Throughout this paper, p will denote a prime and t an integer greater than 1. We consider those finite p-groups P, which satisfy the following conditions:

(1.1) P has subgroups R and Q, both of index p, and there is an isomorphism φ of R onto Q which does not fix any non-identity subgroup of R setwise.

Suppose X is a Banach space and T a continuous linear operator on X. The significance of the asymptotic convergence of T for the approximate solution of the equation (I - T)x = f by means of the Picard iterations was clearly shown in Browder's and Petryshyn's paper [1], The results of [1] have stimulated further investigation of the Picard, and more generally, averaging iterations for the solution of linear and nonlinear functional equations [2; 3; 4; 8; 9]. Kwon and Redheffer [8] analyzed the Picard iteration under the mildest possible condition on T, namely that T be continuous and linear on a normed (not necessarily complete) space X. The results of [8] (still waiting to be extended for the averaging iterations) seem to give the most complete story of the Picard iterations for the linear case. Only when T is subject to some further restrictions, such as asymptotic 4-boundedness and asymptotic A -regularity, one can agree with Dotson [4] that the iterative solution of linear functional equations is a special case of mean ergodic theory for affine operators. This thesis is rather convincingly demonstrated by results of De Figueiredo and Karlovitz [2], and Dotson [3], and most of all by Dotson's recent paper [4], in which the results of [1; 2; 3] are elegantly subsumed under the afrine mean ergodic theorem of Eberlein-Dotson.

From Euclid to Hilbert, and beyond, the primitive terms of geometry have been taken as “point,” “line,” etc., while “distance” plays a secondary role. The reversal of this situation is a modern development. Frechet [4], in 1906 first considered the properties of distance which should be formalized. The most significant contributions to the geometric properties of metric spaces have been by Menger [10] and Blumenthal [2; 3].

In this paper we derive those properties of topologically embedded curves and surfaces in E3 which can be obtained without use of Bing's Side Approximation Theorem [3] for surfaces. The local homology and homotopy properties studied classically play the largest role in the paper, but the final geometrization of some of the results requires theorems such as the PL Schoenflies Theorem, Dehn's Lemma, the Loop Theorem, the Sphere Theorem, and Waldhausen's generalization of the Loop Theorem (n.b., one application of Waldhausen's theorem (in (3.4)) requires use of the nontrivial normal subgroup in the statement of that theorem).

In this paper, we are concerned with proving a formula for the computation of what is variously called the pattern inventory (e.g., see De Bruijn [2]) or the configuration counting series (e.g., see Harary [3]). Rather than redeveloping a large number of definitions, we shall assume the reader is already familiar with the terminology used by De Bruijn [2].

Polya, in a celebrated paper [4], proved a formula for computing the pattern inventory for all functions f defined on a set D (where D is acted on by a permutation group G), and mapping into a set R (which is called the store) for which the “store enumerator” in known.

The general coefficient theorem [2] and the extended general coefficient theorem [3] state that the existence of certain quadratic differentials is a sufficient condition for a function to be a solution of certain extremum problems. The purpose of this paper is to show that in the case of simply connected regions this condition is also necessary.

We shall do this by a variational method of the Schiffer-Golusin-type. The main difficulty is, that the class of admissible functions for the general coefficient theorem is restricted and we must therefore have a method of variation with restrictions.

Forelli in [1] has described the isometries of Hp(U) into Hp(U) for p≠2, 0 < p < ∞. We shall extend his methods to characterize the isometries of Hp(Un) onto Hp(Un).

The notation we shall use can be found in Rudin [3].

Let II represent a permutation that induces a map on functions of n complex variables by

Clearly II is an isometry of Hp(Un) onto Hp(Un).

There is a natural Borel structure on the set F of all factors on a separable Hilbert space [3]. Let denote the algebraic isomorphism classes in F together with the quotient Borel structure. Now that various non-denumerable families of mutually non-isomorphic factors are known to exist [1; 6; 8; 10; 11; 12; 13], the most obvious question to be resolved is whether or not is smooth (i.e. is there a countable family of Borel sets which separate points). We answer this question negatively by an explicit construction.

A graph G is called γ-perfect if ƛ (H) = γ(H) for every vertex-generated subgraph H of G. Here, ƛ(H) is the clique number of H (the size of the largest clique of H) and γ(H) is the chromatic number of H (the minimum number of independent sets of vertices that cover all vertices of H). A graph G is called α-perfect if α(H) = θ(H) for every vertex-generated subgraph H of G, where α (H) is the stability number of H (the size of the largest independent set of H) and θ(H) is the partition number of H (the minimum number of cliques that cover all vertices of H).

