An application of functional operator models to dissipative scattering theory
Daniel Alan
Bondy
1-43
Abstract: An abstract framework for dissipative scattering theory is developed and then applied to two systems previously considered by P. Lax and R. S. Phillips. Results relating the poles and zeroes of the scattering matrix to the spectra of the infinitesimal generators A (which generates the semigroup formed by mapping initial data into solution data at time t) and B (which generates a ``local'' semigroup) are proven. In particular these results are shown to follow from the fact that the characteristic function of A (appropriately defined) and the scattering matrix combine to form the characteristic function of B.
Sets definable over finite fields: their zeta-functions
Catarina
Kiefe
45-59
Abstract: Sets definable over finite fields are introduced. The rationality of the logarithmic derivative of their zeta-function is established, an application of purely algebraic content is given. The ingredients used are a result of Dwork on algebraic varieties over finite fields and model-theoretic tools.
Consistency results concerning supercompactness
Telis K.
Menas
61-91
Abstract: A general framework for proving relative consistency results with regard to supercompactness is developed. Within this framework we prove the relative consistency of the assertion that every set is ordinal definable with the statement asserting the existence of a supercompact cardinal. We also generalize Easton's theorem; the new element in our result is that our forcing conditions preserve supercompactness.
On almost bounded functions
Ruth
Miniowitz
93-102
Abstract: New results are presented with regard to the ``almost bounded functions'' introduced by Goodman [2], including a theorem which contains a proof of Goodman's conjecture for a particular case.
The Remez exchange algorithm for approximation with linear restrictions
Bruce L.
Chalmers
103-131
Abstract: This paper demonstrates a Remez exchange algorithm applicable to approximation of real-valued continuous functions of a real variable by polynomials of degree smaller than n with various linear restrictions. As special cases are included the notion of restricted derivatives approximation (examples of which are monotone and convex approximation and restricted range approximation) and the notion of approximation with restrictions at poised Birkhoff data (examples of which are bounded coefficients approximation, $\varepsilon $-interpolator approximation, and polynomial approximation with restrictions at Hermite and ``Ferguson-Atkinson-Sharma'' data and pyramid matrix data). Furthermore the exchange procedure is completely simplified in all the cases of approximation with restrictions at poised Birkhoff data. Also results are obtained in the cases of general linear restrictions where the Haar condition prevails. In the other cases (e.g., monotone approximation) the exchange in general requires essentially a matrix inversion, although insight into the exchange is provided and partial alternation results are obtained which lead to simplifications.
Hypoelliptic convolution equations in $K'_p$, $p>1$
G.
Sampson;
Z.
Zieleźny
133-154
Abstract: We consider convolution equations in the space $\exp (k\vert x{\vert^p})$ for some constant k. Our main result is a complete characterization of hypoelliptic convolution operators in ${K'_p}$ in terms of their Fourier transforms.
Systems-conjugate and focal points of fourth order nonselfadjoint differential equations
Sui Sun
Cheng
155-165
Abstract: Systems-conjugate points have been studied by John Barrett in relation to selfadjoint fourth order differential equations of the form $({p_2}u'')'' + {p_0}u = 0$. This paper extends his results to the general nonselfadjoint fourth order differential equation via a system of second order equations.
Comparison of eigenvalues associated with linear differential equations of arbitrary order
R. D.
Gentry;
C. C.
Travis
167-179
Abstract: Existence and comparison theorems for eigenvalues of $ (k,n - k)$-focal point and $ (k,n - k)$-conjugate point problems are proved for a class of nth order linear differential equations for arbitrary n.
Sufficient conditions for an operator-valued Feynman-Kac formula
Michael Dale
Grady
181-203
Abstract: Let E be a locally compact, second countable Hausdorff space and let $X(t)$ be a Markov process with state space E. Sufficient conditions are given for the existence of a solution to the initial value problem, $ \partial u/\partial t = Au + V(x) \cdot u,u(0) = f$, where A is the infinitesimal generator of the process X on a certain Banach space and for each $x \in E,V(x)$ is the infinitesimal generator of a $ {C_0}$ contraction semigroup on another Banach space.
On the local stability of differential forms
Martin
Golubitsky;
David
Tischler
205-221
Abstract: In this paper we determine which germs of differential s-forms on an n-manifold are stable (in the sense of Martinet). We show that when $s \ne 1$ or when $s = 1$ and $n \leqslant 4$ Martinet had found almost all of the possible examples. The most interesting result states that for certain generic singularities of 1-forms on 4-manifolds an infinite dimensional moduli space occurs in the classification of the 1-forms with this given singularity type up to equivalence by pull-back via a diffeomorphism.
Periodic homeomorphisms of $3$-manifolds fibered over $S\sp{1}$
Jeffrey L.
Tollefson
223-234
Abstract: Two problems concerning periodic homeomorphisms of 3-manifolds are considered. The first is that of obtaining systems of incompressible surfaces invariant under a given involution. The second problem is the realization by a periodic homeomorphism of an element of finite order in the mapping class group of a 3-manifold. Solutions to both problems are obtained in certain instances.
Hilbert transforms associated with plane curves
Alexander
Nagel;
Stephen
Wainger
235-252
Abstract: Let $(t,\gamma (t))$ be a plane curve. Set ${H_\gamma }f(x,y) =$ p.v.$ \;\smallint f(x - t,y - \gamma (t))dt/t$ for $f \in C_0^\infty ({R^2})$. For a large class of curves, the authors prove ${\left\Vert {{H_\gamma }f} \right\Vert _p} \leqslant {A_p}{\left\Vert f \right\Vert _p},5/3 < p < 5/2$. Various examples are given to show that some condition on the curve $ (t,\gamma (t))$ is necessary.
