数学前沿

For $d,n\in\mathbb{Z}_{\geq1}$ and $p\in[1,\infty)\setminus\{2\}$, we characterize all linear maps $s\colon\ell_d^p\to\ell_n^p$ for which there is $E\subseteq X_d$ such that $\|s(\xi)\|_p=\|\xi|_E\|_p$ for every $\xi\in\ell_d^p$. We call these maps \emph{restricted isometries}. Our Lamperti-type theorem gives a unique description of each such operator using a set transformation, a positive amplitude function, and a unimodular phase function. In particular, it includes noninjective and nonsurjective operators, even when the dimensions differ. We further characterize restricted isometries between finite $\ell^p$-spaces with arbitrary positive atomic weights. Applications include formulas counting set transformations and uniform restricted isometries, as well as a geometric description of all restricted isometries with fixed column supports.

In this article we give an algorithm to produce infinitely many explicit quadratic twists of a given elliptic curve over $\mathbb{Q}$ with a rational point and fixed root number. Moreover, under the Parity Conjecture (or other technical hypotheses), for each $n\geq 1$, we produce explicit multi-quadratic extensions $L_n$ of $\mathbb{Q}$, of degree $2^n$, such that the growth in rank from $\mathbb{Q}$ to $L_n$ is at least $\max\{2n+2^{n-1}-1,\,2^n-1\}$ when the root number of $E$ is $-1$, and at least $\max\{2n,\,2^{n-1}\}$ when it is $+1$. As a result, assuming Parity, we exhibit elliptic curves over $\mathbb{Q}$ and extensions of degree $2^n$ where the rank after base-extension is greater than or equal to $34$ or $30+\max\{2n+2^{n-1},\,2^n\}$.

We classify two-species mass-action networks with at most three internal directed reactions on $\{X,Y,2X,X+Y,2Y\}$, positive inflows, and linear removal of both species. Among the 681 networks up to species exchange, exactly 27 admit two distinct positive hyperbolic attracting equilibria. The same 27 admit bistability with equal removal rates and with independently chosen positive removal rates. Each is minimal under reaction deletion. We give rational equal-removal witnesses for all positive cases and all-parameter exclusions for the remaining 654. The proof combines nullcline geometry, polynomial sign tests, resultant root bounds, and exact flux-cone decompositions. A separate rational-arithmetic implementation verifies the complete classification and its equilibrium certificates.

Continuous-time multiparametric model predictive control solves the optimal control problem once, offline, for every initial state, and divides the initial-state space into critical regions, each with its own control law and sequence of active constraints. A boundary between two regions is either a hyperplane, given by one linear equation, or curved. The curved boundaries have not been computed to date; a classifier has been trained instead to distinguish the regions, so the partition has been known only approximately. In this paper we show that a curved boundary can be computed exactly, without solving any optimal control problem. The method rests on one property. Once the sequence of active constraints is fixed, together with the instants inside it at which a constraint becomes active or inactive (the switching times) and the instant at which the change of the neighboring region first appears (the event time), every condition on a boundary is a linear equation in the initial state, so the computation of a boundary, curved or not, is exact. We fix one of these instants, a switching time or the event time, and solve the linear equations that correspond to it for the corresponding initial state on the boundary; stepping the fixed instant along the horizon yields the whole boundary. A hyperplane is the case in which no instant is left to fix. We illustrate the method on the Newell-Lee evaporator, which has seven critical regions and eight boundaries, four hyperplanes and four curved, computing one boundary of each type in full with the adjacent regions at its ends.

