scholarly journals Towards sharp Bohnenblust–Hille constants

2018 ◽  
Vol 20 (03) ◽  
pp. 1750029 ◽  
Author(s):  
Daniel Pellegrino ◽  
Eduardo V. Teixeira
Keyword(s):  
Exponential Growth ◽  
Optimization Problems ◽  
Best Constants ◽  
Sharp Constants ◽  
Linear Forms ◽  
Entropy Formula ◽  
Optimal Constants

We investigate the optimality problem associated with the best constants in a class of Bohnenblust–Hille-type inequalities for [Formula: see text]-linear forms. While germinal estimates indicated an exponential growth, in this work we provide strong evidences to the conjecture that the sharp constants in the classical Bohnenblust–Hille inequality are universally bounded, irrespectively of the value of [Formula: see text]; hereafter referred as the Universality Conjecture. In our approach, we introduce the notions of entropy and complexity, designed to measure, to some extent, the complexity of such optimization problems. We show that the notion of entropy is critically connected to the Universality Conjecture; for instance, that if the entropy grows at most exponentially with respect to [Formula: see text], then the optimal constants of the [Formula: see text]-linear Bohnenblust–Hille inequality for real scalars are indeed bounded universally with respect to [Formula: see text]. It is likely that indeed the entropy grows as [Formula: see text], and in this scenario, we show that the optimal constants are precisely [Formula: see text]. In the bilinear case, [Formula: see text], we show that any extremum of the Littlewood’s [Formula: see text] inequality has entropy [Formula: see text] and complexity [Formula: see text], and thus we are able to classify all extrema of the problem. We also prove that, for any mixed [Formula: see text]-Littlewood inequality, the entropy do grow exponentially and the sharp constants for such a class of inequalities are precisely [Formula: see text]. In addition to the notions of entropy and complexity, the approach we develop in this work makes decisive use of a family of strongly non-symmetric [Formula: see text]-linear forms, which has further consequences to the theory, as we explain herein.

SHARP CONSTANTS BETWEEN EQUIVALENT NORMS IN WEIGHTED LORENTZ SPACES

2010 ◽  
Vol 88 (1) ◽  
pp. 19-27 ◽  
Author(s):  
SORINA BARZA ◽  
JAVIER SORIA
Keyword(s):  
Lorentz Space ◽  
Lorentz Spaces ◽  
Best Constants ◽  
Sharp Constants ◽  
Dual Norm ◽  
Equivalent Norms ◽  
Weighted Lorentz Spaces ◽  
Weighted Lorentz Space

AbstractFor an increasing weight w in Bp (or equivalently in Ap), we find the best constants for the inequalities relating the standard norm in the weighted Lorentz space Λp(w) and the dual norm.

Identical Distributed Pairwise NQD Random Variables Sequences with a Novel Mutation Operator

2014 ◽  
Vol 989-994 ◽  
pp. 2491-2494
Author(s):  
Jian Cao ◽  
Gang Li ◽  
Cen Rui Ma
Keyword(s):  
Optimization Problems ◽  
Novel Mutation ◽  
Random Variables ◽  
Strong Stability ◽  
Mutation Operator ◽  
Linear Forms ◽  
Grid Portal ◽  
Complex Optimization ◽  
Independent Identically Distributed ◽  
Improved Pso

We study the strong stability of linear forms of pairwise negatively quadrant dependent (NQD) identically distributed random variables sequence under some suitable conditions. To overcome the weakness of collaborative services ability in different grid portal, a new grid portal architecture based on CSGPA (Collaborative Services Grid Portal Architecture), is proposed. This paper aims to enhance the performance of PSO in complex optimization problems and proposes an improved PSO variant by incorporating a novel mutation operator. The results obtained extend and improve the corresponding theorem for independent identically distributed random variables sequence.

scholarly journals Sharp Constants and Optimizers for a Class of Caffarelli–Kohn–Nirenberg Inequalities

2017 ◽  
Vol 17 (3) ◽  
Author(s):  
Nguyen Lam ◽  
Guozhen Lu
Keyword(s):  
Conformal Mapping ◽  
Large Class ◽  
Best Constants ◽  
Extremal Functions ◽  
Sharp Constants

AbstractIn this paper, we use a suitable transform of quasi-conformal mapping type to investigate the sharp constants and optimizers for the following Caffarelli–Kohn–Nirenberg inequalities for a large class of parametersWe compute the best constants and the explicit forms of the extremal functions in numerous cases. When

Best Constants for Moser-Trudinger Inequalities on High Dimensional Hyperbolic Spaces

