Multipliers and integration operators between conformally invariant spaces

In this paper we are concerned with two classes of conformally invariant spaces of analytic functions in the unit disc \D, the Besov spaces $B^p$ $(1\le p<\infty )$ and the $Q_s$ spaces $(0<s<\infty )$. Our main objective is to characterize for a given pair $(X, Y)$ of spaces in these classes, the space of pointwise multipliers $M(X, Y)$, as well as to study the related questions of obtaining characterizations of those $g$ analytic in \D such that the Volterra operator $T_g$ or the companion operator $I_g$ with symbol $g$ is a bounded operator from $X$ into $Y$.Comment: To appear in Rev. R. Acad. Cienc. Exactas F\'is. Nat. Ser. A Mat. RACSA

A Hankel matrix acting on spaces of analytic functions

If $\mu$ is a positive Borel measure on the interval $[0, 1)$ we let $\mathcal H_\mu$ be the Hankel matrix $\mathcal H_\mu =(\mu _{n, k})_{n,k\ge 0}$ with entries $\mu _{n, k}=\mu _{n+k}$, where, for $n\,=\,0, 1, 2, \dots$, $\mu_n$ denotes the moment of order $n$ of $\mu$. This matrix induces formally the operator $$\mathcal{H}_\mu (f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \mu_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\mathbb D$. This is a natural generalization of the classical Hilbert operator. In this paper we improve the results obtained in some recent papers concerning the action of the operators $H_\mu$ on Hardy spaces and on M\"obius invariant spaces.Comment: arXiv admin note: text overlap with arXiv:1612.0830

Dealing with Integer-valued Variables in Bayesian Optimization with Gaussian Processes

Bayesian optimization (BO) methods are useful for optimizing functions that are expensive to evaluate, lack an analytical expression and whose evaluations can be contaminated by noise. These methods rely on a probabilistic model of the objective function, typically a Gaussian process (GP), upon which an acquisition function is built. This function guides the optimization process and measures the expected utility of performing an evaluation of the objective at a new point. GPs assume continous input variables. When this is not the case, such as when some of the input variables take integer values, one has to introduce extra approximations. A common approach is to round the suggested variable value to the closest integer before doing the evaluation of the objective. We show that this can lead to problems in the optimization process and describe a more principled approach to account for input variables that are integer-valued. We illustrate in both synthetic and a real experiments the utility of our approach, which significantly improves the results of standard BO methods on problems involving integer-valued variables.Comment: 7 page

La equidad: de concepto jurídico indeterminado a extensión del arte de lo justo

Artículo de ReflexiónEl artículo aborda las consideraciones de la Corte Constitucional Colombiana respecto del concepto de equidad, su sentido, contenido y alcance en relación con lo justo desde la perspectiva del realismo jurídico clásico. Se propone desde la perspectiva “del jurista”, propia de una teoría de la justicia acorde con la dignidad natural de la persona humana, una interpretación no-exegética de la ley que supere el modelo de la técnica subsuntiva cuando, en los casos concretos, la justicia requiera ser matizada y corregida por la equidad. El trabajo se divide en cuatro partes: 1) una descripción histórica panorámica del desarrollo del concepto desde la epiékeia y la aequitas a las concepciones actuales de equidad; 2) una aproximación a la lectura jurídica colombiana al concepto de equidad; 3) la descripción del papel de la equidad en relación con la justicia desde del realismo jurídico clásico; y 4) algunos elementos de discernimiento, desde la recta razón de prudencia, para oficio del jurista en relación con la adjudicación en equidad.Introducción 1. De la epiékeia griega a la equidad como concepto jurídico indeterminado. 2. El Concepto de equidad en el derecho Colombiano. 3. La equidad en relación con la justicia desde del realismo jurídico clásico. 4. La recta razón de prudencia. Conclusiones ReferenciasPregradoAbogad

