arXiv subjects

Eliseo Sarmiento Rosales

Publications and source records attributed to Eliseo Sarmiento Rosales.

4 recordsLinked to original sources

arXiv2026

Hindman and Owings-like theorems without the Axiom of Choice

We investigate Hindman- and Owings-type Ramsey-theoretic statements in Zermelo-Fraenkel set theory without the Axiom of Choice, with some occasional extra assumptions (such as the Axiom of Dependent Choice and/or the Axiom of Determinacy). We study several variations of Hindman's theorem on $\mathbb Q$-vector spaces; notably, we show that the uncountable analog of Hindman's theorem fails for the additive group of $\mathbb R$ (under ZF), and for $\mathbb Q$-vector spaces of uncountable dimension (under DC if such dimension is not well-orderable), among other results. In contrast, for Owings-type configurations, we obtain several positive results, especially when assuming AD. These results highlight the interaction between determinacy, algebraic structure, and dimension in the study of infinite Ramsey theory without the Axiom of Choice.

math.LO↗

arXiv2026

Geometric Classification of Biased Quantum Capacity via Harmonic Translation

We establish an exact noise-model-derived characterization of quantum error correction under diagonal local phase noise. Under uniform locality, the maximal logical dimension under t-local phase errors equals Aq(n,2t+1), the classical q-ary packing function. Because no affine or stabilizer structure is imposed, nonlinear spectral supports achieve this bound and strictly exceed all affine constructions whenever Aq(n,2t+1)>Bq(n,2t+1). This follows from a harmonic translation principle: diagonal phase operators act as rigid translations in the Fourier domain, reducing the Knill-Laflamme conditions exactly to an additive non-collision constraint (S-S) cap Et={0}. For structured phase noise, exact correction is equivalent to independence in an additive Cayley graph, connecting biased quantum capacity to classical zero-error theory and the Lovasz theta function. Under mixed Pauli noise, simultaneous protection in conjugate domains incurs an intrinsic rate penalty R <= 1-(gamma_X+gamma_Z)/2, exposing a discrete harmonic uncertainty principle. In contrast with stabilizer- or graph-based frameworks, this classical correspondence is derived directly from the phase-noise model itself rather than from an auxiliary algebraic construction.

quant-ph↗

arXiv2024

Owings-like theorems for infinitely many colours or finite monochromatic sets

Inspired by Owings's problem, we investigate whether, for a given an Abelian group $G$ and cardinal numbers $κ,θ$, every colouring $c:G\longrightarrowθ$ yields a subset $X\subseteq G$ with $|X|=κ$ such that $X+X$ is monochromatic. (Owings's problem asks this for $G=\mathbb Z$, $θ=2$ and $κ=\aleph_0$; this is known to be false for the same $G$ and $κ$ but $θ=3$.) We completely settle the question for $κ$ and $θ$ both finite (by obtaining sufficient and necessary conditions for a positive answer) and for $κ$ and $θ$ both infinite (with a negative answer). Also, in the case where $θ$ is infinite but $κ$ is finite, we obtain some sufficient conditions for a negative answer as well as an example with a positive answer.

math.LO↗

arXiv2011

Projective Parameterized Linear Codes Arising from some Matrices and their Main Parameters

In this paper we will estimate the main parameters of some evaluation codes which are known as projective parameterized codes. We will find the length of these codes and we will give a formula for the dimension in terms of the Hilbert function associated to two ideals, one of them being the vanishing ideal of the projective torus. Also we will find an upper bound for the minimum distance and, in some cases, we will give some lower bounds for the regularity index and the minimum distance. These lower bounds work in several cases, particularly for any projective parameterized code associated to the incidence matrix of uniform clutters and then they work in the case of graphs.

cs.IT↗