Some product formulae for nonsimply connected surgery problems
R. J.
Milgram;
Andrew
Ranicki
383-413
Abstract: For an $ n$-dimensional normal map $f :{M^n} \to {N^n}$ with finite fundamental group $ {\pi _1}(N) = \pi$ and PL $ 1$ torsion kernel $ Z[\pi ]$-modules ${K_{\ast}}(M)$ the surgery obstruction $ {\sigma _{\ast}}(f) \in L_n^h(Z[\pi ])$ is expressed in terms of the projective classes $[{K_{\ast}}(M)] \in {\tilde K_0}(Z[\pi ])$, assuming ${K_i}(M) = 0$ if $n = 2i$. This expression is used to evaluate in certain cases the surgery obstruction ${\sigma _ {\ast} }(g) \in L_{m + n}^h(Z[{\pi _1} \times \pi ])$ of the $(m + n)$-dimensional normal map $ g = 1 \times f:{M_1} \times M \to {M_1} \times N$ defined by product with an $ m$-dimensional manifold $ {M_1}$, where ${\pi _1} = {\pi _1}({M_1})$.
Bounds for prime solutions of some diagonal equations. II
Ming Chit
Liu
415-426
Abstract: Let ${b_j}$ and $m$ be certain integers. In this paper we obtain a bound for prime solutions ${p_j}$ of the diagonal equations of order $ k,\;{b_1}p_1^k + \cdots + {b_s}p_s^k = m$. The bound obtained is $ {C^{{{(\log B)}^2}}} + C\vert m{\vert^{1/k}}$ where $B = {\max _j}\{ e,\vert{b_j}\vert\}$ and $ C$ are positive constants depending at most on $k$.
Groups presented by finite two-monadic Church-Rosser Thue systems
J.
Avenhaus;
K.
Madlener;
F.
Otto
427-443
Abstract: It is shown that a group $G$ can be defined by a monoid-presentation of the form $(\Sigma ;T)$, where $T$ is a finite two-monadic Church-Rosser Thue system over $\Sigma$, if and only if $G$ is isomorphic to the free product of a finitely generated free group with a finite number of finite groups.
Homologie de l'espace des sections d'un fibr\'e
Claude
Legrand
445-459
Abstract: For a fiber bundle with a finite cohomology dimension and $1$-connected base $B$ and $1$-connected fiber $F$, we obtain the homology of the section space by an ${E^1}$-spectral sequence. In the "stable" range the $ {E^1}$-terms are the homology of a product of Eilenberg-Mac Lane space of type $ K({H^{p - i}}(B;{\pi _p}F),i)$ (the same as those of the $ {E^1}$-spectral sequences which converges to the homology of the functional space $\operatorname{Hom} (B,F)$ [10]). The differential is the product of two operations: one appears in the ${E^1}$-spectral sequence, which converges to the homology of $\operatorname{Hom} (B,F)$; the second one is a "cup-product" determined by the fiber structure of the bundle. This spectral sequence is obtained by a Moore-Postnikov tower of the fiber, which generalizes Kahn's methods [9].
Random power series generated by ergodic transformations
Judy
Halchin;
Karl
Petersen
461-485
Abstract: Generalizing classical studies of power series with sequences of independent random variables as coefficients, we study series of the forms $\displaystyle {g_{x,\phi }}(z) = \sum\limits_{n = 0}^\infty {\phi ({T^n}x){z^n}... ...m\limits_{n = 1}^\infty {\phi (x)\phi (Tx) \cdots \phi ({T^{n - 1}}x){z^n},} }$ where $T$ is an ergodic measure-preserving transformation on a probability space $(X,\mathcal{B},\mu )$ and $\phi$ is a measurable complex-valued function which is a.e. nonzero. When ${f_{x,\phi }}$ is entire, its order of growth at infinity measures the speed of divergence of the ergodic averages of $\log \vert\phi \vert$. We give examples to show that any order is possible for any $ T$ and that different orders are possible for fixed $\phi$. For fixed $T$, the set of $\phi$ which produce infinite order is residual in the subset of ${L^1}(X)$ consisting of those $\phi$ which are a.e. nonzero and produce entire $ {f_{x,\phi }}$. As in a theorem of Pólya for gap series, if ${f_{x,\phi }}$ is entire and has finite order, then it assumes every value infinitely many times. The functions $ \phi \in {L^1}(X)$ for which ${g_{x,\phi }}$ is rational a.e. are exactly the finite sums of eigenfunctions of $ T$; their poles are all simple and are the inverses of the corresponding eigenvalues. By combining this result with a skew product construction, we can also characterize when ${f_{x,\phi }}$ is rational, provided that $ \phi$ takes one of several particular forms.
The BGG resolution, character and denominator formulas, and related results for Kac-Moody algebras
Wayne
Neidhardt
487-504
Abstract: Let $\mathfrak{g}$ be a Kac-Moody algebra defined by a (not necessarily symmetrizable) generalized Cartan matrix. We construct a BGG-type resolution of the irreducible module $ L(\lambda )$ with dominant integral highest weight $\lambda$, and we use this to obtain character and denominator formulas analogous to those of Weyl. We also determine a condition on the algebra which is sufficient for these formulas to take their classical form, and which implies that the set of defining relations is complete.
