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

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

C-BN Single-Walled Nanotubes from Hybrid Connection of BN/C Nanoribbons: Prediction by ab initio Density Functional Calculations

We demonstrated for the first time by ab initio density functional calculation and molecular dynamics simulation that C0.5(BN)0.5 armchair single-walled nanotubes (NT) are gapless semiconductors and can be spontaneously formed via the hybrid connection of graphene/BN Nanoribbons (GNR/BNNR) at room temperature. The direct synthesis of armchair C0.5(BN)0.5 via the hybrid connection of GNR/BNNR is predicted to be both thermodynamically and dynamically stable. Such novel armchair C0.5(BN)0.5 NTs possess enhanced conductance as that observed in GNRs. Additionally, the zigzag C0.5(BN)0.5 SWNTs are narrow band gap semiconductors, which may have potential application for light emission. In light of recent experimental progress and the enhanced degree of control in the synthesis of GNRs and BNNR, our results highlight an interesting avenue for synthesizing a novel specific type of C0.5(BN)0.5 nanotube (gapless or narrow direct gap semiconductor), with potentially important applications in BNC-based nanodevices.

Potential uses of Bayesian networks as tools for synthesis of systematic reviews of complex interventions

Bayesian networks (BNs) are tools for representing expert knowledge or evidence. They are especially useful for synthesising evidence or belief concerning a complex intervention, assessing the sensitivity of outcomes to different situations or contextual frameworks and framing decision problems that involve alternative types of intervention. Bayesian networks are useful extensions to logic maps when initiating a review or to facilitate synthesis and bridge the gap between evidence acquisition and decision-making. Formal elicitation techniques allow development of BNs on the basis of expert opinion. Such applications are useful alternatives to ‘empty’ reviews, which identify knowledge gaps but fail to support decision-making. Where review evidence exists, it can inform the development of a BN. We illustrate the construction of a BN using a motivating example that demonstrates how BNs can ensure coherence, transparently structure the problem addressed by a complex intervention and assess sensitivity to context, all of which are critical components of robust reviews of complex interventions. We suggest that BNs should be utilised to routinely synthesise reviews of complex interventions or empty reviews where decisions must be made despite poor evidence.

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

Potential Analysis for Hypoelliptic Second Order PDEs with Nonnegative Characteristic Form

In this Thesis we consider a class of second order partial differential operators with non-negative characteristic form and with smooth coefficients. Main assumptions on the relevant operators are hypoellipticity and existence of a well-behaved global fundamental solution. We first make a deep analysis of the L-Green function for arbitrary open sets and of its applications to the Representation Theorems of Riesz-type for L-subharmonic and L-superharmonic functions. Then, we prove an Inverse Mean value Theorem characterizing the superlevel sets of the fundamental solution by means of L-harmonic functions. Furthermore, we establish a Lebesgue-type result showing the role of the mean-integal operator in solving the homogeneus Dirichlet problem related to L in the Perron-Wiener sense. Finally, we compare Perron-Wiener and weak variational solutions of the homogeneous Dirichlet problem, under specific hypothesis on the boundary datum.