Complete topologies on spaces of Baire measure
R. B.
Kirk
1-29
Abstract: Let X be a completely regular Hausdorff space, let L be the linear space of all finite linear combinations of the point measures on X and let ${M_\sigma }$ denote the space of Baire measures on X. The following is proved: If ${M_\sigma }$ is endowed with the topology of uniform convergence on the uniformly bounded, equicontinuous subsets of ${C^b}(X)$, then $ {M_\sigma }$ is a complete locally convex space in which L is dense and whose dual is ${C^b}(X)$, provided there are no measurable cardinals. A complete description of the situation in the presence of measurable cardinals is also given. Let ${M_C}$ be the subspace of $ {M_\sigma }$ consisting of those measures which have compact support in the realcompactification of X. The following result is proved: If ${M_C}$ is endowed with the topology of uniform convergence on the pointwise bounded and equicontinuous subsets of $C(X)$, then ${M_C}$ is a complete locally convex space in which L is dense and whose dual is $C(X)$, provided there are no measurable cardinals. Again the situation if measurable cardinals exist is described completely. Let M denote the Banach dual of ${C^b}(X)$. The following is proved: If M is endowed with the topology of uniform convergence on the norm compact subsets of ${C^b}(X)$, then M is a complete locally convex space in chich L is dense. It is also proved that ${M_\sigma }$ is metrizable if and only if X is discrete and that the metrizability of either $ {M_C}$ or M is equivalent to X being finite. Finally the following is proved: If ${M_C}$ has the Mackey topology for the pair $({M_C},C(X))$, then ${M_C}$ is complete and L is dense in ${M_C}$.
Products of decompositions of $E\sp{n}$
Brian J.
Smith
31-41
Abstract: In this paper we give a sufficient condition for the existence of a homeomorphism $h:{E^m}/G \times {E^n}/H \to {E^{m + n}}$, where G and H are u.s.c. decompositions of Euclidean space. This condition is then shown to hold for a wide class of examples in which the decomposition spaces ${E^m}/G$ and ${E^n}/H$ may fail to be Euclidean.
Prime ideals and sheaf representation of a pseudo symmetric ring
Gooyong
Shin
43-60
Abstract: Almost symmetric rings and pseudo symmetric rings are introduced. The classes of symmetric rings, of almost symmetric rings, and of pseudo symmetric rings are in a strictly increasing order. A sheaf representation is obtained for pseudo symmetric rings, similar to the cases of symmetric rings, semiprime rings, and strongly harmonic rings. Minimal prime ideals of a pseudo symmetric ring have the same characterization, due to J. Kist, as for the commutative case. A characterization is obtained for a pseudo symmetric ring with a certain right quotient ring to have compact minimal prime ideal space, extending a result due to Mewborn.
Restricting a Schauder basis to a set of positive measure
James
Shirey
61-71
Abstract: Let $\{ {f_n}\}$ be an orthonormal system of functions on [0, 1] containing a subsystem $\{ {f_{{n_k}}}\}$ for which (a) ${f_{{n_k}}} \to 0$ weakly in ${L_2}$, and (b) given $E \subset [0,1]$, $\vert E\vert > 0$, $ {\operatorname{Lim}}\;{\operatorname{Inf}}{\smallint _E}\vert{f_{{n_k}}}(x)\vert dx > 0$. There then exists a subsystem $\{ {g_n}\}$ of $ \{ {f_n}\}$ such that for any set E as above, the linear span of $\{ {g_n}\}$ in ${L_1}(E)$ is not dense. For every set E as above, there is an element of ${L_p}(E)$, $ 1 < p < \infty$, whose Walsh series expansion converges conditionally and an element of ${L_1}(E)$ whose Haar series expansion converges conditionally.
