arXiv preprints from January 1, 2026 through September 19, 2026 — 05:32:02 EST
Time series of random objects, such as covariance matrices, probability distributions, and functional data, call for forecasting methods that do not rely on standard arithmetic operations. We introduce geodesic exponential smoothing, a generalization of exponential smoothing to time series in Hadamard spaces: the forecast level moves...
We prove that every traceless real or complex matrix $A$ can be written as $A=BC-CB$, with factors of the same size and over the same field satisfying $\|B\|\,\|C\|\le K\|A\|$, where $K$ is an absolute constant. The proof combines an approximate-rank dichotomy with stable commutator representations obtained from quantum expansion.
We introduce a matrix generalization of the cone of Harvey and Lawson and prove that it is an exact special Lagrangian manifold. We further show that it belongs to a family of exact special Lagrangian manifolds that foliate the balanced manifold arising in deep learning.
We compute the groups of minimal nondegenerate extensions of the real symmetric fusion categories $\mathrm{sVec}_{\mathbb R}$ and $\mathrm{sVec}_{\mathbb H}$. We find that they are both isomorphic to the Klein-four group $(\mathbb{Z}/2\mathbb{Z})^2$. Along the way, we classify all nondegenerately braided fusion categories over...
This is a survey on Pham-Brieskorn varieties and their various properties, targeted for graduate students.
Exceptional-point signatures in first-passage observables are governed by two independent survival mechanisms: nonlinear spectral defectivity must pass from the one-record or transfer description to the physical one-level first-passage ladder, and the resulting ladder Jordan mode must have nonzero overlap with the chosen preparation...
Viscosity solutions of eikonal equations need not enjoy regularity beyond Lipschitz continuity. In contrast, differentiable solutions exhibit stronger structural properties. In the Euclidean setting, building on observations of Caffarelli and Crandall (Comm. Partial Differential Equations $\textbf{35}$ (2010), 391--414), one finds...
The Koopman operator has gained considerable attention in dynamical systems due to its capability to provide a linear viewpoint for nonlinear systems using data-driven methods such as extended dynamic mode decomposition (EDMD). In this work, we suggest an EDMD-variant with a Fourier dictionary on the $d$-dimensional torus, where the...
We prove that, in Zermelo--Fraenkel set theory with the axiom of Foundation removed, the statement that every vector space has a basis implies the Axiom of Choice, concluding that the classical equivalence between $\AC$ and the existence of bases does not require the Axiom of Foundation. More specifically, we prove that if every...
Variance swaps are a convenient instrument for trading vega and convexity, and a listed contract now trades on Cboe. We ask when a trader should put such a position on and when she should take it off, and for a perpetual, continuously settled contract we answer both in closed form: each threshold is the unique root of a smooth-pasting...
This paper studies the regularity of the value function arising from a multidimensional continuous-time principal-agent model with separable, nonquadratic effort costs. The associated stochastic control problem has the output and the agent's continuation utility as state variables, and its Hamilton-Jacobi-Bellman equation is fully...
Randomized trials with limited concurrent control information---including but not limited to unequal-randomization settings---can offer ethical and practical advantages, especially in pediatric and rare diseases, but they often lose power because fewer control patients are available for direct comparison. Borrowing information from...
In physics-informed machine learning, a target function $u^*$ is learned from noisy value observations $y_i=u^*(x_i)+ \varepsilon_i$, together with differential information, given either by noisy observations $d_j=(Du^*)(z_j)+ξ_j$ or by a known physical constraint $Du^*=v$. We consider the setting where $D$ is a linear differential...
Many datasets arise as multivariate time series observed at irregular time intervals, particularly in Earth observation and astronomical measurements. Classical time-series models assume that time is discrete and observations occur at equally spaced intervals, limiting their ability to capture dynamics when observation gaps vary....
Randomized controlled trials provide internally valid estimates of treatment effects, but their results may not directly apply to broader target populations due to differences in baseline covariate distributions, adherence to treatment or variations in outcome mechanisms. Under standard transport and time-to-event identifiability...
Antibody lead optimization calls for a small, bounded set of edits to an existing candidate: substitutions, but also insertions and deletions. Edit-based generative models are the only ones that allocate such an edit budget without fixing the edit positions, the edit count, or the output length in advance. However, the existing...
Parametric copulas capable of capturing full-range tail dependence in both lower and upper tails are useful in many applications, including as pair-copula building blocks in vine copula models. However, it is challenging to create such copulas that have closed-form density functions, which limits their practical applicability in...
Time-to-event outcomes are central in oncology and rare diseases, where treatment effects are often summarized by differences in survival curves or Restricted Mean Survival Time (RMST). In real-world data, estimating these causal effects relies on the absence of unobserved confounding, an assumption that is rarely satisfied. We...
How many compensatory seats are needed to ensure (full) proportionality in two-tier electoral systems with multi-member constituencies? This question is answered by estimating the discrepancy between seats and votes rigorously, probabilistically and empirically. In 1919, Pólya showed that the Jefferson/D'Hondt method favours large...
This paper provides a self-contained proof of the rank of the pathlifting Jacobian of a DAG ReLU network by performing an induction on the network's number of hidden nodes. In fact, the induction is elementary, and the key recipe is to consider the skeleton matrix of the network, a sparse matrix encoding the network paths, and...
Plasmonic excitation of metallic nanostructures can generate energetic carriers capable of transferring charge into nearby adsorbed molecules, providing a possible pathway for driving chemical transformations. In many theoretical descriptions of this process, charge transfer is determined by the equilibrium electronic structure of the...
Magic state cultivation prepares high-fidelity magic states at low expected space-time costs; however, the exact performance of some schemes is unsettled due to the difficulty in simulating non-Clifford circuits. Here, we use matrix-product states (MPS) based methods to compute the exact performance of two types of fold-transversal...
The realization of practical, high-speed magnonic devices requires engineering magnetic materials with low dissipation over wide frequency ranges. While optical and microwave probes are used to infer the damping of low energy/long wavelength modes, the degree to which these $q\sim0$ properties translate into higher-energy,...
Large scale magnetic fields play a crucial role in shaping the dynamics and structure of the inner parts of accretion disks and the resulting jets from the vicinity of black holes and neutron stars. Currently, the primary mechanism considered for generating and amplifying large scale magnetic fields in accretion disks is the...
The Chinese Space-station Survey Telescope (CSST) is a flagship mission designed to carry out a large-area high-resolution multiband imaging and slitless spectroscopic survey over a decade. The survey's science drivers, ranging from precision cosmology to stellar astrophysics, impose stringent requirements on the performance of its...