Let denote the variety of all abelian groups and, for each prime p, let p be the variety of all elementary abelian p-groups. Let be a subvariety of a product of (finitely many) varieties each of which is either soluble or Cross.

A new minimax inequality is first proved. As a consequence, five equivalent fixed point theorems are formulated. Next a theorem concerning the existence of maximal elements for an Lc-majorised correspondence is obtained. By the maximal element theorem, existence theorems of equilibrium points for a non-compact one-person game and for a non-compact qualitative game with Lc-majorised correspondences are given. Using the latter result and employing an “approximation” technique used by Tulcea, we deduce equilibrium existence theorems for a non-compact generalised game with LC correspondences in topological vector spaces and in locally convex topological vector spaces. Our results generalise the corresponding results due to Border, Borglin-Keiding, Chang, Ding-Kim-Tan, Ding-Tan, Shafer-Sonnenschein, Shih-Tan, Toussaint, Tulcea and Yannelis-Prabhakar.

We introduce the concept of pseudo $ \mathcal{S} $-asymptotically periodic functions and study some of the qualitative properties of functions of this type. In addition, we discuss the existence of pseudo $ \mathcal{S} $-asymptotically periodic mild solutions for abstract neutral functional differential equations. Some applications involving ordinary and partial differential equations with delay are presented.

Explicit expressions are given for the noncentral moments of the multivariate normal. Finding the general moment is shown to be equivalent to finding the general derivative of the density of the multivariate normal, that is to finding an expression for the multivariate Hermite polynomial.

We study the concepts of continuous shadowing and continuous inverse shadowing with respect to various classes of admissible pseudo orbits, and characterize hyperbolicity and structural stability using the notion of continuous inverse shadowing.

For the class of meromorphically starlike functions of prescribed order, the concept of type has been introduced. A characterization of meromorphically starlike functions of order α and type β has been obtained when the coefficients in its Laurent series expansion about the origin are all positive. This leads to a study of coefficient estimates, distortion theorems, radius of convexity estimates, integral operators, convolution properties et cetera for this class. It is seen that the class considered demonstrates, in some respects, properties analogous to those possessed by the corresponding class of univalent analytic functions with negative coefficients.

The connection between Clifford analysis and the Weyl functional calculus for an n-tuple of bounded selfadjoint operators is noted. The operators do not necessarily commute with each other.

We establish derivative estimates and existence theorems for the Dirichlet and Neumann problems for nonlinear, degenerate elliptic equations of the form F (D2u) = g in balls. The degeneracy arises through the possible vanishing of the function g and the degenerate Monge-Ampère equation is covered as a special case.

It has recently emerged that a convenient way to establish central limit and iterated logarithm results for processes with stationary increments is to use approximating martingales with stationary increments. Functional forms of the limit results can be obtained via a representation for the increments of the stationary process in terms of stationary martingale differences plus other terms whose sum telescopes and disappears under suitable norming. Results based on the most general form of such a representation are here obtained.

In this paper we show that a contact metric three-manifold is a generalised (k, μ)-space on an everywhere dense open subset if and only if its characteristic vector field ξ determines a harmonic map from the manifold into its unit tangent sphere bundle equipped with the Sasaki metric. Moreover, we classify the contact metric three-manifolds whose characteristic vector field ξ is strongly normal (or equivalently, is harmonic and minimal).

In this paper, we prove the uniqueness of the decaying positive solution on all of Rn for certain second order non linear elliptic equations. This improves earlier work of a number of authors. These problems occur in the theory of peak solutions. In particular, our results apply to a number of non-smooth nonlinearities which occur as limiting equations in population problems.

The characterizations of the closed ranges of the multiplication operators and the composition operators on L2 (λ) are reported in this paper.

We prove that the number of factors of length k in any paperfolding sequence is equal to 4k once k ≥ 7.

Consider the n-th order delay differential equation

In the last few years, the oscillatory behavior of delay differential equations has been the subject of intensive investigations. But much less is known about the equation (A) with small forcing term q(t). The only papers devoted to this problem are by Kartsatos, Kusano, and the present author. The purpose of this paper is to prove some new oscillation theorems which contain the previous results.

Let ℬ be an abstract Segal algebra with respect to 𝒜. For a nonzero character ϕ on 𝒜, we study ϕ-amenability, and ϕ-contractibility of 𝒜 and ℬ. We then apply these results to abstract Segal algebras related to locally compact groups.

In this paper we obtain a sufficient and necessary condition for an analytic function f on D with Hadamard gaps, that is, for satisfying nk+1/nk ≥ λ 1 for all k, to belong to a kind of space consisting of analytic functions on D. The special cases of these spaces are BMOA and VMOA. In view of our result we can answer the open question given recently by Stroethoff.

Given positive integers k, l, m, the (k, l, m) triangle group has presentation δ(k, l, m) = < X, Y, Z | Xk = Yl = Zm = XYZ = 1 >. This paper considers finite permutation representations of such groups. In particular it contains descriptions of graphical and computational techniques for handling them, leading to new results on minimal two-element generation of the finite alternating and symmetric groups and the group of Rubik's cube. Applications to the theory of regular maps and automorphisms of surfaces are also discussed.

We derive interior second derivative estimates for solutions of equations of Monge-Ampère type which extend those of Pogorelov for the case of affine boundary values. A key ingredient in our method is the existence of a strong solution of the homogeneous Monge-Ampère equation.

Gröbner bases have been generalised to polynomials over a commutative ring A in several ways. Here we focus on strong Gröbner bases, also known as D-bases. Several authors have shown that strong Gröbner bases can be effectively constructed over a principal ideal domain. We show that this extends to any principal ideal ring. We characterise Gröbner bases and strong Gröbner bases when A is a principal ideal ring. We also give algorithms for computing Gröbner bases and strong Gröbner bases which generalise known algorithms to principal ideal rings. In particular, we give an algorithm for computing a strong Gröbner basis over a finite-chain ring, for example a Galois ring.

The existence of a negative solution, of a positive solution, and of a sign-changing solution to a Dirichlet eigenvalue problem with p-Laplacian and multi-valued nonlinearity is investigated via sub- and supersolution methods as well as variational techniques for nonsmooth functions.