A subset S of a topological space X is z-embedded in X in case each zero-set of S is the restriction to S of a zero-set of X. (A zero-set is the set of zeros of a real-valued continuous function.) For the basic theory of z-embedding, see [3] and [4] (and see [4] for a comprehensive bibliography of relevant papers).

The (based) homotopy category consists of (based) topological spaces and (based) homotopy classes of maps. In these categories, pull-backs and push-outs do not generally exist. For example, no essential map between Eilenberg-MacLane spaces of different dimensions has a kernel. In this paper we define homotopy pull-backs and push-outs, which do exist and which behave like pull-backs and push-outs, and we give some of their properties. Applications may be found in [3; 5; 6 and 14].

Given a class of topological spaces and a class of mappings of topological spaces, the -résolvant of is denned to be the class of topological spaces all of whose -images lie in . Whenever is closed under composition and includes identity maps, is easily seen to be the largest class of spaces smaller than which is closed under -images.

The Banach space of bounded continuous real or complexvalued functions on a topological space X is denoted C(X). An averaging operator for an onto continuous function ϕ : X → Y is a bounded linear projection of C(X) onto the subspace ﹛ƒ ∈ C(X) : f is constant on each set ϕ -1(y) for y ∈ Y﹜. The projection constant p(ϕ) for an onto continuous map ϕ is the lower bound for the norms of all averaging operators for ϕ ﹛p(ϕ) = ∞ if there is no averaging operator for ϕ).

Let G be a locally compact separable unimodular group. The general theory [18] assigns to G a measure space (Λ, μ) whose points ƛ index a family of unitary factor representations of G in such a way that if U ƛ corresponds to ƛ and then

for all .

A Kaehler manifold of constant holomorphic curvature is called a complex space form. By a Kaehler submanifold we mean a complex submanifold with the induced Kaehler metric. B. Smyth [5] has studied a complete Einstein- Kaehler hypersurface in a complete and simply connected complex space form and classified completely the hypersurface. The local version of this result has been shown to be true by S. S. Chern [1], and partially by T. Takahashi [6] independently.

Let F be a field, Mn(F) be the vector space of all w-square matrices with entries in F and a subset of Mn(F). It is of interest to determine the structure of linear maps T : Mn(F) →Mn(F) such that . For example: Let be GL(n, C), the group of all nonsingular n X n matrices over C [5]; the subset of all rank 1 matrices in MmXn(F) [4] (MmXn(F) is the vector space of all m X n matrices over F) ; the unitary group [2] ; or the set of all matrices X in Mn(F) such that det(X) = 0 [1]. Other results in this direction can be found in [3].

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].

This paper arises in the attempt to solve the following problem related to the Zariski Problem. Let A be a polynomial ring in three variables over a field, . Suppose there is a subring B of A such that k ⊆ B and there is variable t over B such that B[t] = A. Then is it true that B is a polynomial ring over k?

In an interesting, but apparently largely unknown, paper [1] Halin has introduced the concept of a primitive set of vertices of a graph. This concept, or rather a slight modification of it, seems to provide the key to a new approach to the wrell-known and important class of graph-theoretical problems centering on the notion of separation. As is well known, if A, B, C are sets of vertices of a (non-oriented) graph X, C is said to separate A and B if and only if every AB-path in X contains a vertex of C.

For certain classes of singular symmetric differential operators L of order 2n, this paper considers the problem of determining sufficient conditions for L to be of limit point type or of limit circle type. The operator discussed here is defined by

In an earlier paper [7], we have studied the existence, uniqueness and asymptotic behavior of solutions to certain infinite systems of linear differential equations with constant coefficients. In the present paper we are interested in systems of nonlinear equations whose coefficients are not necessarily constants; more specifically, we are concerned with infinite systems of the form

L'objet de cet article est essentiellement de poursuivre la généralisation entreprise par Sabidussi [4] du concept d'ensemble primitif de sommets d'un graphe, introduit par Halin [1].

Let denote the class of cardinal splines S(x) of degree n (n ≧ 1) having their knots at the integer points of the real axis. We assume that the knots are simple so that . Recently Schoenberg [3] has studied cardinal splines such that S(x) interpolates the exponential function tx at the integers and

In the last few years many results have appeared which deal with questions of how various algebraic properties of the symmetric elements of a ring with involution, or the subring they generate, affect the structure of the whole ring. If the ring has an identity, similar questions may be posed by making assumptions about the symmetric units or subgroup they generate. Little seems to be known about the special units which exist in rings with involution, although several questions of importance have existed for some time. For example, given a simple ring with appropriate additional assumptions, is the unitary group essentially simple? Also, what can be said about the structure of subspaces invariant under conjugation by all unitary or symmetric units (see [7])?

In this paper we generalize a theorem of Lyndon's [7], which states that a one-relator group G = F/(r) (F is free and r Ç F) has cohomological dimension cd (F/(r)) ≧ 2 if and only if the relator r is not a proper power in F. His proof relies on the Identity Theorem and recently he has shown [8] how a generalized version of this theorem and a generalized version of the Freiheitsatz can be simultaneously obtained by the methods of combinatorial geometry. These generalizations refer to a situation where the free group F is replaced by a free product of subgroups of the additive group of real numbers.

If H is a (complex) Hilbert space and is a collection of (closed linear) subspaces of H it is easily shown that the set of all (bounded linear) operators acting on H which leave every member of invariant is a weakly closed operator algebra containing the identity operator. This algebra is denoted by Alg . In the study of such algebras it may be supposed [4] that is a subspace lattice i.e. that is closed under the formation of arbitrary intersections and arbitrary (closed linear) spans and contains both the zero subspace (0) and H. The class of such algebras is precisely the class of reflexive algebras [3].

The existence of nonzero fixed points of positive contractions in L1 spaces has received considerable attention in recent years. In 1966, Dean and Sucheston [1] and independently, Neveu [5] showed that a positive contraction has a strictly positive invariant function if and only if for any measurable subset A with positive measure, where 1 is the constant function of value one.

A Latin K-cube A of order n is a k-dimensional array where

runs through the distinct elements 0, 1, …, n — 1 as j runs from 0 to n — 1.

In this paper we introduce a new approximation operator of the Arato-Renyi type and study its properties. Special cases of our operator are the power-series, of W. Meyer-König and K. Zeller (see [9]) and the generalized Berenstein power series introduced by A. Jakimovski and D. Leviatan in [5] and analyzed by them in [6].