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Javier Cabello Sánchez

Publications and source records attributed to Javier Cabello Sánchez.

18 recordsLinked to original sources
arXiv2024

A reflection on Tingley's problem and some applications

In this paper we show how some metric properties of the unit sphere of a normed space can help to approach a solution to Tingley's problem. In our main result we show that if an onto isometry between the spheres of strictly convex spaces is the identity when restricted to some relative open subset, then it is the identity. This implies that an onto isometry between the unit spheres of strictly convex finite dimensional spaces is linear if and only it is linear on a relative open set. We prove the same for arbitrary two-dimensional spaces and obtain that every two-dimensional, non strictly convex, normed space has the Mazur-Ulam Property. We also include some other less general, yet interesting, results, along with a generalisation of curvature to normed spaces.

math.MG↗
arXiv2024

A characterisation of Euclidean normed planes via bisectors

Our main result states that whenever we have a non-Euclidean norm $\|\cdot\|$ on a two-dimensional vector space $X$, there exists some $x\neq 0$ such that for every $λ\neq 1, λ>0$, there exist $y, z\in X$ verifying that $\|y\|=λ\|x\|$, $z\neq 0$, and $z$ belongs to the bisectors $B(-x,x)$ and $B(-y,y)$. Throughout this paper we also state and prove some other simple but maybe useful results about the geometry of the unit sphere of strictly convex planes.

math.MG↗
arXiv2024

TAC Method for Fitting Exponential Autoregressive Models and Others: Applications in Economy and Finance

There are a couple of purposes in this paper: to study a problem of approximation with exponential functions and to show its relevance for the economic science. We present results that completely solve the problem of the best approximation by means of exponential functions and we will be able to determine what kind of data is suitable to be fitted. Data will be approximated using TAC (implemented in the R-package nlstac), a numerical algorithm for fitting data by exponential patterns without initial guess designed by the authors. We check one more time the robustness of this algorithm by successfully applying it to two very distant areas of economy: demand curves and nonlinear time series. This shows TAC's utility and highlights how far this algorithm could be used.

math.NA↗
arXiv2024

A natural correspondence between quasiconcave functions and fuzzy norms

In this note we show that the usual notion of fuzzy norm defined on a linear space is equivalent to that of quasiconcave function, in the sense that every fuzzy norm $N:X\times\mathbb{R}[0,1]$ defined on a (real or complex) linear space X is uniquely determined by a quasiconcave function $f:X\to[0, 1]$. We explore the minimum requirements that we need to impose to some quasiconcave function $f:X\to[0, 1]$ in order to define a fuzzy norm $N:X\times\mathbb{R}[0,1]$. Later we use this equivalence to prove some properties of fuzzy norms, like a generalisation of the celebrated Decomposition Theorem.

math.MG↗
arXiv2023

Chebyshev approximation of exponential data

In this paper we present an algorithm to fit data via exponentials when the error is measured using the max-norm. We prove the necesssary results to show that the algorithm will converge to the best approximation no matter the dataset.

math.NA↗
arXiv2022

Dynamics of the semigroup of contractive automorphisms of Banach spaces

Motivated by some recent twaddles on Mazur rotations problem, we study the "dynamics" of the semigroup of contractive automorphisms of Banach spaces, mostly in finite-dimensional spaces. We focus on the metric aspects of the "action" of such semigroups, the size of the orbits and semitransitivity properties, and their impact on the geometry of the unit ball of the underlying space.

math.FA↗
arXiv2021

The Banach space of quasinorms on a finite-dimensional space

Our main result states that, given a finite-dimensional vector space $E$, the pseudometric defined in the set of continuous quasinorms $\mathcal{Q}_0=\{\|\cdot\|:E\to\mathbb{R}\}$ as $$d(\|\cdot\|_X,\|\cdot\|_Y)=\min\{μ:\|\cdot\|_X \leqλ\|\cdot\|_Y\leqμ\|\cdot\|_X\text{ for some }λ\}$$ induces, in fact, a complete norm when we take the obvious quotient $\mathcal{Q}=\mathcal{Q}_0/\!\sim$ and define the appropriate operations on $\mathcal{Q}$. We finish the paper with a little explanation of how this space and the Banach-Mazur compactum are related.

math.FA↗
arXiv2021

Order isomorphisms between bases of topologies

In this paper we will study the representations of isomorphisms between bases of topological spaces. It turns out that the perfect setting for this study is that of regular open subsets of complete metric spaces, but we have achieved some results about arbitrary bases in complete metric spaces and also about regular open subsets of Hausdorff regular topological spaces.

