Algebraic extensions of difference fields
Peter
Evanovich
1-22
Abstract: An inversive difference field $ \mathcal{K}$ is a field K together with a finite number of automorphisms of K. This paper studies inversive extensions of inversive difference fields whose underlying field extensions are separable algebraic. The principal tool in our investigations is a Galois theory, first developed by A. E. Babbitt, Jr. for finite dimensional extensions of ordinary difference fields and extended in this work to partial difference field extensions whose underlying field extensions are infinite dimensional Galois. It is shown that if $ \mathcal{L}$ is a finitely generated separable algebraic inversive extension of an inversive partial difference field $\mathcal{K}$ and the automorphisms of $\mathcal{K}$ commute on the underlying field of $\mathcal{K}$ then every inversive subextension of $\mathcal{L}/\mathcal{K}$ is finitely generated. For ordinary difference fields the paper makes a study of the structure of benign extensions, the group of difference automorphisms of a difference field extension, and two types of extensions which play a significant role in the study of difference algebra: monadic extensions (difference field extensions $\mathcal{L}/\mathcal{K}$ having at most one difference isomorphism into any extension of $\mathcal{K}$) and incompatible extensions (extensions $ \mathcal{L}/\mathcal{K},\mathcal{M}/\mathcal{K}$ having no difference field compositum).
On embedding set functions into covariance functions
G. D.
Allen
23-33
Abstract: We consider any continuous hermitian kernel $\mathcal{P} \times \mathcal{P}$ where $\mathcal{P}$ is the prering of intervals of [0,1]. Conditions on M are given to find an interval covariance function $K(\Delta ,\Delta ') = M(\Delta ,\Delta ')$ for all nonoverlapping $\Delta$ and $\Delta '$ in $ \mathcal{P}$. The problem is solved by first treating finite hermitian matrices A and finding a positive definite matrix B so that ${b_{ij}} = {a_{ij}},i \ne j$, so that tr B is minimized. Using natural correspondence between interval covariance functions and stochastic processes, a decomposition theorem is derived for stochastic processes of bounded quadratic variation into an orthogonal process and a process having minimal quadratic variation.
The $L^p$ norm of sums of translates of a function
Kanter
Marek
35-47
Abstract: For p not an even integer, $p > 0$, we prove that knowledge of the $ {L^p}$ norm of all linear combinations of translates of a real valued function in $ {L^p}(R)$ determines the function up to translation and multiplication by $ \pm 1$.
On the convergence of best uniform deviations
S. J.
Poreda
49-59
Abstract: If a function f is continuous on a closed Jordan curve $ \Gamma$ and meromorphic inside $\Gamma$, then the polynomials of best uniform approximation to f on $\Gamma$ converge interior to $\Gamma$. Furthermore, the limit function can in each case be explicitly determined in terms of the mapping function for the interior of $ \Gamma$. Applications and generalizations of this result are also given.
Extensions of functions and spaces
Giovanni
Viglino
61-69
Abstract: We investigate, for a given map $\varphi$ from a topological space X to a topological space Y (denoted by $[X,\varphi ,Y]$), those triples $[E,\Phi ,Y]$ where E is an extension of X and $\Phi$ extends $\varphi$ to E. A maximal such extension, similar to the Katětov extension of a topological space, is examined.
Summability of Jacobi series
Richard
Askey
71-84
Abstract: The positivity of some Cesàro mean is proven for Jacobi series $ \Sigma {a_n}P_n^{(\alpha ,\beta )}(x),\alpha ,\beta \geqq - \tfrac{1}{2}$. This has applications to the mean convergence of Lagrange interpolation at the zeros of Jacobi polynomials. The positivity of the $(C,\alpha + \beta + 2)$ means is conjectured and proven for some $ (\alpha ,\beta )$. One consequence of this conjecture would be the complete monotonicity of ${x^{ - c}}{({x^2} + 1)^{ - c}},c \geqq 1$.
Groups of diffeomorphisms and their subgroups
Hideki
Omori
85-122
Abstract: This paper has two purposes. The first is to prove the existence of a normal coordinate with respect to a connection defined on the group of diffeomorphisms of a closed manifold, relating to an elliptic complex. The second is to prove a Frobenius theorem with respect to a right invariant distribution defined on the group of diffeomorphisms of a closed manifold, relating to an elliptic complex. Consequently, the group of all volume preserving diffeomorphisms and the group of all symplectic diffeomorphisms are Fréchet Lie groups.
