Juan Matías Sepulcre

University of Alicante | UA · Department of Mathematics

This paper shows that the conjecture of Lapidus and Van Frankenhuysen on the set of dimensions of fractality associated with a nonlattice fractal string is true in the important special case of a generic nonlattice self-similar string, but in general is false. The proof and the counterexample of this have been given by virtue of a result on exponen...

We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly com...

In this paper we study the distribution of zeros of each entire function of the sequence {Gn(z) ≡ 1 + 2^z + · · · + n^z : n>=2}, which approaches the Riemann zeta function for Re z < −1, and is closely related to the solutions of the functional equations f (z)+ f (2z)+ · · ·+ f (nz) = 0. We determine the density of the zeros of Gn(z) on the critica...

In this paper, we introduce a formula for the exact number of zeros of every partial sum of the Riemann zeta function inside infinitely many rectangles of the critical strips where they are situated.

In this paper we give a new characterization of the closure of the set of the real parts of the zeros of a particular class of Dirichlet polynomials that is associated with the set of dimensions of fractality of certain fractal strings. We show, for some representative cases of nonlattice Dirichlet polynomials, that the real parts of their zeros ar...

As an extension of some classes of generalized almost periodic functions, in this paper we develop the notion of c-almost periodicity in the sense of Stepanov and Weyl approaches. In fact, we extend some basic results of this theory which were already demonstrated for the standard cases. In particular, we prove that every c-almost periodic function...

Let E and F be complex Banach spaces, U be an open subset of E and $$1\le p\le \infty .$$ 1 ≤ p ≤ ∞ . We introduce and study the notion of a Cohen strongly p -summing holomorphic mapping from U to F , a holomorphic version of a strongly p -summing linear operator. For such mappings, we establish both Pietsch Domination/Factorization Theorems and an...

Using Mujica’s linearization theorem, we extend to the holomorphic setting some classical characterizations of compact (weakly compact, Rosenthal, Asplund) linear operators between Banach spaces such as the Schauder, Gantmacher and Gantmacher–Nakamura theorems and the Davis–Figiel–Johnson–Pełczynski, Rosenthal and Asplund factorization theorems.

Let $E$ and $F$ be complex Banach spaces, $U$ be an open subset of $E$ and $1\leq p\leq\infty$. We introduce and study the notion of a Cohen strongly $p$-summing holomorphic mapping from $U$ to $F$, a holomorphic version of a strongly $p$-summing linear operator. For such mappings, we establish both Pietsch domination/factorization theorems and ana...

Using Mujica's linearization theorem, we extend to the holomorphic setting some classical characterizations of compact (weakly compact, Rosenthal, Asplund) linear operators between Banach spaces such as the Schauder, Gantmacher and Gantmacher-Nakamura theorems and the Davis-Figiel-Johnson-Pelczynski, Rosenthal and Asplund factorization theorems.

In this paper, we develop the notion of c -almost periodicity for functions defined on vertical strips in the complex plane. As a generalization of Bohr’s concept of almost periodicity, we study the main properties of this class of functions which was recently introduced for the case of one real variable. In fact, we extend some important results o...

Given two arbitrary almost periodic functions with Fourier exponents which are linearly independent over the rational numbers, we prove that the existence of a common open vertical strip V, where both functions assume the same set of values on every open vertical substrip included in V, is a necessary and sufficient condition for both functions to...

-ISBN: 978-84-368-4610-2 (press), 978-84-368-4611-9 (ebook). -Number of pages: 413. -Webpages (Ediciones Pirámide): i) https://www.edicionespiramide.es/libro.php?id=7038561 ; ii) https://www.edicionespiramide.es/libro.php?id=7038588 -------------------------------------------------------------------------------------------------------------------...

"This paper deals with the sets of real projections of zeros of analytic almost periodic functions defined in a vertical strip. By using our equivalence relation introduced in the context of the complex functions which can be represented by a Dirichlet-like series, this work provides practical results in order to determine whether a real number bel...

