Gracefulness of several unconnected graphs with wheel

In the paper, we study the gracefulness of several unconnected graphs related to wheel. For natural number p ≥1,t ≥1, let n =2t +3, 2t +4, which proved W_n∪K_(p,t)~(1)∪K_(p,t)~(2) is graceful; for p ≥1, t ≥1,let n=2t+3,2t+4, then W_(n,2n+1)∪K_(p,t)~(1)∪K_(p,t)~(2) is graceful and for m≥1,r ≥1, let n =2m +5, W_(n,2n+1) ∪( C_3∨K m) U St( r)is graceful.     

Global Existence and Optimal Decay Rate of the Compressible Magnetohydrodynamic Equation with Potential Force

In this paper, we consider the three dimensional compressible viscous magnetohydrodynamic equations(MHD) with the external potential force. We first derive the corresponding non-constant stationary solutions. Next, we show global wellposedness of the initial value problem for the three dimensional compressible viscous magnetohydrodynamic equations, provided that the initial data is close to the stationary solution. Finally, based on the elaborate energy estimates for the nonlinear system and L~p-L~q decay estimates of the linearized equation, we show the optimal convergence rates of the solution in L~q-norm with 2≤q≤6 and its first derivative in L~2-norm when the initial perturbation is bounded in L~p-norm with 1≤p6/5.     

The periodic solution for a kind odd order delay differential equation

In this paper, we use the coincidence degree theory to research the odd order delay differential equation a(t)χ(2n+1)(t)+b(t)χ(t)+g(t,-τ(t)))=p(t), and we obtain the sufficient conditions for the existence of periodic solutions, which improves and generalizes some related results in the literatures.     

Vertex disjoint cycles in intersection graphs of bases of matroids

The intersection graph of bases of a matroid M=(E, B) is a graph G=G~I(M) with vertex set V(G) and edge set E(G) such that V(G)=B(M) and E(G)={BB′:|B∩B′|≠0, B, B′∈B(M), where the same notation is used for the vertices of G and the bases of M. Suppose that|V(G~I(M))| =n and k_1+k_2+…+k_p=n, where k_i is an integer, i=1, 2,…, p. In this paper, we prove that there is a partition of V(G~I(M)) into p parts V_1 , V_2,…, V_p such that |V_i| =k_i and the subgraph H_i induced by V_i contains a k_i-cycle when k_i ≥3, H_i is isomorphic to K_2 when k_i =2 and H_i is a single point when k_i =1.     

A generalized funk’s section theorem for measures

If L is a star body in R~n whose central(n-i)-slices have the same(n-i)-dimensional measure μ_(n-i)(with appropriate density)as the central(n-i)-slices of an origin-symmetric star body K,then the corresponding n-dimensional measures μ_n of K and L satisfy μ_n(K)≤μ_n(L).This extends a generalized Funk’s section theorem for volumes to that for measures.     

Multi-pattern matching algorithm with wildcards based on bit-parallelism

Multi-pattern matching with wildcards is a problem of finding the occurrence of all patterns in a pattern set{p~1,---,p~k}in a given text t. If the percentage of wildcards in pattern set is not high,this problem can be solved using finite automata. We introduce a multi-pattern matching algorithm with a fixed number of wildcards to overcome the high percentage of the occurrence of wildcards in patterns. In our proposed method,patterns are matched as bit patterns using a sliding window approach. The window is a bit window that slides along the given text,matching against stored bit patterns. Matching process is executed using bit wise operations. The experimental results demonstrate that the percentage of wildcard occurrence does not affect the proposed algorithm's performance and the proposed algorithm is more efficient than the algorithms based on the fast Fourier transform. The proposed algorithm is simple to implement and runs efficiently in O(n+d(n/σ)(m/w))time,where n is text length,d is symbol distribution over k patterns,m is pattern length,and σ is alphabet size.     

