数学前沿

We simplify and improve OpenAI's proof of the propeller conjecture of Khot and Naor from 2008: the sum of the squared lengths of the Gaussian moments of any partition of Euclidean space is at most $9/(8\pi)$. We show equality only occurs for three 120 degree sectors in the plane (up to symmetries, measure zero changes, and taking orthogonal cross products with Euclidean space). We also remove computer assistance from the three-dimensional case of the conjecture, which previously used an epsilon net traversal.

Under quantitative asymptotic hyperbolicity assumptions, we prove the existence of hyperbolic ergodic invariant probability measures satisfying Pesin's entropy formula for $C^1$ diffeomorphisms whose attractors admit extendable continuous invariant splittings. No dominated splitting is required on the attractor. For $C^{1+\alpha}$ diffeomorphisms, the resulting measures are SRB measures. Our results apply to mostly expanding partially hyperbolic systems and to systems with strictly separated Lyapunov exponents under suitable statistical assumptions. We also construct examples with hyperbolic SRB measures of full support on attractors admitting no nontrivial dominated splitting, demonstrating the applicability of our criteria beyond results that require domination.

We present a new method of reproving the best known lower bound of the Erd\H{o}s distinct subset sums problem. Although our lower bound is asymptotically the same as in Dubroff, Fox and Xu, it is slightly better for the case when $n$ is odd. Besides, we also provide a new lower bound on the second largest element $a_{n-1}$.

Continuing our work on the half-integral case, we construct generalized nilpotent $2$-representations of the integral quasi-split type $\operatorname{AIII}$ $\imath$-Kac--Moody $2$-category from affine Brauer actions with dot spectrum exactly $\mathbb Z$ on locally Schurian $\mathbb C$-linear categories. Conversely, nilpotent $2$-representations on such categories yield affine Brauer actions with spectrum contained in $\mathbb Z$. At the fixed vertex $0$, we directly define corrected generators and verify the nil-Brauer and mixed relations. For $\mathbf u$-admissible integral parameters at arbitrary positive level, we prove an isomorphism between the locally unital algebras of the cyclotomic Brauer category and a cyclotomic quotient of the principal $2$-representation. This gives compatible $\mathbb Z$-gradings on the category and its endomorphism algebras, providing a Brauer analogue of the graded realization of type $A$ cyclotomic Hecke algebras. Over algebraically closed fields of odd characteristic $p$, we also obtain both constructions with spectrum contained in $\mathbb F_p$, for the mixed affine datum restricted to its spectral weight subgroup.

The Aldous-Broder algorithm constructs a spanning tree from the first-entrance edges of a covering random walk. Its distribution is known, but deriving it by summing over the orders in which a fixed tree can be discovered remains difficult. In this paper, we evaluate this sum by establishing an order-mass identity for rooted forests. For real matrices with zero row sums, the identity expresses the sum of order weights associated with each root as a principal cofactor of a matrix on the roots, under a nonsingularity condition involving only the sets encountered along the growth orders of the given forest. We prove the identity using linear algebraic tools. With one root, the sum equals one, giving a direct proof of the extended Aldous-Broder theorem for every finite irreducible Markov chain started at a fixed state. With multiple roots, we derive the forest law produced by the stopped covering walk and show how it differs from the weight-proportional forest law. The identity and the resulting tree and forest laws are formalized in Lean 4.

We consider a kinetic model of predator-prey dynamics in which predation is characterized by a sequence of behavioral transitions. Prey and predators are described by distribution functions in phase space, with the predator population further structured into internal states corresponding to inactive, active, and saturated individuals. We show that different macroscopic functional responses emerge from distinct mechanisms governing predator activation. More precisely, activation driven uniquely by pairwise encounters leads to a saturating Holling type II response, whereas activation additionally modulated by local prey density produces a generalized sigmoidal Holling type III response. This is achieved by means of a diffusive scaling and a formal hydrodynamic limit, from which we derive a closed reaction-diffusion system of Rosenzweig-MacArthur type for the prey and total predator spatial densities.

