Let K(y) be a known distribution function on (−∞, ∞) and let {Fn(y), n = 0, 1,…} be a sequence of unknown distribution functions related by subject to the initial condition If the sequence {Fn(y)} converges to a distribution function F(y) then F(y) satisfies the Wiener-Hopf equation

In general, information concerning the distribution of the time to absorption, T, of a simple branching (Galton-Watson) process for which extinction in finite mean time is certain, is difficult to obtain. The process of greatest biological interest is that for which the offspring distribution is Poisson, having p.g.f. F(s) = em(s-1), m < 1.

If S1, S2,… are the times of successive births in a pure birth-process with linear birthrate and i > 0 individuals initially, then given that there are k > 0 births in (0, t), the random variables eλS1—1, eλS2−1,λS1−1 are distributed (conditionally) like the order statistics of k independent random variables, all uniformly distributed on the interval (0, eλt–1), regardless of the initial population size. A similar property holds for the pure death process.

Let K be a field of characteristic 0, G the direct sum of two copies of the additive group of integers. For a total order ≺ on G, which is compatible with the addition, and for any ċ1, ċ2 ∈ K, we define G-graded highest weight modules M(ċ1, ċ2, ≺) over the Virasoro-like algebra , indexed by G. It is natural to call them Verma modules. In the present paper, the irreducibility of M (ċ1, ċ2, ≺) is completely determined and the structure of reducible module M (ċ1, ċ2, ≺)is also described.

Let S be the surface of the unit sphere in three-dimensional euclidean space, and let WN=(x1x2, xN)be an N-tuple of points on S. We consider the product of mutual distances and, for the variable point x on S, the product of distance from x to the points of ωN. We obtain essentially best possible bounds for maxωN p(ΩN) and for minωN maxx∈sp(x, ωN).

In this paper, we derive a number of explicit lower bounds for rational approximation to certain cubic irrationalities, proving, for example, that for any non-zero integers p and q. A number of these irrationality measures improve known results, including those for . Some Diophantine consequences are briefly discussed.

We prove a new energy or Caccioppoli type estimate for minimisers of the model functional ∫Ω|Du|2 + (det Du)2, where Ω ⊂ 2 and u: Ω → 2. We apply this to establish C∞ regularity for minimisers except on a closed set of measure zero. We also prove a maximum principle and use this to establish everywhere continuity of minimisers.

Let X1 and X2 be two independent and nonnegative random variables with distributions F1 and F2, respectively. This paper proves that if both F1 and F2 are of Weibull type and fulfill certain easily verifiable conditions, then the distribution of the product X1X2, called the product convolution of F1 and F2, belongs to the class 𝒮* and, hence, is subexponential.

Let S be a poset with a greatest element 1. We denote order in S by ‘≦’ and, whenever they exist in S, l.u.b and g.l.b by ‘∨’ and ‘∧’ respectively. An orthocomplementation of S is a bijection w: S → S such that x ∨ xω exists for each x in S and (i) xωω = x, (ii) x ≦ y implies yω ≦ xω and (iii) x ∨xω = 1. If a poset S admits an orthocomplementation ω we call the pair (S, ω) an orthoposet.

In this paper, we give multihomogeneous estimates for the group of relations linking multiplicatively dependent algebraic numbers. In the process, we raise a question in the style of Lehmer's problem, concerning multidimensional covolumes in the lattice of units. The proofs are based on the Brill-Gordan duality theorem on orthogonal lattices, and the paper closes with an algebraic version of the theorem, concerning orthogonal abelian subvarieties of an arbitrarily polarized abelian variety.

