Let Mm be a closed PL manifold of dimension m. Then a concordance between two PL-homeomorphisms h0, h1:M → M is a PL-homeomorphismH: M × I → M × I such that H|M × 0 = h0 and H|M × 1 = h. Concordance is an equivalence relation and in his paper [2], M. Kato classifies PL-homeomorphisms of Sp × Sq up to concordance. To do this he treats first the problem of classifying those homeomorphisms that induce the identity in homology, and then describes the automorphisms of the cohomology ring that can arise from homeomorphisms of Sp × Sq. In this paper we show that for sufficiently connected PL-manifolds that embed in codimension 1, one can extend Kato's classification of the homeomorphisms that induce the identity in homology.

It is known that the four-colour problem for the faces of a map on a sphere is isomorphic with the four-colour problem for the vertices of its dual, and the problem is here discussed in the latter form. The isomorphs described below are concerned with codes for the four colours and for a change in colour as we pass along an edge from vertex to vertex. In the first (algebraic) isomorph, the coding involves the four fourth roots of unity, and leads to a graphical representation in the complex plane. In the second (arithmetical) isomorph, the coding involves the integers mod 4, and also the face-edge incidence matrix of the dual.

Let C(I) denote the linear space of continuous functions from the compact interval I = [a, b] into n-dimensional real arithmetic space Rn, and let C′(I) be the subspace of continuously differentiable functions on I. A general boundary value problem for a first-order system of n ordinary differential equations on I is given by

In this paper, we mainly study operator spaces which have the locally lifting property $\left( \text{LLP} \right)$. The dual of any ternary ring of operators is shown to satisfy the strongly local reflexivity, and this is used to prove that strongly local reflexivity holds also for operator spaces which have the $\text{LLP}$. Several homological characterizations of the $\text{LLP}$ and weak expectation property are given. We also prove that for any operator space $V$, ${{V}^{**}}$ has the $\text{LLP}$ if and only if $V$ has the $\text{LLP}$ and ${{V}^{*}}$ is exact.

Using the methods developed in (2 and 3), in this paper we study some properties of the configuration of generators and points of a cone in an w-dimensional finite projective space. The configuration of secants and external points of a quadric in a finite plane of even characteristic is also studied. I t is shown that these configurations lead to several series of partially balanced incomplete block (PBIB) designs. PBIB designs are defined in Bose and Shimamoto (1). A PBIB design with m associate classes is an arrangement of v treatments in b blocks such that.

Let us consider the following problem. Let there be v elements x1 , . . . , xv and v sets S1, . . . , Sv such that every set contains exactly k distinct elements and every pair of sets has exactly λ distinct elements in common. To avoid trivial situations we shall in general assume that 0 < λ < k < v — 1. This is known as a v, k, λconfiguration or design. We can give an equivalent characterization of a configuration in terms of a matrix A = [aij], called its incidence matrix, by writing the elements x1 , . . . , xv row and the sets S1, . . . , Sv in a column and setting aij = 1 if xj is in Si and aij = 0 if xj is not.

The theory of the systems of partial differential equations which arise in connection with the invariant differential operators on a Riemannian manifold may be developed by methods based on those of potential theory. It is therefore natural to consider in the same context the theory of elliptic differential equations, in particular those which are self-adjoint. Some results for a tensor equation in which appears, in addition to the operator Δ of tensor theory, a matrix or double tensor field defined on the manifold, are here presented. The equation may be written

in a notation explained below.

The theory of induced representations has served as a powerful tool in the computations of algebraic K-theory and L-theory ([2], [7], [4, 5], [9], [10, 11, 12, 13], [14], [17], [18]). In this paper we show how to apply this theory to obtain induction theorems for the higher Whitehead groups of Waldhausen. The same technique applies to the analogs of Whitehead groups in unitary K-theory and in L-theory.

