In this work, we study types of undulatory solutions, that we term Directional Wave Fronts (DWF), of non scalar reaction diffusion systems. The DWFs are a natural extension of the well known Plane Wave Fronts (PWFs) solutions. However, the DWFs admit a certain type of boundary conditions. In the present work we show, under suitable conditions on the reaction term, that DWFs also exhibit typical behaviour of PWFs: we just prove the existence of heteroclinic, homoclinic and periodic families of DWFs. Essentially, we require the reaction term to be linearly uncoupled. These results are the generalization of a previous work, concerning the scalar case.

Trace forms have been well studied as invariant quadratic forms on finite dimensional Lie algebras; the best known of these forms in the Cartan-Killing form. All those forms, however, have the ideal [L, L] ∩ R (with the radical R) in the orthogonal L⊥ and thus are frequently degenerate. In this note we discuss a general construction of Lie algebras equipped with non-degenerate quadratic forms which cannot be obtained by trace forms, and we propose a general structure theorem for Lie algebras supporting a non-degenerate invariant quadratic form. These results complement and extend recent developments of the theory of invariant quadratic forms on Lie algebras by Hilgert and Hofmann [2], keith [4], and Medina and Revoy [7].

In this paper, generalised Gaussian Fibonacci numbers are defined and, using the recurrence relation satisfied by them, we obtain a number of summation identities involving the products of combinations of Fibonacci, Pell and Chebyshev polynomials.

Suppose f is a function possessing a kth order derivative, the derivative being Lipschitz continuous of order α, 0 < α ≤ 1, on [−1, 1]. Let Pn be a polynomial of degree ≤ n approximating to f on [−1,1] such that if rn = f − Pn then ∥rn∥∞ ≤ An−k−a Define

where 0 < β ≤ 1. Upper bounds are obtained for Mn(β) when k ≥ 1 thereby generalizing results previously given for functions which are only Lipschitz continuous on [−1,1].

Let f(z) be an analytic function in the unit circle satisfying the functional equation f(z) = a(z) f(zρ) + b(z), where ρ is a natural number and a(z), b(z) are polynomials. If α is an algebraic number, we give a transcendence measure for f(α). This improves earlier results of Galochkin and Miller.

A central theorem in the theory of graphic sequences is due to P. Erdos and T. Gallai. Here, we give a simple proof of this theorem by induction on the sum of the sequence.

It is shown that if R is a semi prime ring in which (xy)2 -xy is central for every x, y ε R, then R is commutative.

It is shown that in the Fitting class of all finite p-by-q groups, where p and q are different primes, there is among the sub-direct product closed sub-Fitting classes a unique maximal one: it consists of the groups whose minimal normal subgroups are central.

Let L = L(X) be the free Lie ring of countable rank and let p be prime. Then L(Vp) = L/[(L')p, L] is the relatively free ring for the variety of Lie rings Vp = [Np−1A, E] and Vp is defined by the identity

The purpose of this note is to establish that there exist elements of order p in the additive group of L(Vp). Previously, the existence of p-torsion was proved by Kuz'min for p = 2 only. Similar results were obtained for varieties of groups by Gupta when p = 2 and by Stöhr when p = 3.

Every hermitian hypergroup structure on the set of nonnegative integers can be generated by a family of real-valued continuous functions defined on a compact interval. we characterise such structures in terms of properties of the generating functions.

It is shown that any sequential subgroup of a free topological group is either sequential of order ω1 or discrete. Hence any metrizable subgroup of a free topological group is discrete.

Let Δ denote the Banach algebra of all conservative triangular matrics, M the maximal group of invertible elements of Δ, B the boundary of M and . In this note little Nörlund means are located with respect to the disjoint decomposition M u B u N of Δ in terms of the zeros of the generating power series. Further, corridor matrices of finite type, that is, conservative methods with finitely many convergent diagonals, are located with respect to M u B u N.

In this paper a new family of quotients of the triangle group < x, y, z | x2 = y3 = z7 = xyz = 1 > is obtained. It is shown that for every positive integer m divisible by 3 there is a Hurwitz group of order 504m6 having a centre of size 3, and as a consequence there is a Riemann surface of genus 6m6 + 1 with the maximum possible number of automorphisms.

An elementary new, simple and purely analytic proof of Laguerre's theorem about the zeros of the polar derivative of a polynomial P(z) with respect to the point α, is given. The proof is based on a lemma which is also of independent interest.

In this paper, we prove the existence of a continuous one-to-one correspondence between bounded solutions of the equation x′(t) = Ax(t) + b(t) which belong to a certain subclass L+ of and bounded mild solutions of the equation x′(t) = Ax(t) + b(t) + f(t, x(t)) of the form u(t) = ø(t) + Ψ(t), with ø(t) ∈ L+ and .

For a homogeneous C*-algebra we identify the quotient of the automorphism group by the locally unitary automorphisms as a subgroup of the homeomorphisms of the spectrum. We sharpen a known criterion on the spectrum that ensures that all locally unitary automorphisms of the algebra are inner.