Let k be an algebraically closed field and let x1, x2, . . . , xn be indeterminates. Denote by Rn the ring k[[x1, x2, … , xn]] of power series in the xi With coefficients in the field k. Let and be two ideals in this ring. Then and will be said to be analytically equivalent if there is an automorphism T of Rn such that T() = . and will be called analytically equivalent under T.

The above situation can be described geometrically as follows. The ideals and can be regarded as defining algebroid varieties V and V' in (x1, x2, … , xn)-space, and these varieties will be said to be analytically equivalent under T.

The automorphism T can be expressed by means of equations of the form :

where the determinant is not zero and the fi are power series of order not less than two (that is to say, containing terms of degree two or more only).

Horadam (11) has established the existence of a locus L in [8] (projective 8-space) having order 45 and dimension 4, which is invariant under a group of order 51840 X 81 (the Clifford similarity transform group CT). Associated with CT are two other groups, the Clifford collineation group CG of order 81, and the Clifford substitution group CS of order 51840. Furthermore, CS may be regarded as either a subgroup of CT, or a symplectic group of index matrices of size 4. Among the matrices of size 9 which perform the operations of CT, there is a set of 81 involutory, symmetric, orthogonal matrices JW. As collineation matrices in [8], these produce 81 pairs of invariant spaces Σ, Π of dimensions 3 and 4 respectively. These [4]'s give rise to a configuration C invariant under the operations of CT, consisting of 360 points, 1080 lines, 120 Jacobian planes, and 81 [4]'s, and their various inter-relationships.

A well-known theorem of Ramsay (8; 9) states that to every n there exists a smallest integer g(n) so that every graph of g(n) vertices contains either a set of n independent points or a complete graph of order n, but there exists a graph of g(n) — 1 vertices which does not contain a complete subgraph of n vertices and also does not contain a set of n independent points. (A graph is called complete if every two of its vertices are connected by an edge; a set of points is called independent if no two of its points are connected by an edge.) The determination of g(n) seems a very difficult problem; the best inequalities for g(n) are (3)

It is not even known that g(n)1/n tends to a limit. The lower bound in (1) has been obtained by combinatorial and probabilistic arguments without an explicit construction.

If a group contains two subgroups A and B such that every element of the group is either in A or can be represented uniquely in the form aba', a, a’ in A, b ≠ 1 in B, we shall call the group an independent ABA-group. In this paper we shall investigate the structure of independent ABA -groups of finite order.

A simple example of such a group is the group G of one-dimensional affine transformations over a finite field K. In fact, if we denote by a the transformation x’ = ωx, where ω is a primitive element of K, and by b the transformation x’ = —x + 1, it is easy to see that G is an independent ABA -group with respect to the cyclic subgroups A, B generated by a and b respectively.

Since G admits a faithful representation on m letters (m = number of elements in K) as a transitive permutation group in which no permutation other than the identity leaves two letters fixed, and in which there is at least one permutation leaving exactly one letter fixed, G is an example of a Frobenius group.

In a recent paper (3)* the following theorem was proved for real matrices.

Theorem 1. If A is a symmetric matrix and Q is a skew-symmetric matrix such that A + Q is non-singular, then

1

is a cogredient automorph (c.a.) of A whose determinant is + 1 and having theproperty that A and I + P span the same row space.

Conversely, if P is a c.a. of A whose determinant is + 1 and if P has theproperty that I + P and A span the same row space, then there exists a skew symmetricmatrix Q such that P is given by equation (1).

Theorem 1 reduces to the well-known Cayley parameterization in the case where A is non-singular. A similar and somewhat simpler result (Theorem 4) was given for the case when the underlying field is the complex field. It was also shown that the second part of the theorem (in either form) is false when the characteristic of the underlying field is 2. The purpose of this paper is to simplify the proof of Theorem 1 and at the same time, to extend these results to matrices over an arbitrary field of characteristic ≠ 2.