We study the importance-tempered random-walk Metropolis (RWM) algorithm on $\mathbb{R}^d$, which targets the tempered density $\pi_\beta(x) \propto \pi(x)^\beta$ with $\beta \in (0,1)$ and corrects for the resulting bias by importance weighting. The dynamics of this sampler can be described by a continuous-time Markov chain whose embedded chain is the tempered RWM. We prove that when the tails of $\pi$ decay at rate $\|x\|^{-\gamma}$, this continuous-time chain is uniformly ergodic if and only if $d/\gamma < \beta < (\gamma - 2)/\gamma$. Further, the global spectral gap of this chain can be controlled via the spectral gap of the chain restricted to a ball with radius growing polynomially in $d$. Our theory and simulation studies show that, with a suitable choice of $\beta$, importance tempering can substantially reduce sensitivity to initialization and significantly improve efficiency over the standard RWM.

We study the $L^q$-regularity of the density of barycenters of $N$ probability measures on $\mathbb{R}^d$ with respect to the $p$-Wasserstein metric ($1<p<\infty$). According to a previous result by the first author and collaborators, if one marginal is absolutely continuous, so is the $W_p$-barycenter. The next natural question is whether the $L^q$- regularity on the marginals is also preserved for any $q > 1$, as in the classical case ($p=2$) of Agueh--Carlier, or for $W_p$-geodesics ($N=2$). Here we prove that this is the case if one marginal belongs to $L^q$ and the supports of all the marginals satisfy suitable geometric assumptions. However, we show that, as soon as $N>2$, it is possible to find examples of $W_p$-barycenters which are not $q$-integrable, even if one marginal is compactly supported and bounded, thus highlighting the role played by the geometry of the supports. Furthermore, we provide a general estimate of the $L^q$-norm, including a detailed study of the sources of singularities, and a characterization of the $W_p$-barycenters \`a la Agueh--Carlier in terms of the associated Kantorovich potentials. Finally, we explicitly compute the $W_p$-barycenters of measures obtained as push-forward of special affine transformations. In this case, regularity holds without any additional requirement on the supports.

We give a short proof of Zhang's conjecture that the number of stack-sortable permutations of length $n + 1$ which end in their smallest element is equal to the $n$-th Motzkin number.

We study the Hausdorff dimension of the closure of the canonical height values associated to finite compositions of two rational maps. For rational maps with common periodic critical points, a nonzero local--global height discrepancy gives a nonempty set of starting points with positive-dimensional spectra. For polynomials, equality of the normalized logarithms of the leading coefficients at every place forces dimension zero; a single-place discrepancy gives positive dimension under boundedness at the other places. We construct rational cubic pairs with spectra of dimension $\log2/\log3$. The spectrum dimension gives a lower bound for the growth exponent of the number of distinct orbit points of bounded height. At the degree exponent, positive Hausdorff measure gives matching growth bounds for multiplicatively dependent degrees. For independent degrees, a restriction of the covering intervals and a Diophantine rotation estimate give a lower bound with logarithmic loss of exponent strictly less than $1/2$. Quadratic monomial examples show that the spectrum dimension can depend on the starting point even when both individual orbits are infinite.

We compute the Chern classes of irreducible representations for the dihedral groups $D_{2n}$ and the generalized quaternion groups $Q_{2^n}$. Utilizing these computations, we investigate whether every even-dimensional class in the integral group cohomology of these families can be realized as a Chern class of some complex representation. It turns out that this realizability question answers in the affirmative for the generalized quaternion groups. In contrast, it turns out that the answer for dihedral groups is erratic.

This paper is the first to study the continuous-time mean-variance portfolio selection problem with no-shorting constraints in a Markov regime-switching market using reinforcement learning (RL). The main challenge stems from the interaction between random regime shifts and portfolio non-negativity constraints, which typically causes the coupled Hamilton-Jacobi-Bellman (HJB) equations to lose smoothness across different state regions. Additionally, directly imposing no-shorting constraints on randomized policies disrupts their analytically tractable Gaussian structure, complicating both exploratory modeling and reinforcement learning. To address these challenges, we propose an auxiliary exploratory approach that limits the policy to a Gaussian distribution and applies the no-shorting constraint solely to its mean. We explicitly characterize the optimal policy for this auxiliary problem and demonstrate that, by appropriately selecting the exploration variance, its optimal mean policy exactly matches the optimal policy of the original constrained mean-variance problem. Building upon this analytical framework, we develop a model-free reinforcement learning algorithm that learns the optimal portfolio policy directly from trajectory data. Numerical experiments support the theoretical results and show that the proposed method can effectively recover the optimal no-shorting portfolio in a regime-switching market.

