Nonstandard analysis of dynamical systems. I. Limit motions, stability.
A. E.
Hurd
1-26
Abstract: The methods of nonstandard analysis are applied to the study of the qualitative theory of dynamical systems. The nonstandard notions connected with limiting behavior of motions (limit sets, etc.) are developed, and then applied to the study of stability theory, including stability of sets, attracting properties, first prolongations and stability of motions.
Representation of holomorphic functions by boundary integrals
Albert
Baernstein
27-37
Abstract: Let $K$ be a compact locally connected set in the plane and let $f$ be a function holomorphic in the extended complement of $K$ with $ f(\infty ) = 0$. We prove that there exists a sequence of measures $\{ {\mu _n}\}$ on $K$ satisfying ${\lim _{n \to \infty }}\vert\vert{\mu _n}\vert{\vert^{1/n}} = 0$ such that $f(z) = \sum\nolimits_{n = 0}^\infty {\int_K {{{(w - z)}^{ - n - 1}}d{\mu _n}(w)(z \in K)} }$. It follows from the proof that two topologies for the space of functions holomorphic on $K$ are the same. One of these is the inductive limit topology introduced by Köthe, and the other is defined by a family of seminorms which involve only the values of the functions and their derivatives on $ K$. A key lemma is an open mapping theorem for certain locally convex spaces. The representation theorem and the identity of the two topologies is false when $K$ is a compact subset of the unit circle which is not locally connected.
New existence theorems for nonlinear equations of Hammerstein type.
W. V.
Petryshyn;
P. M.
Fitzpatrick
39-63
Abstract: Let $X$ be a real Banach space, ${X^ \ast }$ its dual, $A$ a linear map of $X$ into ${X^ \ast }$ and $N$ a nonlinear map of ${X^ \ast }$ into $X$. Using the recent results of Browder and Gupta, Brezis, and Petryshyn, in this paper we study the abstract Hammerstein equation, $ w + ANw = 0$. Assuming suitable growth conditions on $N$, new existence results are obtained under the following conditions on $X,A$ and $N$. In §1: $X$ is reflexive, $A$ bounded with $ f(x) = (Ax,x)$ weakly lower semicontinuous, $N$ bounded and of type $ ($M$)$. In §2: $X$ is a general space, $ A$ angle-bounded, $ N$ pseudo-monotone. In §3: $X$ is weakly complete, $A$ strictly (strongly) monotone, $N$ bounded (unbounded) and of type $($M$)$. In §4: $X$ is a general space, $A$ is monotone and symmetric, $N$ is potential. In §5: $ X$ is reflexive and with Schauder basis, $ {X^ \ast }$ strictly convex, $N$ quasibounded and either monotone, or bounded and pseudo-monotone, or bounded and of type $($M$)$.
Maxima and high level excursions of stationary Gaussian processes
Simeon M.
Berman
65-85
Abstract: Let $X(t),t \geqq 0$, be a stationary Gaussian process with mean 0, variance 1 and covariance function $ r(t)$. The sample functions are assumed to be continuous on every interval. Let $r(t)$ be continuous and nonperiodic. Suppose that there exists $ \alpha , 0 < \alpha \leqq 2$, and a continuous, increasing function $g(t),t \geqq 0$, satisfying $\displaystyle (0.1)\quad \mathop {\lim }\limits_{t \to 0} \frac{{g(ct)}}{{g(t)}} = 1,\quad for\;every\;c > 0,$ such that $\displaystyle (0.2)\quad 1 - r(t) \sim g(\vert t\vert)\vert t{\vert^\alpha },\quad t \to 0.$ For $u > 0$, let $ v$ be defined (in terms of $ u$) as the unique solution of $\displaystyle (0.3)\quad {u^2}g(1/v){v^{ - \alpha }} = 1.