Max-pressure (MP) traffic signal control selects phases using pressure scores computed from estimated traffic states, but these states are uncertain and their consequences depend on whether they alter phase ranking and how much pressure is lost. This paper develops a decision-oriented Regret-Based Max-Pressure framework that represents uncertainty through phase-selection regret, defined as the pressure loss of a candidate phase relative to the pressure-maximizing phase under each plausible latent state. We propose Stochastic Regret MP, which minimizes expected regret under a calibrated residual distribution, and Distributionally Robust (DR) Regret MP, which minimizes worst-case expected regret over a Wasserstein ambiguity set. Robustifying regret rather than absolute pressure makes the formulation invariant to state-dependent pressure components shared by all candidate phases, thereby focusing robustness on relative phase quality. We further derive an explicit queue-linear pressure-loss bound and establish strong queue stability when the net-service reserve exceeds the resulting loss coefficient. Numerical experiments show that both methods reduce normalized true-state pressure regret by about 70% relative to Point-estimate MP, while DR Regret MP yields lower network queues than direct robustification of absolute pressure. In microscopic SUMO experiments on a 28-intersection real-world network, Stochastic Regret MP and DR Regret MP reduce mean delay by 18.9% and 14.9%, respectively, and retain benefits under detector undercount and failure. The results show that representing uncertainty through its decision consequences can improve reliable traffic control under imperfect state information.

We prove that the 2-d cubic nonlinear Schr\"odinger equation with nonlinearity supported on a line or a sphere is globally well-posed in $H^1(\mathbb{R}^2)$. For higher dimensions, we prove that the equation with hyperplane-supported nonlinearity is locally well-posed in suitable anisotropic Sobolev spaces. We also prove that the equation with sphere-supported nonlinearity is globally well-posed in $H^1(\mathbb{R}^n)$ for radial data.

Developing general-purpose algorithms with global asymptotic convergence guarantees for variational inequalities (VIs) remains a longstanding challenge. This paper proposes a path-following framework for finite-dimensional VIs with arbitrary continuous mappings and polytope domains. The approach first transforms a general VI into a smooth VI on a simplex. Its key innovation is to formulate the resulting problem on a fiber bundle, called the fixed-point bundle, whose geometric structure enables the systematic integration of initialization, path-following, and singularity avoidance. Without imposing monotonicity or similar assumptions, the framework guarantees global linear convergence for nonsingular solutions and global linear convergence up to any prescribed precision for singular solutions. In numerical experiments, the algorithm attained a precision of $10^{-5}$ on all 6400 randomly generated instances, with dimensions up to 1024. Code and data are available at https://github.com/shb20tsinghua/FiberBundle_VI.

We give a probabilistic approach to the ultraviolet renormalization of the two-dimensional Sine-Gordon model. The model is formally described by the density $\exp(\alpha\int \mathopen{:} \cos(\beta X) \mathclose{:} )$, where $X$ is a log-correlated Gaussian field. Our construction applies on bounded convex domains for every particle activity $\alpha\in\mathbb R$ and throughout the subcritical range $\beta^2\in[0,4)$, below the Berezinskii-Kosterlitz-Thouless threshold. We apply the martingale method introduced by Lacoin, Rhodes and Vargas to two dimensions by using refined estimates based on charge conjugation symmetries. Our result extends existing constructions by treating the full subcritical regime for a broad class of covariances.

Let $f$ be an isolated hypersurface singularity and let $J(f)$ be its Jacobian ideal. We study the higher Milnor numbers $\mu_k(f)=\dim_{\mathbb C}\mathcal O_n/\mathfrak m^kJ(f).$ Our main result concerns an ordinary semi-homogeneous singularity $f=f_r+f_{r+1}+\cdots$, where the projective hypersurface defined by $f_r$ is smooth. We prove that $\mu_k(f)=\mu_k(f_r) \qquad\text{for every }k\geq0.$ This gives the equal-weight case of a conjecture of Shi, Yau, and Zuo. The main algebraic input is the common-order case of the Valabrega--Valla criterion. From it, we obtain the graded Jacobian module and the exact intersection formula. As a consequence, we obtain a closed formula for $\mu_k$ and determine the sharp stabilization index of $\mathfrak m^kJ(f)$. We also determine the optimal uniform range in which the Tjurina formula holds for all ordinary semi-homogeneous germs with fixed $n$ and $r$. We next consider the log-concavity property. For ordinary semi-homogeneous singularities, the higher Milnor sequence has an initial log-convex part and a log-concave tail. For an arbitrary isolated hypersurface singularity in $n\geq2$ variables, both the higher Milnor and higher Tjurina sequences are eventually strictly log-concave. In the plane curve case, we prove that the log-concavity defects change sign at most once, and we characterize the sequences which are log-concave at every positive index. Finally, we give examples showing that the higher Milnor sequence does not determine the contact-equivalence class.

