We prove an existence theorem for nonlinear differential equations with deviating arguments and with implicit derivatives. The proof is based on the notion of measure of noncompactness and the Darbo fixed point theorem.

This paper is related with the configurations of limit cycles for cubic polynomial vector fields in two variables (χ3).

It is an open question to decide whether every limit cycle configuration in χ3 can be obtained by perturbation of a corresponding Hamiltonian configuration of centres and graphs.

In this work, by considering perturbations of the Hamiltonian vector field XH = (Hy, − Hx), where H(x, y) = [a(x + h)2 + by2 − 1] [a(x − h)2 + by2 − 1], we make a global analysis of the possible cases.

The vector field XH has three centres (C−, C+ and the origin) and two saddles. By means of quadratic perturbations the centres become fine foci where C−and C+ have the same type of stability but opposed to that one of the origin and infinity. Further introducing cubic perturbations changes the stability of C−, C+ and the cycle at infinity and generates limit cycles. Lastly extra linear terms change the stability of the origin and generate another limit cycle.

Finally, we analyse the rupture of saddle connection of the Hamiltonian field under perturbation, via Melnikov's integral, in order to complete the study of the global phase portrait and to consider the possibility of new limit cycles emerging from the Hamiltonian graph.

If a and n are positive integers and if ⌊⌋ is the greatest integer function we obtain upper and lower estimates for stated by Ramanujan in his notebooks.

We determine all additive F, G: ℍ → ℝ and multiplicative M: H → ℝ satisfying the functional equation F(λ) + M(λ)G(λ−1) = 0. As an application we generalise Kurepa's solution of one of Halperin's problem concerning quadratic functionals.

A necessary and sufficient condition for a dual discriminator variety to be the product of a discriminator variety and a dual discriminator variety is given, along with an example of such a variety.

A complete description of holomorphic curves in the Hermitian symmetric space SO(6)/U(3) is given in terms of orthogonal differential invariants.

The elementary nature and simplicity of the theory of continued fractions is mostly well disguised in the literature. This makes one reluctant to quote sources when making a remark on the subject and seems to necessitate redeveloping the theory ab initio. That had best be done succinctly. That is done here and allows the retrieval of some amusing results on pattern in the period of the continued fraction expansion of quadratic integers.

We extend Kollár's result on finitely generated, injectively complete congruence distributive varieties to the congruence modular setting. By doing so we show that, given any finite algebra A of finite type, there is an algorithm to decide whether V(A) is an injectively complete, congruence modular variety.

By using techniques of the calculus of variations in geometric measure theory, we investigate the nonexistence of stable integral currents in Sn1 × Sn2 and its immersed submanifolds. Several vanishing theorems concerning the homology group of these manifolds are established.

A nonlinear vector minimum problem is considered. Duality theorems are proved for Mond-Weir type dual and their application to a certain nonlinear fractional vector minimum problem is discussed.

In this paper we study the number and the relative position of the limit cycles of a plane quadratic system with a weak focus. In particular, we prove the limit cycles of such a system can never have (2, 2)-distribution, and that there is at most one limit cycle not surrounding this weak focus under any one of the following conditions:

(i) the system has at least 2 saddles in the finite plane,

(ii) the system has more than 2 finite singular points and more than 1 singular point at infinity,

(iii) the system has exactly 2 finite singular points, more than 1 singular point at infinity, and the weak focus is itself surrounded by at least one limit cycle.

Let G be a Weyl group of type B, and T a set of representatives of the conjugacy classes of self-inverse elements of G. For each t in T, we construct a (complex) linear character πt of the centraliser of t in G, such that the sum of the characters of G induced from the πt contains each irreducible complex character of G with multiplicity precisely 1. For Weyl groups of type A (that is, for the symmetric groups), a similar result was published recently by Inglis, Richardson and Saxl.

Two consequences of a theorem of Dahlberg are derived. Let f be a holomorphic function in the unit disk D of the complex plane, and let E be an Fσ subset of the unit circle T. Suppose that |f(rw)| ≤ M, ω ∈ T/E, for some constant M.

Then f is bounded in either of the two cases:

(i) if f is in the Bloch space and E is of zero measure with respect to the Hausdorff measure associated with the function ψ(t) = t log log (2πee/t),

(ii) if f is integrable with respect to the planar Lebesgue measure on D and E is of zero measure with respect to the Hausdorff measure associated with the function ψ(t) = t log(2πee/t).