Catalan Generating Functions for Generators of Uni-parametric Families of Operators

In this paper we study solutions of the quadratic equation AY2−Y+I=0 where A is the generator of a one parameter family of operator (C0-semigroup or cosine functions) on a Banach space X with growth bound w0≤14. In the case of C0-semigroups, we show that a solution, which we call Catalan generating function of A, C(A), is given by the following Bochner integral, C(A)x:=∫∞0c(t)T(t)xdt,x∈X, where c is the Catalan kernel, c(t):=12π∫∞14e−λt4λ−1−−−−−√λdλ,t>0. Similar (and more complicated) results hold for cosine functions. We study algebraic properties of the Catalan kernel c as an element in Banach algebras L1ω(R+), endowed with the usual convolution product, ∗ and with the cosine convolution product, ∗c. The Hille–Phillips functional calculus allows to transfer these properties to C0-semigroups and cosine functions. In particular, we obtain a spectral mapping theorem for C(A). Finally, we present some examples, applications and conjectures to illustrate our results

Caputo fractional evolution equations in discrete sequences spaces

In this paper, we treat some fractional differential equations on the sequence Lebesgue spaces ℓp(N0) with p≥1. The Caputo fractional calculus extends the usual derivation. The operator, associated to the Cauchy problem, is defined by a convolution with a sequence of compact support and belongs to the Banach algebra ℓ1(Z). We treat in detail some of these compact support sequences. We use techniques from Banach algebras and a Functional Analysis to explicity check the solution of the problem

Discrete variational calculus for accelerated optimization

Many of the new developments in machine learning are connected with gradient-based optimization methods. Recently, these methods have been studied using a variational perspective (Betancourt et al., 2018). This has opened up the possibility of introducing variational and symplectic methods using geometric integration. In particular, in this paper, we introduce variational integrators (Marsden and West, 2001) which allow us to derive different methods for optimization. Using both Hamilton’s and Lagrange-d’Alembert’s principle, we derive two families of optimization methods in one-to-one correspondence that generalize Polyak’s heavy ball (Polyak, 1964) and Nesterov’s accelerated gradient (Nesterov, 1983), the second of which mimics the behavior of the latter reducing the oscillations of classical momentum methods. However, since the systems considered are explicitly time-dependent, the preservation of symplecticity of autonomous systems occurs here solely on the fibers. Several experiments exemplify the result

A Discrete Variational Derivation of Accelerated Methods in Optimization

Many of the new developments in machine learning are connected with gradient-based optimization methods. Recently, these methods have been studied using a variational perspective. This has opened up the possibility of introducing variational and symplectic methods using geometric integration. In particular, in this paper, we introduce variational integrators which allow us to derive different methods for optimization. Using both, Hamilton's and Lagrange-d'Alembert's principle, we derive two families of respective optimization methods in one-to-one correspondence that generalize Polyak's heavy ball and the well known Nesterov accelerated gradient method, the second of which mimics the behavior of the first reducing the oscillations of classical momentum methods. However, since the systems considered are explicitly time-dependent, the preservation of symplecticity of autonomous systems occurs here solely on the fibers. Several experiments exemplify the result.Comment: 28 pages, 11 figure

EL IMPACTO DEL GASTO DE DEFENSA EN EL TEJIDO EMPRESARIAL ESPAÑOL

El objetivo principal de las Fuerzas Armadas es cumplir con el cometido encomendado en el artículo octavo de la Constitución Española de 1978. En él se encarga al Ejército de Tierra, la Armada y el Ejército del Aire la misión de garantizar la soberanía e independencia de España, defender su integridad territorial y el ordenamiento constitucional. Tal importante cometido sólo podría cumplirse si se disponen de los medios y tecnologías capaces de competir en un mundo que evoluciona a un ritmo arrollador y, cuyas amenazas implican cada vez más riesgos para la población.Por ello, cobra relevancia disponer de una industria de defensa propia, capaz de mantener un nivel competitivo internacionalmente mediante la innovación tecnológica y capacidad de adaptación al entorno dinámico actual. No obstante, además de su trascendencia en la seguridad y defensa nacionales, el sector industrial de defensa se caracteriza por la dualidad de sus tecnologías, pues una gran parte de ellas tiene aplicación tanto en el ámbito militar, como en el civil.Por esa razón, en este proyecto se pretende hacer hincapié en esta capacidad dual, así como conocer los beneficios que tiene la industria de defensa sobre el mundo civil, es decir, averiguar cómo contribuye la industria de defensa sobre el bienestar de los ciudadanos. Para ello, se llevará a cabo un análisis en profundidad de distintas variables relacionadas con aspectos económicos y sociales y, se investigará cómo influye la industria de defensa sobre ellas. De esta forma, se han utilizado métodos estadísticos como la regresión y correlación lineal.Por otra parte, se ha podido contar con la ayuda de un experto en la industria de defensa, como es el Coronel González Casado, al que se ha podido entrevistar y, así, conocer su opinión sobre el tema de este trabajo. Después de este análisis, se tendrán evidencias del impacto de la industria de defensa sobre el bienestar de los ciudadanos y, además, se comprobará en qué medida ocurre este fenómeno. <br /

