arXiv preprints from January 1, 2026 through September 20, 2026 — 18:58:51 EST
The purpose of the paper is to place Whiting's classical growing mode stability argument, extended to real frequencies by Shlapentokh-Rothman for the scalar wave equation and by Andersson, Ma, Paganini and Whiting in general, in the framework of classical PDE theory. The key steps are: a description of the dual or adjoint modes, a...
Sparse domination is a central tool in modern harmonic analysis, offering a unified approach to weighted inequalities for Calderón--Zygmund operators and related operators such as commutator operators, rough singular integrals, square functions etc. In this expository note, we briefly survey the main ideas behind sparse bounds on...
We show that every continuous and isometry invariant valuation on spherical polytopes is a linear combination of the spherical intrinsic volumes. The proof relies on a weak differentiability property satisfied by valuations on polytopes in $\mathbb{R}^n$ with a natural smoothness property with respect to the action of the affine...
Classical Nesterov acceleration employs different choices of damping and inertial parameters in the convex and strongly convex settings, both for continuous-time dynamics and for discrete algorithms. When the strong convexity parameter is small, directly using the strongly convex damping or inertial coefficient may lead to slower...
We compute the local integrals appearing in the refined Gan--Gross--Prasad conjecture for Fourier--Jacobi periods of $\mathrm{Sp}_4$ in new ramified cases and use this to formulate an explicit conjectural identity relating Petersson norms of degree 2 Siegel cusp forms and associated half-integral weight forms. We note consequences of...
We study the connectedness and enumeration of reconfiguration graphs of valid triangulations of convex polygons whose vertices are cyclically colored with $j \ge 3$ colors, where every triangle has vertices of three pairwise distinct colors. For $j = 3$, we settle a conjectural expectation of Acharya, Mütze, and Verciani: we prove...
Let $(R,\mathfrak{m})$ be a one-dimensional Cohen-Macaulay local ring. In this paper, we find the reduction number $r_{z}(\mathfrak{m}^d)$ of $\mathfrak{m}^d$ with respect to a witness $z\in \mathfrak{m}^d \setminus \mathfrak{m}^{d+1}$ to the generalized Loewy length $\text{g}\ell\ell(R)$ for several infinite families of hypersurfaces...
For surfaces of genus $g \geq 3$ with at most one boundary component, we give explicit generating sets for the kernel of the Birman-Craggs-Johnson homomorphism and the commutator subgroup of the Torelli group. As an application, we give a new proof of Johnson's calculation of the abelianization of the Torelli group.
We show that the nilpotent cone associated to a simply connected semisimple algebraic group, together with all its crepant projective partial resolutions, can be constructed as Hamiltonian reductions of the affine closure of the cotangent bundle of base affine space for suitable choices of stability parameter of a maximal torus. We...
Traditional plasticity theory is commonly organized through an elastic law, a yield condition, a flow rule, hardening relations, and loading--unloading conditions. This paper formulates these relations through directional stationarity of one scalar functional evaluated over one-sided admissible plastic paths. The first variation...
Local differential privacy (LDP) protects individuals in a dataset by perturbing each measurement before release. For real-valued data, a widely used mechanism is additive Laplace noise. We study the problem of estimating the distribution of the latent confidential data from the privatized observations via the nonparametric maximum...
We prove optimality of the differentiability bound for isotropic positive definite functions on every even-dimensional sphere. If the even continuation of such a function on the $d$-dimensional sphere is $2k$ times differentiable at zero, then the function has $2k+\lfloor(d-1)/2\rfloor$ continuous interior derivatives; previously,...
Let $F=(f_1,\ldots,f_r)$ be a tuple of holomorphic germs on a smooth complex germ, and let $B_{F,0}$ be its Bernstein--Sato ideal. We develop an iterated-residue obstruction showing that a nonzero coefficient-valued residue class on an SNC stratum forces the corresponding exact affine parameter to lie in $Z(B_{F,0})$. As applications,...
