Stability theorems for some functional equations
R. C.
MacCamy;
J. S. W.
Wong
1-37
Abstract: Functional-differential equations of the form $\displaystyle \dot u(t) = - \int_0^t {A(t - \tau )g(u(\tau ))d\tau + f(t,u(t))}$ are considered. Here $ u(t)$ is to be an element of a Hilbert space $ \mathcal{H},A(t)$ a family of bounded symmetric operators on $\mathcal{H}$ and g an operator with domain in $\mathcal{H}$. g may be unbounded. A is called strongly positive if there exists a semigroup exp St, where S is symmetric and $(S\xi ,\xi ) \leqq - m{\left\Vert \xi \right\Vert^2},m > 0$, such that ${A^ \ast } = A - \exp$ St is positive, that is, $\displaystyle \mathop \int \nolimits_0^T \left( {v(t),\int_0^t {{A^\ast}(t - \tau )v(\tau )} } \right)d\tau \geqq 0,$ for all smooth $v(t)$. It is shown that if A is strongly positive, and g and f are suitably restricted, then any solution which is weakly bounded and uniformly continuous must tend weakly to zero. Examples are given of both ordinary and partial differential-functional equations.
Two point boundary value problems for nonlinear functional differential equations
Paul
Waltman;
James S. W.
Wong
39-54
Abstract: This paper is concerned with the existence of solutions of two point boundary value problems for functional differential equations. Specifically, we consider $L(t,{y_t}) = A(t)y(t)$, that is, when the reduced linear equation is an ordinary rather than a functional differential equation. Several examples are discussed to illustrate the results.
Inverse limits, entropy and weak isomorphism for discrete dynamical systems
James R.
Brown
55-66
Abstract: A categorical approach is taken to the study of a single measure-preserving transformation of a finite measure space and to inverse systems and inverse limits of such transformations. The questions of existence and uniqueness of inverse limits are settled. Sinai's theorem on generators is recast and slightly extended to say that entropy respects inverse limits, and various known results about entropy are obtained as immediate corollaries, e.g. systems with quasi-discrete or quasi-periodic spectrum have zero entropy. The inverse limit $\Phi$ of an inverse system $\{ {\Phi _\alpha }:\alpha \in J\}$ of dynamical systems is (1) ergodic, (2) weakly mixing, (3) mixing (of any order) iff each ${\Phi _\alpha }$ has the same property. Finally, inverse limits are used to lift a weak isomorphism of dynamical systems ${\Phi _1}$ and ${\Phi _2}$ to an isomorphism of systems ${\hat \Phi _1}$ and $ {\hat \Phi _2}$ with the same entropy.
Elementary statements over large algebraic fields
Moshe
Jarden
67-91
Abstract: We prove here the following theorems: A. If k is a denumerable Hilbertian field then for almost all $({\sigma _1}, \ldots ,{\sigma _e}) \in \mathcal{G}{({k_s}/k)^e}$ the fixed field of $\{ {\sigma _1}, \ldots ,{\sigma _e}\} ,{k_s}({\sigma _1}, \ldots ,{\sigma _e})$, has the following property: For any non-void absolutely irreducible variety V defined over $ {k_s}({\sigma _1}, \ldots ,{\sigma _e})$ the set of points of V rational over K is not empty. B. If E is an elementary statement about fields then the measure of the set of $\sigma \in \mathcal{G}(\tilde Q/Q)$ (Q is the field of rational numbers) for which E holds in $ \tilde Q(\sigma )$ is equal to the Dirichlet density of the set of primes p for which E holds in the field ${F_p}$ of p elements.
A note on quadratic Jordan algebras of degree $3$
M. L.
Racine
93-103
Abstract: McCrimmon has defined a class of quadratic Jordan algebras of degree 3 obtained from a cubic form, a quadratic mapping and a base point. The structure of such an algebra containing no absolute zero-divisor is determined directly. A simple proof of Springer's Theorem on isomorphism of reduced simple exceptional quadratic Jordan algebras is given.
Nonresidually finite one-relator groups
Stephen
Meskin
105-114
Abstract: The study of one-relator groups includes the connections between group properties and the form of the relator. In this paper we discuss conditions on the form ${u^{ - 1}}{v^l}u{v^m}$ which force the corresponding one-relator groups to be nonresidually finite, i.e. the intersection of the normal subgroups of finite index to be nontrivial. Moreover we show that these forms can be detected amongst the words of a free group.