Absolutely summing and dominated operators on spaces of vector-valued continuous functions
Charles
Swartz
123-131
Abstract: A. Pietsch has shown that the class of dominated linear operators on $C(S)$ coincides with the class of absolutely summing operators. If the space $C(S)$ is replaced by ${C_X}(S)$, where X is a Banach space, this is no longer the case. However, any absolutely summing operator is always dominated, and the classes of operators coincide exactly when X is finite dimensional. A characterization of absolutely summing operators on $ {C_X}(S)$ is given.
Fundamental theory of contingent differential equations in Banach space
Shui Nee
Chow;
J. D.
Schuur
133-144
Abstract: For a contingent differential equation that takes values in the closed, convex, nonempty subsets of a Banach space E, we prove an existence theorem and we investigate the extendability of solutions and the closedness and continuity properties of solution funnels. We consider first a space E that is separable and reflexive and then a space E with a separable second dual space. We also consider the special case of a point-valued or ordinary differential equation.
The Rudin-Keisler ordering of $P$-points
Andreas
Blass
145-166
Abstract: The Stone-Čech compactification $ \beta \omega$ of the discrete space $\omega$ of natural numbers is weakly ordered by the relation ``D is the image of E under the canonical extension $\beta f:\beta \omega \to \beta \omega$ of some map $f:\omega \to \omega$.'' We shall investigate the structure, with respect to this ordering, of the set of P-points of $\beta \omega - \omega $.
The existence and uniqueness of nonstationary ideal incompressible flow in bounded domains in $R\sb{3}$
H. S. G.
Swann
167-180
Abstract: It is shown here that the mixed initial-boundary value problem for the Euler equations for ideal flow in bounded domains of ${R_3}$ has a unique solution for a small time interval. The existence of a solution is shown by converting the equations to an equivalent system involving the vorticity and applying Schauder's fixed point theorem to an appropriate mapping.
Euclidean $n$-space modulo an $(n-1)$-cell
J. L.
Bryant
181-192
Abstract: This paper, together with another paper by the author titled similarly, provides a complete answer to a conjecture raised by Andrews and Curtis: if D is a k-cell topologically embedded in euclidean n-space ${E^n}$, then ${E^n}/D \times {E^1}$ is homeomorphic to ${E^{n + 1}}$. Although there is at present only one case outstanding ( $ n \geqslant 4$ and $ k = n - 1$), the proof we give here works whenever $n \geqslant 4$. We resolve this conjecture (for $n \geqslant 4$) by proving a stronger result: if $Y \times {E^1} \approx {E^{n + 1}}$ and if D is a k-cell in Y, then $Y/D \times {E^1} \approx {E^{n + 1}}$. This theorem was proved by Glaser for $k \leqslant n - 2$ and has as a corollary: if K is a collapsible polyhedron topologically embedded in ${E^n}$, then ${E^n}/K \times {E^1} \approx {E^{n + 1}}$. Our method of proof uses radial engulfing and a well-known procedure devised by Bing.
Oscillation, continuation, and uniqueness of solutions of retarded differential equations
T.
Burton;
R.
Grimmer
193-209
Abstract: In this paper we present a number of results on continuation and uniqueńess of solutions of the n-dimensional system $ \tau (t) \geq 0$. We then give some necessary, some sufficient, and some necessary and sufficient conditions for oscillation of solutions of the second order equation $x'' + a(t)f(x(t - \tau (t))) = 0$.
A Stong-Hattori spectral sequence
David Copeland
Johnson
211-225
Abstract: Let ${G_ \ast }(\;)$ be the Adams summand of connective K-theory localized at the prime p. Let $B{P_\ast}(\;)$ be Brown-Peterson homology for that prime. A spectral sequence is constructed with $ {E^2}$ term determined by ${G_ \ast }(X)$ and whose ${E^\infty }$ terms give the quotients of a filtration of $B{P_ \ast }(X)$ where X is a connected spectrum. A torsion property of the differentials implies the Stong-Hattori theorem.
Isotopic unknotting in $F\times I$
C. D.
Feustel
227-238
Abstract: This paper is essentially a generalization of Unknotting in ${M^2} \times I$, by E. M. Brown. The major results in this paper concern the existence of ambient isotopies of unknotted arcs (families of arcs) properly embedded in $F \times I$.
The Cauchy problem for degenerate parabolic equations with discontinuous drift
Edward D.
Conway
239-249
Abstract: The coefficient of the gradient is allowed to be discontinuous but is assumed to satisfy a ``one-sided'' Lipschitz condition. This condition insures the pathwise uniqueness of the underlying Markov process which in turn yields the existence of a unique stable generalized solution of the parabolic equation. If the data is Lipschitz continuous, then so is the solution.
