The central distribution of the second largest (smallest) root following the Fisher-Girshick-Hsu-Roy distribution under certain null-hypothesis has been derived in series form by Pillai and Al-Ani [6]. In this paper the noncentral distributions of the second largest roots in the MANOVA situation, the canonical correlation, and equality of two covariance matrices are obtained. Further, the distribution of the second largest root of the covariance matrix is obtained as a limiting case. The largest root and its noncentral distributions have been considered already by Pillai and Sugiyama [7] for the situations stated above. However, in the present paper, the joint densities of the largest and the second largest roots are derived in all the above cases from which the distributions of the largest roots can be obtained, although in more elaborate forms.

The purpose of the present paper is to investigate some of the properties of the coefficient K2n defined by

1.1

We prove

1.2

1.3

1.4

A known multiple integral from the difference calculus is used to derive certain distributions of ordered statistics from the truncated exponential, rectangular, and ordered random intervals populations.

A tournament is a directed graph in which there is exactly one arc between any two distinct vertices. Let denote the automorphism group of T. A tournament T is said to be point-symmetric if acts transitively on the vertices of T. Let g(n) be the maximum value of taken over all tournaments of order n. In [3] Goldberg and Moon conjectured that with equality holding if and only if n is a power of 3. The case of point-symmetric tournaments is what prevented them from proving their conjecture. In [2] the conjecture was proved through the use of a lengthy combinatorial argument involving the structure of point-symmetric tournaments. The results in this paper are an outgrowth of an attempt to characterize point-symmetric tournaments so as to simplify the proof employed in [2].

The problem to be considered in this note, in its most concrete form, is the determination of all quartets f1, f2, g1, g2 of functions analytic on some domain and satisfying

*

where p > 0. When p = 2 the question can be reformulated in terms of finding a necessary and sufficient condition for (two-dimensional) Hilbert space valued analytic functions to have equal pointwise norms, and the answer (Theorem 1) justifies this point of view. If p ≠ 2, the problem is solved by reducing to the case p = 2, and the reformulation in terms of the norm equality of lp valued analytic functions gives no clue to the answer.

The following theorem was conjectured to hold by P. Erdös [1]:

Theorem 1. For each finite directed tree T with no directed path of length 2, there exists a constant c(T) such that if G is any directed graph with n vertices and at least c(T)n edges and n is sufficiently large, then T is a subgraph of G.

In this note we give a proof of this conjecture. In order to prove Theorem 1, we first need to establish the following weaker result.

The purpose of this paper is not to solve a problem but to pose one that may be of some interest, depth, and consequence.

Given that the positive integer n has the canonical representation n = Πhi=1 piαi the problem of finding the number F(n) = f(α1, α2, … αn) of ordered factorizations of n into positive nontrivial integral factors is equivalent to that of finding the number of ordered partitions of the vector (α1, α2, … αn) into nonzero vectors with nonnegative integral components.

The purpose of this exposition is threefold. Firstly, we wish to show that many concepts and arguments carry over from pre-ordered sets to categories. Secondly, we wish to make some propaganda for the notion of "subequalizer" of two functors, which appears to be more fundamental than Lawvere's so-called "commacategory", in the same sense in which equalizers are more fundamental than pullbacks. Thirdly, we wish to give simple proofs of the adjoint functor theorem and related theorems, which appear to be more economic than those in the literature. The author wishes to take this opportunity to refine some arguments that he has published earlier. He is indebted to Michael Barr, whose presentation of similar proofs in his course has provided the stimulation for preparing this note for publication, to John Isbell for his critical reading of the manuscript and to William Schelter for suggesting a shortcut in one of the proofs.

L2(a, b)

1

with the property

2

were originally defined by T. Carleman [4]. Here he imposed on the kernel the conditions of measurability and hermiticity,

3

for all x with the exception of a countable set with a finite number of limit points and

4

where Jδ denotes the interval [a, b] with the exception of subintervals |x - ξv| < δ; here ξv represents a finite set of points for which (3) fails to hold.

Let B be a ring with unity, A a imitai subring of the centre Cof B. Suppose further that B is A-integral. (That is, every element of B satisfies a monic polynomial with coefficients in A.) Under these assumptions, Hoechsmann [2] showed that "contraction to A" is a mapping from:

1. (1) The prime ideals of B onto the prime ideals of A,

2. (2) The maximal ideals of B onto the maximal ideals of A.