A constructive ergodic theorem
J. A.
Nuber
115-137
Abstract: As discussed by Bishop, Birkhoff's Ergodic Theorem is not constructively valid. In this paper we present an hypothesis which is necessary and sufficient for the constructive almost everywhere convergence of the Césaro averages of the translates of an integrable function by a measure preserving transformation. In addition necessary and sufficient conditions are given for the limit function to be constructively integrable. Also we present a necessary and sufficient condition that the averages converge to a constant function and give an equivalent formulation of this condition for finite measure spaces. Several interesting examples are given which satisfy these conditions.
Slices of maps and Lebesgue area
William P.
Ziemer
139-151
Abstract: For a large class of k dimensional surfaces, S, it is shown that the Lebesgue area of S can be essentially expressed in terms of an integral of the $k - 1$ area of a family, F, of $k - 1$ dimensional surfaces that cover S. The family F is regarded as being composed of the slices of F. The definition of the $k - 1$ area of a surface restricted to one of its slices is formulated in terms of the theory developed by H. Federer, [F3].
The size function of abelian varieties
Allen
Altman
153-161
Abstract: The size function is defined for points in projective space over any field K, finitely generated field over Q, generalizing the height function for number fields. We prove that the size function on the K-rational points of an abelian variety is bounded by a quadratic function.
One-dimensional basic sets in the three-sphere
Joel C.
Gibbons
163-178
Abstract: This paper is a continuation of Williams' classification of one-dimensional attracting sets of a diffeomorphism on a compact manifold [Topology 6 (1967)]. After defining the knot presentation of a solenoid in ${S^3}$ and some knottheoretic preliminaries, we prove Theorem: If ${\sum _1},{h_1}$ and $ {\sum _2},{h_2}$ are shift classes of oriented solenoids admitting elementary presentations K, $K,{g_1}$ and K, $K,{g_2}$, resp., where $ {g_1}^ \ast = {({g_2}^ \ast )^t}:{H_1}(K) \to {H_1}(K)$, there is an Anosov-Smale diffeomorphism f of ${S^3}$ such that $\Omega (f)$ consists of a source ${\Lambda ^ - }$ and a sink ${\Lambda ^ + }$ for which ${\Lambda ^ + },f/{\Lambda ^ + }$ and $ {\Lambda ^ - },{f^{ - 1}}/{\Lambda ^ - }$ are conjugate, resp., to ${\sum _1},{h_1}$ and ${\sum _2},{h_2}$. (The author has proved [Proc. Amer. Math. Soc., to appear] that if f is an Anosov-Smale map of $ {S^3},\Omega (f)$ has dimension one, and contains no hyperbolic sets, then f has the above structure.) We also prove Theorem: there is a nonempty ${C^1}$-open set ${F_2}$ in the class of such diffeomorphisms for which $K = {S^1}$ and $ {g_1} = {g_2}$ is the double covering such that each f in ${F_2}$ defines a loop t in $ {S^3}$, stable up to $ {C^1}$ perturbations, for which at every x in t the generalized stable and unstable manifolds through x are tangent at x.
Local to global theorems in the theory of Hurewicz fibrations
James
Arnold
179-188
Abstract: This paper is concerned with the problem of showing a local fibration is a fibration. There are two kinds of local to global theorems proven. The first type of theorem considers local fibrations where local is in terms of closed covers of the base (e.g. the set of closed simplices of a polyhedron, the cones of a suspension). The second type of theorem deals with local in terms of open covers of the total space.
