Birkhoff in [2] poses the following problem:

“Problem 73. Find necessary and sufficient conditions in order that the correspondence between the congruence relations and the (neutral) ideals of a lattice be one-one”.

This problem has been solved by Areškin [l] and Hashimoto [3]. Essentially the conditions reduce to the r e quirement that the lattice be a generalized Boolean algebra.

Hesse's theorem states that “if two pairs of opposite vertices of a quadrilateral are respectively conjugate with respect to a given polarity, then the remaining pair of vertices are also conjugate ”.

In the real projective plane there cannot exist such a quadrilateral, all four sides of which are self-conjugate [1, §5.54]. We shall show that such a quadrilateral exists in PG(2, 3), and that any geometry in which such a quadrilateral exists contains the configuration 134 of PG(2,3). We shall thus provide a synthetic proof of Hesse1 s theorem for a quadrilateral of this type, which, together with [1, § 5.55], constitutes a complete proof of the theorem valid in general Desarguesian projective geometry. We shall also show analytically that a finite Desarguesian geometry which admits a Hessian quadrilateral all of whose sides touch a conic must be of type PG(2, 3n).

This paper continues the work begun in [l] and examines the loss of power when using tests based on the assumption that the variable being sampled has a “complete” normal distribution, when in fact, sampling is from a symmetrically truncated distribution. The hypothesis considered here is the one - sided test for the variance of a normal distribution. Some tables have been computed and they show that appreciable losses in size occur. Some loss occurs in the power too, but this decreases with the alternative value of the variance and the degree of truncation.

In a recent paper [2] Lambek and Moser have introduced the functional equations

1

and

2

in connection with some problems of number theory, in particular in dealing with the sums by pairs of sets of integers. The second may be put into the same form as (1) by the substitutions x = In z, f(ln z) = F(z), h(ln z) = H(z).

In [1] G. Birkhoff stated an algorithm for expressing a doubly stochastic matrix as an average of permutation matrices. In this note we prove two graphical lemmas and use these to find an upper bound for the number of permutation matrices which the Birkhoff algorithm may use.

A doubly stochastic matrix is a matrix of non-negative elements with row and column sums equal to unity and is there - fore a square matrix. A permutation matrix is an n × n doubly stochastic matrix which has n2-n zeros and consequently has n ones, one in each row and one in each column. It has been shown by Birkhoff [1],Hoffman and Wielandt [5] and von Neumann [7] that the set of all doubly stochastic matrices, considered as a set of points in a space of n2 dimensions constitute the convex hull of permutation matrices.

A certain n x n determinant, namely that in (1), provides a rather simple expression for the Tchebychev polynomial Tn(x) = cos n(cos-1x). This determinant also leads to an interesting combinatorial interpretation of the coefficients in that polynomial.

E. Cohen [1], [2] has considered an analogue of the famous Goldbach problem in certain finite rings. In the following another analogue is considered in the ring Jn of residue classes over the integers modulo n.

As 'primes' in Jn we take all residue classes coprime to n. That is, all elements p ϵ Jn which are not factors of zero in Jn. We denote this set by Rn.

Let k be the kernel of a Watson transformation; that is k(x)/x ϵ L2(0,∞), and if x and y are positive,

1

Then if g is the transform of F ϵ L2(0,∞), that is if

2

it i s known that g 2(0,∞), that

3

and that

4

For these results see [1; theorem 79].

It is well known that a symmetric and transitive relation on a set is reflexive wherever it is defined. In this note we show that a converse is true for homomorphic relations in certain classes of algebras.

Consider a class of similar algebras which contains the sub-algebras and quotient algebras of each of its members. Assume also that the direct product A x B of each pair A, B in is also an algebra belonging to . The algebras of , being similar, have the same set of operations. We observe that other operations, called compound operations, may be obtained by composition from the assigned operations.

In Riemannian geometry the autoparallels associated with the affine connexion coincide with the geodesies which arise from the metric. This is not the case in a modification of Riemannian geometry suggested by Lyra. A sufficient condition that the two classes of curves coincide is obtained.

The generalized symmetric group S(n, m) consists of all permutations of mn symbols commutative with

Since each cycle Qi = (1i 2i … mi) is of order m, there are mn permutations within the n cycles, generating an invariant subgroup Q of order mn. Also, there are n! ways of permuting the cycles among themselves, by transformations

where i1, i2, …, in are the symbols 1, 2, …, n in some order [5, p. 39]. The permutations W* form a subgroup Sn* of order n!, isomorphic to the symmetric group Sn.