分数(g,f)-因子覆盖图

一个图称为分烽（g,f)- 因子覆盖图，如果G中的任何一条边e都包含在一个分数（g,f)- 因子中，并且满足h(e)=1，其中h是分数（g,f)- 因子的导出函数。本文给出了一个图是分数（g,f)- 因子覆盖图的充要条件。     

关于分数(g,f)-因子消去图

一个图称为分数（g,f）－因子消去图，如果去掉图G中的任何一条边e图G仍有一个分数（g,f）－因子。本文分别给出了一个力是分数1－因子消去图和分数2－因子消去图的几个充分条件，并给出一个图有一个分数（g,f）－因子不含给定对集中任何一条边的充要条件。     

关于分数可消去图的若干结果

令G=(V(G),E(G))是一个图,并令9和f是两个定义在V(G)上的整数值函数且对所有的x∈V(G)有g(x)≤f(z)成立．若对G的每一条边e都存在G的一个分数(g,f)-因子G_h使得h(e)=0,其中h是G_h的示性函数,则称G是一个分数(g,f)-消去图,若在G中删去E′■E(G),|E′|=k后,所得图有分数完美匹配,则称G是分数k-边-可消去的。本文给出了图是1-可消去,2-可消去和k-边-可消去的与韧度和孤立韧度相关的充分条件。证明了这些结果在一定意义上是最好可能的．     

图中的最大分数(0,f)-因子

本语文给出了图的一个（０,ｆ）因子是最大因子的特征,并得到了一个图有（ｇ,ｆ）－因子的充分条件,从而了关于分数对集和１－因子的有关结果。     

An Extension of the Hirsch Symbolic Calculus

The symbolic calculus developed by Francis Hirsch (in several papers, between 1972 and 1976) is an already classical theory that introduces and studies the operators f(A) associated to a non-negative linear operator A on a Banach space and to the Stieltjes transform f of a Radon measure . It is required that the operator A has a dense domain and that the measure , as well as the value f()), are real and non-negative. These three conditions are essential in the proof of the main results, but they are very restrictive, since important cases are excluded, as the fractional powers A of complex exponent , or of base A non-densely defined. In this paper we present a reconstruction of the Hirsch theory, without using those hypothesis.     

The Extension Degree Conditions for Fractional Factor

In Gao’s previous work, the authors determined several degree conditions of a graph which admits fractional factor in particular settings. It was revealed that these degree conditions are tight if b = f(x) = g(x) = a for all vertices x in G. In this paper, we continue to discuss these degree conditions for admitting fractional factor in the setting that several vertices and edges are removed and there is a difference Δ between g(x) and f(x) for every vertex x in G. These obtained new degree conditions reformulate Gao’s previous conclusions, and show how Δ acts in the results. Furthermore,counterexamples are structured to reveal the sharpness of degree conditions in the setting f(x) =g(x) + Δ.     

A simple proof of the geometric fractional monodromy theorem

A simple proof of the “geometric fractional monodromy theorem” (Broer-Efstathiou-Lukina 2010) is presented. The fractional monodromy of a Liouville integrable Hamiltonian system over a loop γ ? ?² is a generalization of the classic monodromy to the case when the Liouville foliation has singularities over γ. The “geometric fractional monodromy theorem” finds, up to an integral parameter, the fractional monodromy of systems similar to the 1: (?2) resonance system. A handy equivalent definition of fractional monodromy is presented in terms of homology groups for our proof.     

分数因子和分数哈密顿图

本文介绍了图的分数方面,将图中基于整数的定义和变量转化为分数形式．介绍了分数图论的一些新结果,特别是关于分数因子和分数哈密顿图的新结果,其中包括了作者最近得到的一些关于分数(g,f)-因子的若干结果．进而,提出了还没有解决的几个新问题．     

Fixed-point theorems for set-valued mappings and existence of best approximants

The paper gives a proof of the following existence result in best approximation. Let M be a compact, convex, and non-empty subset of a normed space E and let g be a continuous almost affine mapping of M onto M. For each continuous mapping f from M into E there exists a point x in M such that g(x) is a best M- approximation to f(x). The proof uses Bohnenblust and Karlin's extension to normed spaces of Kakutani's Fixed Point Theorem for set-valued mappings on compact, convex, and non-empty subsets of Euclidean n-space.     

