The main object of this paper is to show that the existence of a particular kind of isomorphism between the integral group rings of two finite groups implies that the groups themselves are isomorphic. The proof employs certain types of linear forms which are first discussed in general. These linear forms are in some way related to the bilinear forms used by Weidmann [3] in showing that groups with isomorphic character rings have the same character table, and a shorter and, in a sense, more natural proof of this result is included here as another application of these linear forms.

Le radical R(A) d′un anneau A peut être caract érisé de plusieurs manières (Jacobson [1]). Rappelons seulement ici deux des caractérisations les plus importantes:

(1) R(A) est l′intersection de tous les idéaux à droite (à gauche) modulaires maximaux de A.

(2) R(A) est un idéal quasi - régulier contenant tout idéal à droite (á gauche) quasi - régulier.

L′objet de cet article est de donner quelques caractérisations du radical d′un anneau.

Let R be a ring with an identity 1, and R′ a ring anti - isomorphic to R. Let V be an R-module as well as an R′-module. We assume that 1 a = a for all elements a in V and that V satisfies the minimum condition for R-submodules. Elements of R will be denoted by α, β, …, and those of V by a, b, … Elements of R′ will be a α′, β′, …, where α′ fcorresponds to α by the anti - isomorphism. A mapping f of V x V to R is called a bilinear mapping of V to R if it satisfies the following.

Moser and Zayachkowski [1] have discussed certain planar lattice paths from (0, 0) to (m, n). In this note we consider analogous paths in three dimensional space. For basic definitions see reference [2]. Throughout this note each of m, n and k is a positive integer and if S is a finite set, |S| will denote the number of elements of S.

Each path under consideration here is such that each of its steps is of one of the following types:

If K is a finite geometric (i.e. admitting a rectilinear triangulation) n-complex and σn is an n - simplex of K which is n -1 not a face of any n + 1 simplex of K, and if σn-1 is an n-1 face of σn which is not a face of any other n - simplex in K, then the complex K - σn - σn-1 (the complex whose simplexes are those of K except for σn and σn-1) is called an elementary contraction of K of order n. The correspondence K → K - σn - σn-1 will also be called an elementary contraction, there being no possibility of confusion.

Let A and X be spaces. Then as is wellknown, [∑A, X] is a group where ∑ denotes the suspension. We wish to find conditions on A which will imply that this group is abelian for all spaces X, that is, ∑A is homotopy-commutative. This is equivalent to saying that conii A≤ 1 (see [2] for definition). Our results contain relations between conil A and the generalised Whitehead product of [1]. We work in the category of complexes with base points.

Let (x1, x2, …, xn) denote the coordinates of a point of Euclidean n-space En. Let be a set of n+1 points of En with the property that

form a linearly independent set and define a lattice Λ of points

by allowing u1, …, un to take all integer values.

In this paper we are dealing with differential equations of the forms:

Ei

where the functions pi are positive.

By a solution of an equation of the above forms, we mean a function x(t) ∈ C2 [c, + ∞) where c is a non-negative constant, which satisfies the corresponding equation on the whole interval [c, +∞). By an oscillatory solution of (Ei), we mean a solution with arbitrarily large zeros.

The purpose of this paper is to obtain a set of sufficient conditions for “global asymptotic stability” of the trivial solution x = 0 of the differential equation

1.1

using a Lyapunov function which is substantially different from similar functions used in [2], [3] and [4], for similar differential equations. The functions f1, f2 and f3 are real - valued and are smooth enough to ensure the existence of the solutions of (1.1) on [0, ∞). The dot indicates differentiation with respect to t. We are taking a and b to be some positive parameters.

It could perhaps be reasonably maintained that for most students of calculus the definite integral is in fact the Cauchy Integral. That is to say, where F is a primitive of f.

If done heuristically, the discussion of the Riemann integral serves two purposes. Firstly, it indicates that every continuous function has a primitive.

It is well known that Sobolev′s Lemma on the continuity of functions possessing L2 distributional derivatives of sufficiently high order is a simple consequence of elementary properties of the Fourier transform in L2 (e.g. [1, p. 174]). (In fact this statement remains true if 2 is replaced by p, 1 ≤ p ≤ 2). In this note we show that imbedding theorems of the type Wm, p ⊂Lq can also be obtained using Fourier transforms and an elementary lemma which reduces the cases p > 2 to the case p = 2. The simplicity of this approach is obtained at the expense of a slight loss of generality in the imbedding theorem.

The function was defined by G.N. Watson, [9, (i)]in 1931 by the integral relation∗

+ another term with μ and v interchanged;

The Weierstrass - respectively Weierstrass - Stieltjes transform of a function F(t) or μ(t) is defined by

1.1

and

1.2

for all x for which these integrals converge. In what follows we shall always assume that F(t) is Lebesgue integrable in every finite interval and that μ(t) is a function of bounded variation.

Let f(z) be a function regular for | z | < R, and denote by L(r) the length of the curve Γ onto which the circle | z | = r is mapped by f(z), i. e.

If Dr is the image of | z | ≤ r on the Riemann surface of f(z) then its area S(r) is given by

From the plane R2 we remove the union of the sets Sk (k = 1, 2, …) defined as follows (using the notation z = x + iy):

Sk = {z: arg z = nπ2-k for some integer n; |z|≥k}.

The remaining connected open set Ω we call the spiny urchin.

The “function” δ(x - xo) is known as the Dirac Delta function and may be defined as zero everywhere except at xo, where it is infinite in such a way that

1

having property that for every continuous function φ(x) on (a, b)

2

It is well known [2] δ(x-xo) can be approximated as a limit of a sequence of piecewise continuous functions, and there is an abundance of such sequences.

Let X1 …, Xn be mutually independent random variables with a common continuous distribution function F (t). Let Fn(t) be the corresponding empirical distribution function, that is

Fn(t) = (number of Xi ≤ t, 1 ≤ i ≤ n)/n.

Using a theorem of Manija [4], we proved among others the following statement in [1].

In this note we point out a not easily fillable gap in the proof of a major result in Chebyshev approximations on an interval [α, β]. Tornheim developed a theory for approximation by families unisolvent of degree n (n-parameter families), which was extended by Rice to families unisolvent of variable degree.

A square matrix is said to be stochastic if its elements are non-negative and if each of its row sums is equal to one. Thus λ = 1 is always an eigenvalue of a stochastic matrix.

Recently Gioia and Subbarao [2] studied essentially the following problem: If g(n) is an arithmetic function, and , then what is the behaviour of H(a, n) defined for each fixed integer a ≥ 2 by

1

By using Vaidyanathaswamy′s formula [e.g., 1], they obtain an explicit formula for H(a, n) in case g(n) is positive and completely multiplicative (Formula 2.2 of [2]). However, Vaidyanathaswamy′s formula is unnecessary to the proof of this result, which indeed follows more simply without its use, by exploiting a simple idea used earlier by Subbarao [3] (referred to also in the course of [2]).