The structure of certain unitary representations of infinite symmetric groups
Arthur
Lieberman
189-198
Abstract: Let S be an infinite set, $\beta$ an infinite cardinal number, and ${G_\beta }(S)$ the group of those permutations of S whose support has cardinal number less than $ \beta$. If T is any nonempty set, ${S^T}$ is the set of functions from T to S. The canonical representation $\Lambda _\beta ^T$ of ${G_\beta }(S)$ on $ {L^2}({S^T})$ is the direct sum of factor representations. Factor representations of types ${{\text{I}}_\infty },{\text{II}_1}$, and $ {\text{II}_\infty }$ occur in this decomposition, depending upon S, $ \beta$, and T; the type $ {\text{II}_1}$ factor representations are quasi-equivalent to the left regular representation. Let ${G_\beta }(S)$ have the topology of pointwise convergence on S. $ {G_\beta }(S)$ is a topological group but is not locally compact. Every continuous representation of $ {G_\beta }(S)$ is the direct sum of irreducible representations. Let $ \Gamma$ be a nontrivial continuous irreducible representation of ${G_\beta }(S)$. Then $\Gamma$ is continuous iff $\Gamma$ is equivalent to a subrepresentation of $ \Lambda _\beta ^T$ for some nonempty finite set T iff there is a nonempty finite subset Z of S such that the restriction of $\Gamma$ to the subgroup of those permutations which leave Z pointwise fixed contains the trivial representation of this subgroup.
Weakly wandering vectors and weakly independent partitions
Ulrich
Krengel
199-226
Abstract: We first characterize continuous spectrum and purely discrete spectrum of an isometry U of a Hilbert space geometrically by the existence of a spanning system, resp. by the absence, of vectors with infinitely many orthogonal images under powers of U. We then characterize weak mixing and discrete spectrum of an invertible measure preserving transformation of a probability space in terms of the null sets of the space. Finally for two-fold weakly mixing transformations the result on isometries is strengthened by proving the density of the set of partitions with infinitely many mutually independent images in the set of all finite partitions.
Quotient sheaves and valuation rings
Joel
Cunningham
227-239
Abstract: In this paper a construction of a quotient sheaf of a sheaf of rings is given. This construction is analogous to the Utumi ring of quotients of a ring. For a valuation ring V, a sheaf of rings corresponding to V is introduced and its quotient sheaf is computed. It is shown that this quotient sheaf corresponds to the completion of V in case V is discrete rank one and that V is maximal if and only if its associated sheaf of rings is its own quotient sheaf.
Almost recursively enumerable sets
John W.
Berry
241-253
Abstract: An injective function on N, the nonnegative integers, taking values in N, is called almost recursive (abbreviated a.r.) if its inverse has a partial recursive extension. The range of an a.r. function f is called an almost recursively enumerable set in general; an almost recursive set if in addition f is strictly increasing. These are natural generalizations of regressive and retraceable sets respectively. We show that an infinite set is almost recursively enumerable iff it is point decomposable in the sense of McLaughlin. This leads us to new characterizations of certain classes of immune sets. Finally, in contrast to the regressive case, we show that a.r. functions and sets are rather badly behaved with respect to recursive equivalence.
On a convolution theorem for $L(p,q)$ spaces
A. P.
Blozinski
255-265
Abstract: The principal result of this paper is a proof of the Convolution Theorem based on the definition of a convolution operator as presented by E. M. Stein and R. O'Neil. Closely related are earlier versions and special cases of the Convolution Theorem, which are $L(p,q)$ analogues of an inequality of W. H. Young, given in papers by R. O'Neil, L. Y. H. Yap, R. Hunt, and B. Muckenhoupt and E. M. Stein.
An optimization problem for unitary and orthogonal representations of finite groups
D. Ž.
Djoković;
I. F.
Blake
267-274
Abstract: Let $G \to {\text{GL}}(V)$ be a faithful orthogonal representation of a finite group G acting in an Euclidean space V. For a unit vector x we choose $g \ne 1$ in G so that $\vert gx - x\vert$ is minimal and put $\delta (x) = \vert gx - x\vert$. We study the class of vectors x which maximize $\delta (x)$ and have the additional property that $ \vert gx - x\vert$ depends only on the conjugacy class of $g \in G$. For some special types of representations we are able to characterize completely this class of vectors.
The space of homeomorphisms on a compact two-manifold is an absolute neighborhood retract
R.
Luke;
W. K.
Mason
275-285
Abstract: The theorem mentioned in the title is proved.
Some open mapping theorems for measures
Seymour
Ditor;
Larry Q.