Commutative torsion theory
Paul-Jean
Cahen
73-85
Abstract: This paper links several notions of torsion theory with commutative concepts. The notion of dominant dimension [H. H. Storrer, Torsion theories and dominant dimensions, Appendix to Lecture Notes in Math., vol. 177, Springer-Verlag, Berlin and New York, 1971. MR 44 #1685.] is shown to be very close to the notion of depth. For a commutative ring A and a torsion theory such that the primes of A, whose residue field is torsion-free, form an open set U of the spectrum of A, Spec A, a concrete interpretation of the module of quotients is given: if M is an A-module, its module of quotients $Q(M)$ is isomorphic to the module of sections $\tilde M(U)$, of the quasi-coherent module $ \tilde M$ canonically associated to M. In the last part it is proved that the (T)-condition of Goldman is satisfied [O. Goldman, Rings and modules of quotients, J. Algebra 13 (1969), 10-47. MR 39 #6914.] if and only if the set of primes, whose residue field is torsion-free, is an affine subset of Spec A, together with an extra condirion. The extra, more technical, condition is always satisfied over a Noetherian ring, in this case also it is classical that the (T)-condition of Goldman means that the localization functor Q is exact. This gives a new proof to Serre's theorem [J.-P. Serre, Sur la cohomologie des variétés algébriques, J. Math. Pures Appl. (9) 36 (1957), 1-16. MR 18,765.]. As an application, the affine open sets of a regular Noetherian ring are also characterized.
Smoothness of certain metric projections on Hilbert space
Richard B.
Holmes
87-100
Abstract: A study is made of differential properties of the distance function and the metric projection defined by a closed convex subset of Hilbert space. The former mapping is also considered within the context of more general Banach spaces.
Measurable transformations on compact groups
J. R.
Choksi
101-124
Abstract: For an arbitrary finite Baire measure $\mu$ on an arbitrary compact group G, it is shown that every automorphism of the measure algebra of $\mu$ can be induced by an invertible completion Baire measurable point transformation of G. If $\mu$ is Haar measure, the point transformation is completion Borel measurable.
Topological entropy for noncompact sets
Rufus
Bowen
125-136
Abstract: For $f:X \to X$ continuous and $Y \subset X$ a topological entropy $ h(f,Y)$ is defined. For X compact one obtains results generalizing known theorems about entropy for compact Y and about Hausdorff dimension for certain $Y \subset X = {S^1}$ . A notion of entropy-conjugacy is proposed for homeomorphisms.
On the uniform convergence of quasiconformal mappings
Bruce
Palka
137-152
Abstract: Let D be a domain in extended Euclidean n-space with ``smooth'' boundary and let $ \{ {f_j}\}$ be a sequence of K-quasiconformal mappings of D into $ {R^n}$ which converges uniformly on compact sets in D to a quasiconformal mapping. This paper considers the question: When does the sequence $ \{ {f_j}\}$ converge uniformly on all of D? Geometric conditions on the domains ${f_j}(D)$ are given which are sufficient and, in many cases, necessary for uniform convergence. The particular case where D is the unit ball in ${R^n}$ is examined to obtain analogues to classical convergence theorems for conformal mappings in the plane.
Fixed point structures
T. B.
Muenzenberger;
R. E.
Smithson
153-173
Abstract: A fixed point structure is a triple $ (X,\mathcal{P},\mathcal{F})$ where X is a set, $ \mathcal{P}$ a collection of subsets of X, and $ \mathcal{F}$ a family of multifunctions on X into itself together with a set of axioms which insure that each member of $\mathcal{F}$ has a fixed point. A fixed point structure for noncontinuous multifunctions on semitrees is established that encompasses fixed point theorems of Wallace-Ward and Young-Smithson as well as new fixed point theorems for partially ordered sets and closed stars in real vector spaces. Also two other fixed point structures are presented that subsume fixed point theorems of Tarski-Ward-Smithson on semilattices and, more generally, partially ordered sets. Also the Davis-Ward converse to this last fixed point theorem is obtained.
Analytic functions characterized by their means on an arc
Chin Hung
Ching;
Charles K.