Properties of fixed-point sets of nonexpansive mappings in Banach spaces
Ronald E.
Bruck
251-262
Abstract: Let C be a closed convex subset of the Banach space X. A subset F of C is called a nonexpansive retract of C if either $F = \emptyset$ or there exists a retraction of C onto F which is a nonexpansive mapping. The main theorem of this paper is that if $T:C \to C$ is nonexpansive and satisfies a conditional fixed point property, then the fixed-point set of T is a nonexpansive retract of C. This result is used to generalize a theorem of Belluce and Kirk on the existence of a common fixed point of a finite family of commuting nonexpansive mappings.
Higher dimensional generalizations of the Bloch constant and their lower bounds
Kyong T.
Hahn
263-274
Abstract: A higher dimensional generalization of the classical Bloch theorem depends in an essential way on the ``boundedness'' of the family of holomorphic mappings considered. In this paper the author considers two types of such ``bounded'' families and obtains explicit lower bounds of the generalized Bloch constants of these families on the hyperball in the space $ {{\mathbf{C}}^n}$ in terms of universal constants which characterize the families.
Some stable results on the cohomology of the classical infinite-dimensional Lie algebras
Victor
Guillemin;
Steven
Shnider
275-280
Abstract: In this paper we compute the cohomology of various classical infinite-dimensional Lie algebras generalizing results of Gel'fand-Fuks for the Lie algebra of all formal power series vector fields.
Concerning the shapes of finite-dimensional compacta
Ross
Geoghegan;
R. Richard
Summerhill
281-292
Abstract: It is shown that two ``tamely'' embedded compacta of dimension $ \leq k$ lying in ${E^n}(n \geq 2k + 2)$ have the same (Borsuk) shape if and only if their complements are homeomorphic. In particular, two k-dimensional closed submanifolds of $ {E^{2k + 2}}$ have the same homotopy type if and only if their complements are homeomorphic.
On the classification of metabelian Lie algebras
Michael A.
Gauger
293-329
Abstract: The classification of 2-step nilpotent Lie algebras is attacked by a generator-relation method. The main results are in low dimensions or a small number of relations.
The measure algebra of a locally compact hypergroup
Charles F.
Dunkl
331-348
Abstract: A hypergroup is a locally compact space on which the space of finite regular Borel measures has a convolution structure preserving the probability measures. This paper deals only with commutative hypergroups. §1 contains definitions, a discussion of invariant measures, and a characterization of idempotent probability measures. §2 deals with the characters of a hypergroup. §3 is about hypergroups, which have generalized translation operators (in the sense of Levitan), and subhypergroups of such. In this case the set of characters provides much information. Finally §4 discusses examples, such as the space of conjugacy classes of a compact group, certain compact homogeneous spaces, ultraspherical series, and finite hypergroups.
Localization, homology and a construction of Adams
Aristide
Deleanu;
Peter
Hilton
349-362
Abstract: In recent papers, the authors have developed the technique of using Kan extensions to obtain extensions of homology and cohomology theories from smaller to larger categories of topological spaces. In the present paper, it is shown that the conditions imposed there to guarantee that the Kan extension of a cohomology theory is again a cohomology theory in fact also imply that the Kan extension commutes with stabilization. A construction, due to Adams, for completing a space with respect to a homology theory by using categories of fractions is generalized to triangulated categories, and it is shown that, for any family of primes P, the Adams completion of a space X with respect to the homology theory ${\tilde H_ \ast }( - ;{{\mathbf{Z}}_P})$ is the localization of X at P in the sense of Sullivan. Using this, the Kan extension of the restriction of a homology theory to the category of spaces having P-torsion homotopy groups is determined.
Dual operations on saddle functions
L.
McLinden
363-381
Abstract: Dual operations on convex functions play a central role in the analysis of constrained convex optimization problems. Our aim here is to provide tools for a similar analysis of constrained concave-convex minimax problems. Two pairs of dual operations on convex functions, including addition and infimal convolution, are extended to saddle functions. For the resulting saddle functions much detailed information is given, including subdifferential formulas. Also, separable saddle functions are defined and some basic facts about them established.
On unaveraged convergence of positive operators in Lebesgue space
H.
Fong;
L.
