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.

A periodic wavelet Galerkin method is presented in this paper to solve a weakly singular integral equations with emphasis on the second kind of Fredholm integral equations. The kernel function, which includes of a smooth part and a log weakly singular part, is discretised by the periodic Daubechies wavelets. The wavelet compression strategy and the hyperbolic cross approximation technique are used to approximate the weakly singular and smooth kernel functions. Meanwhile, the sparse matrix of systems can be correspondingly obtained. The bi-conjugate gradient iterative method is used to solve the resulting algebraic equation systems. Especially, the analytical computational formulae are presented for the log weakly singular kernel. The computational error for the representative matrix is also evaluated. The convergence rate of this algorithm is O (N-p log(N)), where p is the vanishing moment of the periodic Daubechies wavelets. Numerical experiments are provided to illustrate the correctness of the theory presented here.

By introducing the function 1/(min{x, y}), we establish several new inequalities similar to Hilbert's type inequality. Moreover, some further unification of Hardy-Hilbert's and Hardy-Hilbert's type integral inequality and its equivalent form with the best constant factor are proved, which contain the classic Hilbert's inequality as special case.

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.

A method of embedding nk configurations into projective space of k–1 dimensions is given. It breaks into the easy problem of finding a rooted spanning tree of the associated Levi graph. Also it is shown how to obtain a “complementary” “theorem” about projective space (over a field or skew-field F) from any nk theorem over F. Some elementary matroid theory is used, but with an explanation suitable for most people. Various examples are mentioned, including the planar configurations: Fano 73, Pappus 93, Desargues 103 (also in 3d-space), Möbius 84 (in 3d-space), and the resulting 74 in 3d-space, 96 in 5d-space, and 107 in 6d-space. (The Möbius configuration is self-complementary.) There are some nk configurations that are not embeddable in certain projective spaces, and these will be taken to similarly not embeddable configurations by complementation. Finally, there is a list of open questions.

Based on the work of Fergola, Zhang and Cerasuolo, a bacteria-immunity model with the mechanism of periodic quorum sensing is formulated, which describes the competition between bacteria and immune cells. A discrete delay is introduced to characterise the time between when bacteria receive signal molecules and then combat with immune cells. In this paper, we focus on a subsystem of the bacteria-immunity model and investigate the existence of a positively periodic solution, and then study its global stability.

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

In this paper, an active set sequential quadratic programming algorithm with non-monotone line search for nonlinear minmax problems is presented. At each iteration of the proposed algorithm, a main search direction is obtained by solving a reduced quadratic program which always has a solution. In order to avoid the Maratos effect, a correction direction is yielded by solving the reduced system of linear equations. Under mild conditions without the strict complementarity, the global and superlinear convergence can be achieved. Finally, some preliminary numerical experiments are reported.

In this paper, we study two-point boundary value problems for the nonlinear second order difference equation We establish the relationship between the number of sign-variation of f on {0,…, T + 2} and the one of the solution u of the above problem.

It is shown that a spacelike maximal surface in the three dimensional Lorentz-Minkowski space can be extended analytically if it meets a spacelike plane at a constant hyperbolic angle.

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.

We give an existence result for a fully nonlinear system consisting of a parabolic equation strongly coupled with an elliptic one. It models in particular miscible displacement in porous media. To this aim, we adapt the tools of Ladyzenskaja, Solon-nikov and Uralćeva [27, 28] to the coupled nonlinear setting. Under some reasonable assumptions on the data, we state the existence of semi-classical solutions for the problem. We also give an existence result of weak solutions for a degenerate form of the problem.

In this note about Genetic Algorithms (GA-), we study the 2-epistasis of a fitness function over a search space. This concept is a natural generalisation of that of epistasis, previously considered by Davidor in 1991 Suys and Verschoren in 1996 and Van Hove and Verschoren in 1994 for example. We completely characterise fitness functions whose 2-epistasis is minimal: these are exactly the second order functions. The validity of 2-epistasis as a measure of hardness with respect to genetic algorithms is checked over some classical laboratory functions. Finally, we obtain an upper bound of the maximal value of the 2-epistasis when we restrict attention to non-negative functions.

Let G be a locally compact group and be the Banach space of all essentially bounded measurable functions on G vansihing an infinity. Here, we study some families of right completely continuous elements in the Banach algebra equipped with an Arens type product. As the main result, we show that has a certain right completely continuous element if and only if G is compact.

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.

The Magnus matrix is an algebraic invariant assigned to each homology cobordism of a surface. We show that this matrix satisfies an equality which can be regarded as a non-commutative symplectic relation.

The S-rank (where ‘S’ abbreviates ‘sandwich’) of a right congruence ρ on a semigroup S is the Cantor-Bendixson rank of ρ in the lattice of right congruences ℛ of S with respect to a topology we call the finite type topology. If every ρ ϵ ℛ possesses S-rank, then S is ranked. It is known that every right Noetherian semigroup is ranked and every ranked inverse semigroup is weakly right Noetherian. Moreover, if S is ranked, then so is every maximal subgroup of S. We show that a Brandt semigroup 0(G, I) is ranked if and only if G is ranked and I is finite.

We establish a correspondence between the lattice of congruences on a chain E, and the lattice of right congruences contained within the least group congruence on any inverse semigroup S with semilattice of idempotents E(S) ≅ E. Consequently we argue that the (inverse) bicyclic monoid B is not ranked; moreover, a ranked semigroup cannot contain a bicyclic -class. On the other hand, B is weakly right Noetherian, and possesses trivial (hence ranked) subgroups.

Our notion of rank arose from considering stability properties of the theory Ts of existentially closed (right) S-sets over a right coherent monoid S. The property of right coherence guarantees that the existentially closed S-sets form an axiomatisable class. We argue that B is right coherent. As a consequence, it follows from known results that TB is a theory of B-sets that is superstable but not totally transcendental.

We construct a transcendental number α whose powers αn!, n = 1, 2, 3,…, modulo 1 are everywhere dense in the interval [0, 1]. Similarly, for any sequence of positive numbers δ = (δn)∞n=1, we find a transcendental number α = α(δ) such that the inequality {αn} < δn holds for infinitely many n ∈ N, no matter how fast the sequence δ converges to zero. Finally, for any sequence of real numbers (rn)∞n=1 and any sequence of positive numbers (δn)∞n=1, we construct an increasing sequence of positive integers (qn)∞n=1 and a number α > 1 such that ‖αqn – τn‖ < δn for each n ≥ 1.

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.