图的孤立韧度与分数k-覆盖图

设G是一个图,若对于图G的任一条边e,都存在一个分数k-因子h,使得h(e)=1,则称图G是分数k-覆盖图.图G的孤立韧度I(a)定义为:若G是完全图,则I(C)= ∞;否则,I(G)=min{|S|/i(G-S):SCV(G),i(G-S)≥2},其中i(G-S)表示G-S中的孤立点数目.本文首次提出并研究了一个图是分数k-覆盖图与它的孤立韧度之间的关系,证明了当I(G)>k,并且δ(G)>k 1时,G是分数k-覆盖图.我们还证明了,这个结果是最好可能的.     

图的联结数与分数κ-消去图

设G是一个图,若对于图G的任一条边e,G-e都存在一个分数k-因子,则称G是一个分数k-消去图.若k=2,则称分数k-消去图为分数2-消去图.本文证明了当bind（G）≥2,并且δ（G）≥3时,G是分数2-消去图.     

图的联结数与分数k-消去图

设G是一个图,若对于图G的任一条边e,G-e都存在一个分数k-因子,则称G是一个分数κ-消去图.若k=2,则称分数κ-消去图为分数2-消去图.本文证明了当bind(G)≥2,并且6(G)≥3时,G是分数2-消去图.     

图的分数因子与孤立韧度

图G的孤立韧度定义为I(G)=min{｜S｜/i(G-S)∶SV(G),i(G-S)≥2},若G不是完全图.否则令I(G)=∞.本文给出了图的分数k因子与图的分数[a,b]因子的存在性与图的孤立韧度的关系.证明了,若δ(G)≥k且I(G)≥k,则G有分数k因子;若δ(G)≥I(G)≥a-1 a/b,则图G有分数[a,b]因子,其中a相似文献   

关于k-消去二分图的一些结果

图 G的一个 k-正则支撑子图称为 G的 k-因子 ,若对 G的任一边 e,图 G- e总存在一个 k-因子 ,则称 G是 k-消去图 .证明了二分图 G=( X,Y) ,且 | X | =| Y|是 k-消去图的充分必要条件是 k| S|≤ r1 + 2 r2 +…+ k( rk+… + rΔ) - ε( S)对所有 S X成立 .并由此给出二分图是 k-消去图的充分度条件 .     

关于k—消去图的若干新结果

设G是一个图．k是自然数．图G的一个k-正则支撑子图称为G的一个k-因子．若对于G的每条边e．G—e都存在一个k-因子，则称G是一个k-消去图．该文得到了一个图是k-消去图的若干充分条件，推广了文［2—4］中有关结论．     

图的孤立韧度与分数因子的存在性

设G是一个简单无向图，若G不是完全图，G的孤立韧度定义为I（G）=min{|S|/I(G-S):S包含于V（G），I(G-S)≥2}。否则，令I（G）=∞。本文引入一个与图的孤立韧度I（G）密切相关的新参数I‘(G)，若G不是完全图时，I‘(G)=min{|S|/(I(G-S)-1):S包含于V（G），I(G-S)≥2}。否则，I‘(G)=∞；本文研究了参数I（G）和I‘(G)的性质以及两者与图的分数k-因子的关系。给出了具有某些约束条件的图的分数因子存在的一些充分条件。并提出进一步的可研究的问题。     

有约束条件的图的分数k-因子

设G是一个简单的无向图，若G不是完全图，G的孤立韧度定义为I(G)=min{｜s｜／i(G-S)：S∈V(G)，i(G-S)≥2)；否则令I(G)=∞．对与图的孤立韧度I(G)密切相关的新参数，I’(G)，若G不是完全图，定义I’(G)=min{｜s｜/i(G-S)-1:S∈V(G)，i(G-S)≥2}；否则I’(G)=∞本文研究了新参数I‘(G)与图的分数κ-因子的关系，给出了具有某些约束条件的图的分数κ-因子存在的一些充分条件．     

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．     

关于图的孤立韧度与分数因子存在性的若干结果

本文讨论了图的孤立韧度I(G)以及与之相关的参数I^1(G)与图的分数因子存在性的关系，给出了I(G)及I^1(G)与图的分数点(边)消去性、分数L-可扩性及分数[1，b]-因子存在性之间关系的一系列结果．     

关于分数k-因子临界图与分数k-可扩图的若干结果

一个简单图G, 如果对于V(G)的任意k元子集S, 子图G-S都包含分数完美匹配, 那么称G为分数k-因子临界图. 如果图G的每个k-匹配M都包含在一个分数完美匹配中, 那么称图G为分数k-可扩图. 给出一个图是分数k-因子临界图和分数k-可扩图的充分条件, 并给出一个图是分数k-因子临界图的充分必要条件.     