We construct quasiperiodic Schr\"odinger operators with Gevrey potentials for which the Lyapunov exponent is discontinuous in the spectral energy. More precisely, let $\alpha\in\R\setminus\mathbb Q$ satisfy the Diophantine condition \(\|k\alpha\|_{\T}\geq\gamma |k|^{-\tau}\) for all $k\in\Z\setminus\{0\}$, with $\gamma>0$ and $\tau\geq1$. For every Gevrey index $\mathfrak s>1+\tau$ and every sufficiently large coupling $\lambda$, we construct a real-valued potential \(v_\lambda\in G^{\mathfrak s}(\T,\R)\) and a sequence of energies $E_j\downarrow0$ such that $$L_{v_\lambda}(0)\geq(1-\varepsilon_\lambda)\log\lambda, \qquad \sup_{j\geq1} L_{v_\lambda}(E_j)\leq(1-c_*)\log\lambda,$$ where $L_{v_\lambda}(E)$ denotes the Lyapunov exponent of the Schr\"odinger operator with potential $\lambda v_\lambda$, $c_*>0$ is an absolute constant, and $\varepsilon_\lambda\to0$ as $\lambda\to\infty$. For bounded-type frequencies ($\tau=1$), the regularity range $\mathfrak s>2$ is sharp up to the endpoint $\mathfrak s=2$, since the Lyapunov exponent is known to be continuous in energy for $1<\mathfrak s<2$.

Let $0<\gamma_1\leq \gamma_2\leq \ldots$ denote the positive ordinates of the non-trivial zeros of the Riemann zeta-function. A result first announced by Selberg states that there exist absolute constants $\Theta, \vartheta>0$ such that for each $r\in \mathbb{N}$, $$\limsup_{n\to \infty}\frac{\gamma_{n+r}-\gamma_n}{2\pi r/\log \gamma_n}\geq 1+\frac{\Theta}{r^\alpha} \qquad \text{and}\qquad \liminf_{n\to \infty}\frac{\gamma_{n+r}-\gamma_n}{2\pi r/\log \gamma_n}\leq 1-\frac{\vartheta}{r^\alpha}$$ where $\alpha$ may be taken as $2/3$, or as $1/2$ if one assumes the Riemann hypothesis. This was recently proved by Conrey and Turnage-Butterbaugh under RH and by Inoue unconditionally. We prove that in fact a positive proportion of $r$-gaps are large (and small) to the above extent, and we provide explicit estimates for the sizes and proportions of these gaps. In the case $r=1$, this quantitatively improves an unconditional result of Simoni\v{c}, Trudgian and Turnage-Butterbaugh.

Background-error covariance models are central to atmospheric data assimilation because they spread partial observational information across space, variables, and dynamically coupled scales. Ensemble and variational methods exploit this covariance geometry efficiently, but their standard analysis formulations are largely built around Gaussian or locally quadratic approximations. Score-based ensemble filtering provides a framework for nonlinear distributional transport, but existing Ensemble Score Filter (EnSF) formulations lack an intrinsic covariance-mediated pathway for propagating likelihood information from observed to unobserved state components. We introduce a covariance-informed ensemble score filter (CI-EnSF) that embeds localized forecast-error covariance geometry directly into the forward diffusion and reverse transport. The proposed forward diffusion is covariance-shaped in standardized state space and becomes isotropic under the corresponding control-variable transformation. This covariance structure continuously propagates observational information from observed to unobserved state components throughout the reverse transport analysis trajectory. Experiments with Lorenz-96 and surface quasi-geostrophic (SQG) dynamics show a clear regime dependence. For linear observations, a well-tuned LETKF is more accurate, while CI-EnSF maintains stable cycling and effective covariance-mediated information propagation. Under strongly nonlinear observations and under climatological initialization, CI-EnSF achieves lower analysis error and greater cycling robustness than Gaussian ensemble benchmarks such as the LETKF, with improvements extending across spatial scales in the nonlinear SQG experiment. These results suggest that the principal benefit of CI-EnSF arises from combining nonlinear distributional transport with the mature covariance geometry used in conventional data assimilation.