Chui
175-183
Abstract: It is known that a function f, holomorphic in the open unit disc U with ${C^{1 + \varepsilon }}$ boundary data for some $\varepsilon > 0$, is uniquely determined by its arithmetic means over equally spaced points on $\partial U$. By using different techniques, we weaken the hypothesis ${C^{1 + \varepsilon }}(\partial U)$ to functions with ${L^p}$ derivatives, $1 < p \leq \infty$. We also prove that a function is determined by its averages over an arc K if f is holomorphic in a neighborhood of $\bar U$, and that this result is false for some functions f in $A \cap {C^\infty }(\bar U)$. On the other hand, we can capture a $ A \cap {C^2}(\bar U)$ function from its means and shifted means on K.
Wandering out to infinity of diffusion processes
Avner
Friedman
185-203
Abstract: Let $\xi (t)$ be a diffusion process in $ {R^n}$, given by $ d\xi = b(\xi )dt + \sigma (\xi )dw$. Conditions are given under which either $\vert\xi (t)\vert \to \infty$ as $t \to \infty$ with probability 1, or $ \xi (t)$ visits any neighborhood at a sequence of times increasing to infinity, with probability 1. The results are obtained both in case (i) $\sigma (x)$ is nondegenerate, and (ii) $\sigma (x)$ is degenerate at a finite number of points and hypersurfaces.
On the singular boundary value problem for elliptic equations
Kazunari
Hayashida
205-221
Abstract: The operator $\mathcal{L}$ is elliptic and of second order in a domain $\Omega$ in ${R^N}$. We consider the following boundary value problem: $ \mathcal{L}u = f$ in $ \Omega$ and $\mathcal{B}u = 0$ on $ \partial \Omega$, where $ \mathcal{B} = ad/dn + \beta$ (d/dn is the conormal derivative on $\partial \Omega$). The coefficient $\alpha$ is assumed to be nonnegative. However, $\alpha$ may vanish partly on $\partial \Omega$. Then the regularity of the weak solutions for the above problem is shown by the variational method.
Finite- and infinite-dimensional representation of linear semisimple groups
James
Lepowsky;
Nolan R.
Wallach
223-246
Abstract: Every representation in the nonunitary principal series of a noncompact connected real semisimple linear Lie group G with maximal compact subgroup K is shown to have a K-finite cyclic vector. This is used to give a new proof of Harish-Chandra's theorem that every member of the nonunitary principal series has a (finite) composition series. The methods of proof are based on finite-dimensional G-modules, concerning which some new results are derived. Further related results on infinite-dimensional representations are also obtained.
Cross-sections of symplectic Stiefel manifolds
François
Sigrist;
Ulrich
Suter
247-259
Abstract: The cross-section problem for the symplectic Stiefel manifolds is solved, using the now-proved Adams conjecture.
The commutant of analytic Toeplitz operators
James A.
Deddens;
Tin Kin
Wong
261-273
Abstract: In this paper we study the commutant of an analytic Toeplitz operator. For $ \phi \;\;{H^\infty }$, let $\phi = \chi F$ be its inner-outer factorization. Our main result is that if there exists $ \lambda \;\epsilon \;{\text{C}}$ such that X factors as $ \chi = {\chi _1}{\chi _2} \cdots {\chi _n}$, each ${\chi _i}$ an inner function, and if $F - \lambda$ is divisible by each $ {\chi _i}$, then $\chi (z) = {z^n},n \geq 1$, then $ \phi \;\epsilon {H^\infty }$ is univalent then $\{ {T_\phi }\} ' = \{ {T_z}\} '$. We are also able to prove that if the inner factor of $ \phi$ is $\chi (z) = {z^n},n \geq 1$, then $ \{ {T_\phi }\} ' = \{ {T_{{z^s}}}\} '$ where s is a positive integer maximal with respect to the property that ${z^n}$ and $F(z)$ are both functions of ${z^s}$. We conclude by raising six questions.
Universal generators for varieties of nuclear spaces
B.
