数学前沿

We study complete embedded translating solitons in R^3 with finite genus and finite entropy. We prove that such collapsed translators whose wings are all of planar type (in a weak topological sense) must be vertical flat planes. In particular, this implies the non-existence result for gluing and doubling constructions involving planes that was conjectured in (14) and announced in (9) as an application of the authors' techniques, under weaker assumptions than previous rigidity results by Khan and Lynch-Tinaglia in (17,19,20). A key ingredient is a sharp new a priori decay estimate in the exterior problem for the horizontal graphical translator equation, showing that bounded solutions to the nonlinear equation, and their gradients, are controlled by multiples of the Green's function of the linear drift Laplacian, regardless of the size of the given L^\infty data. Combined with a graphical description of the infinite ends and the Neves-Tian flux identity, this yields the main rigidity result. As another application, we obtain a full classification of collapsed embedded finite-genus translators of entropy lambda < 3.

Our main purpose is to determine all meromorphic solutions, in explicit form, of the nonlinear differential-difference equations $$\begin{equation*} f^{n}(z)f'(z)+q(z)f(z+c)=p_{1}{\rm e}^{\alpha_{1}z}+p_{2}{\rm e}^{\alpha_{2}z} \end{equation*}$$ and $$\begin{equation*} f^{n}(z)f'(z)+q(z){\rm e}^{Q(z)}f(z+c)=p_{1}{\rm e}^{\alpha_{1}z}+p_{2}{\rm e}^{\alpha_{2}z}, \end{equation*}$$ where $n\geq 3$ is an integer, $q(z)$ and $Q(z)$ are nonzero polynomials with $Q(z)$ nonconstant, and $c,~p_{j},~\alpha_{j}(j=1,2)$ are nonzero constants satisfying $\alpha_{1}\neq \alpha_{2}$. We sharpen the result in the cited literature, as illustrated by specific examples.

For a graph $G$, the $2$-token graph $F_2(G)$ has as vertices all $2$-subsets of $V(G)$, with two such subsets adjacent whenever their symmetric difference is an edge of $G$. In this paper, we determine the domination number of the $2$-token graph of the complete bipartite graph $K_{m,n}$, for all integers $2\le m\le n$. We show that it equals $\lceil m/2\rceil+n-1$, except when $m=3$, or when $m\ge5$ is odd and $n=m+1$; in these two exceptional situations it is one less. Our arguments rely on an explicit characterization of the dominating sets of the $2$-token graph of $K_{m,n}$.

In this paper, we establish a structural compatibility between the Kronecker product of bipartite graphs that admit polarity and their polarity quotients, and provide a sharp upper bound on the diameter of these graphs. We use these properties to construct new families of graphs, all of which have orders that asymptotically approach the Moore bound. We show that the bipartite compatibility graphs have a natural polarity, and that the corresponding polarity quotients comprise a new family of graphs of any even diameter $2k$ with degree $(q+1)^k$ for any prime power $q$, much smaller than those of other known families of that diameter and field order $q$. These families generalize the Erd\H{o}s-R\'enyi-Brown polarity graphs. Kronecker products of these polarity quotients give another new family of graphs of any even diameter for every pair of prime powers $q,r$. Certain generalized polygons are known to admit polarity with polarity quotients approaching the Moore bound asymptotically. Kronecker products of these polarity quotients give another new family of graphs of diameters $2$, $3$ and $5$, including a diameter-$3$ degree-$18$ graph of order larger than any previously known for that degree and diameter.

In this paper, we study the rational realization problem for the classifying space $B\mathrm{aut}_1(X)$. We prove that if $S^{2n}$ is realized as $B\mathrm{aut}_1(X)_{\mathbb{Q}}$ for a simply-connected space $X$, then $X$ is $\pi$-infinite and has vanishing rational Gottlieb elements above degree $2n-1$. In particular, even-dimensional spheres cannot be realized as $B\mathrm{aut}_1(X)_{\mathbb{Q}}$ for any simply-connected $\pi$-finite space $X$. We also prove that, for all $n\geq 2$ and $s,t\geq 1$, the product of Eilenberg--MacLane spaces $K(\mathbb{Q}^s,n)\times K(\mathbb{Q}^t,n+1)$ cannot be realized as $B\mathrm{aut}_1(X)_{\mathbb{Q}}$ for any simply-connected $\pi$-finite space $X$. Moreover, we prove that if $r\geq 2$, $n\geq 3$ and $K(\mathbb{Q}^r,n)$ is realized as $B\mathrm{aut}_1(X)_{\mathbb{Q}}$ for a simply-connected $\pi$-finite space $X$, then $X$ has the rational homotopy type of $K(\mathbb{Q}^r,n-1)$. The proofs are based on two structural results for Gottlieb elements in the derivation Lie algebra of a Sullivan minimal model, which provide a uniform method for these realization problems.

