The purpose of this paper is to consider a representation for the elements of a linear topological space in the form of a σ-integral over a linearly ordered subset of V; this ordered subset is what will be called an L basis. The formal definition of an L basis is essentially an abstraction from ideas used, often tacitly, in proofs of many of the theorems concerning integral representations for continuous linear functionals on function spaces.

The L basis constructed in this paper differs in several basic ways from the integral basis considered by Edwards in [5]. Since the integrals used here are of Hellinger type rather than Radon type one has in the approximating sums for the integral an immediate and natural analogue to the partial sum operators of summation basis theory.

We extend a result of Noritzsch, which describes the orbit sizes in the action of a Frobenius group $G$ on a finite vector space $V$ under certain conditions, to a more general class of finite solvable groups $G$. This result has applications in computing irreducible character degrees of finite groups. Another application, proved here, is a result concerning the structure of certain groups with few complex irreducible character degrees.

Given $A\subseteq GL_2(\mathbb {F}_q)$, we prove that there exist disjoint subsets $B, C\subseteq A$ such that $A = B \sqcup C$ and their additive and multiplicative energies satisfying

$$\begin{align*}\max\{\,E_{+}(B),\, E_{\times}(C)\,\}\ll \frac{|A|^3}{M(|A|)}, \end{align*}$$

where

$$ \begin{align*} M(|A|) = \min\Bigg\{\,\frac{q^{4/3}}{|A|^{1/3}(\log|A|)^{2/3}},\, \frac{|A|^{4/5}}{q^{13/5}(\log|A|)^{27/10}}\,\Bigg\}. \end{align*} $$

$A, B, C\subseteq GL_2(\mathbb {F}_q)$

$$\begin{align*}|AB+C|, ~|(A+B)C|\gg q^4,\end{align*}$$

$|A||B||C|\gg q^{10 + 1/2}$

$Forum Math.$

Any Hausdorff space X is a dense subspace of an H-closed space κX, called the Katětov extension of X, with the property that any H-closed extension Y of X is a continuous image of κX under a mapping which leaves X pointwise fixed [8], [10]. In [8], Liu has shown that the extensions κ(X×Y) and κX×κY of X × Y are equal iff (1) X or Y is finite, or (2) X and Y are H-closed. In this note, we examine whether homeomorphism of these two extensions implies equality. We give a condition under which homeomorphism implies equality and an example to show that this relation does not hold in general.

A ring R (associative ring) is said to be fully ordered provided that R is a linearly ordered set under a relation such that for any a, b and c in R, implies that and if c ε 0 then and . We say a subset K of R is convex provided that if a, b ε K such that then the interval [a, b] is a subset of K. Obviously an additive subgroup K of R is convex if and only if b ε K and b > 0 implies [a, b] ⊆ K.

In [6] B. Sz.-Nagy has proved that every operator on a Hilbert space such that

1

is similar to a unitary operator.

The following problem is an extension of this result: If T and S are two operators such that

1. 1. sup {‖Tn‖, ‖Sn‖}<∞ (n = 0, ±1, ±2,…)

2. 2. TS = ST

then there exists a selfadjoint operator Q such that QTQ-1, QSQ-1 are unitary operators?

Also, in [7] B. Sz.-Nagy has proved that every compact operator T such that

sup ‖Tn‖<∞ (n = 1, 2, 3,…)

is similar to a contraction.

It is shown that if f is an entire transcendental function, l a straight line in the complex plane, and n ≥ 2, then f has infinitely many repelling periodic points of period n that do not lie on l.

It is shown that if 1<n ≡ 0 or 1 (mod 2 m), then the edges of Kn may be partitioned into isomorphic copies of a graph D3(m) and also of a graph D4(m), graphs consisting respectively of a triangle with an attached path of m - 3 edges or a quadrilateral with an attached path of m - 4 edges. If m is a power of 2 then the above condition is shown to be necessary and sufficient for the existence of such a partition.

Upper bounds of second order Dehn functions of asynchronously automatic groups are obtained.

