We are interested in the question of the decideability of the classical knot problem. A knot is the embedded image of a circle S1 in Euclidean 3-space E3. If L1, L2 are knots, then L1 ≈ L2 if there is an orientation-preserving homeomorphism h: E3 ⟶ E3 with h(L1) = L2. By the “knot problem” we mean: given two arbitrary tame knots L1, L2 (a knot is tame if it is equivalent to a polygonal knot), decide in a finite number of steps whether L1 ≈ L2. The object of this paper is to show that the knot problem is ‘'stably equivalent“ to a problem of deciding membership in the double cosets of a distinguished subgroup K2n of the classical braid group B2n [1].

Coercive closed forms on L2-spaces are studied whose associated L2-semigroups are positivity preserving. Earlier work by other authors is extended by further developing the potential theory of such forms and completed by proving an analytic characterization of those of these forms which have a probabilistic counterpart, i.e., are associated with (special standard) Markov processes. Examples with finite and infinite dimensional state spaces are discussed.

In (8, § 3.2) Steinberg proved the following result.

THEOREM. Let K be a finite field, G′ a simple Chevalley group (“normal type1’) over K. Then every automorphism of G’ is the composite of inner, graph, field, and diagonal automorphisms.

For the meaning of these notions, see (8). Our aim in this note is to indicate how the Theorem may be extended to arbitrary infinite fields K, provided we replace G′ by the group denoted G in (5) and ĝ in (8). This amounts to proving the Theorem for automorphisms of G′ which are induced by automorphisms of G; when K is finite, Steinberg's results show that all automorphisms of G′ arise in this way. As Steinberg points out, the sole use made of the finiteness of K in his argument is in the proof of the following statement: Let U be the subgroup of G′ corresponding to the set of positive roots, and let σ be any automorphism of G′; then Uσ is conjugate to U in G′.

Using the triple associated with a Stone algebra L, as introduced in the first part of this paper (1), we will investigate certain problems concerning the structure of a Stone lattice.

The following topics will be discussed: prime ideals, topological representation, completeness, relative Stone lattices, and the reduced triple.

It is assumed that the reader is familiar with §§ 2–4 of (1). For the sake of convenience, we will write L= 〈C, D, ϕ〉 to indicate that 〈C, D, ϕ〉 is the triple associated with L, and whenever convenient we will write the elements of L as ordered pairs 〈x, a〉, as it is given in (1, § 4, the Construction Theorem).

Given a positive continuous function $\mu $ on the interval $0\,<\,t\,\le \,1$, we consider the space of so-called $\mu $-Bloch functions on the unit ball. If $\mu \left( t \right)\,=\,t$, these are the classical Bloch functions. For $\mu $, we define a metric $F_{z}^{\mu }\left( u \right)$ in terms of which we give a characterization of $\mu $-Bloch functions. Then, necessary and sufficient conditions are obtained in order that a composition operator be a bounded or compact operator between these generalized Bloch spaces. Our results extend those of Zhang and Xiao.

In [11] and [12] we investigated the process of localization of right Noetherian rings R at prime ideals. We shall now extend these investigations to semiprime ideals N of R.

In Section 2 we show that localizing at the injective right R-module E(R/N) is the same as localizing with respect to the multiplicative set

Let Specie denote the set of prime ideals of a commutative ring with identity R, ordered by inclusion; and call a partially ordered set spectral if it is order isomorphic to Spec R for some R. What are some conditions, necessary or sufficient, for a partially ordered set X to be spectral? The most desirable answer would be the type of result that would allow one to stare at the diagram of a given X and then be able to say whether or not X is spectral. For example, it is known that finite partially ordered sets are spectral (see [2] or [5]).

Certain phases are associated with the Kronecker squares and cubes of representations of the finite and of the compact semi-simple groups. These phases are important in giving the symmetry properties of the 1 — jm and 3 — jm symbols of the groups [4; 9]. It is our primary purpose to evaluate these phases.

The Frobenius-Schur invariant [12, p. 142] for an irreducible representation of group G

(1.1)

Let

(1)

and set

(2)

Here . Z is the “unique” (modulo multiplication by nonzero functions) holomorphic vector-field which is tangent to the boundary of the “degenerate generalized upper half-plane”

(3)

In our terminology t = Re z1. We note that ℒ is nowhere elliptic. To put it into context, ℒ is of the type □b, i.e. operators like ℒ occur in the study of the boundary Cauchy-Riemann complex. For more information concerning this connection the reader should consult [1] and [2].

