This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

In this paper a procedure is given whereby, from a representation of an ideal contained in the radical, explicit representations of a Lie algebra by matrices can be constructed in an algebraically closed field of characteristic zero. The construction is sufficiently general to permit one arbitrary eigenvalue to be assigned to the representation of each basis element of the radical not in the ideal. The theorem of Ado is proved as an application of the construction. While Ado's theorem has several proofs (1; 3; 5; 6), the present one has a value in its explicitness and in the fact that the degree of the representation can be given.

Let En be the n-dimensional euclidean real space and the group of unimodular affine transformations which operates on it. It is known that the sets of linear h-spaces Lh (0 < h < n) have no invariant measure with respect to u (5). We wish now to consider sets of elements

1.1

composed by q parallel subspaces of dimensions h1, h2, … , hq which transform transitively by .

The purpose of this paper is to discuss conjugate points in symmetric spaces. Although the results are neither surprising nor altogether unknown, the author does not know of their explicit occurrence in the literature.

Briefly, conjugate points in the tangent bundle to the tangent space at a point of a symmetric space are characterized in terms of the algebraic structure of the symmetric space. It is then shown that in the simply connected case the first conjugate locus coincides with the minimum (cut) locus. The interest in this last fact lies in its identification of a more or less locally and analytically defined set with one which includes all the topological interest of the space.

The topological concept of H-space (7) has an analytic counterpart which so far has not been considered in the literature. We define: A complex-analytic manifold S will be called an analytic H-space if it is capable of carrying a continuous binary composition

with the following properties (i) and (ii).

In this paper, we shall be concerned with bounded, holomorphic functions of the form

where

(1)

(2)

and

(3)

B(z{an}) is called a Blaschke product, and any sequence {an} which satisfies (2) and (3) is called a Blaschke sequence. For a general discussion of the properties of Blaschke products, see (18, pp. 271-285) or (14, pp. 49-52).

According to a theorem due to Riesz (15), a Blaschke product has radial limits of modulus one almost everywhere on C = {z: |z| = 1}. Moreover, it is common knowledge that, if a Blaschke product has a radial limit at a point, then it also has an angular limit at the point (see 14, p. 19 and 6, p. 457).

In this paper various concepts intrinsically defined by the differential equation

1.1

are interpreted geometrically by concepts analogous to those in the Minkowski plane. This is carried out in § 2. The point of such a development is that one may apply the techniques or transfer known results in the theory of curves (in particular, convex curves) to (1.1), thereby gaining an additional tool in the investigation of this equation. For an application of a result obtained in this way, namely (3.12), see (4).

Throughout this paper, R(t) is a real-valued, continuous function of t on the real line (— ∞ < t < + ∞) and only the real solutions of (1.1) are considered.

Let

be an ordinary differential operator of order h whose coefficients are (η, η) matrices defined on the interval 0 ≤ x < ∞, hη = n = 2v. Let the operator L0 be formally self adjoint and let v boundary conditions be given at x = 0 such that the eigenvalue problem

(1.1)

has no non-trivial square integrable solution. This paper deals with the perturbed operator L∈ = L0 + ∈q where ∈ is a real parameter and q(x) is a bounded positive (η, η) matrix operator with piecewise continuous elements 0 ≤ x < ∞. Sufficient conditions involving L0, q are given such that L∈ determines a selfadjoint operator H∈ and such that the spectral measure E∈(Δ′) corresponding to H∈ is an analytic function of ∈, where Δ′ is a subset of a fixed bounded interval Δ = [α, β]. The results include and improve results obtained for scalar differential operators in an earlier paper (3).

Real functions

may be identified with elements x = (x0, x1, x2, … ) of the sequence space l1 Since the unit sphere S∞ of l1 is compact under the weak* topology = topology of co-ordinatewise convergence, a countably additive measure on S∞ is induced by a positive linear functional E (integral) on C(S∞), the weak* continuous real-valued functions on S∞. There exists a natural integral over S∞ reducing to

when f is a function of x0 alone. The partial sums Sn = Sn(x) of the power series for x(t) then form a martingale and zero-or-one phenomena appear. In particular, if R(x) is the radius of convergence of the series and e is the base of the natural logarithms, it turns out that R(x) = e for almost all x in S∞. Applications of the integral to the theory of numerical integration, the original motivation, will appear in a later paper.

If A and B are sets then A — B = {x| x £A, x ∉ B}. This notation is also used if A and B are linear spaces. If X and Y are Banach spaces an embedding of X into Y is a continuous linear mapping u of X onto a closed subspace of F which is 1 — 1. In this case X is said to be embedded in Y. If |ux| = |x| for every x ∈ X (| … | stands for norm), then u embeds Xisometrically into Y. If u is onto then X and Y are isomorphic and if, in addition, |ux| = |x| for every x ∈ X, then X and Y are isometric. Then an embedding u has a continuous inverse u-1 (4, p. 36) defined on uX and this fact is used below without further reference. The conjugate space of X is denoted by X′. Unless otherwise noted, all topological spaces considered are Hausdorff spaces.

