A graph Γ is said to be a G-locally primitive graph, for G ≥ Aut Γ, if for every vertex, α, the stabiliser Gα induces a primitive permutation group on Γ (α) the set of vertices adjacent to α. In 1978 Richard Weiss conjectured that there exists a function f: ℕ →ℕ such that for any finite connected vertex-transitive G-locally primitive graph of valency d and a vertex α of the graph, |Gα| ≥ f(d). The purpose of this paper is to prove that, in the case Soc(G) = Sz(q), the conjecture is true.

Let ℚp denote the p-adic numbers. We consider curves in defined by p-adic polynomials of one p-adic variable. We show that maximal averages along these curves are bounded, where 1 < q < ∞.

We study complete submanifolds of Euclidean space where every geodesic passing through a fixed point is the normal section along it. We prove that all such geodesics are independent of the direction at the point and such submanifolds are pointed Blaschke manifolds or diffeomorphic to a Euclidean space.

We prove that every bounded representation of the tensor product of two C*-algebras, one of which is nuclear and contains matrices of any order, is similar to a *-representation.

A group is called capable it if is a central factor group. For each prime p and positive integer c, we prove the existence of a capable p-group of class c minimally generated by an element of order p and an element of order p1+⌊c−1/p−1⌋. This is best possible.

In this paper, we show that the ratio of preinvex functions is invex. Hence, we give a positive answer to the open question which was proposed in a paper of Yang, Yang and Teo in (2003).

I prove general combinatorial properties which apply to singular Artin monoids and examine their relationship with the Vassiliev homomorphism ≠. I show that ≠ preserves the Intermediate Property, discovered by Corran, which holds in positive singular Artin monoids of finite type. From this it follows that ≠ is injective for a class of monoids which include singular Artin monoids of type I2(p), generalising a result of East.

In this paper, two necessary and sufficient conditions of Möbius subgroups to be g-discontinuous are obtained. These are generalisations of Lehner's and Larcher's corresponding results.

We prove the existence of a positive solution to an equation of the form (1/Φ(t)) (Φ(t)u′(t))′ = f(u(t)) with Dirichlet conditions where the friction term Φ′/Φ is increasing. Our method combines variational and topological arguments and provides an L∞ estimate of the solution.

Polynomial near-rings in k-commuting indeterminates are our object of study. We illustrate out work for k = 2, that is, N[x, y] as an extension to N[x], while the case for arbitrarily k follows easily. Our approach is different from the recursive definition N[x][y]. However, it can be shown that N[x, y] is isomorphic to N[x][y]. Several important tools such as the degree, the least degree, et cetera are defined with respect to N[x, y]. We also clarify some notations involved in defining polynomial near-rings.

Some quadratic reverses of the continuous triangle inequality for the Bochner integral of vector-valued functions in Hilbert spaces are given. Applications of complex-valued functions are provided as well.

A contractive mapping on a complete metric space may fail to have a fixed point. Diametrically contractive mappings are introduced and it is shown that a diametrically contractive self-mapping of a weakly compact subset of a Banach space always has a fixed point.

We prove that, for n ≥ 4 and arbitrary infinite dimensional Banach spaces X1,…Xn, there exists an extendible n-linear form T: X1 x…x Xn →  that is not integral.

In this note, we discuss the exceptional set E ⊆ [−1, 1] of points x0 satisfying the inequality where λ > 0, λ ≠ 2, 4, … and Ln(fλ,.) is the Lagrange interpolation polynomial of degree at most n to fλ(x):= |x|λ on the interval [−1, 1] associated with the equidistant nodes. It is known that E has Lebesgue measure zero. Here we show that E contains infinite families of rational and irrational numbers.

The question of which Abelian groups can be the inner mapping group of a loop has been considered by Niemenmaa, Kepka and others. We give a construction which shows that many finite Abelian groups can be the inner mapping group of a loop.

Ramanujan recorded many beautiful continued fractions in his notebooks. In this paper, we derive several indentities involving the Ramanujan continued fraction c(q), including relations between c(q) and c(qn). We also obtain explicit evaluations of for various positive integers n.

One of the most important examples of conditional Cauchy equations with the condition dependent on the unknown function is Dhombres' equation This paper is devoted to the stability of the above equation.

We study properly embedded and immersed p(pseudohermitian)-minimal surfaces in the 3-dimensional Heisenberg group. From the recent work of Cheng, Hwang, Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There are two types of such surfaces: band type and annulus type according to their topology. We givn an explicit expression for these surfaces. Among band types there is a class of properly embedded p-minimal surfaces of so called helicoid type. We classify all the helicoid type p-minimal surfaces. This class of p-minimal surfaces includes all the entire p-minimal graphs (except contact planes) over any plane. Moreover, we give a necessary and sufficient condition for such a p-minimal surface to have no singular points. For general complete immersed p-minimal surfaces, we prove a half space theorem and give a criterion for the properness.

In this paper we consider inequalities of the form , Where M is a mean. The main results of the paper offer sufficient conditions on M so that the above inequality holds with a finite constant C. The results obtained extend Hardy's and Carleman's classical inequalities together with their various generalisations in a new dirction.