Based on an equivalence relation that was established recently on exponential sums, in this paper we study the class of functions that are equivalent to the Riemann zeta function in the half-plane $$\{s\in {\mathbb {C}}:\mathrm{Re}\, s>1\}$$ { s ∈ C : Re s > 1 } . In connection with this class of functions, we first determine the value of the maxim...

In this paper, we consider an equivalence relation on the space $AP(\mathbb {R},X)$ of almost periodic functions with values in a prefixed Banach space X . In this context, it is known that the normality or Bochner-type property, which characterizes these functions, is based on the relative compactness of the family of translates. Now, we prove tha...

In this paper we introduce the notion of almost equality (or, more specifically, almost equality by translations) of complex functions of an unrestricted real variable in terms of the new concept of ϵ-translation number of a function with respect to other one, which is inspired by Bohr’s notion of ϵ-translation number associated with an almost peri...

The development of fractal (and multifractal) geometry, as a theoretical framework for the formal study of a certain type of geometric objects and natural phenomena associated with highly irregular shapes or structures, has reported in recent decades great advances in the field of medicine, with special emphasis on the cellular biological context....

In this paper we establish a new equivalence relation on the spaces of almost periodic functions which allows us to prove a result like Bohr's equivalence theorem extended to the case of all these functions.

Based on Bohr’s equivalence relation for general Dirichlet series, in this paper we connect the families of equivalent exponential polynomials with a geometrical point of view related to lines in crystal-like structures. In particular we characterize this equivalence relation, and give an alternative proof of Bochner’s property referring to these f...

Quantum computing is a revolutionary computational method that tries to enhance the way in which nature operates at microscopic scales, where particles -such as electrons and photons- are governed by the rules of quantum mechanics, and therefore have special properties that allow for processing large datasets in a faster manner (in comparison with...

In this paper we will get a class of functional equations involving a countable set of terms, summed by the well known Bochner–Fejér summation procedure, which are closely associated with the set of almost periodic functions. We will show that the zeros of a prefixed almost periodic function determine analytic solutions of such a functional equatio...

A construction analogous to that of Godefroy-Kalton for metric spaces allows to embed isometrically, in a canonical way, every quasi-metric space $(X,d)$ to an asymmetric normed space $\mathcal{F}_a(X,d)$ (its quasi-metric free space, also called asymmetric free space or semi-Lipschitz free space). The quasi-metric free space satisfies a universal...

Our paper is focused on spaces of generalized almost periodic functions which, as in classical Fourier analysis, are associated with a Fourier series with real frequencies. In fact, based on a pertinent equivalence relation defined on the spaces of almost periodic functions in Bohr, Stepanov, Weyl and Besicovitch’s sense, we refine the Bochner-type...

Based on an equivalence relation that was established recently on exponential sums, in this paper we study the class of functions that are equivalent to the Riemann zeta function in the half-strip $\{s\in\mathbb{C}:\operatorname{Re}s>1\}$. In connection with this class of functions, we first determine the value of the maximum abscissa from which th...

A construction analogous to that of Godefroy-Kalton for metric spaces allows to embed iso-metrically, in a canonical way, every quasi-metric space (X, d) to an asymmetric normed space Fa(X, d) (its quasi-metric free space, also called asymmetric free space or semi-Lipschitz free space). The quasi-metric free space satisfies a universal property (li...

Based on Bohr's equivalence relation which was established for general Dirichlet series, in this paper we introduce a more general equivalence relation on the space of almost periodic functions in the sense of Besicovitch, $B(\mathbb{R},\mathbb{C})$, defined in terms of polynomial approximations. From this, we show that in an important subspace $B^...

This report aims to convince readers that there are clear indications that society is increasingly taking a greater interest in science and particularly in mathematics, and thus society in general has come to recognise, through different awards, privileges, and distinctions, the work of many mathematicians. We provide examples of recognition accord...