Eifler
287-293
Abstract: Given a compact Hausdorff space X, let $C(X)$ be the Banach space of continuous real valued functions on X with sup norm and let $M(X)$ be its dual considered as finite regular Borel measures on X. Let $U(X)$ denote the closed unit ball of $ M(X)$ and let $ P(X)$ denote the nonnegative measures in $M(X)$ of norm 1. A continuous map $ \varphi$ of X onto another compact Hausdorff space Y induces a natural linear transformation $\pi$ of $M(X)$ onto $M(Y)$ defined by setting $\pi (\mu )(g) = \mu (g \circ \varphi )$ for $\mu \in M(X)$ and $ g \in C(Y)$. It is shown that $\pi$ is norm open on $U(X)$ and on $ A \cdot P(X)$ for any subset A of the real numbers. If $\varphi$ is open, then $\pi$ is $ \mathrm{weak}^*$ open on $A \cdot P(X)$. Several examples are given which show that generalization in certain directions is not possible. The paper concludes with some remarks about continuous selections.
Tensor products of locally convex modules and applications to the multiplier problem
Roger
Rigelhof
295-307
Abstract: In this paper we present a representation theorem for the tensor product of locally convex modules. This theorem has a number of consequences in the study of the multiplier problem in harmonic analysis, and the remainder of the paper is devoted to these applications.
Local behaviour of solutions of stochastic integral equations
William J.
Anderson
309-321
Abstract: Let X denote the solution process of the stochastic equation $ dX(t) = a(X(t))dt + \sigma (X(t))dW(t)$. In this paper, conditions on $a( \cdot )$ and $ \sigma ( \cdot )$ are given under which the sample paths of X are differentiate at $t = 0$ with probability one. Variations of these results are obtained leading to a new uniqueness criterion for solutions of stochastic equations. If $\sigma ( \cdot )$ is Hölder continuous with exponent greater than $ \tfrac{1}{2}$ and $a( \cdot )$ satisfies a Lipschitz condition, it is shown that in the one-dimensional case the above equation has only one continuous solution.
Entropy-expansive maps
Rufus
Bowen
323-331
Abstract: Let $f:X \to X$ be a uniformly continuous map of a metric space. f is called h-expansive if there is an $ \varepsilon > 0$ so that the set $ {\Phi _\varepsilon }(x) = \{ y:d({f^n}(x),{f^n}(y)) \leqq \varepsilon$ for all $n \geqq 0$} has zero topological entropy for each $x \in X$. For X compact, the topological entropy of such an f is equal to its estimate using $\varepsilon :h(f) = h(f,\varepsilon )$. If X is compact finite dimensional and $ \mu$ an invariant Borel measure, then ${h_\mu }(f) = {h_\mu }(f,A)$ for any finite measurable partition A of X into sets of diameter at most $ \varepsilon$. A number of examples are given. No diffeomorphism of a compact manifold is known to be not h-expansive.
Mapping cylinder neighborhoods of one-complexes in four-space
J. L.
Bryant;
R. C.
Lacher
333-339
Abstract: We prove the following theorem: If K is a 1-complex topologically embedded in ${S^4}$, and if K has mapping cylinder neighborhoods in ${S^4}$ at almost all of its points, then K is tame. The proof uses engulfing and the theory of proper, one-acyclic mappings of 3-manifolds onto the real line.
A dominance theorem for partitioned Hermitian matrices
Russell
Merris
341-352
Abstract: Let $A = ({A_{ij}})$ be a partitioned positive semidefinite hermitian matrix, where ${A_{ij}}$ is n-square, $1 \leqq i,j \leqq m$. A class of ordered pairs of functions $ ({f_1},{f_2})$ is given such that $({f_1}({A_{ij}})) - ({f_2}({A_{ij}}))$ is positive semidefinite hermitian. Applications are given.
On the irreducibility of nonunitary induced representations of certain semidirect products
Ernest
Thieleker
353-369
Abstract: Let G be a connected Lie group which is a semidirect product of a compact subgroup K and a normal solvable subgroup S. Let $\Lambda$ be a character of S, and let ${M_\Lambda }$ be the stabilizer of $\Lambda$ in K. Let $[H,{\Lambda _\mu }]$ be a finite-dimensional irreducible representation of the subgroup $S{M_\Lambda }$ on the complex vector space H. In this paper we consider the induced representations of G on various Banach spaces, and study their topological irreducibility. The basic method used consists in studying the irreducibility of the Lie algebra representations which arise on the linear subspaces of K-finite vectors. The latter question then can be reduced to the problem of determining when certain modules over certain commutative algebras are irreducible. The method discussed in this paper leads to two theorems giving sufficient conditions on the character $\Lambda$ that the induced representations be topologically irreducible. The question of infinitesimal equivalence of various induced representations is also discussed.
