数学前沿

We prove that the kissing number in 19 dimensions is at least 11948, improving the bound of Cohn and Li by 256. Their odd-sign construction reduces the problem to finding a binary code of length 19 and minimum distance 5 inside a 5-punctured extended binary Golay code. In an explicit model $D$ of this ambient code, the 21 words of weights 3 and 4 are exactly the pairwise differences that such a code must avoid. They span a 10-dimensional subcode $K$. We exhibit a 6-dimensional subcode $M\le K$ for which all 21 words lie in only five cosets of $M$. After identifying $K/M$ with $\mathbb F_2^4$, these five cosets are represented by $\Sigma=\{e_1,e_2,e_3,e_4,e_1+e_2+e_3+e_4\}$. In the graph on $\mathbb F_2^4$ in which $u$ and $v$ are adjacent when $u+v\in\Sigma$, the set $\Sigma$ itself is independent. Lifting its five vertices gives 320 words in $K$, and translating this set through the four cosets of $K$ in $D$ gives a nonlinear code of size 1280 and minimum distance 5.

This paper is devoted to the study of the long time behavior of Nash equilibria in Mean Field Games within the framework of displacement monotonicity. We first show that any two equilibria defined on the time horizon $[0,T]$ must be close as $T \to \infty$, in a suitable sense, independently of initial/terminal conditions. The way this stability property is made quantitative involves the $L^2$ distance between solutions of the associated Pontryagin system of FBSDEs that characterizes the equilibria. Therefore, this implies in particular the stability in the 2-Wasserstein distance for the two flows of probability measures describing the agent population density and the $L^2$ distance between the co-states of agents, that are related to the optimal feedback controls. We then prove that the value function of a typical agent converges as $T \to \infty$, and we describe this limit via an infinite horizon MFG system, involving an ergodic constant. All of our convergence results hold true in a unified way for deterministic and idiosyncratic noise driven Mean Field Games, in the case of strongly displacement monotone non-separable Hamiltonians. All these are quantitative at exponential rates.

We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces, and show that almost every such locally Hamiltonian flow with only simple saddles has singular spectrum. Furthermore, we prove that for almost every pair of such flows, the elements of the pair are spectrally disjoint. More generally, the results from which these statements are deduced are singularity of the spectrum and pairwise spectral disjointness for special flows over full measure sets of interval exchange transformations under a roof with symmetric logarithmic singularities. Spectral singularity is proved using a criterion based on tightness of Birkhoff sums with exponential tail decay. The assumptions of the criterion are verified exploiting the cancellations proved by the last author to prove the absence of mixing in this class of flows, by showing that the latter can be combined with rigidity by exploiting the local product structure of Rauzy-Veech induction. Pairwise spectral disjointness then follows by producing mixing times (for the second flow), using a new mechanism for shearing based on what we call resonant rigidity times.

The lattice point problems of the $p$-circle (for example, the astroid), which a generalized circle for positive real numbers $p$, have been solved for approximately $p$ more than 3, based on the series representation of the error term using the generalized Bessel functions by E. Kr\"{a}tzel and the results of G. Kuba. On the other hand, for the cases $0<p<2$, the method via this series representation cannot make progress. Therefore, in such cases, it is necessary to consider another method. In this paper, we prove that certain functions closely related to the problems can be displayed as series by newly generalized Bessel functions based on the property $p$-radial, generalization of spherical symmetry, and highlight the possibility that attempts to solve the problems via this display are suitable especially for the cases $0<p\leq1$. This study is based on the harmonic-analytic method by S. Kuratsubo and E. Nakai, using certain functions generalizing the error term of the circle problem by variables and series representation of the functions by the Bessel functions.