Research on a Class of Equations over Finite Fields

Let F_q stand for the finite field of odd characteristic p with q elements(q=p~n,n∈N)and F_q~* denote the set of all the nonzero elements of F_q.In this paper,by using the augmented degree matrix and the result given by Cao,we obtain a formula for the number of rational points of the following equation over F_q:f(x _1,x _2,...,x _n)=(a_1 x_1 x_2~d+a_2 x_2 x_3~d...+a_(n-1)x_(n-1)x_n~d+a_n x_n x_1~d)~λ-bx_1~(d1)x_2~d2...x_n~(dn),with a_i,b∈F_q~*,n≥2,λ0 being positive integers,and d,d_i being nonnegative integers for 1≤i n.This technique can be applied to the polynomials of the form h_1~λ=h_2 with λ being positive integer and h_1,h_2∈F_q[x _1,x _2,...,x _n].It extends the results of the Markoff-Hurwitz-type equations.     

An algorithm for the spectral immunity of binary sequence with period 2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

To resist the fast algebraic attack and fast selective discrete Fourier transform attacks, spectral immunity of a sequence or a Boolean function was proposed. At the same time, an algorithm to compute the spectral immunity of the binary sequence with odd period N was presented, here N is a factor of 2 ⁿ ? 1, where n is an integer. The case is more complicated when the period is even. In this paper, we compute linear complexity of every orthogonal sequence of a given sequence using Chan-Games algorithm and k - error linear complexity algorithm. Then, an algorithm for spectral immunity of binary sequence with period N = 2 ⁿ is obtained. Furthermore, the time complexity of this algorithm is proved to be O(n).     

Ranks of repeated-root constacyclic code

By the discussion of division in \(F_{^{^{^{_2 m} } } } \left[ u \right]/\left\langle {u^4 } \right\rangle \), the minimal spanning set and the rank of a (1 + u + u ²) - constacyclic code with an arbitrary length N = 2 ^e n over \(\Phi \) are determined based on the factorization of (x ⁿ - 1) over \({F_{{2^m}}}\).     

Fault-tolerant cycles embedding in folded hypercubes

The generalized conditional fault-tolerant embedding is investigated, in which the n-dimensional folded hypercube networks(denoted by FQ_n) acts as the host graph, and the longest fault-free cycle represents the guest graph. Under the conditions looser than that of previous works, it is shown that FQ_n has a cycle with length at least 2~n-2︱F_v︱ when the number of faulty vertices and non-critical edges is at most 2n-4; where ︱F_v︱ is the number of faulty vertices. It provides further theoretical evidence for the fact that FQ_n has excellent node-fault-tolerance and edge-fault-tolerance when used as a topology of large scale computer networks.     

Parameter-Dependent Third-Order Susceptibility of InxGa1-xN/GaN Parabolic Quantum Dots

The electron states confined in wurtzite InxGa1-xN/GaN strained quantum dots （QDs） have been investigated in the effective -mass approximation by solving the Schrtdinger equation, in which parabolic confined potential and strong built-in electric field effect （due to the piezoelectricity and spontaneous polarization） have been taken into account. The real part Rex^（3）（0,0,ω） and the imaginary part Imx^（3）（0,0,ω） of the third-order susceptibil- ity describe quadratic electro-optic effects and electro-absorption process of the QDs respectively. And both of them have been calculated in directions parallel and vertical to z axis. Furthermore, the study shows Rex^（3）（0,0,ω） and Imx^（3）（0,0,ω） increase under resonant conditions with the QDs＇ radius and height increase, and the same results occur when the content increase. In addition, the resonant position shift to the lower energy region when the parabolic frequencies increase.     

A new second-order Mehrotra-type predictor-corrector algorithm for SDO

In Zhang's recent works,a second-order Mehrotra-type predictor-corrector algorithm for linear optimization was extended to semidefinite optimization and derived that the algorithm for semidefinite optimization had O(n~(3/2)log(X~0)~T·S~0/ε) iteration complexity based on the NT direction as Newton search direction. In this paper, we extend the second-order Mehrotra-type predictor-corrector algorithm for linear optimization to semidefinite optimization and discuss the polynomial convergence of the algorithm by modifying the corrector direction and new iterates. It is proved that the iteration complexity is reduced to O(n~(3/2)log(X~0)~T·S~0/ε), which coincides with the currently best iteration bound of Mehrotra-type predictor-corrector algorithm for semidefinite optimization.     