A characterization of the group ${\rm U}\sb{3}\,(4)$
Richard
Lyons
371-387
Abstract: Let T be a Sylow 2-subgroup of the projective special unitary group ${U_3}(4)$, and let G be a finite group with Sylow 2-subgroups isomorphic to T. It is shown that if G is simple, then $G \cong {U_3}(4)$; if G has no proper normal subgroup of odd order or index, then $G \cong {U_3}(4)$ or T.
Cyclic vectors and irreducibility for principal series representations. II
Nolan R.
Wallach
389-396
Abstract: This paper is a continuation of the author's paper Cyclic vectors and irreducibility for principal series representations. In this paper the nonunitary principal series is studied. Using a theorem of Kostant, a sufficient condition is found for irreducibility of nonunitary principal series representations.
Asymptotic behavior of functions with bounded boundary rotation
James W.
Noonan
397-410
Abstract: For $k \geqq 2$ denote by ${V_k}$ the class of normalized functions, analytic in the unit disc, which have boundary rotation at most $k\pi$. Let ${a_n}$ be the nth Taylor coefficient of $f(z) \in {V_k}$. Let ${I_\lambda }(r,f)$ be the $\lambda$-integral mean of $f'(z)$ and $f(z)$ respectively. We determine asymptotic formulas for $f'(z)$, and these formulas are then applied to study the behavior of $\vert{a_n}\vert$ as $ n \to \infty$, and the behavior of ${I_\lambda }(r,f)$ as $r \to 1$.
A divergence theorem for Hilbert space
Victor
Goodman
411-426
Abstract: Let B be a real separable Banach space. A suitable linear imbedding of a real separable Hilbert space into B with dense range determines a probability measure on B which is known as abstract Wiener measure. In this paper it is shown that certain submanifolds of B carry a surface measure uniquely defined in terms of abstract Wiener measure. In addition, an identity is obtained which relates surface integrals to abstract Wiener integrals of functions associated with vector fields on regions in B. The identity is equivalent to the classical divergence theorem if the Hilbert space is finite dimensional. This identity is used to estimate the total measure of certain surfaces, and it is established that in any space B there exist regions whose boundaries have finite surface measure.
Weakly almost periodic functionals carried by hypercosets
Charles F.
Dunkl;
Donald E.
Ramirez
427-434
Abstract: For G a compact group and H a closed normal subgroup, we show that a weakly almost periodic (w.a.p.) linear functional on the Fourier algebra of G/H lifts to a w.a.p. linear functional on the Fourier algebra of G.
$C\sp{\ast} $-algebras generated by Fourier-Stieltjes transforms
Charles F.
Dunkl;
Donald E.
Ramirez
435-441
Abstract: For G a locally compact group and $\hat G$ its dual, let ${\mathcal{M}_d}(\hat G)$ be the ${C^ \ast }$-algebra generated by the Fourier-Stieltjes transforms of the discrete measures on G. We show that the canonical trace on $ {\mathcal{M}_d}(\hat G)$ is faithful if and only if G is amenable as a discrete group. We further show that if G is nondiscrete and amenable as a discrete group, then the only measures in ${\mathcal{M}_d}(\hat G)$ are the discrete measures, and also the sup and lim sup norms are identical on $ {\mathcal{M}_d}(\hat G)$. These results are extensions of classical theorems on almost periodic functions on locally compact abelian groups.
Associo-symmetric algebras
Raymond
Coughlin;
Michael
Rich
443-451
Abstract: Let A be an algebra over a field F satisfying $(x,x,x) = 0$ with a function $g:A \times A \times A \to F$ such that $(xy)z = g(x,y,z)x(yz)$ for all x, y, z in A. If $ g({x_1},{x_2},{x_3}) = g({x_{1\pi }},{x_{2\pi }},{x_{3\pi }})$ for all $ \pi$ in ${S_3}$ and all ${x_1},{x_2},{x_3}$ in A then A is called an associo-symmetric algebra. It is shown that a simple associo-symmetric algebra of degree $ > 2$ or degree $= 1$ over a field of characteristic $\ne 2$ is associative. In addition a finite-dimensional semisimple algebra in this class has an identity and is a direct sum of simple algebras.