Hankel matrices acting on the Hardy space H1H^1 and on Dirichlet spaces

If $\,\mu \,$ is a finite positive Borel measure on the interval $\,[0,1)$, we let $\,\mathcal H_\mu \,$ be the Hankel matrix $\,(\mu _{n, k})_{n,k\ge 0}\,$ with entries $\,\mu _{n, k}=\mu _{n+k}$, where, for $\,n\,=\,0, 1, 2, \dots$, $\mu_n\,$ denotes the moment of order $\,n\,$ of $\,\mu$. This matrix induces formally the operator $\,\mathcal{H}_\mu (f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \mu_{n,k}{a_k}\right)z^n\,$ on the space of all analytic functions $\,f(z)=\sum_{k=0}^\infty a_kz^k\,$, in the unit disc $\,\mathbb D$. When $\,\mu \,$ is the Lebesgue measure on $\,[0,1)\,$ the operator $\,\mathcal H_\mu\,$ is the classical Hilbert operator $\,\mathcal H\,$ which is bounded on $\,H^p\,$ if $\,1<p<\infty$, but not on $\,H^1$. J. Cima has recently proved that $\,\mathcal H\,$ is an injective bounded operator from $\,H^1\,$ into the space $\,\mathscr C\,$ of Cauchy transforms of measures on the unit circle. \par The operator $\,\mathcal H_\mu \,$ is known to be well defined on $\,H^1\,$ if and only if $\,\mu \,$ is a Carleson measure and in such a case we have that $\mathcal H_\mu (H^1)\subset \,\mathscr C$. Furthermore, it is bounded from $\,H^1\,$ into itself if and only if $\,\mu\,$ is a $1$-logarithmic $1$-Carleson measure. \par In this paper we prove that when $\,\mu\,$ is a $1$-logarithmic $1$-Carleson measure then $\,\mathcal H_\mu \,$ actually maps $\,H^1\,$ into the space of Dirichlet type $\,\mathcal D^1_0\,$. We discuss also the range of $\,\mathcal H_\mu\,$ on $\,H^1\,$ when $\,\mu \,$ is an $\alpha$-logarithmic $1$-Carleson measure ($0<\alpha <1$). We study also the action of the operators $\,\mathcal H_\mu \,$ on Bergman spaces and on Dirichlet spaces.Comment: 21 page

A generalized Hilbert operator acting on conformally invariant spaces

If μ is a positive Borel measure on the interval [0,1), we let H_μ be the Hankel matrix with entries μ_{n,k}=μ_{n+k}, where μ_n denotes the moment of order n of the measure μ. This matrix formally induces an operator on the space of all analytic functions in the unit disk D. This is a natural generalization of the classical Hilbert operator. The action of the operators H_μ on Hardy spaces has been recently studied. This article is devoted to a study of the operators H_μ acting on certain conformally invariant spaces of analytic functions on the disk such as the Bloch space, the space BMOA, the analytic Besov spaces, and the Q_s-spaces.- Proyecto del Ministerio de Economía y Competitividad MTM2014-52865-P. - Proyecto de la Junta de Andalucía FQM-210. - Ayuda FPU del Ministerio de Educación, Cultura y Deporte. FPU2013/01478

Pointwise multipliers between spaces of analytic functions.

Política de acceso abierto tomada: https://v2.sherpa.ac.uk/id/publication/305A Banach space X of analytic function in D, the unit disc in C, is said to be admissible if it contains the polynomials and convergence in X implies uniform convergence in compact subsets of D. If X and Y are two admissible Banach spaces of analytic functions in D and g is a holomorphic function in D, g is said to be a multiplier from X to Y if g · f is in Y for every f in X. The space of all multipliers from X to Y is denoted M(X; Y ), and M(X) will stand for M(X;X). The closed graph theorem shows that if g is in M(X; Y ) then the multiplication operator Mg, defi ned by Mg(f) = g · f, is a bounded operator from X into Y. It is known that M(X) c H^inf and that if g is in M(X), then ∥g∥_H^inf <= ∥Mg∥. Clearly, this implies that M(X; Y ) c H^inf if Y c X. If Y is not contained in X, the inclusion M(X; Y ) c H^inf may not be true. In this paper we start presenting a number of conditions on the spaces X and Y which imply that the inclusion M(X; Y ) c H^inf holds. Next, we concentrate our attention on multipliers acting an BMOA and some related spaces, namely, the Qs-spaces (0 < s < 1)."El Ministerio de Economía y Competitividad", España (PGC2018-096166-B-I00) y ayudas de "la Junta de Andalucía (FQM-210 y UMA18-FEDERJA-002)