$ Let $ {I_A}$ be the indicator of the event $A$; then $\displaystyle \int_0^T {{I_{[X(s) > u]}}ds}$ represents the time spent above $u$ by $ X(s),0 \leqq s \leqq T$. It is shown that the conditional distribution of $\displaystyle (0.4)\quad v\int_0^T {{I_{[X(s) > u]}}ds,}$ given that it is positive, converges for fixed $T$ and $u \to \infty $ to a limiting distribution ${\Psi _\alpha }$, which depends only on $ \alpha$ but not on $ T$ or $g$. Let $ F(\lambda )$ be the spectral distribution function corresponding to $ r(t)$. Let ${F^{(p)}}(\lambda )$ be the iterated $p$-fold convolution of $F(\lambda )$. If, in addition to (0.2), it is assumed that $\displaystyle (0.5)\quad {F^{(p)}}\;is\;absolutely\;continuous\;for\;some\;p > 0,$ then $ \max (X(s):0 \leqq s \leqq t)$, properly normalized, has, for $t \to \infty$, the limiting extreme value distribution $ \exp ( - {e^{ - x}})$. If, in addition to (0.2), it is assumed that $\displaystyle (0.6)\quad F(\lambda )\;is\; absolutely \;continuous\; with\; the\; derivative\; f(\lambda ),$ and $\displaystyle (0.7)\quad \mathop {\lim }\limits_{h \to 0} \log h\int_{ - \infty }^\infty {\vert f(\lambda } + h) - f(\lambda )\vert d\lambda = 0,$ then (0.4) has, for $ u \to \infty$ and $T \to \infty$, a limiting distribution whose Laplace-Stieltjes transform is $\displaystyle (0.8)\quad \exp [{\text{constant}}\int_0^\infty {} ({e^{ - \lambda \xi }} - 1)d{\Psi _\alpha }(x)],\quad \lambda > 0.$
$\Gamma $-compact maps on an interval and fixed points
William M.
Boyce
87-102
Abstract: We characterize the $ \Gamma$-compact continuous functions $f:X \to X$ where $X$ is a possibly-noncompact interval. The map $f$ is called $\Gamma$-compact if the closed topological semigroup $\Gamma (f)$ generated by $f$ is compact, or equivalently, if every sequence of iterates of $f$ under functional composition $(\ast)$ has a subsequence which converges uniformly on compact subsets of $X$. For compact $X$ the characterization is that the set of fixed points of $f\ast f$ is connected. If $X$ is noncompact an additional technical condition is necessary. We also characterize those maps $ f$ for which iterates of distinct orders agree ( $ \Gamma (f)$ finite) and state a result on common fixed points of commuting functions when one of the functions is $\Gamma$-compact.
Metric properties of transformations of $G$-spaces
R. K.
Thomas
103-117
Abstract: The measure-preserving transformation $T$ acts on a Lebesgue space $(M,\mathcal{B},\mu )$ which is also a $ G$-space for a compact separable group $G$. It is proved that if the factor-transformation on the space of $G$-orbits has completely positive entropy and a certain condition regarding the relations between the actions of $G$ and $T$ is satisfied, then $T$ weakly mixing implies $T$ has completely positive entropy.
The addition theorem for the entropy of transformations of $G$-spaces
R. K.
Thomas
119-130
Abstract: For a measure-preserving transformation $T$ which is a skew-product of a measure-preserving transformation $S$ and a topological group endomorphism $ \sigma$, it is shown that the entropy $h$ satisfies the following ``addition theorem": $ h(T) = h(S) + h(\sigma )$.
Bounds for solutions to a class of nonlinear integral equations
Robert H.
Martin
131-138
Abstract: In this paper a bound is obtained for solutions to a class of nonlinear Stieltjes integral equations. The class of equations under consideration have been previously studied by J. S. Mac Nerney. It is shown that the bound obtained here is in some sense a best bound, and in the linear case, reduces to a previous result of the author.
Identities involving the coefficients of a class of Dirichlet series. V
Bruce C.
Berndt
139-156
Abstract: We derive various forms of the Voronoï summation formula for a large class of arithmetical functions. These arithmetical functions are generated by Dirichlet series satisfying a functional equation with certain gamma factors. Using our theorems, we establish several arithmetical identities.
Identities involving the coefficients of a class of Dirichlet series. V, VI
Bruce C.
Berndt
157-167
Abstract: In 1949 Chowla and Selberg gave a very useful formula for the Epstein zeta-function associated with a positive definite binary quadratic form. Several generalizations of this formula are given here. The method of proof is new and is based on a theorem that we formerly proved for ``generalized'' Dirichlet series. An easy proof of Kronecker's second limit formula is also given.
Products with closed projections. II
N.
