数学前沿

We establish the stability of the optimal non-endpoint Fourier restriction range under smooth flat perturbations of curved-root surfaces in $\mathbb R^3$. Specifically, we consider graphs of $$\phi(x_1,x_2)=(x_2-x_1^m)^n+R(x_1,x_2),$$ where $m,n\ge2$ are integers and $R$ is smooth and flat at the origin. On a sufficiently small neighborhood depending on the perturbation, we prove the local $L^p(\mathbb R^3)\to L^2(S,d\sigma)$ restriction estimate for $1\le p<p_c$, where $$p_c=\frac{q_c}{q_c-1}, \qquad q_c=\max\left\{2+\frac{2mn}{m+1},2n\right\}.$$ If the nonnegative localizing density is positive at the origin, an adapted-box example shows that the estimate fails for $p>p_c$. Thus arbitrary smooth flat perturbations preserve the optimal restriction range away from the critical endpoint. For $n\ge3$, the main local estimate combines planar curve decoupling with a one-dimensional finite-type extension estimate, whose principal scale factors cancel exactly. For $n=2$, we prove a uniform $C^4$ perturbation estimate using a weighted convolution bound and an anisotropic Whitney decomposition. Anisotropic rescaling and summation complete the proof.

For every right-c.e.\ real $p\in(1,2)$, $L^p[0,1]$ has a computable presentation. The proof combines an effective stability modulus for almost isometric embeddings with a Fra\"{\i}ss\'e-style construction in which the exponent varies. In the same range, if $\ell^p$ has a computable presentation, then $p$ is computable. Combining these with the known results gives the computable presentability classification for all separable Lebesgue spaces. We also formulate an effective Fra\"{\i}ss\'e theorem for Banach spaces and apply it to obtain a computable presentation of the Gurarij Banach space.

In this paper we study the convergence of solutions of the Stokes equations in a domain perforated by slender bodies placed on a fixed periodic grid of size $2\epsilon$ as the size of the bodies goes to zero and the number of bodies goes to infinity. We assume that the width $b_\epsilon$ of the bodies vanishes faster than the length $a_\epsilon$ of the particles. We identify the critical scaling regimes for $a_\epsilon$ and $b_\epsilon$ and prove that in this case the system is effectively described by the Brinkman equations.

For the symmetric pair $(\mathrm{GL}_n(\mathbb C),\mathrm{GL}_n(\mathbb R))$, we compute the exact Flicker--Rallis period of a distinguished vector in a cohomological minimal $U(n)$-type, normalized by the Gel'fand--Tsetlin pairing of Hara--Miyazaki--Namikawa. Our main tool is a weight-independent identity between the Whittaker period and a compact $O(n)$-average. We determine the proportionality constant using the archimedean newform formula of Humphries--Jo, and evaluate the compact functional on the normalized distinguished vector. This yields an explicit formula for the Flicker--Rallis period for every cohomological weight.

We introduce a spatially adaptive proper orthogonal decomposition (POD) method for error-bounded lossy compression of snapshot data in finite-dimensional function spaces. The method combines hierarchical graph partitioning with independent low-rank approximations on spatial subdomains, allowing the local approximation ranks to adapt to the spatial distribution of information in the data. For partitions that are orthogonal in the discrete inner product, we establish an exact decomposition of the global squared reconstruction error into local contributions and characterize the optimality of the resulting local POD approximations for fixed partitions and ranks. These results provide a rigorous mechanism for allocating a prescribed global error tolerance among subdomains and selecting the smallest admissible local ranks. We further develop a sufficient error bound for the coefficient-space normalized root-mean-square error (NRMSE). A greedy hierarchical refinement algorithm accepts a spatial subdivision only when it reduces the storage required by the local low-rank factors while preserving the prescribed reconstruction-error bound. The method is assessed using synthetic data and snapshot sequences from two- and three-dimensional compressible-flow simulations. The numerical results illustrate how spatially adaptive ranks and partitioning can improve compression relative to global POD and fixed spatial partitions, particularly when the data exhibit spatially heterogeneous dynamics.

