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Stability Estimates for Commutativity Properties of the Dirichlet-to-Neumann Operator
Romain Speciel
https://arxiv.org/abs/2510.08822 https://arxiv.org/pdf…

Stability Estimates for Commutativity Properties of the Dirichlet-to-Neumann Operator
The Laplacian $Δ_{\mathbb{S}^{n-1}}$ on the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$ has the property that it can explicitly be expressed in terms of $Λ$, the Dirichlet-to-Neumann map of the unit ball, as $Δ_{\mathbb{S}^{n-1}}=Λ^2+(n-2)Λ$. In this paper, we seek to characterize those manifolds for which such an exact relationship holds, and more generally measure the discrepancy of such a relationship holding in terms of geometric data. To this end, we obtain a stability estimate…

Gr\"obner bases and the second generalized Hamming weight of a linear code
Hern\'an de Alba (SECIHTI, Universidad Aut\'onoma de Zacatecas), Cecilia Mart\'inez-Reyes (Universidad Aut\'onoma de Zacatecas)
https://arxiv.org/abs/2510.09917

Gröbner bases and the second generalized Hamming weight of a linear code
It is known that for binary codes one can use Gröbner bases to obtain a subset of codewords of minimal support that can be used to determine the second generalized Hamming weight of the code. In this paper we establish conditions on a nonbinary code under which the same property holds. We also construct a family of codes over any nonbinary finite field where the property does not hold. Furthermore, we prove that whenever the subset obtained via Gröbner basis suffices to determine the second g…

Totally paracompact spaces and the Menger covering property
Davide Giacopello, Maddalena Bonanzinga, Piotr Szewczak
https://arxiv.org/abs/2511.10252 https://arxiv.org/pdf/2511.10252 https://arxiv.org/html/2511.10252
arXiv:2511.10252v1 Announce Type: new
Abstract: A topological space is totally paracompact if any base of this space contains a locally finite subcover. We focus on a problem of Curtis whether in the class of regular Lindel\"of spaces total paracompactness is equivalent to the Menger covering property. To this end we consider topological spaces with certain dense subsets. It follows from our results that the above equivalence holds in the class of Lindel\"of GO-spaces defined on subsets of reals. We also provide a game-theoretical proof that any regular Menger space is totally paracompact and show that in the class of first-countable spaces the Menger game and a partial open neighborhood assignment game of Aurichi are equivalent. We also show that if $\mathfrak{b}=\omega_1$, then there is an uncountable subspace of the Sorgenfrey line whose all finite powers are Lindel\"of, which is a strengthening of a famous result due to Michael.
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A sparse canonical van der Waerden theorem
Jos\'e D. Alvarado, Yoshiharu Kohayakawa, Patrick Morris, Guilherme O. Mota, Miquel Ortega
https://arxiv.org/abs/2510.03084 https:…

A sparse canonical van der Waerden theorem
The canonical van der Waerden theorem asserts that, for sufficiently large $n$, every colouring of $[n]$ contains either a monochromatic or a rainbow arithmetic progression of length $k$ ($k$-AP, for short). In this paper, we determine the threshold at which the binomial random subset $[n]_p$ almost surely inherits this canonical Ramsey type property. As an application, we show the existence of sets $A\subseteq [n]$ such that the $k$-APs in $A$ define a $k$-uniform hypergraph of arbitrarily hig…

Haar random codes attain the quantum Hamming bound, approximately
Fermi Ma, Xinyu Tan, John Wright
https://arxiv.org/abs/2510.07158 https://arxiv.org/pdf/2…

Haar random codes attain the quantum Hamming bound, approximately
We study the error correcting properties of Haar random codes, in which a $K$-dimensional code space $\boldsymbol{C} \subseteq \mathbb{C}^N$ is chosen at random from the Haar distribution. Our main result is that Haar random codes can approximately correct errors up to the quantum Hamming bound, meaning that a set of $m$ Pauli errors can be approximately corrected so long as $mK \ll N$. This is the strongest bound known for any family of quantum error correcting codes (QECs), and continues a li…

