arXiv preprints from January 1, 2026 through September 19, 2026 — 09:31:16 EST
This paper shows that the optimal gain of the risk-sensitive linear quadratic regulator (LQR) problem is invariant under input randomization, i.e., when the controller deliberately injects noise into the nominal control input. This appears counterintuitive at first glance because certainty equivalence does not hold for risk-sensitive...
In this lecture note, we present a concise and self-contained derivation of the discrete-time Kalman filter equations that requires only a basic understanding of least squares estimation. The treatment is designed to minimize mathematical overhead while preserving both rigor and generality.
Recent advances in equation discovery methods such as SINDy have highlighted the growing interest in identifying governing parameters and models directly from data. In this work, we take a complementary approach grounded in analysis and numerical PDE methods: we recover an unknown viscosity in steady Navier-Stokes equations (NSE) from...
We develop a localized persistent theory of commutative algebra for Stanley-Reisner rings, based on local cohomology supported at a coordinate prime rather than at the maximal ideal. The construction is modeled on the persistent Stanley-Reisner theory of Suwayyid and Wei (arXiv:2503.23482) and its functorial development for graphs and...
We study how far gradient descent (GD) can be accelerated by predetermined stepsizes in smooth convex optimization. Going beyond the classical $Ω(n^{-2})$ first-order oracle lower bound of Nemirovsky and Yudin, we prove an $Ω(n^{-1.6342})$ non-anytime lower bound and an $Ω(n^{-1.2408})$ anytime lower bound. These improve the recent...
We construct realizable-volume models over any field for the factorially normalized homogeneous Lascoux, Lascoux-atom, and positive Grothendieck packets. Their volume minors include normalized key polynomials, Demazure atoms, Schubert polynomials, and all sign-corrected homogeneous Grothendieck components. Over $\mathbb C$, these...
This article equips the universal moduli spaces of non-degenerate $\mathbb{Z}/2\mathbb{Z}$ harmonic spinors with the structure of a tame Fréchet manifold and proves that the projection map to the parameter space is uniformly Fredholm. This recovers earlier work of Donaldson, Parker, and Takahashi in a unified framework.
A $B$-coloring of a graph is a proper edge-coloring in which every $4$-cycle is rainbow, and $q_B(G)$ denotes the minimum number of colors in such a coloring. Let $Δ_2(G)$ denote the maximum number of common neighbors of two distinct vertices of $G$. We prove that, for integers $1\le d\leΔ$, every finite simple $d$-degenerate graph...
The first paper worked out one family of four lattices in full -- $|\det| = 12, 24, 36$ and $72$. This paper generalizes it. We enumerate the 44 symmetrizable rank-three hyperbolic Cartan matrices and their 56 depth-one edits, 100 root bases in all, and compute the obstruction space $S_{5/2}(ρ_L)$ of the 98 within our weight-$5/2$...
We prove the global existence and uniqueness of strong solutions to a constrained version of the damped nonlinear wave equation $$ \vartheta_{tt}+γ\vartheta_t-Δ\vartheta+|\vartheta|^{p-2}\vartheta=0 $$ on a smooth bounded domain $\varOmega\subset\mathbb{R}^d$, where the evolution is projected onto the tangent space of the Hilbert...
This survey paper provides a comprehensive study of unitarily invariant norms on the algebra of $n \times n$ matrices. This investigation leads naturally to the theory of symmetric gauge functions, a class of norms on $\mathbb{R}^n$ characterized by invariance and monotonicity properties. We develop the necessary framework by...
We give an example showing that the dual of an algebraic matroid need not be algebraic. Our example is a matroid of rank 6 on 10 elements which is algebraic in characteristic 2. It is obtained by gluing an algebraic realization of the non-Fano matroid to a realization of the Fano matroid. Its dual is shown not to be algebraic using...
The divisible sandpile on $\mathbb{Z}^d$ starts from i.i.d. masses at each site, and, in each discrete time step, a site with mass above one keeps one unit and sends the excess equally to its neighbors. Levine, Murugan, Peres and Ugurcan (2016) showed that at mean one this process explodes, with every site emitting infinite mass. We...
