It is shown that a positive measure μ on the Borel subsets of Rk is translation-bounded if and only if the Fourier transform of the indicator function of every bounded Borel subset of Rk belongs to L2(μ).

A ring R is called an H-ring if for every x ∈ R there exists an interger n=n(x)> 1 such that xn-x ∈C, where C is center of R. I. N. Herstein proved that H-rings must be commutative [See 3 pp. 220–221]. We now introduce the following definition.

We say that a function has locally small Riemann sums on an interval if for each point x in the interval, and for each positive number ε, all sufficiently fine partitions of intervals lying in neighborhood of x but not containing x have Riemann sums of absolute value less than ε. The main result is then as the title states. We use the generalized Riemann approach to Perron integration, assuming that functions are measurable only to insure that conditions involving the positive and negative parts of the functions are satisfied.

In [4] S. Ulam asks the following question.‘Does there exist a universal compact semigroup; i.e., a semigroup U such that every compact topological semigroup is continuously isomorphic to a subsemigroup of it?’The author has not been able to answer this question. However, in this paper, a proof is given for the following related result.

Let Q denote the Hilbert cube of countably infinite dimension and C(Q) the Banach space of continuous real-valued functions on Q with the usual norm. Let U denote the semigroup consisting of all bounded linear operators T: C(Q)→ C(Q) with ∥T∥ ≦ 1 and let U be endowed with the strong topology. Then, for every compact metric semigroup S with the property:(1.1) for all x, y ∈ S, with x ≠ y, there exists a z ∈ S, such that xz ≠ yz or zx ≠ zy;

there exists a 1 − 1 mapping φ of S into U such that φ is both a semigroup isomorphism and a homeomorphism.

U is metrizable, but is not compact; hence it does not provide an answer to the question of Ulam. The proof of the above statement leans heavily on a result of S. Kakutani [1].

Let R be an artinian ring. A family, ℳ, of isomorphism types of R-modules of finite length is said to be canonical if every R-module of finite length is a direct sum of modules whose isomorphism types are in ℳ. In this paper we show that ℳ is canonical if the following conditions are simultaneously satisfied: (a) ℳ contains the isomorphism type of every simple R-module; (b) ℳ has a preorder with the property that every nonempty subfamily of ℳ with a common bound on the lengths of its members has a smallest type; (c) if M is a nonsplit extension of a module of isomorphism type II1 by a module of isomorphism type II2, with II1, II2 in ℳ, then M contains a submodule whose type II3 is in ℳ and II1 does not precede II3. We use this result to give another proof of Kronecker's theorem on canonical pairs of matrices under equivalence. If R is a tame hereditary finite-dimensional algebra we show that there is a preorder on the family of isomorphism types of indecomposable R-modules of finite length that satisfies Conditions (b) and (c).

This paper gives characterizations of functionally compact spaces in terms of filterbases and nets. Also, a topological property that is weaker than countable compactness but stronger than first countable minimality is investigated.

The special Denjoy-Bochner integral (the D*B-integral) which are generalisations of Lebesgue-Bochner integral are discussed in [7, 6, 5]. Just as the concept of numerical almost periodicity was extended by Burkill [3] to numerically valued D*- or D-integrable function, we extend the concept of almost periodicity for Banach valued function to Banach valued D*B-integrable function. For this purpose we introduce as in [3] a distance in the space of all D*B-integrable functions with respect to which the D*B-almost periodicity is defined. It is shown that the D*B-almost periodicity shares many of the known properties of the almost periodic Banach valued function [1, 4].

We study a convolution semigroup satisfying Gaussian estimates on a group G of polynomial volume growth. If Q is a subgroup satisfying a certain geometric condition, we obtain high order regularity estimates for the semigroup in the direction of Q. Applications to heat kernels and convolution powers are given.

