In this survey, we compare Roth’s theorem with abc-conjecture, and introdu ce an approach routine for the abc-conjecture.
Let G=(V1,V2;E) be a balanced bipartite graph with |V1|=|V2|=3k and k≥1. If the minimum degree δ≥2k, then G either contains k vertices d isjoint 6-cycles or contains k-1 vertices disjoint 6-cycles and a 4-cycle.
The Merrifield-Simmons index for a 5-leaf tree of order n is investiga ted, and the minimal Merrifield-Simmons index for a 5-leaf tree is characteri zed.
Let γ′ws(G) be the signed edge total domination number of a grap h G. The low bound of the signed edge total domination number of a graph G is given,and the signed edge total domination number of Kn is determined.
It is proved that Graham’s conjecture holds for a thorn graph of the complete graph by a graph with the two-pebbling property. As a corollary, Graham's conjecture is true when G and H are a thorn graph of the complete graphs.
The L(2,1,1)-labeling numbers for comp lete graphs, complete bipartite graphs, paths and cycles are given. An arithmetic method for L(2,1,1)-labeling to a graph G with maximum degree Δ is given. It is proved that λ2,1,1(G)≤Δ3-Δ2+2Δ.
The relationships between the semi cover-avoiding property of subgroups and the solvability of groups are investigated. As an application, some sufficient and necessary conditions for finite groups to be solvable are obtained. It is proved that a finite group G is solvable if and only if for every maximal subgroup M of G, either M is semi cover-avoiding in G, or M has a solvable m aximal completion C such that C is semi cover-avoiding in G.
The existence of a tensor product is proved in the left quasi-normal ban d category, and the relationship with the tensor product in the semi-group category is provided. Furthermore, the relationship between the tensor product of semi-lattices in the left quasi-normal band category and in the semi-lattice category is given.
A new kind of generalized matrix function is defined, which is with classical generalized matrix functions for exceptions . Some properties of this function are obtained, and some application are investigated.
Lower bounds for the rank and the estimation for eigenvalues of matrices are discussed. Two lower bounds for the rank and estimation for the real part and imaginary part of eigenvalues are obtained. Also, we prove that all the eigenvalues of any complex matrix are located in one disk. Some numerical examples show the effectiveness of our results.
The definition of strong FS-Posets is given. Some properties of strong FS-Posets are discussed, and it is proved that strong FS -Posets not only are continuous, but also are Scott. Based on strong FS-Poset s, the concept of strong FS-Lattices is given. Some properties of strong FS-La tt ices are studied. Strong FS-Lattices are characteristic of the function space. The sufficient and necessary conditions for a continuous lattice to be a s trong FS-Lattice are obtained.
We give the regular condition for the congruence class of an idempotent in some semirings, and prove that the congruence class of an idempotent in a completely regular semiring and an inverse semiring is regular. At the same time,we discuss the eventually regularity on the congruence class of an idem potent in an eventually regular semiring.
Let A⊆Bbe an extension of commutative rings with identity, x an indeterminate over B, R:=A+xB［［ x］］ and SA a given multiplicative set . It is shown that if ,(∩snA,n≥1)∩S≠Фfor each s∈S where Scons ists of nonzerodivisors, then Ris SNoetherian if and only if A is S-Noetherian and B is an S -finite A-module.
Let α be a ring endomorphism α. Weak α-reversible rings are introduced whi ch generalized α-reversible rings， some extension of weak α-reversible rings and their properties are investigated. The relations between weak α-reversible rings and weak α-skew Armendariz rings are obtained.
An M／M／1／N queuing system was considered with negative customers and a single working vacation. The server works at a lower rate rather than completel y stops service during the vacation period. Negative customers remove positive customers only one by one at the head (if present). When a negative customer arrives, if the system is empty, it will disappear. Negative customers need no services. The matrix form solution of the steadystate probability is derive d by the Markfov process method and the matrix solution method. Some performance measures of the system such as the expected number of customers the system or in the queue and the loss probability of the customer are also presented. Finally the effects of the parameters of the system are investigated, such as the va cati on service rate μvand the vacation rate θon the expected waiting queue length and the loss probability of customers by numerical examples.
By constructing a closed convex set and using the fixed point theory of completely continuous operators, the existence of positive solutions for an initial value problems of first order nonlinear impulsive singular integro-differential equations in a Banach space are obtained.
The fuzzy modal logic system M?uk is introduced, where the evaluation lattice is taken to be the unite interval and the binary relation R is fuzzif ied, then it is discretized to be the multi-valued fuzzy modal logic system M? n. It is proved that for any possible value α in M?n there exists an exact α-tautology; and in M?uk, for any rational α∈［0,1］, there exists an exac t α-tautology. It is pointed out that the lift-algorithm which plays a key role in R0 systems is not suitable for the system M?n, and the reasons are analyzed.
Under the integrated considerations about the possibilistic mean and the ideal points, the fuzzy numbers are mapped to the possibilistic interval by a αcut. In order to form a synthesized index for ranking fuzzy numbers, the distances between the possibilistic mean and the ideal point are calculated.
The method of characteristics is combined with dynamic finite element spaces based on the moving grids method to create a fully discrete characteristic dynamic finite element procedure for the solution of second order linear convectiondominated diffusion problems. The procedure is proved to be stable. Convergence analysis and error estimates are established, which show that the error estimate in the energy norm is optimal when Mh4／Δt is bounded. Both the L2-norm and the energy norm error estimate are optimal when Mh2／Δt is bounded, where M is the total number of gridschanging. h and Δt are the spatial and temporal mesh parameters respectively.
