数学前沿

In this paper, we first derive the Bochner-type formulas of the Hermitian-Tanno connection on contact Riemannian manifolds. Second, we prove a sub-Laplacian comparison theorem on complete noncompact contact Riemannian manifolds with bounded geometric conditions. Finally, we obtain the gradient estimates of a positive eigenfunction on a complete noncompact contact Riemannian manifold which satisfies similar geometric conditions.

We describe a new family of Dirichlet L-functions that provably have an unusual value distribution. Experimentally, we find that the L-functions in this family have a significant and previously undetected bias in the distribution of gaps between their zeros. These phenomena appear to have an arithmetic explanation that corresponds to the nonvanishing of a certain Gauss-type sum. We give a complete classification of the characters for which this sum is nonzero and count them. This sum vanishes for 100% of primitive Dirichlet characters, so L-functions in our newly discovered family are rare (a set of density zero among all primitive characters). The nonvanishing of this sum implies that there is a main term when one integrates the Z-function associated to a Dirichlet L-function over a long interval. We prove that the L-functions in this family are the only GL(1)/Q or GL(2)/Q cuspidal automorphic L-functions whose Z-functions have a "large" mean value when integrated. We discuss analogous results for the Riemann zeta-function.

This paper proposes an energy-variational framework for the well-posedness of the initial boundary value problem of the parabolic equation with variable exponents on an annulus, under the multiple-regime setting consisting of the subcritical, critical and supercritical regimes of the variable exponents. The multiple-regime setting leads to the appearance of the critical and supercritical regions of positive measure, which cause the failure of the classical Sobolev embedding and the lack of compactness. To address the challenges posed by multiple regimes to well-posedness within the energy framework, we recover the compact Sobolev embedding for a class of radially symmetric functions by the Strauss inequality and obtain the local existence of the solution via the combination of the subdifferential technique and the Principle of Symmetric Criticality for any given radially symmetric initial data. Additionally, the uniqueness result is also given with the recovered Sobolev embedding. Moreover, examining the impact on well-posedness of solutions caused by the variable exponents crossing three sensitive regions, we establish a variational structure of the potential well theory for the evolution equation under the multiple-regime setting on the annulus. For subcritical and critical initial energy levels, we consider the non-positive and positive initial energy cases separately. We obtain the finite-time blow-up of the solution with non-positive initial energy and the threshold condition for global existence and finite-time blow-up of the solution with positive initial energy. For supercritical initial energy level, several sufficient criteria for the global existence and finite-time blow-up of the solution are given. Finally, the above results, including global existence and finite-time blow-up, are generalized to a more general locally symmetric domain containing an annulus.

The Li-Yau and Aronson-B\'enilan estimates are fundamental differential inequalities for heat and porous-medium equations, yielding differential Harnack inequalities and sharp smoothing and decay rates. We develop analogues for the critical parabolic-elliptic Keller-Segel system on $\mathbb{R}^d$, where diffusion and aggregation act at the same scale. Our main tool is the effective pressure $v=p-u$, with $p=\log\rho$ in $d=2$ and $p=\frac{m}{m-1}\rho^{m-1}$ for $d>2$, $m=2-\frac{2}{d}$. We prove quantitative lower bounds on $\Delta v$, showing that classical pressure estimates subsist in a suitable form despite the nonlocal interaction. These estimates unify previous results and yield new regularity. For nonnegative finite-measure initial data of sufficiently small mass, assuming only a logarithmic moment in $d=2$, we construct global solutions smooth for positive times and satisfying $\|\rho_t\|_{L^\infty}\le C/t$. For finite-free-energy data, we obtain global $L^\infty$ bounds of order $1+t^{-1}$ in the subcritical and critical mass regimes. We also introduce a new characterization of the critical mass as the minimal mass $M_c^{\mathrm{sub}}(d)$ of subsolutions of the elliptic Lane-Emden equation (Liouville in $d=2$). We prove that it equals $8\pi$ in dimension $2$ and, for $d\ge3$, coincides with the Keller-Segel critical mass given by the HLS inequality, by reducing the problem to a one-dimensional optimal-control problem. Along the way we prove tools of independent interest: $L^\infty$ estimates from lower bounds on $\Delta v$, critical log-HLS/HLS estimates away from Dirac masses, Hessian bounds for Newtonian potentials, and Lipschitz and Hessian estimates for bounded porous-medium solutions with drifts on the whole space.