The main purpose of this paper is to prove that every automorphism of the semigroup of all Hadamard-differentiable mappings of a separable real Banach space into itself is inner. This generalizes the results of [7] which is a generalization of a result proved by Magill, Jr. [5].

Let f(x) and g(x) be two polynomials with arbitrary complex coefficients that are relatively prime. Hence the maximum is positive for all complex x. Since m(x) is continuous and tends to infinity with |x|, the quantity E(f, g) = min m(x) is therefore also positive.

We present the definition of crossed products of Hilbert C*-bimodules by Hilbert bundles with commuting finite group actions and finite dimensional fibers. This is a general construction containing the bundle construction and crossed products of Hilbert C*-bimodule by finite groups. We also study the structure of endomorphism algebras of the tensor products of bimodules. We also define the multiple crossed products using three bimodules containing more than 2 bundles and show the associativity law. Moreover, we present some examples of crossed product bimodules easily computed by our method.

An entire function f(z) is of bounded index if there exists a non-negative integer N such that . The least such integer N is called the index of f (se Lepson (1966)).

Properties of solutions of , are studied in the cause uH(t, u)>0 for u≠0. It is shown that two inequalities may always be associated with (I) in such a way that if one of these inequalities has a small positive solution and the other inequality has a small negative solution, then (I) is oscillatory. Further asymptotic properties of (I) are studied under assumptions involving intermediate antiderivatives P(i)(t), with P(n) = Q. Several results of this type ensure the non-existence of positive solutions of (I).

All groups in this paper are Abelian. The notation is similar to Fuchs (1970) and Maclane (1963).

Some results on fixed points of certain involutions in Banach spaces have been obtained, and whence a few coincidence theorems are also derived. These are indeed generalization of previously known results due to Browder, Goebel-Zlotkiewicz and Iséki. Illustrative examples are also given.

Let and .

In 1951 Mohanty proved the following theorem: .

In this paper a general theorem on summability |R,l (w), 1 | of Σ An(x) has been given which improves upon Mohanty's result in different ways (see Corollaries 1, 2 and 3) and it is also shown that some of the results of this note are the best possible.

One proof of Minkowski's fundamental theorem on lattice points in an n-dimensional parallelopiped depends upon a multiple Fourier expansion. Its terms involve multiple integrals which are easily evaluated and so are shown to have positive values. Then the expansion takes only positive values and the desired proof follows at once. It seems of interest to note a multiple integral which assumes only positive values.

A hyperfinite von Neumann algebra satisfies the condition that every o.d. homomorphism is a normal operator if and only if it is a factor of type In.

We study the existence, uniqueness and regularity of solutions to an exterior elliptic free boundary problem. The solutions model stationary solutions to nonlinear diffusion reaction problems, that is, they have compact support and satisfy both homogeneous Dirichlet and Neumann-type boundary conditions on the free boundary ∂{u > 0}. Then we prove convexity and symmetry properties of the free boundary and of the level sets {u > c} of the solutions. We also establish symmetry properties for the corresponding interior free boundary problem.

We shall take for granted the basic terminology currently in use in the theory of varieties of groups. Kovács, Newman, Pentony [2] and Levin [3] prove that if m is an integer greater than 2, then the variety Νm of all nilpotent groups of class at most m is generated by its free group Fm-1(Νm) of rank m – 1 but not by its free group Fm–2(Νm) of rank m — 2. That is, the free groups Fk(Nm), 2≦k ≦ m – 2, do not generate Nm. In general little is known of the varieties generated by them. The purpose of the present paper is to record the varieties of the free groups Fk(Nm) of the nilpotent varieties Nm of all nilpotent groups of class at most m for 2 ≦ k ≦ m – 2 and 5 ≦ m ≦ 6. This is done by describing a basis for the laws in these groups, that is a set of laws the fully invariant closure of which is the set of all laws for Fk(Nm). The set of laws, which, together with the appropriate nilpotency law, form a basis for the relevant groups Fk(Nm) are listed below: .