Binding number,minimum degree and bipancyclism in bipartite graphs

Let G =(V_1,V_2,E) be a balanced bipartite graph with2 n vertices.The bipartite binding number of G,denoted by B(G),is defined to be n if G =K_n and min i∈{1,2}|N(S)|n min |N(S)|/|S|otherwise.We call G bipancyclic if it contains a cycle of every even length m for 4 ≤ m ≤ 2n.A theorem showed that if G is a balanced bipartite graph with 2n vertices,B(G) 3 / 2 and n 139,then G is bipancyclic.This paper generalizes the conclusion as follows:Let 0 c 3 / 2 and G be a 2-colmected balanced bipartite graph with 2n(n is large enough) vertices such that B(G) c and δ(G)(2-c)n/(3-c)+2/3.Then G is bipancyclic.     

Lüroth Expansion Digits and Maclaurin's Inequality

It is well known that for almost all real number x, the geometric mean of the first n digits d_i(x) in the Lüroth expansion of x converges to a number K_0 as n→∞. On the other hand, for almost all x, the arithmetric mean of the first n Lüroth expansion digits d_i(x) approaches infinity as n→∞. There is a sequence of refinements of the AM-GM inequality, Maclaurin's inequalities, relating the 1/k-th powers of the k-th elementary symmetric means of n numbers for 1≤k≤n. In this paper, we investigate what happens to the means of Lüroth expansion digits in the limit as one moves f(n) steps away from either extreme. We prove sufficient conditions on f(n) to ensure divergence when one moves away from the arithmetic mean and convergence when one moves f(n) steps away from geometric mean.     

Molecular phylogenetics of <Emphasis Type="Italic">Gymnocypris</Emphasis> (Teleostei: Cyprinidae) in Lake Qinghai and adjacent drainages

149 complete mitochondrial DNA (mtDNA) cytochrome b (Cyt b) genes (1140 bp) of Gymnocypris przewalskii, Gymnocypris eckloni and Gymnocypris scolistomus from the Lake Qinghai, Yellow River and Qaidam Basin were sequenced and analyzed. Consistent dendrogram indicated that the samples collected from the same species do not constitute a separate monophyletic group and all the samples were grouped into three highly divergent lineages (A, B and C). Among them, Lineage A contained all samples of G. przewalskii from the Lake Qinghai and partial samples of the G. eckloni from the Yellow River. Lineage B contained the remaining samples of G. eckloni from the Yellow River. Lineage C was composed of a monophyletic group by G. eckloni from the Qaidam Basin. Analysis of molecular variance (AMOVA) indicated that most of genetic variations were detected within these three mtDNA lineages (93.12%), suggesting that there are three different lineages of Gymnocypris in this region. Our Cyt b sequence data showed that G. przewalskii was not a polytypic species, and G. scolistomus was neither an independent species nor a subspecies of G. eckloni. The divergent mtDNA lineages of G. eckloni from the Yellow River suggested that gene flow between the different populations was restricted to a certain extent by several gorges on the upper reach of the Yellow River. Lineage B of G. eckloni might be the genetic effect from the ancestor which was incorporated with the endemic schizothoracine fishes when the headward erosion of the Yellow River reached to its current headwaters of late. The G. eckloni from Basin Qaidam was a monophyletic group (lineage C) and F_st values within G. eckloni from the Yellow River were higher than 0.98, suggesting that the gene flow has been interrupted for a long time and the G. eckloni from Basin Qaidam might have been evolved into different species by ecology segregation. The correlation between the rakers number of Gymnocypris and population genetic variation was not significant. All Gymnocypris populations exhibited a low nucleotide diversity (π = 0.00096–0.00485). Therefore the Gymnocypris populations from Basin Qaidam could have experienced severe bottleneck effect in history. Our result suggested Gymnocypris populations of Basin Qaidam should give a high priority in conservation programs.     