Construction of a family of non-self-dual gauge fields
Ignacio
Sols
505-508
Abstract: Using the generalization of vector bundles by reflexive sheaves recently introduced by R. Hartshorne in [2] we construct a $ 15$-dimensional family of nontrivial complex gauge fields $(U,E,\nabla )$ which are not self-dual nor anti-self-dual. ($U$ is an affine neighborhood in $ {Q_4} = \operatorname{Gr} (2,{{\mathbf{C}}^4})$ of the stereographic compactification ${S^4}$ of $ {\mathbb{R}^4}$, $ E$ is a vector bundle on $ U$ and $\nabla$ is a connection on it whose curvature $\phi$ satisfies the inequalities ${}^{\ast}\phi \ne \phi$ and ${}^{\ast}\phi \ne - \phi $.)
The intersection topology w.r.t. the real line and the countable ordinals
G. M.
Reed
509-520
Abstract: If ${\Upsilon _1}$ and $ {\Upsilon _2}$ are topologies defined on the set $X$, then the intersection topology w.r.t. ${\Upsilon _1}$ and ${\Upsilon _2}$ is the topology $\Upsilon$ on $X$ such that $\{ {U_1} \cap {U_2}\vert{U_1} \in {\Upsilon _1}\;{\text{and}}\;{U_2} \in {\Upsilon _2}\}$ is a basis for $ (X,\Upsilon )$. In this paper, the author considers spaces in the class $\mathcal{C}$, where $(X,\Upsilon ) \in \mathcal{C}$ iff $X = \{ {x_\alpha }\vert\alpha < {\omega _1}\} \subseteq {\mathbf{R}}$, ${\Upsilon _{\mathbf{R}}}$ is the inherited real line topology on $X$, $ {\Upsilon _{{\omega _1}}}$ is the order topology on $X$ of type $ {\omega _1}$, and $ \Upsilon$ is the intersection topology w.r.t. ${\Upsilon _{\mathbf{R}}}$ and ${\Upsilon _{{\omega _1}}}$. This class is shown to be a surprisingly useful tool in the study of abstract spaces. In particular, it is shown that: (1) If $ X \in \mathcal{C}$, then $ X$ is a completely regular, submetrizable, pseudo-normal, collectionwise Hausdorff, countably metacompact, first countable, locally countable space with a base of countable order that is neither subparacompact, metalindelöf, cometrizable, nor locally compact. (2) $(\operatorname{MA} + \neg \operatorname{CH} )$ If $X \in \mathcal{C}$, then $ X$ is perfect. (3) There exists in ZFC an $ X \in \mathcal{C}$ such that $X$ is not normal. (4) $(\operatorname{CH} )$ There exists $X \in \mathcal{C}$ such that $X$ is collectionwise normal and ${\omega _1}$-compact but not perfect.
On the rational homotopy Lie algebra of a fixed point set of a torus action
Christopher
Allday;
Volker
Puppe
521-528
Abstract: Let $X$ be a simply connected topological space, and let ${\mathcal{L}_{\ast}}(X)$ be its rational homotopy Lie algebra. Suppose that a torus acts on $X$ with fixed points, and suppose that $ F$ is a simply connected component of the fixed point set. If ${\mathcal{L}_{\ast}}(X)$ is finitely presented and if $ F$ is full, then it is shown that ${\mathcal{L}_{\ast}}(F)$ is finitely presented, and that the numbers of generators and relations in a minimal presentation of ${\mathcal{L}_{\ast}}(F)$ do not exceed the numbers of generators and relations (respectively) in a minimal presentation of ${\mathcal{L}_{\ast}}(X)$. Various other related results are given.
Shape properties of Whitney maps for hyperspaces
Hisao
Kato
529-546
Abstract: In this paper, some shape properties of Whitney maps for hyperspaces are investigated. In particular, the following are proved: (1) Let $X$ be a continuum and let $ \mathfrak{H}$ be the hyperspace ${2^X}$ or $C(X)$ of $X$ with the Hausdorff metric. Then if $ \omega$ is any Whitney map for $ \mathfrak{H}$, for any $0 \leqslant s \leqslant t \leqslant \omega (X){\omega ^{ - 1}}(t)$ is an approximate strong deformation retract of ${\omega ^{ - 1}}([s,t])$. In particular, $\operatorname{Sh} ({\omega ^{ - 1}}(t)) = \operatorname{Sh} ({\omega ^{ - 1}}([s,t]))$. (2) Pointed $1$-movability is a Whitney property. (3) For any given $ {\text{n}} < \infty$, the property of (cohomological) dimension $\leqslant n$ is a sequential strong Whitney-reversible property. (4) The property of being chainable or circle-like is a sequential strong Whitney-reversible property. (5) The property of being an FAR is a Whitney property for $1$-dimensional continua. Property (2) is an affirmative answer to a problem of J. T. Rogers [16, 112]. Properties (3) and (4) are affirmative answers to problems of S. B. Nadler [20, (14.57) and 21].