Noble
169-183
Abstract: Conditions under which some or all of the projections on a product space will be closed or $z$-closed are studied, with emphasis on infinite products. These results are applied to characterize normal products up to countably many factors, to characterize closed product maps up to finitely many factors, and to give conditions under which products will be countably compact, Lindelöf, paracompact, $\mathfrak{m} - \mathfrak{n}$-compact, etc. Generalizations of these results to $\mathfrak{n}$-products and box products are also given. Our easily stated results include: All powers of a ${T_1}$ space $X$ are normal if and only if $X$ is compact Hausdorff, all powers of a nontrivial closed map $p$ are closed if and only if $p$ is proper, the product of countably many Lindelöf $P$-spaces is Lindelöf; and the product of countably many countably compact sequential spaces is countably compact sequential.
Critical Markov branching processes with general set of types
H.
Hering
185-202
Abstract: This paper is concerned with the asymptotic behaviour of critical, quasi-positively regular Markov branching processes. Several results which have been established with restrictions on the set of types or on the parameter are proven on slightly different moment assumptions for an arbitrary set of types and continuous as well as discrete parameter. The methods employed are analytic and rest upon the properties of probability-generating functionals constructed from the given transition function.
Comparison and oscillation theorems for matrix differential inequalitites
E. S.
Noussair
203-215
Abstract: Strong comparison theorems of Sturm's type are established for systems of second order quasilinear elliptic partial differential equations. The technique used leads to new oscillation and nonoscillation criteria for such systems. Some criteria are deduced from a comparison theorem, and others are derived by a direct variational method. Some of our results constitute extensions of known theorems to nonselfadjoint quasilinear systems.
On the concentration and extension of cylinder measures
Alejandro D.
de Acosta
217-228
Abstract: It is shown that with respect to certain set-theoretic operations-- directed decreasing or even arbitrary intersections of certain families of convex, balanced, weakly closed sets--cylinder measures behave almost as regular Borel measures do. A refinement is proved when the cylinder measure satisfies a scalar concentration condition. These results are applied to obtain stronger versions of Prohorov's theorem and an extension theorem for complete Hausdorff locally convex spaces generalizing a result of Dudley, Feldman, and Le Cam.
Maximal and minimal topologies
Douglas E.
Cameron
229-248
Abstract: A topological space $ (X,T)$ with property ${\text{R}}$ is maximal $ {\text{R}}$ (minimal ${\text{R}}$) if $T$ is a maximal (minimal) element in the set $ {\text{R}}(X)$ of all topologies on the set $X$ having property $ {\text{R}}$ with the partial ordering of set inclusions. The properties of maximal topologies for compactness, countable compactness, sequential compactness, Bolzano-Weierstrass compactness, and Lindelöf are investigated and the relations between these spaces are investigated. The question of whether any space having one of these properties has a strictly stronger maximal topology is investigated. Some interesting product theorems are discussed. The properties of minimal topologies and their relationships are discussed for the quasi-$ P$, Hausdorff quasi-$ P$, and $P$ topologies.
Rings of invariant polynomials for a class of Lie algebras
S. J.
Takiff
249-262
Abstract: Let $G$ be a group and let $\pi :G \to GL(V)$ be a finite-dimensional representation of $G$. Then for $g \in G,\pi (g)$ induces an automorphism of the symmetric algebra $ S(V)$ of $V$. We let $ I(G,V,\pi )$ be the subring of $S(V)$ consisting of elements invariant under this induced action. If $G$ is a connected complex semisimple Lie group with Lie algebra $L$ and if Ad is the adjoint representation of $ G$ on $L$, then Chevalley has shown that $I(G,L,$Ad$)$ is generated by a finite set of algebraically independent elements. However, relatively little is known for nonsemisimple Lie groups. In this paper the author exhibits and investigates a class of nonsemisimple Lie groups $G$ with Lie algebra $L$ for which $ I(G,L,$Ad$)$ is also generated by a finite set of algebraically independent elements.
A theorem and a counterexample in the theory of semigroups of nonlinear transformations
Michael G.
Crandall;
Thomas M.
Liggett
263-278
Abstract: This paper studies the basic method in current use for constructively obtaining a generator from a given semigroup of nonlinear transformations on a Banach space. The method is shown to succeed in real two-dimensional Banach spaces and to fail in a particular three-dimensional example. Other results of independent interest are obtained. For example, it is shown that the concepts of ``maximal accretive'' and ``hyperaccretive'' (equivalently, $m$-accretive or hypermaximal accretive) coincide in ${R^n}$ with the maximum norm.
