The genera of amalgamations of graphs
Seth R.
Alpert
1-39
Abstract: If $p \leq m$, n then ${K_m}{ \vee _{{K_p}}}{K_n}$ is the graph obtained by identify ing a copy of ${K_p}$ contained in ${K_m}$ with a copy of ${K_p}$ contained in ${K_n}$ . It is shown that for all integers $p \leq m$, n the genus $ g({K_m}{ \vee _{{K_p}}}{K_n})$ of $ {K_m}{ \vee _{{K_p}}}{K_n}$ is less than or equal to $g({K_m}) + g({K_n})$. Combining this fact with the lower bound obtained from the Euler formula, one sees that for $2 \leq p \leq 5,g({K_m}{ \vee _{{K_p}}}{K_n})$ is either $ g({K_m}) + g({K_n})$ or else $ g({K_m}) + g({K_n}) - 1$. Except in a few special cases, it is determined which of these values is actually attained.
Asymptotic abelianness of infinite factors
M. S.
Glaser
41-56
Abstract: Studying Pukánszky's type III factor, ${M_2}$, we show that it does not have the property of asymptotic abelianness and discuss how this property is related to property L. We also prove that there are no asymptotic abelian ${\text{II}_\infty }$ factors. The extension (by ampliation) of central sequences in a finite factor, N, to $ M \otimes N$ is shown to be central. Also, we give two examples of the reduction (by equivalence) of a central sequence in $M \otimes N$ to a sequence in N. Finally, applying the definition of asymptotic abelianness of $ {C^\ast}$-algebras to $ {W^\ast}$-algebras leads to the conclusion that all factors satisfying this property are abelian.
Geodesic flows on negatively curved manifolds. II
Patrick
Eberlein
57-82
Abstract: Let M be a complete Riemannian manifold with sectional curvature $K \leq 0$, SM the unit tangent bundle of M, ${T_t}$ the geodesic flow on SM and $ \Omega \subseteq SM$ the set of nonwandering points relative to ${T_t}$. ${T_t}$ is topologically mixing (respectively topologically transitive) on SM if for any open sets 0, U of SM there exists $A > 0$ such that $\left\vert t \right\vert \geq A$ implies $ {T_t}(O) \cap U \ne \emptyset$ (respectively there exists $t\;\varepsilon \;R$ such that ${T_t}(O) \cap U \ne \emptyset$). For each vector $ v\;\varepsilon \;SM$ we define stable and unstable sets $ {W^s}(v),{W^{ss}}(v),{W^u}(v)$ and $ {W^{uu}}(v)$, and we relate topological mixing (respectively topological transitivity) of ${T_t}$ to the existence of a vector $v\; \in \;SM$ such that $ {W^{ss}}(v)$ (respectively $ {W^s}(v)$) is dense in SM. If M is a Visibility manifold (implied by $K \leq c < 0$) and if $\Omega = SM$ then ${T_t}$ is topologically mixing on SM. Let $ {S_n} =$ {Visibility manifolds M of dimension n such that $ {T_t}$ is topologically mixing on SM}. For each $n \geq 2$, ${S_n}$ is closed under normal (Galois) Riemannian coverings. If $ M\; \in \;{S_n}$ we classify { $ v\; \in \;SM:\;{W^{ss}}(v)$ is dense in SM}, and M is compact if and only if this set = SM. We also consider the case where $\Omega$ is a proper subset of SM.
Algebraic cohomology of topological groups
David
Wigner
83-93
Abstract: A general cohomology theory for topological groups is described, and shown to coincide with the theories of C. C. Moore [12] and other authors. We also recover some invariants from algebraic topology.
Alternating Chebyshev approximation
Charles B.
Dunham
95-109
Abstract: An approximating family is called alternating if a best Chebyshev approximation is characterized by its error curve having a certain number of alternations. The convergence properties of such families are studied. A sufficient condition for the limit of best approximation on subsets to converge uniformly to the best approximation is given: it is shown that this is often (but not always) a necessary condition. A sufficient condition for the Chebyshev operator to be continuous is given: it is shown that this is often (but not always) a necessary condition.
Decomposable braids and linkages
H.
Levinson
111-126
Abstract: An n-braid is called k-decomposable if and only if the removal of k arbitrary strands results in a trivial $(n - k)$-braid. k-decomposable n-linkages are similarly defined. All k-decomposable n-braids are generated by an explicit geometric process, and so are all k-decomposable n-linkages. The latter are not always closures of k-decomposable n-braids. Many examples are given.
Elementary properties of free groups
George S.
