New kink-shaped solutions and periodic wave solutions for the (2+1)-dimensional Sine-Gordon equation

New exact solutions including the kink-shaped solutions, bell-shaped solutions, periodic solutions, singular soliton solutions and rational solution for the (2 + 1)-dimensional Sine-Gordon equation are obtained using the tanh method, the -expansion method and the auxiliary function method, respectively.     

Multiple solutions for elliptic problems with asymmetric nonlinearity

In this paper we establish the existence of multiple solutions for the semilinear elliptic problem

     

New exact homoclinic wave and periodic wave solutions for the Ginzburg-Landau equation

New exact solutions including homoclinic wave and periodic wave solutions for the 2D Ginzburg-Landau equation are obtained using the auxiliary function method and the -expansion method, respectively. The solutions are expressed by the hyperbolic functions and the trigonometric functions. There result shows that there exists a kink wave solution which tends to one and the same periodic wave solution as time tends to infinite.     

Some properties for a class of interchange graphs

The Wiener number is the sum of distances between all pairs of vertices of a connected graph. In this paper, we give an explicit algebraic formula for the Wiener number of a class of interchange graphs. Moreover, distance-related properties and cliques of this class of interchange graphs are investigated.     

Condensed Cramer rule for some restricted quaternion linear equations

In this paper, we derive an alternative more condensed Cramer rule for the unique solution of some restricted left and right systems of quaternion linear equations. The findings of this paper extend some known results in the literature.     

A note on exact complex travelling wave solutions for (2+1)-dimensional B-type Kadomtsev-Petviashvili equation

The -expansion method can be used for constructing exact travelling wave solutions of real nonlinear evolution equations. In this paper, we improve the -expansion method and explore new application of this method to (2+1)-dimensional B-type Kadomtsev-Petviashvili (BKP) equation. New types of exact complex travelling wave solutions of (2+1)-dimensional BKP equation are found. Some exact solutions of (2+1)-dimensional BKP equation obtained before are special cases of our results in this paper.     

Improved decay rates for solutions to one-dimensional linear and semilinear dissipative wave equations in all space

Better decay estimates to the 1-dimensional Cauchy problem on to the linear equation □u+u_t=0 can be discussed under rather restricted conditions on the initial data. Furthermore, as applications we derive the small data global existence result to the equation □u+u_t=|u|^p−1u, which has the “odd” functions as the initial data. Furthermore, the new method (see R. Ikehata, T. Matsuyama, Sci. Math. Japon. 55 (2002) 33-42) used in the first half will be applied to the problem coming from Ehrenpreis (Sugaku 26 (1974) 168).     

DFT calculation for the {2}-inverse of a polynomial matrix with prescribed image and kernel

In this paper, we first give the finite algorithm for generalized inverse of a matrix A over an integral domain, and, based on it and the discrete Fourier transform, present an algorithm for calculating {2}-inverses of a polynomial matrix with prescribed image and kernel. And the algorithm is implemented in the Mathematica programming language and expands the algorithms in [13].     

Rationalization and the Gray index of phantom maps

In this paper we study a homotopy invariant of phantom maps called the Gray index. We give a new interpretation of the Gray index of a phantom map f:X→Y, in terms of the rationalization of X. We use this interpretation, in order to detect phantom maps of a specific Gray index. Finally, we examine the set of phantom maps with infinite Gray index in a tower theoretic way.     

Full-rank representations of {2, 4}, {2, 3}-inverses and successive matrix squaring algorithm

We present the full-rank representations of {2, 4} and {2, 3}-inverses (with given rank as well as with prescribed range and null space) as particular cases of the full-rank representation of outer inverses. As a consequence, two applications of the successive matrix squaring (SMS) algorithm from [P.S. Stanimirovi?, D.S. Cvetkovi?-Ili?, Successive matrix squaring algorithm for computing outer inverses, Appl. Math. Comput. 203 (2008) 19-29] are defined using the full-rank representations of {2, 4} and {2, 3}-inverses. The first application is used to approximate {2, 4}-inverses. The second application, after appropriate modifications of the SMS iterative procedure, computes {2, 3}-inverses of a given matrix. Presented numerical examples clarify the purpose of the introduced methods.     