Relative types of points in $\beta N-N$
A. K.
Steiner;
E. F.
Steiner
279-286
Abstract: Using the concepts of type and relative type for points in $\beta N - N$, as introduced by W. Rudin, M. E. Rudin, and Z. Frolik, an inductive method is presented for constructing types. The relative types are described for points having these constructed types and a point in $\beta N - N$ is found which has exactly $ c$ relative types.
Construction of measures and integrals
B. S.
Thomson
287-296
Abstract: The systems introduced by R. Henstock and later by E. J. McShane to provide powerful generalizations of the Riemann integral are used to construct outer measures and upper integrals and to develop a Lebesgue type theory in quite general settings.
An uncountable collection of chainable continua
David P.
Bellamy
297-304
Abstract: We demonstrate the existence of an uncountable collection of chainable continua, no member of which can be mapped onto any other member. Path connectedness considerations and techniques involving compactifications play an important role in the construction and argument.
Semigroups that are the union of a group on $E\sp{3}$ and a plane
Frank
Knowles
305-325
Abstract: In Semigroups on a half-space, Trans. Amer. Math. Soc. 147 (1970), 1-53, Horne considers semigroups that are the union of a group $G$ and a plane $L$ such that $G \cup L$ is a three-dimensional half-space and $G$ is the interior. After proving a great many things about half-space semigroups, Horne introduces the notion of a radical and determines all possible multiplications in $L$ for a half-space semigroup with empty radical. (It turns out that $S$ has empty radical if and only if each $ G$-orbit in $L$ contains an idempotent.) An example is provided for each configuration in $ L$. However, no attempt was made to show that the list of examples actually exhausted the possibilities for a half-space semigroup without radical. Another way of putting this problem is to determine when two different semigroups can have the same maximal group. In this paper we generalize Horne's results, for a semigroup without zero, by showing that if $S$ is any locally compact semigroup in which $ L$ is the boundary of $ G$, then $S$ is a half-space. Moreover, we are able to answer completely, for semigroups without radical and without a zero, the question posed above. It turns out that, with one addition (which we provide), Horne's list of half-space semigroups without radical and without zero is complete.
Finitely generated nilpotent groups with isomorphic finite quotients
P. F.
Pickel
327-341
Abstract: Let $\mathcal{F}(G)$ denote the set of isomorphism classes of finite homomorphic images of a group $ G$. We say that groups $ G$ and $H$ have isomorphic finite quotients if $ \mathcal{F}(G) = \mathcal{F}(H)$. In this paper we show that if $G$ is a finitely generated nilpotent group, the finitely generated nilpotent groups $ H$ for which $ \mathcal{F}(G) = \mathcal{F}(H)$) lie in only finitely many isomorphism classes. This is done using some finiteness results from the theory of algebraic groups along with some heretofore unpublished results of A. Borel.
Real orthogonal representations of algebraic groups
Frank
Grosshans
343-352
Abstract: The purpose of this paper is to determine explicitly, nondegenerate real symmetric bilinear forms invariant under a real absolutely irreducible representation of a real semisimple algebraic group, $G$. If $G$ is split, we construct an extension ${G^ \ast }$ containing $G$ and those outer automorphisms of $ G$ fixing the highest weight of the representation. The representation is then extended to ${G^ \ast }$ and the form is described in terms of the character of this extension. The case of a nonsplit algebraic group is then reduced to the above. The corresponding problem for representations by matrices over the real quaternion division algebra is also considered using similar methods.
An asymptotic theorem for systems of linear differential equations.
Allen
Devinatz
353-363
Abstract: Asymptotic estimates are obtained for a complete linearly independent set of solutions of a linear system of differential equations of the form $n \times n$ matrix with $n$ distinct eigenvalues, $R(t)$ is an integrable matrix valued function on $(0,\infty )$ and $V(t)$ is an $n \times n$ matrix valued function having certain differentiability properties. The theorem that is obtained generalizes a theorem of N. Levinson, Duke Math. J. 15 (1948), 111-126.