In this note we show that, under additional assumptions, a noncommutative version of the rest of the Cohen-Seidenberg "going up theorem" can be established.

Recent efforts to generalize a classic result of Hajos [3] on the decomposition of finite abelian groups into direct sums of subsets (see Fuchs [1, Chap. XV]) led B. Gordon [2] to the following conjecture. If are r-dimensional row vectors over GF(3) such that: (i) Any weighted (±) sum of any even number of 's is nonzero, (ii) For each r-dimensional , there exists an s such that

Then there exists a subset of either 1 or 4 's which satisfies the same conditions.

This paper proves Gordon's conjecture.

Although a nontopological concept, boundedness seems to be of considerable importance in a topological space. 'There are many topological problems in which it is essential to be able to make this distinction' (between bounded and unbounded sets) [1]. Boundedness and in particular boundedness-preserving' uniform spaces appear to have applications to topological dynamics [4].

In spite of this importance, there have been only isolated attempts at developing the concept. Alexander [1] and Hu [7] tried the axiomatic approach. Hu, for example, calls a nonempty family of sets a boundedness if is hereditary and closed under finite union.

Let (P, ≥) be a p.o. set. The basis graph of (P, ≥) is defined to be the directed graph whose vertex set is P and in which the ordered pair 〈a, b〉 is an edge if and only if b covers a in (P, ≥).

Let D be a directed graph. All graphs considered in this note are finite and are free of loops and multiple edges.

A graph G is a couple (X, R) where X is a set, R ⊂ X × X. If G is an undirected graph without loops (R a symmetric irreflexive relation), we can interpret G as a couple (X, R), where R is a set of two-element subsets of X, i.e. . This interpretation is generalized in the notion of society.

A society is a couple (X, R), where ; a k-society is a society (X, R) with |A| = k for each A ∊ R.

In what follows, the transposed complex conjugate of a complex rectangular matrix D is denoted by D* and the rank of D by r(D). Meyer [1] proved the following result using generalized inverses:

Theorem. Let A and B be complex m × n matrices such that AB*=B*A=0. Then r(A+B) = r(A)+r(B).

Below we prove this result by repeated use of the fact that for every complex m × n matrix D we have r(D) = r(D*D) = r(DD*) (e.g. See [2] Theorem 5.5.4).

Soit R un anneau primitif à droite et soit A un idéal à droite de R. Dans un récent article [2], Ménard a montré que A est minimal si et seulement si abA = baA ∀ a, b ∊ A. Le but de cette note est de montrer que A est minimal si et seulement si aA = a2A ∀ a ∊ A.

Lemme. Soit R un anneau semi-simple [1], soit A un idéal à droite de R et soit S = Rad(A) le radical de Vanneau A. Alors A est un idéal à droite minimal de R si et seulement si Ā = A/S est un corps.

The purpose of this note is to prove that an involutionf on a cyclic Peano space S leaves some simple closed curve in Ssetwise invariant.

We shall first define the required terms. A Peano space is a locally compact, connected and locally connected metric space. A connected space is called cyclic if it has no cut-point. An involution on a space is a periodic mapping whose period is 2; it is necessarily a homeomorphism. A mapping f: X → X is said to leave a subset E of S setwise invariant if f(E) = E. These definitions may be found, for example, in [2].

As has long been known, the irreducible tensor representations of GL(d) of rank n may be labeled by means of the irreducible representations of Sn, i.e., by means of the Young diagrams [λ], where λ1 + λ2 + … λr = n. We denote such a tensor representation by 〈λ〉. Using Young's raising operator Rij we can write [1, p. 42]

1.1

where the dot denotes the inducing process. For example, [3] . [2] is that representation of S5 induced by the identity representation of its subgroup S3 × S2.

Lehmann [1] in his lecture notes on estimation shows that for estimating the unknown mean of a normal distribution, N(θ, 1), the usual estimator is neither minimax nor admissible if it is known that θ belongs to a finite closed interval [a, b] and the loss function is squared error. It is shown that , the maximum likelihood estimator (MLE) of θ, has uniformly smaller mean squared error (MSE) than that of . It is natural to ask the question whether the MLE of θ in N(θ, 1) is admissible or not if it is known that θ ∊ [a, b]. The answer turns out to be negative and the purpose of this note is to present this result in a slightly generalized form.