For any ring A with unit, let K(A) be the spectrum of the algebraic K-theory of A ([8, p. 343]). Given a discrete group Γ and a subring R of the rational numbers, Loday defines a map of spectra:

*

where (BΓ) is the classifying space of Γ union with a disjoint base point and RΓ is the group-ring of Γ over R.

Let $p$ be an odd prime number, and let $F$ be a field of characteristic not $p$ and not containing the group ${{\mu }_{p}}$ of $p$-th roots of unity. We consider cyclic $p$-algebras over $F$ by descent from $L\,=\,F\left( {{\mu }_{p}} \right)$. We generalize a theorem of Albert by showing that if ${{\mu }_{{{p}^{n}}}}\,\subseteq \,L$, then a division algebra $D$ of degree ${{p}^{n}}$ over $F$ is a cyclic algebra if and only if there is $d\,\in \,D$ with ${{d}^{{{P}^{n}}}}\,\in \,F\,-\,{{F}^{P}}$. Let $F(p)$ be the maximal $p$-extension of $F$. We show that $F(p)$ has a noncyclic algebra of degree $p$ if and only if a certain eigencomponent of the $p$-torsion of $\text{Br(F(p)(}{{\mu }_{p}}\text{))}$ is nontrivial. To get a better understanding of $F(p)$, we consider the valuations on $F(p)$ with residue characteristic not $p$, and determine what residue fields and value groups can occur. Our results support the conjecture that the $p$ torsion in $\text{Br}(F(p))$ is always trivial.

Let $\mathbb{F}$ be a finite real abelian extension of $\mathbb{Q}$. Let $M$ be an odd positive integer. For every squarefree positive integer $r$ the prime factors of which are congruent to 1 modulo $M$ and split completely in $\mathbb{F}$, the corresponding Kolyvagin class ${{k}_{r}}\in {{\mathbb{F}}^{\times }}/{{\mathbb{F}}^{\times M}}$ satisfies a remarkable and crucial recursion which for each prime number $\ell $ dividing $r$ determines the order of vanishing of ${{k}_{r}}$ at each place of $\mathbb{F}$ above $\ell $ in terms of ${{k}_{r/\ell }}$. In this note we give the recursion a new and universal interpretation with the help of the double complex method introduced by Anderson and further developed by Das and Ouyang. Namely, we show that the recursion satisfied by Kolyvagin classes is the specialization of a universal recursion independent of $\mathbb{F}$ satisfied by universal Kolyvagin classes in the group cohomology of the universal ordinary distribution à la Kubert tensored with $\mathbb{Z}/M\mathbb{Z}$. Further, we show by a method involving a variant of the diagonal shift operation introduced by Das that certain group cohomology classes belonging (up to sign) to a basis previously constructed by Ouyang also satisfy the universal recursion.

There are several notions of positive definiteness for functions on topological groups, the two of which are: Bochner type positive definite functions and integrally positive definite functions. The class P(F) of positive definite functions for the class F can be defined more generally and it is interesting to observe that a change in F produces a different class P(F) of positive definite functions. The purpose of this paper is to study the functions in P(LP(G)) which are positive definite for the class LP(G) (1 ≦ p < ∞), where G is a compact or locally compact group. The relevant information about the class P(F) can be found in [1; 2; 3 and 8].

In this section we fix some notations and give a definition of an area measure on a differentiate manifold, where throughout the paper the word differentiable implies differentiability of class C∞. Let M denote a differentiate manifold of dimension n and call a set of m linearly independent vectors {e1, … , em} at a point of M an m-frame of M. The set E′ of all such m-frames can be given the structure of a differentiable fibre bundle over M and we denote the projection of E' onto M by π′.

We study Tychonoff spaces $X$ with the property that, for all topological embeddings $X\,\to \,Y$, the induced map $C(Y)\,\to \,C(X)$ is an epimorphism of rings. Such spaces are called absolute $\mathcal{C}\mathcal{R}$-epic. The simplest examples of absolute $\mathcal{C}\mathcal{R}$-epic spaces are $\sigma $-compact locally compact spaces and Lindelöf $P$-spaces. We show that absolute $\mathcal{C}\mathcal{R}$-epic first countable spaces must be locally compact.

