Let = A0u + μA1u + J (u) be a Navier-Stokes parameterized evolution equation in a Hilbert space H and let F1 ⊂ F2 ⊂ F3 ⊂ … be an increasing sequence of finite dimensional spaces such that every Fn ⊕ ℝ contains the center-unstable linear subspace Hu ⊕ ℝ ⊂ H ⊕ ℝ of the system = A0u + μA1u + J (u), = 0. Then each Fn ⊕ ℝ determines a Galerkin approximant of the original system, with the same center-unstable linear subspace Hu ⊕ ℝ The flow on the center-unstable manifold of the original system may be identified with a parameterized flow on Hu given by x = f∞ (x,μ). The flow on the center-unstable manifold of the Galerkin approximant determined by Fn ⊕ ℝ may be identified with a parameterized flow on Hu given by ẋ = fn (x,μ). It is proved that Theorem I holds: in the Cktopology on a compact neighborhood of the origin in Hu ⊕ ℝ From this theorem it is concluded that Theorem 2 holds: If a certain priori bound holds relating f∞ and fn and an asymptotically stable set A of ẋ = fn (x,μ) near the origin, then ẋ = f∞ (x,μ) has an asymptotically stable set near the origin with the same Borsuk shape as A. Conversely, for each asymptotically stable set near the origin of ẋ = f∞(x,μ), there is one of the same Borsuk shape for ẋ = fn (x,μ) provided n is large enough. Informally, these results amount to the statement that asymptotically stable sets of the Navier-stokes equation, bifurcating from a steady solution, are recovered up to Borsuk shape by those of large enough Galerkin approximants.

The paper classifies all distance transitive graphs Γ such that An ≤ Aut Γ ≤ Aut An for some alternating group An, and Aut Γ acts primitively on the vertices of Γ. This result forms part of our programme for determining all finite primitive distance transitive graphs.

In this paper we study the following problem: To what extent does the type of the Gauss map of a submanifold of Em determine the submanifold? Several results in this respect are obtained. In particular, submanifolds with 1-type Gauss map are characterized. Surfaces with 1-type Gauss map and minimal surfaces of Sm−1 with 2-type Gauss map are completely classified. Some applications are also given.

For each λ ≥ 1 a class of Banach spaces φλ is defined. Isometric results are obtained on the equivalence between a measure of non-compactness and the essential norm of a linear operator defined on a φλ space. Best values of λ for the classical Banach spaces and for spaces with unconditional basis are investigated. For the space c of convergent sequences the non-existence of a λ-unconditional basis with λ < 2 is deduced.

Let A and B be unital JB-algebras. We study the extremal structure of the convex set S (A,B) of all identity preserving positive linear maps from A to B. We show that every unital Jordan homomorphism from A to B is an extreme point of S (A,B). An extreme point of S (A,B) need not be a homomorphism and we show that, given A, every extreme point of S (A,B) is a homomorphism for any B if, and only if, dim A ≤ 2. We also determine when S (A,B) is a simplex.

Sargent and later Bullen and Mukhopadhyay obtained a definition of absolutely continuous functions, functions, that is related to kth Peano derivatives. The generalised notions of ACkG*, [ACkG*], ACkG* above, etcetera functions led Bullen and Mukhopadhyay to define certain general integrals of the kth order.

The present work is concerned with a further simplification of the definitions of such functions by the use of divided differences but still retaining similar fundamental properties. These concepts lead to the introduction of Denjoy and Ridder type integrals which are shown to be equivalent to a Perron type integral that corresponds to kth Riemann* derivatives. All three of these integrals are shown to be equivalent to the three integrals of Bullen and Mukhopadhyay.

Let H∞ be the Banach algebra of all bounded analytic functions in the unit disc. We present a complete description of the closed primary (respectively prime) ideals contained in a maximal ideal of the Shilov boundary of H∞. The paper is also concerned with chains of prime ideals in H∞.

A consequence of the main proposition includes results of Tacon, and John and Zizler and says: If a Banach space X possesses a continuous Gâteaux differentiable function with bounded nonempty support and with norm-weak continuous derivative, then its dual X* admits a projectional resolution of the identity and a continuous linear one-to-one mapping into c0 (Γ). The proof is easy and selfcontained and does not use any complicated geometrical lemma. If the space X is in addition weakly countably determined, then X* has an equivalent dual locally uniformly rotund norm. It is also shown that l∞ admits no continuous Gâteaux differentiable function with bounded nonempty support.

R.S.L. Srivastava and O.P. Juneja (1967) proved an existence theorem for the proximate type T (r) of an entire function with classical growth. For an interesting generalization of this theorem for an entire function with index-pair (p,q), which is due essentially to H.S. Kasana and S.K. Vaish (1984), a remarkably simple (and markedly different) construction of T (r) is presented here. The main theorem established here applies to a much larger class of entire functions with index-pair (p,q) than that considered earlier.

It is a classical result, deriving from the Gaussian theory of genera of integral binary quadratic forms, that there exist only finitely many imaginary quadratic fields for which the ideal class group is a group of exponent two. This finiteness has been shown to extend to all those totally imaginary quadratic extensions of any fixed totally real algebraic number field. In this paper we put forward the conjecture that there exist only finitely many imaginary abelian algebraic number fields which have ideal class groups of exponent two, and we examine the extent to which existing methods can be brought to bear on this conjecture. One consequence of the validity of the conjecture would be a proof of the existence of finite abelian groups which do not occur as the ideal class group of any imaginary abelian field.

This paper shows that the smallest size of a set for which a finite fundamental inverse semigroup can be faithfully represented by partial transformations of that set is the number of join irreducible elements of its semilattice of idempotents.

The Weierstrass elliptic function satisfies a nonlinear first order and a nonlinear second order differential equation. It is shown that these differential equations can be discretized in such a way that the solutions of the resulting difference equations exactly coincide with the corresponding values of the elliptic function.

Let P (z) be a polynomial of degree n which does not vanish in the disk |z| < K. For K = 1, it is known that

In this paper we consider the two cases K ≥ 1 and K < 1, and present certain generalizations of these results.

This paper concerns the problem of description of the set of rings containing a given ring as an essential ideal. The results obtained are applied to some problems of ring theory and radicals.

Lower bounds for the tau coefficients and operator norms are derived by using composite matrix norms. For a special class of matrices B, our bounds on ‖B‖p (the operator norm of B induced by the ℓp norm) improve upon a general class of Maitre (1967) bounds for p ≥ 2.

In this paper we establish sufficient or necessary conditions for the nonlinear differential equation u″ + f (t, u) = 0 to have solutions which are asymptotic to lines with non-zero slopes and correct some formulations in theorems obtained by D.S. Cohen and J. Tong.

In this paper we obtain some inequalities for some regular functions f defined on the unit disc. Our results include or improve several previous results.

In this paper we introduce regular topological distributively generated (d.g.) near-rings (distinct from d.g. regular near-rings) as the d.g. near-ring analogue of regular rings and develop a structure theory for this class of near-rings.

The definition of division d.g. near-rings given by the author in 1976 has been found to be too restrictive. In this paper we generalise the definition of division d.g. near-rings and extend the results obtained earlier for division d.g. near-rings to the new wider class of near-rings.

We show that every compact convex set in a Banach space X is an intersection of balls provided the cone generated by the set of all extreme points of the dual unit ball of X* is dense in X* in the topology of uniform convergence on compact sets in X. This allows us to renorm every Banach space with transfinite Schauder basis by a norm which shares the mentioned intersection property.