数学前沿

An identifying code of a graph is a dominating set $C$ such that the sets $N[v]\cap C$ are pairwise distinct. The minimum density of an identifying code has been studied extensively for several infinite grids. It is known to be $1/4$ for the triangular grid and $7/20$ for the square grid, whereas for the hexagonal grid the problem is still open, between $5/12$ and $53/126$. These three grids are the regular members of the family of the eleven Archimedean grids. In this paper, we start a systematic study of identifying codes in this family. We determine the minimum density for five of the nine Archimedean grids for which it was not known: it is $2/3$ for the truncated hexagonal grid 3.12.12, $3/7$ for the truncated square grid 4.8.8, and $1/3$ for the trihexagonal (kagome) grid 3.6.3.6, the elongated triangular grid $3^3.4^2$ and the snub hexagonal grid $3^4.6$. All lower bounds are proved by the discharging method. For the three remaining non-regular Archimedean grids, we also obtain some bounds: the minimum density lies between $3/7$ and $13/30$ for the truncated trihexagonal grid 4.6.12, between $1/3$ and $11/30$ for the rhombitrihexagonal grid 3.4.6.4, and between $2/7$ and $9/28$ for the snub square grid $3^2.4.3.4$.

Estimating a camera pose from 3D-2D correspondences - Perspective-n-Point - is one of the most fundamental problems in computer vision. Existing methods either solve a minimal problem or replace the objective with an easier one. Surprisingly, no scalable certifiable solver for the actual Maximum Likelihood Estimation (MLE) exists. We propose PnP*, the first practical solver that guarantees efficient computation and returns a certifiable globally optimal solution. We prove that the underlying convex relaxation is exact under moderate noise, which to our knowledge is the first stability guarantee for any relaxation for the MLE of PnP. Furthermore, we reduce the relaxation by exploiting chordal sparsity pattern and quotient ring reduction such that the relaxation is tractable and scalable. The stability and scalability results are verified numerically, as PnP* returns a provably correct camera pose at over 10 Hz.

We study \emph{prismatical} polytopes, those all of whose facets are combinatorially isomorphic to the simplicial prism $\Delta^{d-2}\times\Delta^1$. In dimension three these are exactly the cubical polytopes, so the class sits naturally between simplicial and cubical polytopes. In dimension four, $\Delta^2\times\Delta^2$ is the simplest prismatical polytope and we can construct infinitely many by using connected sums. By fibreing a triangle over a triangulated polygon we construct infinitely many connected-sum-prime prismatical $4$-polytopes. Finally, we show that there are no prismatical $d$-polytopes for $d\ge5$, answering a question of Perles and Shephard.

We construct closed smooth $4$-manifolds with finite fundamental group which are homotopy equivalent but not simple homotopy equivalent, even after taking a connected sum with copies of $S^2\times S^2$. Our explicit examples have remarkably large fundamental groups; in particular, the smallest examples we obtain have fundamental groups of order roughly $10^{10^{42}}$.

A minimal finite model of a topological space is a finite $T_0$-space of least cardinality that is weak homotopy equivalent to it. We determine, up to homeomorphism, the minimal finite models of eight wedges of one- and two-dimensional spheres. The spaces $S^{2}\vee S^{1}$ and $S^{2}\vee S^{2}$ have exactly two and three minimal finite models, respectively, all with seven points, whereas $S^{1}\vee S^{1}\vee S^{2}$ has exactly seven minimal finite models, $S^{1}\vee S^{1}\vee S^{1}\vee S^{2}$ has one, $S^{1}\vee S^{2}\vee S^{2}$ has six, $S^{2}\vee S^{2}\vee S^{2}$ has five, $S^{2}\vee S^{2}\vee S^{2}\vee S^{2}$ has three, and $S^{1}\vee S^{1}\vee S^{2}\vee S^{2}$ has two, all with eight points. To prove this we classify the connected finite $T_0$-spaces of height $2$ without beat points that have seven or eight points, and we identify their weak homotopy types by means of McCord's theorem and explicit collapse and suspension arguments for their two-dimensional order complexes. Together with the known results for spheres and wedges of circles, this determines, up to weak homotopy equivalence, all path-connected spaces whose minimal finite models have at most eight points.

