Fractional Fokker–Planck equation

By using the definition of the characteristic function and Kramers–Moyal Forward expansion, one can obtain the Fractional Fokker–Planck Equation (FFPE) in the domain of fractal time evolution with a critical exponent α (0<α⩽1). Two different classes of fractional differential operators, Liouville–Riemann (L–R) and Nishimoto (N) are used to represent the fractal differential operators in time. By applying the technique of eigenfunction expansion to get the solution of FFPE, one finds that the time part of eigenfunction expansion in terms of L–R represents the waiting time density Ψ(t), which gives the relation between fractal time evolution and the theory of continuous time random walk (CTRW). From the principle of maximum entropy, the structure of the distribution function can be known.     

乘积图的Fractional控制

Let G =(V,E) be a simple graph.For any real function g :V-→ R and a subset S V,we write g(S) =∑v∈Sg(v).A function f :V-→ [0,1] is said to be a fractional dominating function(F DF) of G if f(N [v]) ≥ 1 holds for every vertex v ∈ V(G).The fractional domination number γf(G) of G is defined as γf(G) = min{f(V)|f is an F DF of G }.The fractional total dominating function f is defined just as the fractional dominating function,the difference being that f(N(v)) ≥ 1 instead of f(N [v]) ≥ 1.The fractional total domination number γ0f(G) of G is analogous.In this note we give the exact values ofγf(Cm × Pn) and γ0f(Cm × Pn) for all integers m ≥ 3 and n ≥ 2.     

Multilinear Singular and Fractional Integrals

In this paper, we treat a class of non-standard commutators with higher order remainders in the Lipschitz spaces and give （L^v, L^q）, （H^p, L^q） boundedness and the boundedness in the Triebel- Lizorkin spaces. Our results give simplified proofs of the recent works by Chen, and extend his result.     

On Compositions of Generalized Fractional Integrals

Compositions of generalized fractional integral operators involving Gauss hy-pergeometric function with power weights are studied.Com position formulas forsuch integrals which are the operators of the same type are obtained.In parti-cular,compositions of two identical operators are given.     

几类图的Fractional星控制数

设G=(V,E)是一个无孤立点的图,一个实值函数f:E(G)→[0,1]若对所有的点u∈V(G),均有∑uv∈Ef(uv)≥1成立,则称f为图G的一个Fractional星控制函数.图G的Fractional星控制数定义为γ_(fs)(G)=min{∑uv∈Ef(uv)|f为图G的一个Fractional星控制函数}.研究了几类乘积图的Fractional星控制问题,给出了一些常见特殊图的Fractional星控制数,主要确定了积图P_m×P_n和C_m×P_n的Fractional星控制数.     

Dimensional Properties of Fractional Brownian Motion

Let B^α = {B^α（t）,t E R^N} be an （N,d）-fractional Brownian motion with Hurst index α∈ （0, 1）. By applying the strong local nondeterminism of B^α, we prove certain forms of uniform Hausdorff dimension results for the images of B^α when N 〉 αd. Our results extend those of Kaufman for one-dimensional Brownian motion.     

Boundary Value Problems for Riemann–Liouville Fractional Differential Inclusions with Nonlocal Hadamard Fractional Integral Conditions

In this paper, we study a new class of boundary value problems from a fractional differential inclusion of Riemann–Liouville type and nonlocal Hadamard fractional integral boundary conditions. Some new existence results for convex as well as non-convex multi-valued maps are obtained using standard fixed point theorems. The obtained results are illustrated by examples.     

Fractional Sobolev-Poincaré Inequalities in Irregular Domains

This paper is devoted to the study of fractional(q, p)-Sobolev-Poincar′e inequalities in irregular domains. In particular, the author establishes(essentially) sharp fractional(q, p)-Sobolev-Poincar′e inequalities in s-John domains and in domains satisfying the quasihyperbolic boundary conditions. When the order of the fractional derivative tends to 1, our results tend to the results for the usual derivatives. Furthermore, the author verifies that those domains which support the fractional(q, p)-Sobolev-Poincar′e inequalities together with a separation property are s-diam John domains for certain s, depending only on the associated data. An inaccurate statement in [Buckley, S. and Koskela, P.,Sobolev-Poincar′e implies John, Math. Res. Lett., 2(5), 1995, 577–593] is also pointed out.     