We study rotationally symmetric graphical solutions to mean curvature flow that are initially close to a homothetic reference expander, in particular near infinity. Based on a computer algebra aided proof, we show that under appropriate conditions, at large times, the solutions are closer to a different expander than to the reference expander.

Adversarial sharpness penalty per iteration of a base optimizer, known as sharpness-aware minimization (SAM), is an accepted technique to improve generalization in deep learning. Recently, Muon has emerged as a dominant optimization algorithm for training large language models (LLMs). It achieves lower sample complexity for heavy-tailed gradient updates - a phenomenon common in LLM training. However, sharpness penalties with Muon updates are yet largely unexplored in the literature. Moreover, the behavior of SAM under heavy-tailed noise is unknown both theoretically and empirically. A potential reason for this gap is the additional cost of extra gradient computation per iteration for penalty perturbations in SAM, which becomes prohibitive for LLMs. This paper introduces Sharpness-aware Orthogonalized Momentum Algorithm (SOMA) to combine a computationally efficient and theoretically grounded perturbation scheme with the Muon algorithm. The key idea is to use stale (purely) stochastic gradient noise for adversarial perturbations. Theoretically, we show that SOMA's sample complexity is invariant to the perturbation direction. We provide explorations and insights into SOMA's robustness to heavy-tailed gradient noise. We extensively explore SOMA's efficacy in (a) mitigating quality degradation due to post-training quantization, and (b) catastrophic forgetting during fine-tuning of LLMs.

We study the asymptotic behaviour, as $\varepsilon\to 0^+$, of the $p$-Dirichlet energy of maps $u\colon\Omega\subset\mathbb{R}^2 \to \mathbb{R}^2$ constrained to take values in the rescaled periodic network $\varepsilon\mathcal Z$, where $\mathcal Z$ is a standard square network in $\mathbb{R}^2$. The oscillations are carried by the constraint in the target, not by the energy density, and their effect depends on $p$. For $1<p<2$ the constraint disappears in the limit: the $\Gamma$-limit is an integral functional on the whole of $\rm{W}^{1,p}(\Omega , \mathbb{R}^2 )$, whose density is given by an asymptotic cell formula on tilted cells. This density is equal to $(|a_1|+|a_2|)^p$ on rank-one matrices $a\otimes\nu$, $|\nu|=1$, and exceeds its convex envelope by a quantity comparable to $|\det A|\,|A|^{p-2}$, with constants of order $(2-p)^{-1}$. Consequently, although the original energy density is convex, the limit density is quasiconvex but neither convex nor polyconvex: quasiconvexity is generated by the geometry of the constraint. For $p\ge2$ the constraint survives in the limit as $\det\nabla u=0$, and the $\Gamma$-limit is the degenerate energy $\int_\Omega(|\nabla u_1|+|\nabla u_2|)^p\, \rm{d} x$, that is, the $p$-energy of $u$ considered as a map into the normed plane $(\mathbb{R}^2,|\cdot|_1)$, the metric limit of the networks.

We prove a power saving upper bound for the sum of Fourier coefficients $\rho_f(\cdot)$ of a fixed cubic metaplectic cusp form $f$ over primes. Our result is the cubic analogue of a celebrated 1990 Theorem of Duke and Iwaniec, and the cuspidal analogue of a Theorem due to the author and Radziwill for the bias in cubic Gauss sums. The proof has two main inputs, both of independent interest. Firstly, we prove a new large sieve estimate for a bilinear form whose kernel function is $\rho_f(\cdot)$. The proof of the bilinear estimate uses a number field version of circle method due to Browning and Vishe, Voronoi summation, and Gauss-Ramanujan sums. Secondly, we use Voronoi summation and the cubic large sieve of Heath-Brown to prove an estimate for a linear form involving $\rho_f(\cdot)$. Our linear estimate overcomes a bottleneck occurring at level of distribution $2/3$.