Recovery of Integer Images from Limited DFT Measurements with Lattice Methods
Howard W Levinson, Isaac Viviano
https://arxiv.org/abs/2510.11949 https://arx…

Recovery of Integer Images from Limited DFT Measurements with Lattice Methods
Exact reconstruction of an image from measurements of its Discrete Fourier Transform (DFT) typically requires all DFT coefficients to be available. However, incorporating the prior assumption that the image contains only integer values enables unique recovery from a limited subset of DFT coefficients. This paper develops both theoretical and algorithmic foundations for this problem. We use algebraic properties of the DFT to define a reduction from two-dimensional recovery to several well-chosen…

Establishing strong 1-boundedness via non-microstates free entropy techniques
Benjamin Major, Dimitri Shlyakhtenko
https://arxiv.org/abs/2510.07558 https://

Establishing strong 1-boundedness via non-microstates free entropy techniques
We show that, for many choices of finite tuples of generators $X = (x_1, \dots , x_d)$ of a tracial von Neumann algebra $(M, τ)$ satisfying certain decomposition properties (non-primeness, possessing a Cartan subalgebra, or property $Γ$), one can find a diffuse, hyperfinite subalgebra $N \subseteq (W^*(X))^ω$ (often in $W^*(X)$ itself), such that $W^*(N,X+\sqrt{t}S) = W^*(N,X,S)$ for all $t > 0$. (Here $S$ is a free semicircular family, free from $\{X\} \cup N$). This gives a short non-micro…

Concentrated sets and the Hurewicz property
Valentin Haberl, Piotr Szewczak, Lyubomyr Zdomskyy
https://arxiv.org/abs/2511.09320 https://arxiv.org/pdf/2511.09320 https://arxiv.org/html/2511.09320
arXiv:2511.09320v1 Announce Type: new
Abstract: A set of reals $X$ is $\mathfrak{b}$-concentrated if it has cardinality at least $\mathfrak{b}$ and it contains a countable set $D\subseteq X$ such that each closed subset of $X$ disjoint with $D$ has size smaller than $\mathfrak{b}$. We present ZFC results about structures of $\mathfrak{b}$-concentrated sets with the Hurewicz covering property using semifilters. Then we show that assuming that the semifilter trichotomy holds, then each $\mathfrak{b}$-concentrated set is Hurewicz and even productively Hurewicz. We also show that the appearance of Hurewicz $\mathfrak{b}$-concentrated sets under the semifilter trichotomy is somewhat specific and the situation in the Laver model for the consitency of the Borel Conjecture is different.
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Quantitative Gaffney and Korn inequalities
Wadim Gerner
https://arxiv.org/abs/2510.05870 https://arxiv.org/pdf/2510.05870

Quantitative Gaffney and Korn inequalities
We prove a homogeneous, quantitative version of Ehrling's inequality for the function spaces $H^1(Ω)\subset\subset L^2(\partialΩ)$, $H^1(Ω)\hookrightarrow L^2(Ω)$ which reflects geometric properties of a given $C^{1,1}$-domain $Ω\subset\mathbb{R}^n$. We use this result to derive quantitative homogeneous versions of Gaffney's inequality, of relevance in electromagnetism as well as Korn's inequality, of relevance in elasticity theory. The main difference to the corresponding classical re…

Gradient regularity for widely degenerate parabolic equations
Michael Strunk
https://arxiv.org/abs/2510.07999 https://arxiv.org/pdf/2510.07999

Gradient regularity for widely degenerate parabolic equations
In this paper, we are interested in the regularity of weak solutions $u\colonΩ_T\to\mathbb{R}$ to parabolic equations of the type \begin{equation*} \partial_t u - \mathrm{div} \nabla \mathcal{F}(x,t,Du) = f\qquad\mbox{in $Ω_T$}, \end{equation*} where $\mathcal{F}$ is only elliptic for values of $Du$ outside a bounded and convex set $E\subset \mathbb{R}^n$ with the property that $0\in \mathrm{Int}{E}$. Here, $Ω_T :=Ω\times(0,T)\subset\mathbb{R}^{n+1}$ denotes a space-time cylinder taken ov…