In this paper we consider the problem of an upper bound estimate for the least quadratic non-residue modulo prime number on special arithmetic sequences such as f(n) = [αn] and f(n) = [nc].

Let Ω ⊂ ℝN be a bounded domain. We study existence and uniqueness of principal eigenvalues for the Dirichlet periodic parabolic problem with weight Lu = λmu in Ω × ℝ when the independent coefficient of the differential operator L is not necessarily positive.

Let Bk denote the Euclidean unit ball in ℝκ equipped with the k-dimensional Lebesgue measure and let φ: ℝ+ → ℝ+ be a convex function satisfying φ(0) = 0, φ(t) > 0 for some t > 0. Denote by Eφ = Eφ(Bk) the Orlicz space of finite elements (see (1.6)) generated by φ. The aim of this paper is to show that there exists a retraction of the closed unit ball in Eφ onto the unit sphere in Eφ being a (2 + ɛ)γφ;-set contraction (Theorem 3.6), which generalises [9, Corollary 6] proved for the case of Lp[−1, 1], 1 ≤ p < ∞. Here γφ, denote the Hausdorff measure of noncompactness. This theorem is proved both for the Amemiya and the Luxemburg norms. Also some related results concerning the case of s-convex (0 < s ≤ 1) functions are presented.

We study Kalton's theorem on the unconditional convergence of series of compact operators and we use some matrix techniques to obtain sufficient conditions, weaker than previous ones, on the convergence and unconditional convergence of series of compact operators. Finally, we characterise weak unconditionally Cauchy series in Cℒ(X, Y) in the terms of certain spaces of vector sequences.

The purpose of this paper is to give some necessary and sufficient conditions for p-nilpotent groups. We extend some results, including the well-known theorems of Burnside and Frobenius as well as some very recent theorems. We also apply our results to determine the structure of some finite groups in terms of formation theory.

The present paper proposes new criteria for compactness of a composition operator Cφf = f ∘ φ on the Bloch space and the little Bloch space. For the case when φ is univalent, a criterion given by K. Madigan and A. Matheson is generalised.

In this paper we describe a construction of a large class of hyperconvex metric spaces. In particular, this construction contains well-known examples of hyperconvex spaces such as ℝ2 with the “river” metric or with the radial one.

Further, we investigate linear hyperconvex spaces with extremal points of their unit balls. We prove that only in the case of a plane (and obviously a line) is there a strict connection between the number of extremal points of the unit ball and the hyperconvexity of the space.

Some additional properties concerning the notion of hyperconvexity are also investigated.

In this paper we give a non-existence property of real hypersurfaces in complex two-plane Grassmannians G2(ℂm+2) which have a shape operator A commuting with the structure tensors {φ1, φ2, φ3}. From this view point we give a characterisation of real hypersurfaces of type B in G2(ℂm+2).

In this paper, we investigate a public key cryptosystem which is derived from a third order linear recurrence relation and is analogous to the RSA and LUC cryptosystems. The explicit formulation involves a generalisation of the rule for composition of powers and of the calculus of the Euler totient function which underlie the algebra of the RSA cryptosystem. The security of all these systems appears to be comparable and to depend on the intractability of factorization but the systems do not seem to be mathematically equivalent.

The abstract concept of an elementary operator was recently introduced and studied by other authors. In this paper, we describe the general form of elementary operators between standard subalgebras of 𝒥-subspace lattice algebras. The result can apply to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras.

On montre que la partie nilpotente d'une solveriété d'Einstein standard, dont le type de valeur propre est égal à (1 < 2; d1, d2), est nécessairement indécomposable si d1 et d2 sont premiers entre eux.

Weighted Hadamard product characterisations of Bloch functions in the unit ball of ℂn are studied. In particular, we prove that f belongs to the Bloch space ℬ(Bn) if and only if the non-weighted Hadamard products of f and g belong to BMOA(Un) for all f in , a subspace of the Hardy space H1(Bn).

Various n-th derivative characterisations involving different kinds of oscillations of F(p,q,s) functions are established, and the mean growth of derivatives of F(p,q,s) functions is considered. Moreover, inclusion relations between certain analytic function spaces are discussed.

This paper is devoted to the study of a class of generalised P-nilpotent groups in the universe c of all radical locally finite groups satisfying min-q for every prime q. Some results of finite groups are extended and a characterisation of the injectors associated with this class is given.

We give an example of an Engel structure on the 4-dimensional Euclidean space which admits a compact characteristic leaf. We also show that every Engel structure on an open 4-manifold can be modified so that the resulting structure has a compact characteristic leaf.

In 1988 Howie, Robertson and Schein characterised the transformations of a finite set X that can be written as a product of two or of three idempotent transformations of X; and in 1989 Saito did the same for products of four idempotents. In 1998 Thomas extended the characterisation of two idempotents to arbitrary sets, and here we characterise products of three idempotents in general. We also define a notion of complexity for transformations of any set and use it to provide a different solution to the three-idempotent problem.

We introduce n-Prüfer domains which axe generalisations of Prüfer domains and give several characterisations of them in terms of generalisations of purity, flatness and absolute purity.

Let  be a certain set of nonnegative symmetric matrices, such as the set of symmetric doubly stochastic matrices or the set, of symmetric permutation matrices. It is proven that a linear transformation mapping  onto  must be of the form X ↦ PtX P for some permutation matrix P except for several low dimensional cases.

This note relates polynomial remainders with polynomial automorphisms of the plane. It also formulates a conjecture, equivalent to the famous Jacobian Conjecture. The latter provides an algorithm for checking when a polynomial map is an automorphism. In addition, a criterion is presented for a real polynomial map to be bijective.

In this paper we prove that the affine diameter of any closed uniformly convex hypersurface in Euclidean space enclosing finite volume is bounded from above.