Chromatic capacity and graph operations

The chromatic capacityχ_cap(G) of a graph G is the largest k for which there exists a k-coloring of the edges of G such that, for every coloring of the vertices of G with the same colors, some edge is colored the same as both its vertices. We prove that there is an unbounded function f:N→N such that χ_cap(G)?f(χ(G)) for almost every graph G, where χ denotes the chromatic number. We show that for any positive integers n and k with k?n/2 there exists a graph G with χ(G)=n and χ_cap(G)=n-k, extending a result of Greene. We obtain bounds on that are tight as r→∞, where is the complete n-partite graph with r vertices in each part. Finally, for any positive integers p and q we construct a graph G with χ_cap(G)+1=χ(G)=p that contains no odd cycles of length less than q.     

流面上的Navier-Stokes方程及其维数分裂方法

该文首先提出了流面和流层的概念,然后推导出了半测地坐标系下流层内的三维NS (Navier-Stokes)方程,以及流面上的二维NS方程.通过引入流面上的流函数,得到了流函数方程的非线性初边值问题,并讨论了方程解的存在性和唯一性.基于以上讨论,提出了求解三维NS方程的维数分裂方法, 并给出了算例.     

Implicit iterative algorithms of the split common fixed point problem for Bregman quasi-nonexpansive mapping in Banach spaces

In this paper, we study a modified implicit rule for finding a solution of split common fixed point problem of a Bregman quasi-nonexpansive mapping in Banach spaces. We propose a new iterative algorithm and prove the strong convergence theorem under appropriate conditions. As an application, the results are applied to solving the zero problem and the equilibrium problem.     

Basic aspects of stop-flow split injection

Summary Major concerns in the development of the stop-flow split injection are discussed. Split ratio fluctuation caused by the pressure wave, solvent recondensation and gas viscosity change can be substantially diminished through the formation of a nearly uniform gaseous sample plug. Instrumental variables exerting influence on the accuracy of quantitation were studied. Higher injector temperature and the use of a carrier gas with higher heat conductivity benefit the quick vaporization of the sample liquid, and thus, help in the formation of a sample plug with more unformity.     

The principal Erdős–Gallai differences of a degree sequence

The Erd?s–Gallai criteria for recognizing degree sequences of simple graphs involve a system of inequalities. Given a fixed degree sequence, we consider the list of differences of the two sides of these inequalities. These differences have appeared in varying contexts, including characterizations of the split and threshold graphs, and we survey their uses here. Then, enlarging upon properties of these graph families, we show that both the last term and the maximum term of the principal Erd?s–Gallai differences of a degree sequence are preserved under graph complementation and are monotonic under the majorization order and Rao's order on degree sequences.     

On endomorphism-regularity of zero-divisor graphs

The paper studies the following question: Given a ring R, when does the zero-divisor graph Γ(R) have a regular endomorphism monoid? We prove if R contains at least one nontrivial idempotent, then Γ(R) has a regular endomorphism monoid if and only if R is isomorphic to one of the following rings: Z₂×Z₂×Z₂; Z₂×Z₄; Z₂×(Z₂[x]/(x²)); F₁×F₂, where F₁,F₂ are fields. In addition, we determine all positive integers n for which Γ(Z_n) has the property.     

Galois theory over rings of arithmetic power series

Let R be a domain, complete with respect to a norm which defines a non-discrete topology on R. We prove that the quotient field of R is ample, generalizing a theorem of Pop. We then consider the case where R is a ring of arithmetic power series which are holomorphic on the closed disc of radius 0<r<1 around the origin, and apply the above result to prove that the absolute Galois group of the quotient field of R is semi-free. This strengthens a theorem of Harbater, who solved the inverse Galois problem over these fields.     

Split Newton iterative algorithm and its application

Inspired by some implicit-explicit linear multistep schemes and additive Runge-Kutta methods, we develop a novel split Newton iterative algorithm for the numerical solution of nonlinear equations. The proposed method improves computational efficiency by reducing the computational cost of the Jacobian matrix. Consistency and global convergence of the new method are also maintained. To test its effectiveness, we apply the method to nonlinear reaction-diffusion equations, such as Burger’s-Huxley equation and fisher’s equation. Numerical examples suggest that the involved iterative method is much faster than the classical Newton’s method on a given time interval.     

On split common fixed point problems

Based on the convergence theorem recently proved by the second author, we modify the iterative scheme studied by Moudafi for quasi-nonexpansive operators to obtain strong convergence to a solution of the split common fixed point problem. It is noted that Moudafi's original scheme can conclude only weak convergence. As a consequence, we obtain strong convergence theorems for split variational inequality problems for Lipschitz continuous and monotone operators, split common null point problems for maximal monotone operators, and Moudafi's split feasibility problem.     

One-point extensions of generalized hexagons and octagons

In this note, we prove the uniqueness of the one-point extension S of a generalized hexagon of order 2 and prove the non-existence of such an extension S of any other finite generalized hexagon of classical order, different from the one of order 2, and of the known finite generalized octagons provided the following property holds: for any three points x, y and z of S, the graph-theoretic distance from y to z in the derived generalized hexagon S_x is the same as the distance from x to z in S_y.