The purpose of the present note is to give a partial answer to a question raised by Professor Coxeter, namely, if an orthogonal transformation is expressed as a product of orthogonal involutions, how many involutions do we need? Our answer is partial because we are going to consider only non-degenerate symmetric bilinear forms of index 0 and fields of characteristic ≠2. Under these conditions we prove that any orthogonal transformation is the product of at most two orthogonal involutions, which implies that we can write any orthogonal transformation as the product of two involutions.

There are several statements in [3] which require clarification.

Theorem 1 [3, p. 246] states that U3= U2U4 U2-1U4 U2 U4. In fact this is (essentially) the relation O = PUPU-1PU given in [1, p. 91, (7.35)]. To see this we note that 0 = U3, P = U4, U = U4U2U4 (as explained in [1, p. 88]); since U42, = E,

Melzak [2] has shown that there exists a convex pseudopolyhedron Q (the convex hull of a convergent sequence of points together with its limit point) in E3 which is s-universal for triangles, that is, all possible triangles occur (up to similarity) as plane sections of Q. He conjectured that no polyhedron P has this property. In this short note we give an elementary proof of this conjecture.

The purpose of this expository note is to establish the fact mentioned in the title. While this is not difficult and requires no new ideas, it seems worth doing, as such a simple characterization does not appear explicitly in a recent treatise on the subject of flat modules [Bourbaki XXVII, Chapter 1].

In Theorem 4 of [5] Stone stated that the theory of Boolean rings was "mathematically equivalent" to the theory of Boolean spaces without, however, properly defining the phrase "mathematically equivalent". It is the main purpose of this note to establish a precise reformulation of Theorem 4 in [5]. This is accomplished by introducing special classes of maps between Boolean lattices, Boolean rings and Boolean spaces respectively, and showing the categories arising in conjunction with these maps to be equivalent in the sense of Grothendieck [2]. Thus the notion of equivalence of categories will replace the phrase "mathematically equivalent" in [5]. In addition the well-known axiomatic characterization of meet and complementation of Boolean lattices with unit is discussed in analogous terms.

Let G(n;m) denote a graph of n vertices and m edges. Vertices of G will be denoted by x1, …, y1 …; edges will be denoted by (x, y) and triangles by (x, y, z). (G - x1-x2 - … - xk) will denote the graph G from which the vertices x1, …, xk and all edges incident to them have been omitted G - (xi, xj) denotes the graph G from which the edge (xi, xj) has been omitted.

A system of circles is said to form a k-fold packing of the plane if every point of the plane is an interior point of at most k circles. In this paper we consider only lattice packings, that is, we suppose that the centers of the circles form a lattice. We further restrict our consideration to circles of equal radius. Without loss of generality, the circles may have unit radius.

The polytope {5, 3, 3} is one of the six convex regular four-dimensional polytope s, and in some ways is the most complicated of the six, being bounded by 120 dodecahedra. The symbol {p, q, r} to denote a regular figure in four dimensions originated with L. Schláfli: p is the number of edges bounding each face, q is the number of edges of a cell which meet at a vertex of this ceil, and r is the number of ceils that share each edge. The Schláfli symbol can be generalized to denote regular figures in any number of dimensions; in particular, a regular p-gon is symbolized by {p}, and a dodecahedron by {5, 3}.

The permanent of an n-square matrix A = (aij) is defined by

where the summation extends over all permutations σ of the symmetric group Sn. A matrix is said to be a (0, 1)-matrix if each of its entries is either 0 or 1. A (0, 1)-matrix of n-1 the form , where θj = 0 or 1, j = 1,…, n, and Pn is the n-square permutation matrix with ones in the (1, 2), (2, 3),…, (n-1, n), (n, 1) positions, is called a (0, 1)-circulant. Denote the (0, 1)-circulant . It has been conjectured that

1

This paper considers the consistency of the two-sample empty cell test suggested by S. S. Wilks [2]. A description of this test is as follows: Let a sample of n1 independent observations be taken from a population whose cumulative distribution function F1(x) is continuous, but 1 otherwise unknown. Let X(1) < X(2) < … < X(n1) be their order statistics. Let a second sample of n2 observations be taken from a population whose cumulative distribution function is F2(x), assumed continuous, but otherwise unknown.

Let be a polynomial of degree n. Then clearly

(1.1).

(1.2).

and for R > 1

(1.3).