Anderson Localization for a Class of Models with a Sign-Indefinite Single-Site Potential via Fractional Moment Method

A technically convenient signature of Anderson localization is exponential decay of the fractional moments of the Green function within appropriate energy ranges. We consider a random Hamiltonian on a lattice whose randomness is generated by the sign-indefinite single-site potential, which is however sign-definite at the boundary of its support. For this class of Anderson operators, we establish a finite-volume criterion which implies that the fractional moment decay property holds. This constructive criterion is satisfied at typical perturbative regimes, e.g. at spectral boundaries which satisfy “Lifshitz tail estimates” on the density of states and for sufficiently strong disorder. We also show how the fractional moment method facilitates the proof of exponential (spectral) localization for such random potentials.     

二阶奇异非线性微分方程边值问题的正解

分别在0≤f₀+<M₁,m₁<f_∞-≤∞和0≤f_∞+<M₁,m₁<f₀-≤∞的情形下研究了非线性奇异边值问题u″+g(t)f(u)=0,0<t<1,αu(0)-βu′(0)=0,γu(1)+δu′(1)=0正解的存在性,其中f₀+=₀f(u)/u,f_∞-=_∞f(u)/u,f₀-=₀f(u)/u,f_∞+=_∞f(u)/u,g在区间[0,1]的端点可以具有奇性。     

图的分数(g,f)-因子

本文研究了图的分数因子的性质,特别给出了图的弧立韧度这一新概念,研究了孤立韧度与分数因子的关系,文中给出了一个图具有某些约束条件的（g,f）-分数因子的一些充分条件,得到了若干新结果,并提出 了一些可供进一步研究的问题。     

Martingale Transforms and the Hardy-Littlewood-Sobolev Inequality for Semigroups

We give a representation of the fractional integral for symmetric Markovian semigroups as the projection of martingale transforms and prove the Hardy-Littlewood-Sobolev (HLS) inequality based on this representation. The proof rests on a new inequality for the fractional Littlewood-Paley g–function.     

Changing the variable in convolution of distributions

In this paper the author has extended the concept of changing the variables in distributions to the convolution of distributions. For an infinitely differentiable functionh(x), he has first defined the convolution of two distributions f(h(x)) and g(h(x)) and then proved some of its properties. Finally, he has applied his results to the fractional integral and fractional differential operators     

Rough bilinear fractional integrals with variable kernels

We study the rough bilinear fractional integral

Parametric approaches to fractional programs

The fractional program P is defined by maxf(x)/g(x) subject tox∈X. A class of methods for solving P is based on the auxiliary problem Q(λ) with a parameter λ: maxf(x)?λg(x) subject tox∈X. Starting with two classical methods in this class, the Newton method and the binary search method, a number of variations are introduced and compared. Among the proposed methods. the modified binary search method is theoretically interesting because of its superlinear convergence and the capability to provide an explicit interval containing the optimum parameter value \(\bar \lambda \) . Computational behavior is tested by solving fractional knapsack problems and quadratic fractional programs. The interpolated binary search method seems to be most efficient, while other methods also behave surprisingly well.     

Monotonic solutions of a quadratic integral equation of fractional order

In the paper we indicate an error made in the proof of the main result of the paper [M.A. Darwish, On quadratic integral equation of fractional orders, J. Math. Anal. Appl. 311 (2005) 112-119]. Moreover, we provide correct proof of a slightly modified version of the mentioned result. The main tool used in our proof is the technique associated with the Hausdorff measure of noncompactness.     

Sharp Weak Type Inequality for Fractional Integral Operators Associated with d-Dimensional Walsh–Fourier Series

Suppose that d ≥ 1 is an integer, ${\alpha \in (0,d)}$ is a fixed parameter and let I _α be the fractional integral operator associated with d-dimensional Walsh–Fourier series on [0, 1) ^d . The paper contains the proof of the sharp weak-type estimate $$||I_\alpha(f)||_{L^{d/(d-\alpha),\infty}([0,1)^d)}\leq\frac{2^d-1}{(2^{d-\alpha}-1)(2^\alpha-1)}||f||_{L^1([0,1)^d)}.$$ The proof rests on Bellman-function-type method: the above estimate is deduced from the existence of a certain family of special functions.     

Weighted estimates for fractional integral operators on central Morrey spaces

We prove weighted estimates for fractional integral operators on central Morrey spaces. Our result covers the weighted theorem by De Napoli, Drelichman and Durán (2011). Our proof is different from theirs.     

图的分数k-因子

给定图G=(V,E).设a和b是两个非负整数.fE→［0,1］是一个函数.如果