Let X1X2,… be i.i.d. r.v. with EX=μ>0, and E(X-μ)2 = σ2<∞.

Let Sk=X1+…+Xk and vx=max{k:Sk≤x}, x≥0 and vx=0 if X1>x. Billingsley [1] proved if X1≥0 then

converges weakly to the Wiener measure W.

Let τx(ω)=inf{k≥1|Sk>x}. In §2 we prove that

converges weakly to the Wiener measure when the X's may not necessarily be nonnegative. Also we indicate that this result can be extended to the nonidentical case.

Let Pm, k denote the set of pyramidal numbers

1.1

The question has been asked whether there exist elements p, q, r in Pm, k such that p+q = r or, as the problem is usually posed,

1.2

The case k=2 has been studied by Sierpinski [6] and Khatri [3]; the case k=3 by Oppenheim [4] and Segal [5]; recently Fraenkel [2] has generalized (1.1) to the larger set

1.3

We shall call a semigroup S a σ-reflexive semigroup if any subsemigroup H in S is reflexive (i.e. for all a, b ∈ S, ab ∈ H implies ba∊H ([2], [5]). It can be verified that any group G is a σ-reflexive semigroup if and only if any subgroup of G is normal. In this paper, we characterize subdirectly irreducible o-reflexive semigroups. We derive the following commutativity result: any generalized commutative ring R ([1]) in which the integers n=n(x, y) in the equation (xy)n = (yx)m can be taken equal to 1 for all x, y ∈ R must be a commutative ring.

By "multiplicative inverse eigenvalue problem" (m.i.e.p., for short) we mean the following. Let A be an n×n matrix and let s1,…, sn be n given numbers. Under what conditions does there exist an n×n diagonal matrix V such that VA has eigenvalues s1,…,sn?

In the "additive inverse eigenvalue problem" (a.i.e.p., for short) we seek the diagonal matrix V so that A + V has eigenvalues s1,…, sn?.

In the present paper we extend to the m.i.e.p. the ideas used in [7] for the a.i.e.p.

By per X we denote the permanent of the square matrix X.

J. Ch. Boland suggested, and Mrs. Sheehan named, the idea of the slimming number of a graph G, i.e. the minimum number, s(G), of edges, e1e2,…, es, which must be removed from G in order that G—∪ ei be planar.

For the complete graph, Kn (n≥3), it may be seen by Euler's formula that a planar subgraph contains at most 3n—6 edges; moreover one may construct such a subgraph inductively, starting from K3, and adding points successively, joining them to the three vertices of the region in which they lie, so

1

An application of graph theory to automatic traffic control [2] gave rise to the problem of deciding which connected graphs have points of degree 2 which can be successively suppressed until only a single edge remains.

A linear connexion ∇ on a smooth manifold M with vanishing torsion and curvature is called flat. It is well known that such a connexion need not be the Levi-Civita connexion of a Riemannian (nor pseudo-Riemannian) structure (cf. [1]).

The following example with M homeomorphic to a cylinder is particularly simple. Using polar coordinates the punctured plane M=R2\{0} is covered by a smooth atlas with two charts

The covering problem. Let Si(i = 1, 2,…,n) be given sets containing mi elements respectively and let

1

be their cartesian product. The elements of S(n) will be called vectors. The vector (x1 x2,…, xn) covers (y1 y2,…, yn) if xi =yi for at least n—1 values of i. A subset M of S(n) is said to be a covering (perfect covering) of S(n) if each member of S(n) is covered by at least (exactly) one member of M. A covering M is said to be linear if the sets Si are groups Gi and M is a subgroup of G(n) = S(n) Denote by σ(n; m1 m2,…, mn) the value of min |M| when M runs through all coverings of S(n) and by σ(n; m1 m2,…, mn) the value of min |M| when the sets Si are given groups Gi and M runs through all linear coverings of G(n).

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.

Let D denote the open right half plane and

a Stolz domain in D with vertex at the origin. If h is a minimal harmonic function on D with pole at the origin then E⊂D is minimally thin at the origin iff where is the reduced function of h on E in the sense of Brelot. We now define

where s shall be fixed to be 1/e. For the set E∩In we shall let cn denote the outer ordinary capacity (see [1, pp. 320-321]), An the outer logarithmic capacity, and on the outer Green capacity with respect to D. If E⊂K, Mme. Lelong [3, p. 131] was able to prove that E is minimally thin at the origin Since one cannot easily relate the classical measure theoretic properties of a plane set with its Green capacity, it would appear desirable to find some other criteria for minimal thinness.

Let L be a positive definite symmetric matrix having a noncentral multivariate beta density of an arbitrary rank, see, e.g. Hayakawa ([2, p. 12, Equation 38]). Then an explicit procedure is given for decomposing the density of L in terms of densities of independent beta variates.