Let K* be an associative algebra over a field F with identity u, and let u, e1, e2, … , be a basis for K*. Denote by K the linear space, over F, spanned by the ei,i = 1, 2, … . Then for x, y in K, xy = αu + a, where a ∈ K. Define h(x, y) = α and x.y = a. With respect to the operation thus defined, K becomes an algebra over F satisfying

1

Further, the bilinear form h(x, y) is associative on K. Any algebra, over a field F, which possesses an associative bilinear form h(x, y) and satisfies (1) will be called a algebra. It is not difficult to show that any algebra K can be obtained from a unique associative algebra K* with identity by the process described above. The algebra K* will be called the associated associative algebra of K.

Let G be a finite group of order g, and

be an absolutely irreducible representation of degree fμ over a field of characteristic zero. As is well known, by using Schur's lemma (1), we can prove the following orthogonality relations for the coefficients :

1

It is easy to conclude from (1) the following orthogonality relations for characters:

where

and is 1 or 0 according as t and s are conjugate in G or not, and n(t) is the order of the normalize of t.

Let Mn denote the algebra of n-square matrices over the complex numbers; and let Un, Hn, and Rk denote respectively the unimodular group, the set of Hermitian matrices, and the set of matrices of rank k, in Mn. Let ev(A) be the set of n eigenvalues of A counting multiplicities. We consider the problem of determining the structure of any linear transformation (l.t.) T of Mn into Mn having one or more of the following properties:

• (a) T(Rk) ⊆ for k = 1, …, n.

• (b) T(Un) ⊆ Un

• (c) det T(A) = det A for all A ∈ Hn.

• (d) ev(T(A)) = ev(A) for all A ∈ Hn.

We remark that we are not in general assuming that T is a multiplicative homomorphism; more precisely, T is a mapping of Mn into itself, satisfying

T(aA + bB) = aT(A) + bT(B)

for all A, B in Mn and all complex numbers a, b.

We consider the group of rotations in three-dimensional Euclidean space, leaving the origin fixed. These rotations are represented by real orthogonal third-order matrices with positive determinant. It is known that this rotation group contains free non-abelian subgroups of continuous rank (see 1).

In this paper we shall prove the following conjectures of J. de Groot (1, pp. 261-262):

Theorem 1. Two rotations with equal rotation angles a and with arbitrary but different rotation axes are free generators of a free group, if cos α is transcendental.

Theorem 2. A free product of at most continuously many cyclic groups can be isomorphically represented by a rotation group.

Let R be a commutative ring with identity 1. A character of an R-algebra E is a homomorphism from E onto R, regarded as an algebra over itself. If (Eα) is a family of R-algebras indexed by a set A and if

then for every β ∈ A and every character vβ of Eβ, vβ o prβ is a character of E where prβ is the projection homomorphism from E onto Eβ. Further if A is finite and if the only idempotents of R are 0 and 1 (equivalently, if R is not the direct sum of two proper ideals), it is easy to see that every character of E is of this form.

It is well known (2, 4) that the ring of all real (complex) continuous functions on a compact Hausdorff space can be characterized algebraically as a Banach algebra which satisfies certain additional intrinsic conditions. It might be expected that rings of all continuous functions on other topological spaces also have algebraic characterizations. The main purpose of this note is to discuss two such characterizations. In both cases the characterizations are given in the terms of the theory of F-brings (1). In one case a characterization is given for the ring of all (real) continuous functions on a generalized P-space, that is, a zero-dimensional topological space in which the class of open-closed sets forms a σ-algebra. A Hausdorff generalized P-space is a P-space in the terminology of (3). In the other case a theorem of Sikorski (6) is employed to give a characterization of the ring of all (real) continuous functions on an upper X1-compact P-space.

The present note is concerned with the existence and properties of certain types of extensions of Banach algebras which allow a faithful representation as the normed ring C(E) of all bounded continuous real functions on some topological space E. These Banach algebras can be characterized intrinsically in various ways (1); they will be called function rings here. A function ring E will be called a normal extension of a function ring G if E is directly indecomposable, contains C as a Banach subalgebra and possesses a group G of automorphisms for which C is the ring of invariants, that is, the set of all elements fixed under G. G will then be called a group of automorphisms of E over C. If E is a normal extension of C with precisely one group of automorphisms over C, which is then the invariance group of C in E, then E will be called a Galois extension of C. Such an extension will be called finite if its group is finite.

