数学前沿

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.

Let $p$ be a prime and $n\geq 1$. We prove that the Tate-Frobenius of the $\operatorname{K}(n)$-local sphere induces an equivalence between $L_{n-1}L_{\operatorname{K}(n)}Y$ and $(L_{\operatorname{K}(n)}Y)^{t\Sigma_p} \cong (L_nY)^{t\Sigma_p}$ for any finite $p$-local type $n-1$ spectrum $Y$ with trivial $\Sigma_p$-action.

We give a direct proof of the equivalence between the Muckenhoupt $\mathcal{A}_2$ and the Helson--Szeg\H{o} conditions on the unit circle $T$. A positive weight $w$ belongs to $\mathcal{A}_2(T)$ if and only if $\log w=u+Hv$, where $u,v$ are real bounded functions, $\|v\|_\infty<\frac{\pi}{2}$, and $H$ denotes the periodic Hilbert transform. The classical proof combines the Helson--Szeg\H{o} theorem with the Hunt--Muckenhoupt--Wheeden characterization of weighted boundedness of the Hilbert transform. Our proof derives the Helson--Szeg\H{o} representation directly from the Muckenhoupt $\mathcal{A}_2$ condition, without using boundedness of $H$ on $L^2(w)$.

We give two alternate formulations for the second moment formula of the Siegel transform over $\mathrm{Sp}(2n, \mathbb Z) \backslash \mathrm{Sp}(2n,\mathbb R)$, previously established by Athreya-Konstantoulas and Kelmer-Yu. We also provide several applications that demonstrate their usefulness.

In their seminal paper introducing the theory of random graphs, Erd\H{o}s and R\'{e}nyi considered the evolution of the structure of a random subgraph of $K_n$ as the density increases from $0$ to $1$, identifying two key points in this evolution -- the percolation threshold, where the order of the largest component seemingly jumps from logarithmic to linear in size, and the connectivity threshold, where the subgraph becomes connected. Similar phenomena have been observed in many other random graph models, and in particular, works of Ajtai, Koml\'{o}s and Szemer\'{e}di and of Erd\H{o}s and Spencer determine corresponding thresholds for random subgraphs of the hypercube. The permutahedron, like the hypercube, has many equivalent representations, and arises as a natural object of study in many areas of combinatorics. In particular, as a highly-symmetric simple polytope, like the $n$-simplex and $n$-cube, percolation on the permutahedron arises naturally as an analogue to the Erd\H{o}s-R\'{e}nyi random graph and the percolated hypercube. We determine the percolation threshold and the connectivity threshold for random subgraphs of the permutahedron. Along the way we develop a novel graph exploration technique which can be used to find exponentially large clusters after percolation in high-dimensional geometric graphs, and we initiate the study of the isoperimetric properties of the permutahedron.

We prove the Hardy--Littlewood asymptotic for prime pairs of difference four from a restricted uniform two-over-two zeta-ratios formula. The formula has error $O_\varepsilon(U^{1/2+\varepsilon})$, real parts of all four shifts in $[3/64,5/64]$, and imaginary parts of absolute value at most $U^{1/4}$. For a fixed real smooth weight supported in $(1,101/100)$, we obtain an error $O_\varepsilon(x^{-1/16+\varepsilon})$ in the correlation sum with weight $W(n/x)^2/n$. Adding the Riemann hypothesis improves this error to $O_\varepsilon(x^{-3/8+\varepsilon})$. The proof retains the crossed zero residues in Mellin inversion and estimates their mixed term with the logarithmic-derivative integral. Two nonnegative time weights have the same diagonal term and opposite coefficients for the difference four. We deduce the logarithmic asymptotic for the weighted prime-pair sum $B_4$ and an asymptotic for the second finite difference of integrals of squared increments of the truncated Chebyshev function. The latter is connected to a logarithmically weighted M\"obius approximant in the Nyman--Beurling criterion by finite identities.

We propose an approximate interpolation construction and a practical coarsening algorithm for algebraic multigrid (AMG) method when applied to H(curl) problems. Building on the generalized AMG framework, we introduce an interior/exterior splitting that yields both a refinement-based and a fully algebraic construction of the interpolation. The refinement-based approach follows a geometric hierarchy, while the purely algebraic interpolation is constructed through a coarsening process that first coarsens a nodal dual problem and then builds coarse and fine variables using a matching algorithm. Under a uniform fine-space smoothing assumption, we prove a jump-independent two-grid convergence bound for canonical finite-element spaces. We also establish an exact commuting relation for the unshifted construction. Two-dimensional numerical experiments demonstrate sparse multilevel hierarchies and robust convergence under strong coefficient jumps in the curl--curl term. Three-dimensional experiments provide a two-level proof of concept, although the interpolation operators remain dense. Developing a practical sparse three-dimensional multilevel method is left to future work.

This paper focuses on optimization problems constrained by Parametric Variational Inequalities (PVI) defined on a moving set. Unlike most existing works on mathematical programs with equilibrium constraints, the equilibrium constraints have parameters not only in the function but also in the related set. We show that the solution function of the PVI is Lipschitz continuous with respect to the upper-level decision variables and the solution set of the optimization problem is nonempty and bounded. Moreover, we prove that the metric regularity of the constraints holds automatically, which allows us to characterize stationary points without any additional assumptions. A Smoothing Implicit Gradient Algorithm (SIGA) is proposed based on the smoothing approximation of the PVI with a novel scheme for updating the smoothing parameter. We prove the convergence of SIGA to a stationary point and validate the efficiency of SIGA by numerical examples from synthetic problems and portfolio management with real data.

We show that each hyperbolic once-punctured torus bundle has exactly one veering triangulation. We also find manifolds for which the exact number of veering structures is known and is greater than one. Finally, we give new examples of manifolds without veering triangulations.

We prove Penner's theorem on horocycles and theorems of Ptolemy and Casey, all with full converses, in hyperbolic space of several dimensions. Recently Waddle observed that the equations underpinning these three theorems are related, and it is this viewpoint that we advance, using the Lorentzian model of hyperbolic space. We show that all three theorems can be derived from a common Gram-matrix calculation applied to lightlike, timelike, and spacelike vectors. Remarkably, our approach gives a version of Casey's theorem in the plane with a full converse, involving three geometric alternatives, which to our knowledge has not previously been recorded.

We study the preservation of spatial concavity and log-concavity for stochastic parabolic equations posed on bounded convex domains. For the linear noise, by using Minkowski convolutions, we establish the convexity preservation of solution almost surely. For the nonlinear noise, by using It\^{o} formula, the impact of noise is given. Moreover, we establish the relationship between the weighted (stochastic) Pr\'{e}kopa-Leindler inequality and properties of (stochastic) parabolic equation on bounded domain, which is new even for deterministic case.

We propose a globally convergent Newton method for convex composite optimization problems with polyhedral regularizers. The method rests on the observation that the proximal gradient step yields not only a new point but also a subgradient, which determines the face on which the linearization of the regularizer is exact. Minimizing a second-order model of the objective over this face produces a Newton direction that involves no generalized second-order information about the regularizer and is a descent direction for the objective itself. The method is therefore globalized by an Armijo line search on the original objective which, unlike forward-backward envelope techniques, requires only evaluations of the smooth component. We establish global convergence and local quadratic convergence to tilt-stable minimizers without any nondegeneracy condition, and show that the unit step is always accepted when the smooth component is quadratic, which yields a line-search-free method. The subproblems require only projections onto faces, which are explicit for the Lasso and nonnegative least squares and computable in linear time for the fused Lasso. Numerical experiments show that the method compares favorably with state-of-the-art solvers.