Smith theory for$p$-groups
James A.
Maiorana
253-266
Abstract: When a p-group G acts on a manifold, the behavior of the cohomology of the subgroups of G singles out a special collection of fixed point sets of these subgroups. A bound on the size of the spaces in this collection is derived using equivariant cohomology. For a special class of nonabelian p-groups this bound is strong enough to require that certain fixed point sets must vanish. Application of this bound to a linear representation of G yields a lower bound for the cohomology of G.
A quasi-Anosov diffeomorphism that is not Anosov
John
Franks;
Clark
Robinson
267-278
Abstract: In this note, we give an example of a diffeomorphism f on a three dimensional manifold M such that f has a property called quasi- Anosov but such that f does not have a hyperbolic structure (is not Anosov). Mañé has given a method of extending f to a diffeomorphism g on a larger dimensional manifold V such that g has a hyperbolic structure on M as a subset of V. This gives a counterexample to a question of M. Hirsch.
A Banach algebra of functions with bounded $n$th differences
John T.
Daly;
Philip B.
Downum
279-294
Abstract: Several characterizations are given for the Banach algebra of $ (n - 1)$-times continuously differentiable functions whose $(n - 1)$st derivative satisfies a bounded Lipschitz condition. The structure of the closed primary ideals is investigated and spectral synthesis is shown to be satisfied.
On bounded functions satisfying averaging conditions. II
Rotraut Goubau
Cahill
295-304
Abstract: Let $S(f)$ denote the subspace of ${L^\infty }({R^n})$ consisting of those real valued functions f for which $\displaystyle \mathop {\lim }\limits_{r \to 0} \frac{1}{{\vert B(x,r)\vert}} {\int} _{B(x,r)}f(y)dy = f(x)$ for all x in ${R^n}$ and let $L(f)$ be the subspace of $S(f)$ consisting of the approximately continuous functions. A number of results concerning the existence of functions in $S(f)$ and $L(f)$ with special properties are obtained. The extreme points of the unit balls of both spaces are characterized and it is shown that $L(f)$ is not a dual space. As a preliminary step, it is shown that if E is a ${G_\delta }$ set of measure 0 in ${R^n}$, then the complement of E can be decomposed into a collection of closed sets in a particularly useful way.
Boundary value problems for functional differential equations with $L\sp{2}$ initial functions
G. W.
Reddien;
G. F.
Webb
305-321
Abstract: Existence results are given for boundary value problems for vector systems of functional differential equations with $ {L^2}$ initial functions. The proofs are essentially constructive and lead to computational methods in important cases.
On nullity distributions
Sin Leng
Tan
323-335
Abstract: The nullity concept of Riemannian manifolds is extended to affine manifolds. Results obtained by Chern and Kuiper and Maltz on Riemannian manifolds are generalized to affine manifolds. A structure theorem for affine symmetric spaces is obtained. Finally, the nullity concept is generalized to study the partial integrability of certain geometric structures.
On the second Hankel determinant of areally mean $p$-valent functions
J. W.
Noonan;
D. K.
Thomas
337-346
Abstract: In this paper we determine the growth rate of the second Hankel determinant of an areally mean p-valent function. This result both extends and unifies previously known results concerning this problem.
Existence theorems for parametric problems in the calculus of variations and approximation
Robert M.
Goor
347-365
Abstract: In this paper, we investigate the parametric growth condition which arises in connection with existence theorems for parametric problems of the calculus of variations. In particular, we study conditions under which the length of a curve is dominated in a suitable sense by its ``cost". We show that we may restrict our attention to local growth conditions on a particular set. Then we link the growth conditions to a certain approximation problem on this set. Finally, we prove that under suitable topological restrictions related to dimension theory, the local and global problems can be solved.
Spherical distributions on Lie groups and $C\sp{\infty }$ vectors
R.
Penney
367-384
Abstract: Given a Lie group G (not necessarily unimodular) and a subgroup K of G (not necessarily compact), it is shown how to associate with every finite-dimensional unitary irreducible representation $ \delta$ of K a class of distributions analogous to the class of spherical functions of height $\delta$ familiar from the unimodular-maximal compact case. The two concepts agree as nearly as possible. A number of familiar theorems are generalized to our situation. As an application we obtain a generalization of the Frobenius reciprocity theorem and of Plancherel's theorem to arbitrary induced representations of Lie groups.
Necessary and sufficient conditions for the derivation of integrals of $L\sb{\psi }$-functions
C. A.
Hayes
385-395
Abstract: It has been shown recently that a necessary and sufficient condition for a derivation basis to derive the $\mu $-integrals of all functions in $ {L^{(q)}}(\mu )$, where $1 < q < + \infty$, and $ \mu$ is a $\sigma $-finite measure, is that the basis possess Vitali-like covering properties, with covering families having arbitrarily small ${L^{(p)}}(\mu )$-overlap, where ${p^{ - 1}} + {q^{ - 1}} = 1$. The corresponding theorem for the case $ p = 1,q = + \infty$ was established by R. de Possel in 1936. The present paper extends these results to more general dual Orlicz spaces. Under suitable restrictions on the dual Orlicz functions $\Phi$ and $\Psi$, it is shown that a necessary and sufficient condition for a basis to derive the $\mu $-integrals of all functions in ${L_\Psi }(\mu )$ is that the basis possess Vitali-like covering families whose ${L_\Phi }(\mu )$-overlap is arbitrarily small. Certain other conditions relating ${L_\Phi }(\mu )$-strength and derivability are also discussed.
Compact nilmanifold extensions of ergodic actions
Robert J.
Zimmer
397-406
Abstract: We study extensions of dymanical systems defined by co-cycles into nilpotent Lie groups.