In [1] Tang proved that the generalized direct product of a finite set of cyclic groups amalgamating subgroups which satisfy certain compatibility conditions always exists. In the proof, Theorem 4.1 is made use of. However, this theorem is not correct since we can construct examples of groups which satisfy the conditions of Theorem 4.1, but whose generalized direct product does not exist. Therefore, a modification of this result as pointed out by Professor Tang is given here, together with the resulting modification of the proof of the result stated above.

The theory of Plücker coordinates and Grassmann varieties is well-developed and well-known among the algebraic geometers. It gives a one-to-one correspondence between the set of all subspaces of a given dimension in the ambient projective space and the set of points on a certain projective algebraic variety called a Grassmann variety. The unacquainted can find the theory discussed in detail in Hodge-Pedoe [1, Chapters VII and XIV].

Let f be a real valued function defined in some neighbourhood of a point x. If there are numbers α1, α2, … αr-1, independent of h such that

then the number αk is called the kth Peano derivative (also called kth de la Vallée Poussin derivative [6]) of f at x and we write αk = fk(x). It is convenient to write α0 = f0(x) = f(x). The definition is such that if the mth Peano derivative exists so does the nth for 0 ≦ n ≦ m.

Let k and A denote commutative rings with identity and assume that A is a k-algebra. A qth order k-derivation δ of A into an A -module V is an element of Homk(A, V) such that for any q + 1 elements a0, … , aq of A, the following identity holds:

Thus, a 1st-order derivation is just an ordinary derivation of A into V.

A singular partial differential equation which occurs frequently in mathematical physics is given by

where is the Laplacian operator on Rn of which the generic point is denoted by x = (x1, … , xn) and s and k are real numbers. The study of solutions of this equation for the case k = 0 was initiated by A. Weinstein [5], who named it ‘Generalized Axially Symmetric Potential Theory'. Numerous references to the literature on this equation can be found in [1; 3; 6]. The analytic theory of equations of the type mentioned above has extensively been treated in [2].

L'enveloppe de translations d'un demi-groupe présente un intérêt particulier pour les deux raisons suivantes : d'une part, elle apparaît d'une façon naturelle dans la construction des extensions idéales des demi-groupes, et d'autre part, elle est souvent utilisée pour des caractérisations abstraites des demi-groupes. Afin de l'appliquer aux problèmes particuliers, il est important de fournir des constructions qui, pour un demi-groupe donné sous une forme ou une autre, soient aussi explicites que possible. Un exemple d'une telle construction est celle de l'enveloppe de translations d'un demi-groupe de matrices de Rees (voir [4]) et ses nombreuses applications [5 ; 6 ; 7 ; 8].

This paper continues the study, begun in [7], of the spectral theory of non-self-ad joint second order ordinary differential operators on a half-line. The case of a ‘Very small” potential was studied in [4; 5; 6]. The case considered in [7], and in the present paper, is that where the potential is not so small.

In the present paper a general form of integral limit laws for additive functions is obtained. Our limit law contains Kubilius’ results [5] on his class H. In the proof we make use of characteristic functions (Fourier transforms), which reduces our problem to finding asymptotic formulas for sums of multiplicative functions. This requires an extension of previous results in order to enable us to take into consideration the parameter of the characteristic function in question. We call this extension a parametric mean value theorem for multiplicative functions and its proof is analytic on the line of [4].

In [4] it was conjectured that a light map p : E → B for which paths can be lifted and lifting of arcs is unique is a Serre fibration. As is well-known this implies that paths have unique liftings. In this paper we shall prove several special cases of this conjecture.

The two main theorems are: (3.5) Let p be a light compact map of a metric space E onto a connected semi-locally contractible along arcs metric space B. If arcs can be lifted uniquely then p is locally trivial.

For graph theoretic terms, see Tutte [1], A rank 2 tactical configuration of girth 2g and order (s, t) may be regarded as a (1 + s, 1 + t)-regular bipartite graph of girth 2g. We assume s ≦ t. Using a technique of Friedman [2] we show (i) and (ii).

Let ωn(x) denote the Jacobi polynomials with the weight Function

If we denote the corresponding normalized Jacobi polynomials by we Have

(1.1)

Now let

be the nth partial sum of the Fourier series of Jacobi polynomials of a function f(x).