Design and rationale of a multicentre, randomised, double-blind, placebo-controlled clinical trial to evaluate the effect of vitamin D on ventricular remodelling in patients with anterior myocardial infarction: the VITamin D in Acute Myocardial Infarction (VITDAMI) trial

Introduction:Decreased plasma vitamin D (VD) levels are linked to cardiovascular damage. However, clinical trials have not demonstrated a benefit of VD supplements on left ventricular (LV) remodelling. Anterior ST-elevation acute myocardial infarction (STEMI) is the best human model to study the effect of treatments on LV remodelling. We present a proof-of-concept study that aims to investigate whether VD improves LV remodelling in patients with anterior STEMI. Methods and analysis:The VITamin D in Acute Myocardial Infarction (VITDAMI) trial is a multicentre, randomised, double-blind, placebo-controlled trial. 144 patients with anterior STEMI will be assigned to receive calcifediol 0.266 mg capsules (Hidroferol SGC)/15 days or placebo on a 2:1 basis during 12 months. Primary objective:to evaluate the effect of calcifediol on LV remodelling defined as an increase in LV end-diastolic volume >= 10\% (MRI). Secondary objectives:change in LV end-diastolic and end-systolic volumes, ejection fraction, LV mass, diastolic function, sphericity index and size of fibrotic area; endothelial function; plasma levels of aminoterminal fragment of B-type natriuretic peptide, galectin-3 and monocyte chemoattractant protein-1; levels of calcidiol (VD metabolite) and other components of mineral metabolism (fibroblast growth factor-23 (FGF-23), the soluble form of its receptor klotho, parathormone and phosphate). Differences in the effect of VD will be investigated according to the plasma levels of FGF-23 and klotho. Treatment safety and tolerability will be assessed. This is the first study to evaluate the effect of VD on cardiac remodelling in patients with STEMI. Ethics and dissemination: This trial has been approved by the corresponding Institutional Review Board (IRB) and National Competent Authority (Agencia Espanola de Medicamentos y Productos Sanitarios (AEMPS)). It will be conducted in accordance with good clinical practice (International Council for Harmonisation of Technical Requirements for Pharmaceuticals for Human Use-Good Clinical Practice (ICH-GCP)) requirements, ethical principles of the Declaration of Helsinki and national laws. The results will be submitted to indexed medical journals and national and international meetings.The VITDAMI trial is an investigator initiated study, sponsored by the Instituto de Investigacion Sanitaria Fundacion Jimenez Diaz (IIS-FJD). Funding has been obtained from Fondo de Investigaciones Sanitarias (PI14/01567; http://www.isciii.es/) and Spanish Society of Cardiology (http://secardiologia.es/). In addition, the study medication has been provided freely by the pharmaceutical Company FAES FARMA S.A. (Leioa, Vizcaya, Spain; http://faesfarma.com/). This company was the only funder who collaborated in study design (IG-H).S

Teoremas de reordenamiento de series

La suma de una cantidad infinita de números reales puede depender del orden en el que se sumen los números. En este trabajo hacemos un recorrido por varios resultados que involucran reordenamiento de los términos de una serie, desde series en R hasta en espacios de Banach pasando por los euclidianos (Rn). No incluimos demostraciones de los teoremas, solo las ideas básicas de éstas. Primero vemos el caso de las series de números reales, donde presentamos el teorema de reordenamiento de Riemann junto con otros resultados. Continuaremos con el teorema de Lévy-Steinitz, un resultado análogo al de Riemann para series de vectores en Rn. En particular, consideraremos la serie de Eisenstein, definida en los complejos, que tiene la propiedad de que al reordenar sus términos obtenemos un cambio en el valor de su suma; esta serie es útil al estudiar formas modulares. Por último, presentamos el teorema de Pechersky sobre reordenamiento de series en espacios de Hilbert, un resultado útil para probar la universalidad de la función de Riemann.The sum of an infinite number of real numbers can depend on the arranging of these numbers. In this paper we will take you through several results about rearranging the terms of series; from series of real numbers to series in Rn; even results about series in Banach spaces. We do not include proofs of theorems but only their main ideas. First, we study the real numbers series case, in which we see the Riemann rearrangement theorem together with other results. We will continue with the Lévy- Steinitz theorem, an analogous result of Riemmans theorem for vector series inRn. In particular, we will consider the Eisenstein series defined in the complex field. Also, this series has the property that rearrangement in the order of summations results in a predictable change in the value of the series. This series is useful in the study of modular form. Finally, we show Pechershys theorem on rearrangement of series in Hilbert space