Let A = (aij) be an m × n matrix and let K = {s1, …, sk} be a k-subset from {1, 2, …, n}. For 0≤t≤k≤n define the (t, K)-permanent of A to be

(1)

the summation taken over all m-tuples (i1, i2, …, im) (repetitions allowed) of 1, 2, …, n each containing exactly t distinct entries from K and any number of distinct entries from the remaining n-k integers. For example, (4, 4, 7, 1, 1, 2), (4, 4, 6, 6, 6, 5) are 6-tuples, each containing exactly two distinct entries from K={2, 4, 5} for n ≥ 7.

Let X be a Banach space. Consider the family I(α) of linear continuous operators from X into itself, depending on the parameter α ≥ 0. Suppose that I(α1+α2)=I(α1)I(α2)∀α1, α2 ≥ 0 and I(0)=I (semigroupal property). Such a semigroup is said to be strongly continuous for α > 0, if limh→0I(α+h)x =I(α)x, ∀x ∊ X.

P. Erdös and G. Szekeres [1] proved that from any points in the plane one can always choose n + 1 of them which are the vertices of a convex polygon, thus answering a question due to Miss Esther Klein (who later became Mrs. G. Szekeres).

Various definitions of capacity of a subset of a domain in Euclidean space have been used in recent times to shed light on the solvability and spectral theory of elliptic partial differential equations and to establish properties of the Sobolev spaces in which these equations are studied. In this paper we consider two definitions of the capacity of a closed set E in a domain G. One of these capacities measures, roughly speaking, the amount by which the set of function in C∞(G) which vanish near E fails to be dense in the Sobolev space Wm, p(G).

Let R be a ring with 1 and let Q denote the maximal left quotient ring of R [6]. In a recent paper [12], Wei called a (left). R-module M divisible in case HomR (Q, N)≠0 for each nonzero factor module N of M. Modifying the terminology slightly we call such an R-module a Q-divisible R-module. As shown in [12], the class D of all Q-divisible modules is closed under factor modules, extensions, and direct sums and thus is a torsion class in the sense of Dickson [5].

The extension of the Rellich-Kondrachov theorem on the complete continuity of Sobolev space imbeddings of the sort

1

to unbounded domains G has recently been under study [1–5] and this study has yielded [4] a condition on G which is necessary and sufficient for the compactness of (1). Similar compactness theorems for the imbeddings

2

are well known for bounded domains G with suitably regular boundaries, and the question naturally arises whether any extensions to unbounded domains can be made in this case.

This paper was inspired by [4]. The main result there suggests a close relationship between injective hulls of C* algebras as studied in [2] and [3] and rational completions as studied in [1]. We shall prove an analogue of Theorem 1 in [3] for rational completions. The latter theorem states that the injective hull of the algebra C(X) of all complex valued functions on the compact Hausdorff space X is the algebra B(X) of all bounded Borel functions modulo sets of first category.

Let be two foliations on a Cr manifold M. We say and are Ck-conjugate if there exists a Ck diffeomorphism h:M→M such that h maps the leaves of onto the leaves of .

Twenty-five years ago McCoy published a characterization of commutative subdirectly irreducible rings. This result was generalized by Thierrin to duo rings with the word “field” which appeared in McCoy's theorem replaced by “division ring”. The purpose of this note is to give another generalization in which the words “division ring” will be replaced by “simple ring with 1 ”. The techniques resemble those of McCoy and Thierrin.

The purpose of this note is to give some characterizations of collectionwise normality by means of the normality of certain product topological spaces; they are a consequence of a theorem proved by M. Katětov, but they appear to be new.

Let A be a finite dimensional commutative and associative algebra with identity, over a field K. We assume also that A is generated by one element and consequently, isomorphic to a quotient algebra of the polynomial algebra K[X]. If A=K[a] and bi=fi(A), fi(X) ∊ K[X], 1≤i≤r we find necessary and sufficient conditions which should be satisfied by fi(X) in order that A = K[b1, …, br].

The result can be stated as a theorem about matrices. As a special case we obtain a recent result of Thompson [4].

In fact this last result was established earlier by Mirsky and Rado [3]. I am grateful to the referee for supplying this reference.

In [2] we showed how, for a certain class of outer measures on a metric space, a measurable cover could be constructed for each subset A of the space. The function is a closure operator, and in this note some of the properties of the resulting topology are investigated. In particular, we obtain a sufficient condition for the space to be connected.

This is the continuation of our paper [1] and includes the results promised there. As in [1], we consider a homogeneous elliptic equation in two variables. In [1] we showed that all solutions of such equations can be written in a specific form, viz. in the form of an infinite series in certain specific polynomials. Here we first establish that a common solution of any two positive powers of any two linearly independent, linear elliptic polynomials can be expressed as a polynomial (Lemma 2).

We consider the following functional equation

1

where ƒ = ƒ(x, y) is a real-valued function of two real variables x, y on the whole xy-plane and t is a real variable.

With regard to the geometric meaning of (1), the equation is called the “square” functional equation.

Let us consider the linear system in the two parameters λ and μ; i. e.,

1.1

1.2

and where for the moment we shall assume both b(x) and q(x) are real-valued, continuous functions in [0, 1].

In this paper we use some known transformations available in the Theory of Multiple Integrals to give straightforward, simpler, and elegant proofs of some distributions of ordered values from Burr and beta distributions. The exact distribution (under the null hypothesis) of Wilks' ∧ criterion is obtained by considering it as a certain minimum value distribution problem.

The following theorem is well known [1, p. 66]: Suppose that, in a ballot, candidate P scores p votes and candidate Q scores q votes, where p > q. The probability that throughout the counting there are always more votes for P than for Q, equals (p-q)/(p+q).

Let H be a cyclic group, K⊂H a subgroup and x, y generators of H, K. We shall say that x, y are related if y=xa where a is the index of K in H, in other words, y is the smallest positive power of x in K. The main purpose of this note is to show that for any group G one may, by means of the axiom of choice, choose for each cyclic group H⊂G a generator xH such that when K⊂H then xK, xH are related.