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.

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).

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.

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).

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].

Let H be a complex Hilbert space and let K = H ⊕ H. Then K can be identified with the set of all column matrices

equipped with componentwise addition and scalar multiplication and the scalar product

By a tournament Tn on n vertices, we shall mean a directed graph on n vertices for which every pair of distinct vertices form the endpoints of exactly one directed edge (e.g., see [5]). If x and y are vertices of Tn we say that x dominates y if the edge between x and y is directed from x to y. In 1962, K. Schütte [2] raised the following question: Given k > 0, is there a tournament Tn(k) such that for any set S of k vertices of Tn(k) there is a vertex y which dominates all k elements of S. (Such a tournament will be said to have property Pk.)

Let X be a topological space. We say that cat X ≤ n if there exists a map ϕ: X → T1(X, …, X) such that jϕ≃Δ: X → Xn+1, where T1(X, …, X) is the “fat wedge”, j is the inclusion and Δ is the diagonal map. This is an example of a right structure system. This right structure system leads to an associated weak structure system, namely weak category in this particular case.

A three-term recurrence

1

where An (n ≥ 0), Bn (n ≥ 0) and Cn (n ≥ 0) are real numbers for which AnCn+1 ≠0 (n ≥ 0), generates a sequence {Pn} of real polynomials in which Pn has degree exactly n.

In [4], A. Sade defines the singular direct product for quasigroups. In this paper we use the singular direct product to construct quasigroups satisfying the identity x(xy)=yx. In particular, we show that the singular direct product preserves the identity x(xy)=yx.

In [2], A. Sade gives a construction for quasigroups which he calls the singular direct product. In this paper we generalize Sades' construction. As an application we obtain a recursive construction for quasigroups orthogonal to their transposes. All quasigroups considered in this paper will be finite.

For a sequence σ = (x1, …, xn) of real numbers, let σi and respectively denote the cyclic permutation (xi, xi+1, …, xi-1) and the reverse cyclic permutation (xj, xj-1, …, xj+1), and let . Also denote by Mrj(σ) and mrj(σ) the rth largest and the rth smallest numbers respectively, among the first j partial sums s1, s2, …, Sj for 1≤r≤j≤n. As usual, let the superscripts + and — respectively mean maximize and minimize with zero.

A function f(x) is said to be completely monotonic on (0, ∞) if

1

Familiar examples of such functions are given by f(x) = exp(—αx) and f(x) = (x+β)-α, where α ≥ 0, β ≥ 0. A discussion of completely monotonic functions is given in [5, Ch. IV].

Given an additive group G and nonempty subsets S, T of G, let S+T denote the set ﹛s + t | s ∊ S, t ∊ T﹜, S the complement of S in G and |S| the cardinality of S. We call S a sum-free set in G if (S+S) ⊆ S. If, in addition, |S| ≥ |T| for every sum-free set T in G, then we call S a maximal sum-free set in G. We denote by λ(G) the cardinality of a maximal sum-free set in G.

Stone lattices can be considered as forming a category of abstract algebras and thus there is a forgetful functor from this category to the category of distributive lattices with zero and unit. In this note we consider Stone lattices in this light (cf. [3], [4]) and describe an adjoint to the forgetful functor. The Stone extension of a distributive lattice with zero unit which we obtain differs markedly from the one given in [1].

The theory of integration in division spaces introduced by Henstock ([3], [4]) serves to unite and simplify much of the classical material on nonabsolute integration as well as to provide a new approach to Lebesgue integration. In this paper we sketch a simplified approach to the division space theory and show how it can lead rapidly to the standard Lebesgue-type theory without a substantial departure from the usual methods; some applications to integration in locally compact spaces are briefly developed.

Let (a1, a2, …, am), ai ∊ {l, 2, …, n}, be an m-permutation of n (repetitions allowed) with exactly kj of the a's equal to j, j = 1, 2, …, n, m = k1+ …+kn, fixed nonnegative integers. An inversion is a pair i, j such that i < j, ai > aj. Denote by N(r; k, …, kn) the number of such permutations with exactly r inversions.

The purpose of this note is to show that there is one and only one coloring of the sides of a dodecahedron with four colors, under the condition that no two adjacent sides share the same color.

In [1] Busacker and Saaty give a graphical solution to the game known as “Instant Insanity”.

By compactness, the unit ball Bn in Rn has a finite covering by translates of rBn, for any r > 0. The main theorem of this note shows that a weaker covering property does not hold in any infinite-dimensional normed space.

In a recent publication [2], D. J. Lewis stated that the Diophantine equation x3+117y3 = 5 has at most 18 integer solutions, but the exact number is unknown. In this paper we shall solve this problem by proving the following

Theorem. The equationx3+117y3 = 5 has no integer solutions.