We present new inversion formulas for the unattenuated and attenuated magnetic ray transforms in the Poincar\'e disk, in the case of a constant, supercritical magnetic field. Our approach mirrors earlier complexification approaches by Novikov (Ark. Mat., 2002) and Bal (J. Math. Pures Appl., 2005), and a second feature of this article is to connect the approach with the Transport Twistor framework recently introduced by Bohr-Paternain (J. \'Ec. polytech. Math., 2023; arXiv:2606.25173).

In this paper, we propose an efficient subgradient-based framework, referred to as the sequential subgradient-aligned rays (SSAR) method, for improving the convergence performance of the dual subgradient method. In particular, the proposed method identifies and exploits a finite sequence of rays, when available, as the dual subgradient method evolves, along each of which a subgradient of the negative dual function remains constant and aligns with the ray. While the dual variable iterates remain on each identified ray, a constant stepsize associated with the ray is employed, with the associated constant stepsizes decreasing from one ray to the next. This approach yields constant decrements in the negative dual function values while the dual variable iterates remain on the associated rays, enabling the dual variable iterates to progress more quickly toward a dual optimal solution. We then show that the proposed SSAR method can be leveraged to improve the performance of the dual decomposition method for minimizing the average of local loss functions associated with multiple agents that share a common decision variable, a widely used problem formulation in many large-scale real-world applications. In particular, the SSAR method improves the convergence performance of the corresponding dual decomposition method while also reducing its communication cost. Numerical experiments on a least squares regression problem and a logistic regression problem using real datasets are conducted to evaluate the performance of the SSAR method.

We show that if a contractible Stein manifold $X$ with the density property admits a proper holomorphic embedding into $\mathbb{C}^k$, then $X \times \mathbb{C}^k$ is biholomorphic to $\mathbb{C}^{n+k}$ with $n=\dim X$. We also prove that McLean's exotic affine variety $K_n$, $n\ge4$, a smooth contractible affine variety which is not biholomorphic to $\mathbb{C}^n$, has the density property. Hence for $n \ge 4$ there exist an integer $m \in \{ n, \dots, 2n\}$ and an affine variety $X$ such that $X \times \mathbb{C}$ is biholomorphic to $\mathbb{C}^{m+1}$ and $X$ is not biholomorphic to $\mathbb{C}^m$. This settles the Holomorphic Cancellation Problem in the negative.

Let $S(Q,M,N,(a_n))=\sum_{q\le Q}\sum_{a\bmod q^2,(a,q)=1}|\sum_{M<n\le M+N}a_ne(an/q^2)|^2$ be Zhao's large sieve sum with square moduli. Zhao conjectured that $S\ll(QN)^\epsilon(Q^3+N)\sum|a_n|^2$. The bound of Baier and Zhao (2008) exceeds this conjecture by a factor which reaches $Q^{1/2}$ at the critical point $N=Q^3$, and there the exponent $1/2$ has not been lowered since. We prove that, under Hooley's Hypothesis $R^*$ for short Sali\'e sums, $S\ll(QN)^\epsilon(Q^3+N+Q^{4/3}N^{2/3})\sum|a_n|^2$. This replaces the excess over Zhao's conjecture by its $2/3$-th power throughout the range $Q^{5/2}<N<Q^4$; at $N=Q^3$ it gives $S\ll Q^{1/2-1/6+\epsilon}N\sum|a_n|^2$. The proof is short. Following Zhao's original treatment, we approximate the point under consideration by fractions $b/r$ with denominators up to $Q^{2/3}N^{1/3}$, which for $N<Q^4$ is beyond the customary $\sqrt{N}$. If the approximation is poor, an elementary count suffices. If it is good, the analytic part of the problem disappears: after Poisson summation in both variables the count is expressed through quadratic Gauss sums, and their exact evaluation at every modulus turns the sum over the dual variable into short Kloosterman and Sali\'e sums, to which Hypothesis $R^*$ applies and gives the square root of the elementary count. At odd moduli the hypothesis is needed only in the form of Conjecture (C*) of Friedlander and Iwaniec. The Gauss-sum evaluations, including even moduli and coefficients sharing a factor with the modulus, are proved in full. This version supersedes earlier versions, which obtained the exponent $1/2-1/28$ at $N=Q^3$ by a much longer route through sums of modular square roots. The paper was prepared in collaboration with Claude (Anthropic).