Properties of a class of groups on <Emphasis Type="Italic">Z</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript>

In this paper, we obtain the factorization of direct production and order of group GL(n, Z _m ) in a simple method. Then we generalize some properties of GL(2, Z _p ) proposed by Huppert, and prove that the group \(GL\left( {2,{Z_{{z^\lambda }}}} \right)\) is solvable. We also prove that group GL(n, Z _p ) is solvable if and only if GL(n, Z _p ) is solvable, and list the generators of groups GL(n, Z _p ) and SL(n, Z _p ). At last, we prove that PSL(2, Z _p )(p > 3) and PSL(n, Z _p )(n > 3) are simple.     

Analysis of diversity of CO I gene and system generation of nine species of chitons

Cytochrome c oxidase subunit Ⅰ(COⅠ) genes from 9 chiton species in China’s coast area were sequenced. A phylogeny tree about these gene sequences were reconstructed together with other 4 COⅠgene sequences of chitons from Genbank. The affinity genetic relationship and the taxonomic status of molecular evolution for these 13 chiton species were analyzed. The results show that Liolophura japonica and Onithochiton hirasei belong to Chitonidae, Acanthochiton rubrolineatus and Acanthochiton dissimilis belong to Acanthochitonidae, and Placiphorella japonica and Mopalia retifera belong to the same family Mopliidae. However, Lepidozona coreanica, Ischnochiton comptus, and Ischnochiton hakodadensis are not supposed to be referred to Ischnochi-tonidae according to the genetic distance analysis (L. coreanica, I. comptus, and I. hakodadensis are usually classed as Ischnochitonidae according to their morphological character). Furthermore, I. hakodadensis could not been classed as Ischnochiton, and it is more likely to be treated as a close relative of Lepidozona.     

The Reverse Hoelder Classes in Martingale Setting

In martingale setting, it has been shown that Ap weights can be factorized in terms of A1 weights. This factorization benefits many problems very much. ]n this paper, the new class of RH~ plays the same role for RHs, which makes the reverse HOlder inequalities hold with exponent s＞1, that the class A1 does for Ap class. Therefore, the Jones‘ factorization theorem for Ap weights was extended to include some information about the reverse HOlder classes. And it is the most convenient obiect in weight theory indeed.     

On the Arithmetic of Endomorphism Ring End$$({Z_p} \times {Z_{{p^m}}})$$

For a prime p, let \({E_{P,{P^m}}} = \left\{ {\left( {\begin{array}{*{20}{c}}aamp;b \\ {{p^{m – 1}}c}amp;d \end{array}} \right)\left| {a,b,c \in {Z_p},d} \right. \in {Z_{{p^m}}}} \right\}\). We first establish a ring isomorphism from \({Z_{p,{p^m}}}\)onto \({E_{p,{p^m}}}\). Then we provide a way to compute -d and d^–1 by using arithmetic in Z_p and \({Z_{{p^m}}}\), and characterize the invertible elements of \({E_{p,{p^m}}}\). Moreover, we introduce the minimal polynomial for each element in \({E_{p,{p^m}}}\)and give its applications.     

On the Dual <Emphasis Type="Italic">p</Emphasis>-Measures of Asymmetry for Star Bodies

Recently, the connection between p-measures of asymmetry and the L_p -mixed volumes for convex bodies was found soon after the p-measure of asymmetry was proposed, and the Orlicz-measures of asymmetry was proposed inspired by such a kind of connection. In this paper, by a similar way the dual p -measures of asymmetry for star bodies (naturally for convex bodies) is introduced first. Then the connection between dual p -measures of asymmetry and L_p -dual mixed volumes is established. Finally, the best lower and upper bounds of dual p-measures and the corresponding extremal bodies are discussed.