We develop stability and error estimates for multistep convolution quadrature (CQ) under general radial bounds on the Laplace symbol. Instead of the classical single-power condition $|F(z)|\leq M|z|^{-\mu}$, we assume $|F(z)|\leq C\Phi(|z|)$, where $\Phi$ is positive, continuous and regularly varying at zero and infinity. For strongly $A(\alpha)$-stable multistep methods, we derive error estimates and stability estimates for the CQ weights, as well as error bounds for power-type inputs $g_\beta(t)=t^{\beta1}/\Gamma(\beta)$ with $\beta>0$. These estimates explicitly retain the dependence on the memory law through the factor $\Phi(t_n^{-1})$. The framework accommodates symbols whose behavior cannot be described by a global single-power. It also preserves logarithmic and multiscale factors that would be lost under a coarser power bounds. Then we apply the to fractional viscoelastic, distributed-order, and logarithmic-power memory models. Numerical experiments with backward Euler and second-order backward differentiation CQ confirm the expected temporal convergence orders for direct responses. A distributed-order PDE further illustrates temporal convergence for smooth and nonsmooth initial data.

Decoupling inequalities for the parabola are obtained for weights in physical space. The decoupling iteration relies on a fractal bilinear Kakeya estimate. We record applications to exponential sums.

Let $\mathcal{P}$ be a finite collection of cells. Two non-attacking rook configurations on $\mathcal{P}$ are switching-equivalent if one can be obtained from the other by a sequence of switches inside rectangles contained in $\mathcal{P}$. The switching rook polynomial of $\mathcal{P}$ enumerates the resulting equivalence classes according to the number of rooks. We characterize the collections of cells whose switching rook polynomial is palindromic. More precisely, we introduce a new combinatorial property, called domino-stability, and we prove that the switching rook polynomial of $\mathcal{P}$ is palindromic if and only if $\mathcal{P}$ is domino-stable.

In this paper, we study the global injectivity of local diffeomorphisms in $\mathbb{R}^n$, extending the criterions of Braun and Llibre in [{\it Pacific J. Math.} {\bf 333} (2024) 1--15] from the planar radial case to arbitrary dimension. Then we extend the results of Llibre and Mello in [{\it Bull. Sci. Math.} {\bf 213} (2026) 103889] to arbitrary dimension, obtaining two necessary and sufficient conditions for the real Jacobian conjecture in $\mathbb{R}^n$ by qualitative theory of differential equations. Finally, using weighted Poincar\'e compactification in $\mathbb{R}^n$, we provide a sufficient condition for the real Jacobian conjecture.

Takemura conjectured that the Conway polynomials $\sum a_\ell z^\ell$ of the Turk's head link $Th(p,q)$ and $\sum b_\ell z^\ell$ of $Th(q,p)$ satisfy $a_\ell=b_\ell$ for $\ell\equiv0,1\pmod 4$ and $a_\ell=-b_\ell$ for $\ell\equiv2,3\pmod4$, and proved it for $q=2,3$. We prove it for all $p,q\ge1$, knots and links. Equivalently, $\nabla_{Th(q,p)}(z)=c\,\nabla_{Th(p,q)}(iz)$ with $c=1$ if $\gcd(p,q)$ is odd and $c=-i$ if it is even. The proof combines a product formula $\nabla_{Th(p,q)}(z)^2=\prod_{j=1}^{p-1}\prod_{k=1}^{q-1}(z^2+2\cos\frac{2\pi k}{q}-2\cos\frac{2\pi j}{p})$, which comes from the reduced Burau representation, with the leading coefficients, read off the Seifert surface of the closed braid.

We study the quantitative codimension-two estimate for the singular set in the boundary neighborhoods of the solutions of $$\begin{equation*} \Delta u+V(x)u=0\qquad\text{in }\Omega, \qquad u=0\qquad\text{on }\partial\Omega, \end{equation*}$$ where $\Omega\subset\mathbb R^n$ is a bounded $C^{1,\mathrm{Dini}}$ domain and $V\in L^p(\Omega)$ for some $p>n$. We first prove the explicit upper bound for the doubling index is given by $C(n,p,\Omega)(1+\|V\|_{L^p(\Omega)}^{\frac{2p}{3p-2n}})$.The analytic input is an interior volume estimate for solutions of second order elliptic equations with uniformly elliptic Dini leading coefficients and $V\in L^p$. Using the quantitative doubling index bound, boundary flattening, we show an explicit upper bound for singular sets in the neighborhood of the boundary of the $C^{1,\mathrm{Dini}}$ domain.

We consider a near-critical version of loop-erased random walk (LERW) by boosting the weight of the underlying random walk according to its length before loop-erasure. We construct the scaling limit in natural parametrization in the two-dimensional unit disk and the three-dimensional unit ball. The boosted scaling limit undergoes a phase transition in mutual absolute continuity with the classical LERW scaling limit; in the disk the transition occurs at boost rate $m^2=\pi^2/4$. As the boost increases, the curve becomes macroscopically space-filling, in the sense that unvisited cubes have side length $O(m^{-1})$, while its Hausdorff dimension at each finite boost is unchanged. These results are motivated by Makarov--Smirnov's questions on near-critical lattice models. In the planar chordal setting, we also construct a continuum loop tilt below the spectral threshold of the domain and calculate its Loewner driving function.