The Unadjusted Langevin Algorithm (ULA) is widely used for high-dimensional Bayesian inference. Classical analyses in Wasserstein-2 distance predict an asymptotic bias and a mixing time that grow with dimension, but this metric is stronger than necessary for many statistical tasks. We instead analyze the bias in subspace robust Wasserstein distance, which controls the mean and arbitrary low-dimensional marginals and is therefore well suited to observables typically estimated in practice. For ridge-separable potentials formed with i.i.d. data, we prove high-probability bounds showing a bell-shaped mixing profile as the number of data points $N$ varies relative to the dimension $d$. In the overparameterized regime, the effective dimension is the rank $N$ of the data-dependent perturbation, leading to complexity depending on $N$ rather than $d$. In the large-sample regime, Gaussian-perturbation estimates yield dimension-free complexity once $N$ is sufficiently large. In contrast, near the proportional regime $\asymp d$, we prove that dimension-free guarantees are impossible in general.

Let A be a unital *-subalgebra of $M_n(C)$ and let $M = U(n)/U(A)$, with the quotient Finsler metric induced by a unitarily invariant norm on $u(n)$, normalized so that $||iP|| = 1$ for rank-one projections $P$. This family contains the unitary orbits of Hermitian matrices, in particular flag manifolds and Grassmann manifolds. We prove that if $z_0$ is a minimal lifting of a tangent vector $X$ with $||X||_{\infty} < \pi/2$, then the one-parameter group $t \mapsto e^{t z_0}\cdot o$ has minimal length among all piecewise smooth curves in $M$ joining its endpoints. The same holds in manifolds $N=SU(n)/(SU(n) \cap A)$ for every Ad-invariant Finsler norm on $su(n)$, not necessarily symmetric (i.e. the unit sphere does not need to be balanced). The proof relies on the convexity of the distance from the identity along geodesic segments, valid for all these norms. We also prove that the ball of radius $\pi/2$ around the identity, for the spectral norm, is convex along one-parameter segments. This extends known results for the p-Schatten norms with p even, and gives a new proof for the operator norm, where the result is due to Duran, Mata-Lorenzo and Recht.

In this paper, we study the multifractal formalism for Lyapunov exponents of typical one-step cocycles. We establish a variational relation between the Legendre transform of topological pressure of the generalized singular value function and measure-theoretic entropies. As a consequence, we show that the restricted variational principle of Lyapunov exponents holds for typical one-step cocycles. Furthermore, we prove a variational principle formula for the generalized singular value function.

We provide a detailed analysis of the quantum diffusive Stochastic Master Equation~(SME) for qubit/cavity systems with dispersive coupling, incorporating classical input signals and output signals (measurement outcomes via homodyne/heterodyne detection). We show that the dynamics of the qubit/cavity density operator converges exponentially towards a slow invariant manifold. This manifold is parameterized by the density operator of a \emph{fictitious} qubit governed by a quantum SME that incorporates the classical input/output signals while preserving complete positivity and trace. The reduced cavity (resp.\ qubit) operator, obtained by partial trace over the qubit (resp.\ cavity), is then expressed as a time-varying deterministic quantum channel acting on the fictitious-qubit density operator. This formulation avoids the non-Markovian descriptions---with temporarily negative dephasing rates or detection efficiencies exceeding one---that appear in several publications. Extensions are considered in which the qubit is replaced by a qudit dispersively coupled to an arbitrary set of cavity modes driven by collective classical input/output signals.