Rosenberger
275-290
Abstract: It is shown that a product of several copies of $\Lambda ({\beta ^\phi })$ is a universal $ \phi$-nuclear space if the power series space $\Lambda ({\beta ^\phi })$ with $\beta _k^\phi = - \log ({\phi ^{ - 1}}(1/\sqrt {k + 1} )),k\;\epsilon \;\{ 0,1,2, \cdots \}$, is $\phi$-nuclear; here $\phi = [0,\infty ) \to [0,\infty )$ is a continuous, strictly increasing subadditive function with $\phi (0) = 0$. In case $\Lambda ({\beta ^\phi })$ is not $\phi $-nuclear the sequence space $ \Lambda (l_\phi ^ + )$ is a $\phi$-nuclear space with the property that every $ \phi$-nuclear space is isomorphic to a subspace of a product of $\Lambda (l_\phi ^ + )$ if ${\lim\;\sup _{t \to 0}}{(\phi (t))^{ - 1}}\phi (\sqrt t ) < \infty $.
Absolute convergence of series of Fourier coefficients
James R.
McLaughlin
291-316
Abstract: In this article the author unifies and generalizes practically all known sufficiency results for absolute convergence of series of Fourier coefficients that are given in terms of the integrated modulus of continuity, best approximation, or bounded pth variation. This is done for the trigonometric, Walsh, Haar, Franklin, and related systems as well as general orthonormal systems. Many of the original proofs of previous results relied upon special properties of the trigonometric, Haar, and other systems and were done independently of one another. Also, several authors have proved results which at the time they believed to be generalizations of past results, but are, in fact, corollaries of them. The present author will expose underlying principles and illustrate their usefulness.
A metric characterization of cells
Ellard
Nunnally
317-325
Abstract: We examine finite dimensional compact convex metric spaces each having the property that the union of two line segments in the space, having more than one point in common, is a line segment. The question has been asked (Borsuk; Bing) whether each such space is a cell. The answer is yes if the dimension of the space is $\leq 2$ (Lelek and Nitka) or 3 (Rolfsen). Here we provide an affirmative answer for arbitrary finite dimension provided the space has the additional property that the join of any point to any line segment in the space is a convex set.
Decomposition theorems of Riemannian manifolds
Pyng
Wang
327-341
Abstract: Given two complementary orthogonal parallel foliations on a complete connected Riemannian manifold M, a necessary and sufficient condition for the direct product of the two leaves through a point m being a covering manifold of M is obtained. It is shown that the direct product of the two leaves through m of the two foliations is a Riemannian covering of M if the two leaves are regular at m. Moreover, if one of the foliations is regular and the intersection of the two leaves through m contains only the point m, then M is isometric to the direct product of the two leaves.
The representation of norm-continuous multipliers on $L\sp{\infty }$-spaces
Gregory A.
Hively
343-353
Abstract: If G is a group and ${\mathcal{L}^\infty }(G,\mathcal{S})$ is an appropriate space of bounded measurable functions on G, a representation is obtained for the algebra of norm-continuous multipliers on ${\mathcal{L}^\infty }(G,\mathcal{S})$ as an algebra of bounded additive set functions on G. If G is a locally compact group, a representation of the norm-continuous multipliers on the quotient space ${\mathcal{L}^\infty }(G)$ is obtained in terms of a quotient algebra of bounded additive set functions on G.
Pseudo-differential estimates for linear parabolic operators
David
Ellis
355-371
Abstract: In recent papers, S. Kaplan and D. Ellis have used singular integral operator theory, multilinear interpolation and forms of the classical ``energy inequality'' to obtain results for linear parabolic operators. For higher order linear parabolic operators the local estimates were globalized by a Gårding type partition of unity. In the present paper it is shown how the theory of pseudo-differential operators can be used to study linear parabolic operators without recourse to multilinear interpolation. We also prove that the Gårding type partition of unity is square summable in the Sobolev type spaces ${H^S}$ and $ {\mathcal{K}^{r,S}}$.
