Asymptotic formulae for the eigenvalues of a two-parameter ordinary differential equation of the second order
M.
Faierman
1-52
Abstract: We consider a two-point boundary value problem associated with an ordinary differential equation defined over the unit interval and containing the two parameters $\lambda$ and $\mu$. If for each real $\mu$ we denote the $m$th eigenvalue of our system by ${\lambda _m}(\mu )$, then it is known that ${\lambda _m}(\mu )$ is real analytic in $- \infty < \mu < \infty$. In this paper we concern ourselves with the asymptotic development of ${\lambda _m}(\mu )$ as $\mu \to \infty$, and indeed obtain such a development to an accuracy determined by the coefficients of our differential equation. With suitable conditions on the coefficients of our differential equation, the asymptotic formula for ${\lambda _m}(\mu )$ may be further developed using the methods of this paper. These results may be modified so as to apply to ${\lambda _m}(\mu )$ as $\mu \to - \infty$ if the coefficients of our differential equation are also suitably modified.
One-parameter inverse semigroups
Carl
Eberhart;
John
Selden
53-66
Abstract: This is the second in a projected series of three papers, the aim of which is the complete description of the closure of any one-parameter inverse semigroup in a locally compact topological inverse semigroup. In it we characterize all one-parameter inverse semigroups. In order to accomplish this, we construct the free one-parameter inverse semigroups and then describe their congruences.
Two Hilbert spaces in which polynomials are not dense
D. J.
Newman;
D. K.
Wohlgelernter
67-72
Abstract: Let $S$ be the Hilbert space of entire functions $f(z)$ such that $\vert\vert f(z)\vert{\vert^2} = \iint {\vert f(z){\vert^2}}dm(z)$, where $m$ is a positive measure defined on the Borel sets of the complex plane. Two Hilbert spaces are constructed in which polynomials are not dense. In the second example, our space is one which contains all exponentials and yet in which the exponentials are not complete. This is a somewhat surprising result since the exponentials are always complete on the real line.
Polynomials defined by generating relations
Hari Ballabh
Mittal
73-84
Abstract: Various authors have made efforts for finding new generating functions for known polynomial sets. In the present paper, by making use of the operator ${T_k} = x(k + xD)$, a number of generating functions and characterizations have been obtained for various polynomials in a systematic manner.
A representation theorem and approximation operators arising from inequalities involving differential operators
D.
Leviatan
85-99
Abstract: A representation of functions as integrals of a kernel $\psi (t;x)$, which was introduced by Studden, with respect to functions of bounded variation in $[0,\infty )$ is obtained whenever the functions satisfy some conditions involving the differential operators $(d/dt)\{ f(t)/{w_i}(t)\} ,i = 0,1,2, \ldots$. The results are related to the concepts of generalized completely monotonic functions and generalized absolutely monotonic functions in $ (0,\infty )$. Some approximation operators for the approximation of continuous functions in $ [0,\infty )$ arise naturally and are introduced; some sequence-to-function summability methods are also introduced.
Covering relations in the lattice of $T\sb{1}$-topologies
Roland E.
Larson;
W. J.
Thron
101-111
Abstract: A topology ${\mathcal{T}_1}$ is said to cover another topology $ {\mathcal{T}_2}$ if ${\mathcal{T}_2} \subsetneq {\mathcal{T}_1}$ and no other topology may be included between the two. In this paper, we characterize the relationship between a $ {T_1}$-topology and its covers. This characterization is used to prove that the lattice of ${T_1}$-topologies is both upper and lower semimodular. We also prove that the sublattice generated by the covers of a ${T_1}$-topology is isomorphic to the Boolean lattice of all subsets of the set of covers.
Semigroups satisfying variable identities. II
Mohan S.
Putcha;
Julian
Weissglass
113-119
Abstract: The concept of a semigroup satisfying an identity $xy = f(x,y)$ is generalized by considering identities in $n$-variables and letting the identity depend on the variables. The property of satisfying a ``variable identity'' is studied. Semigroups satisfying certain types of identities are characterized in terms of unions and semilattices of groups.
