数学前沿

Let $f: \mathbb{N} \rightarrow (0,\infty)$ be a multiplicative function, and let $\omega > 0$. We consider the frequency with which patterns of the form $$f(n) > \omega \max\{f(n-1),f(n+1)\}, \quad f(n) < \frac{1}{\omega}\min\{f(n-1),f(n+1)\}$$ occur in the sequence $(f(n))_n$, and give necessary conditions on $f$ so that the set of occurrences of at least one of these patterns has logarithmic density $0$. This includes a complete classification of all such completely multiplicative functions. As a particular application, we characterise (up to prime powers $p^k$, $k \geq 2$) all multiplicative functions $f: \mathbb{N} \rightarrow \mathbb{N}$ that have either finitely many local maxima, or finitely many local minima. This answers, in a strong form, two conjectures of A. S\'{a}rk\"{o}zy from 2001. \\ The proof is based on a natural ``pretentiousness'' dichotomy for additive functions. Key tools in the analysis include an analogue of the Matom\"{a}ki-Radziwi\l\l theorem for additive functions, due to the author, as well as a new bound for the concentration of the gaps $g(n+1)-g(n)$ of an additive function $g$, which may be of independent interest.

Toeplitz matrices are among the simplest structured models of translation invariance, arising naturally in convolution, stationary covariance models, and discretizations of translation-invariant operators. For a pair of real matrices, the B\"ottcher--Wenzel form (BW form) is a nonnegative quartic measuring the gap in the corresponding commutator inequality. L\'aszl\'o conjectured that, when both matrices are Toeplitz, this quartic is a sum of squares for every matrix order. We show that the conjecture is false. More precisely, we identify an explicit family of corner-supported Toeplitz pairs, retaining only the outermost diagonals on the two sides, for which the associated BW form is not a sum of squares at all sufficiently large orders. The proof reduces the complete family of Gram representations to a finite complex test: its Gram-invariant part develops a strictly negative limiting direction, while the remaining Gram-dependent contribution vanishes. We also give an explicit sufficient threshold for the matrix order. The negative result is complemented by two positive regimes. If either Toeplitz factor is symmetric or skew-symmetric, then the corresponding BW form is SoS for every order, with the other factor allowed to be an arbitrary real Toeplitz matrix. For unrestricted Toeplitz pairs, we further prove SoS representability for every $2\le N\le50$, with formal Lean~4 verification through order 20. The large-order non-SoS theorem, including the explicit sufficient threshold, has also been formally verified in Lean 4.

We prove smoothness up to the boundary for $k$-Hessian eigenfunctions on smooth bounded uniformly convex domains when $2\leq k<n$. This resolves an open problem in the regularity theory of degenerate fully nonlinear elliptic equations and extends the boundary smoothness theorem for Monge--Amp\`ere eigenfunctions. In contrast to the Monge--Amp\`ere case, the linearized $k$-Hessian operator remains uniformly elliptic up to the boundary.

Let $p(n)$ be the ordinary partition function and let $c(n)=\sum_{k=0}^{n-1}p(k)$, with $c(0)=0$. We prove that, for every $a\geq1$, $b\geq0$, and $m\geq2$, the sequence $c(an+b)$ is not eventually constant modulo $m$, thereby answering a question posed by Chen. The proof combines the algebraicity over $\mathbb C(j)$ of modular functions for finite-index subgroups of $\mathrm{SL}_2(\mathbb Z)$ with Voloch's theorem on the transcendence of elliptic modular functions in positive characteristic.

We develop three retraction-free Landing methods for nonsmooth composite optimization over the Stiefel manifold, combining an (approximately) tangential direction with an infeasibility-reducing normal direction. The subgradient Landing method, with normal stepsizes parameterized by $\alpha\in[0,1]$, has best-iterate complexity bounds $\mathcal{O}(\epsilon^{-4})$ for surrogate stationarity and $\mathcal{O}(\epsilon^{-2/(2-\alpha)})$ for feasibility. For $\alpha=0$, it converges locally $R$-linearly under local weak sharpness and mild assumptions. The proximal gradient Landing method computes a tangential direction by solving a proximal subproblem and combines it with the same normal direction. Moreover, the inexact proximal gradient Landing method allows the proximal subproblem to be solved approximately and applies an explicit correction to control violations of the tangency condition. Both proximal methods have outer-iteration complexity bounds $\mathcal{O}(\epsilon^{-2})$ for stationarity and $\mathcal{O}(\epsilon^{-1})$ for feasibility, and every accumulation point is first-order stationary. Under an additional asymptotic Bregman-error condition and the Kurdyka-Lojasiewicz property of a suitable auxiliary function, the primal sequence of the inexact method converges to a first-order stationary point. Numerical experiments illustrate the effectiveness of all three methods. On the tested sparse principal component analysis instances, the inexact method has favorable running times, and both proximal methods maintain small feasibility residuals.

