporate risk sensitivity is to optimize a specific quantile of the cumulative reward distribution. However, exact quantile objectives are non-smooth and can change abruptly under small perturbations of the return distribution, making them difficult to optimize reliably when the transition model must be learned from data. Motivated by this instability, we develop UCB-BQRL, a model-based optimistic learning algorithm that maintains confidence sets for the transition kernel and plans using a lower-buffered quantile criterion. The buffered criterion smooths the exact quantile objective by averaging nearby lower quantiles, thereby improving stability under transition-estimation error. To compute the buffered-quantile policy at each episode, we introduce EVI-BQ, an exact dynamic-programming procedure. We establish a high-probability regret bound for UCB-BQRL, which up to logarithmic factors scales as $\mathcal{O}(\mathrm{e}^{\tau/\rho_\tau}+H^2\sqrt{SAT})$, where $\rho_\tau$ is denoted as the root-level left-plateau threshold, which is a problem-dependent constant. Further, we establish an information-theoretic lower bound of $\Omega(H/\rho_\tau\sqrt{AT})$ for the regret of any algorithm dealing with a quantile objective function. Finally, we prove that the exact point-quantile evaluation and exact lower-buffered quantile evaluation are PP-hard under polynomial-time Turing reductions, even for a fixed policy in a two-state, one-action finite-horizon MDP.

We prove that in the anti-Gaussian space $\bigl(\R^{m+1},\bar g_{\Euc},e^{|x|^2/4}\,dx\bigr)$, $m\ge2$, round spheres centered at the origin are the only closed, connected, two-sided immersed hypersurfaces with constant weighted mean curvature that are stable under weighted-volume-preserving variations.

In this paper, by implementing the generalized inverse, known as the Moore-Penrose inverse, for Hilbert space operators, several inner product bounds that are of Cauchy-Schwarz type are presented. As a standard application, some new refined and generalized versions of numerical radius-norm bounds are also stated, in a generalized form that involves different combinations of operators.

A non-technical exposition of the Kakeya conjecture, why it matters, and the road to the solution of this conjecture in three dimensions by Hong Wang and Joshua Zahl.

We establish formulae for point counts of cluster varieties of cluster algebras of marked surfaces, possibly with punctures. We then establish formulae for the number of non-deep points in these cluster varieties over $\mathbb{F}_2$, which gives us the number of algebraic tori necessary to cover the cluster manifold over $\mathbb{F}_2$. We also show that these formulae satisfy certain recurrence relations.