Let $R$ be a discrete valuation ring. We prove that every proper flat finitely presented commutative $R$-group scheme $P$ fits into an exact sequence \[ 0\longrightarrow E\longrightarrow P\longrightarrow B\longrightarrow0, \] where $E$ is finite flat and $B$ is an abelian scheme. For a quasiabelian model---that is, a proper flat...
We study the category of dual operator spaces equipped with an action of a locally compact quantum group $\mathbb{G}$. The Fubini crossed product functor $-\rtimes^\mathcal{F} \mathbb{G}$ and the weak$^*$-crossed product functor $-\bar{\rtimes}\mathbb{G}$ are shown to be equal if and only if $\mathbb{G}$ has the approximation property...
In the parking model, each site of the $d$-dimensional lattice independently starts with one car with probability $p$ or one parking spot with probability $1-p$. Cars move according to independent discrete-time simple random walks and park at the first spot they find free. We prove that in the critical regime $p=1/2$, the expected...
We study a parabolic obstacle partial integro-differential equation (PIDE) with a dynamically moving bilateral free boundary. This type of problem arises in the mathematical modeling of speculative asset bubbles with Lévy jump processes. We consider the existence of viscosity supersolutions within the class of functions exhibiting...
We investigate the dynamics of Chebyshev's method applied to the polynomial family $p_n(z)=z(z^n-1)$ for $n>1$. The resulting map is denoted by $C_n$. It is proved that the immediate basins corresponding to the non-zero roots are unbounded and simply connected. We also show that the Julia set of $C_n$ is connected. It is proved that...
We obtain a new, conceptually simpler, unconditional proof that more than $67.25\%$ of the non-trivial zeros of the Riemann zeta function are simple and on the critical line, and that at least $83.62\%$ of the non-trivial zeros are distinct. A proof of these results was very recently produced by an internal research version of Claude...
For a semiconcave function with linear modulus, we introduce the cut locus through a touching approach and prove that it coincides with the variational cut locus defined via the Lax--Oleinik semigroup, independently of the Hamiltonian. Cut points are characterized by the emptiness of the proximal subdifferential of $φ$, which we use...
We study collapsing finite-time singularities of the unnormalized Kähler--Ricci flow on Fano bundles arising in the analytic minimal model program. For a Fano bundle $X^n\rightarrow Y^m$, we prove a maximal splitting theorem by the Kähler-Ricci flow that every tangent space at a fixed limiting point splits globally as \((\C^m,g_{\rm...
We obtain upper bounds for the number of real zeros of functions of the form $$ f(x) = \sum_{k=1}^{n} c_k \bigl(P_k(x)\bigr)^{α_k}, $$ where $c_k, α_k \in \mathbb{R}$ and each $P_k$ is a real polynomial of degree at most $d$ that is non-negative on an interval $I\subset \mathbb{R}$. We improve previously known exponential upper bounds...
We study Gaussian-restricted barycenters for quadratic two-sided Kullback--Leibler unbalanced optimal transport with independent marginal penalties and no coupling entropy. Exact profiling of the barycenter mass reduces the problem to a smooth Gaussian shape functional with endogenous Gibbs weights. We establish global attainment,...
For a bipartite graph $G$ and an integer $t\geq2$, let $\NM_t(G)$ be its $t$-non-matching complex. We prove that adding an edge while preserving bipartiteness induces an injection on reduced homology in degree $2t-3$. Combined with the cyclic-polytope model for non-matching complexes of cycles, this shows that $\NM_t(G)$ has Leray...
We study the classical $XY$-model on random geometric graphs $\mathcal{G}_{n, \varepsilon}$, which are obtained by sampling $n \in \mathbb{N}$ independent points in a finite domain $Ω\subset \mathbb{R}^d$, $d \geq 2$, and connecting two points by and edge if their distance is of order $\varepsilon > 0$. We refer to $\mathcal{G}_{n,...