Sacerdote
127-138
Abstract: In this paper we show that several classes of elementary properties (properties definable by sentences of a first order logic) of groups hold for all nonabelian free groups. These results are obtained by examining special embeddings of these groups into one another which preserve the properties in question.
Iterated limits in $N\sp{\ast} (U\sp{n})$
Carl Stephen
Davis
139-146
Abstract: It is shown that if f is in $ {N^\ast}({U^n})$, then the iterated limits of f are almost everywhere independent of the order of iteration. In fact, the iterated limit and the radial limit are equal almost everywhere.
Free products of von Neumann algebras
Wai Mee
Ching
147-163
Abstract: A new method of constructing factors of type ${\text{II}_1}$, called free product, is introduced. It is a generalization of the group construction of factors of type $ {\text{II}_1}$ when the given group is a free product of two groups. If $ {A_1}$ and ${A_2}$ are two von Neumann algebras with separating cyclic trace vectors and ortho-unitary bases, then the free product ${A_1} \ast {A_2}$ of ${A_1}$ and ${A_2}$ is a factor of type ${\text{II}_1}$ without property $\Gamma$.
Surjective stability in dimension $0$ for $K\sb{2}$ and related functors
Michael R.
Stein
165-191
Abstract: This paper continues the investigation of generators and relations for Chevalley groups over commutative rings initiated in [14]. The main result is that if A is a semilocal ring generated by its units, the groups $L({\mathbf{\Phi }},A)$ of [14] are generated by the values of certain cocycles on ${A^\ast} \times {A^\ast}$. From this follows a surjective stability theorem for the groups $L({\mathbf{\Phi }},A)$, as well as the result that $ L({\mathbf{\Phi }},A)$ is the Schur multiplier of the elementary subgroup of the points in A of the universal Chevalley-Demazure group scheme with root system ${\mathbf{\Phi }}$, if ${\mathbf{\Phi }}$ has large enough rank. These results are proved via a Bruhat-type decomposition for a suitably defined relative group associated to a radical ideal. These theorems generalize to semilocal rings results of Steinberg for Chevalley groups over fields, and they give an effective tool for computing Milnor's groups ${K_2}(A)$ when A is semilocal.
Cobordism invariants, the Kervaire invariant and fixed point free involutions
William
Browder
193-225
Abstract: Conditions are found which allow one to define an absolute version of the Kervaire invariant in ${Z_2}$ of a $ {\text{Wu - }}(q + 1)$ oriented 2q-manifold. The condition is given in terms of a new invariant called the spectral cobordism invariant. Calculations are then made for the Kervaire invariant of the n-fold disjoint union of a manifold M with itself, which are then applied with $M = {P^{2q}}$, the real protective space. These give examples where the Kervaire invariant is not defined, and other examples where it has value $1 \in {{\mathbf{Z}}_2}$. These results are then applied to construct examples of smooth fixed point free involutions of homotopy spheres of dimension $4k + 1$ with nonzero desuspension obstruction, of which some Brieskorn spheres are examples (results obtained also by Berstein and Giffen). The spectral cobordism invariant is also applied directly to these examples to give another proof of a result of Atiyah-Bott. The question of which values can be realized as the sequence of Kervaire invariants of characteristic submanifolds of a smooth homotopy real projective space is discussed with some examples. Finally a condition is given which yields smooth embeddings of homotopy $ {P^m}$'s in ${R^{m + k}}$ (which has been applied by E. Rees).
Kernels in dimension theory
J. M.
Aarts;
T.
Nishiura
227-240
Abstract: All spaces are metrizable. A conjecture of de Groot states that a weak inductive dimension theory beginning with the class of compact spaces will characterize those spaces which can be extended to a compact space by the adjunction of a set of dimension not exceeding n. Nagata has proposed a variant of this conjecture as a means of finding insights into the original conjecture. (See Internat. Sympos. on Extension Theory, Berlin, 1967, pp. 157-161.) The proposed variant replaces compact with $ \sigma$-compact. The present paper concerns a study of strong inductive dimension theory beginning with an arbitrary class of spaces. The study is motivated by the above two conjectures. It indicates that a theory of kernels is a more natural by-product of inductive theory than a theory of extensions. An example has resulted which, with the aid of the developed theory and the Baire category theorem, resolves the second conjecture in the negative. The original conjecture is still unresolved. It is also shown that the notion of kernels results in a further generalization of Lelek's form of the dimension lowering map theorem (Colloq. Math. 12 (1964), 221-227. MR 31 #716).
Equivariant cobordism and duality
Edward C.