A unital C*-algebra A is called extremally rich if the set of quasi-invertible elements A-1 ex (A)A-1 (= A-1q) is dense in A, where ex(A) is the set of extreme points in the closed unit ball A1 of A. In [7, 8] Brown and Pedersen introduced this notion and showed that A is extremally rich if and only if conv(ex(A)) = A1. Any unital simple C*-algebra with extremal richness is either purely infinite or has stable rank one (sr(A) = 1). In this note we investigate the extremal richness of C*-crossed products of extremally rich C*-algebras by finite groups. It is shown that if A is purely infinite simple and unital then A xα, G is extremally rich for any finite group G. But this is not true in general when G is an infinite discrete group. If A is simple with sr(A) =, and has the SP-property, then it is shown that any crossed product A xαG by a finite abelian group G has cancellation. Moreover if this crossed product has real rank zero, it has stable rank one and hence is extremally rich.

Let G be a locally compact Abelian Hausdorff group (abbreviated LCA group); let X be its character group and dx, dx be the elements of the normalised Haar measures on G and X respectively. If 1 < p, q < ∞, and Lp(G) and Lq(G) are the usual Lebesgue spaces, of index p and q respectively, with respect to dx, a multiplier of type (p, q) is defined as a bounded linear operator T from Lp(G) to Lq(G) which commutes with translations, i.e. τxT = Tτx for all x ∈ G, where τxf(y) = f(x+y). The space of multipliers of type (p, q) will be denoted by Lqp. Already, much attention has been devoted to this important class of operators (see, for example, [3], [4], [7]).

For any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation

$$\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*}$$

$x, y, z$

$f(n)\gt 0$

$n\geq 2$

$f(n)$

$f(p)$

$n$

$p$

$$\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*}$$

Shrinkande and Bhagwan Das (1970) showed how to extend a (4t – 1, 4t) row-orthogonal matrix with entries ± 1 to a Hadamard matrix of order 4t. Using a slightly different approach we consider extensions of (4t – k, 4t) row-orthogonal matrix to a Hadamard matrix of order 4t.

We show that a bounded linear operator on a uniformly convex space may be perturbed by a compact operator of arbitrarily small norm to yield an operator which attains its numerical radius.

We consider a single server queueing system in which customers arrive at the instants t0, t1, …, tm, …. We write τm = tm+1 − tm, m ≧ 0. There is a single server with distribution of service times B(x) given by where k is an integer not less than unity.

For totally real algebraic integers x of degree D(x), we examine the stucture of the set of values M(x)1D(x). where M(x) is the measure of x. We find a small limit point λ of this set, and show that the set is everywhere dense in (λ x).

Let M=G/K be a generalized flag manifold, that is, an adjoint orbit of a compact, connected and semisimple Lie group G. We use a variational approach to find non-Kähler homogeneous Einstein metrics for flag manifolds with two isotropy summands. We also determine the nature of these Einstein metrics as critical points of the scalar curvature functional under fixed volume.

It is the aim of this paper to introduce a new concept relating various subgroups of the automorphism group of a graph to corresponding subgraphs. Throughout g will denote a (Michigan) graph on a vertex set V(¦V¦ =n) and Γ(g)=G will be the automorphism group of G considered as a permutation group on V.En, Cn, Dn and Sn are the identity, cyclic, dihedral, and symmetric groups acting on a set of size n, while Sp(q) is the permutation group of pq objects which is isomorphic to Sp but is q-fold in the sense that the objects are permuted q at a time [6]. H ≦ G means that H is a subgroup of G. Other group concepts can be found in Wielandt [7]. The graphs G1 ∪ G2, G1 + G2, G1 × G2, and G1[G2] along with their corresponding groups are as defined in, for example, Harary [4]. Finally we use Kn for the complete graph on n vertices.

A composite integer N is said to be a strong pseudoprime for the base C if with N – 1 = 2sd, (2, d) = 1 either Cd = 1, or C2r ≡ 1 (mod N) some r, 0 ≤ r < s. It is shown that every arithmetic progression ax+b (x = 0,1, …) where a, b are relatively prime integers contains an infinite number of odd strong pseudoprimes for each base C ≤ 2.

1980 Mathematics subject classification (Amer. Math. Soc.): 10 A 15.