Let $S\subset\Bbb R_+^n$ be compact and convex with $0\in S$, and let $\mathcal{P}^S$ be the algebra spanned by monomials with exponents in $\bigcup_{m\ge0}(mS\cap\Bbb N_0^n)$. We define the associated polynomial hull and prove a Kallin lemma for the union of two sets convex with respect to this algebra. A weighted approximation lemma then shows that, under a sector or ray condition, two admissible monomials generate exactly the functions of the unrestricted algebra that are constant on a specified zero fibre. Applications include an exact density criterion for mixed monomials, a transfer principle for relative perturbations, and density and hull computations on stars of totally real discs.

We prove a conjecture of Conlon, Fox, Gunby, He, Mubayi, Suk, Verstra\"{e}te, and Yu that for a $3$-graph $H$, the off-diagonal Ramsey number $R(H,n)$ is polynomial in $n$ if and only if $H$ is isomorphic to a subgraph of the "depth-$t$ iterated blowup" $\textsf{B}_t$ for an integer $t\geq 0$. Here, the $3$-graphs $\textsf{B}_t$ are defined recursively: $\textsf{B}_0$ has one vertex, and $\textsf{B}_t$ for $t\geq 1$ is obtained from the disjoint union of three copies of $\textsf{B}_{t-1}$ by adding all transversal hyperedges.

We prove that every quasi-elementary sub-Hopf algebra of the polynomial part of the odd primary Steenrod algebra must lie in a certain sub-Hopf algebra called $D$.

In this paper, we study sequences of perfect t-embeddings of a uniformly weighted family of graphs we call generalized tower graphs. We show that the embeddings of these graphs satisfy certain technical assumptions, in particular, the rigidity assumption of Berggren-Nicoletti-Russkikh. As a result, we confirm the convergence of the gradients of the height function fluctuations of these graphs to those of the Gaussian free field.

Let $\mathcal{H}$ be a complex Hilbert space, and let $T=H+iK\in\mathcal{B}(\mathcal{H})$ be the Cartesian decomposition of $T$, where $H=H^{*}$ and $K=K^{*}$. We introduce the $L^2$-Cartesian numerical radius and study its basic properties. We compare it with the classical numerical radius $\omega(T)$. In particular, we establish the sharp estimate $ \frac{1}{\sqrt{2}}\omega(T)\le \wcart{2}(T)\le \omega(T), $ and establish lower and upper bounds in terms of the Cartesian parts $H,K$. We also give an equality characterization and explicit sharpness examples. As a consequence, we obtain $\frac{1}{4}\left\|T^{*}T+TT^{*}\right\|\le \wcart{2}(T)^2\le \omega(T)^2. $ More generally, we study the $L^p$-Cartesian numerical radius and prove the sharp comparison $ c_p\,\omega(T)\le \wcart{p}(T)\le \omega(T),\qquad 1\le p<\infty. $ Finally, we present a block-operator analogue of the classical refinement of the triangle inequality.

Ion correlations set the structure of polymer electrolytes and depend on the host permittivity, which is low and varies between polymers. A functional that carries these correlations from molecular simulations to larger inhomogeneous systems must transfer across domain sizes, geometries and coupling strengths. We introduce a neural density functional for the ions of a salt-doped polymer melt with built-in integrability, continuous spatial symmetry, the Noether identities, perfect screening in stable bulk states, and parameters that refer to no grid or box. The permittivity enters only through the Bjerrum length, an explicit argument constrained by an exact relation: the free-energy derivative with respect to the Bjerrum length is the Coulomb correlation energy. Trained by force matching on planar profiles at five Bjerrum lengths spanning 8-30 monomer diameters, the functional predicts, without retraining, density profiles in fourfold longer boxes and under two-dimensional fields, bulk structure factors and the long-wavelength number response, also at a concentration and a permittivity left out of training. It reproduces this energy identity on Coulomb energies that never enter the training. Its nonlinear density dependence captures correlations that a quadratic pair closure misses: the number response grows several-fold with concentration, and at low concentration the coupling lowers it through ion association, below the ideal value at the strongest coupling, while the host sets it in the concentrated melt. At fixed coupling, it is more accurate than three neural architectures and loses little accuracy with a quarter of the training data.