We study the unweighted error-sum function $\mathcal{E}(x) \coloneqq \sum_{n \geq 0} (x - p_n(x)/q_n(x))$, where $p_n(x)/q_n(x)$ denotes the $n$th convergent of the continued fraction expansion of $x \in \mathbb{R}$, and we prove that the graph of $\mathcal{E}$ has Hausdorff dimension $1$. The key ingredient is an exact formula for the oscillation of $\mathcal{E}$ on fundamental intervals, which we obtain in a symbolic space of continued fraction sequences containing non-realizable sequences. Using the same framework, we also give a new derivation of the upper bound $3/2$ of Ridley and Petruska (2000) for the Hausdorff dimension of the graph of the relative error-sum function $P(x) \coloneqq \sum_{n \geq 0} (q_n(x)x - p_n(x))$. Moreover, we present an alternative argument based on M\"obius inversion and summation over coprime convergent denominators, and we explain why the square coverings used for $P$ cannot yield an upper bound smaller than $3/2$.

We study the extremal function $S^k_d(n)$, defined as the maximum number of regular $(k-1)$-simplices spanned by $n$ points in $\mathbb{R}^d$. For any fixed $d\geq2k\geq6$, we determine the asymptotic behavior of $S^k_d(n)$ up to a lower-order term. In particular, when $k=3$, we determine the exact value of $S^3_d(n)$, for all even dimensions $d\geq6$ and sufficiently large $n$. This resolves a conjecture of Erd\H{o}s in a stronger form. The proof leverages techniques from hypergraph Tur\'an theory and linear algebra.

We prove the corrected Bessis--Moussa--Villani pinching conjecture for all symmetrized word averages. For Hermitian matrices $A,B$ and $D=E_A(B)$, we construct a positive compactly supported measure $\zeta_{A,B}$ on $\mathbb R^2$ such that $$\Tr f(xA+yB)-\Tr f(xA+yD) =y^2\int_{\mathbb R^2}f''(xa+yb)\,d\zeta_{A,B}(a,b).$$ Its total mass is $\frac12\|B-D\|_2^2$. This representation proves the corrected conjecture, including its equality cases, and gives a positive bivariate moment representation for the coefficient differences. We also derive complete monotonicity of normalized exponential pinching defects and a comparison with Lie--Trotter approximations. A one-dimensional second-order loss measure is obtained for trace-preserving conditional expectations on finite tracial von Neumann algebras. Finally, we relate $\zeta_{A,B}$ to Hein\"avaara's tracial joint spectral measure by an explicit positive radial transform.

The basic question in perturbation analysis of Markov chains is how small changes in their transition kernels affect their stationary distributions. Classical perturbation bounds typically require the kernel error to be much smaller than $1/\tau$, where $\tau$ is a mixing or relaxation time. Although this scaling is sharp for Markov chains in general, we investigate a general "square-rooting" phenomenon in which one-step errors of order roughly $1/\sqrt{\tau}$ can be sufficient for local updates. We proved a form of this phenomenon in Lin, Liu and Smith (2025) under strong assumptions. Here we substantially weaken these assumptions, and prove this phenomenon occurs using three distinct approaches. First, block factorization applies under structural assumptions on the stationary measures of both chains. Second, approximate block-update arguments extend the result to statistically relevant Markov chain Monte Carlo (MCMC) settings, where structural guarantees are available for the exact posterior and its associated sampler, but not for the perturbed posterior. Third, we use direct calculations for a class of models with hard constraints where neither general result is directly available. We illustrate these results in three MCMC settings and show how they directly inform the tuning of approximate MCMC algorithms.

A vertex-centroid move slides one vertex of a polygon along the line through it and the centroid of all vertices; Problem 60 of The Open Problems Project asks whether such moves can make every polygon regular. We prove that two simple labeled $n$-gons, with straight angles allowed, are joined through simple polygons by finitely many positive moves exactly when they have the same orientation; a nonconstant move is positive if the vertex remains on the same side of the centroid of the other vertices. The same holds for flows toward or away from the instantaneous area centroid. Inward-only or outward-only moves reach every target of the same orientation up to translation and positive scaling. The common mechanism is a first-order criterion: a symmetric definite center response makes the one-vertex fields of a smooth affine-equivariant center bracket generating. The response is a weighted covariance for point barycenters and $(d+2)$ times the volume covariance for polyhedral centroids. Once translation and scale are factored out, the fields sum to zero, giving exact one-sided control. We resolve Problem 60 under both conventions for a vertex at the centroid. For the vertex-centroid system on $n$ affinely spanning points in $\mathbb{R}^d$, we classify positive-move reachability and prove a local $3dn$-move bound.