An extension of the Weyl-von Neumann theorem to normal operators
I. David
Berg
365-371
Abstract: We prove that a normal operator on a separable Hilbert space can be written as a diagonal operator plus a compact operator. If, in addition, the spectrum lies in a rectifiable curve we show that the compact operator can be made Hilbert-Schmidt.
Complex Fourier analysis on a nilpotent Lie group
Roe
Goodman
373-391
Abstract: Let $G$ be a simply-connected nilpotent Lie group, with complexification $ {G_c}$. The functions on $ G$ which are analytic vectors for the left regular representation of $ G$ on ${L_2}(G)$ are determined in this paper, via a dual characterization in terms of their analytic continuation to ${G_c}$, and by properties of their ${L_2}$ Fourier transforms. The analytic continuation of these functions is shown to be given by the Fourier inversion formula. An explicit construction is given for a dense space of entire vectors for the left regular representation. In the case $G = R$ this furnishes a group-theoretic setting for results of Paley and Wiener concerning functions holomorphic in a strip.
Automorphisms of a free associative algebra of rank $2$. I
Anastasia J.
Czerniakiewicz
393-401
Abstract: Let ${R_{\left\langle 2 \right\rangle }} = R\left\langle {x,y} \right\rangle$ be the free associative algebra of rank 2, on the free generators $x$ and $y$, over $R$ ($R$ a field, a Euclidean domain, etc.). We prove that if $\varphi$ is an automorphism of $ {R_{\left\langle 2 \right\rangle }}$ that keeps $(xy - yx)$ fixed (up to multiplication by an element of $R$), then $\varphi$ is tame, i.e. it is a product of elementary automorphisms of ${R_{\left\langle 2 \right\rangle }}$. This follows from a more general result about endomorphisms of ${R_{\left\langle 2 \right\rangle }}$ via a theorem due to H. Jung [6] concerning automorphisms of a commutative and associative algebra of rank 2.
Uniqueness of solutions of the Dirichlet and Neumann problems for hyperbolic equations
Eutiquio C.
Young
403-409
Abstract: Conditions for uniqueness of solutions of the Dirichlet and Neumann problems are obtained for a singular hyperbolic equation involving a real parameter.
On the remainders of certain metric spaces
Kenneth D.
Magill
411-417
Abstract: Let $X$ be completely regular and Hausdorff. The family $ \mathcal{R}(X)$ of all those spaces which are remainders of $X$ in compactifications has attracted a considerable amount of interest. In this paper we determine, in a certain sense, the family $\mathcal{R}(X)$ for a fairly large number of spaces $X$.
When is $\mu \ast L\,\sb{1}$ closed?
I.
Glicksberg
419-425
Abstract: For a finite measure $\mu$ on a locally compact abelian group, we partially answer the question of when $\mu \ast {L_1}$ is closed in ${L_1}$.
Some new classes of kernels whose Fredholm determinants have order less than one
Dale W.
Swann
427-435
Abstract: Let $K(s,t)$ be a complex-valued $ {L_2}$ kernel on the square $a \leqq s,t \leqq b$ and $\{ {\lambda _v}\}$, perhaps empty, denote the set of finite characteristic values (f.c.v.) of $ K$, arranged according to increasing modulus. Such f.c.v. are complex numbers appearing in the integral equation $ {\phi _v}(s) = {\lambda _v}\int_a^b {K(s,t){\phi _v}(t)dt}$, where the ${\phi _v}(s)$ are nontrivial ${L_2}$ functions on $[a,b]$. Further let ${k_1} = \int_a^b {K(s,s)} $ be well defined so that the Fredholm determinant of $K,D(\lambda )$, exists, and let $\mu$ be the order of this entire function. It is shown that (1) if $K(s,t)$ is a function of bounded variation in the sense of Hardy-Krause, then $ \mu \leqq 1$; (2) if in addition to the assumption (1), $K(s,t)$ satisfies a uniform Lipschitz condition of order $ \alpha > 0$ with respect to either variable, then $\mu < 1$ and ${k_1} = {\Sigma _v}1/{\lambda _v}$; (3) if $ K(s,t)$ is absolutely continuous as a function of two variables and $ {\partial ^2}K/\partial s\partial t$ (which exists almost everywhere) belongs to class ${L_p}$ for some $p > 1$, then $\mu < 1$ and ${k_1} = {\Sigma _v}1/{\lambda _v}$. In (2) and (3), the condition $ {k_1} \ne 0$ implies $ K(s,t)$ possesses at least one f.c.v.