Smooth extensions in infinite dimensional Banach spaces
Peter
Renz
121-132
Abstract: If $B$ is $ {l_p}(\omega )$ or ${c_0}(\omega )$ we show $B$ has the following extension property. Any homeomorphism from a compact subset $M$ of $B$ into $B$ may be extended to a homeomorphism of $ B$ onto $B$ which is a $ {C^\infty }$ diffeomorphism on $B\backslash M$ to its image in $ B$. This is done by writing $B$ as a direct sum of closed subspaces $ {B_1}$ and ${B_2}$ both isomorphically isometric to $ B$ so that the natural projection of $K$ into ${B_1}$ along ${B_2}$ is one-to-one (see H. H. Corson, contribution in Symposium on infinite dimensional topology, Ann. of Math. Studies (to appear)). With $ K,B,{B_1}$ and $ {B_2}$ as above a homeomorphism of $B$ onto itself is constructed which leaves the $ {B_1}$-coordinates of points in $B$ unchanged, carries $K$ into ${B_1}$ and is a $ {C^\infty }$ diffeomorphic map on $ B\backslash K$. From these results the extension theorem may be proved by standard methods.
A local spectral theory for operators. III. Resolvents, spectral sets and similarity
J. G.
Stampfli
133-151
Abstract: Let $T$ be a bounded linear operator on a Hilbert space and assume $T$ has thin spectrum. When is $T$ similar to a normal operator? This problem is studied in a variety of situations and sufficient conditions are given in terms of characteristic functions, resolvents, spectral sets, and spectral resolutions. By contrast, the question ``When is $ T$ normal?'' has a relatively simple answer since in that case a necessary and sufficient condition can be given in terms of the resolvent alone.
The variation of singular cycles in an algebraic family of morphisms
Joel
Roberts
153-164
Abstract: (1) Let $g:{V^r} \to {W^m}(m \geqq r)$ be a morphism of nonsingular varieties over an algebraically closed field. Under certain conditions, one can define a cycle ${S_i}$ on $V$ with $\operatorname{Supp} ({S_i}) = \{ x\vert{\dim _{k(x)}}(\Omega _{X/Y}^1)(x) \geqq i\} $. The multiplicity of a component of ${S_i}$ can be computed directly from local equations for $g$. If $ {V^r} \subset {P^n}$, and if $g:V \to {P^m}$ is induced by projection from a suitable linear subspace of ${P^n}$, then ${S_1}$ is ${c_{m - r + 1}}(N \otimes \mathcal{O}( - 1))$, up to rational equivalence, where $N$ is the normal bundle of $V$ in ${P^n}$. (2) Let $f:X \to S$ be a smooth projective morphism of noetherian schemes, where $S$ is connected, and the fibres of $f$ are absolutely irreducible $ r$-dimensional varieties. For a geometric point $\eta :\operatorname{Spec} (k) \to S$, and a locally free sheaf $E$ on $X$, let ${X_\eta }$ be the corresponding geometric fibre, and ${E_\eta }$ the sheaf induced on ${X_\eta }$. If ${E_1}, \ldots ,{E_m}$ are locally free sheaves on $ X$, and if ${i_1} + \cdots + {i_m} = r$, then the degree of the zero-cycle $ {c_{{i_1}}}({E_{1\eta }}) \cdots {c_{{i_m}}}({E_{m\eta }})$ is independent of the choice of $\eta$. (3) The results of (1) and (2) are used to study the behavior under specialization of a closed subvariety ${V^r} \subset {P^n}$.
Abstract evolution equations and the mixed problem for symmetric hyperbolic systems
Frank J.
Massey
165-188
Abstract: In this paper we show that Kato's theory of linear evolution equations may be applied to the mixed problem for first order symmetric hyperbolic systems of partial differential equations.
Solvable fundamental groups of compact $3$-manifolds
Benny
Evans;
Louise
Moser
189-210
Abstract: A classification is given for groups which can occur as the fundamental group of some compact $3$-manifold. In most cases we are able to determine the topological structure of a compact $3$-manifold whose fundamental group is known to be solvable. Using the results obtained for solvable groups, we are able to extend some known results concerning nilpotent groups of closed $3$-manifolds to the more general class of compact $3$-manifolds. In the final section it is shown that each nonfinitely generated abelian group which occurs as a subgroup of the fundamental group of a $ 3$-manifold is a subgroup of the additive group of rationals.
