It is well known that in many cases the solutions of a linear differential equation can be expressed as definite integrals, different solutions of the same equation being represented by integrals which have the same integrand, but different paths of integration. Thus, the various solutions of the hypergeometric differential equation

can be represented by integrals of the type

the path of integration being (for one particular solution) a closed circuit encircling the point t = 0 in the positive direction, then the point t = 1 in the positive direction, then the point t = 0 in the negative direction, and lastly the point t = 1 in the negative direction; or (for another particular solution) an arc in the t-plane joining the points t = 1 and t = ∞.

A subgroup H is called c-normal in a group G if there exists a normal subgroup N of G such that HN = G and H∩N ≤ HG, where HG =: Core(H) = ∩g∈GHg is the maximal normal subgroup of G which is contained in H. We use a result on primitive groups and the c-normality of maximal subgroups of a finite group G to obtain results about the influence of the set of maximal subgroups on the structure of G.

In this paper we show that for each n∈{2,3,4,5,} the topological separationproperty Tn can be decomposed

where C, N2,…,Nn are purely lattice theoretic properties with the expected im-plications holding between them.

This paper concerns a Jacobson-type radical for the near-ring N. This radical, denoted by Js(N) has an external representation on a type-0N-group of a very special kind. Such N-groups are said to be of type-s. The main objective of this paper is to decompose Js(N) as a sum Js(N) = J1/2(N) + A + B for N satisfying the descending chain condition for N-subgroups. In this decomposition J1/2(N) is nilpotent and A is the unique minimal ideal modulo which Js(N) is nilpotent.

A k-tuple of members of a group is free if it freely generates a free group. In an earlier paper [1] the authors have shown:

1·1. The two functions F (λ, θ) and G (λ, θ), defined by the infinite integrals (1) and (2) respectively, below, occur in Kottler's theoretical discussion of the diffraction of a monochromatic plane wave by a perfectly black half plane. Some properties of these functions have been investigated by several recent writers.

If P=a+b−c−d, and K=cd−ab, the solution of the equation

may be most readily found by putting (x−a)(x−b)=z2; whence (x−c)(x−d)=z2+Px+K, and therefore =z2−2ez+e2, and finally x is given by the equation

We propose a geometric method to measure the wild ramification of a smooth étale sheaf along the boundary. Using the method, we study the graded quotients of the logarithmic ramification groups of a local field of characteristic p > 0 with arbitrary residue field. We also define the characteristic cycle of an ℓ-adic sheaf, satisfying certain conditions, as a cycle on the logarithmic cotangent bundle and prove that the intersection with the 0-section computes the characteristic class, and hence the Euler number.

The admissible representations of a real reductive group G are known by work of Langlands, Knapp, Zuckerman and Vogan. This paper describes an effective algorithm for computing the irreducible representations of G with regular integral infinitesimal character. The algorithm also describes structure theory of G, including the orbits of K(ℂ) (a complexified maximal compact subgroup) on the flag variety. This algorithm has been implemented on a computer by the second author, as part of the ‘Atlas of Lie Groups and Representations’ project.

Henry Richard Dowson – known to everyone as Harry – was born in Newcastle upon Tyne in England on 2 March 1939. His father Matthew Ridley Dowson came from near Penrith, Cumbria, England and was a park gardener who rose to be head gardener of Newcastle's hospitals. His mother Frances Walker Dowson (whose maiden name was McQuet, pronounced mac-yew-it) came from Bo'ness, near Linlithgow in Scotland and was trained as a lady's maid. They met when they were in the service of the same landowner.

We find all solutions of in integers m, n1, n2, n3, n4 for all relatively prime integers x, y below 100.

We continue the study of a mathematical model for a forest ecosystem which has been presented by Y. A. Kuznetsov, M. Y. Antonovsky, V. N. Biktashev and A. Aponina (A cross-diffusion model of forest boundary dynamics, J. Math. Biol. 32 (1994), 219–232). In the preceding two papers (L. H. Chuan and A. Yagi, Dynamical systemfor forest kinematic model, Adv. Math. Sci. Appl. 16 (2006), 393–409; L. H. Chuan, T. Tsujikawa and A. Yagi, Aysmptotic behavior of solutions for forest kinematic model, Funkcial. Ekvac. 49 (2006), 427–449), the present authors already constructed a dynamical system and investigated asymptotic behaviour of trajectories of the dynamical system. This paper is then devoted to studying not only the structure (including stability and instability) of homogeneous stationary solutions but also the existence of inhomogeneous stationary solutions. Especially it shall be shown that in some cases, one can construct an infinite number of discontinuous stationary solutions.

A closed Riemann surface which can be realized as a three-sheeted covering of the Riemann sphere is called trigonal, and such a covering is called a trigonal morphism. If the trigonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic trigonal Riemann surface. Using the characterization of cyclic trigonality by Fuchsian groups, we find the structure of the space of cyclic trigonal Riemann surfaces of genus 4.

We prove that the class number of the imaginary quadratic field is divisible by n for any positive integers k and n with 22k < 3n, by using Y. Bugeaud and T. N. Shorey's result on Diophantine equations.