Let $\mathrm{F}/\mathbb{Q}_p$ be a finite extension with $p\ne 2$, and let $V$ be a quadratic space over $\mathrm{F}$ of dimension at least two admitting a self-dual lattice. We prove the local arithmetic Siegel-Weil formula for GSpin Rapoport-Zink spaces formulated by Li and Zhang. We use the global-to-local strategy of Mihatsch and Zhang and its application by Luo. Using a generalization of the prismatic realization functor of Imai, Kato, and Youcis, we construct and study derived special cycles on integral models of SO Shimura varieties and their Rapoport-Zink uniformizations. We derive the local intersection formula from the modularity of special cycles on the generic fiber. As a first application, we prove the corresponding semi-global arithmetic Siegel-Weil identity for orthogonal Shimura varieties over a totally real field $F\ne\mathbb{Q}$.

We introduce and analyze a round-robin simultaneous resource allocation contest, motivated by sports leagues and other competitive environments where agents must distribute limited resources across multiple encounters. Players allocate resources in advance to matches against all opponents, and match outcomes are determined by relative allocations. We study two payoff schemes: total points, where players accumulate rewards for each win, and winner-takes-all, where only the tournament winner is rewarded. For three players, we provide complete equilibrium characterizations under both schemes. For four or more players, we show that the total points payoff admits Blotto-type equilibria, while the winner-takes-all payoff requires numerical and algorithmic analysis. To address this, we develop a best-response dynamics algorithm with stochastic inertia and use it to approximate equilibria in games with four and five players.

We introduce a family of five-dimensional non-flexible quadratic real algebras $\{\mathcal A_\gamma\}_{0\leq\gamma\leq1}$ with canonical basis $1,i,j,k,\ell$, in which $\gamma$ changes the multiplication but not their norm, and study how this change affects geometry and dynamics. The algebras are pairwise non-isomorphic, $\mathcal A_0$ contains Hamilton's quaternion algebra, and $\mathcal A_1$ is the Pentarnion algebra. The alternating part of the trilinear form $\langle u\times_\gamma v,w\rangle$ induced by the multiplication of $\mathcal{A}_\gamma$ defines a three-dimensional subspace $H_\gamma\subset\operatorname{span}_{\mathbb R}\{i,j,k,\ell\}$, tilted by the angle $\arctan(\gamma/\sqrt3)$, whose rotation group $G_\gamma\cong\mathrm{SO}(3)$ contains the automorphism group of $\mathcal A_\gamma$. For $\gamma_1\ne\gamma_2$ the groups $G_{\gamma_1}$ and $G_{\gamma_2}$ generate $\mathrm{SO}(V)\cong\mathrm{SO}(4)$. For the six ways of ordering and bracketing the product of $i,p,p$ in $\mathcal A_\gamma$, two map the unit circle in ${\rm span}\{j,k\}$ to a point and, for $\gamma>0$, four map it to ellipses. As an application, we derive a Hamiltonian model of a classical spin on $H_\gamma$ whose multiplication-induced quadratic energy is isotropic for $\gamma=0$ and anisotropic for $\gamma>0$. It conserves spin length and, for fixed $\gamma$, energy, and satisfies an exact work identity when $\gamma$ varies in time. Numerical experiments with periodically varying $\gamma$ indicate both regular and irregular spin dynamics.

We consider a one-dimensional symmetric simple exclusion process on the segment $\{-N,\ldots,N\}$, coupled to boundary dynamics that inject particles from the right reservoir and remove particles from the left reservoir, each acting on a window of size $K$. We derive sharp bounds on the space and space-time $v$-functions of arbitrary degree associated with the centered occupation variables. In particular, we establish bounds of order $N^{-1+2\zeta}$ for spatial $v$-functions of degrees $1$ and $2$, of order $N^{-1}$ for degrees $3,\ldots,K$, and of order $N^{-1-\zeta}$ for degrees at least $K+1$, for some sufficiently small $\zeta>0$. For space-time $v$-functions, we obtain bounds of order $N^{-1}$ for total degrees $2$ and $3$, together with improved bounds for higher degrees, incorporating the dependence on the time separation. These estimates provide the quantitative control needed to study the non-equilibrium fluctuations of the model.