For each $d \ge 2$ we prove that the unit $d$-cube $\square_d \subset \mathbb{R}^d$ is, on the class $\mathcal{P}$ of convex polytopes combinatorially equivalent to a $d$-cube, (i) a strict local minimizer of the scale-invariant isoperimetric ratio $I(\Omega) = \tfrac{P(\Omega)^d}{|\Omega|^{d-1}}$, with $P$ the perimeter, (ii) a strict local minimizer of the volume-normalized first Dirichlet eigenvalue $\bar\lambda(\Omega) = |\Omega|^{2/d}\lambda_1(\Omega)$, and (iii) a strict local maximizer of the volume-normalized first Neumann eigenvalue $\bar\mu(\Omega) = |\Omega|^{2/d}\mu_1(\Omega)$. The method is a shape-derivative Hessian analysis adapted to the hyperoctahedral symmetry of the cube. In symmetry-adapted coordinates on the parameter space --- the decomposition into irreducible representations of the hyperoctahedral group --- each Hessian is block-diagonal, with four scalar eigenvalues, which we give in closed form in $\pi$, rational coefficients, and a few universal series-constants. The first Neumann eigenvalue is not differentiable at $\square_d$, since its lowest cluster splits under perturbation. The Neumann case therefore works with a smooth auxiliary functional, whose Hessian is computed by second-order perturbation theory of the cluster; the coupling to modes outside the cluster enters at the same order. Finally, we show that extremality of $I$, $\bar\lambda$, and $\bar\mu$ does not hold in high dimensions for a particular multilinear perturbation that leaves $\mathcal{P}$ by breaking convexity. All results are local; we discuss why a global statement, available for quadrilaterals (Endo and Osting, 2026), appears out of reach for $d \ge 3$.

This survey provides a concise overview of matrix triangle inequalities. Classical topics, such as inequalities for unitarily invariant norms, are revisited from a different perspective, with a particular emphasis on unitary orbit inequalities and various notions of matrix moduli. The exposition progresses from elementary results to much more recent developments, highlighting the evolution of the subject and its ongoing directions of research.

We study weighted sampling of operator orbit frames in infinite dimensional separable Hilbert spaces $H$, keeping the operator and generating vectors fixed. Let $N_\mu(R)$ count the sampling times in $[0,R]$. For a single orbit of a positive operator, let $m_A(R)$ count the eigenvalues in $(0,e^{-1/R}]$. Every real sampling design that preserves the frame property must satisfy the lower bound $N_\mu(R)\ge m_A(R)-O_\mu(1)$. Conversely, for every $\eta>0$, we construct an integer sampling design that preserves the frame property and satisfies $N_\mu(R)\le(1+\eta)m_A(R)+O_\eta(1)$. Thus the coefficient $1$ is optimal. Under additional spectral conditions, we attain $N_\mu(R)=m_A(R)+O(1)$ and characterize the weights that preserve the frame property on the constructed sets. For finitely or countably many orbits of a positive operator, let $P_j$, $j\ge1$, be the spectral projection corresponding to $2^{-j-1}<-\log\lambda\le2^{-j}$ and define $K_A(R)=\#\{j\ge1:2^j\le R,\ P_j\ne0\}$. Every real sampling design that preserves the frame property satisfies $N_\mu(R)\ge c_\mu K_A(R)-C_\mu$ for $R\ge2$, with constants $c_\mu>0$ and $C_\mu\ge0$. Conversely, we construct an integer sampling design that preserves the frame property and satisfies $N_\mu(R)\le L K_A(R)+L$ for $R\ge2$, where $L$ depends only on the bounds of the given orbit frame. For normal operators of norm one, we prove that the smallest frame condition number is attained among integer sampling designs that preserve the frame property and satisfy a prescribed counting bound, whenever this class is nonempty. Finally, we construct one normal operator with two families of generating vectors for which the optimal growth orders of $N_\mu(R)$ are $\log R$ and $R$, respectively, for both real and integer sampling.

We provide a short probabilistic proof of the classical characterization of multivariate quasi-copulas, addressing the call for a simpler and more transparent proof expressed by Sempi (2017). Our approach is based on a coupling construction for random variables with fixed marginal distributions and a prescribed distribution of their maximum. In our main result, we show that such a coupling exists if and only if the maximum is stochastically no smaller than each component and the distribution of the maximum is dominated by the sum of the marginal measures. The construction applies to arbitrary distributions and relies only on a decomposition of the maximum law and monotone quantile couplings. A direct consequence of our main result is the existence of copulas with prescribed values along an increasing track.