In this paper we deal with trivalent planar maps in which the boundary of each country (or “face“) is a simple closed curve. One vertex is distinguished as the root and its three incident edges are distinguished as the first, second, and third major edges. We determine the average number of Hamiltonian polygons, passing through the first and second major edges, in such a “rooted map” of 2n vertices. Next we consider the corresponding problem for 3-connected rooted maps. In this case we obtain a functional equation from which the average can be computed for small values of n.

Let A = (ank) and x = {sn} (n,k = 0,1,2, … ) be a matrix and a sequence of complex numbers, respectively. We write formally

(1.1)

and say that the sequence x is summable A to the sum t or that the A matrix sums the sequence x to the value t if the series in (1.1) converges and

exists and equals t. We say that the matrix A is regular provided it sums every convergent sequence to its limit.

We shall consider finite groups of order of g which satisfy the following condition:

(*) There exists a prime p dividing g such that if P ≠ 1 is an element of p-Sylow group ofthen the centralizer(P) of P incoincides with the centralizer() of in.

This assumption is satisfied for a number of important classes of groups. It also plays a role in discussing finite collineation groups in a given number of dimensions.

Of course (*) implies that is abelian. It is possible to obtain rather detailed information about the irreducible characters of groups in this class (§ 4).

It is well known (1, p. 162) that the lattice of subalgebras of a finite Boolean algebra is dually isomorphic to a finite partition lattice. In this paper we study the lattice of subalgebras of an arbitrary Boolean algebra. One of our main results is that the lattice of subalgebras characterizes the Boolean algebra. In order to prove this result we introduce some notions which enable us to give a characterization and representation of the lattices of subalgebras of a Boolean algebra in terms of a closure operator on the lattice of partitions of the Boolean space associated with the Boolean algebra. Our theory then has some analogy to that of the lattice theory of topological vector spaces. Of some interest is the problem of classification of Boolean algebras in terms of the properties of their lattice of subalgebras, and we obtain some results in this direction.

Let Δn (n > 0) denote the subset of the Euclidean (n + 1)-dimensional space defined by

A subset σ of Δn is called a face if there exists a sequence 0 ≤ i1 ≤ i2 ≤ … < im ≤ n such that

and the dimension of σ is defined to be (n — m). Let denote the union of all faces of Δn of dimensions less than n. A topological space Y is called solid if any continuous map on a closed subspace A of a normal space X into Y can be extended to a map on X into Y. By Tietz's extension theorem, each face of Δn is solid. The present paper is concerned with a generalization of the following theorem which seems well known.

The relation between the notions of measurability and continuity of a function has received a great deal of attention. The best-known result in this connection is the Vitali-Lusin theorem. Various versions of it can be found in (2; 3; 5; 8; 9). We prove one in this paper (Theorem 3.5) under very weak assumptions and state the classical one in Corollary 3.6. Functions satisfying the property stated in the theorem are frequently called quasicontinuous.

Here we are interested in the notion of approximate continuity, which we call μ-continuity for short and which is closely connected to differentiation. It was first introduced by Lebesgue (4) and it has so far required knowledge of density theorems in order to prove its relation to measurability. In this paper, making use of a certain type of Vitali property, we prove first the relation between μ-measurable functions and those μ-continuous almost everywhere (Theorems 3.8 and 3.9). Then, in §4, with Theorem 3.8 as the main tool, we derive several theorems about density and differentiation which extend results found in (1; 3; 7; 8).

In § 1 we give a characterization of a lattice L that is freely α-generated by a given partially ordered set P. In § 2 we obtain a representation of an element of such a lattice as a sum (product) of additively (multiplicatively) irreducible elements which, although not unique, has some of the desirable features of the canonical representation, in Whitman (2), of an element of a free lattice. The usefulness of this representation is illustrated in § 3, where some further arithmetic properties of these lattices are derived.

We use + and . for the binary operations of lattice addition and multiplication, and Σ and II for the corresponding operations on arbitrary sets and sequences of lattice elements. In other respects the terminology will be the same as in Crawley and Dean (1).

In this paper we shall derive a concise formula for the number of Euler graphs on n labelled nodes and k edges. An Euler graph is a connected graph in which every node has even valency, where by the valency of a node is meant the number of edges which are incident with that node. Throughout most of the paper we shall be dealing with graphs whose nodes have even valencies but which may or may not be connected. For convenience we shall refer to these graphs as Euler graphs, although the usage is not, strictly speaking, correct. We shall impose the condition of connectedness in § 4.

It is known that every sublattice A of a free lattice satisfies the following conditions:

(W) For all a, b, c, d ∈ A, if ab ≤ c + d, then ab ≤ c or ab ≤ d or a ≤ c+dor b ≤ c + d.

(SD) For all u, a, b, c ∈ A, if u = a + b = a + c, then u = a + bc.

(SD′) For all u, a, b, c ∈ A, if u = ab = ac, then u = a(b + c).

In fact, (W) is one of the four conditions used in Whitman (4) to characterize free lattices, and in Jónsson (3) it was shown that (SD) and (SD′) follow from Whitman's canonical representations of elements of a free lattice.

This note is concerned with lattices that satisfy one or more of the above conditions, and especially with finite lattices that satisfy all three conditions.