数学前沿

The covering number $C(v,k,t)$ is the least number of $k$-subsets, called blocks, of a $v$-set such that every $t$-subset lies in one of them. We reduce the question whether a covering with $L$ blocks exists to the classification of links. If every point of such a covering lies in at least $r$ blocks and $(r+1)v>kL$, some point lies in exactly $r$ blocks, and its link, the blocks through it with the point removed, is a $(v-1,k-1,t-1)$-covering with $r$ blocks. It then suffices to list these links up to isomorphism, to prove the list complete, and to show that no link in the list extends to a covering with $L$ blocks. Completeness is certified by one unsatisfiable formula that excludes the listed classes, so the program that lists the links need not be trusted. In general the last two steps are decided by propositional satisfiability (SAT) solving, with every formula written by two separately implemented encoders and every refutation checked by a formally verified proof checker. For links with four blocks and $t=3$ we give the classification by hand, under a condition on the parameters. As an application we determine twenty-five covering numbers with $t\in\{3,4,5\}$ in which the published lower bound was one less than the size of a known covering, among them $C(39,24,3)=9$.

We establish sharp strong error estimates for a fully discrete approximation of semilinear stochastic convection--diffusion equations driven by additive noise subject to nonsmooth initial data. The problem is discretized in space using a standard Galerkin finite element method and in time via the linearly implicit Euler scheme. By decoupling the noise regularity index $\beta \in (0, 2]$ from the initial data regularity index $\mu \in (0, 2]$, we derive time-weighted spatial and temporal regularity bounds that isolate the initial singularity at $t = 0$ into an integrable temporal weight. By allowing the drift operator $F$ to map into negative-order fractional spaces to incorporate the convection term, we prove sharp strong convergence rates of order $O(h^\beta + k^{\beta/2})$ for $\beta \in (0, 2)$ and $O\bigl(h^2 + k(1+\ln(t_n/k))^{1/2}\bigr)$ at the critical endpoint $\beta = 2$ for any fixed $t_n > 0$, without requiring the regularity of the initial data to match that of the noise. Furthermore, these convergence rates are achieved under a relaxed differentiability condition requiring only a Lipschitz continuous first G\^ateaux derivative of $F$. Numerical experiments validate the theoretical results across both trace-class and space-time white noise regimes.

We study the Abelian sandpile model on a sequence of finite fractal graphs: the hierarchical, or Migdal--Kadanoff, lattices. Our first main result consists of matching power-law tail bounds for the avalanche and the uniform wave started at certain "pivotal" vertices. We also establish a limiting result for the pivotal reach of the avalanche and, finally, a scaling limit for the size of a uniform wave started at a vertex of high degree.

We prove that Atiyah duality holds in the $\infty$-category of non-$\mathbb A^1$-invariant motivic spectra over arbitrary derived schemes: every smooth projective scheme is dualizable with dual given by the Thom spectrum of its negative tangent bundle. The Gysin maps recently constructed by L. Tang are a key ingredient in the proof. We then present several applications. First, we study $\mathbb A^1$-colocalization, which transforms any module over the $\mathbb A^1$-invariant sphere into an $\mathbb A^1$-invariant motivic spectrum without changing its values on smooth projective schemes. This can be applied to all known $p$-adic cohomology theories and gives a new elementary approach to "logarithmic" or "tame" cohomology theories; it recovers for instance the logarithmic crystalline cohomology of strict normal crossings compactifications over perfect fields and shows that the latter is independent of the choice of compactification. Second, we prove a motivic Landweber exact functor theorem, associating a motivic spectrum to any graded formal group law classified by a flat map to the moduli stack of formal groups. Using this theorem, we compute the ring of $\mathbb P^1$-stable cohomology operations on the algebraic K-theory of qcqs derived schemes, and we prove that rational motivic cohomology is an idempotent motivic spectrum.

We prove that, for any five fixed strictly positive masses, the Newtonian central configurations of exact affine dimension three form finitely many similarity classes. The proof uses the Dziobek equations, two constant identities, and local analysis of zeros and poles on a compact complex curve.

