Upper and lower bounds are obtained for the left tail of the normed limit W0 of a supercritical branching process with varying environment, that is, for P(W0 < x) for small x. Two types of process are dealt with—öttcher type and Schröder type—which between them cover “most” processes with zero extinction probability.

For years it has remained a problem to find suitable functionals satisfying certain maximum principles for solutions of the equation Δu + f(x, u) = 0. In this paper, maximum principles for certain functionals which are defined on solutions of semilinear elliptic equations subject to mixed or Robin boundary conditions are obtained. The principles derived may be used to deduce bounds on important quantities in physical problems of interest.

We discuss the evolution of plane curves which are described by entire graphs with prescribed opening angle. We show that a solution converges to the unique self-similar solution with the same asymptotics.

Responding to a question on right weakly semisimple rings due to Jain, Lopez-Permouth and Singh, we report the existence of a non-right-Noetherian ring R for which every uniform cyclic right it-module is weakly-injective and every uniform finitely generated right R-module is compressible. We show that a ring R is a right Noetherian ring for which every cyclic right R-module is weakly R-injective if and only if R is a right Noetherian ring for which every uniform cyclic right R-module is compressible if and only if every cyclic right R-module is compressible. Finally, we characterise those modules M for which every finitely generated (respectively, cyclic) module in σ[M] is compressible.

We study the class of Tychonoff topological spaces such that the free Abelian topological group A(X) is reflexive (satisfies the Pontryagin-van Kampen duality). Every such X must be totally path-disconnected and (if it is pseudocompact) must have a trivial first cohomotopy group π1(X). If X is a strongly zero-dimensional space which is either metrisable or compact, then A(X) is reflexive.

We show that the Murasugi conditions for the Alexander polynomial of a cyclically periodic knot imply a modified form of the Burde-Trotter condition.

The spectral C*-algebra of the discrete product systems of H.T. Dinh is shown to be a twisted semigroup crossed product whenever the product system has a twisted unit. The covariant representations of the corresponding dynamical system are always faithful, implying the simplicity of these crossed products; an application of a recent theorem of G.J. Murphy gives their nuclearity. Furthermore, a semigroup of endomorphisms of B(H) having an intertwining projective semigroup of isometries can be extended to a group of automorphisms of a larger Type I factor.

It is shown that an amenable algebra of operators on Hibert space which is generated by its normal elements is necessarily self-adjoint, so it is a C*-algebra.

Let L = Q[α] be a cyclic number field of odd prime degree q over the field Q of rationals. In this paper we give an algorithm to compute the discriminant of L/Q, which relies upon a fast method to find Eisenstein elements in L. The algorithm accepts as input the minimal polynomial of α over Q and a complete factorisation of the discriminant of α, and computes, in time polynomial in the size of the input, a list consisting of all the ramified primes with corresponding Eisenstein elements.

The well-known Schur inequality relates the sum of the squares of the absolute values of the eigenvalues of A to the elements of A. This was recently generalised to powers between one and two. Here we show that the inequality holds for powers between zero and two.