Based on a generalization of Bohr's equivalence relation for general Dirichlet series, in this paper we study the sets of values taken by classes of equivalent almost periodic functions in their strips of almost periodicity. In fact, the main result of this paper consists of a generalization of Bohr's equivalence theorem. Moreover, we also improve...

The authors wish to draw the attention to a mistake which appears in the proof of Proposition 3 of the above quoted paper [4].

Given two arbitrary almost periodic functions with associated Fourier exponents which are linearly independent over the rational numbers, we prove that the existence of a common open vertical strip $V$, where both functions assume the same set of values on every open vertical substrip included in $V$, is a necessary and sufficient condition for bot...

We study the global distribution of zeros of exponential polynomials with complex coefficients and frequencies. For any $P(z)$ in some class of such polynomials, we show that the closure of the projection of $P^{−1}(0)$ on a certain line is a finite union of disjoint segments. We describe this set, in particular we discuss the case where it consist...

This paper deals with the set of the real projections of the zeros of an arbitrary almost periodic function defined in a vertical strip $U$. It provides practical results in order to determine whether a real number belongs to the closure of such a set. Its main result shows that, in the case that the Fourier exponents $\{\lambda_1,\lambda_2,\lambda...

In this paper we introduce an equivalence relation on the classes of almost periodic functions of a real or complex variable which is used to refine Bochner’s result that characterizes these spaces of functions. In fact, with respect to the topology of uniform convergence, we prove that the limit points of the family of translates of an almost peri...

In this paper, we study the distribution of zeros of the ordinary Dirichlet polynomials which are generated by an equivalence relation introduced by Harald Bohr. Through the use of completely multiplicative functions, we construct equivalent Dirichlet polynomials which have the same critical strip, where all their zeros are situated, and satisfy th...

Sumario: >INTRODUCCIÓN >CAPÍTULO 1 Los inicios de Karl Weierstrass >CAPÍTULO 2 El padre del análisis moderno >CAPÍTULO 3 El proceso de aritmetización del análisis >CAPÍTULO 4 La repercusión de Weierstrass como maestro >LECTURAS RECOMENDADAS >ÍNDICE

This work attempts to provide an overview of some of the most important mathematicians who have combined their mathematical knowledge with other academic and non-academic specialities. The various examples given, many of them included in the well-known MacTutor History of Mathematics archive, corroborate the fact that although the idea of the typic...

Esta obra por capítulos pretende difundir y divulgar la importancia de las matemáticas en el desarrollo de la actividad humana desde diferentes contextos y puntos de vista, señalando al mismo tiempo sus relaciones con otras disciplinas o campos de conocimiento. En particular, este libro muestra algunas conexiones de las matemáticas con la arqueolog...

We provide the proof of a practical pointwise characterization of the set R P defined by the closure set of the real projections of the zeros of an exponential polynomial P ( z ) = ∑ j = 1 n c j e w j z with real frequencies w j linearly independent over the rationals. As a consequence, we give a complete description of the set R...

Segunda edición: Mulero, J.; Sepulcre, J.M.: LaTeX con palabras clave. ISBN: 978-84-9717-704-7, Publicaciones Universidad de Alicante, 2020 Este manual está especialmente dirigido a aquellos estudiantes, profesores o investigadores que deseen iniciarse y adquirir soltura en el manejo de LaTeX, un sistema de composición de textos que es en la actua...

This paper proves that the real projection of each zero of any function P(z) in a large class of exponential polynomials is an interior point of the closure of the set of the real parts of the zeros of P(z). In particular it is deduced that, for each integer value of n>=17, if z_0=x_0+iy_0 is an arbitrary zero of the nth partial sum of the Riemann...

Through several equivalence binary relations, in this paper we identify, on the one hand, groups of exponential polynomials with the same set of zeros, and on the other hand, groups of functional equations of the form that lead to equivalent exponential polynomials with the same set of zeros.