Fractional Poincaré inequalities for general measures

We prove a fractional version of Poincaré inequalities in the context of ${\mathbb{R}}^n$ endowed with a fairly general measure. Namely we prove a control of an $L^2$ norm by a non-local quantity, which plays the role of the gradient in the standard Poincaré inequality. The assumption on the measure is the fact that it satisfies the classical Poincaré inequality, so that our result is an improvement of the latter inequality. Moreover we also quantify the tightness at infinity provided by the control on the fractional derivative in terms of a weight growing at infinity. The proof goes through the introduction of the generator of the Ornstein–Uhlenbeck semigroup and some careful estimates of its powers. To our knowledge this is the first proof of fractional Poincaré inequality for measures more general than Lévy measures.     

Orlicz–Morrey Spaces and Fractional Operators

By taking an interest in a natural extension to the small parameters of the trace inequality for Morrey spaces, Orlicz–Morrey spaces are introduced and some inequalities for generalized fractional integral operators on Orlicz–Morrey spaces are established. The local boundedness property of the Orlicz maximal operators is investigated and some Morrey-norm equivalences are also verified. The result obtained here sharpens the one in our earlier papers.     

The Riesz–Bessel Fractional Diffusion Equation

This paper examines the properties of a fractional diffusion equation defined by the composition of the inverses of the Riesz potential and the Bessel potential. The first part determines the conditions under which the Green function of this equation is the transition probability density function of a Lévy motion. This Lévy motion is obtained by the subordination of Brownian motion, and the Lévy representation of the subordinator is determined. The second part studies the semigroup formed by the Green function of the fractional diffusion equation. Applications of these results to certain evolution equations is considered. Some results on the numerical solution of the fractional diffusion equation are also provided.     

The Fractional Derivatives of a Fractal Function

The present paper investigates the fractional derivatives of Weierstrass function, proves that there exists some linear connection between the order of the fractional derivatives and the dimension of the graphs of Weierstrass function.     

Jump Type Cahn-Hilliard Equations with Fractional Noises

The authors explore a class of jump type Cahn-Hilliard equations with fractional noises. The jump component is described by a （pure jump） Lévy space-time white noise. A fixed point scheme is used to investigate the existence of a unique local mild solution under some appropriate assumptions on coefficients.     

Weighted Lorentz Norm Inequalities for Riemann-Liouville Fractional Integra

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Semi-Linear Fractional σ-Evolution Equations with Nonlinear Memory

In this paper we study the local or global (in time) existence of small data solutions to semi-linear fractional σ−evolution equations with nonlinearmemory. Our main goals is to explain on the one hand the influence of the memory term and on the other hand the influence of higher regularity of the data on qualitative properties of solutions.     

Fractional Square Functions and Potential Spaces, Ⅱ

In this paper, we study fractional square functions associated with the Poisson semigroup for Schrdinger operators. We characterize the potential spaces in the Schrdinger setting by using vertical, area and g*λ fractional square functions.     

Fuzzy Fractional Monodromy and the Section–Hyperboloid

A theorem about the matrix of fractional monodromy will be formulated. The monodromy corresponds to going around a fiber with a singular point of oscillator type with arbitrary resonance. The reason of fractional monodromy and fuzziness of such a monodromy is explained. Some ideas for the proof of the theorem are given. A few remarks about the semi-global structure of singular lagrangian fibration are made. Lecture held in the Seminario Matematico e Fisico di Milano on November 8, 2004 Received: June 2007     

Fluctuation Limit of a Fractional Brownian Particle System

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