The Cesaro operator is cyclic on $H^p$
In this paper, we show that the Cesàro operator on the Hardy space $H^p$, $0 < p < \infty$, is cyclic. Our techniques will involve semigroups.
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Mathematics
arXiv preprints from January 1, 2026 through September 19, 2026 — 23:33:14 EST
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Naturality characterizes product measures and relative product measures
We show that the product measure is the only natural way to assign to each pair of probability measures on measurable spaces a probability measure on their product. Here, naturality is meant in the sense of category theory and amounts to the condition that the assignment commutes with pushforward along measurable maps. In the standard...
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Sharp decay thresholds for eigenvalues of discrete Schrödinger operators on $\mathbb{Z}^2$
We study eigenvalues of discrete Schrödinger operators $H=Δ+V$ on $\mathbb{Z}^2$, where $Δ$ is the uncentered Laplacian, i.e., the un-normalized adjacency operator of $\mathbb{Z}^2$ and $V$ decays at infinity. By Weyl's theorem, the essential spectrum of $H$ is $[-4,4]$. We determine the sharp decay thresholds for existence of...
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Scalar-mean rigidity theorem and Llarull's theorem for nonspin manifolds
We prove Llarull's scalar curvature rigidity theorem for spheres and scalar-mean rigidity for strictly convex Euclidean domains in all dimensions without the spin assumption.
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Strategic Index Reconstitution: Differential Games, Closed-Loop Equilibria and Mean-Field Dynamics
We study strategic trading around index reconstitution in a continuous-time, multiasset game with transient cross-asset price impact and heterogeneous beliefs about future index membership. Opportunistic traders position before a public announcement, adjust to the revealed composition, and trade around an indexer following a...
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Cyclic Haagerup-Izumi fusion categories at every odd order
We construct a complex spherical fusion category with cyclic Haagerup-Izumi fusion rules for every odd n >= 3, and deduce pseudo-unitary existence. The proof has four parts: explicit real coefficients and their quadratic identities; a contour calculation for a range of cubic Fourier coefficients; algebraic completion of all cubics;...
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Convergence for Small-Order Derivatives of Random Polynomials with Independent Roots
Let $X_1,X_2,\ldots$ be i.i.d. complex-valued random variables with arbitrary Borel probability law $μ$, and set $P_n(z)=\prod_{j=1}^n(z-X_j)$. For every deterministic sequence $k_n=o(n)$, we prove that the empirical zero measure of the $k_n$-th derivative $P_n^{(k_n)}$ converges weakly to $μ$ almost surely.
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Fusion-stable endosplit $p$-permutation resolutions
Let $k$ be a field of characteristic $p$. We show that if $\mathcal{F}$ is a saturated fusion system over a finite $p$-group $P$, and $V$ is an $\mathcal{F}$-stable indecomposable capped endopermutation $kP$-module whose Dade class $[V]$ belongs to the subgroup $D_k^Ω(P)$ of the Dade group $D_k(P)$ generated by all the relative...
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Wasserstein Metric Normalization of Projective Cycle Spaces
Let $X$ be an irreducible reduced projective subvariety of a Chow variety. On a generic smooth parameter locus, normal motions of resolved components define a quadratic Wasserstein metric, even when the cycles are singular, reducible, or carry multiplicities. Its metric completion is canonically $X^ν$: resolution fibers collapse,...
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Infinite Horizon Mean-Field Terminal Value Problem
We introduce a class of finite- and infinite-horizon discrete-time control problems for studying macroscale systems in discrete time, which we refer to as mean-field terminal value problems (MFTVPs). An MFTVP takes a terminal \(Q\)-function as input and, starting from this terminal \(Q\)-function, seeks an indefinite backward...
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High-Probability Convergence of SGD via Batched Updates
Stochastic gradient descent (SGD) is the primary workhorse for large-scale optimization. While the average behavior of its iterates, typically characterized by mean-squared error bounds, is well-understood, obtaining high-probability guarantees for the last iterate remains challenging. Prior approaches to this problem have either...
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Jacquet--Rallis transfer for GL(2)
We study archimedean smooth transfer for the Jacquet--Rallis relative trace formula comparison, in particular, identities between orbital integrals on GL(n) and its unitary forms. We work with Lie algebras and a (g,K)-module setting. Our main result states that for n=2, meaning GL(2) acting on gl(3), every polynomial type Schwartz...
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The zero set of the Zak transform of B-splines with applications to Gabor frames
We study the zero sets of the Zak transform $Z_λB_n(x,ν)$ of B-splines for $λ> 0$. Specifically, we provide a full characterization of the zero set for the hat spline for all positive values of the parameter $λ$ and for higher order B-splines when $λ> 1$. Finally, we apply these results to establish the frame property of...
