Let X be a Banach space and let A be a uniformly closed algebra of compact operators on X, containing the finite rank operators. We set up a general framework to discuss the equivalence between Banach space approximation properties and the existence of right approximate identities in A. The appropriate properties require approximation in the dual X* by operators which are adjoints of operators on X. We show that the existence of a bounded right approximate identity implies that of a bounded left approximate identity. We give examples to show that these properties are not equivalent, however. Finally, we discuss the well known result that, if X* has a basis, then X has a shrinking basis. We make some attempts to generalize this to various bounded approximation properties.

In this note we give examples of convex functions whose subdifferentials have unpleasant properties. Particularly, we exhibit a proper lower semicontinuous convex function on a separable Hilbert space such that the graph of its subdifferential is not closed in the product of the norm and bounded weak topologies. We also exhibit a set whose sequential normal cone is not norm closed.

A partial homomorphism ϕ from B into A will be called a fractional homomorphism if dom ϕ ≤ B (A). Every extension of a fractional homomorphism is again fractional, and each fractional homomorphism has a unique irreducible extension.

The relationships between synchronous and asynchronous reversible Markov systems are investigated. It is shown that the invariant measure of such systems is a second order Markov random field. The conditions under which the invariant measure is a first order Markov random field are obtained.

We consider a piecewise monotonie and piecewise continuous map T on the interval. If T has a derivative of bounded variation, we show for an ergodic invariant measure μ with positive Ljapunov exponent λμ that the Hausdorff dimension of μ equals hμ / λμ.

The following question has been asked by J. Dénes [2]: If n - 1 elements of the diagonal of an n × n array are prescribed, is it possible to complete the array to form an n × n latin square?" It is known that if n diagonal elements are given such a completion is not always possible.

That the answer to Dénes' question is yes follows directly from a theorem of M. Hall Jr. [1].

If S is an ordered set we write tp S to denote the order type of S and |5| for the cardinal of S. We also write [S]k for the set {X:X ⊂ S, |X|=k}. The partition symbol

(1)

In (4), pp. 25−27, Coxeter divided the regular maps on a surface of genus 1 into three infinite families. They are:

1. (i) Maps of type {4, 4}.

2. (ii) Maps of type {6, 3}.

3. (iii) Maps of type {3, 6} (the duals of (ii)).

We consider the family (iii). By adjoining an element to the group of any map in (iii) we shall derive the group of a regular map of type {6, 6}. Thus we produce a 1−1 correspondence between the members of the family (iii) and of the new family. Corresponding members in the two families have certain properties in common, the most interesting of which is the property of reflexibility. Our results are summarized in Theorems 1 and 2.

The eigenvalues of a second order self-adjoint elliptic differential operator on Riemannian n-space R will be considered. Our purpose is to obtain asymptotic variational formulae for the eigenvalues under the topological deformations of (i) removing an ɛ -cell (and adjoining an additional boundary condition on the boundary component thereby introduced); and (ii) attaching an ɛ -handle, valid on a half-open interval 0 < ɛ ≤ ɛo. In particular the formulae will exhibit the non-analytic nature of the variation. Similar variational problems for singular ordinary differential operators have been considered by the writer in [3] and [4].

We prove that the elliptic surface ${{y}^{2}}={{x}^{3}}+2\left( {{t}^{8}}+14{{t}^{4}}+1 \right)x+4{{t}^{2}}\left( {{t}^{8}}+6{{t}^{4}}+1 \right)$ has geometric Mordell–Weil rank 15. This completes a list of Kuwata, who gave explicit examples of elliptic $K3$-surfaces with geometric Mordell–Weil ranks 0, 1, … , 14, 16, 17, 18.

In this note we consider some problems related to the following question: What is the smallest integer m(n) for which there exists a family Fn of sets A1, A2,…, Am(n) with the following properties, (i) each member of Fn has n elements and (ii) if S is a set which meets each member of Fn, then S contains at least one member of Fn?

Erdős and Hajnal [1] observed that

and that m(l) = 1, m(2) = 3, m(3) = 7.

In answer to a question raised by Leo Moser, A. Meir proved some years ago that every plane arc of unit length lies in some closed semidisk of radius ½. His elegant, unpublished argument is reproduced here with his kind permission.

Let G(x,s) be the Green's function for the boundary value problem y(n) = 0, Ty = 0, where Ty = 0 represents boundary conditions at two points. The signs of G(x,s) and certain of its partial derivatives with respect to x are determined for two classes of boundary value problems. The results are also carried over to analogous classes of boundary value problems for difference equations.

The action by multiplication of the class of transpositions of the symmetric group on the other conjugacy classes defines a graded poset as described by Birkhoff ([2]). In this paper, the edges of this poset are given a weight and the structure obtained is called the poset of conjugacy classes of the symmetric group. We use weights of chains in the posets to obtain new formulas for the decomposition of products of conjugacy classes of the symmetric group in its group algebra as linear combinations of conjugacy classes and we derive a new identity involving partitions of n.

In [l], Moser derives a recurrence relation and studies the limiting behaviour of the Expectations En of the following game. “A real number is drawn at random from [0,1]. We may either keep the number selected, or reject it and draw again. We can then either keep the second number chosen or reject it, and draw again, and so on. Suppose we have at most n choices. What stopping rule gives the largest En and how can we estimate En?”

We give a characterization of products of projective spaces using unsplit covering families of rational curves.

In 1936 A. M. Turing published his analysis of the notion of effective computability. Very roughly speaking, its object was to distinguish between numbers defined by existential statements and those defined by purely constructive ones. Guided by his schematizing of what goes on when a person calculates using paper and pencil, Turing introduced the concept of an A-machine, eventually to be called a Turing machine. Such a machine consists of a box and a linear tape divided into squares, indefinitely long in both directions. The tape passes through the box and exactly one square of it is always within the box. The system changes only at certain discrete times: t = 0, 1, …. Each square can carry exactly one of the finite set of symbols: s1, s2, …, sn, one of which may be the blank.

This note is concerned with some new integral inequalities which are extensions of the results in [2]. The method by which these results are obtained is due to D. C. Benson [1]. Throughout the present note we shall assume 1<p<∞ and f(x) a non-negative measurable function.

It is shown that the nilpotency of a derivation on a 2-torsion free semiprime ring is always an odd number. Examples are provided to show the necessity of the assumptions.

A new oscillation criterion is given for the delay differential equation ${x}'\,(t)\,+\,p(t)x\,(t\,-\,\text{ }\!\!\tau\!\!\text{ (t)})\,=\,0$, where $p,\,\text{ }\!\!\tau\!\!\text{ }\,\in \,\text{C}\,\text{( }\!\![\!\!\text{ 0,}\,\infty ),\,\text{ }\!\![\!\!\text{ 0,}\,\infty )\text{)}$ and the function $T$ defined by $T(t)\,=\,t-\,\text{ }\!\!\tau\!\!\text{ }\,\text{(t),}\,\text{t}\ge \,\text{0}$ is increasing and such that ${{\lim }_{t\to \infty }}\,T(t)\,=\,\infty $. This criterion concerns the case where $\lim \,{{\inf }_{t\to \infty }}\int _{T(t)}^{t}p(s)ds\le \frac{1}{e}$.