However, a “bad” class of absolute $\mathcal{C}\mathcal{R}$-epic spaces is exhibited whose pathology settles, in the negative, a number of open questions. Spaces which are not absolute $\mathcal{C}\mathcal{R}$-epic abound, and some are presented.

Let X be a given complex matrix of order n. If f(z) is analytic at the eigenvalues of X, one may define the primary matrix function f(X) with stem function f(z) by using any of several well-known methods: for instance, canonical forms, power series, or interpolating polynomials [9].

In recent years there has been some interest in trying to improve the behaviour of maps by extending their domains (see Whyburn [10], Baur [3], Krolevec [8], Dickman [5], Franklin and Kohli [6]). It was shown in [6] that every map can be extended to an open map so that certain properties of the domain and range are preserved in the new domain. In [6] and [7] we also related the topological properties of the domain and range of the mapping with the new domain; also these results were then used to obtain analogues and improvements of recent theorems of Arhangelskii, Čoban, Hodel, Keesling, Nagami, Okuyama, and Proizvolov. In this note we give a method of unifying the domain and range of a mapping so as to yield a meaningful open extension.

In [6] Walter Taylor investigated the relationship between the algebraic structure of a topological algebra A and the group structure of its fundamental group π1(A) and of the higher homotopy groups πn(A),n > 1. The main result is that a variety satisfies a group law λ in homotopy (that is, π1) if and only if every group in the idempotent reduct of obeys λ. (The relevant definitions are in [6] and also § 2 of this paper.) A similar result is stated for the higher homotopy groups. As Taylor points out in the introduction, the hard part of the theorem is constructing a topological algebra in whose fundamental group may fail to obey λ; indeed, in [6] this is only done in detail for the commutative law, and the proof is rather computational.

Let X be a complex Banach space and A a linear operator from X into X with dense domain. We construct the minimal and maximal operators of the operator A and prove that they are equal under reasonable hypotheses on the space X and operator A. As an application, we obtain the existence and regularity of weak solutions of linear equations on the space X. As another application we obtain a criterion for a symmetric operator on a complex Hilbert space to be essentially self-adjoint. An application to pseudo-differential operators of the Weyl type is given.

Let VN(G) be the von Neumann algebra of a locally compact group G. We denote by μ the initial ordinal with |μ| equal to the smallest cardinality of an open basis at the unit of G and X = ﹛α ; α < μ﹜.We show that if G is nondiscrete then there exist an isometric *-isomorphism of l∞(X) into VN(G) and a positive linear mapping π of VN(G) onto l∞(X) such that π o = idl∞(X) and and π have certain additional properties. Let UCB((Ĝ)) be the C*–algebra generated by operators in VN(G) with compact support and F(Ĝ) the space of all T∈ VN(G) such that all topologically invariant means on VN(G) attain the same value at T. The construction of the mapping π leads to the conclusion that the quotient space UCB((Ĝ))/F((Ĝ)) ∪UCB((Ĝ)) has l∞(X) as a continuous linear image if G is nondiscrete. When G is further assumed to be non-metrizable, it is shown that UCB((Ĝ))/F((Ĝ)) ∪UCB((Ĝ)) contains a linear isomorphic copy of l∞(X). Similar results are also obtained for other quotient spaces.

This study of r-separating sets was originally motivated by the maximum-flow-minimum-cut theorem of finite networks [1 ; 2]. In working toward a continuous version of the maximum-flow-minimum-cut theorem from the usual discrete version, one is led directly to the notion of r-connectedness of a set as defined below. This notion of r-connectedness also has a very simple intuitive interpretation. Intuitively speaking, a set of points in the plane is r-connected if a person starting at any one point of the set is able to reach any other point of the set by jumping from point to point within the set, but never jumping a distance exceeding r in any one jump.