Let $2\le m < n$ be integers, and let $K(\mathcal{A})$ be the Bedford--McMullen carpet generated by $\{(x,y)\mapsto((x+i)/m,(y+j)/n):(i,j)\in\mathcal{A}\}$, where $\mathcal{A}$ is a subset of $\{0,1,\ldots,m-1\}\times\{0,1,\ldots,n-1\}$. Write $\mathcal{B}$ for the set of occupied columns, and assume that $\#\mathcal{B}\ge2$ and each column contains at least two symbols of $\mathcal{A}$. At each vertex of the column tree $\mathcal{B}^*$, we independently choose one forbidden symbol from $\mathcal{A}$ according to a prescribed distribution. Let $\mathbb{P}$ denote the resulting product measure on $\mathcal{A}^{\mathcal{B}^*}$. This gives a random survivor set $K_\omega(\mathcal{A})$ with $\omega\in\mathcal{A}^{\mathcal{B}^*}$. We show that, for $\mathbb{P}$-a.e. $\omega$, $$\begin{equation*} \dim_H K_\omega(\mathcal{A})=\frac{1}{\log m}\log\left(\int \sum_{b\in\mathcal{B}}\ell_b(\omega)^{\frac{\log m}{\log n}} d\mathbb{P}(\omega)\right), \end{equation*}$$ where $\ell_b(\omega)$ is the number of surviving symbols in column $b$ at the root.

We investigate the Typical Periodic Optimization (TPO) Conjecture and freezing phase transitions for Dini potentials and beyond. We show that the TPO conjecture fails already at Dini regularity. We identify a sharp regularity threshold for freezing phase transitions, which we call weak log-log Dini. Both results reveal the stability of aperiodic ground states. Moreover, we identify Dini regularity as the sharp threshold for a quantitatitve analog of the freezing phase transition.

Bergeron, Garsia, Haiman, and Tesler (1999) introduced two Schur positivity conjectures concerning the action of the $\nabla$ operator on modified Hall--Littlewood polynomials. Although the proofs of the celebrated compositional shuffle conjecture and its rational generalization, along with various symmetric function identities, have largely resolved these conjectures in the affirmative, we provide an elegant alternative proof driven by $q$-series techniques. Through this approach, we derive explicit Schur function expansions for two-column partitions and extend our results to arbitrary positive powers $\nabla^k$.

A thrackle is a drawing of a graph in the plane in which every pair of edges meets exactly once, either at a common endpoint or at a proper crossing. Conway conjectured that every thrackle has at most as many edges as vertices. We prove that every thrackle on $n\ge4$ vertices has at most $\left\lfloor4(n-1)/3\right\rfloor$ edges. The proof combines Pammer's computer-assisted verification of the conjecture for graphs on at most twelve vertices with a discharging argument in suitable plane and projective-plane embeddings.

Let $B^H$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in(0,1/2)$. We prove the existence of a strong solution to $$d X_t=b(t,X_t)d t+d B_t^H, X_0=x_0\in R^d,$$when $b\in L^q([0,T];L^p(\mathbb R^d;\mathbb R^d)),2\le p,q<\infty$ and the subcritical Ladyzhenskaya--Prodi--Serrin condition $\frac1q+\frac{Hd}{p}<1-H$ is satisfied. Thus, within the finite-exponent range $p,q\ge2$, the strong-existence result covers the full subcritical region, including the regime $p<2Hd$, where a uniform strong Novikov condition may fail. The proof is based on a compactness criterion for Wiener functionals. Its main ingredients are an exponentially quantitative derivative estimate for the finite-dimensional densities of fractional Brownian motion, a conditional moment estimate obtained through a martingale representation of fractional Brownian motion, and a frequency-localized truncation of the Cameron--Martin shift. These estimates yield uniform Malliavin--Sobolev bounds for smooth approximations of the equation.

In this paper, we propose a series of theoretical results for prescribed-time control to rigorously prove that the closed-loop system will not reach the equilibrium point before the prescribed time. Thus, it is convenient to prove that the settling time is exactly equal to the prescribed time instant. In other words, conservative estimation of the settling time, which often accompanies classical finite-/fixed-time or prescribed-time control results, is theoretically avoided in this paper. Specifically, we introduce a local Lipschitz condition, a local homogeneous linear-growth condition, and an Osgood condition in sequence. Both reduction to absurdity and the Lyapunov method are applied to demonstrate that the system cannot converge to zero before the prescribed time instant. Finally, the proposed framework is applied to prescribed-time controllers based on adaptive backstepping, parametric Lyapunov equations, and distributed leader-following consensus, demonstrating the generality of the proposed results.

