A variety B is called solutionally complete if any system of algebraic equations over an algebra A in B has a solution in A provided it is solvable in B and has at most one solution in any extension of A in B. B is called solutionally compatible if every solvable system of equations over an algebra in B is also solvable over any extension of that algebra. It is shown that solutional compatibility is equivalent with the amalgamation property and that a weaker form of the strong amalgamation property is sufficient but not necessary for equational completeness.

Let k be a complex number and let u be an element of a finite group G. Suppose that u does not belong to O(G), the maximal normal subgroup of G of odd order. It is shown that G satisfies X(1) – X(u) = k for every complex nonprincipal irreducible character X in the principal 2-block of G if and only if G/O(G) is isomorphic either to C2, a cyclic group of order 2, or to PSL (2, 2n), n ≧ 2.

J. F. C. Kingman (1967) proved that if {φn}, n = 1, 2, …, N is any set of Radon-Nikodym derivatives of λn w.r.t. μn, when λn, μn are the restrictions of λ, μ respectively to a sub σ-algebra Gn of G then where bn ≧ 0 ∀n and .

We investigate the problem of the existence of filters with some properties. This leads to a solution of two problems of Ulam concerning σ-fields on the real line.

An integer n >0 is called a CLT-number if any group of order n has subgroups of order d for every divisor d of n. The Set of CLT-numbers n is characterized by properties of the prime factorization of n. In addition, if G has order dividing a CLT-number then the structure of G/CG(U) is given where U is a chief factor of G. As a consequence, it is shown that G is solvable of Fitting height at most 3.

Let R be any ring and α any automorphism of R. We determine here all the automorphisms of the skew polynomial ring R[x, α] which fix R elementwise. We then deal with isomorphisms between different skew polynomial rings whose underlying rings are isomorphic.

In this article, we define the concept of a Malcev ideal in an alternative ring in a manner analogous to Lie ideals in associative rings. By using a result of Kleinfield's we show that a nonassociative alternative ring of characteristic not 2 or 3 is a ring sum of Malcev ideals Z and [R, R] where Z is the center of R and [R, R] is a simple non-Lie Malcev ideal of R. If R is a Cayley algebra over a field F of characteristic 3 then [R, R] is a simple 7 dimensional Lie algebra. A similar result is obtained if R is a simple associative ring.

We show that for any self-consistent sequentially maximal system {μα} of infinite (perhaps non-σ-finite) Radon measures, the projective limit of {μα} exists.

If {Xn: 1 ≦ n < ∞} are independent, identically distributed random variables having E(X1) = 0 and Var(X1) = 1, the most elementary form of the central limit theorem implies that P(n-½Sn≧ zn) → 0 as n → ∞, where Sn = Σnk=1 X,k, for all sequences {zn:1 ≧ n gt; ∞} for which zn → ∞. The probability P(n-½ Sn ≧ zn) is called a “large deviation probability”, and the rate at which it converges to 0 has been the subject of much study. The objective of the present article is to complement earlier results by describing its asymptotic behavior when n-½zn → ∞ as n → ∞, in the case of absolutely continuous random variables having moment-generating functions.

The higher order chain rule for Fréchet and Hadamard differentiable mappings on topological linear spaces is proved and various formulae for (g°f)(n) (x) are given. Leibniz' theorem (in a very general form) is also proved.

It is known that there is a finite affine (or projective) plane of order n if and only if there is a sharply two-transitive set of permutations of degree n. This paper deals with a generalization of this theorem, in which finite sets are replaced by isolated sets, cardinalities by isols and certain effectiveness conditions are imposed on the two systems involved which are trivially satisfied in the finite case.

Subject classification (Amer. Math. Soc. (MOS) 1970): primary 02F40, secondary 05B25.

An approximation method based on a certain Sonnenschein matrix is studied. Results are obtained for approximation in an interval and in the complex plane. A connection between convergence of the approximation process and regularity of the matrix is also discussed.