Expandibility and collectionwise normality
J. C.
Smith;
L. L.
Krajewski
437-451
Abstract: In 1958 M. Katětov proved that in a normal space $X,X$ is expandable if and only if $ X$ is collectionwise normal and countably paracompact. This result has since been used to answer many questions in various areas of general topology. In this paper Katětov's theorem is generalized for nonnormal spaces and various characterizations of collectionwise normality are shown. Results concerning metrization, paracompactness, sum theorems, product theorems, mapping theorems and $M$-spaces are then obtained as applications of these new theorems.
Linear representations of certain compact semigroups
D. R.
Brown;
Michael
Friedberg
453-465
Abstract: In this paper we initiate the study of representation theory of compact, not necessarily commutative, uniquely divisible semigroups. We show that a certain class of semigroups are all topologically isomorphic to real matrix semigroups. The proof utilizes a group embedding theorem and the standard results on homomorphisms of Lie groups into matrix groups.
On knots with nontrivial interpolating manifolds
Jonathan
Simon
467-473
Abstract: If a polygonal knot $ K$ in the $3$-sphere ${S^3}$ does not separate an interpolating manifold $ S$ for $K$, then $S - K$ does not carry the first homology of either closed component of ${S^3} - S$. It follows that most knots $ K$ with nontrivial interpolating manifolds have the property that a simply connected manifold cannot be obtained by removing a regular neighborhood of $K$ from ${S^3}$ and sewing it back differently.
Homomorphisms on groups and induced maps on certain algebras of measures
Charles F.
Dunkl;
Donald E.
Ramirez
475-485
Abstract: Suppose that $ \varphi$ is a continuous homomorphism of a locally compact group $G$ into another such group, $ H$, then $\varphi$ induces in a natural way a homomorphism $ {\varphi ^ \ast }$ of the measure algebra of $G$, called $M(G)$, into $M(H)$. The action of $ {\varphi ^ \ast }$ on the subspace ${M_0}(G)$ is studied in this paper. The space $ {M_0}(G)$ is the nonabelian analogue to the space of measures on a locally compact abelian group whose Fourier-Stieltjes transforms vanish at infinity, and is defined herein. We prove that if $\varphi$ is an open homomorphism then $ {\varphi ^ \ast }({M_0}(G)) \subset {M_0}(H)$. If $G$ and $H$ are abelian and $\varphi$ is not open, then ${\varphi ^ \ast }(M(G)) \cap {M_0}(H) = \{ 0\}$. The main tool for this theorem is the fact, proved herein, that $\varphi$ is open if and only if its adjoint, $\hat \varphi :\hat H \to \hat G$, is proper (where $\hat G,\hat H$ are the character groups of $G,H$ resp.). Further properties of ${M_0}(G)$ for abelian or compact groups $ G$ are derived.
Minimal convergence spaces
D. C.
Kent;
G. D.
Richardson
487-499
Abstract: We are primarily concerned with minimal $ {\text{P}}$ convergence spaces, where $ {\text{P}}$ is one of the following convergence space properties: Hausdorff, ${{\text{T}}_2}, \lambda$-regular, $ \lambda$-Urysohn, and first countable, $\lambda$ an infinite cardinal number. Our conclusions usually resemble the corresponding topological results, but with some deviations ; for instance, a minimal Hausdorff convergence space is always compact, whereas a countable minimal regular convergence space need not be compact.
Structure of the solution set of some first order differential equations of comparison type
T. G.
Hallam;
J. W.
Heidel
501-512
Abstract: The comparison principle is a powerful tool that has a wide variety of applications in ordinary differential equations. The results of this article describe the geometric structure of the solution space of some first order scalar differential equations that may arise in the comparison method. A quite general class of differential equations is found to have a similar solution set configuration as the differential equation of separable variable type. One of the main results establishes, under certain conditions, that there is a unique unbounded solution of the first order differential equation which exists on an interval of the form [ ${t_0},\infty$). Furthermore, this unbounded solution separates the solutions that are bounded on [ ${t_0},\infty $) from those that are not continuable to all $t > {t_0}$.