Let X1, X2, … be a sequence of random variables and let {ajk}, j,k = 1, 2, …, be a matrix of real numbers. Write We establish the following result.

Let E be a real Banach space. If f: E→E is (Fréchet-) differentiable at every point of E, the derivative of f at x is denoted by f'(x), which is an element of the Banach algebra ℒ=ℒ(E) of all linear continuous mappings of E into itself with the usual upper bound norm, and, if we put , we have .

Let S be a compact Hausdorff space; let C(S) be the algebra of all continuous complex valued functions on S; and let M(S) be the dual space of (S) (the space of all regular Borel measures on S). In [2] Grothendieck gave a description of weak sequential convergence in M(S) in terms of uniform convergence on sequences of disjoint open sets in S. In this note we give a condition on the carriers of measures to guarantee that weak zero convergent sequences are norm zero convergent. While this condition is interesting in its own right, it can also be used to obtain immediately some well-known results about compact operators from C(S) to c0.

The object of this note is to give an aspect to the problem of the functional equation of the generalized gamma function and Dirichlet series which are defined in [1]. In general, we cannot answer the problem yet. But it is worthy to attack this problem for some special cases.

If V is a variety of metabelian Lie algebras then V has a finite basis for its laws [3]. The proof of this result is similar to Cohen's proof that varieties of metabelian groups have the finite basis property [1]. However there are centre-by-metabelian Lie algebras of characteristic 2 which do not have a finite basis for their laws [4] this contrasts with McKay's recent result that varieties of centre-by-metabelian groups do have the finite basis property [2]. The rollowing theorem shows that once again “2” is the odd man out.

This paper introduces two new separation axioms, point paracompactness and point countable paracompactness, both somewhat weaker than regularity, and shows that they can replace regularity in several standard theorems about paracompact or absolutely H-closed or Lindelöf spaces. Thus we obtain sharpened versions of these theorems. We also show that under certain hypotheses the new properties are equivalent to regularity.

Recently Brown [1] gave two new classes of generating functions which include the generating functions for the polynomials of Gegenbauer, Jacobi and Laguerre. The aim of the paper is to give a new class of generating functions which includes both sets of generating functions given by Brown and provides a new class of generating functions for the polynomials of Gegenbauer, Jacobi and Laguerre.

In [1] it was shown if S is a stable semigroup and H an ℋ-class of S then there is a congruence (H) on S in which H is a single class. After considering some consequences of this result for abstract semigroups, we consider some analogous questions for compact semigroups.

In the notation of Riordan ([5], p. 33), the Stirling numbers, s(n, k) and S(n, k), of the first and second kind respectively are defined by the relation where (x)n = x(x − 1) … (x − n + 1) is the factorial power function. They have been used by Jordan ([3], p. 184) to define the numbers C(m, k) and D(m, k), as linear combinations of s(n, k) and S(n, k) respectively, given by where and and where and D(m, O.

In this paper, we show how to generate orthomodular posets from sesquilinear forms on a vector space.

The concepts of order and type associated with an entire function over the complex plane admit different “natural” extensions, most of which are in vogue, for the case of an entire function over 𝒞k, the cartesian product of k copies of the complex plane. This work is concerned with the relations among such extended concepts and with the analogous properties of the concepts.

If R is a commutative semiprime ring with identity Kist [4], [5] has shown that R can be embedded into a commutative Baer ring B(R), and has given some properties of this embedding. More recently Mewborn [7] has given a construction which embeds R into a commutative Baer ring with the stronger property that every annihilator is generated by an idempotent. Both of these constructions involve a representation of R as a ring of global sections of a sheaf over a Boolean space.

It is known that a divisor d>0 of the positive integer n is called unitary [3;§ 1] if dδ = n and (d, δ) = 1, where (d, δ) denotes the g.c.d. of d and δ. We write d¶. Let k be a fixed positive integer. A positve integer n is called k-free if n is not divisible by the kth power of any prime.

