The two-dimensional flow around an infinite cylinder at low Reynolds number has interested fluid dynamicists for many years. In this paper it is shown that an infinite cylinder can propel itself through a viscous fluid (for example micro-organisms) if it has certain undulations on its surface.

By considering the concept of weak minima, different scalar duality results are extended to multiple objective programming problems. A number of weak, strong and converse duality theorems are given under a variety of generalised convexity conditions.

The concept of invexity is extended to nondifferentiable functions. Characterisations of nonsmooth invexity are derived as well as results in unconstrained and constrained optimisation and duality. The principal analytic tool is the generalised gradient of Clarke for Lipschitz functions.

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.

Dependence structures (in the finite case, matroids) arise when one tries to abstract the properties of linear dependence of vectors in a vector space. With the help of a theorem due to P. Hall and M. Hall, Jr concerning systems of distinct representatives of families of finite sets, it is proved that if B1 and B2 are bases of a dependence structure, then there is an injection σ: B1 → B2 such that (B2 / {σ(e)}) ∩ {e} is a basis for all e in B1. A corollary is the theorem of R. Rado that all bases have the same cardinal number. In particular, it applies to bases of a vector space. Also proved is the fact that if B1 and B2 are bases of a dependence structure then given e in B1 there is an f in B2 such that both (B1 / {e}) ∩ {f} and (B2 / {f}) ∩ {e} are bases. This is a symmetrical kind of replacement theorem.

For a constrained minimization problem with cone constraints, lagrangean necessary conditions for a minimum are well known, but are subject to certain hypotheses concerning cones. These hypotheses are now substantially weakened, but a counter example shows that they cannot be omitted altogether. The theorem extends to minimization in a partially ordered vector space, and to a weaker kind of critical point (a quasimin) than a local minimum. Such critical points are related to Kuhn-Tucker conditions, assuming a constraint qualification; in certain circumstances, relevant to optimal control, such a critical point must be a minimum. Using these generalized critical points, a theorem analogous to duality is proved, but neither assuming convexity, nor implying weak duality.

New inequalities for the general case of convex functions defined on linear spaces which improve the famous Jensen's inequality are established. Particular instances in the case of normed spaces and for complex and real n-tuples are given. Refinements of Shannon's inequality and the positivity of Kullback-Leibler divergence are obtained.

A Lie algebra over a field of characteristic 0 splits over its soluble radical and all complements are conjugate. I show that the splitting theorem extends to Leibniz algebras but that the conjugacy theorem does not.

Permutation polynomials of the form xτf (x3) over a finite field give rise to group permutation polynomials. We give a group theoretic criterion and some other criteria in terms of symmetric functions and power functions.

In this paper, we announce results on the Dirichlet and periodic boundary-value problems for the equation -x″(t) = g(x(t)) – f(t) on [0, π]. We consider degenerate cases not covered by the author's earlier work.

With a view to further applications, we give a self-contained account of indexed limits for 2-categories, including necessary and sufficient conditions for 2-categorical completeness. Many important 2-categories fail to be complete but do admit a wide class of limits. Accordingly, we introduce a variety of particular 2-categorical limits of practical importance, and show that certain of these suffice for the existence of indexed lax- and pseudo-limits. Other important 2-categories fail to admit even pseudo-limits, but do admit the weaker bilimits; we end by discussing these.

In this work, we give some theorems on (mild) weighted pseudo-almost periodic solutions for some abstract semilinear differential equations with uniform continuity. To facilitate this we give a new composition theorem of weighted pseudo-almost periodic functions. Our composition theorem improves the known one by making use of a uniform continuity condition instead of the Lipschitz condition.

The purpose of this paper is to present a brief proof of the well known result of Herstein which states that any Jordan derivation on a prime ring with characteristic not two is a derivation.

Given a closed and convex set K in Rn and two continuous maps F: K → Rn and η: K × K → Rn, the problem considered here is to find ε K such that

.

We call it a variational-like inequality problem, and develop a theory for the existence of a solution. We also show the relationship between the variational-like inequality problem and some mathematical programming problems.

In this note we present a new result which is equivalent to the celebrated Ekeland's variational principle, and a set of implications which includes a new non-convex minimisation principle due to Takahashi.

We shall discuss boundedness and compactness of the products of composition and differentiation between Hardy spaces.

In this paper, we first prove an improved version of the selection theorem of Yannelis-Prabhakar and next prove a fixed point theorem in a non-compact product space. As applications, an intersection theorem and two equilibrium existence theorems for a non-compact abstract economy are given.

In this note, we prove certain inequalities for elementary symmetric funtions that are relevant to the study of partial differential equations associated with curvature problems.

We prove that the solution set of a generalized differential equation is connected and that points on the boundary of the solution funnel are peripherally attainable. This is done without the additional assumption of continuity in the state variable required in previous results. The result on upper semicontinuity of the solution set with respect to initial conditions is extended to include variations of initial time.

An approach to analysing the family of Cayley graphs for a finite group G is given which identifies normal edge-transitive Cayley graphs as a sub-family of central importance. These are the Cayley graphs for G for which a subgroup of automorphisms exists which both normalises G and acts transitively on edges. It is shown that, for a nontrivial group G, each normal edge-transitive Cayley graph for G has at least one homomorphic image which is a normal edge-transitive Cayley graph for a characteristically simple quotient group of G. Moreover, given a normal edge-transitive Cayley graph ΓH for a quotient group G/H, necessary and sufficient conditions are obtained for the existence of a normal edge-transitive Cayley graph Γ for G which has ΓH as a homomorphic image, and a method for obtaining all such graphs Γ is given.