The fix-points and factorization of meromorphic functions
Fred
Gross;
Chung-chun
Yang
211-219
Abstract: In this paper, we use the Nevanlinna theory of meromorphic functions and a result of Goldstein to generalize some known results in factorization and fixpoints of entire functions. Specifically, we prove (1) If $f$ and $g$ are nonlinear entire functions such that $ f(g)$ is transcendental and of finite order, then $f(g)$ has infinitely many fix-points. (2) If $ f$ is a polynomial of degree $\geqq 3$, and $g$ is an arbitrary transcendental meromorphic function, then $f(g)$ must have infinitely many fix-points. (3) Let $p(z),q(z)$ be any nonconstant polynomials, at least one of which is not $c$-even, and let $a$ and $b$ be any constants with $a$ or $b \ne 0$. Then $h(z) = q(z)\exp (a{z^2} + bz) + p(z)$ is prime.
On residually finite knot groups
E. J.
Mayland
221-232
Abstract: The residual finiteness of the class of groups of fibred knots, or those knot groups with finitely generated and, therefore, free commutator subgroups, has been known for some time. Using Baumslag's results on absolutely parafree groups, this paper extends the result to twist knots (Whitehead doubles of the trivial knot) and certain other classes of nonfibred knots whose minimal spanning surface has complement with free fundamental group. As a by-product more explicit finite representations, namely cyclic extensions of certain $p$-groups, are obtained for these knot groups and the groups of fibred knots. Finally composites of two such knots also have residually finite groups.
Absolute Tauberian constants for Ces\`aro means
Soraya
Sherif
233-241
Abstract: This paper is concerned with introducing two inequalities of the form $\sum\nolimits_{n = 0}^\infty {\vert{\tau _n}} - {a_n}\vert \leqq KA$ and ${\tau _n} = C_n^{(k)} - C_{n - 1}^{(k)},C_n^{(k)}$ denote the Cesàro transform of order $ k,K$ and $K'$ are absolute Tauberian constants, $A = \sum\nolimits_{n = 0}^\infty {\vert\Delta (n{a_n}} )\vert < \infty ,B = \s... ...ty {\vert\Delta ((1/n)\sum\nolimits_{v = 1}^{n - 1} {v{a_v}} } )\vert < \infty$ and $ \Delta {u_k} = {u_k} - {u_{k + 1}}$. The constants $K,K'$ will be determined.
A generalized area theorem for harmonic functions on hermitian hyperbolic space
Robert Byrne
Putz
243-258
Abstract: Let $D$ be the noncompact realization of hermitian hyperbolic space. We consider functions on $ D$ which are harmonic with respect to the Laplace-Beltrami operator. The principal result is a generalized area theorem which gives a necessary and sufficient condition for the admissible convergence of harmonic functions.
Equivariant bordism and Smith theory. III
R. E.
Stong
259-263
Abstract: The bordism classes of a manifold with involution is determined by means of Wu type classes in the Smith cohomology.
The asymptotic behavior of a Fourier transform and the localization property for eigenfunction expansions for some partial differential operators
Burton
Randol
265-271
Abstract: The asymptotic behavior of a certain Fourier transform is investigated, and the result is applied to obtain a localization theorem for elliptic operators on the torus.
Some degeneracy theorems for entire functions with values in an algebraic variety
James A.
Carlson
273-301
Abstract: In the first part of this paper we prove the following extension theorem. Let $P_q^ \ast$ be a $q$-dimensional punctured polycylinder, i.e. a product of disks and punctured disks. Let ${W_n}$ be a compact complex manifold such that the bundle of holomorphic $q$-forms is positive in the sense of Grauert. Let $ f:P_q^ \ast \to {W_n}$ be a holomorphic map whose Jacobian determinant does not vanish identically. Then $f$ extends as a rational map to the full polycylinder $ {P_q}$. In the second half of the paper we prove the following generalization of the little Picard theorem to several complex variables: Let $V \subset {P_n}$ be a hypersurface of degree $d \geqq n + 3$ whose singularities are locally normal crossings. Then any holomorphic map $f:{C^n} \to {P_n} - V$ has identically vanishing Jacobian determinant.
The connectedness of the collection of arc cluster sets
Peter
Lappan
303-310
Abstract: Let $f$ be a continuous complex-valued function defined on the unit disk and let $p$ be a boundary point of the disk. A very natural topology on the collection of all arc cluster sets of $f$ at the point $p$ has been investigated by Belna and Lappan [1] who proved that this collection is a compact set under certain suitable conditions. It is proved here that this collection is an arcwise connected set under the topology in question, but is not in general locally arcwise connected or even locally connected. It is also shown by example that it is generally not possible to map the real line onto the collection of arc cluster sets at $p$ in a continuous manner.
