Nonlocal Effective Field Equations for Quantum Cosmology

The possibility that the strength of gravitational interactions might slowly increase with distance, is explored by formulating a set of effective field equations, which incorporate the gravitational, vacuum-polarization induced, running of Newton's constant $G$. The resulting long distance (or large time) behaviour depends on only one adjustable parameter $\xi$, and the implications for the Robertson-Walker universe are calculated, predicting an accelerated power-law expansion at later times $t \sim \xi \sim 1/H$.Comment: 9 page

Fractional Fokker-Planck Equation for Fractal Media

We consider the fractional generalizations of equation that defines the medium mass. We prove that the fractional integrals can be used to describe the media with noninteger mass dimensions. Using fractional integrals, we derive the fractional generalization of the Chapman-Kolmogorov equation (Smolukhovski equation). In this paper fractional Fokker-Planck equation for fractal media is derived from the fractional Chapman-Kolmogorov equation. Using the Fourier transform, we get the Fokker-Planck-Zaslavsky equations that have fractional coordinate derivatives. The Fokker-Planck equation for the fractal media is an equation with fractional derivatives in the dual space.Comment: 17 page

Pinning control of fractional-order weighted complex networks

In this paper, we consider the pinning control problem of fractional-order weighted complex dynamical networks. The well-studied integer-order complex networks are the special cases of the fractional-order ones. The network model considered can represent both directed and undirected weighted networks. First, based on the eigenvalue analysis and fractional-order stability theory, some local stability properties of such pinned fractional-order networks are derived and the valid stability regions are estimated. A surprising finding is that the fractional-order complex networks can stabilize itself by reducing the fractional-order q without pinning any node. Second, numerical algorithms for fractional-order complex networks are introduced in detail. Finally, numerical simulations in scale-free complex networks are provided to show that the smaller fractional-order q, the larger control gain matrix D, the larger tunable weight parameter , the larger overall coupling strength c, the more capacity that the pinning scheme may possess to enhance the control performance of fractional-order complex networks

Electromagnetic field of fractal distribution of charged particles

Electric and magnetic fields of fractal distribution of charged particles are considered. The fractional integrals are used to describe fractal distribution. The fractional integrals are considered as approximations of integrals on fractals. Using the fractional generalization of integral Maxwell equation, the simple examples of the fields of homogeneous fractal distribution are considered. The electric dipole and quadrupole moments for fractal distribution are derived.Comment: RevTex, 21 pages, 2 picture

Homogeneous fractional embeddings

Fractional equations appear in the description of the dynamics of various physical systems. For Lagrangian systems, the embedding theory developped by Cresson ["Fractional embedding of differential operators and Lagrangian systems", J. Math. Phys. 48, 033504 (2007)] provides a univocal way to obtain such equations, stemming from a least action principle. However, no matter how equations are obtained, the dimension of the fractional derivative differs from the classical one and may induce problems of temporal homogeneity in fractional objects. In this paper, we show that it is necessary to introduce an extrinsic constant of time. Then, we use it to construct two equivalent fractional embeddings which retain homogeneity. The notion of fractional constant is also discussed through this formalism. Finally, an illustration is given with natural Lagrangian systems, and the case of the harmonic oscillator is entirely treated.Comment: 14 page

Approach of a class of discontinuous dynamical systems of fractional order: existence of the solutions

In this letter we are concerned with the possibility to approach the existence of solutions to a class of discontinuous dynamical systems of fractional order. In this purpose, the underlying initial value problem is transformed into a fractional set-valued problem. Next, the Cellina's Theorem is applied leading to a single-valued continuous initial value problem of fractional order. The existence of solutions is assured by a P\'{e}ano like theorem for ordinary differential equations of fractional order.Comment: accepted IJBC, 5 pages, 1 figur

Anomalous spatial diffusion and multifractality in optical lattices

Transport of cold atoms in shallow optical lattices is characterized by slow, nonstationary momentum relaxation. We here develop a projector operator method able to derive in this case a generalized Smoluchowski equation for the position variable. We show that this explicitly non-Markovian equation can be written as a systematic expansion involving higher-order derivatives. We use the latter to compute arbitrary moments of the spatial distribution and analyze their multifractal properties.Comment: 5 pages, 3 figure

The fractional Schr\"{o}dinger operator and Toeplitz matrices

Confining a quantum particle in a compact subinterval of the real line with Dirichlet boundary conditions, we identify the connection of the one-dimensional fractional Schr\"odinger operator with the truncated Toeplitz matrices. We determine the asymptotic behaviour of the product of eigenvalues for the $\alpha$-stable symmetric laws by employing the Szeg\"o's strong limit theorem. The results of the present work can be applied to a recently proposed model for a particle hopping on a bounded interval in one dimension whose hopping probability is given a discrete representation of the fractional Laplacian.Comment: 10 pages, 2 figure

Maximal, potential and singular operators in the local "complementary" variable exponent Morrey type spaces

We consider local "complementary" generalized Morrey spaces M-c({x0})p(.).omega (Omega) in which the p-means of function are controlled over Omega \ B(x(0), r) instead of B(x(0), r), where Omega subset of R-n is a bounded open set, p(x) is a variable exponent, and no monotonicity type condition is imposed onto the function omega(r) defining the "complementary" Morrey-type norm. In the case where omega is a power function, we reveal the relation of these spaces to weighted Lebesgue spaces. In the general case we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev type M-c({x0})p(.).omega (Omega) -> M-c({x0})p(.).omega (Omega)-theorem for the potential operators I-alpha(.), also of variable order. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities-on omega(r), which do not assume any assumption on monotonicity of omega(r).Science Development Foundation under the President of the Republic of Azerbaijan [EIF-2010-1(1)-40/06-1]; Scientific and Technological Research Council of Turkey (TUBITAK) [110T695]info:eu-repo/semantics/publishedVersio