Alp\"oge constructed a complex structure on $S^6$ through a torus fibration with exactly three special fibres. We prove that every member of the associated admissible period family is Oka and that the family projection over each relatively compact parameter disc is an Oka map. The Oka property survives the deletion or simultaneous blow-up of any finite set of distinct points in any member. Blowing up a single point therefore produces an Oka complex manifold diffeomorphic, but not biholomorphic, to $\mathbb{P}^3$. The proofs combine explicit cyclic coverings with analytic extension and the invariance of the Oka property under holomorphic coverings.

We give a computable criterion of the triviality of the unramified Iwasawa module over the maximal multiple $\mathbb Z_p$-extension of a number filed in which an odd prime number $p$ splits completely. This criterion is obtained from the description of the presentation matrix of the Galois coinvariant of the Iwasawa module by arithmetic linking numbers. The description is based on the Koch type presentation of a pro-$p$ Galois group with restricted ramification, which is analogous to the Milnor presentation of a link group from a viewpoint of arithmetic topology.

Let $S$ be a regular projective surface over a field $k$ of characteristic $p>0$, with $H^0(S,\mathcal{O}_S)=k$ and $-K_S$ nef. We prove that $S$ is geometrically integral over $k$ when $p\geq 7$, and we also find an example of $S$ that is not geometrically integral when $p=5$.

A left brace is cofinite integral (CIB) if its additive group is free abelian of finite rank and its quotient by the socle is finite. Rump introduced such braces and showed that the multiplicative group of every CIB is a crystallographic group. We study CIBs from the point of view of crystallographic groups. We prove an analogue of the Bieberbach theorems: in each dimension there are only finitely many isomorphism classes of CIBs. For a fixed crystallographic group, we show that its CIB structures correspond to the orbits of a finite group action on a set of bijective 1-cocycles. We use this description to develop algorithms that enumerate them. Finally, we give a complete census of CIBs on crystallographic and Bieberbach groups up to dimension 6.

We define simple tilings in the general context of a $G$-tiling on a Riemannian homogeneous space $M$ to be tilings by Riemannian simplices. As evidence that this definition is natural, we prove that a large class of tilings of $M$ are MLD to simple ones. We further show that there is an isomorphism between \u{C}ech cohomology and pattern-equivariant cohomology of simple tilings of homogeneous Riemannian Spaces. This demonstrates the utility of our definition by generalizing an established result on the topology of tiling spaces of Euclidean space.

For every $n\ge3$, we construct compact Type II ancient Yamabe flows in the round conformal class of the unit $n$-sphere $\Ss^n$. After stereographic projection and rescaling of time and conformal factor, the solutions are asymptotic backward in time to finite superpositions of Aubin--Talenti bubbles with mutually comparable scales. For integers $h\ge1$ and $k\ge2$, the limiting centers form $h$ coaxial regular $k$-gons; when $h\ge2$, their distinct layer heights may be prescribed arbitrarily in the interval $(-1,1)$. If $hk\ge3$, the resulting flows are not geometrically equivalent to any rotationally symmetric ancient flow. For each flow, there is a fixed point of $\Ss^n$ at which the Ricci tensor has both positive and negative eigenvalues on every time slice. The corresponding volume-normalized Yamabe flows are also ancient. After rescaling each time slice to a fixed volume, the metrics converge to a wedge of $hk$ isometric round $n$-spheres in the Gromov--Hausdorff topology, as time tends to $-\infty$. For fixed $h,k\ge2$, the concentration configurations admit an explicit $(h-1)$-parameter description up to M\"obius transformations. Consequently, the construction yields geometrically inequivalent flows indexed by a nonempty open subset of $\R^{h-1}$, all with the same fixed-volume backward Gromov--Hausdorff limit. The proof uses inner--outer gluing for the quasilinear parabolic equation in $n$-dimensional space that exhibits Kelvin symmetry. Weighted Euclidean and spherical H\"older estimates control the nonlinear terms and, together with parabolic regularity, yield smooth extension of the conformal metrics across the north pole of $\Ss^n$.