Note that if w = p(z) maps |z|<1 on a domain D of the w-plane then the area of D is given by

According to Pluecker's equations a plane quartic can have at most 28 bitangents. These can be all real as has been shown by Pluecker himself with the help of the example (see [l]):

Several authors have investigated "rings of quotients" of a given ring R. Johnson showed that if R has zero right singular ideal, then the injective hull of RR may be made into a right self injective,- regular (in the sense of von Neumann) ring (see [7] and [12]). In articles by Utumi [10], Findlay and Lambek [6], and Bourbaki [2], various structures which correspond to sub-modules of the injective hull of R are made into rings in a natural manner, in [8], Lambek points out that in each of these cases the rings constructed are subrings of Utumi' s maximal ring of right quotients, which is the maximal rational extension of R in its injective hull. Lambek also shows that Utumi's ring is canonically isomorphic to the bicommutator of the injective hull of RR if R has 1. It thus appears that a "natural" definition of the injective hull of RR as a ring extending module multiplication by R has been carried out only in the case that the injective hull is a rational extension of R. (See [12], [10], or [6] for various definitions of this concept.)

This expository note consists of two parts: In the first we present a homological formulation of Goursat's theorem. In the second we indicate how some rudimentary concepts of homological algebra can be introduced with the help of this form of Goursat's theorem. The second part is addressed to those readers with an algebraic background who wish to be initiated into homological algebra as painlessly as possible. The ideas developed here are close to the spirit of [5, Chapter II, § 6], where further references may be found. In fact, these ideas are very much in the air, and any originality in the present note is purely coincidental.

It is well-known in elementary calculus that if a differentiable function has a monotone increasing curvature, then its curvature is continuous and the circles of curvature at distinct points have no points in common. In particular, two one-sided osculating circles at distinct points of an arc A3 of cyclic order three have no points in common; cf. [l], [2], [3]. The conformai proof given here that any two general osculating circles at distinct points of A3 are disjoint (Theorem 1), may be of interest. We also prove that all but a countable number of points of A3 are strongly conformally differentiable (Theorem 2).

In his 1922 article [l] on functions of bounded variation, Vitali gave a method for constructing monotone non-absolutely continuous functions, generalizing ideas from the ternary set introduced in another connection by Cantor. In [2], Hille and Tamarkin gave a full account of the "middle-third" function, showing it to be a singular distribution function, and finding its characteristic function. In [3], Evans obtained a generalization of the middle - third function by discarding middle intervals of length other than one-third, and obtained algorithms by which the moments of his function could be calculated. Invarious papers, among them [4], Wintner studied infinite convolutions of symmetric Bernoulli distributions, finding a great variety of distributions whose characteristic functions were of the form

A divisible abelian group D can be characterized by the following property: Every homomorphism from an abelian group A to D can be extended to every abelian group B containing A. This together with the result that every abelian group can be embedded in a divisible group is a crucial point in many investigations on abelian groups. It was Baer, [1], who extended this result to modules over an arbitrary ring, replacing divisible groups by injective modules, that is, modules with the property mentioned above. Another proof was found later by Eckmann and Schopf, [3]. This proof assumes the proposition to hold for abelian groups and transfers it in a very simple and elegant manner to modules. In the sequel, we shall refer to this proof as to the Eckmann-Schopf proof.

The principle of superposition of states requires that the states of a dynamical system form a linear vector space. This hypothesis of linearity has usually been regarded as a fundamental postulate in quantum mechanics, of a kind that cannot be explained by classical concepts. Indeed, Dirac [2, p. 14] comments that "the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory, as is shown by the fact that the quantum superposition principle demands indeterminacy in the results of observations in order to be capable of a sensible physical interpretation."

A category with zero-maps is called "quasi-exact" in the sense of D. Puppe (see [4], page 8, 2. 4), if it satisfies the following axioms:

1. (Q1) Every may f is a product f=με of an epimorphisrn εfollowed by a monomorphism μ

1. (Q2) a) Every epimorphism ε has a kernel k = ker ε

b) Every monomorphism μ has a cokernel γ = Coker ε, where Ker and Coker are characterized by the familiar universality properties (see [3], page 252, (1. 10) and (1. 11)).

The subject of alternants and alternating functions was widely studied during the last century (cf. Muir [6]). One of the best-known alternants is actually a double alternant (rows and columns) defined by Cauchy [2] in 1841. Cauchy's result may be stated as follows:If D = [dpq] p, q = l, …, n, where dpq = (xp+yq)-1, then

1.