Predictive entropy search for multi-objective Bayesian optimization with constraints

This work presents PESMOC, Predictive Entropy Search for Multi-objective Bayesian Optimization with Constraints, an information-based strategy for the simultaneous optimization of multiple expensive-to- evaluate black-box functions under the presence of several constraints. Iteratively, PESMOC chooses an input location on which to evaluate the objective functions and the constraints so as to maximally reduce the entropy of the Pareto set of the corresponding optimization problem. The constraints considered in PESMOC are assumed to have similar properties to those of the objectives in typical Bayesian optimization problems. That is, they do not have a known expression (which prevents any gradient computation), their evaluation is considered to be very expensive, and the resulting observations may be corrupted by noise. Importantly, in PESMOC the acquisition function is decomposed as a sum of objective and constraint specific acquisition functions. This enables the use of the algorithm in decoupled evaluation scenarios in which objectives and constraints can be evaluated separately and perhaps with different costs. Therefore, PESMOC not only makes intelligent decisions about where to evaluate next the problem objectives and constraints, but also about which objective or constraint to evaluate next. We present strong empirical evidence in the form of synthetic, benchmark and real-world experiments that illustrate the effectiveness of PESMOC. In these experiments PESMOC outperforms other state-of-the-art methods for constrained multi-objective Bayesian optimization based on a generalization of the expected improvement. The results obtained also show that a decoupled evaluation scenario can lead to significant improvements over a coupled one in which objectives and constraints are evaluated at the same input.The authors acknowledge the use of the facilities of Centro de Computaci on Cient ca (CCC) at Universidad Aut onoma de Madrid, and nancial support from the Spanish Plan Nacional I+D+i, Grants TIN2016-76406-P and TEC2016-81900- REDT, and from Comunidad de Madrid, Grant S2013/ICE-2845 CASI-CAM-CM

Improved max-value entropy search for multi-objective bayesian optimization with constraints

We present MESMOC+, an improved version of Max-value Entropy search for Multi-Objective Bayesian optimization with Constraints (MESMOC). MESMOC+ can be used to solve constrained multi-objective problems when the objectives and the constraints are expensive to evaluate. It is based on minimizing the entropy of the solution of the optimization problem in function space (i.e., the Pareto front) to guide the search for the optimum. The cost of MESMOC+ is linear in the number of objectives and constraints. Furthermore, it is often significantly smaller than the cost of alternative methods based on minimizing the entropy of the Pareto set. The reason for this is that it is easier to approximate the required computations in MESMOC+. Moreover, MESMOC+’s acquisition function is expressed as the sum of one acquisition per each black-box (objective or constraint). Therefore, it can be used in a decoupled evaluation setting in which it is chosen not only the next input location to evaluate, but also which black-box to evaluate there. We compare MESMOC+ with related methods in synthetic, benchmark and real optimization problems. These experiments show that MESMOC+ has similar performance to that of state-of-the-art acquisitions based on entropy search, but it is faster to execute and simpler to implement. Moreover, our experiments also show that MESMOC+ is more robust with respect to the number of samples of the Pareto frontThe authors acknowledge financial support from Spanish Plan Nacional I + D+i, grant PID2019-106827 GB-I00/ AEI/ 10.13039/50110001103