We study the sharp homogeneous Dirichlet Gagliardo--Nirenberg quotient on a closed Riemannian manifold with holes of small capacity and connected complements. For each fixed exponent $2<q<2^*$, the nonnegative $L^2$-normalized extremal exists and is unique when the first Dirichlet eigenvalue $\lambda_D$ is sufficiently small. If the ambient volume is $V$ and $\beta=n(q-2)/2$, the sharp constant satisfies $C_D=V^{1-q/2}\lambda_D^{-\beta/2}(1+O(\lambda_D))$. For the normalized first eigenfunction $\varphi_D$, we obtain $\|u_D-\varphi_D\|_{H^1}=O(\lambda_D^{3/2})$ and $C_D/W_{M_D}(\varphi_D)=1+O(\lambda_D^2)$, together with a quantitative stability estimate in terms of the relative deficit. The constants depend only on the ambient manifold and the exponent. On general compact manifolds with Dirichlet boundary conditions, we prove compactness above the Euclidean sharp constant and describe concentration at equality. Positive scalar curvature at one interior point implies attainment. For the $L^2$-critical nonlinear Schr\"odinger energy, the mass threshold for negative energy tends to zero as the capacity of the holes tends to zero, whereas the threshold for unboundedness below is unchanged.

This paper presents a novel optimal control approach for Distributed Space Systems (DSS) to minimize the total delta-v cost for large, time-constrained reconfigurations. Reconfiguration cost is decreased by blending differential drag and propulsive control within the same control window. By parameterizing the problem with Relative Orbital Elements (ROE) and leveraging reachable set theory, a maneuver planning methodology is derived to produce provably-optimal, full roto-translational control solutions in closed-form, such that they can be implemented on hardware-limited spacecraft.

We study equality cases in Laffey's lower bound for local maximal commutative subalgebras of matrix algebras with cube-zero radical. We introduce a stacking operation that combines algebras with common outer Loewy layers while preserving maximal commutativity. This yields standard equality bricks $B_{a,c}=(a,ac+1,c;a+c)$ for all $a,c\ge2$ and reduces further constructions to smaller building blocks. We establish the existence of all minimal equality bricks on the balanced branch and obtain sharp realizations throughout the $(4,5)$-sector. We also construct the first nontrivial nearly balanced root over arbitrary fields. We determine the balanced--nearly-balanced envelope of the numerical Laffey profile and give an arithmetic criterion for preservation of sharpness under stacking. As a consequence, the profile is attained in every matrix dimension $14\le n\le52$ and $93\le n\le114$, as well as along five infinite families over arbitrary fields. The main existence and interval results are proved in characteristic zero; the field-independent constructions are stated explicitly.

We give a formula for quasi-minuscule monopole operators in $K$-theoretic quantized Coulomb branch algebras of cotangent type. We then study signed quivers with involutions, with orthogonal or symplectic gauge factors at the involution-fixed vertices. We construct a GKLO-type homomorphism from the corresponding shifted affine $\mathrm{i}$quantum group to an algebra of $q$-difference operators and prove that it factors through the quantized Coulomb branch algebra. These homomorphisms give geometric realizations of the shifted affine $\mathrm{i}$quantum groups in terms of dressed minuscule and quasi-minuscule monopole operators.

We introduce a generalized metric space endowed with a weighted graph and establish sufficient conditions for the existence of fixed points for set-valued mappings that satisfy a contractive condition only along the links of the graph. To demonstrate the applicability and significance of our results, several illustrative examples are presented. As an application, we establish the solvability of a class of integral problems involving set-valued data.