We construct a factorizable Markov map $T$ on a hyperfinite finite von Neumann algebra and a positive element $x\in L\log L$ for which the sequence $T^n(T^*)^n x$ does not converge bilaterally almost uniformly. This answers a question of Anantharaman-Delaroche \cite{AD2006}.

The topology of a separable metrizable space $M$ is \emph{generated} by a family $\mathcal{C}$ of its subsets provided that a set $A\subseteq M$ is closed in $M$ if and only if $A\cap C$ is closed in $C$ for each $C\in \mathcal{C}$. The \emph{sequentiality number}, $\mathop{seq}(M)$, and \emph{$k$-ness number}, $\mathop{k}(M)$, of $M$, are the minimum size of a generating family of convergent sequences, respectively compact subsets. Let $\mathfrak{b}$ be the minimum size of an unbounded set in $\omega^\omega$ with the mod finite order. For a cardinal $\kappa$, the \emph{covering number}, $\mathop{cov}(\kappa)$, is the minimum size of a family of countable subsets of $\kappa$ so that every countable subset of $\kappa$ is contained in an element of the family. It is shown using the Tukey order on relations that (1) $\mathop{seq}(M)=\mathop{cov}(|M|)\cdot \mathfrak{b}$, unless $M$ is locally small (every point of $M$ has a neighborhood of size strictly less than $|M|$) in which case $\mathop{seq}(M)=\lim_{\mu <|M|} \mathop{cov}(\mu)\cdot \mathfrak{b}$ and (2) $k(M)$ is in the interval $[kc(M)\cdot\mathfrak{b},\mathop{cov}(kc(M))\cdot \mathfrak{b}]$, where $kc(M)$ is the minimum number of compact sets that cover $M$. Solutions to problems of van Douwen's on the $k$-ness number of analytic and of co-analytic spaces are deduced.

In this paper, we study the quermassintegral inequalities for capillary hypersurfaces in a horoball in hyperbolic space. We construct a locally constrained nonlinear curvature flow that preserves the $k$-th quermassintegral while monotonically increasing the $l$-th quermassintegral for any $(1\leq l<k\leq n)$. Using this flow, we establish a full family of the quermassintegral inequalities for horo-convex capillary hypersurfaces in a horoball with contact angle $\theta\in(0,\pi/2]$. Moreover, we weaken the horo-convexity to convexity and obtain the quermassintegral inequalities for convex capillary hypersurfaces between the $n$-th and $l$-th quermassintegrals.

We prove asymptotic completeness of the wave operators in time-dependent scattering theory which intertwine the ordinary or fractional Laplacian and the corresponding Hardy operator, i.e., the fractional Laplacian with an added subcritical or critical Hardy potential, whenever the Hardy potential is short-range. Moreover, we prove asymptotic completeness of the wave operators intertwining Hardy operators with and without external short-range perturbations. We also show the nonexistence of the considered wave operators when the Hardy or external potentials are long-range. We arrive at these results using three different approaches. The first approach uses an abstract semigroup framework for asymptotic completeness of wave operators for Fourier multipliers with short-range potentials. The second approach uses Kato smoothness theory. In the third approach, we develop and apply new propagation estimates for Hardy operators with and without perturbations.

We study objective-function-free (OFF) Newton-type methods for smooth unconstrained optimization with inexact gradients and Hessians. We propose two methods for first- and second-order stationarity, coupling explicit step-size rules with derivative-accuracy and linear-solver residual conditions to guarantee sufficient decrease without evaluating the objective function. The OFF regularized Newton method finds an $\varepsilon$-first-order stationary point in $\mathcal{O}(\varepsilon^{-2})$ iterations. Under strong convexity, a local Newton variant converges with $Q$-order at least $1+\theta$, where $\theta \in (0, 1]$. For second-order stationarity, we combine regularized Newton steps with approximate eigenvectors associated with the minimum eigenvalue. With high probability, the resulting method finds an $(\varepsilon, \sqrt{L_h\varepsilon})$-second-order stationary point in $\mathcal{O}(\varepsilon^{-3/2})$ iterations, using a total of $\widetilde{\mathcal{O}}(\varepsilon^{-7/4})$ approximate gradient evaluations and Hessian-vector products. For finite-sum problems, we develop subsampling schemes that yield best-iterate convergence in expectation in the nonconvex first-order setting and local $R$-linear convergence with high probability in the strongly convex setting, followed by $Q$-order at least $1+\theta$ once full sampling is reached. The subsampled second-order method achieves a high-probability $\widetilde{\mathcal{O}}(\varepsilon^{-7/2})$ total sample-operation bound, counting component gradient evaluations and component Hessian-vector products.

