Semi-F-spaces are spaces such that given any pair of disjoint cozero sets, every countable subset of one is completely separated from the other. This generalizes the notion of an F-space and is stronger than the property that every countable subset is C*-embedded. Semi-F-spaces are studied and several examples are given.

The paper characterizes the set of all possible values for the number of lines determined by n points for n sufficiently large. For the lower bound of Kelly and Moser for the number of lines in a configuration with n — k collinear points is shown to be sharp and it is shown that all values between Mmin(k) and Mmax(k) are assumed with the exception of Mmax — 1 and Mmax — 3. Exact expressions are obtained for the lower end of the continuum of values leading down from In particular, the best value of c = 1 is obtained in Erdös’ previous expression for this lower end of the continuum.

Suppose b(t) decreases to 0 on [1, ∞). Define the singular integral operator Cb at periodic f of period 1 in L1 (T),T = ( - 1 / 2, 1/2), by

Then, for a large class of b one has the rearrangement inequality

This inequality is used to construct a rearrangement invariant function space X corresponding to a given such space Y so that Cb maps X into Y.

The enumeration of rooted maps (embedded graphs), by number of edges, on the torus and projective plane, is studied. Explicit expressions for the generating functions are obtained. From these are derived asymptotic expressions and recurrence relations. Numerical tables for the numbers with up to 20 edges are presented.

Generalized boundary value problems are considered for hyperbolic equations of the form utt — uss + λp(s, t)u = 0. By constructing symmetric Green's functions appropriate to such problems the existence of eigenvalues is established.

R. A. Brualdi [1] presents a construction yielding matrices whose Birkhoff representation consists of the maximum number of permutation matrices and having 0(2n2) line sum. In this note a counterexample to such a construction is given. Furthermore, a new construction is presented, yielding matrices with lower line sums.

Anderson and Ohm have introduced valuations of monoid rings k[Γ] where k is a field and Γ a cancellative torsion-free commutative monoid. We study the residue class fields in question and solve a problem concerning the pure transcendence of the residue fields.

Let X and Y be metrizable compact spaces and μ and v be nonzero continuous measures on X and Y, respectively. Then there is no bounded operator from the space of bimeasures BM(X, Y) onto the closed subspace of BM(X, Y) generated by L1 (μ X v); in particular, if X and Fare nondiscrete locally compact groups, then there is no bounded projection from BM(X, Y) onto the closed subspace of BM(X, Y) generated by L1(X X Y).

In this paper we prove the following result: let R be a prime ring with no non-zero nil left ideals whose characteristic is different from 2 and let U be a non central Lie ideal of R.

If d ≠ 0 is a derivation of R such that d(u) is invertible or nilpotent for all u ∈ U, then either R is a division ring or R is the 2 X 2 matrices over a division ring. Moreover in the last case if the division ring is non commutative, then d is an inner derivation of R.

A Σ-group is an abelian group on which is given a collection of infinite sums having properties suggested by those of absolutely convergent series in R or C. It is shown that the usual decomposition of a torsion abelian group into its p-components carries over to the case of Σ-groups when the property of being torsion is replaced by an appropriate uniform version. For a certain class of Σ-groups, it turns out that being torsion is already sufficient for primary decomposition to hold.

We prove that the conjugate function operator is bounded in Lp(Г, wds), 1 < p < ∞, if and only if w ∈ Ap(Г), where Г is a quasiregular curve.

In this paper some cardinal inequalities for Urysohn spaces are established. In particular the following two theorems are proved:

(i)If where [A]θ denotes the θ-closed hull of A, i.e., the smallest θ-closed subset of X containing A;

(ii), where aL(X, X) is the smallest cardinal number m such that for every open cover of X there is a subfamily for which

Let ﹛zn﹜ be a sequence in the open unit disc and write In the case of pn for all n, the interpolation problems are considered.

Let be the category of all homomorphisms (i.e. functions preserving satisfaction of atomic formulas) between models of a set of sentences T in a finitary first-order language L. Functors between two such categories are said to be canonical if they commute with the forgetful functors. The following properties are characterized syntactically and also in terms of closure of for some algebraic constructions (involving products, equalizers, factorizations and kernel pairs): There is a canonical isomorphism from to a variety (resp. quasivariety) in a finitary expansion of L which assigns to a model its (unique) expansion. This solves a problem of H. Volger.

In the case of a purely algebraic language, the properties are equivalent to:“ is canonically isomorphic to a finitary variety (resp. quasivariety)” and, for the variety case, to “the forgetful functor of is monadic (tripleable)”.

A collection of measure preserving mappings having Bernoulli generators is considered. Only three conditions are required to be satisfied, and they are quite easy to check.

The Sobolev inequality of order m asserts that if p ≧ 1, mp < n and 1/q = 1/p — m/n, then the Lq-norm of a smooth function with compact support in Rn is bounded by a constant times the sum of the Lp-norms of the partial derivatives of order m of that function. In this paper we show that that sum may be reduced to include only the completely mixed partial derivatives or order m, and in some circumstances even fewer partial derivatives.

A field F satisfies n-linkage on a subset of Ḟ if whenever the quaternion algebras

are equal in Br(F) there exist z ∈ Ḟ with

for i = 1, 2, . . ., n. This linkage of quaternion algebras is examined and its relationship to the torsion freeness of I2(F) and to the strong approximation property is investigated.

Let n = 4t +- 2, where the integer t ≧ 2. A necessary condition is given for a particular Latin square L of order n to have a complete set of n — 2 mutually orthogonal Latin squares, each orthogonal to L. This condition extends constraints due to Mann concerning the existence of a Latin square orthogonal to a given Latin square.

An estimate for Kloosterman sums with oscillating coefficients is presented. Precisely we show: for any ϵ > 0 and a, b positive integers with (a, b) — 1 we have,

Similar techniques may be used to estimate other Kloosterman sums with oscillating coefficients which are not smooth.