Seven-dimensional supergravity poses a uniquely intricate supersymmetry problem, combining topologically massive higher-form gauge dynamics, distinct Clifford algebra properties, and rich geometric structures. We first establish the complete formulation of the maximal Cartan-gauged theory in terms of the gauge-invariant higher-form flux, including both the bosonic action and supersymmetry variations. This formulation, which also makes the smooth ungauged limit manifest, is achieved by requiring a consistent truncation to the minimal Cartan-gauged theory in the equal-charge limit. We construct the complete bilinear system for a timelike Killing vector, where additional internal orientation of the symplectic-Majorana spinor arises already in the scalar sector, uniquely among pure minimal ungauged theories. In the equal-charge sector, we decouple the algebraic constraints into SU$(3)$-isotypic components, and analyse the differential bilinear relations, determining the permitted dilaton gradient, fluxes, fibration twist and intrinsic torsion of the base. We prove that with gauging and active Cartan flux, the complex scalar bilinear vanishes identically. We also construct a Killing gauge for the higher-form potential compatible with the self-duality condition. We show that with gauging but no active Cartan flux, the Killing vector also acts as a generator of the gauged R-symmetry, and the base structure is enlarged from SU$(3)$ to U$(3)$. We find that BPS-flow solutions realize this setting. With supersymmetry, we isolate a single independent component of the 3-form equation across an exhaustive four-class partition for both theories within this setup. Applied to known and proposed solutions in both theories, this classification exposes unexplored sectors and provides an accessible toolbox for constructing and testing solutions and developing new geometric and holographic applications.

This paper addresses exact controllability and rapid stabilization issues for linear difference delay systems with finitely many pointwise delays. We first establish a criterion for exact controllability in $L^r$, $r \in [1, +\infty)$, characterized by a necessary and sufficient Hautus-type frequency-domain condition. The sufficiency of this condition in $L^r$ was already known [https://hal.science/hal-04830757/], whereas its necessity had previously been established only in $L^1$ [arXiv:2210.13590]. We answer [arXiv:2210.13590, Conjecture 3.8] by proving that this condition is also necessary for $r \in (1, +\infty)$. The analysis is based on rewriting the system as an evolution equation in a suitable Banach space, which allows one to apply duality arguments in the spirit of [https://doi.org/10.1137/S036301299119795X]. Building on this controllability analysis and the abstract stabilization framework developed in [https://doi.org/10.1137/S0363012996301609, arXiv:1301.5744], we prove that our exact controllability criterion in $L^r$ is also a sufficient condition for rapid stabilization in $L^2$, i.e., that the system of interest can be stabilized by a family of state feedback control laws ensuring that the closed-loop system is exponentially stable in $L^2$ with an arbitrarily large decay rate. As an application, we extend our results to linear one-dimensional hyperbolic partial differential equations, which can be reduced to linear difference delay systems via the method of characteristics, obtaining also a necessary and sufficient frequency-domain criterion for exact controllability in $L^r$, $r\in [1, +\infty)$, which, as before, is a sufficient condition for rapid stabilization in $L^2$.

We establish sharp quantitative stability of the Ashbaugh--Benguria inequality on bounded convex domains in $\mathbb R^n$. After volume and centroid normalization, the eigenvalue-ratio deficit controls the squared $H^{-1/2}(\mathbb S^{n-1})$ norm of the radial displacement from the unit sphere. This yields the optimal exponent $5/2$ for the Fraenkel asymmetry and, for small deficits, the optimal exponent $(n+4)/2$ for the normalized Hausdorff distance. All three exponents are sharp even among smooth strictly convex domains whose principal curvatures lie in $[1/2,2]$. In particular, the quadratic Fraenkel-asymmetry estimate proposed by Brasco and De Philippis (2017, Open Problem 7.65) fails. We also determine the optimal stability exponents under higher-order radial regularity bounds.

