arXiv preprints from January 1, 2026 through September 19, 2026 — 06:25:34 EST
The Erd\H os--Sós conjecture states that every $n$-vertex graph with more than $(k-2)n/2$ edges contains every $k$-vertex tree. We prove that for every $γ$ there is an $n_0$ such that for all $n\ge n_0$ and $k \ge γn$ the conjecture holds. As a corollary of our result, we obtain a solution of a 51-year-old problem of Erd\H os and...
By Tutte's classic theorem of 1956 that every 4-connected planar graph is Hamiltonian, every planar graph of order at least three with toughness greater than $\frac{3}{2}$ is Hamiltonian. In 1999, Owens constructed a sequence of maximal planar graphs whose toughness approaches $\frac{3}{2}$ from below and which do not contain even a...
The Erd\H os--Sós conjecture states that every $n$-vertex graph $G$ with more than $(k-2)n/2$ edges contains every $k$-vertex tree. We solve the extremal cases of this conjecture, showing that for some fixed $μ>0$, the conjecture holds for each $G$ that minimally satisfies the assumptions of the conjecture and has a subgraph~$H$...
Let $(R,\mathfrak m)$ be a Noetherian local ring and $J$ be an arbitrary ideal of $R$. Suppose that $f_1,\ldots,f_r$ is a $J$-filter regular sequence in $R$ and $I=(f_1,\ldots,f_r)$. In this paper, we establish an improved upper bound for the perturbation index of the associated graded ring $\mathrm{gr}_J(R/I)$, refining the bound...
We consider the adjacency matrix of a random directed $d$-regular graph on $N$ vertices. For fixed $d\geq 2$, we prove that the empirical eigenvalue density converges in probability to the oriented Kesten--McKay law as $N\to \infty$. The key technical input is the small-ball probability estimate for the smallest singular value. The...
We apply the variational method of Weighted Energy Dissipation (WED) to obtain a global-in-time approximation of solutions to fourth order degenerate parabolic equations of Cahn-Hilliard type. Differently from the standard approach to the existence theory, WED induces an elliptic regularization in time, not in space. The confinement...
Let an infinite sequence of independent and identically distributed random variables over a countable alphabet generate a graph by joining consecutive symbols and suppressing repeated edges. We determine the exact tail and local asymptotics of the limiting degree distributions of this range-renewal graph. If the ordered sampling...
Suppose that the eventual use of data is not known when the data are reduced or collected. This note considers two simple boundary cases. In a finite statistical experiment, a statistic preserves the Bayes risk for every finite later decision problem if and only if it is sufficient. Hence, when the minimal sufficient statistic is...
Inspired by questions related to inverse problems and shape optimization, we derive an asymptotic expansion of the voltage potential, solution to a model elliptic second-order partial differential equation, under small perturbations of its boundary conditions. More precisely, the homogeneous Dirichlet or homogeneous Neumann boundary...
We complete the local classification, up to isometry, of $2$-dimensional pseudo-Riemannian metrics (i.e. both Riemannian and Lorentzian), admitting a projective symmetry algebra of dimension at least two. The new contribution is the treatment of the non-regular case: we obtain a complete list of mutually non-isometric normal forms in...
We compute the limit of the eigenvalue of the $k$-Hessian operator on the unit ball in $\mathbb{R}^n$ when $k\rightarrow \infty$, assuming that the ratio $\frac{n}{k}$ remains fixed. When $n< 2ke$, we moreover identify the limit of the corresponding eigenfunctions. We also derive monotonicity results and consider when the ratio varies in $n$.
Relative ($τ$) is equivalent to a statement that the sequence of Cayley graphs associated to group quotients $Γ_q=Γ/N_q$, $q\in\mathbb{N}$, form an expander family. There is a philosophy that expander graphs give rise to good mixing; for instance, one has exponential mixing for the geodesic flow uniformly the along a family of...
We study a natural extension of Ramsey theory relative to the classes of maximally planar and maximally outerplanar graphs. This can be seen as a continuation of the study of `Planar Ramsey theory', introduced by Axenovich et al. The question we ask is the following: For a fixed family $\mathcal{K}$ of graphs and a pair of graphs...
We investigate the best $L_2$ approximation of mixed Sobolev spaces by shallow neural networks with $n$ neurons and general activation functions. We first establish an activation-independent Fourier-block principle: if an activation has univariate approximation order $ρ$ in the sense of the Fourier-block property, then the global...
A total coloring of a graph is an assignment of colors to its vertices and edges so that adjacent or incident elements receive distinct colors, and it is equitable when the cardinalities of any two color classes differ by at most one. Stemock conjectured that every $4$-total coloring of a cubic graph of order less than $20$ is...
Large language models (LLMs) are increasingly used to formulate optimization models from natural-language problem descriptions, yet realistic operations research (OR) requests are often incomplete: missing objectives, constraints, or business rules can change the resulting mathematical program. Existing evaluations largely assume a...
This paper proves a generalisation of Coleman's isomorphism (between norm compatible cyclotomic units and a group of invertible power series) to various cohomology theories evaluated on proper regular $p$-adic formal schemes, for future applications to Iwasawa theory. This generalisation is an immediate consequence of a description,...
We develop an energy-optimal generalized scalar auxiliary variable (EOP-GSAV) framework for nonlinear index-one port-Hamiltonian differential-algebraic equations (pH-DAEs). Exploiting the port-Hamiltonian structure, we separate the nonlinear effort, interconnection, and dissipation terms from a constant implicit core. The resulting...
We consider a rectangular random matrix formed by adding a rank-one term to a matrix with independent entries. The direction on the left side of that term is estimated by the leading left singular vector, and the error is multiplied by the square of the size of the added term. For entries with mean zero and variance one, we identify...
We investigate the Weyl representation of the canonical commutation relations for a model describing a quantized massive scalar field under the influence of an external potential. The main problem is to determine whether the Weyl representation remains equivalent to or becomes inequivalent to the original one when the mass and/or...
We study regression under bounded responses in terms of excess mean squared error. When the comparator class is finite, this setting is known as model selection aggregation, and achieving minimax excess risk requires improper learning algorithms. Contrary to this, in the universal learning framework no improperness is needed, as...
We consider density estimation under the relaxed local differential privacy condition that the privatized distributions are $α$-close in total variation distance. We show that adding independent noise with a convenient symmetrized Gamma distribution to each sensitive observation attains the $α$-TV-LDP. We prove that the deconvolution...
Let $C$ be an $[n,k]_q$ maximum distance separable (MDS) code with $4\le k\le q-3$, and suppose that it has a $2$-transitive permutation automorphism group. In this paper we show that $n\le q+1$, so the MDS conjecture holds for this class of codes.
In this letter, we study finite-horizon linear-quadratic Stackelberg difference games with coupled-affine state-control inequality constraints. Under the stated assumptions, we show that generalized open-loop Stackelberg equilibria admit an exact reformulation as an optimal control problem subject to a discrete-time linear...
In this work, we study policy optimization under domain randomization for linear quadratic control, focusing on learning a single state-feedback controller that minimizes the average cost across systems with uncertain dynamics. We propose a policy iteration algorithm with a step-size rule that preserves stability across all sampled...