In this short note, we prove the global well-posedness of the modified dispersion-generalized Benjamin-Ono equation with dispersion $1<\alpha<2$, $$\partial_t u + D^\alpha \partial_x u + u^2\partial_x u=0,$$ over the Sobolev space $H^s(\mathbb{R})$, for any regularity $s$ greater or equal than the sharp analytic well-posedness regularity. As a byproduct, we also derive nonlinear smoothing estimates and give a simple proof of local well-posedness.

A classical theorem states that the independence number of any connected graph is at least its radius. In 2007, DeLaVi\~na, Pepper, and Waller proved that every connected graph with equal independence number and radius is traceable. We first give a new short proof of this theorem and then prove two results for graphs of higher connectivity. (1) Every 2-connected graph with equal independence number and radius is Hamiltonian. (2) Every 3-connected graph with equal independence number and radius is Hamiltonian-connected and pancyclic. Several lemmas used in the proofs are of independent interest.

In this paper we prove that plane waves always scatter from bounded, strictly convex planar inhomogeneities with constant material parameters. Our approach involves the introduction of a modified Pompeiu property, whose failure corresponds to the vanishing of the Fourier transform of the indicator function of the inhomogeneity on a complexified circle not centered at the origin. We then show that such vanishing is impossible for bounded, strictly convex planar sets.

We propose a new regression method named Direct Inference on Admissible Sets (Diona) that directly uses samples to estimate the state and its derivatives at a query point, along with their admissible bounds, without long global training. By casting the values and derivatives at the query and neighbouring samples as latent variables in a constrained optimisation problem, the burden of long training loops becomes one of on-demand local solves. We establish Diona's theoretical properties including finite-sample coverage and, in the noiseless case, convergence to the true state and derivatives under dense sampling. In a nonlinear regression benchmark on six functions of three to eight dimensions, Diona was more accurate than a Gaussian process (GP) on noiseless data especially with denser data sets. Furthermore, as with GP, it produced uncertainty bounds that contained the true value at more than 95% of query points in four functions. Diona's time to a first prediction after a data set update was three to four orders of magnitude shorter than that of the GP, and two to three orders shorter than that of a neural network, which was more accurate but offers no uncertainty quantification. This low latency is paid for by on-demand predictions, so this latency advantage applies when relatively few predictions are needed between data updates.

Autonomous multi-agent control systems (MACSs) enable distributed coordination in emerging applications such as smart power grids and autonomous vehicle or drone networks. In many of these applications, inter-agent communication relies on wireless networks. However, conventional MACS designs neglect the interaction between the communication and control subsystems. At the same time, conventional wireless designs reserve communication resources based on worst-case MACS requirements. Consequently, control performance is degraded and wireless resources are utilized inefficiently. To address these challenges, we propose a consensus-aware co-design approach for MACSs and wireless communication systems. We introduce a system model in which agents achieve consensus by exchanging information through device-to-device (D2D) links that share spectrum with cellular users under limited bandwidth resources. Based on this model, we formulate a joint optimization problem for consensus-topology selection and wireless resource allocation to minimize bandwidth consumption while guaranteeing consensus and satisfying the quality-of-service (QoS) requirements of both MACSs and cellular users. Although the resulting problem is a mixed-integer nonlinear program (NP-hard in general), a structural property of the optimal solution enables a sequential two-stage optimization algorithm with polynomial complexity. Simulations on a smart power grid case study demonstrate that the proposed approach achieves 24-27% bandwidth savings compared with the two baselines while maintaining consensus and satisfying all QoS requirements of the control agents and the communication users.

We prove non-uniqueness for the forced one-dimensional Vlasov-Poisson equation on the whole space. More precisely, for every finite $p$, we construct a nonnegative ion density, smooth for positive times, and a source $S\in C([0,1];L^1(\R^2))$ with $S(0)=0$, for which there exist two distinct nonnegative distributional solutions $f_\pm\in C([0,1];L^1(\R^2))$ with the same zero initial datum and with spatial densities bounded in $L^p(\R)$ uniformly in time. The result is sharp in terms of the integrability of the spatial densities: our solutions have Lipschitz electric fields, and we show that in this class uniqueness holds as soon as the spatial density is bounded.