The regularity of solutions to the two-dimensional radiation transport equation plays a central role in understanding the behavior of scalar flux near geometric corners. In a previous study of Kaper's corner singularity problem, the physical transport kernel was decomposed into its principal inverse-distance singularity and a lower-order remainder, reducing the corner asymptotic analysis to the question of whether the scalar flux possesses sufficient continuity to justify differentiation of the associated singular potential. The required regularity, however, remained unresolved. In this paper, we establish a sharp modulus-of-continuity estimate for the scalar flux generated by the physical transport kernel and prove the estimate $\omega_{\phi}(h)=O\!\left(h\log(D/h)\right)$, which implies Dini continuity. This estimate yields absolute convergence of the fluctuation term arising in the differentiated principal potential and permits a rigorous differentiation formula for transport solutions. Using this regularity framework, we derive the fixed-angle asymptotic behavior of the scalar flux and its first derivatives near a rectangular corner. For every fixed interior ray, the gradient exhibits logarithmic growth, while the scalar flux admits an $r\log(1/r)$ correction with an explicitly identified coefficient determined by the corner values of the flux and source. The analysis shows that a rectangular corner does not generate a singularity stronger than the logarithmic first-derivative singularity already present in smooth-boundary transport theory. Numerical calculations on benchmark problems are consistent with the theoretical predictions. These results provide a rigorous resolution of the regularity issue underlying Kaper's corner singularity problem and establish a mathematical framework for corner asymptotics in two-dimensional radiation transport.

Multi-state HLL-type approximate Riemann solutions of ideal magnetohydrodynamics (MHD) typically assume that the medium within the Riemann fan is incompressible, and thus the slow magnetoacoustic mode cannot be included in their space-time wave configurations for the intermediate states. We propose a new strategy to design multi-state HLL-type solutions, allowing for the medium within the Riemann fan to be compressible. In particular, for the complete seven-wave configuration of the MHD Riemann problem, we estimate the slow magnetoacoustic speeds and a longitudinal flow speed between the slow modes, whose compressibility is estimated based on the relations between fast, Alfv\'en, and slow speeds. Then, following the conservation laws and Rankine-Hugoniot jump relations across the fast, Alfv\'en, and slow waves, we calculate all the intermediate states within the Riemann fan. Moreover, we discuss the solutions when certain wave modes degenerate, ensuring well-posedness and smooth transitions between complete and degenerate wave configurations. Numerical simulations using a Finite Volume solver show that the proposed approximate Riemann solution is less diffusive than the classic HLLD scheme, particularly for slow mode waves. For example, in a 1D test case with a strong longitudinal magnetic field, the new scheme needs one order of magnitude fewer grid cells compared to the classic HLLD scheme to resolve all wave modes.

We study a family of affine algebraic varieties over a finite field $\mathbb{F}_q$ defined by systems of power sum polynomials. Our main goal is to investigate their geometric and arithmetic properties, with particular emphasis on the structure and dimension of their singular loci. Using tools from the theory of symmetric polynomials and generalized Vandermonde determinants, we establish explicit conditions under which these varieties are complete intersections and prove that, in a broad range of parameters, their singular loci have codimension at least two. As a consequence, we derive normality and irreducibility results for both the affine varieties and their projective closures. Building on this geometric analysis, we obtain estimates for the number of $\mathbb{F}_q$-rational points and deduce effective existence results for solutions of systems of symmetric power sum equations over finite fields. Finally, we apply these results to the study of Waring's problem and to the analysis of the cost of computing a $\mathbb{F}_q$-rational point of such systems.

