数学前沿

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.

In this paper, we prove that an annular capillary minimal surface outside the unit ball in $\mathbb{R}^3$ with one complete embedded end and finite total curvature must be either part of the plane or part of the catenoid. The proofs are based on the use of the Weierstrass data through the Hopf differential and associated holomorphic functions. For a planar end, we control the second fundamental form, while for a catenoidal end, we analyze the Gauss map in more detail using the boundary condition. We also prove nonexistence results for annular free boundary minimal surfaces with one end asymptotic to the end of Enneper's surface, under appropriate conditions. Finally, we establish an exterior analogue of Choe's result on capillary minimal surfaces with vertices.

We prove bilinear Strichartz estimates for the Schr\"odinger equation on the hyperbolic space. These bilinear estimates are then applied, as part of an I-method argument, to establish global well-posedness for the cubic defocusing nonlinear Schr\"odinger equation on the hyperbolic plane for regularities above 4/7. The I-method is implemented in physical space using a heat-based modified energy.

Full-waveform inversion (FWI) reconstructs sound-speed fields from measured waveforms, by solving an optimization problem that minimizes the discrepancy between measured waveforms and those predicted by a numerical wave-propagation model. However, choosing suitable regularization for this optimization is challenging because it must stabilize the inversion while preserving relevant structures, prompting us to investigate whether priors learned from representative sound-speed images can improve reconstruction. In this work, we adapt plug-and-play flow matching (PnP-Flow) to FWI by alternating waveform-misfit updates with updates from a pretrained flow prior. We explore the effectiveness of PnP-Flow reconstruction by comparing it with total-variation (TV) regularization using noiseless transmission-only plane-wave data. Separate flow priors are trained on two circular-inclusion model families with constant and spatially varying backgrounds. We evaluate the method both on these in-distribution circle models and under substantial distribution shift by transferring the Gaussian-circle prior and reconstruction settings to breast phantom patches without retraining or retuning. Using the same number of waveform-misfit updates, the proposed PnP-Flow framework generally improves circle reconstruction accuracy and recovers breast lesion contrast more accurately than TV baselines.

For every field $k$, we construct a commutative coherent $k$-algebra admitting a Gorenstein projective module that is not Gorenstein flat.

We prove that the minimum Hausdorff dimension of a continuous curve joining $W_0$ to $W_1$ within the range $W([0,1])$ of a three-dimensional Brownian motion is almost surely a deterministic constant $\gamma_3\in(1,5/3]$. Moreover, we construct continuum many simple curves with pairwise distinct images that attain this minimum. Each curve has box dimension $\gamma_3$ and admits a parametrization that is $\beta$-H\"older for every $0<\beta<1/\gamma_3$. To identify $\gamma_3$, we partition Brownian time into $n$ equal intervals and form a graph by joining pairs with intersecting ranges. We show that its endpoint graph distance and diameter have a common deterministic almost sure growth exponent $\theta$ and that $\gamma_3=2\theta$. We also establish $n^{p\theta+o(1)}$ growth for their $p$-th moments, for every $p>0$.

We prove the following two statements: (1) If $X$ is a compact K\"ahler fourfold with pseudo-effective canonical bundle and no subvarieties of codimension one or two, then $K_X$ is a torsion line bundle; (2) If $X$ is a compact K\"ahler fourfold with no subvarieties of codimension one or two, then $X$ is not uniruled. Using the Beauville-Bogomolov decomposition and a characterization of Uniruled manifolds by Wenhao Ou, we get that if $X$ is a compact K\"ahler fourfold with no subvarieties of codimension one or two, then either $X$ is a torus or an irreducible holomorphic symplectic manifold.

For $\beta\ge0$, let $A_1^{(\beta)}$ be the class of weights satisfying $$[w]_{A_1^{(\beta)}}:= \sup_{0<\varepsilon\le1} \varepsilon^\beta[w]_{A_{1+\varepsilon}}<\infty.$$ These classes satisfy $A_1\subsetneq A_1^{(\beta)}\subsetneq\bigcap_{p>1}A_p$ for $\beta>0$. For every $\alpha\ge0$, we prove that $w\in A_1^{(\beta)}$ if and only if $$w\left(\left\{M_{L(\log L)^\alpha}f>\lambda\right\}\right) \le C\int_{\mathbb{R}^n} \frac{|f(x)|}{\lambda} \left[ \log\left(e+\frac{|f(x)|}{\lambda}\right) \right]^{\alpha+\beta} w(x)\,\mathrm{d}x.$$ The least constant $C$ is comparable to $[w]_{A_1^{(\beta)}}$. Also, we obtain corresponding characterizations for Calder\'on--Zygmund operators and their commutators.

The multiplicative ergodic theorem (MET) describes the asymptotic growth rates of $q$-volumes under the iterates of an operator cocycle, i.e., compositions of operators driven by a stationary process, organizing the underlying space into equivariant filtrations or splittings into so called Oseledets subspaces. For operator cocycles in finite dimensions, and more generally in Hilbert spaces, one can control the geometry of the Oseledets subspaces given by the MET via the convergence rates of finite-time volume growth data. This makes the MET "effective" for matrix and Hilbert-space operator cocycles. We extend this effective theory to cocycles of Banach-space operators, obtaining quantitative angle estimates that are new even in the finite-dimensional setting. We take a sequential approach \`{a} la Raghunathan and Ruelle, allowing us to separate the geometric argument of the MET from measure-theoretic considerations. This permits weaker cocycle measurability hypotheses for the non-invertible and semi-invertible METs on a broad class of Banach spaces, including nonseparable ones. As an application, we establish the MET for transfer operator cocycles on $\operatorname{BV}[0,1]$ associated with jointly measurable random Lasota-Yorke maps.

Let $Y$ be a smooth projective threefold with $H^2(Y, \mathcal{O}_Y)=0$ and unobstructed deformations. We define reflection groups $W_Y^{sm}$ and $W_Y^{big}$ canonically associated to $Y$, acting on the nef, resp. movable cone of a very general deformation of $Y$. If $-K_Y$ is nef, we show that $\mathrm{Nef}(Y)\cap\mathrm{Big}(Y)$ (resp. $\overline{\mathrm{Mov}}(Y)\cap\mathrm{Big}(Y)$) is a fundamental domain for the action of the corresponding group on the big part of the generic cone. This generalizes work of Wilson and Lutz on Calabi--Yau threefolds. The group $W_Y^{big}$ recovers and gives a deformation-theoretic explanation for the Weyl group action on divisor classes of point blowups of $\mathbb P^3$ studied by Dolgachev--Ortland and Mukai.