In this paper, it is showed that, given an integer number n ≥ 2, each zero of an exponential polynomial of the form w_1a_1^z+w_2a_2^z+...+w_na_n^z, with non-null complex numbers w_1, w_2,…,w_n and a_1, a_2,…,a_n, produces analytic solutions of the functional equation w_1f(a_1z)+ w_2f(a_2z) +...+w_nf(a_nz) = 0 on certain domains of \mathbb{C}, which...

The importance of mathematics in the context of the scientific and technological development of humanity is determined by the possibility of creating mathematical models of the objects studied under the different branches of Science and Technology. The arithmetisation process that took place during the nineteenth century consisted of the quest to d...

With the arrival of the nineteenth century, a process of change guided the treatment of three basic elements in the development of mathematics: rigour, the arithmetization and the clarification of the concept of function, categorised as the most important tool in the development of the mathematical analysis. In this paper we will show how several p...

In this paper it is shown that a conjecture of Lapidus and van Frankenhuysen of 2003 on the existence of a vertical line such that the density of the complex dimensions of nonlattice fractal strings with \(M\) scaling ratios off this line vanishes in the limit as \(M\rightarrow \infty \), fails on the class of nonlattice self-similar fractal string...

Las matemáticas constituyen un lenguaje universal y la base de cualquier tipo de desarrollo científico y tecnológico. Es incuestionable que no podríamos entender el mundo sin ellas. Esta monografía, compuesta de trece capítulos escritos con un lenguaje cercano y ameno por destacados especialistas, supone un recorrido por diversos conceptos matemát...

For pointed compact metric spaces $(X,d)$, we address the biduality problem as to when the space of Lipschitz functions $\mathrm{Lip}_0(X,d)$ is isometrically isomorphic to the bidual of the space of little Lipschitz functions $\mathrm{lip}_0(X,d)$, and show that this is the case whenever the closed unit ball of $\mathrm{lip}_0(X,d)$ is dense in th...

El campus de la Universidad de Alicante reúne una serie de características que hacen de él uno de los mejores de Europa. La sensación de espacio abierto salpicado de zonas verdes ajardinadas inunda al visitante ofreciendo una perspectiva acorde a la actividad docente e investigadora realizada en el interior de los edificios que lo conforman. Más aú...

In this paper we give an example of a nonlattice self-similar fractal string such that the set of real parts of their complex dimensions has an isolated point. This proves that, in general, the set of dimensions of fractality of a fractal string is not a perfect set.

In this paper, we prove that infinite-dimensional vector spaces of -dense curves are generated by means of the functional equations f(x)+f(2x)++f(nx) = 0, with , which are related to the partial sums of the Riemann zeta function. These curves -densify a large class of compact sets of the plane for arbitrary small , extending the known result that t...

La matemática elemental es una rama atractiva e interesante para un gran número de estudiantes de Matemáticas. Estos alumnos son vocacionales y disfrutan resolviendo tipos de problemas que requieren el uso de conceptos básicos. Han aprendido a entender un teorema, una definición rigurosa o un contraejemplo, lo que les permite abordar problemas cuya...

Let vv be a weight sequence on ZZ and let ψ,φψ,φ be complex-valued functions on ZZ such that φ(Z)⊂Zφ(Z)⊂Z. In this paper we study the boundedness, compactness and weak compactness of weighted composition operators Cψ,φCψ,φ on predual Banach spaces c0(Z,1/v)c0(Z,1/v) and dual Banach spaces ℓ∞(Z,1/v)ℓ∞(Z,1/v) of Beurling algebras ℓ1(Z,v)ℓ1(Z,v).

This paper proves that the real projection of each simple zero of any partial sum of the Riemann zeta function \({\zeta_{n}(s) := \sum_{k=1}^{n} \frac{1}{k^{s}}, n > 2}\) , is an accumulation point of the set {Res : ζ n (s) = 0}.