Let $1\le h\le n-2$ and $g=\gcd(n,h+1)$. We prove that, for every prime power $q$, a maximum $h$-scattered $\mathbb{F}_q$-subspace of $V(r,q^n)$ exists whenever $(h+1)\mid rn$ and $$r\ge\frac{h+1}{g}\bigl(h(n-h-2)+1\bigr).$$ In particular, for fixed $n$ and minimum distance $\delta\in\{2,\ldots,n-1\}$, $\mathbb{F}_{q^n}$-linear MRD codes of length $N$ exist for every $q$ and every sufficiently large $N$ such that $n$ divides $(n-\delta+1)N$, with a bound on $N$ depending only on $n$ and $\delta$. We take the $\mathbb{F}_{q^s}$-span of the columns of a random matrix $A$ over the algebra $R=\mathbb{F}_{q^n}\otimes_{\mathbb{F}_q}\mathbb{F}_{q^s}$, for a suitable $s$. Such a subspace is maximum $h$-scattered if and only if the kernel of $A$ contains no nonzero vector whose expansion with respect to an $\mathbb{F}_q$-basis of $\mathbb{F}_{q^s}$ has rank weight at most $h+1$. When $\gcd(n,s)=1$, $R$ is a field, and a first-moment argument, counting these vectors up to multiplication by $\mathbb{F}_{q^n}^*\mathbb{F}_{q^s}^*$, shows that with positive probability the kernel contains none of them if $s>h(n-h-2)$. When $d=\gcd(n,s)>1$, $R$ is a product of $d$ fields, and vectors vanishing in some factors lie in the kernel with larger probability. We control them by proving that, for an $(h+1)$-dimensional $\mathbb{F}_q$-subspace $D$ of $\mathbb{F}_{q^n}$, the elements of $D\otimes_{\mathbb{F}_q}\mathbb{F}_{q^s}$ vanishing outside $k$ given factors, $0<k<d$, form an $\mathbb{F}_{q^s}$-space of dimension at most $(h+1)k/d$, with equality only if $D$ is linear over a subfield of $\mathbb{F}_{q^n}$ larger than $\mathbb{F}_q$. Taking $s=gr/(h+1)$ gives the bound on $r$. Finally, if $(n-h-1)\mid(h+1)$, then for every $r$ satisfying the above inequality there is an $h$-scattered $\mathbb{F}_q$-subspace of $V(r,q^n)$ of dimension $\lfloor rn/(h+1)\rfloor$, without a divisibility condition on $rn$.

For a graph $G = (V, E)$ with a vertex set $V$ and an edge set $E$, a function $f : V \rightarrow \{0, 1, 2, . . . , diam(G)\}$ is called a \emph{broadcast} on $G$. For each vertex $u \in V$, if there exists a vertex $v$ in $G$ (possibly, $u = v$) such that $f (v) > 0$ and $d(u, v) \leq f (v)$, then $f$ is called a dominating broadcast on $G$. The cost of the dominating broadcast $f$ is the quantity $\sum_{v\in V}f(v)$. The minimum cost of a dominating broadcast is the broadcast domination number of $G$, denoted by $\gamma_{b}(G)$. A multipacking is a set $M \subseteq V$ in a graph $G = (V, E)$ such that for every vertex $v \in V$ and for every integer $r \geq 1$, the ball of radius $r$ around $v$ contains at most $r$ vertices of $M$, that is, there are at most $r$ vertices in $M$ at a distance at most $r$ from $v$ in $G$. The multipacking number of $G$ is the maximum cardinality of a multipacking of $G$ and is denoted by $mp(G)$. It is known that $mp(G)\leq\gamma_b(G)$. In 2014, Hartnell and Mynhardt proved that $\gamma_b(G)\leq 3mp(G)-2$ whenever $mp(G)\geq2$. In 2019, Beaudou, Brewster, and Foucaud improved this bound to $\gamma_b(G)\leq 2 mp(G)+3$ and conjectured that $\gamma_b(G)\leq 2 mp(G)$. We solve their conjecture by proving that $\gamma_b(G)\leq 2 mp(G)$ for every graph $G$. Our proof is constructive and yields a linear-time $2$-approximation algorithm for Maximum Multipacking problem which improves the earlier approximation factor $2+o(1)$.

We show that multifractional Brownian motions (mBms) can be lifted to geometric rough paths. One of the main ingredients in the proof is a decomposition of mBms via the Lemari\'e-Meyer orthogonal wavelet basis, which is introduced in the context of multiresolution analysis. As a consequence, we are able to construct solutions to rough differential equations driven by mBms.