The not-a-knot condition provides a simple spline interpolation scheme that requires no derivative data at the endpoints. Whereas the classical not-a-knot scheme using odd-degree splines selects knots that coincide with the data points, this approach does not extend to the even-degree case. In this paper we establish the well-posedness of the not-a-knot interpolation problem for even-degree splines, where knots are selected at midpoints of data values. We then show that the quadratic case, introduced in [M. J. D. Powell, Approximation Theory and Methods, Cambridge University Press, 1981], achieves $\mathcal{O}(h^3)$ convergence for $C^3$ functions on arbitrary partitions without mesh-ratio restrictions. The proof derives a linear system defining the interpolating spline, normalizes the system to bound the matrix norm of its inverse using a Neumann-series argument, and establishes global error bounds.

Gorsky, B. Liu and Moore provided a formula for the $d$-invariants of integral surgeries on $2$-component $L$-space links with linking number $0$. This paper generalizes these results to the nonzero linking number case. We describe how the $d$-invariant depends on the definiteness of the framing matrix and provide formulae for certain families of positive definite framings, as well as a skein relation for non-algebraically split links.

Even a carefully proofread paper can leave small gaps in its proofs or its stated assumptions. We introduce PaperFormalizer, a Lean-based agentic pipeline that helps authors find and close such gaps in their own papers and turn each paper into a Lean-verified version. Starting from the LaTeX source, PaperFormalizer transcribes the paper into a dependency graph of statements, audits each statement in natural language for faithfulness, ambiguity and correctness, and then formalizes and proves the statements in Lean against Mathlib, optional domain libraries and the author's personal toolkit. Only the author knows what a statement is meant to say and how a formalization gap in a result is best fixed. The pipeline therefore stops and asks the author whenever a statement admits more than one reading, presents every repair, in natural language and TeX symbols, as a proposal that the author can accept, decline or redirect, and leaves to the author every decision that could potentially change what the paper claims. The output is an annotated and revised version of the paper, a report of the changes, and Lean code in which each declaration is linked to the statement it formalizes. The author's toolkit grows as more papers are formalized, and authors can share their verified papers so that researchers who build on them can reuse their Lean code. By making verified papers practical to produce, PaperFormalizer aims to make research that rests on mathematical proofs more robust. PaperFormalizer runs inside a coding agent (Claude Code or Codex) and is available at https://github.com/Jiyuan-Tan/PaperFormalizer. Papers with Lean code shared by users are available at https://paperformalizer.org.

Time delays are inherent in many physical and engineered systems and can significantly affect their stability and performance. In this work, we investigate the dissipativity of a class of nonlinear time-delay systems with multiple discrete delays and derive sufficient conditions for both delay-dependent and delay-independent dissipativity using Bessel-Legendre inequalities. For linear systems, the resulting dissipativity conditions are expressed in terms of linear matrix inequalities (LMIs) which can be solved numerically to obtain Lyapunov-Krasovskii-type storage functions.

In modern e-commerce and service operations, firms jointly manage inventory replenishment and real-time order fulfillment under demand uncertainty. We study their interaction in a single-resource lost-sales system with heterogeneous rewards, positive lead times, and replenishment cycles of length $T$. A central difficulty is that low regret in one fulfillment cycle need not imply a small profit gap under repeated replenishment, since fulfillment decisions change inventory and, under base-stock replenishment, subsequent orders. We consider base-stock and constant-order replenishment and compare online fulfillment with a myopic offline benchmark that knows current-cycle demand but not future demand. We give conditions under which $O(T^\alpha)$ one-cycle regret yields an $O(T^\alpha)$ upper bound on the expected average profit gap, uniformly over all finite horizons $N$. For base stock, the bound remains valid when the online and offline targets are optimized separately. For constant orders, a uniform quadratic count-loss bound gives an unrestricted two-sided comparison between separately optimized systems. For the cycle-restarted Bayes Selector, the finite-horizon bounds are $O(1)$. The optimized long-run constant-order profit gap is also $O(1)$. We use these results to compare optimized base stock with greedy fulfillment against optimized constant order with the Bayes Selector and decompose the profit difference into replenishment and fulfillment components. A representative finite-horizon simulation shows that the ranking of the two policy combinations can change with cycle length. We also extend the transfer analysis to multiple resources. Under single-resource base-stock replenishment, a mean-LP quota policy has uniform $O(\sqrt T)$ regret against full-information benchmark.

