数学前沿

We give a complete characterization of the kernels $\Omega\in \Lup^1(\Sp)$ for which the rough singular integral $\T_\Omega$ and the associated maximal operator $\M_\Omega$ satisfy pointwise bounds by $\Int_1(|\nabla \cdot|)$ and $\M_1(|\nabla \cdot|)$, respectively, where $\Int_1$ is the Riesz potential and $\M_1$ is the fractional maximal operator. These bounds hold if and only if $\Omega$ belongs to the homogeneous dual Sobolev space $\dot \W^{-1,\infty}(\Sp)$. Equivalently, $\Omega=\text{div}_\Sp \mathbf{V}$ for some bounded vector field $\mathbf{V}$ on the sphere, or $\big|\int_E\Omega\big|\leq C\,\mathcal H^{d-2}(\partial E)$ for every open set $E\subseteq\Sp$ with smooth boundary. On the Lebesgue scale, we show that for $d\geq3$ the condition $\Omega\in\Lup^{d-1,\infty}(\Sp)$, together with mean zero, suffices, and that the exponent $d-1$ is sharp. Finally, in dimension $d=2$, we use complex-analytic methods to prove a new pointwise bound in terms of the complex derivative $\bar\partial f$ that is valid for every mean-zero $\Omega\in\Lup^1(\mathbb S^1)$. As a consequence, we obtain new Sobolev bounds for rough singular integrals in the plane.

Given a dual Coxeter system (W, T) of rank n, the Hurwitz action of the braid group on the reduced T-decompositions of a Coxeter element is transitive. When we choose a geometric representation of W, the reflections with respect to the n walls of one given chamber form a generating set S that we call cameral. An ordering of S is called a cameral decomposition and the product a cameral Coxeter element, which depends only on the quiver obtained by orienting the Coxeter diagram with vertices S according to the ordering. We show that the reduced decompositions of a cameral Coxeter elements for which the underlying set S satisfies that (W, S ) is a Coxeter system, are necessarily cameral, and we count the number of chambers from which a cameral Coxeter element can be obtained. We show that one can go through all these chambers using the restricted Hurwitz action on the set of cameral decompositions, which is generated by transpositions of adjacent commuting reflections and by the Garside element {\delta} of the dual Garside structure, which acts by a quiver mutation of the orientation of the Coxeter diagram.

We present an incidence-based method for product bounds on cross-intersecting families in finite graded meet-semilattices. The two families may lie in different rank or type fibres and may have different design strengths. The criterion separates global incidence parameters from two directional local counts and yields the sharp product bound, with equality only for full stars with a common centre. Applications include mixed subspace types relative to a fixed subspace, a type-restricted generator family in $Q^-(7,q)$, and mixed form ranks in symplectic spaces. The same short criterion gives several sharp results not obtained directly from the specific theorems and general criteria compared here. In the singular-space case, it also applies in an infinite parameter range beyond the natural star-ratio and weak-spreadness tests.

Contextual decision making uses observed covariates to guide decisions under uncertainty. Two natural approaches are predict-then-optimize (PtO), which learns the conditional expected objective, and generate-then-optimize (GtO), which learns conditional uncertainty and optimizes using generated scenarios. We study when each approach is preferable by examining how the choice of learning target affects statistical accuracy and optimization tractability. Although PtO learns a narrower target, our regret bounds show how differences in smoothness can offset GtO's additional dimensional burden. Exploiting smoothness also raises an optimization tradeoff: within the class of nonnegative-weight PtO methods studied, a lower bound limits the gains from additional smoothness, while more flexible methods may introduce signed weights and lose the guarantee of preserving convexity. GtO preserves scenario-wise convexity, but requires learning a richer object and generating scenarios for optimization. Synthetic experiments and a real case study illustrate how these tradeoffs vary with data availability and problem structure. Together with the scope for reusing a model across tasks, our findings provide practical guidance for choosing between the approaches without identifying a universal winner.

This paper counts curves, and multicurves, on hyperbolic surfaces, providing explicit upper bounds that depend polynomially on length and exponentially in terms of an upper bound on self-intersection.