Descriptive properties of families of autohomeomorphisms of the unit interval

Let H⊂C[0,1] stand for the Polish space of all increasing autohomeomorphisms of [0,1]. We show that the family of all strictly singular autohomeomorphisms is -complete. This confirms a suggestion of Graf, Mauldin and Williams. Some related results are also included.     

Minimal rankings and the arank number of a path

Given a graph G, a function f:V(G)→{1,2,…,k} is a k-ranking of G if f(u)=f(v) implies every u-v path contains a vertex w such that f(w)>f(u). A k-ranking is minimal if the reduction of any label greater than 1 violates the described ranking property. The arank number of a graph, denoted ψ_r(G), is the largest k such that G has a minimal k-ranking. We present new results involving minimal k-rankings of paths. In particular, we determine ψ_r(P_n), a problem posed by Laskar and Pillone in 2000.     

Short proofs of summation and transformation formulas for basic hypergeometric series

We show that several terminating summation and transformation formulas for basic hypergeometric series can be proved in a straightforward way. Along the same line, new finite forms of Jacobi's triple product identity and Watson's quintuple product identity are also proved.     

A minimax theorem for vector-valued functions in lexicographic order

In this paper, a minimax theorem and a saddle point theorem are obtained for vector-valued functions in the sense of lexicographic order, respectively. An equivalent relationship between the minimax inequality and the saddle point is established. Some examples are given to illustrate our results.     

Remarks on the formula for γ-interior

The paper presents a formula for the γ-interior of a set under special conditions for , more general than those in the previous paper [Acta Math. Hungar. 80 (1998) 89-93]. There are also some applications.     

Combinatorial constructions of fault-tolerant routings with levelled minimum optical indices

The design of fault-tolerant routings with levelled minimum optical indices plays an important role in the context of optical networks. However, not much is known about the existence of optimal routings with levelled minimum optical indices besides the results established by Dinitz, Ling and Stinson via the partitionable Steiner quadruple systems approach. In this paper, we introduce a new concept of a large set of even levelled -design of order v and index 2, denoted by -LELD, which is equivalent to an optimal, levelled (v−2)-fault-tolerant routing with levelled minimum optical indices of the complete network with v nodes. On the basis of the theory of three-wise balanced designs and partitionable candelabra systems, several infinite classes of -LELDs are constructed. As a consequence, the existence problem for optimal routings with levelled minimum optical indices is solved for nearly a third of the cases.     

Identities between q-hypergeometric and hypergeometric integrals of different dimensions

Given complex numbers m₁,l₁ and nonnegative integers m₂,l₂, such that m₁+m₂=l₁+l₂, for any a,b=0,…,min(m₂,l₂) we define an l₂-dimensional Barnes type q-hypergeometric integral I_a,b(z,μ;m₁,m₂,l₁,l₂) and an l₂-dimensional hypergeometric integral J_a,b(z,μ;m₁,m₂,l₁,l₂). The integrals depend on complex parameters z and μ. We show that I_a,b(z,μ;m₁,m₂,l₁,l₂) equals J_a,b(e^μ,z;l₁,l₂,m₁,m₂) up to an explicit factor, thus establishing an equality of l₂-dimensional q-hypergeometric and m₂-dimensional hypergeometric integrals. The identity is based on the duality for the qKZ and dynamical difference equations.     

Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains

Using several complex variables techniques, we investigate the interplay between the geometry of the boundary and compactness of Hankel operators. Let β be a function smooth up to the boundary on a smooth bounded pseudoconvex domain Ω⊂Cn. We show that, if Ω is convex or the Levi form of the boundary of Ω is of rank at least n−2, then compactness of the Hankel operator Hβ implies that β is holomorphic “along” analytic discs in the boundary. Furthermore, when Ω is convex in C² we show that the condition on β is necessary and sufficient for compactness of Hβ.