The statement from the title is proved.

Solutions of the differential equation

for large positive values of the parameter u, are considered for ζ in some domain Δ which includes the turning-point ζ = 0. The functions ψ(ζ) and ω(ζ) are holomorphic for ζ є Δ

In typical linear programming problems, we are concerned with finding non-negative integers {x1,…, xn} that maximize a linear form c1x1 + … + cnxn, subject to a number of linear inequalities, for The maximum is necessarily attained at one of the vertices of the convex hull of integer points defined by the inequalities, so we have an interest in estimating the number M of these vertices. We give two results; one improving an upper bound result for M of Hayes and Larman concerning the Knapsack polytope, the other an example showing that, in 3-dimensions, it is possible to choose the coefficients aij to obtain a lower bound for M.

I investigate what can be said about a set E in a probability space X when the “square” E x E can be covered by the squares of stochastically independent sets of given measure.

There are two principal ways of decomposing knots and links into simpler ones: (1) a sphere intersecting the knot in two points gives a connected sum decomposition; (2) an incompressible torus in the knot complement gives a satellite decomposition. If a knot K is such that in every connected sum decomposition one of the factors is an unknotted arc spanning the sphere then K is called a prime knot. In [L] Raymond Lickorish explored the possibility of using 2-string tangles to construct and detect prime knots. He defined prime tangles and showed that the sum of prime tangles is always a prime knot or link. Later, Quach Cam Van studied partial sums of tangles and gave necessary and sufficient conditions for the resulting tangle to be prime. In this paper, similar results are established which relate to the satellite decomposition rather than to the connected sum.

Fifty years ago Marcinkiewicz and Zygmund studied the circular structure of the limit points of the partial sums for (C, 1) summable Taylor series. More specifically, let

be a power series with complex coefficients, let

be the partial sums, and let

be the Cesàro averages. When the sequence σn(z) converges to a finite limit σ(Z), we say that the Taylor series is (C, 1) summable and σ(z) is the (C, 1) sum of the series. Concerning (C, 1) summable Taylor series Marcinkiewicz and Zygmund ([5], [6] Vol. II, p. 178) established the following theorem.

The main result of this paper is the following theorem. If P is a convex polytope of Ed with affine symmetry, then P can be illuminated by eight (d - 3)-dimensional affine subspaces (two (d- 2)-dimensional affine subspaces, resp.) lying outside P, where d ≥ 3. For d = 3 this proves Hadwiger's conjecture for symmetric convex polyhedra namely, it shows that any convex polyhedron with affine symmetry can be covered by eight smaller homothetic polyhedra. The cornerstone of the proof is a general separation method.

Introduction. Throughout the paper K(x) is a simple transcendental extension of a field K; v is a valuation of K and w is an extension of v to K(x). Also koÍk and GoÍG denote respectively the residue fields and the value groups of the valuations v and w. A well-known theorem conjectured by Nagata asserts that either k is an algebraic extension of feo or k is a simple transcendental extension of a finite extension of ko (cf [4] or [6] or [1, Corollary 2.3]). We prove here an analogous result for the value groups viz. either G/ Go is a torsion group or there exists a subgroup G1 of G containing Go with [G1: Go] > ∞ such that G is the direct sum of G1 and an infinite cyclic group. Incidentally we obtain a description of the valuation w as well as of its residue field in the second case. Thus a characterization of all those extensions w of v to K(x), for which w(K(x)\{0})/Go is not a torsion group, is given. Corresponding to such a valuation w, we define three numbers N, S and T which satisfy the inequality N ≥= ST. This is analogous to the fundamental inequality established by Ohm (cf. [5, 1.2]) for residually transcendental extensions of v to K(x). We also investigate the conditions under which N = ST

All manifolds in this paper are assumed to be closed, oriented and smooth.

A contact structure on a (2n + l)-dimensional manifold M is a maximally non-integrable hyperplane distribution D in the tangent bundle TM, i.e., D is locally denned as the kernel of a 1-form α satisfying α ۸ (da)n ۸ 0. A global form satisfying this condition is called a contact form. In the situations we are dealing with, every contact structure will be given by a contact form (see [5]). A manifold admitting a contact structure is called a contact manifold.

A. Bezdek and W. Kuperberg constructed a nonlattice packing of congruent ellipsoids in Euclidean 3-space E3 with density 0·7459 …, which exceeds the density σL2 = 0·74048… of the densest lattice packing of spheres and hence of ellipsoids in E3. G. Kuperberg improved this to 0·7533… We improve this slightly to 0·7549…. In our case the quotient of the largest and the smallest halfaxis of the ellipsoids is <42, so the ellipsoids are not too degenerate. If one combines G. Kuperberg's refinement and ours, one obtains a packing density of 0·7585…

The n-dimensional cross polytope, |x|+|x2|+…+|xn≤1, can be lattice packed with density δ satisfying

but proofs of this, such as the Minkowski-Hlawka theorem, do not actually provide such packings. That is, they are nonconstructive. Here we exhibit lattice packings whose density satisfies only

but by a highly constructive method. These are the densest constructive lattice packings of cross polytopes obtained so far.

Let J = (s1, s2, … ) be a collection of relatively prime integers, and suppose that π(n) = |J∩{1,2,…, n}| is a regularly varying function with index a satisfying 0 < α < l. We investigate the “stationary random sieve” generated by J, proving that the number of integers less than k which escape the action of the sieve has a probability mass function with approximate order k-α/2 in the limit as k → ∞. This result may be used to deduce certain asymptotic properties of the set of integers which are divisible by no s є J, in that it gives new information about the usual deterministic (that is, non-random) sieve. This work extends previous results valid when si=pi2, the square of the ith prime.