This paper studies how to relax pointwise feasibility in continuous-time dynamics for inequality-constrained nonlinear programming while retaining a quantitative feasible-recovery guarantee. Existing safe gradient flows enforce forward invariance of the feasible set, which yields an anytime-feasible dynamics but can be overly restrictive for optimization because it suppresses transient infeasible motion that may be essential for effective descent. To address this limitation, we propose a timed recovery safe gradient flow that replaces pointwise feasibility with fixed-horizon feasible recovery. The proposed construction augments the state with a timer that records the elapsed infeasible duration and computes a minimum-norm correction to the nominal gradient flow through a pointwise convex quadratic program. We show that the resulting augmented safe set is forward invariant and that every trajectory initialized from the feasible set returns to it within every time window of length $\tau$. We further show that, in a strict interior inactive regime, the closed-loop dynamics reduces exactly to the nominal gradient flow and recovers the corresponding local convergence behavior. Numerical examples illustrate how the recovery horizon shapes the closed-loop behavior and corroborate the theoretical analysis.

On stacks there is a richer version of $\mathcal{D}$-modules---called $\textit{potent }\mathcal{D}\textit{-modules}$---which decompletes at equivariant parameters, in the same sense that genuine equivariant $K$-theory decompletes equivariant cohomology. In this paper, we prove a ``genuine'' version of Kashiwara's lemma for $\mathcal{D}$-modules. Namely, we show that for a broad class of geometric stacks $X$, if $Z \subset X$ is a closed substack, the category $\mathcal{D}^{\mathrm{pot}}(Z)$ of potent $\mathcal{D}$-modules on $Z$ is equivalent under pushforward to the category $\mathcal{D}^{\mathrm{pot}} _Z(X)$ of potent $\mathcal{D}$-modules on $X$ with support contained in $Z$.

Let $T_n$ be the number of triangles in the Erd\H{o}s-R\'enyi random graph $G(n,p)$ with $n^{-1/2}\ll p\ll1$ and fix $\eta\in(0,1)$. We study the large deviation rate of the lower tail event $T_n\le\eta\mathbb{E}[T_n]$, and the typical structure of $G(n,p)$ conditioned on this event. By earlier work [16,19,21,49], both the rate and the structure are governed by a variational problem over graphons. In this work we solve this variational problem for every $\eta\in(0,1)$. We show that every minimizer is either a constant graphon or a step graphon with two blocks of equal size. Correspondingly, the conditioned graph is close to an Erd\H{o}s-R\'enyi graph or to a symmetric stochastic block model with two blocks. These two regimes are separated by a first order phase transition at the threshold $\eta_*=0.009217\ldots$. This resolves a conjecture of Zhao [78] and an open problem of Janson and Warnke [39] on triangles. It thereby gives a complete description of the lower tail of triangle counts in this range of $p$.

Update rules of a one dimensional Cellular Automaton can be expressed using an edge-labeled deBruijn graph. We explore various questions about the image of a cellular automaton given by a randomly selected update rule by considering a random deBruijn graph.

The goal of this paper is to compute the motivic local density of an isolated singularity of an algebraic variety, in order to explain its link with algebraic multiplicity. We compute the density as the residue in 1 of a Zeta function associated to normalized local volumes and we use a resolution of the singularity to get the formula.

We develop a new tree-search framework for mixed-integer linear programming problems, which incorporates a filter mechanism as a node-acceptance device. The introduced branch-and-filter algorithm combines proper branching actions with a filter on integrality violation and an incumbent-dependent objective coordinate. Inspired by classical filter methods in nonconvex constrained optimization, the proposed method moves to a restoration phase whenever sufficient progress cannot be provided by the filter and funnel mechanisms, the latter of which is anchored at global relaxation lower bounds. Under mild assumptions, the introduced method either certifies infeasibility in finitely many iterations, or converges to global optimality. Moreover, we establish a total outer iteration complexity bound of order $\mathcal{O}(\varepsilon^{-1})$ to certification of an $\varepsilon$-optimal mixed-integer solution through fixed-integer linear programming recovery. We finally perform several numerical experiments to study the performance of branch-and-filter against benchmark solvers, under a branch-and-bound, a branch-and-cut, and a novel cell-deletion realization of the proposed restoration oracle.