Sucheston
383-397
Abstract: Let T be a power-bounded positive conservative operator on $ {L_1}$ of a $\sigma $-finite measure space. Let e be a bounded positive function invariant under the operator adjoint to T. Theorem. (1) $\smallint \vert{T^n}f\vert \cdot \;e \to 0$ implies (2) $\smallint \vert{T^n}f\vert \to 0$. If T and all its powers are ergodic, and T satisfies an abstract Harris condition, then(l) holds by the Jamison-Orey theorem for all integrable f with $ \smallint f \cdot e = 0$, and hence also (2) holds for such f. A new proof of the Jamison-Orey theorem is given, via the 'filling scheme'. For discrete measure spaces this is due to Donald Ornstein, Proc. Amer. Math. Soc. 22 (1969), 549-551. If T is power-bounded, conservative and ergodic, and $0 < {f_0} = T{f_0}$, then $ {f_0}\; \cdot \;e \in {L_1}$ implies $ {f_0} \in {L_1}$, hence (2) implies that ${T^n}f$ converges for each $f \in {L_1}$. Theorem. Let T be a positive conservative contraction on $ {L_1}$; then the class of functions $\{ f - Tf,f \in L_1^ + \} $ is dense in the class of functions $\{ f - Tf,f \in {L_1}\} $.
Differential equations on closed subsets of a Banach space
R. H.
Martin
399-414
Abstract: In this paper the problem of existence of solutions to the initial value problem $A:[a,b) \times D \to E$ is continuous, D is a closed subset of a Banach space E, and $z \in D$. With a dissipative type condition on A, we establish sufficient conditions for this initial value problem to have a solution. Using these results, we are able to characterize all continuous functions which are generators of nonlinear semigroups on D.
Envelopes of holomorphy and holomorphic convexity
Robert
Carmignani
415-431
Abstract: This paper is primarily a study of generalized notions of envelope of holomorphy and holomorphic convexity for special (algebraically restricted) subsets of ${{\mathbf{C}}^n}$ and in part for arbitrary subsets of $ {{\mathbf{C}}^n}$. For any special set S in ${{\mathbf{C}}^n}$, we show that every function holomorphic in a neighborhood of S not only can be holomorphically continued but also holomorphically extended to a neighborhood in ${{\mathbf{C}}^n}$ of a maximal set $\tilde{S}$, the ``envelope of holomorphy'' of S, which is also a special set of the same type as S. Formulas are obtained for constructing $\tilde{S}$ for any special set S. ``Holomorphic convexity'' is characterized for these special sets. With one exception, the only topological restriction on these special sets is connectivity. Examples are given which illustrate applications of the theorems and help to clarify the concepts of ``envelope of holomorphy'' and ``holomorphic convexity."
A characterization of the invariant measures for an infinite particle system with interactions
Thomas M.
Liggett
433-453
Abstract: Let $p(x,y)$ be the transition function for a symmetric, irreducible, transient Markov chain on the countable set S. Let ${\eta _t}$ be the infinite particle system on S with the simple exclusion interaction and one-particle motion determined by p. A characterization is obtained of all the invariant measures for $ {\eta _t}$ in terms of the bounded functions on S which are harmonic with respect to $p(x,y)$. Ergodic theorems are proved concerning the convergence of the system to an invariant measure.
The \v Silov boundary of $M\sb{0}(G)$
William
Moran
455-464
Abstract: Let G be a locally compact abelian group and let ${M_0}(G)$ be the convolution algebra consisting of those Radon measures on G whose Fourier-Stieltjes transforms vanish at infinity. It is shown that the Šilov boundary of ${M_0}(G)$ is a proper subset of the maximal ideal space of ${M_0}(G)$. The measures constructed to prove this theorem are also used to obtain a stronger result for the full measure algebra $M(G)$.
On the quasi-simple irreducible representations of the Lorentz groups
Ernest
Thieleker
465-505
Abstract: For $n \geq 2$, let $G(n)$ denote the generalized homogeneous Lorentz group of an $n + 1$-dimensional real vector space; that is, $ G(n)$ is the identity component of the orthogonal group of a real quadratic form of index $( + , - - \ldots - )$. Let $ \hat G(n)$ denote a two-fold covering group of $G(n)$, and let $ \hat S(n)\hat M(n)$ be a parabolic subgroup of $\hat G(n)$. We consider the induced representations of $\hat G(n)$, induced by the finite-dimensional irreducible representations of $\hat S(n)\hat M(n)$. By an extension of the methods used in a previous paper, we determine precise criteria for the topological irreducibility of these representations. Moreover, in the exceptional cases when these representations fail to be irreducible, we determine the irreducible subrepresentations of these induced representations. By means of some general results of Harish-Chandra together with the main results of this paper, we obtain a complete classification, up to infinitesimal equivalence, of the quasi-simple irreducible representations of the groups $\hat G(n)$.