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Guozhen Lu ◽  
Hanli Tang
Keyword(s):  
Riemannian Manifolds ◽  
Best Constants ◽  
Hyperbolic Spaces ◽  
High Dimensional ◽  
Sharp Constant ◽  
Heisenberg Groups ◽  
Euclidean Spaces ◽  
Sharp Constants ◽  
Trudinger Inequality

AbstractThough there have been extensive works on best constants for Moser-Trudinger inequalities in Euclidean spaces, Heisenberg groups or compact Riemannian manifolds, much less is known for sharp constants for the Moser-Trudinger inequalities on hyperbolic spaces. Earlier works only include the sharp constant for the Moser-Trudinger inequality on the twodimensional hyperbolic disc. In this paper, we establish best constants for several types of Moser-Trudinger inequalities on high dimensional hyperbolic spaces ℍ

scholarly journals Complexity and fractal dimensions for infinite sequences with positive entropy

2019 ◽  
Vol 21 (06) ◽  
pp. 1850068
Author(s):  
Christian Mauduit ◽  
Carlos Gustavo Moreira
Keyword(s):  
Exponential Growth ◽  
Fractal Dimensions ◽  
Combinatorial Structure ◽  
Finite Alphabet ◽  
Combinatorial Proof ◽  
Infinite Words ◽  
Entropy Formula ◽  
Complexity Function ◽  
Infinite Word ◽  
Infinite Sequences

The complexity function of an infinite word [Formula: see text] on a finite alphabet [Formula: see text] is the sequence counting, for each non-negative [Formula: see text], the number of words of length [Formula: see text] on the alphabet [Formula: see text] that are factors of the infinite word [Formula: see text]. The goal of this work is to estimate the number of words of length [Formula: see text] on the alphabet [Formula: see text] that are factors of an infinite word [Formula: see text] with a complexity function bounded by a given function [Formula: see text] with exponential growth and to describe the combinatorial structure of such sets of infinite words. We introduce a real parameter, the word entropy [Formula: see text] associated to a given function [Formula: see text] and we determine the fractal dimensions of sets of infinite sequences with complexity function bounded by [Formula: see text] in terms of its word entropy. We present a combinatorial proof of the fact that [Formula: see text] is equal to the topological entropy of the subshift of infinite words whose complexity is bounded by [Formula: see text] and we give several examples showing that even under strong conditions on [Formula: see text], the word entropy [Formula: see text] can be strictly smaller than the limiting lower exponential growth rate of [Formula: see text].

Design emergence in Morocco as an African country: a pending institutionalization

2019 ◽  
Vol 1 (1) ◽  
Author(s):  
Nabil EL HILALI
Keyword(s):  
Institutional Theory ◽  
Exponential Growth ◽  
African Country ◽  
Political Stability ◽  
The State ◽  
Design Management ◽  
Specific Context ◽  
Developed Economies ◽  
Design Value ◽  
Specific Country

If design management is worldwide institutionalized especially in developed economies, little is known about African design even though the continent is becoming an attractive economy thanks to his exponential growth and more political stability. Oriented toward one specific country: Morocco, this study through a questioning embedded in institutional theory brings an overview about design in a specific context. This research captures design management emergence in Morocco by spotting the light on the state of design institutionalization toward the creation of design value.

Comparison of Meta-heuristic Algorithms on Benchmark Functions

2019 ◽  
Vol 2 (3) ◽  
pp. 508-517
Author(s):  
FerdaNur Arıcı ◽  
Ersin Kaya
Keyword(s):  
Heuristic Algorithms ◽  
Optimization Problems ◽  
Search Algorithm ◽  
Optimization Algorithms ◽  
Gravitational Search Algorithm ◽  
Differential Evolution Algorithm ◽  
Population Based ◽  
Time Interval ◽  
Bee Colony ◽  
Different Characteristics

Optimization is a process to search the most suitable solution for a problem within an acceptable time interval. The algorithms that solve the optimization problems are called as optimization algorithms. In the literature, there are many optimization algorithms with different characteristics. The optimization algorithms can exhibit different behaviors depending on the size, characteristics and complexity of the optimization problem. In this study, six well-known population based optimization algorithms (artificial algae algorithm - AAA, artificial bee colony algorithm - ABC, differential evolution algorithm - DE, genetic algorithm - GA, gravitational search algorithm - GSA and particle swarm optimization - PSO) were used. These six algorithms were performed on the CEC’17 test functions. According to the experimental results, the algorithms were compared and performances of the algorithms were evaluated.

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