An operational category is a category of models for an equational theory where the interpretation of some operations is predetermined. Examples include the equational and co-equational categories of Linton, categories of functors preserving some class of limits, and algebras for a prop as defined by MacLane. The chief result is a characterisation of the operational categories and functors in terms of their internal structure.

This paper shows the existence of an infinite family of cyclic balanced ternary designs where the block size is 4, the index 2 and each block contains precisely one repeated element.

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].

Recently in this journal we have shown the similarity between the Kleinecke-Shirokov theorem for subnormal operators and the Fuglede-putnam theorem. The purpose of this paper is to show that this similarity can be generalized to operators which belong to some classes of non-normal operators wider than the class of subnormal operators.

A symmetry of an ordered Banach space is an order and norm isomorphism which commutes with its ideal centre. A class of ordered Banach spaces is introduced to show that, for a space in this class, the group of symmetries is trivial if and only if the space is lattice-ordered. When this group becomes larger, the space approaches an antilattice. This phenomenon is also investigated.

The purpose of this note is to give extensions of an inequality of Jensen for analytic functions in an unit disc. We investigate functions which satisfy equalities in the inequalities.

We extend three well-known facts of Fourier series on the disc to Fourier series on the torus, a theorem of Riesz, a theorem of Szegö, and the fact that any function in H1 can be factored as the product of two functions in H2. Here the rôle of negative integers is played by the lattice points in the third quadrant. In earlier extensions of these theorems this rôle was played by half-planes.

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.

We give a complete description of holomorphic curves in the complex two quadric via the method of moving frames. For compact curves a Morse theory type integral formula is derived.

Let 〈an〉 be an increasing sequence of real numbers and 〈bn a sequence of positive real numbers. We deal here with the Dirichlet series and its Laurent expansion at the abscissa of convergence, λ, say. When an and bn behave like

as N → ∞, where P2(x) is a certain polynomial, we obtain the Laurent expansion of f (s) at s = λ, namely

where P1(x) is a polynomial connected with P2(x) above. Also, the connection between P1 and P2 is made intuitively transparent in the proof.

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].

It is proved that a ring satisfying the variable indentity [[xn, y], y] = 0, n = n (x,y) ≥ 1, has nil commutator ideal.

A review of the development of estimates for Lebesgue constants associated with Lagrange interpolation on the one hand, and estimates for the rate of convergence of Hermite-Fejér interpolation on the other hand, provides a historical perspective for the following surprising, close link between these apparently diverse concepts. Denoting by Λn (T) the Lebesgue constant of order n and by Δn (T) the maximum interpolation error for functions of class Lip 1 by Hexmite-Fejér interpolation polynomials of degree not exceeding 2n − 1, based on the zeros of the Chebyshev polynomial of first kind, we discover that, for even values of n, Λn(T) = n Δn(T).

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.

For an arbitrary field K there are two related questions that can be asked:

(1) Is there a proper subfield, L, of K such that K is countably generated over L?

(2) Given a proper subfield M of K is there a proper subfield, L, of K containing M such that K is countably generated over L?

We give an affirmative answer to (1) in characteristic p ≠ 0 and provide counterexamples to (2) for arbitrary characteristic ≠ 2.

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.

Recently, Visintin gave conditions under which weak convergence in L1 (T;RN) implies strong convergence. Here we analyze such results in terms of associated young measures and present an extension to L1 (T;Ξ), where Ξ is a separable reflexive Banach space.

It is well-known that if W is a set of costant width λ then the circumsphere and the insphere of W are concentric and their radii sum to λ. Here this fact is generalized to sets of constant relative width and it is shown that the result does not depend no W being of constant width, but rather on W satisfying the spherical intersection property; that is, W = ∩{ω + λB:ω ɛ W}.

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.