In (1) the lattice of all equivalence relations on a set S was studied and many important properties were established. In (2) and (3) the lattice of all geometries on a set S was studied and it was shown to be a universal lattice which shares many properties with the lattice of equivalence relations on S. In this paper we shall give the definition of a partition of type n and investigate the lattice formed by all partitions of type n on a fixed set S. It will be seen that a partition of type one on S can be considered as an equivalence relation on S and similarly a partition of type two on S can be considered as a geometry on S as defined in (2). Thus we shall obtain a unified theory of lattices of equivalence relations, lattices of geometries and partition lattices of higher types.

In a recent paper Mostert and Shields (4) showed that if a space homeomorphic to the non-negative real numbers is a certain type of topological semigroup, then the semigroup must be that of the non-negative real numbers with the usual multiplication. Somewhat earlier Faucett (2) showed that if a compact connected ordered space is a suitably restricted topological semigroup, then it must be both topologically and algebraically the same as the unit interval of real numbers with its usual multiplication.

In studying certain binary relations on topological spaces there have become known (see, in particular, Wallace (5) and the author (3)) a number of properties analogous to those possessed by topological semigroups. Because of these analogous properties between relations and semigroups the author was motivated by the general nature of the Faucett and Mostert-Shields results (that is, that the multiplication assumed turned out to be the same as the usual multiplication) to feel that certain relations on a connected ordered space should turn out to be the same as the orders whose order topologies are the topology on the space.

Let T, E be two sets and I ⊂ β(T),l, ⊂ β (E) two tribes. For every n ∈ N* denote by En the product E{1....,n} and by the tribe. For every x ∈ E let ux be a mapping of T into T. For x = (x1,… , xn) ∈ ∈En define ux = uxo … oxl and suppose that {(t, xi1, …, xn) ∣u(x1.....,xn)(t) ∈ A} .

Let m be the Banach space of functions defined on T, real-valued, bounded and I measurable with norm

A function

an real, is called typically-real of order one in the closed region ∣z∣ ≦ R if it satisfies the following conditions (6).

(1) f(z) is regular in ∣z∣ ≦ R

(2) ∮{f(z)} > 0 if and only if ∮{z} >0.

The same function is called typically-real of order p, p a positive integer greater than one, if it satisfies condition (1) above and in addition the followingcondition (4; 5):

(2’) there exists a constant ρ, 0 < p < R, such that on every circle ∣z∣ = r, ρ, 0 < p < R, ∮{f(z)} changes sign exactly 2p times.

The purpose of this paper is to give conditions under which the absolute convergence of the subsequence of odd or of even approximants to a continued fraction implies convergence of the continued fraction. In § 2 we consider the problem in general, and in § 3 we impose a condition which gives absolute convergence of the odd or of the even part of the continued fraction and state conditions which imply convergence of the continued fraction.

Hermite's function Hn(x) is denned for all complex values of x and n by

where F (α; γ; x) is Kummer's function with the customary indices omitted. It satisfies the differential equation

1.1

of which

is a second solution. Every solution of (1.1) is an entire function.

On replacing the parameter n in Bessel's differential equation

1.1

by the operator y(∂/∂y), the partial differential equation Lu = 0 is constructed, where

1.2

This operator annuls u(x, y) = v(x)yn if, and only if, v(x) satisfies (1.1) and hence is a cylindrical function of order n. Thus every generating function of a set of cylindrical functions is a solution of Lu = 0.

It is shown in § 2 that the partial differential equation Lu = 0 is invariant under a three-parameter Lie group. This group is then applied to the systematic determination of generating functions for Bessel functions, following the methods employed in two previous papers (4; 5).

The Bessel polynomials, which arise as solution of the classical wave equation in spherical co-ordinates, are defined by Krall and Frink (3) by the equation

1

The purpose of this paper is to present some series of products of these polynomials when the two arguments are different as in the case of Legendre and Hermite polynomials. Such an explanation was given by Brafman (2), namely :

2

These series will be stated and proved in § 2.