This paper proves that every zero of any nth, n ≥ 2, partial sum of the Riemann zeta function provides a vector space of basic solutions of the functional equation f(x) + f(2x) + · · · + f(nx) = 0, x ∈ R. The continuity of the solutions depends on the sign of the real part of each zero.

This paper shows, by means of Kronecker’s theorem, the existence of infinitely many privileged regions called \(r\)-rectangles (rectangles with two semicircles of small radius \(r\)) in the critical strip of each function \(L_{n}(z)\!:=\!\) \(1-\sum _{k=2}^{n}k^{z}\), \(n\!\ge \!2\), containing exactly \(\left[ \dfrac{T\log n}{2\pi }\right] +1\) ze...

Tradicionalmente, la enseñanza de las matemáticas se suele presentar alejada de la vida cotidiana y como una disciplina tediosa y aburrida no vinculada a actividades tales como las bellas artes, la literatura, el cine o la televisión. Es por ello que consideramos oportuno tratar de analizar, desde este punto de vista, las relaciones existentes entr...

Purpose This paper aims to introduce a new class of entire functions whose zeros ( z k ) k ≥1 satisfy ∑ k =1 ∞ Im z k = O (1). Design/methodology/approach This is done by means of a Ritt's formula which is used to prove that every partial sum of the Riemann Zeta function, ζ n ( z ):=∑ k =1 n 1/ k z , n ≥2, has zeros ( s n k ) k ≥1 verifying ∑ k =1...

Purpose ‐ This paper aims to present a new method for obtaining points of the set determined by the closure of the real projections of the zeros of each partial sum 1+2^s+...+n^s, n>=2, s=sigma+it, of the Riemann zeta function and to show several applications of this result. Design/methodology/approach ‐ The authors utilize an auxiliary function...

Desde el Departamento de Análisis Matemático de la Facultad de Ciencias de la Universidad de Alicante, los distintos profesores que han estado impartiendo docencia en los primeros cursos, han detectado carencias de conocimientos, diversidad en las formas de acceso y problemas relativos a metodología y planificación de estudio en los alumnos de nuev...

RESUMEN (ABSTRACT) Tradicionalmente, la enseñanza de las matemáticas se suele presentar alejada de la vida cotidiana y como una disciplina tediosa y aburrida no vinculada a actividades tales como las bellas artes, la literatura, el cine o la televisión. Es por ello que consideramos oportuno tratar de analizar, desde este punto de vista, las relacio...

We give a partition of the critical strip, associated with each partial sum 1+2^z+· · · +n^z of the Riemann zeta function for Re z < −1, formed by infinitely many rectangles for which a formula allows us to count the number of its zeros inside each of them with an error, at most, of two zeros. A generalization of this formula is also given to a lar...

RESUMEN (ABSTRACT) Desde nuestra experiencia docente en los primeros cursos de licenciatura y actuales estudios de grado, los profesores hemos podido constatar que los alumnos de nuevo ingreso no tienen bien asentados, o incluso desconocen en algunos casos, muchos de los conceptos elementales sobre los cuales se construyen los principales contenido...

Work presented at III WINTER SCHOOL IN COMPLEX ANALYSIS AND OPERATOR THEORY (2010)

En 1782 Lagrange relacionó el problema astronómico de la perturbación de los planetas grandes con la variación del argumento de ciertos polinomios exponenciales, emergiendo un problema matemático conocido como Mean Motion. Para extender el problema de Lagrange de los polinomios exponenciales a una clase más general, H. Bohr en 1924 introdujo la noc...

(Tesina) La idea principal que se pretende llevar a cabo en las distintas ecuaciones funcionales analizadas en esta memoria es la de utilizar diversos resultados importantes que encontramos en la variable compleja con tal de obtener todas sus soluciones posibles y, al mismo tiempo, intentar sistematizar y modelizar las técnicas de resolución de dic...