In this paper we construct the prolongation of a linear connection Γ on a manifold Μ to the bundle space of its frame bundle, and show that such prolongated connection coincides with the so-called complete lift of Γ to .

For a given ground ring L, let dimL(X) denote the sheaf theoretic cohomological dimension of a space X. In this paper we prove an inverse limit theorem for this dimension function. Then we apply this theorem to show that for a large classs of rings the Freudenthal's expansion theorem, expressing a compact metric space X of dimL(X) as the inverse limit of an system of compact polyhedra Kα of dimL(Kα), is not valid.

The purpose of this paper is to introduce two new classes of locally convex spaces which contain the classes of semi-Montel and Montel spaces. Further we give some examples and study some properties of these classes. As to permanence properties, these classes have similar properties to semi-Montel and Montel spaces except strict inductive limits and these classes are not always preserved under their completions. We shall call these two classes as β–semi–Montel and β–Montel spaces. A β–semi–Montel space is obtained by replacing the word “bounded” by “strongly bounded” in the definition of a semi–Montel space. If a β–semi–Montel space is infra-barrelled, we call the space β–Montel.

A classification is begun of varieties of algebras with the property that each algebra in the variety has a modular lattice of subalgebras. This turns out to be a very restrictive condition. Such a variety is hamiltonian. If the algebras in it are idempotent, then it is a variety of sets. A variety is subalgebra-modular if and only if it is hamiltonian and satisfies certain conditions on the terms in its three generator free algebras.

The simple twills on n harnesses can be classified according to their maximum float length. The number of n-harness twills with specified maximum float length is determined both by Burnside enumeration and, for n ≤ 20 , by an adaptation of a sieve algorithm for twills.

In this paper we obtain some classical results by using the general integral operators which transform Jakubowski's class K(m, M) into itself and K(μ) × S(m, M) into K(μ). Our results generalize some recent known results due to Causey and Reade, Patil and Thakare.

The degree of disjunction, δ(F), of a family F of functions is the least cardinal τ such that every pair of functions in F agree on a set of cardinality less than τ.

Suppose θ, μ, λ, κ are non-zero cardinals with θ ≤ μ ≤ λ. This paper is concerned with functions which map μ-sized subsets of λ into κ. We first show there is always a ‘large’ family F of such functions satisfying δ(F) ≤ θ. Next we determine the cardinalities of families F of such functions that are maximal with respect to δ(F) ≤ θ.

Let n(w, f) be the number of w-points of f meromorphic in D = {|z| < 1}. Beurling defined the quantity and called w an ordinary value of f if < ∞. We shall consider the intermediate quantity ṉ(w, f) in the sense that n(w, f) ≤ ṉ(w, f) , and construct two bounded holo-morphic functions f1 and f2 of finite Dirichlet integrals in D for which

and

.

We construct a connected subspace M of the euclidean plane R2 containing two points A and B such that, for every pair of points {P, Q} of M\{A, B}, the three real numbers d(A, P), d(P, Q) and d(Q, B) are not the same. This solves a question posed by Väisälä.

We prove the local triviality of module bundles over semisimple Lie algebra bundles and using this result we establish the complete reducibility of module bundles over semisimple Lie algebra bundles.

This paper deals with approximation of certain operators defined on the space C(X) of real-valued continuous functions on an arbitrary compact metric space (X, d). In particular the problem of giving quantitative Korovkin type theorems for approximation by positive linear operators is solved. This is achieved by using a smoothing approach and the least concave majorant of the modulus of continuity of a function f in C(X). Several new estimates are given as applications, including such for Shepard's method of metric interpolation.

Every finite group G satisfies a law [x, ry] = [x, sy] for some positive integers r < s. The minimal value of r is called the depth of G. It is well known that groups of depth 1 are abelian. In this paper we prove the following. Let G be a finite group of depth 2. Then G/F(G) is supersoluble, metabelian and has abelian Sylow p-subgroups for all odd primes p. Moreover, lp(G) ≤ 1 for p odd and l2(G2) ≤ 1.

The authors determine the sharp radius of convexity for functions analytic in the unit disc having power series representation of the form f(z) = z + an+1zn+1 + an+2zn+2 + … where an+1 is fixed and such that zf′(z)/f(x) = (1 + Aw(z))/(1 + Bw(z)), −1 ≤ B < 0 < A ≤ 1 where w(z) is an analytic function satisfying the conditions of Schwarz's lemma, in the case A + B ≥ 0. The estimate obtained is an improvement over the corresponding result obtained by Mogra and Juneja for functions analytic and starlike in the unit disc, with missing coefficients where the initial non-vanishing coefficient is fixed.

A new set of polynomials in powers of xh (h > 0) which are related to the Laguerre polynomials , have been defined by a Rodrigues type formula involving the forward difference operator Δh of increment h. Besides obtaining shift operator formulae for , a definition of fractional power of Δh has been given. The definition has been illustrated by establishing known relations of the confluent hypergeometric function.

In the present note we prove some results for p-valent functions with initial gaps by means of N-subordination.