math.GN↗
arXiv2021

On the concept of non-ultrametric non-Archimedean analysis

Given some non-Archimedean field $\mathbb{K}$ and some $\mathbb{K}$-linear space $X$, the usual way to define a norm over $X$ involves the {\em ultrametric inequality} $\|x+y\|\leq\max\{\|x\|,\|y\|\}$. In this note we will try to analyse the convenience of considering a wider variety of norms. The main result of the present note is a characterisation of the isometries between finite-dimensional linear spaces over some valued field endowed with the norm $\|\,\cdot\,\|_1$, a result that can be seen as the closest to a Mazur--Ulam Theorem in non-Archimedean analysis.

math.GT↗
arXiv2021

Isometries of ultrametric normed spaces

We show that the group of isometries of an ultrametric normed space can be seen as a kind of a fractal. Then, we apply this description to study ultrametric counterparts of some classical problems in Archimedean analysis, such as the so called Problème des rotations de Mazur or Tingley's problem. In particular, it turns out that, in contrast with the case of real normed spaces, isometries between ultrametric normed spaces can be very far from being linear.

math.MG↗
arXiv2021

Every non-smooth $2$-dimensional Banach space has the Mazur-Ulam property

A Banach space $X$ has the $Mazur$-$Ulam$ $property$ if any isometry from the unit sphere of $X$ onto the unit sphere of any other Banach space $Y$ extends to a linear isometry of the Banach spaces $X,Y$. A Banach space $X$ is called $smooth$ if the unit ball has a unique supporting functional at each point of the unit sphere. We prove that each non-smooth 2-dimensional Banach space has the Mazur-Ulam property.

math.FA↗
arXiv2021

Linearity of isometries between convex Jordan curves

In this paper, we show that the $C^1$-differentiability of the norm of a two-dimensional normed space depends only on distances between points of the unit sphere in two different ways. As a consequence, we see that any isometry between the spheres of normed planes $τ:S_X\to S_Y$ is linear, provided that there exist linearly independent $x,\overline{x}\in S_X$ where $S_X$ is not differentiable and that $S_X$ is piecewise differentiable. We end this work by showing that the isometry $τ:C_X\to C_Y$ is linear even if it is not an isometry between spheres: every isometry between (planar) Jordan piecewise $C^1$-differentiable convex curves extends to $X$ whenever $X$ and $Y$ are strictly convex and the amount of non-differentiability points of $S_X$ and $S_Y$ is finite and greater than 2.

math.MG↗
arXiv2020

Voidness of strict convexity in non-Archimedean fuzzy normed spaces

In this short note, we show by elementary computations that the notion of non-Archimedean fuzzy normed (and 2-normed) spaces is void. Namely, there are no strictly convex spaces at all --not even the zero-dimensional linear space. Before this, we also study the case of strictly convex non-Archimedean normed spaces; in this setting we see that the only nonzero linear space (defined over an arbitrary non-Archimedean field) that satisfies this property is the one-dimensional linear space over $\mathbb{Z}/3\mathbb{Z}$. Consequently, the results that have been proven for this class of spaces, like the Mazur-Ulam Theorem, are either trivial or empty statements.

math.FA↗
arXiv2019

Quiz your maths: do the uniformly continuous functions on the line form a ring?

The paper deals with the interplay between boundedness, order and ring structures in function lattices on the line and related metric spaces. It is shown that the lattice of all Lipschitz functions on a normed space $E$ is isomorphic to its sublattice of bounded functions if and only if $E$ has dimension one. The lattice of Lipschitz functions on $E$ carries a "hidden" $f$-ring structure with a unit, and the same happens to the (larger) lattice of all uniformly continuous functions for a wide variety of metric spaces. An example of a metric space whose lattice of uniformly continuous functions supports no unital $f$-ring structure is provided.

math.FA↗
arXiv2017

A New Algorithm to Fit Exponential Decays

This paper deals with some nonlinear problems which exponential and biexponential decays are involved in. A proof of the quasiconvexity of the error function in some of these problems of optimization is presented. This proof is restricted to fitting observations by means of exponentials having the form $f (t) = λ_1 \exp(kt) + λ_2.$ Based on its quasiconvexity, we propose an algorithm to estimate the best approximation to each of these decays. Besides, this algorithm does not require an initial guess.

math.OC↗
arXiv2017

When is U(X) a ring?

In this short paper, we will show that the space of real valued uniformly continuous functions defined on a metric space $(X,d)$ is a ring if and only if every subset $A\subset X$ has one of the following properties: $A$ is Bourbaki-bounded, i.e., every uniformly continuous function on $X$ is bounded on $A$. $A$ contains an infinite uniformly isolated subset, i.e., there exist $δ>0$ and an infinite subset $F\subset A$ such that $d(a,x)\geq δ$ for every $a\in F, x\in X$.

math.FA↗