Dual characterization of fractional capacity via solution of fractional p-Laplace equation

In terms of weak solutions of the fractional p-Laplace equation with measure data, this paper offers a dual characterization for the fractional Sobolev capacity on bounded domain. In addition, two further results are given: one is an equivalent estimate for the fractional Sobolev capacity; the other is the removability of sets of zero capacity and its relation to solutions of the fractional p-Laplace equation.     

Existence of fractional differential chains and factorizations based on transformations

In this work, we deal with the existence of the fractional integrable equations involving two generalized symmetries compatible with nonlinear systems. The method used is based on the Bä cklund transformation or B‐transformation. Furthermore, we shall factorize the fractional heat operator in order to yield the fractional Riccati equation. This is done by utilizing matrix transform Miura type and matrix operators, that is, matrices whose entries are differential operators of fractional order. The fractional differential operator is taken in the sense of Riemann–Liouville calculus. Copyright © 2014 John Wiley & Sons, Ltd.     

SOME PROBLEMS OF FRACTIONAL PARTIAL DIFFERENCE DIFFUSION EQUATIONS

This paper mainly discusses the problems of fractional variational problems and fractional diffusion problems using fractional difference and summation. Through the Euler finite difference method we get a variational formulation of the variation problem and the discrete solution to the time-fractional and space-fractional difference equation using separating variables method and two-side Z-transform method.     

A fractional partial differential equation based multiscale denoising model for texture image

Traditional integer‐order partial differential equation based image denoising approach can easily lead edge and complex texture detail blur, thus its denoising effect for texture image is always not well. To solve the problem, we propose to implement a fractional partial differential equation (FPDE) based denoising model for texture image by applying a novel mathematical method—fractional calculus to image processing from the view of system evolution. Previous studies show that fractional calculus has some unique properties that it can nonlinearly enhance complex texture detail in digital image processing, which is obvious different with integer‐order differential calculus. The goal of the modeling is to overcome the problems of the existed denoising approaches by utilizing the aforementioned properties of fractional differential calculus. Using classic definition and property of fractional differential calculus, we extend integer‐order steepest descent approach to fractional field to implement fractional steepest descent approach. Then, based on the earlier fractional formulas, a FPDE based multiscale denoising model for texture image is proposed and further analyze optimal parameters value for FPDE based denoising model. The experimental results prove that the ability for preserving high‐frequency edge and complex texture information of the proposed fractional denoising model are obviously superior to traditional integral based algorithms, as for texture detail rich images. Copyright © 2013 John Wiley & Sons, Ltd.     

分数阶微分方程的比较定理

本文给出了非线性Riemann—Liouville分数阶微分方程和Caputo分数阶微分方程与相应的非线性Volterra积分方程的等价性,并在此基础上建立了分数阶微分方程的比较定理．     

Caputo–Hadamard Fractional Derivatives of Variable Order

In this article, we present three types of Caputo–Hadamard derivatives of variable fractional order and study the relations between them. An approximation formula for each fractional operator, using integer-order derivatives only, is obtained and an estimation for the error is given. At the end, we compare the exact fractional derivative of a concrete example with some numerical approximations.     

分数k-因子临界图的条件

设G是-个连通简单无向图,如果删去G的任意k个项点后的图有分数完美匹配,则称G是分数k-因子临界图．给出了G是分数k-因子临界图的韧度充分条件与度和充分条件,这些条件中的界是可达的,并给出G是分数k-因子临界图的一个关于分数匹配数的充分必要条件．     

New approach to a generalized fractional integral

The paper presents a new fractional integration, which generalizes the Riemann-Liouville and Hadamard fractional integrals into a single form. Conditions are given for such a fractional integration operator to be bounded in an extended Lebesgue measurable space. Semigroup property for the above operator is also proved. We give a general definition of the fractional derivatives and give some examples.     

分数阶尺度函数的分解与重构算法

引入分数阶多分辨分析与分数阶尺度函数的概念.运用时频分析方法与分数阶小波变换,研究了分数阶正交小波的构造方法,得到分数阶正交小波存在的充要条件.给出分数阶尺度函数与小波的分解与重构算法,算法比经典的尺度函数与小波的分解与重构算法更具有一般性.