Let Hn denote the set of complex n-square positive semidefinite hermtian matrices. We partially order Hn: If A, B, A–B єHn, write A > B. For A єHn, Write P(A) for the permanental adjoint of A, i.e., P(A) is the n-suare matrix whose i, j entry is per A(j/i), where A(j/i) is the submatrix of A obtained by deleting row j and column i. Now, P(A) is a principal submatrix of t he (n–1)st induced power matrix of AT. Hence, P(A) ∈Hn. Also D(A), the classical adjoing, is in Hn.

Let μ and ν be measures defined on some σ-algebrs with values in locally convex topological vector spaces X and Y, repectively. It is possible [1] to construct their product λ = μ × ν as a measure on a σ-algebra if λ is allowed to take its values in X ⊗ε Y, the completion of X ⊗ Y in the topology of bi-equicontinuous convergence. The reason is, roughly speaking, that the topology of biequicontinuous convergence on X ⊗ Y is coarse enough to make λ σ-additive and the completion X ⊗ε Y is big enough to accommodate all values of λ. Here we are going to improve the result by introducing a finer topology on X ⊗ Y in which λ will be σ-additive and such that all values of λ will belong to the completion of X ⊗ Y under that topology. The topology in question is obtained by a slight modification from a topology considered for the first time in the work [3] of Jacobs. Curiously enough, the proof of the improved result is simpler than that of [1] and reduces almost to a direct observation avoiding duality arguments.

I establish here a criterion for a sequence of ordinals to be generic over a transitive model of ZFC with respect to a notation of forcing first considered by Prikry in his Doctoral dissertation [2]. In Section 0 I review some notation, in Section 1 I list some facts about measurable cardinals, and in Section 2, after giving Prikry's result, I state and prove mine.

It is the aim of this paper to introduce a new concept relating various subgroups of the automorphism group of a graph to corresponding subgraphs. Throughout g will denote a (Michigan) graph on a vertex set V(¦V¦ =n) and Γ(g)=G will be the automorphism group of G considered as a permutation group on V.En, Cn, Dn and Sn are the identity, cyclic, dihedral, and symmetric groups acting on a set of size n, while Sp(q) is the permutation group of pq objects which is isomorphic to Sp but is q-fold in the sense that the objects are permuted q at a time [6]. H ≦ G means that H is a subgroup of G. Other group concepts can be found in Wielandt [7]. The graphs G1 ∪ G2, G1 + G2, G1 × G2, and G1[G2] along with their corresponding groups are as defined in, for example, Harary [4]. Finally we use Kn for the complete graph on n vertices.

Let n be a positive integer and let T be any nonempty set of positive integers. By (n, T) = 1 we men n is relatively prime to each element of T. Hence n can be written as n = n1n2 where n1 is the largest divisor of n such that (n1, T)= 1. Let P be a property associated with positive integers. We shall say that a positive integer n is a P-number if it satisfies the property P. If in the above representation of n, the integer n1 is a P-number. then we shall say that n is a quasi P-number relative to T, or, simply, a quasi P-number. In particular, for k a positive integer >1, n is quasi k-free for given set T) if n1 is k-free.

Suppose S is a regular semigroup and E is its set of idempotents. If E is subsemigroup of S, then S has been called orthodox and studied recently by Hall [3], Meakin [6], and Yamada [8]. In this paper we assume that E is not (necessarily) a subsemigroup of S and consider the subsemigroup generated by E, denoted <E>. If E denotes the set of all elements of S which can be written E, denoted <E>. If E denotes the set of all elements of S which can be written as the product of n (not necessarily distinct) idempotents of S, then . We show that <E> is always a regular subsemigroup of S and investigate relationships between it and S. The case where <E> = S is of particular interest to us; such semigroups will be referred to as idempotent-generated regular semi- groups.

Let S be a compact semitopological semigroup and let P(S) be the convolution semigroup of probability measures on S. An idempotent measure μ in P(S) is defined to be primitive if and only idempotent measures in μP(S)μ are μ and the zero element m of P(S). In a previous paper [2] we give some characterization of primitive idempotent measures on S. Let Π(P(S)) be the set of primitive idempotents in P(S) and let Πc be the set of central primitive idempotents in P(S). It is shown in [1] that Π(P(S)) is neither an ideal nor even a subsemigroup of P(S) in general. The purpose of this paper is to investigate the structure of Πc.