We give a normal form of the cuspidal edge that uses only diffeomorphisms on the source and isometries on the target. Using this normal form, we study differential geometric invariants of cuspidal edges that determine them up to order three. We also clarify relations between these invariants.

Let R be a principal ideal domain, i.e., a commutative ring without zero divisors in which every ideal is principal. The invariant factors of a matrix A with entries in R are the diagonal elements when A is converted to a diagonal form D = UAV, where U, V have entries in R and are unimodular (invertible over R), and the diagonal entries d1 …, dn of D form a divisibility chain: d1|d2| … |dn. Very little has been proved about how invariant factors may change when matrices are added. This is in contrast to the corresponding question for matrix multiplication, where much information is now available [6].

In a paper (5) published in the Proceedings of the Cambridge Philosophical Society, Primrose obtained the formulae for the number of points contained in a non-degenerate quadric in PG(n, s), the finite projective geometry of n dimensions based on a Galois field GF(s). In § 3 of the present paper the formulae for the number of p-flats contained in a non-degenerate quadric in PG(n, s) are obtained. In § 4 an interesting property of a non-degenerate quadric in PG(2k, 2m) is proved. These properties of a quadric will be used in solving some combinatorial problems of statistical interest in a later paper.

We produce a complete description of the lattice of gauge-invariant ideals in ${{C}^{*}}(\Lambda )$ for a finitely aligned $k$-graph $\Lambda $. We provide a condition on $\Lambda $ under which every ideal is gauge-invariant. We give conditions on $\Lambda $ under which ${{C}^{*}}(\Lambda )$ satisfies the hypotheses of the Kirchberg–Phillips classification theorem.

Given a finite positive measure on the Borel subsets of the complex plane with compact support containing infinitely many points, we deduce some formulas for the sequence of monic orthogonal polynomials associated to a rational modification of the measure. These expressions depend on so called functions of the second kind. Some examples for particular Jordan curves are given

A method, due to Élie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2, 2) signature are Einstein of which one is Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is simply connected, then it is shown to be diffeomorphic to ${{\mathbb{R}}^{4}}$. All metrics for the simply connected non-reductive Einstein spaces are given explicitly. There are no non-reductive pseudo-Riemannian homogeneous spaces of dimension two and none of dimension three with connected isotropy subgroup.

The purpose of this note is to characterize those distributive lattices that can be isomorphically embedded in free lattices. If it is known (cf. (2)) that in a free lattice every element is either additively or multiplicatively irreducible, and consequently every sublattice of a free lattice must also have this property. We therefore begin by studying the class of all those distributive lattices in which this condition is satisfied.

We define an iterative interpolation process for data spread over a closed discrete subgroup of the Euclidean space. We describe the main algebraic properties of this process. This interpolation process, under very weak assumptions, is always convergent in the sense of Schwartz distributions. We find also a convenient necessary and sufficient condition for continuity of each interpolation function of a given iterative interpolation process.

Every tree T determines a set of distinct maximal proper subtrees Ti = T — vi, which are obtained by the deletion of an endpoint of T. In this paper we prove that a tree is almost always uniquely determined by this set of its subtrees, and point out two interesting consequences of this result.

In [5], Ulam proposed the following conjecture, which we state in a slightly stronger form due to Harary [1].

ULAM'S CONJECTURE. A graph G with at least three points is uniquely determined up to isomorphism by the subgraphs Gi = G — vi.

Kelly [4] proved the conjecture for trees and Harary and Palmer [3] showed that not all of the Gi are needed in that case by proving Corollary 1 below. If we remove from the list of subgraphs Gi of a graph G all but one graph of each isomorphism type, we obtain a set of Gi which are distinct up to isomorphism.

In 1927, J. H. Redfield (12), discussed some of the links between combinatorial analysis and permutation groups, including such topics as group transitivity, the enumeration of certain geometrical configurations, and the construction of various permutation isomorphs of a given group. Except for a revision of Redfield's treatment of transitivity by D. E. Littlewood (6), this 1927 paper appears to have been overlooked. However, it has recently been described (5) as a remarkable pioneering paper which appears to contain or anticipate virtually all of the enumeration results for graphs which have been discovered and developed during the last thirty years.

Conditions to be imposed on q(x) which ensure the uniqueness of the Green's function associated with the linear second-order differential equation