We consider the value-distribution of the Euler--Zagier type multiple zeta-functions with identical arguments on the critical line. Mishou showed that the set of the values of the Euler--Zagier double sum with identical arguments on the critical line is dense in the complex plane under the Riemann Hypothesis. In this paper, we generalize Mishou's result to the $r$-fold Euler--Zagier multiple zeta-functions with identical arguments for $r geq 4$.

The dominant contemporary paradigm of direct state-led urban environmental regulation entails significant risks, including high administrative burden, frequent conflicts with polluting entities, and economic vulnerabilities for those sources. To address these deficiencies, this paper integrates third-party service provider (TPSP) into China's existing management framework as a beneficial extension beyond conventional government intervention. Utilizing an evolutionary game approach, we compare two distinct institutional settings, a non-integration management mode and an integration management mode, and scrutinize the evolutionary dynamics across both two- and three-dimensional systems. To enhance system robustness, we embed a strategic behavior-based dynamic punishment mechanism that effectively eliminates the perturbing effects of varied penalty rules, thereby steering all stakeholders toward an ideal stable state. This comparative institutional analysis isolates the marginal contribution of third-party intermediation, specifies when outsourcing environmental services outperforms direct government enforcement, and reframes evolutionary game theory as a design device for institutional self-correction. Furthermore, a predefined variable importance analysis is conducted to highlight critical factors warranting attention in practical policy implementation. Extensive simulation experiments, carried out under diverse system-level and parameter-level conditions, validate the theoretical properties of our model. The findings offer concrete managerial insights to inform the formulation of more effective urban environmental supervision regulations.

Since Strain's work \cite{Strain-2012-KRM} on almost optimal time decay rates for the microscopic part of perturbative classical solutions to the Boltzmann system with soft potentials via the time-velocity splitting method, how to obtain the optimal decay of the microscopic part has remained unsolved. In this paper, we resolve this problem by developing a nonlinear energy estimate method different from the time-velocity splitting approach. The proof relies on a combination of a hierarchy of time-weighted energy estimates, the damping factor $e^{-\nu(v)(t-\tau)}$, and a dyadic decomposition of velocity space.

This third paper in a series on a parabolic-elliptic chemotaxis model studies bifurcation of positive constant equilibria in one space dimension. We consider $$\left\{ \begin{array}{ll} u_t=\Delta u-\chi_0\nabla\cdot\left(\frac{u^m}{(1+v)^\beta}\nabla v\right)+au-bu^{1+\alpha}, & x\in\Omega,\\ 0=\Delta v-\mu v+\nu u^\gamma, & x\in\Omega,\\ \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0, & x\in\partial\Omega, \end{array} \right.$$ where $\Omega$ is a bounded interval, $\alpha,\gamma,m,\mu,\nu>0$, $\chi_0\in\mathbb R$, and $a,b,\beta\ge0$. Building on earlier results on global existence, boundedness, persistence, and stabilization, we take $\chi_0$ as the bifurcation parameter and examine the effects of $\beta$ and the domain size on the bifurcation. Center manifold reduction yields a coefficient $\beta_{n_0}$ at a simple bifurcation mode $n_0$ of a positive constant equilibrium. The pitchfork bifurcation is supercritical if $\beta_{n_0}>0$ and subcritical if $\beta_{n_0}<0$. Stronger desensitization raises the sensitivity required for loss of linear stability. Bifurcation may be supercritical for small $\beta$ but is always subcritical for sufficiently large $\beta$. Thus, desensitization makes patterns harder to trigger and favors abrupt rather than continuous emergence. The bifurcation mode increases with domain size, allowing more complex spatial patterns. Under suitable conditions, we prove unbounded continuation of local bifurcation branches. We also show that large-sensitivity stationary sequences satisfying a peak-gap hypothesis develop spike-like profiles. Numerical continuation and time-evolution experiments illustrate the bifurcating branches and spike-like solutions and suggest unbounded continuation and increasingly sharp stationary profiles.