For every irrational rotation number $\alpha$ of constant type and every $q>1$, we construct a $C^2$ circle diffeomorphism $F$ arbitrarily close to the rigid rotation $R_\alpha$ in the combined $C^2$ and $B^2_{2,q}$ topology whose normalized conjugacy is not Lipschitz. We also prove that $F''\in B^0_{2,1}$ implies global $C^1$ conjugacy. Thus the thresholds in the Besov summability and weak Dini scales are $q=1$ and $\gamma=1/2$, respectively. These results complement Herman's Baire category theorem on the residual existence of $C^2$ diffeomorphisms without $C^1$ conjugacy by providing quantitative counterexamples and sufficient conditions for differentiable conjugacy.

We study faithful realisations of Coxeter groups over fusion rings and study Vinberg systems associated to them. We show that they induce embeddings of hyperplane complements, which provide geometrical realisations of certain types of strong admissible (LCM) homomorphisms between Artin--Tits groups.

This work extends least-squares support vector regression (LS-SVR) based on the generalized moving least squares (GMLS) approximation to the numerical solution of partial differential equations on spheres. We show that the proposed approach constructs a kernel-based approximant in its dual form and establishes its connection with the associated reproducing kernel Hilbert space (RKHS). Furthermore, solving the PDE through the proposed LS-SVR framework is shown to be equivalent to solving a smoothing problem or a kernel ridge regression problem in the corresponding RKHS. As applications, the method is employed to solve the Allen-Cahn equation and its conservative counterpart on spheres. An implicit first-order operator-splitting scheme is used to separate each model into linear and nonlinear subproblems. The linear subproblem is discretized by the backward differentiation formula and approximated at each time step using LS-SVR derived from a GMLS-based collocation formulation, while the nonlinear part is treated analytically. The resulting scheme is shown to admit a unique numerical solution at every time step. For the conservative Allen-Cahn model, a correction procedure is incorporated to preserve mass. Numerical experiments demonstrate the accuracy and effectiveness of the proposed framework.

We use Stein's method to prove general upper bounds on the second order Wasserstein distance in the CLT for general dependent partial sums $S_N=X_1+\ldots +X_N$. Then we use these estimates to provide optimal $O(N^{-1/2})$ rates for a variety of chaotic dynamical systems and $\phi$- and $\psi$-mixing sequences. Our results for $\psi$-mixing sequences are obtained under the fourth moment condition, which matches the best known results in the iid case. This is the first time that optimal CLT rates of any kind are obtained for unbounded weakly stationary variables under general mixing conditions. We then apply this result and obtain optimal CLT rates $O\left((\text{Var}(S_N))^{-1/2}\right)$ for non-stationary $\psi$-mixing sequences without any growth rates on $\text{Var}(S_N)$. We also obtain close to optimal CLT rates for nonconventional sums and some classes of random dynamical systems with non-uniform decay of correlations

The learning dynamics of Stochastic Gradient Descent (SGD) are believed to play an important role in influencing the generalization properties of trained models. While the precise dynamics of how online (one-pass) SGD optimizes the population risk are now understood in a broad range of high-dimensional problems, a comparable understanding of multi-pass SGD for empirical risk minimization is still largely lacking. In this work, we develop an asymptotically exact characterization of the dynamics of multi-pass SGD for a non-convex multi-index model, in the high-dimensional regime where sample size $n$ and data dimension $d$ increase proportionally. We consider any sub-linear batch size $\kappa \asymp n^\alpha$ with $\alpha \in [0,1)$ and a commensurate ``critical'' scaling of the learning rate, and show that the coordinate-wise dynamics of SGD admit a novel continuous-time scaling limit, given by a system of dynamical mean-field equations that are driven by a scalar Poisson jump process representing the asymptotic limit of SGD sampling noise. This scaling limit connects several previously disparate theories around SGD, namely (i) the classical interpretation of SGD as an approximation to gradient flow on the empirical risk, (ii) the characterization of online SGD via an autonomous ODE in its overlap statistics, and (iii) the coincidence between multi-pass SGD dynamics and diffusion approximations in least-squares problems. We illustrate how these perspectives are recovered by specializing our result to small learning rates, large sample sizes, and linear models, respectively. Outside of these specializations, the scaling limit of SGD is distinct both from diffusion approximations and gradient flow, and may enable SGD to converge to local optima of the non-convex empirical risk with different statistical properties.