Ciet, Quisquater, and Sica have recently shown that every elliptic curve E over a finite field 𝔽p is isomorphic to a curve y2 = x3 + ax + b with a and b of size O (p¾). In this paper, we show that almost all elliptic curves satisfy the stronger bound O (p⅔). The problem is motivated by cryptographic considerations.

The radial distribution of Julia sets and non-existence of unbounded Fatou components of transcendental meromorphic functions are investigated in this paper.

It is well-known that warped products play some important roles in differential geometry as well as in physics. In this article we extend the notion of warped product to the notion of convolution of Riemannian manifolds. We study the basic properties of convolutions of Riemannian manifolds. We also apply the notion of convolution to establish and characterise the Euclidean version of Segre embedding.

Let  be a commutative family of nonexpansive self-mappings of a closed convex subset C of a uniformly smooth Banach space X such that the set of common fixed points is nonempty. It is shown that if a certain regularity condition is satisfied, then the sunny nonexpansive retraction from C to F can be constructed in an iterative way.

A characterisation of a smoothing spline is sought in a convex closed set C of Hilbert space: , T and A are linear operators. A representation of the solution is obtained in the terms of the kernels of the above operators, of the dual operators T*, A* and of the dual cone C0. A particular case is considered when T is the differential operator and A is the operator-trace of a function.

The term “rigid set” appears often in the literature in different contexts and with different meanings. In many cases the notions are unrelated, while in other they refer to related facts. Here we study infinite sets which are rigid according to [2] and analyse the relationship with the notion of rigidity used in [1]. We make several remarks, summarise different results in the area and prove a few new results.

In this paper, a sharp version of the Schwarz–Pick Lemma for hyperbolic derivatives is provided for holomorphic selfmappings on the unit disk with fixed multiplicity for the zero at the origin. This extends a recent result due to Beardon. A property of preserving hyperbolic distances also studied by Beardon is here completely characterised.

A charactersiation of topologically weak mixing is given by using the topological sequence entropy.

Sierpiński graphs S (n, κ) generalise the Tower of Hanoi graphs—the graph S (n, 3) is isomorphic to the graph Hn of the Tower of Hanoi with n disks. A 1-perfect code (or an efficient dominating set) in a graph G is a vertex subset of G with the property that the closed neighbourhoods of its elements form a partition of V (G). It is proved that the graphs S (n, κ) possess unique 1-perfect codes, thus extending a previously known result for Hn. An efficient decoding algorithm is also presented. The present approach, in particular the proposed (de)coding, is intrinsically different from the approach to Hn.

In this short note, we obtain a concrete description of rank-one operators in Alg(ℒ1 ⊗…⊗ ℒn). Based on this characterisation, we give a simple proof of the tensor product formula: if Alg(ℒ1 ⊗…⊗ ℒn) is weakly generated by rank-one operators in itself and ℒi(i = 1,…,n) are subspace lattices.

In this paper we study the boundedness of the weighted Bergman projections on the weighted subspaces of Bergman spaces and the Lipschitz spaces on the unit ball and the unit polydisc.

In this paper we prove the existence of solutions to an inverse semilinear elliptic partial differential equation. Physically, solutions represent stream functions of steady planar flows with bounded vortices. The kinetic energy functional is maximised over the set of rearrangements of a given function.

Some conditions under which a derivation on some operator algebras can be completely determined by the action on operators of zero product are given.

We prove Lp -estimates for the Littlewood–Paley g-function associated with a complex elliptic operator L = − div A∇ with bounded measurable coefficients in ℝn.

In this article, the existence of traveling wave fronts for a one step chemical reaction with a natural discontinuous reaction rate function is studied. This discontinuity occurs because of the cold boundary difficulty and implies a discontinuous system of ordinary differential equations. By some general topological arguments in ordinary differential equations, the Chapman-Jouguet detonation for exothermic reactions is shown to exist. In addition, the uniqueness of this wave is considered.

We study the behaviour of the hitting probabilities of conditional Brownian motion in a domain D in Euclidean space when we apply polarization to D. We also study how polarization affects the probability that conditional Brownian motion meets a subset of D.

In this paper we introduce the notion of asymptotical smoothness of a Banach space and show that it is strongly related to the Kadec-Klee property. This notion is then applied to obtain new theorems about weak convergence of almost orbits of three various types of semigroups of mappings.

The main result of this paper is a sufficient condition for the minimax relation to hold for the canonical bilinear form on X × Y, where X is a nonempty convex subset of a real locally convex space and Y is a nonempty convex subset of its dual. Using the known “converse minimax theorem”, this result leads easily to a nonlinear generalisation of James's (“sup”) theorem. We give a brief discussion of the connections with the “sup-limsup theorem” and, in the appendix to the paper, we give a simple, direct proof (using Goldstine's theorem) of the converse minimax theorem referred to above, valid for the special case of a normed space.

We introduce a concept of metric space valued mappings of two variables with finite total variation and define a counterpart of the Hardy space. Then we establish the following Helly type selection principle for mappings of two variables: Let X be a metric space and a commutative additive semigroup whose metric is translation invariant. Then an infinite pointwise precompact family of X-valued mappings on the closed rectangle of the plane, which is of uniformly bounded total variation, contains a pointwise convergent sequence whose limit is a mapping with finite total variation.

In this note we prove that the group G of infinite dimensional upper unitriangular matrices over a finite field contains an explicit countable subgroup ‘full’ of free subgroups. We deduce from this fact that, in a suitable sense, almost all k–generator subgroups of G are free groups of rank k.