Soient p un nombre premier ≥ 7 et d un entier naturel sans puissances cinquièmes. Nous mettons en œuvre les différentes méthodes modulaires connues pour l'étude de l'équation diophantienne x5+y5 = dzp. Nous montrons en particulier qu'elle n'admet aucune solution propre et non triviale pour p ≥ 7 ou pour une infinité de nombres premiers, dans certains cas où d est de la forme 2α.3β.5γ. Pour d = 3, on énonce un critère permettant de vérifier, notamment, que tel est le cas lorsque p est ≤ 106.

Let p be a prime number ≥ 7 and d be a positive integer fifth power free. We use the known modular methods for the study of the diophantine equation x5+y5 = dzp. We prove that this equation has no non trivial proper solution for p ≥ 7 or for infinitely many prime numbers, in some cases where d is of the form 2α.3β.5γ. For d = 3, we give a criterion which allows us to verify that this holds if p is less than 106.

We prove several stability results for Atkinson linear relations under additive perturbation by small norm, strictly singular and strictly cosingular multivalued linear operators satisfying some additional conditions.

Fredholm, injective, isometric and surjective composition operators on Lorentz spaces L(p, q) are characterised in this paper.

We construct a counterexample to a theorem of Jon Wolfson concerning the existence of globally smooth solutions of the second boundary value problem for Monge-Ampère equations in two dimensions, or equivalently, on the existence of minimal Lagrangian diffeomorphisms between simply connected domains in R2.

D.P. Baturov proved in ‘Subspaces of function spaces’ Vestnik Moskov University Series I (1987) that Lindelöf degree equals extent for subspaces of Cp(Χ) when Χ is a Lindelöf Σ-space. We prove that if the Lindelöf degree of the subspace is “big enough” the equality is true for a topological space Χ not necessarily Lindelöf Σ.

In this paper we study the existence of mild solutions for Cauchy problem

We derive conditions under which the mild solutions exist, and also get the relative compactness of the solution set, which extend and improve some existing results in this area.

We consider two systems of real analytic partial differential equations, related by a holomorphic contact map H. We study how the generalised symmetries of the first equation are mapped into those of the second one, and determine under which conditions on H such a map is invertible. As an application of these results, an example of physical interest is discussed.

For a supernilpotent radical α and a special class σ of rings we call a ring R (α, σ)-essential if R is α-semisimple and for each ideal P of R with R/P ε σ, P ∩ I ≠ 0 whenever I is a nonzero two-sided ideal of R. (α, σ)-essential rings form a generalisation of prime essential rings introduced by L. H. Rowen in his study of semiprime rings and their subdirect decompositions and they have been a subject of investigations of many prominent authors since. We show that many important results concerning prime essential rings are also valid for (α, σ)-essential rings and demonstrate how (α, σ)-essential rings can be used to determine whether a supernilpotent radical is special. We construct infinitely many supernilpotent nonspecial radicals whose semisimple class of prime rings is zero and show that such radicals form a sublattice of the lattice of all supernilpotent radicals. This generalises Yu.M. Ryabukhin's example.

In this paper we obtain criteria of stability for ηEinstein k-contact manifolds, for Sasakian manifolds of constant ϕ-sectional curvature and for 3-dimensional Sasakian manifolds. Moreover, we show that a stable compact Einstein contact metric manifold M is Sasakian if and only if the Reeb vector field ξ minimises the energy functional. In particular, the Reeb vector field of a Sasakian manifold M of constant ϕ-holomorphic sectional curvature +1 minimises the energy functional if and only if M is not simply connected.

We show that a suitable adaptation of the so-called method of trajectories can be used to construct an exponential attractor for a very general class of nonlinear reaction-diffusion systems with a bounded delay.

In particular, we assume that the dependence on the past history is controlled via convolution with a possibly singular measure. Assuming a priori that the solutions are bounded, a simple proof of the existence of an exponential attractor is given under very little regularity requirements.

In this paper we prove partial regularity for a weakly stable p-harmonic map from Ω into Sk when k > 2p – 1.

Let G be a graph of order n, and let a, b, k be nonnegative integers with 1 ≤ a < b. An [a, b]-factor of graph G is defined as a spanning subgraph F of G such that a ≤ dF(x) ≤ b for each x ϵ V (F). Then a graph G is called an (a, b, k)-critical graph if after deleting any k vertices of G the remaining graph of G has an [a, b]-factor. In this paper, it is proved that G is an (a, b, k)-critical graph if the binding number and

Furthermore, it is showed that the result in this paper is best possible in some sense.

This paper looks at simple rings which have right ideals satisfying various types of injectivity conditions. We characterise when a simple regular ring is right self-injective and show that if R is a simple ring in which every right ideal is the direct sum of quasi-continuous right ideals then R is either Artinian or a non-selfinjective right Goldie ring in which every right ideal is a direct sum of uniform right ideals.