We prove relative Langlands duality for $X = GL_{p+q}/(GL_p \times GL_q)$. These symmetric spaces arise in the setting of the linear period of Friedberg-Jacquet when $p=q$, and exhibit non-tempered behavior when $q>p+1$. Via the real-symmetric correspondence, we establish a derived Satake equivalence for the real group $G_{\mathbb R}=U(p,q)$, and use it to obtain a constructible realization of coherent sheaves on Kato's exotic nilpotent cones. Along the way, we describe the dual variety of $X$ in terms of the Whittaker cotangent bundle and regular centralizers, viewed as Coulomb branches.

We characterize the generating function of bipartite planar maps counted according to the degree distribution of their black and white vertices. This result is applied to the solution of the hard particle and Ising models on random planar lattices. We thus recover and extend some results previously obtained by means of matrix integrals. Proofs are purely combinatorial and rely on the idea that planar maps are conjugacy classes of trees. In particular, these trees explain why the solutions of the Ising and hard particle models on maps of bounded degree are always algebraic.

Combinatorial optimization is a leading target for quantum algorithms, but finite-depth QAOA can suffer from strong diabatic errors when the interpolation Hamiltonian has small, or exponentially small, spectral gaps. We propose a Pauli-sparse counterdiabatic extension of linear-ramp QAOA based on the regularised adiabatic gauge potential $$\bigl(\mathcal L_H^2+\eta I\bigr)A_\lambda^{(\eta)} = -\mathrm{i}\mathcal L_H(\partial_\lambda H), \qquad \mathcal L_H(X)=[H,X].$$ Instead of computing a dense AGP, we solve this equation approximately by an inexact conjugate-gradient method in Pauli coordinates, truncating the Pauli expansion during the iteration to obtain a gate-budget-aware set of implementable rotations. The selected support is then improved by a Galerkin refit and certified by an a posteriori residual bound. The regularization parameter $\eta$ acts as an energy-resolution scale: it suppresses transitions below $\sqrt{\eta}$ while retaining larger-gap transitions. Thus, the method can avoid resolving exponentially small splittings inside a low-energy solution manifold while reducing leakage away from it. Numerical experiments on Ferromagnetic Chain (FC) and perturbed FC--MaxCut/MarketSplit instances show that the resulting LR-CD-QAOA ansatz improves approximation ratios over the uncorrected linear ramp, especially in regimes where LR-QAOA remains far from the optimum. Overall, the proposed regularized LR-CD-QAOA framework substantially broadens the practical applicability of QAOA to QUBO optimization by improving its robustness across heterogeneous problem landscapes, including instances with near-degenerate low-energy structures and small spectral gaps.

We reformulate average-cost (ergodic) control of a Markov jump process as a finite-horizon stochastic optimal transport (SOT) problem that jointly optimizes over the controlled evolution and the marginal law from which it starts and returns to (i.e., a self-transport). The constraint relating the marginal flow to the controlled generator is the basic adjoint relationship (BAR), so we call the resulting problem BAR-SOT. For any time horizon T>0, its optimal value is a constant scaling of the long-run average-cost rate. We give three equivalent formulations (through controlled processes, a Fokker--Planck constraint, and relaxed marginal measures) and show that the optimal dual is a stationary potential plus a term linear in time, with slope equal to the rate. A relative-entropy penalty on the control yields a cost-tilted Schrodinger bridge problem, computable by a Sinkhorn-type iteration when every transition rate is controlled, whose value converges to the unregularized optimum as the penalty vanishes. We develop the theory for finite Markov decision processes and then for general controlled Markov jump processes. A neural parametrization of the dual, trained as a physics-informed neural network (PINN), that encodes Harrison's equivalent-workload formulation (see Harrison 2000) matches the strong reinforcement-learning baseline of Dai and Gluzman (2022); numerically conditioning the fit of a sub-dominant transverse correction then improves on both that baseline and the best priority heuristic. On the input-queued switch, a graph-attention parametrization of the dual improves on the strongest matching heuristic we are aware of, with a parameter count that does not grow with the switch size.

SHACL shapes enable data graph validation, making automatic shape learning essential for knowledge graph applications. We investigate the well-known fitting approach to this task: given sets P and N of positive and negative example nodes from an input graph, compute a shape expression C, possibly using shape names defined in a recursive shape catalogue, that validates at every node in P and none in N. We focus on the case where C is written in a core fragment of SHACL corresponding to the Description Logic ELI*. For the catalogue, we consider the well-founded, stable, and supported semantics. We address fitting existence and most specific fitting computation, establish tight exponential-time upper bounds for both problems, and obtain polynomial bounds for relevant special cases.