For two even integers $r_1,r_2$ with $4\leq r_1\leq r_2\leq N/2$, a graph $G$ of even order $N\geq8$ is \emph{2-DCC edge $[r_1,r_2]$-bipancyclic} if for every ordered pair $(e,f)$ of independent edges and every even integer $\ell\in[r_1,r_2]$, there exist two vertex-disjoint cycles $C_1$ and $C_2$ with $V(C_1)\cup V(C_2)=V(G)$, $|C_1|=\ell$, $|C_2|=N-\ell$, $e\in E(C_1)$, and $f\in E(C_2)$. We study this property for graphs defined by Cartesian products. Let $G=H_1\square\cdots\square H_d$, where $d\geq2$ and each factor $H_i$ is either $K_2$ or a bipartite graph of even order at least four that has a Hamiltonian cycle and in which every edge lies on a Hamiltonian cycle. We prove that $G$ is 2-DCC edge $[4,|V(G)|/2]$-bipancyclic if one of the following holds: (i) $d\geq4$; (ii) $d=3$ and at least two factors have order at least four; (iii) $d=3$, exactly one factor has order at least four, and every set of at most two edges of that factor lies on a common Hamiltonian cycle; (iv) $d=2$, both factors have order at least four, and every set of at most two edges of each factor lies on a common Hamiltonian cycle; (v) $G\cong K_2\square K_2\square K_2$. The theorem applies to several families of interconnection networks. In this paper, we give a sharp characterization for undirected toroidal meshes, star-cube networks, and hyperstar networks.

Modern deep learning has been shown to operate using learning rates far larger than those justified by classical optimization theory. Most prior analyses of this phenomenon focus on deterministic gradient descent, leaving the stochastic setting largely unexplored. In this work, we provide sharp convergence guarantees for Stochastic Gradient Descent (SGD) applied to the multiclass cross-entropy loss, for both linear classifiers and two-layer neural networks. We show that the stochasticity of SGD may cause the dynamics to alternate between an edge-of-stability regime that is dominated by curvature-driven oscillations, and a stable regime in which the expected loss decreases at a controlled rate. Despite that, we prove that SGD stabilizes the dynamics, ensuring that the iterates return to stability in a fixed number of iterations and allowing convergence in the best-iterate sense even with large learning rates. Experiments validate our theoretical findings and illustrate the benefits of SGD in the large-stepsize regime.

In this paper, we study complete minimal surfaces in $\mathbb{R}^4$ with three embedded planar ends parallel to those of the union of the Lagrangian catenoid and the plane containing its waist circle. We prove that no such complete embedded minimal surface of genus $g\geq 2$ with finite total curvature admits at least $4(g+1)$ symmetries. This contrasts sharply with the genus-one case, where highly symmetric examples do exist and are uniquely determined up to scaling and rigid motion.

We study higher complements of pseudohyperplane arrangements and the specialization maps between them, induced by degenerations. We develop a presheaf formalism on oriented matroids and weak maps, which model degenerations. Our main constructions are the posets of weighted and weightwise topes, which provide functorial combinatorial models for higher complements and their specialization morphisms. Their presheaf structure yields a deletion-contraction homotopy pushout, from which we derive a uniform combinatorial proof of the integral cohomology presentation for all $n \geq 2$. This presentation is natural with respect to weak maps and identifies cohomology rings with higher Orlik-Solomon algebras extending the known formulas for linear arrangements to non-realizable oriented matroids. Finally, we equip Deligne groupoids with a presheaf structure and identify them functorially with the fundamental groupoids of the complexified complements.