Riesz potential operators and inverses via fractional centred derivatives

International Journal of Mathematics and Mathematical Sciences, Vol.2006

Riesz potential versus fractional Laplacian

This paper starts by introducing the Grünwald–Letnikov derivative, the Riesz potential and the problem of generalizing the Laplacian. Based on these ideas, the generalizations of the Laplacian for 1D and 2D cases are studied. It is presented as a fractional version of the Cauchy–Riemann conditions and, finally, it is discussed with the n-dimensional Laplacian.

Riesz potential versus fractional Laplacian

This paper starts by introducing the Grünwald–Letnikov derivative, the Riesz potential and the problem of generalizing the Laplacian. Based on these ideas, the generalizations of the Laplacian for 1D and 2D cases are studied. It is presented as a fractional version of the Cauchy–Riemann conditions and, finally, it is discussed with the n-dimensional Laplacian.

Sobolev-Morrey Type Inequality for Riesz Potentials, Associated with the Laplace-Bessel Differential Operator

2000 Mathematics Subject Classification: 42B20, 42B25, 42B35

On Limiting Case of the Stein-Weiss Type Inequality for the B-Riesz Potentials

Mathematics Subject Classification: Primary 42B20, 42B25, 42B35

Potential theory on metric spaces

This work revolves around potential theory in metric spaces, focusing on applications of dyadic potential theory to general problems associated to functional analysis and harmonic analysis. In the first part of this work we consider the weighted dual dyadic Hardy's inequality over dyadic trees and we use the Bellman function method to characterize the weights for which the inequality holds, and find the optimal constant for which our statement holds. We also show that our Bellman function is the solution to a stochastic optimal control problem. In the second part of this work we consider the problem of quasi-additivity formulas for the Riesz capacity in metric spaces and we prove formulas of quasi-additivity in the setting of the tree boundaries and in the setting of Ahlfors-regular spaces. We also consider a proper Harmonic extension to one additional variable of Riesz potentials of functions on a compact Ahlfors-regular space and we use our quasi-additivity formula to prove a result of tangential convergence of the harmonic extension of the Riesz potential up to an exceptional set of null measure

What is a fractional derivative?

This paper discusses the concepts underlying the formulation of operators capable of being interpreted as fractional derivatives or fractional integrals. Two criteria for required by a fractional operator are formulated. The Grünwald–Letnikov, Riemann–Liouville and Caputo fractional derivatives and the Riesz potential are accessed in the light of the proposed criteria. A Leibniz rule is also obtained for the Riesz potential.

Best Constant in the Weighted Hardy Inequality: The Spatial and Spherical Version

Mathematics Subject Classification: 26D10.

On Multi-Dimensional Random Walk Models Approximating Symmetric Space-Fractional Diffusion Processes

Mathematics Subject Classification: 26A33, 47B06, 47G30, 60G50, 60G52, 60G60.

The Stein-Weiss Type Inequality for Fractional Integrals, Associated with the Laplace-Bessel Differential Operator

2000 Math. Subject Classification: Primary 42B20, 42B25, 42B35

A family of implementation-friendly BN elliptic curves

For the last decade, elliptic curve cryptography has gained increasing interest in industry and in the academic community. This is especially due to the high level of security it provides with relatively small keys and to its ability to create very efficient and multifunctional cryptographic schemes by means of bilinear pairings. Pairings require pairing-friendly elliptic curves and among the possible choices, Barreto-Naehrig (BN) curves arguably constitute one of the most versatile families. In this paper, we further expand the potential of the BN curve family. We describe BN curves that are not only computationally very simple to generate, but also specially suitable for efficient implementation on a very broad range of scenarios. We also present implementation results of the optimal ate pairing using such a curve defined over a 254-bit prime field. (C) 2001 Elsevier Inc. All rights reserved.

Greenhouse evaluation of the potential of sorghum, pearl millet and crotalaria in the management of Meloidogyne javanica

Crop rotation can play a valuable role in managing plant parasitic nematodes, depending on the availability of profitable non-host or poor host crops. Alternatively, non-host cover crops or green manures can be used in succession to summer cash Crops for this Purpose. The aim of the current study was to evaluate, under greenhouse conditions, the host status of commercial hybrids and cultivars of grain and silage sorghum (Sorghum bicolor) for Meloidogyne javanica, and to assess the effect of sorghum on nematode population in comparison with pearl millet (poor host for M. javanica), showy crotalaria and sunn hemp (both non-hosts). Based on two experiments, it was stated that, as a rule, grain sorghum is a poor host for M. javanica, but silage sorghum is a good host. Silage sorghum `BRS 601` was an exception. In other experiments, grain sorghum, pearl millet (Pennisetum glaucum `BN 2`), showy crotalaria (Crotolaria spectabilis `Comum`) and sunn hemp (C. juncea `IAC-KR-1`) reduced M. javanica population level, while silage sorghum increased the nematode density.

Quasiconformal distortion of Riesz capacities and Hausdorff measures in the plane

In this paper we prove the sharp distortion estimates for the quasiconformal mappings in the plane, both in terms of the Riesz capacities from non linear potential theory and in terms of the Hausdorff measures.

Efeito do campo elétrico em nanocones formados por aln e BN: um estudo por primeiros princípios

The development of computers and algorithms capable of making increasingly accurate and rapid calculations as well as the theoretic foundation provided by quantum mechanics has turned computer simulation into a valuable research tool. The importance of such a tool is due to its success in describing the physical and chemical properties of materials. One way of modifying the electronic properties of a given material is by applying an electric field. These effects are interesting in nanocones because their stability and geometric structure make them promising candidates for electron emission devices. In our study we calculated the first principles based on the density functional theory as implemented in the SIESTA code. We investigated aluminum nitride (AlN), boron nitride (BN) and carbon (C), subjected to external parallel electric field, perpendicular to their main axis. We discuss stability in terms of formation energy, using the chemical potential approach. We also analyze the electronic properties of these nanocones and show that in some cases the perpendicular electric field provokes a greater gap reduction when compared to the parallel field