In this paper, we study the asymptotic behaviour for distribution of supercritical branching process in a Markovian environment with linear fractional offspring distributions. Phase transitions in the behaviour of the process is discovered, through investigating the strongly, intermediately and weakly supercritical cases.

In 1992, \L uczak, Ruci\'{n}ski and Voigt showed that for any graph $H$, a parameter $m_1(H)$ - called the 1-density - governs the threshold at which the binomial random graph $G(n,p)$ inherits the vertex-Ramsey property. In 2000, Friedgut and Krivelevich showed this threshold is sharp for graphs satisfying a natural balancedness condition. In 1996, Kreuter showed that for any graphs $H_1, \ldots, H_r$ with $m_1(H_1) \geq \cdots \geq m_1(H_r)$, a parameter dependent only on $H_1$ and $H_2$ governs the threshold at which $G(n,p)$ inherits the `asymmetric' vertex-Ramsey property. We prove this threshold is sharp for graphs respecting natural balancedness conditions, including in the analogous hypergraph setting.

The interest in studying the size of the intersection of multiple $q$-ary Hamming balls has grown due to the recent advances in DNA-based data storage systems. We present an exact formula for the cardinality of the intersection of $s$ Hamming balls of varying radii over a $q$-ary alphabet. It is known that the distances between the center points of the Hamming balls are not enough, in general, to determine the size of the intersection. Based on our formula, we are able to find more refined structural properties of the center points for determining the exact size of the intersection. Moreover, we also analyze the size of the intersection of Hamming balls with equal radii $t$ for sufficiently large $n$ when $t$ and $q$ are constants. When $s=3$, we give the necessary and sufficient conditions (for all $q\ge 2$, $q\neq 6$ and sufficiently large $n$) to obtain the maximum size of the intersection when the centers of the Hamming balls have a given minimum distance and demonstrate how to compute it using our general formula.

We study the fixed-$q$ right-edge asymptotics of the ordinary $q$-Krawtchouk ensemble. For $0<q<1$, finite polynomial duality gives a direct proof of convergence of the correlation kernels to an explicit limiting kernel. The local Jacobi limit is a shifted continuous $q^{-1}$-Hermite expression with an indeterminate moment problem, so the limiting Jacobi expression does not determine a unique self-adjoint operator. A single spectral atom, asymptotically saturating the mass bound, selects the realization and yields strong-resolvent convergence. The limiting kernel represents its positive spectral projection and admits spectral-series, Christoffel--Darboux, and partial-theta representations. After a ground-state transformation, the associated dual operator agrees with a two-step spatial Markov operator of Borodin--Corwin. For $q>1$, the limiting Jacobi operator is of Al-Salam--Carlitz~I type and is essentially self-adjoint.

Let $K$ be a number field of degree $n$ and discriminant $\Delta$. The estimate $|\zeta_K(1+it)|\le C_K\log{t}$, where $C_K$ is an unspecified constant depending on $K$, is a well--known result in the literature. We obtain an explicit upper bound for $|\zeta_K(1+it)|$, depending only on $n$ and $\Delta$. We also show how improvements on the bound depend on improvements on estimates of the ideal counting function $I_K$ and the residue of $\zeta_K$ at $s=1$.

We provide an example of a normalized absolutely continuous probability whose location-scatter family (LSF) does not contain all the 2-Wassertein geodesics passing through its elements. In particular, we find a counterexample for Corollary 4.5 in [2]. Finally, we prove that LSFs with a (possibly singular) radially symmetric generator are closed under 2-Wasserstein barycenters.

