arXiv preprints from January 1, 2026 through September 23, 2026 — 11:01:11 EST
This paper studies the convergence of stochastic gradient descent (SGD) when the implemented updates are subject to a persistent and state-dependent bias, in which the desired update is scaled by response functions component-wise. Our first contribution is to demonstrate that SGD in this setting implicitly optimizes a penalized...
In electromagnetic heating and other applications, we need to compute the volume enclosed by an isosurface of the 3D temperature distribution that is numerically represented on a rectangular grid. This situation arises naturally when the temperature distribution is obtained by solving a partial differential equation numerically using...
We consider the two-species particle system introduced by Stevens (2000) related to the doubly parabolic Keller-Segel equation. It consists of $N$ cells and of a varying number of chemoattractant particles. Cells diffuse in the plane and follow the (mollified) empirical gradient of concentration of chemoattractant. Chemoattractant...
For a finite subgroup $G\subset \operatorname{SL}(3,\mathbb{C})$, the Cautis--Logvinenko conjecture states that for each nontrivial irreducible representation $ρ$ of $G$, the image of the sheaf $\mathcal{O}_0\otimes ρ$ under the derived equivalence of Bridgeland--King--Reid is a pure sheaf on the $G$-Hilbert scheme. We prove this when...
We study the quasilinear Liouville equation \[ -Δ_n u=e^u \] on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. Our first result shows that, if a solution $u$ satisfies the optimal logarithmic lower bound \[ u(x)\ge -\frac{n^2}{n-1}\log r(x)+o(\log r(x)) \quad \text{as }r(x)\to+\infty, \] then the underlying...
The Strong Szegő Limit Theorem is a theorem about the asymptotics of the determinants of large Toeplitz matrices. It can be reformulated as a probabilistic statement about eigenvalue statistics of random unitary matrices. We prove a multivariate generalization of the theorem in this latter form, replacing a single unitary with a...
For a ladder-system space $X_L$ with carrier $S\subseteq E^{ω_1}_ω$, the finite-label uniformization property $M_{<ω}$ characterizes countable metacompactness, and countable metacompactness is equivalent to the $Δ$-property. Both equivalences are known for stationary carriers. For arbitrary carriers, an active-tail formulation gives a...
We connect modal set theory with Boolean-valued models by developing an \emph{internal} Kripke semantics for modal formulas whose atomic propositions are set-theoretic sentences. Given a complete Boolean algebra $B$, we view its elements as ``local perspectives on truth'' inside the Boolean-valued universe $V^{(B)}$ and interpret the...
We have previously given a factorization of a symplectic column under the action of the parity involution which enabled to explicitly have written the inverse of the reduction map in the quantum Littlewood-Richardson bijection. Watanabe has written the reduction map as a composition of several maps, among them, combinatorial...
Algebraic concordance of knots can be understood from the perspective of Seifert matrices, Blanchfield forms, and homology surgery. We initiate a systematic study of algebraic concordance for links from each of these viewpoints. The present article is concerned with algebraic concordance from the perspective of homology surgery and...
Influence matrices encode the action exerted on local subsystems by the rest of an extended quantum many-body system during their evolution. Thus, knowledge of the influence matrix facilitates computationally efficient simulations of local dynamics. Here we propose a new systematic approach to generating quantum circuits with complex...
Singular Value Decomposition (SVD) underlies matrix factorisation tasks across computational imaging, with medical applications increasingly demanding real-time processing. Yet SVD algorithms are inherently sequential, constraining real-time GPU throughput and limit online deployment in clinical pipelines. This study introduces...
We prove a series of new results in Hamiltonian dynamics on a general closed symplectic manifold $(M, ω)$, including: 1. If $M$ admits a Hamiltonian diffeomorphism which is either a pseudo-rotation or has finite order, then $M$ is geometrically uniruled. This resolves a variant of Problem 24 in McDuff--Salamon's list, which predicts...
We prove Pólya's conjectured lower bound for the Neumann counting function of every Euclidean ball. If $d\ge2$, $R>0$, and $E\ge0$, then \begin{equation*} N_{B_R^d}^{<}(E) \ge \frac{ω_d}{(2π)^d}|B_R^d|E^{d/2} = \frac{(R\sqrt E)^d}{2^dΓ(\frac d2+1)^2}. \end{equation*} The radial boundary condition in dimensions $d\ge3$ is a Dini...
An antichain is a family of rectangles in which no member contains another. Given a family $\mathcal{E}$ of rectangles, let $h_{\mathcal{E}}$ be the sum of the indicator functions of its members. We show that there exist constants $c, C > 0$ such that for every sparse antichain $\mathcal{E}$ of dyadic rectangles in $\mathbb{R}^2$ one...
Multi-warehouse inventory allocation is typically formulated as a mixed-integer programming (MIP) problem, yet no single formulation consistently matches heterogeneous instance-level regimes induced by demand concentration, inventory imbalance, replenishment scale, service constraints, and forecast volatility. We study this issue as...
We establish necessary and sufficient conditions for the uniform boundedness of the multi-parameter singular exponential sum $$ \sum_{|t_1|\le N_1,\dots,|t_k|\le N_k} \frac{e^{2πi P(t_1,\dots,t_k)}}{t_1\cdots t_k}, $$ where $P:\mathbb{Z}^k\to\mathbb{R}$ is a polynomial of the form $ P(t)=\sum_{\mathfrak{m}\in Λ} c_{\mathfrak{m}}\,...
In this work, we develop a weak-form version of Extended Dynamic Mode Decomposition that we call WEDMD. We establish a number of analytic results about the method and show explicitly how the weak form is able to mitigate the impacts of noise in linear stochastic differential equations. In nonlinear systems, we likewise show how the...
We develop an analytic Fourier cosine (COS) method for the valuation of compound options. By deriving closed-form expressions for the cosine coefficients at all compound stages, the proposed method eliminates the need for numerical quadrature in intermediate exercise stages while retaining the convergence properties of the underlying...
Mechanistic interpretability has largely focused on language models and deterministic toy tasks. Much less is known about how sequence models internally represent latent stochastic dynamics under noisy, partially observed observations. We study this question in a controlled multivariate stochastic volatility setting, where models...
A finite panel of option quotes with bid-ask spreads generally does not point-identify either the risk-neutral probability of breaching a specified threshold or the expected shortfall below that threshold conditional on a breach. Sharp marginal bounds characterize each quantity in isolation but not their jointly attainable...
Hedging a derivative position under transaction costs and market frictions requires a trading rule that adapts to changing conditions. Deep hedging trains a neural policy for this task but policy training does not determine whether a trading desk can afford to run the policy. We apply robust hedging valuation adjustment (HVA) as a...
Automated code generation is becoming an important tool in quantitative finance, where large language models can generate option pricing implementations directly from mathematical model specifications. Validating such implementations, however, requires considerably more than conventional software testing: numerical pricing methods...
This paper proposes a GARCH-type volatility model in which level-and-slope updates of a latent power-law kernel generate state-dependent decay of past shocks within a two-dimensional Markov state. We derive a joint Foster--Lyapunov condition and establish positive Harris recurrence and uniqueness of the invariant distribution....
Solving high-dimensional parabolic partial differential equations (PDEs) is important in engineering, physics, and stochastic control. Deep BSDE methods reformulate semilinear PDEs as backward stochastic differential equations and admit a model-based reinforcement learning interpretation, where trajectories are generated from known...