Hook
241-258
Abstract: We consider equivariant cobordism theory, defined by means of an equivariant Thorn spectrum; in particular, we investigate the relationship between this theory and the more geometric equivariant bordism theory, showing that there is a Poincaré-Lefschetz duality theorem which is valid in this setting.
The trace-class of a full Hilbert algebra
Michael R. W.
Kervin
259-270
Abstract: The trace-class of a full Hilbert algebra A is the set $ \tau (A) = \{ xy\vert x \in A,y \in A\}$. This set is shown to be a $\ast$-ideal of A, and possesses a norm $\tau$ defined in terms of a positive hermitian linear functional on $\tau (A)$. The norm $\tau$ is in general both incomplete and not an algebra norm, and is also not comparable with the Hilbert space norm $ \left\Vert\right\Vert$ on $ \tau (A)$. However, a one-sided ideal of $\tau (A)$ is closed with respect to one norm if and only if it is closed with respect to the other. The topological dual of $\tau (A)$ with respect to the norm $\tau$ is isometrically isomorphic to the set of left centralizers on A.
Multiplicities of second order linear recurrences
Ronald
Alter;
K. K.
Kubota
271-284
Abstract: A second order linear recurrence is a sequence $ \{ {a_n}\}$ of integers satisfying a ${a_{n + 2}} = M{a_{n + 1}} - N{a_n}$ where N and M are fixed integers and at least one $ {a_n}$ is nonzero. If k is an integer, then the number $m(k)$ of solutions of ${a_n} = k$ is at most 3 (respectively 4) if ${M^2} - 4N < 0$ and there is an odd prime $q \ne 3$ (respectively q = 3) such that $ q\vert M$ and $ q\nmid kN$. Further $M = {\sup _k}{\;_{{\text{integer}}}}m(k)$ is either infinite or $\leq 5$ provided that either (i) $(M,N) = 1$ or (ii) $6\nmid N$.
Submanifolds and a pinching problem on the second fundamental tensors
Masafumi
Okumura
285-291
Abstract: This paper gives a sufficient condition for a submanifold of a Riemannian manifold of nonnegative constant curvature to be totally umbilical. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature.
Local and asymptotic approximations of nonlinear operators by $(k\sb{1},\,\ldots k\sb{N})$-homogeneous operators
R. H.
Moore;
M. Z.
Nashed
293-305
Abstract: Notions of local and asymptotic approximations of a nonlinear mapping F between normed linear spaces by a sum of N $ {k_i}$-homogeneous operators are defined and investigated. It is shown that the approximating operators inherit from F properties related to compactness and measures of noncompactness. Nets of equi-approximable operators with collectively compact (or bounded) approximates, which arise in approximate solutions of integral and operator equations, are studied with particular reference to pointwise (or weak convergence) properties. As a by-product, the well-known result that the Fréchet (or asymptotic) derivative of a compact operator is compact is generalized in several directions and to families of operators.
Infinite particle systems
Sidney C.
Port;
Charles J.
Stone
307-340
Abstract: We consider a system of denumerably many particles that are distributed at random according to a stationary distribution P on some closed subgroup X of Euclidean space. We assume that the expected number of particles in any compact set is finite. We investigate the relationship between P and the distribution Q of particles as viewed from a particle selected ``at random'' from some set. The distribution Q is called the tagged particle distribution. We give formulas for computing P in terms of Q and Q in terms of P and show that, with the appropriate notion of convergences, $ {P_n} \to P$ implies ${Q_n} \to Q$ and vice versa. The particles are allowed to move in an appropriate translation invariant manner and we show that the tagged particle distribution Q' at a later time 1 is the same as the distribution of particles at time 1 as viewed from a particle selected ``at random'' from those initially in some set. We also show that Q' is the same as the distribution of particles at time 1 as viewed from a particle selected at random from those at the origin, when initially the particles are distributed according to Q. The one-dimensional case is treated in more detail. With appropriate topologies, we show that in this case there is a homeomorphism between the collection of stationary distributions P and tagged particle distributions Q. A stationary spacings distribution ${Q_0}$ related to Q is introduced, and we show that with the appropriate topology the map taking Q to ${Q_0}$ is a homeomorphism. Explicit expressions are found for all these maps and their inverses. The paper concludes by using the one-dimensional results to find stationary distributions for a class of motions of denumerably many unit intervals and to establish criteria for convergence to one of these distributions.
Additive set functions on lattices of sets
Gene A.