Superlinear density-dependent viscosity presents a fundamental obstacle of global regularity to the compressible Navier-Stokes equations in the presence of far-field vacuum: the viscous diffusion vanishes at infinity, so excluding vacuum on bounded sets does not suffice to control the flow globally. For $\alpha\in (1,11.7)$ and $\gamma\in [2\alpha-1,3\alpha-1)$ , we establish global regularity for arbitrarily large, strong, finite-mass initial data with positive density at every finite point and vacuum at infinity. First, we introduce new rescalsed variables regarding the velocity, effective velocity and mass, then we exploit several novel quantities near the center and employ a refined maximum principle to establish the Lipschitz continuity of the velocity at the center-thereby resolving the singularity of the velocity gradient. Another key mechanism is a quantitative control of material volume distortion that remains effective in the vacuum tail. Conservative estimates for the physical and effective velocity gradients yield uniformly vanishing exterior variation tails. These estimates identify a material trajectory with small cumulative Jacobian variation and lead to two-sided bounds for the exterior volume Jacobian, making the evolving density comparable to its initial value along particle paths. On spatial scales determined by the initial density, the resulting degenerate equations admit uniform parabolic estimates that recover control of the physical deformation without a global density lower bound. Combined with compact positivity estimates and a one-sided Glaeser inequality, this closes a homogeneous continuation argument without imposing $L^2$-integrability on the density powers themselves. The solutions conserve mass, preserve far-field vacuum, and remain smooth and strictly positive on every compact spacetime cylinder.

Higher order sharp Sobolev and local Sobolev type inequalities for log-concave functions are established. The new inequalities directly imply the isoperimetric inequalities by Bollob\'as and Thomason in 1995, by Liakopoulos in 2019, and the recently discovered local isoperimetric inequalities from [1]. In order to do so, we make use of three ingredients. First, sets of functional inequalities proven by Alonso-Guti\'errez, Bernu\'es, Brazitikos and Carbery in 2021. Second, a multidimensional polynomial optimization result that originated in [1] and was initiated by Bollob\'as and Thomason. Third, a new integral formula for the projection of a log-concave function that links the geometric part with the more functional versions via some dependence on the Hessian matrix restricted to subspaces.

For an arbitrary congruence group $\Gamma$, we study the zeros of a generating set of the space of Eisenstein series of varying even weight for all $\Gamma(1)$-conjugates of $\Gamma$. These zeros form a $\Gamma(1)$-invariant set that descends to the modular orbifold $Y(1)$. It is known that, as the (even) weight tends to infinity, the set of zeros converges in the Hausdorff metric to finitely many geodesic segments. In this paper, we find the weak limit of the normalised counting measure of the zeros in the (natural) orbifold sense: if a geodesic occurring in the Hausdorff limit is parametrised by the unit circle $S^1$ via a M\"obius transformation then the limit measure on this segment is the pullback of the Haar measure on $S^1$ by that transformation. For the proof, we first show a version of the Poincar\'e--Lelong theorem for orbifold curves, and then apply it on $Y(1)$ to reduce the problem to computing the scaled limit of the subharmonic functions $\log |E|$ associated to Eisenstein series $E$. This turns out to be the maximum of the negative logarithm of the modulus of finitely many linear functions supported on the relevant geodesic segments, and we finish the proof by showing that the associated measure has no atoms and making explicit, on each open segment, the distributional Laplacian of this function in terms of the argument measure. The non-atomicity allows us to conclude a sublinear growth rate in the weight for the total order of all Eisenstein series at a given point in $Y(1)$, and weak convergence also of the non-orbifold measure to the same limit.

We provide a convergence result for sequences of random variables taking values in a metric space that satisfy a stochastic quasi-Fej\'er monotonicity condition, in the context of a (local) compactness assumption. Our result is quantitative in that we derive an explicit and effective construction which, in terms of only a few moduli representing quantitative witnesses to key properties of the sequence of random variables and the underlying metric space involved, provides a metastable rate of pointwise convergence, a type of generalized fluctuation bound. That quantitative result in particular relies on the development of a finitary theory of martingales, culminating in a fully finitary Robbins-Siegmund theorem. We outline how this result particularises to the circumstances of the seminal work of Combettes and Pesquet on stochastic quasi-Fej\'er monotone sequences in separable Hilbert spaces, and we provide an initial application by illustrating how these results can be used to provide a metastable rate of pointwise convergence for a stochastic Krasnoselskii-Mann scheme solving a stochastic common fixed point problem for nonexpansive maps over proper Hadamard spaces. This work is set in the context of recent applications of the logic-based methodology of proof mining to probability theory, and represents its most sophisticated case study to date.