Wordle result prediction involves both daily participation and the difficulty of the solution word. We model these quantities with ARIMA forecasts of report counts, XGBoost regressors using word frequency, letter entropy, and repetition, and a decision tree trained on K-means difficulty tiers. Experiments use 355 daily outcome distributions, with a 107-word chronological holdout and four expanding-window tests. XGBoost reduces holdout mean absolute error from 4.08 to 3.84 percentage points and improves over the training mean in every temporal window. Ridge and random forests remain competitive. XGBoost systematically underestimates failure rates on difficult words. The lexical decision tree achieves 54.2% tier accuracy, compared with 40.2% when predicted histograms are assigned to the same training centroids. Bootstrap analysis shows that cluster stability does not imply that tiers are predictable from lexical attributes. For report counts, comparisons at matched horizons reveal substantial overdispersion in full-history ARIMA intervals. These results characterize the predictive value and limitations of a compact lexical model of reported Wordle performance.

Let $M_\psi$ be the maximal operator associated to $\psi(x)=-\ln(1-|x|^2)$. We obtain dimension-free bounds on $L^p(\mathbb R^d)$ for $M_\psi$, $1<p\le\infty$. We show that the weak $L^1$ estimate does not hold for $M_\psi$, but does hold for the corresponding lacunary maximal operator $M_\psi^D$. An weak $L\log L$ type estimate for $M_\psi$ is obtained, which seems to be the right substitution for the weak $L^1$ estimate. Estimates for $M_\psi$ are applied to dimension-free estimates for maximal Riesz transforms obtained by Mateu, Verdera, Orobitg and P\'{e}rez.

Interactions between ultrasound and microbubbles underlie a range of biomedical and industrial applications, yet predicting how the physical parameters of the medium and bubbles govern nonlinear wave propagation remains difficult. This study proposes a parameter estimation framework that combines a Korteweg-de Vries-Burgers (KdVB) equation, describing weakly nonlinear wave propagation in a bubbly liquid, with a physics-informed neural network (PINN). The contributions are threefold. First, the PINN reproduces the reference solution of the governing equation in the forward problem, with a near-zero prediction error at the initial time that confirms the initial condition is enforced exactly as intended. Second, prior to inverse estimation, a sensitivity analysis is introduced that quantifies, at both the coefficient and solution levels, how each of eight physical parameters influences the KdVB equation, allowing the expected difficulty of estimating each parameter to be assessed in advance. Third, based on this assessment, single-parameter inverse estimation is performed for seven of the eight parameters using the PINN. Liquid viscosity, bubble initial radius, liquid density, sound speed, and liquid pressure are estimated with errors of 0.8-2.1%, the polytropic exponent converges only partially, and the surface tension coefficient cannot be estimated at all, consistent with the sensitivity analysis. These results establish a general framework for identifying and estimating the physical parameters governing nonlinear wave propagation in bubbly liquids from observed waveform data.

An explanation may exist in every complete context yet be impossible to select from the information observed. We study this distinction in finite propositional Horn abduction with integrity constraints. Observations are sufficient when each report admits one hypothesis set that is consistent and entails the manifestation in every compatible context. We characterize sufficiency by minimal context families with no common explanation. Their variable spans form a canonical hypergraph whose transversals are exactly the sufficient observation sets. The characterization recovers classical discernibility for unique explanations and supports composition across independent components. Minimal obstructions have unbounded cardinality even for acyclic rules, while rectangular explanation families admit a tight bound given by their largest choice domain. With circuit-defined contexts, sufficiency is $\Pi_3^P$-complete and selecting at most $k$ observable variables is $\Sigma_4^P$-complete, already for unrestricted contexts and acyclic rules. The selection reduction makes observation choices enforce globally fixed existential commitments. Explicit contexts reduce both tasks to NP-complete problems; tabulated explanation families make sufficiency polynomial. These results connect explanatory choice, information requirements, and representation cost.