We study the Robin, or impedance, Laplacian on a half-space of dimension $n \ge 3$ which is perturbed inside a fixed ball. The boundary may be deformed, and obstacles may be present in the interior or attached to the boundary, in any combination and with no smallness assumption. Outside the ball the configuration is the flat impedance half-space of Part I, with constant impedance $\alpha > 0$, where a bulk channel and a surface channel propagate at every positive energy and only the surface channel propagates on the band $(-\alpha^2,0)$. We prove a limiting absorption principle, valid at every positive energy and outside a discrete set of embedded eigenvalues in the band. We construct the generalized Fourier transform and prove its completeness by a stationary argument which constructs no wave operator and reduces completeness to the uniqueness of the scattering solution at each energy. We then obtain the scattering matrix, its unitarity and a two-channel optical theorem. The main result is a uniqueness theorem for the inverse problem. Under two geometric conditions on the pair, the scattering amplitude at a single positive energy determines the perturbation completely, and the argument does not distinguish a deformation of the boundary from an obstacle. It follows that no nontrivial perturbation satisfying the same conditions is invisible at any positive energy. On the surface band the corresponding statement is reduced to one explicitly stated question about the exterior Robin problem.

We show how regulator constants of a finitely generated $\mathbb{Z}[G]$-module can be related to $G$-cohomology, where $G$ is a finite group. We then derive consequences of such relation for modules naturally arising in number theory, such as ring of integers and units of number fields, $K$-theory groups of ring of integers and Mordell-Weil groups of elliptic curves.

In this paper, we study matrix valued orthogonal polynomials whose block Jacobi matrix is $\Theta(J)$, viewed in blocks of size $m=\deg\Theta$, where $J$ is a tridiagonal Jacobi matrix and $\Theta$ is a scalar polynomial. We identify their matrix measure of orthogonality intrinsically as the push-forward, under $\Theta$, of a matrix measure constructed from the original scalar orthogonal polynomials. We formulate reducibility in the push-forward variable and prove that the $2\times2$ Hermite family associated with $\Theta(x)=x^2+\alpha_1x$ is irreducible for $\alpha_1\neq0$ and reducible for $\alpha_1=0$. This problem is motivated by random walks with transition matrix $\Theta(J)$. For these models, we derive scalar and block Karlin--McGregor representations for the powers of $\Theta(J)$. We apply the construction to the mixed one- and two-ball Ehrenfest chain, obtaining explicit spectral measures, spectral multiplicities, and transition formulas, and we conclude with the corresponding $k$-ball and multi-ball extensions.

Stochastic Galerkin schemes (SGS) are used to approximate solutions to differential equations with stochastic parameters. In particular, they express the solution with an infinite orthogonal functional basis of the parameter space and derive a deterministic system of PDEs; the solution of this deterministic system can be used to produce a {surrogate} approximation of the original stochastic equations. When the parameters are correlated, the infinite series representation of the solution does not provably converge for all joint distributions; further, classic decorrelating transformations (e.g., Roseblatt) often introduce known numerical instabilities. Rahman (2017) established convergence in probability and distribution for correlated {Gaussian} parameters when the basis functionals are {multivariate Hermite polynomials} (MHPs), where these polynomials are only {weakly orthogonal} with respect to the underlying joint density. Due to weak orthogonality, the $N$th moments of MHPs introduced during Galerkin projection do not trivially decompose into products of one-dimensional expectations and have no closed-form expression for $N > 2$; consequently, these moments are typically estimated via brute-force sampling and quickly become prohibitively expensive to compute. In this paper, we introduce and prove several foundational results for MHPs. These results culminate in an equation that can be used to calculate the $N = 3$ moments (triple products) exactly, whose implementation dramatically reduces the computational bottleneck otherwise inherent to SGS with correlated Gaussian parameters.