In this note we study dependence in matroids as an exercise in combinational algebra. Because the work seems to have little connection with graph theory we will not use Tutte′s approach (1) which uses dual concepts. To define a matroid we use Edmunds (2).

We write ω for the set of natural numbers (including zero) and for the set of all finite sequences of natural numbers. If n ∊ ω we write . If x is a function which takes its values in cu and whose domain of definition contains then we write for the element (x(0), …, x(n)) of .

In his 1922 article [l] on functions of bounded variation, Vitali gave a method for constructing monotone non-absolutely continuous functions, generalizing ideas from the ternary set introduced in another connection by Cantor. In [2], Hille and Tamarkin gave a full account of the "middle-third" function, showing it to be a singular distribution function, and finding its characteristic function. In [3], Evans obtained a generalization of the middle - third function by discarding middle intervals of length other than one-third, and obtained algorithms by which the moments of his function could be calculated. Invarious papers, among them [4], Wintner studied infinite convolutions of symmetric Bernoulli distributions, finding a great variety of distributions whose characteristic functions were of the form

In certain boundary value problems associated with two-parameter ordinary differential equations defined and having 2p, (p>1), turning points in a given interval, there arises certain equations in integers whose solutions determine the coefficients in the asymptotic expansions for the eigenvalues [1,2, 3, pp. 134-139].

In [2], Ladas and Sficas made two conjectures about the asymptotic behavior of solutions of some neutral differential equations. In this paper we confirm that these conjectures are indeed correct.

Let K be a compact subset of the real Euclidean space En. We say that K has constant width if the distance between each pair of distinct parallel hyperplanes which support K is constant. The collection of all compact convex subsets of En which have constant width is denoted .

The concept of a firm fixed point of a selfmap of a metric space is introduced. Loosely speaking a fixed point is firm if it cannot be moved to a point nearby with the help of a map which is arbitrarily close to the given map. It is shown that a continuum always admits a selfmap with a firm fixed point if the continuum contains a triod and if the vertex of the triod has a neighbourhood which is a dendrite. This condition holds in particular for local dendrites. Hence a local dendrite is an arc or a simple closed curve if and only if it does not admit a selfmap which has a firm fixed point.

Let

1

be the simple continued fraction (SCF) of an irrational number x. The partial quotients ai which we shall sometimes refer to as the "terms" of the SCF are integers and, for i ≥ 2, they are positive. If x is a quadratic irrationality then, by Lagrange's Theorem, the right side of (1) becomes periodic from some point on.

Under certain conditions the reduced product space JX of a space X has the same homotopy type as ΩΣX, the loop-space on the suspension of X. Several proofs can be found in the literature. The original proof [6] made unnecessarily strong assumptions. Later, in the last chapter of [3], torn Dieck, Kamps and Puppe gave a proof under much weaker conditions and showed that they could not be further weakened. The question then arose as to whether the reduced product construction could be generalized to provide combinatorial models not only of ΩΣX but also of Ω2Σ2X, Ω3Σ3X and so on. This was answered in the affirmative by May [10], using ideas of Boardman and Vogt [1], and the construction was further developed by Segal [11] and others.

The present note, however, is not concerned with these generalizations. Its purpose is to give a simple proof of the original result, without striving for maximum generality, and to show that the same method can be used to prove an equivariant version of the reduced product theorem and hence a fibrewise version. Fibrewise versions of the reduced product theorem have previously been given by Eggar [5] and, more recently, by one of us [8] but it may be useful to have a relatively simple treatment which is adequate for the majority of applications.

Let f be an additive arithmetical function having a distribution F. For any sequence let

In this note, we determine the slowest growing function b so that Qn{b, f) tends weakly to F, for various f.

We study the existence of solutions of the semilinear equations (1) in which the non-linearity g may grow superlinearly in u in one of directions u → ∞ and u → −∞, and (2) −Δu + g(x, u) = h, in which the nonlinear term g may grow superlinearly in u as |u| → ∞. The purpose of this paper is to obtain solvability theorems for (1) and (2) when the Landesman-Lazer condition does not hold. More precisely, we require that h may satisfy are arbitrarily nonnegative constants, . The proofs are based upon degree theoretic arguments.