Simplified spectral models predict dimension-dependent superlinear convergence for gradient methods, but the corresponding behavior of actual iterations remains only partially understood. Historical methods introduce an additional difficulty: the gradients defining the step sizes differ from those receiving them. We study this offset for cyclic reuse of historical Ritz step sizes on strictly convex quadratics. Within the existing limited memory steepest descent with retards (LMSDR) family, a rolling cyclic Ritz (RCR) rule reuses $m$ historical Ritz values $\ell$ times per cycle, ending with a complete sweep. At $q=m+1$ active distinct eigenvalues, an exact delayed recurrence gives almost-everywhere doubly exponential convergence with an explicit rate and exceptional weights. Nonempty open basins also exist for $m+1<q\le2m$ and $\ell\ge4$. Rebuilding the history from each cycle-start gradient costs $m$ additional updates and gives sampled cyclic Ritz reuse (SCR) a cycle-end map determined by the current weights. Under uniform sampling bounds, SCR converges doubly exponentially almost everywhere when $(2\ell-1)^m>\binom{q-1}{m}$. Increasing $\ell$ admits larger $q$ for fixed $m$. The guarantee extends to history-dependent choices from finite sampling menus under a bound on the number of admissible sampling sequences. Geometrically spaced decisions retain the SCR condition and rate range. Every update is counted, including sampling; a numerical illustration compares finite-tolerance update counts.

The hard-core model can be used to understand the number of independent sets in graphs in extremal graph theory. The occupancy and variance fractions are the expectation and variance, respectively, of the independent-set size, normalized by the number of vertices. We provide upper bounds on the occupancy and variance fractions of linear $d$-regular $r$-uniform hypergraphs for small positive fugacities. In proving these results, we use the cluster expansion and the abstract polymer models, two tools from statistical physics, to analyse auxiliary functions associated with independence polynomials, comparing their low-order coefficients and controlling the remainders uniformly in the number of vertices.

The higher Bruhat orders are partial orders originally constructed by Manin and Schechtman, with an equivalent construction from Ziegler. These partial orders generalize the weak order on the symmetric group $\mathfrak{S}_n$, and the second higher Bruhat order is a poset on commutation classes of reduced words for the longest element in ${\mathfrak{S}}_n$, where covering relations correspond to braid relations. Like the weak order, each of the higher Bruhat orders is a ranked poset with unique minimal and maximal elements. Constructing analogs of the higher Bruhat orders in other settings is an area of recent interest, including recent work of Hothem for permutations in $S_{n}$. We give a construction that generalizes any interval $[\mathrm{id},w]$ in the weak order where $w$ is an element of the affine symmetric group $\widetilde{\mathfrak{S}}_n$. Our construction recovers the classical higher Bruhat orders when $w$ is the longest element in $\mathfrak{S}_n$. Paralleling the classical case, we show that the second higher Bruhat order is a ranked poset on commutation classes of reduced words for $w\in \widetilde{\mathfrak{S}}_n$, allowing us to prove a conjecture due to Elias. When $w\in \mathfrak{S}_n$, we also establish results for our higher Bruhat orders that are direct analogs of numerous results in the classical higher Bruhat orders.

The existence of sets of balanced incomplete block (BIB) designs with pairwise additivity, called pairwise additive BIB designs, has been studied through direct and recursive constructions in the literature. This paper presents a new method of constructing such designs from affine resolvable semi-regular group divisible (ARSRGD) designs. Since many known ARSRGD designs are obtained from generalized Hadamard matrices, we also describe the corresponding construction of pairwise additive BIB designs from generalized Hadamard matrices. Using the construction together with known existence results for generalized Hadamard matrices, we obtain several new infinite series of pairwise additive BIB designs, including designs with parameter sets whose existence was previously unknown.

We analyse the boundary behaviour of the gradient of the torsion function $w_\Om$ in convex domains $\Om\subset\mathbb{R}^d$. We prove that its $L^\infty$-norm is attained at a boundary point and establish the stability of this norm under general (convex) perturbations. For problems of the form $$\sup\{\|\nabla w_{\Om}\|_\infty: \Om \subset \R^d,\textup{open, bounded and convex}, G(\Om)=m\},$$ where $G$ is either volume or perimeter, we show the existence of maximisers which are $C^1$-smooth, rotationally symmetric about an axis and contain a $(d-1)$- dimensional ball (of estimated size) in their boundary. Other constraints, whether geometric (such as diameter or circumradius) or variational (such as torsion or first Dirichlet eigenvalue), are also briefly discussed, and some of their characteristic behaviours are highlighted.