Bounded continuous functions on a completely regular space
F. Dennis
Sentilles
311-336
Abstract: Three locally convex topologies on $C(X)$ are introduced and developed, and in particular shown to coincide with the strict topology on locally compact $X$ and yield dual spaces consisting of tight, $ \tau$-additive and $ \sigma$-additive functionals respectively for completely regular $ X$.
On the H. Lewy extension phenomenon
Ricardo
Nirenberg
337-356
Abstract: We prove local approximation and extension theorems for ${C^\infty }$ submanifolds $M$ of ${C^n}$ (CR submanifolds). Under some conditions on $ M$, any smooth solution of the induced Cauchy-Riemann equations can be extended holomorphically to bigger (and sometimes open) sets.
Knots whose branched cyclic coverings have periodic homology
C. McA.
Gordon
357-370
Abstract: Let ${M_k}$ be the $k$-fold branched cyclic covering of a (tame) knot of $ {S^1}$ in ${S^3}$. Our main result is that the following statements are equivalent: (1) ${H_1}({M_k})$ is periodic with period $ n$, i.e. ${H_1}({M_k}) \cong {H_1}({M_{k + n}})$ for all $k$, (2) ${H_1}({M_k}) \cong {H_1}({M_{(k,n)}})$ for all $k$, (3) the first Alexander invariant of the knot, ${\lambda _1}(t) = {\Delta _1}(t)/{\Delta _2}(t)$, divides ${t^n} - 1$.
Integral decomposition of functionals on $C\sp{\ast} $-algebras
Herbert
Halpern
371-385
Abstract: The spectrum of the center of the weak closure of a ${C^ \ast }$-algebra with identity on a Hilbert space is mapped into a set of quasi-equivalence classes of representations of the $ {C^ \ast }$-algebra so that every positive $\sigma$-weakly continuous functional on the algebra can be written in a central decomposition as an integral over the spectrum of a field of states whose canonical representations are members of the respective quasi-equivalence classes except for a nowhere dense set. Various questions relating to disjointness of classes, factor classes, and uniformly continuous functionals are studied.
Classification of all hereditarily indecomposable circularly chainable continua
Lawrence
Fearnley
387-401
Abstract: In a recent paper the author has established an affirmative solution to a well-known and previously unsolved problem raised by R. H. Bing in 1951 concerning whether or not the pseudo-circle is topologically unique. Now in this present paper, as the natural culmination of the study initiated in the earlier paper, complete topological and mapping classification theorems are established for all hereditarily indecomposable circularly chainable continua. The principal topological classification results of this paper are the theorems that hereditarily indecomposable circularly chainable continua are characterized set-theoretically by their equivalence classes of fundamental sequences and are characterized algebraic-topologically by their Čech cohomology groups. These topological classification theorems are then used in establishing mapping classification theorems for all hereditarily indecomposable circularly chainable continua and in proving that the mapping hierarchy of hereditarily indecomposable circularly chainable continua constitutes a lattice. Among the consequences of the foregoing primary results of the paper are the additional theorems that two hereditarily indecomposable circularly chainable continua are topologically equivalent if and only if each of them is a continuous image of the other, and that every $ k$-adic pseudo-solenoid is topologically unique.
Orbits in a real reductive Lie algebra
L. Preiss
Rothschild
403-421
Abstract: The purpose of this paper is to give a classification of the orbits in a real reductive Lie algebra under the adjoint action of a corresponding connected Lie group. The classification is obtained by examining the intersection of the Lie algebra with the orbits in its complexification. An algebraic characterization of the minimal points in the closed orbits is also given.
Morita theorems for functor categories
D. C.
Newell
423-433
Abstract: We generalize the Morita theorems to certain functor categories using properties of adjoint functors.