We consider the initial-boundary value problem for the parabolic partial differential equation

1.1

in the bounded domain D, contained in the upper half of the xy-plane, where a part of the x-axis lies on the boundary B(see Fig.1).

Relatively little is known about the ideal structure of A⊗RA' when A and A' are R-algebras. In [4, p. 460], Curtis and Reiner gave conditions that imply certain tensor products are semi-simple with minimum condition. Herstein considered when the tensor product has zero Jacobson radical in [6, p. 43]. Jacobson [7, p. 114] studied tensor products with no two-sided ideals, and Rosenberg and Zelinsky investigated semi-primary tensor products in [9].

All rings considered in this paper are assumed to be commutative with identity. Furthermore, R will always denote a field.

Let A be an additive abelian group and Dn(A) the set of those n x n matrices over A all of whose row and column sums are equal. Such matrices can be regarded as a possible generalization of doubly stochastic real matrices; alternatively, if A is a commutative ring, it turns out that Dn(A) is exactly the image of the permutation representation of Sn, the symmetric group of degree n, over A.

The classical mean value theorem asserts that if f is a real, bounded, Riemann integrable function defined on a finite real interval a≤t≤b, then , where infa≤t≤bf(t)≤y0supa≤t≤bf(t). The extensions of Choquet [3], Price [15], and of this paper generalize the fact that y0 belongs to the closure of the convex hull of f([a, b]). The version of Choquet ([3, p. 38]) applies to a continuous function on a compact interval with values in a Banach space; that of Price ([15, p. 24]) applies to a bilinear integral of a special type containing the Birkhoff integral [2]. The m-integral of Dinculeanu [6] (specialization of Bartle's *- integral [1]) leaves intact the Lebesgue dominated convergence theorem and is strong enough to support an extended development. The paper is organized as follows: the object of §2 is to express the integral of a bounded m-integrable function as a limit of Riemann sums; §3 gives Price's generalization of "convex hull" [15]; the theorem of the paper is established in §4; §5 gives applications to vector differentiation which, for continuously differentiable functions, contain results of Dieudonné [5] and McLeod [13].

We are concerned with the nth (n≥3) order linear differential equation

1

where the coefficients are continuous on (∞, ∞). Our main result is to give conditions under which the two-point boundary value function rij(t) (see Definition 2) are strictly increasing continuously differentiable functions of t. Levin [1] states without proof a similar theorem concerning just the monotone nature of the rij(t) but assumes that the coefficients in (1) satisfy the standard differentiability conditions when one works with the formal adjoint of (1). Bogar [2] looks at the same problem for an nth-order quasi-differential equation where he makes no assumption concerning the differentiability of the coefficients in the quasi differential equation that he considers. Bogar gives conditions under which the rij(t) are strictly increasing and continuous. The different approach of the author to this problem also enables the author to establish the continuous differentiability of the rij(t) and to express the derivatives in terms of the principal solutions Uj(x, t), j=0, 1,…,n-1 (see Definition 4).

The Kantorovitch polynomials of a summable function s, defined on [0, 1], are

where

and

They are the analogue for summable functions of the Bernstein polynomials Bnf(x), and they possess similar properties [1].

In this paper we characterize the 2 × 2 invertible matrices over a Boolean ring, and, using this characterization, show that every invertible matrix has order dividing 6. This suggests that GL2 of a Boolean ring is built up out of copies of the symmetric group S3. Indeed, if B is a finite Boolean ring, then GL2(B) turns out to be a direct sum of copies of S3. If B is infinite, then GL2(B) is more difficult to calculate; we present here descriptions of GL2(B) for the "extreme" cases of countable Boolean rings—namely, the Boolean ring which is generated by its atoms and the atomless Boolean ring. The former provides a negative answer to the question of whether the functor GL2(⋅) preserves inverse limits; the latter is a corollary of a theorem which states that, under certain circumstances, GL2(⋅) preserves direct limits. It turns out, in addition, that every invertible matrix is a product of elementary ones, as is the case for matrices over a Euclidean domain.

A simple but useful result in the measure theory for product spaces can be stated as follows:

Theorem A. A necessary and sufficient condition that a measurable subset E of X×Y has measure zero is that almost every X-section (or almost every Y-section) has measure zero (see [1, §36]).

We will show, in this short note, that a similar result also holds for the exponential of measure spaces. Before proceeding any further, we describe briefly here the exponential construction of a measure space.

Let V[0, 2π] denote the class of all normalized functions F of bounded variation in [0, 2π] such that F(x) = 2-1{F(x+0)+F(x-0)} and F(x+2π)-F(x) = F(2π) — F(0) for all x and let {Cn} be the sequence of Fourier-Stieltjes coefficients of F. Wiener [9] (cf. Bari [1, p. 212], Zygmund [10, p. 108]) proved the following theorem.