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.     

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.     

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.     

Wiener Indices in Random Cyclooctane Chains

The Wiener index of a connected graph (molecule graph) G is the sum of the distances between all pairs of vertices of G. In this paper, simple exact formula are established for the expected value of Wiener index in random cyclooctane chain. Moreover, we obtain the average value of the Wiener indices with respect to the set of all cyclooctane chains with n octagons.     

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.     

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.     

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.     

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.     

<Emphasis Type="Italic">a</Emphasis>-Wythoff’s game with <Emphasis Type="Italic">n</Emphasis> players

Letting a = 1 in a- Wythoff’s game introduced by Fraenkel yields Wythoff’s game which is a well-known 2-player impartial combinatorial game introduced by Wythoff in 1907. A method of solving n-player impartial games was presented by Krawec in 2012. In this paper, we employ Krawec’s function to analyze n > 2 players a-Wythoff’s game and obtain game values for all a 1. The results obtained cover n-player Wythoff’s game, a special case 1-Wythoff’s game.     

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).     

Further results on generalized LCM matrices

Let S={x_1,x_2,...,x_n } be a set of n distinct positive integers and f be an arithmetic function.By(f[S])(resp.( f[S])),we denote the n*n matrix whose i,j entry is Σ[x_i,x_j]|l l∈S f(l) (resp.Σx∈Sf(x)-Σ x_i,|l l∈S f(l)-Σ x_j,|l l∈S f(l)+Σ[x_i,x_j]|l l∈S f(l)).In this paper,we first investigate the structures of the matrices ( f[S]) and( f[S]),then we give the formulae for the determinants of these matrices.These extend the results obtained by Bege in 2011.Finally,we give two examples to demonstrate the validity of our main results.     

Critical Chords of Convex Bodies of Constant Width

In this paper, we show that when Minkowski measure of asymmetry of convex body K of constant width is bigger than α (n-1), K has at least n+1 critical chords, where \(\alpha (n) = \frac{{n + \sqrt {2n(n + 1)} }}{{n + 2}}\).     

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 \({F_{{2^m}}}\).     

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.     

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.     

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.     

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.     

Generalized Intersection Bodies with Parameter

AIn 2005, the classical intersection bodies and L_p intersection bodies were extended. Afterwards, the concept of general L_p intersection bodies and the generalized intersection bodies was introduced. In this paper, we define the generalized bodies with parameter. Besides, we establish the extremal values for volume, Brunn-Minkowski type inequality for radial combination and L_p harmonic Blaschke combination of this notion.     

Super-Edge-Connectivity of G(k,d,s)(s≥k/2)

A graph G is super-edge-connected,for short super-λ,if every minimum edge-cut consists of edges adjacent to a vertex of minimum degree.Alphabet overlap graph G(k,d,s)is undirected,simple graph with vertex set V={v|v=1()kv…v;vi∈{1,2,…,d},i=1,…,k}.Two vertices u=(u1…uk)and v=(v1…vk)are adjacent if and only if us+i=vi or vs+i=ui(i=1,…,k-s).In particular G(k,d,1)is just an undirected de Bruijn graph.In this paper,we show that the diameter of G(k,d,s)is k s,the girth is 3.Finally,we prove that G(k,d,s)(s≥k/2)is super-λ.     

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.