Connections on semisimple Lie groups
Robert E.
Beck
453-460
Abstract: The plus and minus connections of Cartan and Schouten, which exist on any Lie group, have the following three properties: (1) the connection is left invariant, (2) the curvature of the connection is zero, (3) the set of maximal geodesics through the identity of the Lie group is equal to the set of one-parameter subgroups of the Lie group. It is shown that the plus and minus connections are the only ones with these properties on a real simple Lie group. On a real semisimple Lie group the connections with these properties are in one-to-one correspondence with the ways of choosing an ideal of the Lie algebra and then choosing a complementary subspace to it.
Integral operators and the compactness of induced representations
Robert C.
Busby;
Irwin
Schochetman;
Harvey A.
Smith
461-477
Abstract: Integral operators are investigated which are compositions of multiplication by bounded vector valued functions and convolutions with vector valued functions. All of the functions are based on a fixed locally compact group. Conditions are given under which certain of these operators are compact. As an application of these conditions we consider induced representations of twisted group algebras (these are generalizations of representations of groups induced from closed normal sub-groups in the sense of Mackey and include these as special cases) and we give necessary and sufficient conditions for these representations to be compact (that is, to consist entirely of compact operators).
Convolution operators on Lebesgue spaces of the half-line
Victor W.
Daniel
479-488
Abstract: In this paper we determine the lattice of closed invariant subspaces for certain convolution operators on Lebesgue spaces ${L^p}(d\sigma )$ where $\sigma$ is a suitable weighted measure on the half-line. We exploit the rather close relationship between convolution operators and the collection of right translation operators ${\{ {T_\lambda }\} _{\lambda \geqq 0}}$ on ${L^p}(d\sigma )$. We show that a convolution operator K and the collection $ {\{ {T_\lambda }\} _{\lambda \geqq 0}}$ have the same lattice of closed invariant subspaces provided the kernel k of K is a cyclic vector. The converse also holds if we assume in addition that the closed span of $ {\{ {T_\lambda }k\} _{\lambda \geqq 0}}$ is all of ${L^p}(d\sigma )$. We show that the lattice of closed right translation invariant subspaces of ${L^p}(d\sigma )$ is totally ordered by set inclusion whenever $\sigma$ has compact support. Thus in this case a convolution operator K is unicellular if and only if its kernel is a cyclic vector. Finally, we show for suitable weighted measures $\sigma$ on the half-line that the convolution operators on $ {L^p}(d\sigma )$ are Volterra.
Reflection principle for systems of first order elliptic equations with analytic coefficients
Chung Ling
Yu
489-501
Abstract: Let T be a simply connected domain of the $z = x + iy$ plane, whose boundary contains a portion $ \sigma$ of the x-axis. Also let $A(z,\zeta ),B(z,\zeta ),F(z,\zeta ),\alpha (z),\beta (z)$ and $\rho (z)$ be holomorphic functions for $z,\zeta \in T \cup \sigma \cup \bar T$, with $ \alpha (z) - i\beta (z) \ne 0$ for $z \in \bar T \cup \sigma ,\alpha (z) + i\beta (z) \ne 0$ for $z \in T \cup \sigma$. Furthermore, we assume that $ \alpha (x)$ and $ \beta (x)$ are real valued functions for $ x \in \sigma$. Our reflection principle states that for any solution $w = u + iv$ of an equation of the type $\partial w/\partial \bar z = A(z,\bar z)w + B(z,\bar z)\bar w + F(z,\bar z)$ in T under the boundary condition $\alpha (x)u + \beta (x)v = \rho (x)$ on $ \sigma ,w$ can be continued analytically across the x-axis, onto the entire mirror image $\bar T$.
Decompositions of $3$-manifolds and pseudo-isotopies
William
Voxman
503-508
Abstract: In this paper we construct pseudo-isotopies which realize certain cellular decompositions of 3-manifolds. In general we show that the pseudo-isotopy may be defined so as to leave points fixed outside of a given open set containing the nondegenerate elements of the decomposition. For nondegenerately continuous decompositions it is shown that the pseudo-isotopy does not move the nondegenerate elements far from their original positions.