Convolution equations and harmonic analysis in spaces of entire functions
D. G.
Dickson
373-385
Abstract: If H is the topological space of functions analytic in the simply connected open set $\Omega$ of the plane with the topology of compact convergence, its dual may be identified with the space E of functions of exponential type whose Borel transforms have their singularities in $ \Omega$. For f in H and $\phi$ in E, $ (f \ast \phi )(z) \equiv \left\langle {f,{\phi _z}} \right\rangle$ where $ {\phi _z}$ is the z-translate of $\phi$. If $ f{\nequiv}0$ in any component of $ \Omega ,f \ast \phi = 0$ if and only if $\phi$ is a finite linear combination of monomial-exponentials $ {z^p} \exp (\omega z)$ where $\omega$ is a zero of f in $\Omega$ of order at least $ p + 1$. For such f and $\psi$ in E, $f \ast \phi = \psi$ is solved explicitly for $ \phi$. If E is assigned its strong dual topology and $\tau (\phi )$ is the closed linear span in E of the translates of $\phi$, then $ \tau (\phi )$ is a finite direct sum of closed subspaces spanned by monomial-exponentials. Each closed translation invariant subspace of E is the kernel of a convolution mapping $\phi \to f \ast \phi$; there is a one-to-one correspondence between such subspaces and the closed ideals of H with the correspondence that of annihilators.
On arbitrary sequences of isomorphisms in $R\sp{m}\rightarrow R\sp{m}$
Charles C.
Pugh
387-400
Abstract: In this paper a new, clean proof of an algebraic theorem needed in ordinary differential equations is presented. The theorem involves the existence and uniqueness of a ``complete splitting'' for some subsequence of an arbitrary sequence of isomorphisms of Euclidean m-space. In the positive-definite case, a complete splitting is a limit condition on eigenspaces and eigenvalues.
On the points of Weierstrass in dimensions greater than one
Roy H.
Ogawa
401-417
Abstract: In this paper, the classical concept of Weierstrass points on a Riemann surface is generalized to the consideration of similar points associated with a holomorphic vector bundle E over a compact complex manifold M. These points are invariants of the pair (E, M). The study of these general Weierstrass points is then initiated in this paper by deriving some results about the relationship of the points to singular sets of holomorphic mappings of the manifold to Grassmann spaces associated with the vector space of sections of the vector bundle. The accessibility of the point sets are demonstrated with some examples.
Stability of foliations
Harold I.
Levine;
Michael
Shub
419-437
Abstract: Let X be a compact manifold and let k be an integer. It is shown that the set of homeomorphism conjugacy classes of germs at X of foliations of codimension k and the set of homeomorphism conjugacy classes of (holonomy) representations of ${\prod _1}(X)$ in the group of germs at 0 of 0-fixed self-diffeomorphisms of ${{\text{R}}^k}$ are homeomorphic when given appropriate topologies. Stable foliation germs and stable holonomy representations correspond under this homeomorphism. It is shown that there are no stable foliation germs at a toral leaf if the dimension of the torus is greater than one.
The periods of Eichler integrals for Kleinian groups
Hiroki
Sato
439-456
Abstract: We shall give period relations and inequalities for Eichler integrals for Kleinian groups $\Gamma$ which have simply connected components of of the region of discontinuity of $ \Gamma$. These are a generalization of those for abelian integrals. By using the period inequality, we shall give an alternate proof of a result of Kra.
A free boundary problem connected with the optimal stopping problem for diffusion processes
Daniel B.
Kotlow
457-478
Abstract: This paper deals with a free boundary problem for a parabolic equation in one space variable which arises from the problem of selecting an optimal stopping strategy for the diffusion process connected with the equation. It is shown that a solution of the free boundary problem yields the solution of a minimum problem concerning supersolutions of the parabolic equation as well as the solution of the optimal stopping problem. Theorems regarding the existence, uniqueness, regularity, and approach to the steady state of solutions of the free boundary problem are established.
A completely mitotic nonrecursive r.e. degree
Richard E.
Ladner
479-507
Abstract: A nonrecursive r.e. degree d is constructed that has the property that every r.e. set of degree d is mitotic. The degree d has several other interesting properties including the property that any two r.e. sets of degree d are weak truth table equivalent.
Erratum to ``The nonstandard theory of topological vector spaces'' (Trans. Amer. Math. Soc. {\bf 172} (1972), 405--435)
C. Ward
Henson;
L. C.
Moore
509