If is a variety of groups which are nilpotent of class c then is generated by its free group of rank c. It is proved that under certain general conditions cannot be generated by its free group of rank c - 2, and that under certain other conditions is generated by its free group of rank c - 1. It follows from these results that if is the variety of all groups which are nilpotent of class c, then the least value of k such that the free group of of rank k generates is c - 1. This extends known results of L.G. Kovács, M.F. Newman, P.P. Pentony (1969) and F. Levin (1970).

Some of the results in the theory of Möbius functions of finite partially ordered sets are extended to arbitrary non-singular binary relations on finite sets.

It is of relevance to studies in the logic of quantum mechanics whether or not every separable completely orthomodular poset admits a normed σ-ortho-valuation. A finite orthomodular poset is constructed which is a counter-example to this proposition.

This note studies cardinal numbers κ which have a partition property which amounts to the following. Let ν be a cardinal, η an ordinal limit number and m a positive integer. Let the m-length sequences of finite subsets of κ be partitioned into ν parts. Then there is a sequence H1, … Hm of subsets of κ, each having order type η, such that for each choice of non-zero numbers n1, …, nm there is some class of the partition inside which fall all sequences having in their i-th place (for i = 1, …, m) a subset of Hi which contains exactly ni elements. The case when m = 1 is thus seen to be the well known property κ . The most interesting results obtained relate to the ordering of the least cardinals with the appropriate properties as m and η vary.

In 1951 Mohanty established the following theorem.

If Φ(t)log is of bounded variation in (0, π), wherek ≥ πe2and, then is summable, for however large positive Δ. In this present note we have generalised the above theorem by taking a more general type of Riesz means and under the condition, is of bounded variation in (0, π), where c is finite, imposed upon the generating function of Fourier series.

By generalizing a technique of Landau, the authors prove that the excess of the number of primes of the form 10x ± 3 over the number of primes of the form 10x ± 1 is infinite.

The problem of unicity in the uniform approximation of vector-valued functions is considered. A recent result of Cheney and Wulbert concerning unicity in the uniform approximation of real-valued functions is extended and some “point-wise” criteria for unicity are given.

This paper examines a large class of common numerical methods for computing the eigenvectors of a compact linear operator. Special cases include all projection methods and the Weinstein intermediate problems method. Simple sufficient conditions are established for the sequence of approximate eigenvectors obtained by any of these methods to converge to an exact eigenvector. In the most general case only the convergence of a subsequence was proviously known.

One of the more important concepts in the study of universal algebras is that of a free algebra. It is our purpose in this communication to describe the structure of the free algebra Kk() of k generators (k a positive integer) determined by a categorical algebra, and to indicate how this information encompasses results in such diverse areas as the study of Post algebras, boolean rings, p-rings, pk-rings, finite commutative rings with unity, etc.

A finite algebra is called categorical if every algebra in its equational class is isomorphic to a sub-direct power of A. If has n elements, permutable identities, no non-identical automorphism and exactly m distinct one-element subalgebras, then .

This paper contains two theorems. The first theorem treats the |R, r, l| summability of Fourier series and their associated series of functions of bounded variation. The second concerns the |R, r, l| summability of Fourier series of functions f such that φ(t)m(l/t) is of bounded variation where m(u) increases to infinity as u → ∞. These theorems generalize Mohanty's theorems.

We consider varieties with m prime to p. We show that the subvariety lattice of distributive and has descending chain condition and that is its only just non-Cross subvariety. When m is prime we determine the join-irreducible subvarieties of . The method involves fairly detailed description of the structure of non-nilpotent critical groups in .

An existence theorem is given for differential equations which fail the simple Lipschitz condition on some surfaces but whose solution curve is not tangential to any of these surfaces.

In this note we show that if H is any subgroup of the finite group G and if D is a normal subgroup of H such that H/D is soluble and the order of H/D is relatively prime to the index of B in G then the existence of a normal subgroup N of G such that NH = G and N ∩ H is contained in D is equivalent to the condition that every irreducible character of H/D can be extended to one of G. This is a generalization of a result due to Sah for the case when D is the identity subgroup.

Let G be a finite group with a nilpotent maximal subgroup S and let P denote the 2-Sylow subgroup of S. It is shown that if P ∩ Q is a normal subgroup of P for any 2-Sylow subgroup Q of G, then G is solvable.

Let {Xn} be a sequence of independent identically distributed random variables and let The rate of convergence of probabilities , where 2 > r > 1, is studied.

The results in the first parts of Theorems 2 and 3 of the paper in the title (see [2]) have been previously obtained by B.M. Schein in Theorem 1.12, page 299 [4], and in Proposition 1.13 (combined with the last paragraph of page 300) [4], respectively. To deduce the first part of Theorem 2 [2] from Theorem 1.12 [4] one merely uses the fact that a binary relation R on a set X satisfies RR−1R ⊆ R if if and only if it satisfies: R{x) ∩ R(y) ≠ □ implies R(x) = R(y), for any x, y ∈ X (see Proposition 9, page 132 [3]).