Locally uniform spaces
James
Williams
435-469
Abstract: The axioms for a locally uniform space $(X,\mathcal{U})$ may be obtained by localizing the last axiom for a uniform space to obtain $ \forall x \in X,\forall U \in \mathcal{U},\exists V \in \mathcal{V}:(V \circ V)[x] \subseteq U[x]$. With each locally uniform space one may associate a regular topology, just as one associates a completely regular topology with each uniform space. The topologies of locally uniform spaces with nested bases may be characterized using Boolean algebras of regular open sets. As a special case, one has that locally uniform spaces with countable bases have pseudo-metrizable topologies. Several types of Cauchy filters may be defined for locally uniform spaces, and a major portion of the paper is devoted to a study and comparison of their properties. For each given type of Cauchy filter, complete spaces are those in which every Cauchy filter converges; to complete a space is to embed it as a dense subspace in a complete space. In discussing these concepts, it is convenient to make the mild restriction of considering only those locally uniform spaces $(X,\mathcal{V})$ in which each element of $\mathcal{V}$ is a neighborhood of the diagonal in $X \times X$ with respect to the relative topology; these spaces I have called NLU-spaces. With respect to the more general types of Cauchy filters, some NLU-spaces are not completable; this happens even though some completable NLU-spaces can still have topologies which are not completely regular. Examples illustrating these completeness situations and having various topological properties are obtained from a generalized construction. It is also shown that there is a largest class of Cauchy filters with respect to which each NLU-space has a completion that is also an NLU-space.
Nested sequences of local uniform spaces
James
Williams
471-481
Abstract: A locally uniform space is a pair of sets $ (X,\mathcal{V}),\mathcal{V}$ being a filter on $X \times X$ such that $\forall U \in \mathcal{V},\Delta (X) = \{ \langle x,x\rangle :x \in X\} \subse... ...,{U^{ - 1}} = \{ \langle y,x\rangle :\langle x,y\rangle \in U\} \in \mathcal{V}$, and $ \forall x \in X,\exists V \in \mathcal{V}:(V \circ V)[x] \subseteq U[x]$. We shall say that a sequence $\{ ({X_n},{\mathcal{V}_n}):n \in \omega \}$ is nested iff $ \forall n \in \omega ,{X_n} \subseteq {X_{n + 1}}$ and $ {\mathcal{V}_{n + 1}}\vert{X_n} = {\mathcal{V}_n}$. By a limit for a nested sequence $\{ ({X_n},{\mathcal{V}_n}):n \in \omega \}$, we shall mean any locally uniform space $ (X,\mathcal{V})$ such that $ X = \cup \{ {X_n}:n \in \omega \}$ and $ \forall n \in \omega ,\mathcal{V}\vert{X_n} = {\mathcal{V}_n}$. Our first task will be to consider when a nested sequence of locally uniform spaces has a limit; in order to do this, we shall introduce a weak generalization of pseudo-metric functions. We shall also show that, in contrast to locally uniform spaces, each nested sequence of uniform spaces has a limit. With each locally uniform space one can associate a regular relative topology in the obvious fashion. E. Hewitt and J. Novak have constructed regular spaces of the type on which every real-valued continuous function is constant; we shall use our results about limits to give a relatively simple general construction for producing locally uniform spaces which have this type of relative topology. The construction may be done in such a way that the spaces produced have several pleasant topological properties.
Approximation of strictly singular and strictly cosingular operators using nonstandard analysis
J. W.
Brace;
R. Royce
Kneece
483-496
Abstract: The strictly singular operators and the strictly cosingular operators are characterized by the manner in which they can be approximated by continuous linear operators of finite-dimensional range. We make use of linear convergence structures to obtain each class as limit points of the operators with finite-dimensional range. The construction of a nonstandard model makes it possible to replace convergence structures by topologies. Our nonstandard models are called nonstandard locally convex spaces.
Open mappings of the universal curve onto continuous curves
David C.
Wilson
497-515
Abstract: A criterion for the existence of an open mapping from one compact metric space onto another is established in this paper. This criterion is then used to establish the existence of a monotone open mapping of the universal curve onto any continuous curve and the existence of a light open mapping of the universal curve onto any nondegenerate continuous curve. These examples show that if $f$ is a monotone open or a light open mapping of one compact space $X$ onto another $Y$, then it will not necessarily be the case that $ \dim Y \leqq \dim X + k$, where $k$ is some positive integer.
Addendum to: ``On a problem of Tur\'an about polynomials with curved majorants''
Q. I.
Rahman
517-518
Erratum to: ``On entropy and generators of measure-preserving transformations''
Wolfgang
Krieger
519
Erratum to ``Improbability of collisions in Newtonian gravitational systems''
Donald Gene
Saari
521