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.     

The hilbert boundary value problem for generalized analytic functions in clifford analysis

Let ?_0,n be the real Clifford algebra generated by e₁, e₂,…, e_n satisfying e_ie_j+e_je_i=−2δ_ij, i, j=1,2,…, n. e₀ is the unit element. Let Ω be an open set. A function f is called left generalized analytic in Ω if f satisfies the equation

equation(0.1)

Lf=0,$Lf=0,$

where

L=q₀e₀∂x₀+ q₁e₁∂x₁+…+qnen_∂xn,$L=q_0{e}_0\partial x_0+\hspace{0.17em}{q}_1{e}_1\partial x_1+\dots +q_n{e}_n{\partial }_{x_n},$

qi <0, i=0,1,…, n. In this article, we first give the kernel function for the generalized analytic function. Further, the Hilbert boundary value problem for generalized analytic functions in ?ⁿ⁺¹₊ will be investigated.     

The Neumann boundary problem for a nonlocal Cahn-Hilliard equation

We study the existence, uniqueness and continuous dependence on initial data of the solution to a nonlocal Cahn-Hilliard equation on a bounded domain. The equation generates a gradient flow for a free energy functional with nonlocal interaction. Also we apply a nonlinear Poincaré inequality to show the existence of an absorbing set in each constant mass affine space.     

Six-variable generalization of Ramanujan's reciprocity theorem and its variants

By virtue of Shukla's well-known bilateral summation formula and Watson's transfor-mation formula, we extend the four-variable generalization of Ramanujan's reciprocity theorem due to Andrews to a six-variable one. Some novel variants of Ramanujan's reciprocity theorem and q-series identities are presented.     

A cubic spline approximation and application of TAGE iterative method for the solution of two point boundary value problems with forcing function in integral form

In this article, we report an efficient high order numerical method based on cubic spline approximation and application of alternating group explicit method for the solution of two point non-linear boundary value problems, whose forcing functions are in integral form, on a non-uniform mesh. The proposed method is applicable when the internal grid points of solution interval are odd in number. The proposed cubic spline method is also applicable to integro-differential equations having singularities. Computational results are given to demonstrate the utility of the method.     

Analytic discs in the boundary and compactness of Hankel operators with essentially bounded symbols

We derive conditions for compactness of Hankel operators () with bounded, holomorphic symbols f for a large class of convex and bounded domains Ω with Ω⊆Dk.     

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.     

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.     

Multiple solutions for elliptic problems with asymmetric nonlinearity

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

     

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.     

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.     

Monotonicity characteristic of Köthe-Bochner spaces

Estimates of the characteristic of monotonicity in Köthe-Bochner function spaces E(X) with some consequences are given. Characterizations of strict and uniform monotonicity of the sequence lattice are obtained.     

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.     

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.     

Generalized Navier-Stokes equations with initial data in local Q-type spaces

In this paper, we establish a link between Leray mollified solutions of the three-dimensional generalized Navier-Stokes equations and mild solutions for initial data in the adherence of the test functions for the norm of . This result applies to the usual incompressible Navier-Stokes equations.     

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).     

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.