We introduce a framework for analyzing and designing linear multirate control systems and their interconnections through the lens of robust control. The proposed approach models multirate systems as impulsive systems with multiple independent jump maps and corresponding jump-time sequences. We tackle analysis and design problems with stability, dissipativity, and robustness objectives, relying on a multi-clock-dependent Lyapunov function parameterization. Further, we develop a compositional approach for analyzing interconnections of multirate systems based on individual dissipativity properties. The theoretical conditions can be verified via linear matrix inequalities and they require weak assumptions on the sampling rates, allowing for asynchronous and time-varying sampling rates with or without separated time scales. The practicality of the framework is demonstrated with numerical examples.

The Module of Integral Absolute Complex Error (MIACE) was created by me 23 years ago. It gave very nice behaviour to the transient response of the controlled systems I tested. It combines absolute tracking-error and equilibrium-centered command areas. For a stable unit-gain plant with known pure delay, an exact identity separates squared MIACE excess into a negative-area penalty and a weighted area imbalance. The resulting lower bound depends on the plant's first low-frequency moment and the delay. Every attaining response of a positive cascade is monotone, independently of controller structure. For a first-order plant, output variation and peak overshoot also admit quantitative bounds. Exact conditions identify the unique positive PI controller attaining the delayed bound; filtered PID extends attainment beyond this PI range. For cascades of arbitrary order, Hermite interpolation and a finite-support delay remainder construct proper rational controllers of the same order, with internal exponential stability proved by positive convolution representations. An exponential-moment characterization then closes the attainment question when internal delay blocks are allowed: an ordinary command attains the bound if and only if $\alpha>\mu+\delta-1$. An explicit internal-model controller realizes every feasible weight and delay. A sharp amplitude-cap condition determines the least command peak among all attaining responses and the complete capped sign-preserving area front. A small-gain estimate supplies an explicit MIACE deviation bound under delay error. A storage balance also extends the exact MIACE identity and monotone attainment to a nonlinear first-order class without delay. Independent physical integrations retain delays exactly; quadratic comparisons provide complementary benchmarks.

We study stationary fronts for a bistable reaction-diffusion equation with the McKean reaction on a periodic necklace metric graph. For profiles that agree on each pair of parallel edges, the problem reduces to a scalar equation on the real line with a periodic, piecewise constant diffusion coefficient. The diffusion contrast of the necklace is not small, so the coefficient cannot be treated as a small perturbation of a constant. Because the McKean reaction is piecewise linear, the equation is affine between threshold crossings and coefficient interfaces, and a front is determined explicitly by its crossing points. We find an explicit interval of reaction thresholds, the pinning interval, such that stationary fronts, monotone or not, exist exactly for thresholds in this interval. Up to translations by the spatial period, there are exactly two decreasing fronts at each interior threshold and one front at each endpoint. For each fixed interior threshold, one decreasing front is Lyapunov stable and the other is unstable in the $W^{1,\infty}(\mathbb{R})$ norm, among mild solutions with one continuously moving threshold crossing. The crossing phases of the two decreasing fronts also determine nonmonotone fronts. For each interior threshold and odd $m\geq3$, choose $m$ periods of the coefficient that are sufficiently far apart. There are then exactly $2^m$ fronts with one threshold crossing in each chosen period and none elsewhere. Varying the chosen periods gives infinitely many inequivalent fronts with $m$ crossings.

We study reduction and reconstruction of reflected backward stochastic differential equations (RBSDEs) across random horizons. We establish optional and predictable contact-measure transfer criteria by dual projection, prove a Cox survival identity for arbitrary integrable reflection measures (including atoms), and give a Brownian Cox reconstruction theorem with continuous obstacles and a converse within a specified reduction class. We also isolate componentwise minimality criteria for irregular obstacles and predictable reflection. The unresolved semimartingale and obstacle transport problems for general non-Cox invariance times are formulated as open questions rather than asserted as theorems. The paper distinguishes established results from conditional constructions.