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.

We consider here the equation

1

This equation was first studied by Hammerstein [4] under the assumption that the linear operator

2

is selfadjoint and completely continuous. V. Nemytsky [5] and M. Golomb [3] dropped the assumption that A be selfadjoint and positive. M. Vainberg [6] considered (among other cases) the case in which A is a bounded operator generated by a Carleman kernel. The kernels considered in this work do not necessarily generate bounded, completely continuous or selfadjoint, operators.

Let L be the linear, elliptic, self-adjoint partial differential operator given by

where Dj denotes partial differentiation with respect to xj, 1 ≤ j ≤ n, b is a positive, continuous real-valued function of x = (x1,…,xn) in n-dimensional Euclidean space En, the aij are real-valued functions possessing uniformly continuous first partial derivatives in En and the matrix {aij} is everywhere positive definite. A solution u of Lu = 0 is assumed to be of class C1.

A well known theorem by S. Banach states that a contractive function f: X → X on a complete metric space X has a fixed point, and that this fixed point is unique. This result has a partial extension to multi-functions: every contractive compact-valued multi-function on a complete metric space has a fixed point (see Definition 1 and Theorem 1 below). But simple examples show that this fixed point is no longer unique. We investigate some questions concerned with the properties of the fixed point set Φ of a contractive multi-function φ. Is, e.g., Φ connected if φ is connected-valued? Is Φ convex if φ is convex-valued? The answer is yes if X is the real line (§2), but examples in §3 and §4 show that in general the answer is no.

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

In this paper we provide new proofs of some interesting results of Firey [2] on isoperimetric ratios of Reuleaux polygons. Recall that a Reuleaux polygon is a plane convex set of constant width whose boundary consists of a finite (odd) number of circular arcs. Equivalently, it is the intersection of a finite number of suitably chosen congruent discs. For more details, see [1, p. 128].

If a Reuleaux polygon has n sides (arcs) of positive length (where n is odd and ≥ 3), we will refer to it as a Reuleaux n-gon, or sometimes just as an n-gon. If all of the sides are equal, it is termed a regular n-gon.

In this paper, we consider real linear spaces. By (V:‖ ‖) we mean a normed (real) linear space V with norm ‖ ‖. By the statement "V has the (Y, X) norm preserving (Hahn-Banach) extension property" we mean the following: Y is a subspace of the normed linear space X, V is a normed linear space, and any bounded linear function f: Y → V has a linear extension F: X → V such that ‖F‖ = ‖f‖. By the statement "V has the unrestricted norm preserving (Hahn-Banach) extension property" we mean that V has the (Y, X) norm preserving extension property for all Y and X with Y ⊂ X.

Let M be an n-dimensional connected C∞ manifold with a linear connection Γ. M is said to be of recurrent curvature with respect to Γ if the corresponding curvature tensor R satisfies [1], [4]

where Δ denotes covariant derivative with respect to Γ and W is a nonzero covector called the recurrence co-vector. Let T be the torsion of Γ.

The kernels of the transforms in the class that we shall treat satisfy

1.1

where Re ak = ak Re ck = ck, 0 ≤ ak/ck and Σa-2k (see also [1], [2], [3], [6], [7] and [8]).

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. M. Macbeath, in November 1965, communicated the following theorem to me which he proved with the aid of the Lefschetz fixed point formula.

Theorem. If Γ is a Fuchsian group and N a torsion free normal subgroup, then the rank of N/[Γ, N] is twice the genus of the orbit space D/Γ where D denotes the hyperbolic plane which Γ acts.

This theorem will follow from a consideration of the exact sequence

*

Let R be a commutative Noetherian ring with identity, and let M be a fixed ideal of R. Then, trivially, ring multiplication is continuous in the ilf-adic topology. Let S be a multiplicative system in R, and let j = js: R → S-1R, be the natural map. One can then ask whether (cf. Warner [3, p. 165]) S-1R is a topological ring in ihe j(M)-adic topology. In Proposition 1, I prove this is the case if and only if M ⊂ p(S), where

Hence S-1R is a topological ring for all S if and only if M ⊂ p*(R), where

T. Evans in [4] introduced a concept generalizing the associative law for groups. Certain very restricted cases of this generalization had occurred, somewhat as side results, in the considerations of A. K. Suškevič [7], D. C. Murdoch [5], [6], and R. H. Bruck [2] associated with quasigroups which obey theorems generalizing certain results in group theory.

In this paper we consider quasigroups obeying an instance of Evans' generalization. We show two means of constructing such quasigroups, and we demonstrate these constructions with several examples. The principal result of this paper is a theorem which reveals a peculiar trait of quasigroups obeying an instance of the generalized associative law we are considering. Such a quasigroup obeys a law expressed with permutations in which the nontrivial permutations involved either are both automorphisms of the quasigroup or else both fail to distribute over any product from the quasigroup.

Let f(τ) be a complex valued function, defined and analytic in the upper half of the complex τ plane (τ=x+iy, y > 0), such that f(τ+λ) = f(τ) where λ is real and f(-1/τ) = γ(-iτ)k f(τ), k being a complex number. The function (—iτ)k is defined as ek log(-iτ) where log(—iτ) has the real value when — iτ is positive and γ is a complex number with absolute value 1. Such functions have been studied by E. Hecke [4] who calls them functions with signature (λ, k, γ). We further assume that f(τ) = O(|y| -c) as y tends to zero uniformly for all x, c being a positive real number. It then follows that f(τ) has a Fourier expansion of the type f(τ) = a0 + Σ an exp(2πinτ/λ) (n = 1,2,…), the series being convergent absolutely in the upper half plane.

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 purpose of this paper is to consider a representation for the elements of a linear topological space in the form of a σ-integral over a linearly ordered subset of V; this ordered subset is what will be called an L basis. The formal definition of an L basis is essentially an abstraction from ideas used, often tacitly, in proofs of many of the theorems concerning integral representations for continuous linear functionals on function spaces.

The L basis constructed in this paper differs in several basic ways from the integral basis considered by Edwards in [5]. Since the integrals used here are of Hellinger type rather than Radon type one has in the approximating sums for the integral an immediate and natural analogue to the partial sum operators of summation basis theory.

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.

In this work we consider the equation

1

where K(x, y) is singular in the sense that it does not properly belong to L2 and f(x) is an arbitrary L2 function.

A Lebesgue measurable function K(x, y) of two variables, having real values on [0.1] × [0.1] is called a singular normal kernel of

1. (i)

   There exists approximating kernels Km(x, y) satisfying

2. (ii)

3. (iii)

4. (iv)

One very interesting and important problem in ring theory is the determination of the position of the singular ideal of a ring with respect to the various radicals (Jacobson, prime, Wedderburn, etc.) of the ring. A summary of the known results can be found in Faith [3, p. 47 ff.] and Lambek [5, p. 102 ff.]. Here we use a new technique to obtain extensions of these results as well as some new ones.

Throughout we adopt the Bourbaki [2] conventions for rings and modules: all rings have 1, all modules are unital, and all ring homomorphisms preserve the 1.

In a recent study of a specific class of quasi-Frobenius rings, Feller has found it useful to introduce the X-rings ([3]). He suggested among others the following topics:

1. (A) Determine the properties of completely indecomposable rings and matrix rings over completely indecomposable rings.

2. (B) Determine the properties of modules over quasi-Frobenius X-rings.

We point out that the completely indecomposable rings are the local quasi-Frobenius rings. Problems (A) and (B) then lead naturally to semi-local quasi-Frobenius rings, and to matrix algebra over local quasi-Frobenius rings. These types of rings are discussed in sections 1 and 2.