DeBoth
341-355
Abstract: This paper is concerned with properties of additive set functions defined on lattices of sets. Extensions of results of Brunk and Johansen, Darst, Johansen, and Uhl are obtained. Two fundamental approximation properties for lattices of sets (established in another paper) permit us to translate the setting and consider countably additive set functions defined on sigma lattices of sets. Thereby results for countably additive set functions defined on sigma lattices of sets are used to obtain alternate derivations and extensions of Darst's results for additive set functions defined on lattices of sets, i.e., we consider the Radon-Nikodym derivative, conditional expectation, and martingale convergence for lattices of sets.
A nonlinear optimal control minimization technique
Russell D.
Rupp
357-381
Abstract: Hestenes' method of multipliers is applied to a nonlinear optimal control problem. This requires that a differentially constrained problem be embedded in a family of unconstrained problems so as to preserve standard sufficiency criteria. Given an initial estimate of the Lagrange multipliers, a convergent sequence of arcs is generated. They are minimizing with respect to members of the above family, and their limit is the solution to the differentially constrained problem.
Decreasing rearrangements and doubly stochastic operators
Peter W.
Day
383-392
Abstract: In this paper generalizations to measurable functions on a finite measure space $ (X,\Lambda ,\mu )$ of some characterizations of the Hardy-Littlewood-Pólya preorder relation $\prec$ are considered. Let $\rho$ be a saturated, Fatou function norm such that ${L^\infty } \subset {L^\rho } \subset {L^1}$, and let $ {L^\rho }$ be universally rearrangement invariant. The following equivalence is shown to hold for all $f \in {L^\rho }$ iff $(X,\Lambda ,\mu )$ is nonatomic or discrete: $g \prec f$ iff g is in the $\rho$-closed convex hull of the set of all rearrangements of f. Finally, it is shown that $g \prec f \in {L^1}$ iff g is the image of f by a doubly stochastic operator.
On the zeros of power series with Hadamard gaps-distribution in sectors
I Lok
Chang
393-400
Abstract: We give a sufficient condition for a power series with Hadamard gaps to assume every complex value infinitely often in every sector of the unit disk.
Some integral inequalities with applications to the imbedding of Sobolev spaces defined over irregular domains
R. A.
Adams
401-429
Abstract: This paper examines the possibility of extending the Sobolev Imbedding Theorem to certain classes of domains which fail to have the ``cone property'' normally required for that theorem. It is shown that no extension is possible for certain types of domains (e.g. those with exponentially sharp cusps or which are unbounded and have finite volume), while extensions are obtained for other types (domains with less sharp cusps). These results are developed via certain integral inequalities which generalize inequalities due to Hardy and to Sobolev, and are of some interest in their own right. The paper is divided into two parts. Part I establishes the integral inequalities; Part II deals with extensions of the imbedding theorem. Further introductory information may be found in the first section of each part.
On fibering of cobordism classes
R. E.
Stong
431-447
Abstract: This paper studies the problem posed by Conner and Floyd of finding which cobordism classes are represented by the total space of a fibering with prescribed base or fiber.
Equivariant bordism of maps
R. E.
Stong
449-457
Abstract: This note computes the bordism classification of equivariant maps between closed manifolds with action of a cyclic group of prime order.
Exit properties of stochastic processes with stationary independent increments
P. W.
Millar
459-479
Abstract: Let $\{ {X_t},t \geq 0\}$ be a real stochastic process with stationary independent increments. For $x > 0$, define the exit time ${T_x}$ from the interval $( - \infty ,x]$ by ${T_x} = \inf \{ t > 0:{X_t} > x\}$. A reasonably complete solution is given to the problem of deciding precisely when ${P^0}\{ {X_{{T_x}}} = x\} > 0$ and precisely when $ {P^0}\{ {X_{{T_x}}} = x\} = 0$. The solution is given in terms of parameters appearing in the Lévy formula for the characteristic function of ${X_t}$. A few applications of this result are discussed.
Monotonically normal spaces
R. W.
Heath;
D. J.
Lutzer;
P. L.
Zenor
481-493
Abstract: This paper begins the study of monotone normality, a common property of linearly ordered spaces and of Borges' stratifiable spaces. The concept of monotone normality is used to give necessary and sufficient conditions for stratifiability of a ${T_1}$-space, to give a new metrization theorem for p-spaces with $ {G_\delta }$-diagonals, and to provide an easy proof of a metrization theorem due to Treybig. The paper concludes with a list of examples which relate monotone normality to certain familiar topological properties.
Generalized Dedekind eta-functions and generalized Dedekind sums
Bruce C.
Berndt
495-508
Abstract: A transformation formula under modular substitutions is derived for a very large class of generalized Eisenstein series. The result also gives a transformation formula for generalized Dedekind eta-functions. Various types of Dedekind sums arise, and reciprocity laws are established.