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Diameter-free reverse inequalities and superorthogonality
We prove three results as part of the program of diameter-free estimates initiated in [CDW26]. The first two are reverse square function estimates for the light cone in $\mathbb{R}^3$. We first establish an abstract $L^4$ inequality of independent interest, under an ordered superorthogonality hypothesis: for every four distinct...
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From $\mathrm{BV}^\mathcal A$ to $\mathrm{BV}$: An endpoint Korn estimate
In this short note we prove that, given a $\mathbb{C}$-elliptic operator $\mathcal{A}$ and a map $u\in \mathrm{BV}^{\mathcal{A}}(Ω;V)$ satisfying \( \nabla_{\mathrm{ap}}u\in L^1(Ω;V\otimes\mathbb{R}^d), \) then $u\in \mathrm{BV}(Ω;V)$. The result is quantitative and follows from the Korn-type estimate \[ |Du|(Ω) \leq...
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Categorical genus for log del Pezzo surfaces with cyclic quotient singularities
For the canonical stack $\mathcal{X}$ of a log del Pezzo surface with cyclic quotient singularities, we compute its categorical genus as \[ g_{\mathrm{cat}}(\mathcal{X})= 1+\frac{1}{2}\sum_j w_j(\ell_j-1), \] where $w_j=\gcd(n_j,q_j+1)$ and $\ell_j=n_j/w_j$ are the local widths and Gorenstein indices of the singularities...
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Knot Floer homology of boundary Dehn twists
We prove that the knot Floer complex, along with some restrictions on the flip map, determines the dual knot to $\pm 1$ surgery on the Borromean knot. In particular, this implies that Heegaard Floer homology detects boundary Dehn twists.
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The Cauchy problem for Manton's Chern-Simons-Schrödinger equation
The Chern-Simons-Schrödinger equation (in the temporal gauge) arises as the Hamiltonian flow of the abelian Higgs energy on $\mathbb R^2$. Manton (arXiv:hep-th/9701027) introduced the equation as a model for the dynamics of the critical points of the energy, known as vortices. He conjectured that, for a certain range of coupling...
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An asymptotic formula for a cubic moment of $GL_2$ $L$-functions
We obtain an asymptotic formula for a cubic moment of self-dual $GL_2$ $L$-functions studied by Petrow and Young, with a small extra averaging over characters. Our asymptotic consists of the main term conjectured by Conrey--Farmer--Keating--Rubinstein--Snaith and a power-saving error term whose strength depends on the amount of extra...
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How natural of a geometric operation is Murasugi sum?
Gabai proved that any Murasugi sum of $π_1$-essential Seifert surfaces is also $π_1$-essential, and Ozawa extended this result to unoriented spanning surfaces. We show, however, that the analogous statement about geometrically essential surfaces is untrue. (A spanning surface is geometrically essential if it cannot be compressed or...
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On Skabelund's Ray Class Field Covers of the Suzuki and Ree Curves
Let $\Sm_q$ and $\Rm_q$ denote the Suzuki and Ree curves. Motivated by the Giulietti--Korchmáros curve, Skabelund constructed cyclic covers $\tSm_q$ and $\tRm_q$ of these curves and proved that they are maximal over $\F_{q^4}$ and $\F_{q^6}$, respectively. In the same paper he associated to the Suzuki and Ree curves certain ray class...
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A Unified Two-Grid Framework for Anderson Acceleration and Nonlinear GMRES
In this work, we recast two widely used acceleration methods for fixed-point iterations, Anderson acceleration (AA) and the nonlinear generalized minimal residual method (NGMRES), as two-grid methods. By explicitly deriving the error propagation matrices for AA and NGMRES on linear problems, we show that both methods, together with...
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Premaniplexes of rank $3$ and $4$ as symmetry type graphs of maniplexes
We show that every finite premaniplex of rank $3$ or $4$ is the symmetry type graph of a finite maniplex, settling the finite rank $3$ and $4$ case of the maniplex version of the symmetry type graph problem. The proof uses the fact that the universal string Coxeter groups of rank $3$ and $4$ are amalgams (of finite groups), hence...
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Kolyvagin's conjecture at non-ordinary primes
Let $K$ be an imaginary quadratic field and let $p \ge 5$ be a prime that is unramified in $K$. Let $\mathcal{A}_f/\mathbb{Q}$ be an abelian variety of $\mathrm{GL}_2$-type associated with a weight-two modular form $f$, with good non-ordinary reduction at $p$, and suppose that $(f,K)$ satisfies the generalized Heegner hypothesis. In...
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Intrinsic Bohnenblust--Hille Inequalities for Local Qudit Systems
We study dimension-free Bohnenblust--Hille inequalities for local operators on systems of $K$-level qudits. For the operator space of support at most $d$, uniformly over all tensor-product orthonormal operator bases, we prove a Bohnenblust--Hille inequality with the optimal exponent $2d/(d+1)$ and a constant of order $O(\sqrt K)^d$....