Esakia duality is the dual equivalence between Heyting algebras and Esakia spaces. However, the traditional proof uses the Prime Ideal Theorem to recover the algebra from its spectrum, a choice principle that is not constructively valid. We build on Townsend's localic Priestley duality to describe a fully constructive, localic...
Let $X$ be an $n$-dimensional generalized Grassmannian not isomorphic to $\mathbb{P}^n$. We prove that $k(X)\le n-1$, where $k(X)$ denotes the maximal integer such that every uniform bundle on $X$ of rank at most $k(X)$ is homogeneous. In particular, for smooth quadrics $\mathbb{Q}^n$, we have $k(\mathbb{Q}^n)=n-1$ for odd $n$, and...
In this paper, we investigate the neighbourhood convexity ($n$-convexity) on graphs, a new finite convexity space grounded in the common closed neighbourhood closure operator. Unlike standard path-based graph convexities, $n$-convexity shows a non-canonical behaviour, giving rise to compelling structural properties and being almost...
This note supplies examples for the manuscript: Rigidity on the Two-Torus and Sarnak's Conjecture. For every $0<δ<\tfrac12$, we give two constructions of semi-irrational $C^\infty$ diffeomorphisms of $\mathbb{T}^2$ that satisfy the hypotheses of both Theorems 1 and 2 of that manuscript but do not have bounded mean motion, together...
Thousands of sangaku tablets, inscribed with colorful geometry problems, were posted in shrines and temples across the Japanese archipelago starting in the Edo period (1603--1868), representing an intersection of artistic, religious, and mathematical traditions. The Sangaku Archive is an attempt to carefully document these objects...
Let $s_2(n)$ be the binary sum-of-digits function and let $c_t$ be the natural density of the integers $n\ge0$ for which $s_2(n+t)\ge s_2(n)$. Earlier work of the author proved the universal exponential bound $$c_t-\frac12\ge 2^{-2s_2(t)-1},$$ thereby resolving Cusick's conjecture for every $t$. This estimate, however, does not...
Lacey used sparse domination to study the sharp weighted norm estimate of the maximal function of predictable multipliers in discrete time filtration spaces. Domelevo, Petermichl, and Škreb developed the self similarity argument known as sparse domination in an abstract martingale setting with a continuous time parameter. In our...
A celebrated result of Bayer and Stillman from 1987 states that for a homogeneous ideal $I$ of a polynomial ring $S$, the regularities of $S/I$ and $S/\textrm{GIN}(I)$ are the same under the reverse lexicographic monomial ordering, where $\textrm{GIN}(I)$ is the generic initial ideal. If $R$ is a polynomial ring whose variables are...
We construct infinitely many new pairwise nondiffeomorphic smooth structures on infinitely many closed simply connected spin 4-manifolds with positive signature. Our construction builds on an infinite family of simply connected complex surfaces of general type due to Roulleau and Urzúa that populate points arbitrarily near the...
We determine, up to absolute constants, the worst-case number of point evaluations required for a weighted $L_2$ Marcinkiewicz-Zygmund inequality for an $m$-dimensional complex function space. If $0<\varepsilon<1$ is the relative distortion, this number is $$Θ\Big(\min\Big\{m^2,\frac{m}{\varepsilon^2}\Big\}\Big),$$ and exact...
The Serre problem asked if a locally trivial holomorphic fibration with Stein base and Stein fiber is itself Stein. \v Skoda constructed the first counterexample for this problem. Later, Cœuré and Loeb constructed another remarkable counterexample, with bounded domain of holomorphy as fiber. The total space of their fibration has...
In this paper, we establish the boundedness of the bilinear Bochner-Riesz means $\mathcal{B}^α_R$ on the complex sphere $\mathbb{S}$. More precisely, we prove that $\mathcal{B}^α_R$ is bounded from $L^{p_1}(\mathbb{S}) \times L^{p_2}(\mathbb{S}) \to L^p(\mathbb{S})$ where $1/p_1+1/p_2=1/p$ and $1\leq p_1, p_2 \leq \infty$, for an...