A formula for Casson's invariant
Jim
Hoste
547-562
Abstract: Suppose $ H$ is a homology $ 3$-sphere obtained by Dehn surgery on a link $L$ in a homology $3$-sphere $M$. If every pair of components of $L$ has zero linking number in $ M$, then we give a formula for the Casson invariant, $ \lambda (H)$, in terms of $\lambda (M)$, the surgery coefficients of $ L$, and a certain coefficient from each of the Conway polynomials of $ L$ and all its sublinks. A few consequences of this formula are given.
On ${\bf R}\sp \infty\;(Q\sp \infty)$-manifold bundles over CW complexes
Vo Thanh
Liem
563-585
Abstract: Let $\Lambda \in \mathcal{C}\mathcal{W}(\mathcal{C}) \cup \mathcal{C}\mathcal{W}(\mathcal{M})$ be a pseudo CW complex generated either by Hausdorff compact spaces or by metric spaces. In the theory of manifolds modeled on $ {R^\infty }$ or ${Q^\infty }$, we will prove the $\Lambda $-fiber-preserving versions of the following: Equivalences among the notions of $D$-sets, $ {D^{\ast}}$-sets and infinite deficient sets; relative stability theorems; relative deformation of homotopy equivalences to homeomorphisms; strong unknotting theorem for $ D$-embeddings; and $ \alpha$-approximation theorems.
Stratification of continuous maps of an interval
L. S.
Block;
W. A.
Coppel
587-604
Abstract: We define the motion of turbulence for a continuous map of an interval into the line and study its relation with periodic and homoclinic points. We define also strongly simple orbits and show, in particular, that they represent periodic orbits with minimum entropy. Further results are obtained for unimodal maps with negative Schwarzian, which sharpen recent results of Block and Hart.
A topological proof of the equivariant Dehn lemma
Allan L.
Edmonds
605-615
Abstract: An elementary topological proof is given for a completely general version of the Equivariant Dehn Lemma, in the spirit of the original proof of the nonequivariant version due to C. D. Papakyriakopolous in 1957.
Smooth maps, pullback path spaces, connections, and torsions
Kuo Tsai
Chen
617-627
Abstract: By generalizing the local version of the usual differential geometric notion of connections and that of torsions, a model for the pullback path space of a smooth map is constructed from the induced map of the de Rham complexes. The pullback path space serves not only as a homotopy fiber but also as a device reflecting differentiable properties of the smooth map. Applications are discussed.
Range transformations on a Banach function algebra
Osamu
Hatori
629-643
Abstract: We study the range transformations $\operatorname{Op} ({A_{D,}}\operatorname{Re} B)$ and $\operatorname{Op} ({A_D},B)$ for Banach function algebras $A$ and $B$. As a special instance, the harmonicity of functions in $\operatorname{Op} ({A_D},\operatorname{Re} A)$ for a nontrivial function algebra $A$ is established and is compared with previous investigations of $\operatorname{Op} ({A_D},A)$ and $\operatorname{Op} ({(\operatorname{Re} A)_I},(\operatorname{Re} A))$ for an interval $I$. In $\S2$ we present some results on $\operatorname{Op} ({A_D},B)$ and use them to show that functions in ${\operatorname{Op} ^C}({A_D},B)$ are analytic for certain Banach function algebras.
Boundary behavior of a nonparametric surface of prescribed mean curvature near a reentrant corner
Alan R.
Elcrat;
Kirk E.
Lancaster
645-650
Abstract: Let $\Omega$ be an open set in ${{\mathbf{R}}^2}$ which is locally convex at each point of its boundary except one, say $(0,0)$. Under certain mild assumptions, the solution of a prescribed mean curvature equation on $ \Omega$ behaves as follows: All radial limits of the solution from directions in $ \Omega$ exist at $ (0,0)$, these limits are not identical, and the limits from a certain half-space $ (H)$ are identical. In particular, the restriction of the solution to $\Omega \cap H$ is the solution of an appropriate Dirichlet problem.
Cauchy problem for nonlinear hyperbolic systems of partial differential equations
Victoria
Yasinovskaya
651-668
Abstract: We proved the sharp Sobolev estimate for Cauchy data for the general type of hyperbolic systems of nonlinear partial differential equations, which leads to a local existence and uniqueness theorem for solutions of the Cauchy problem in Sobolev spaces.
Poisson integrals of regular functions
José R.
Dorronsoro
669-685
Abstract: Tangential convergence of Poisson integrals is proved for certain spaces of regular functions which contain the spaces of Bessel potentials of ${L^p}$ functions, $1 < p < \infty$, and of functions in the local Hardy space ${h^1}$, and the corresponding tangential maximal functions are shown to be of strong $ p$ type, $p \geqslant 1$.
Dimension-free quasiconformal distortion in $n$-space
G. D.
Anderson;
M. K.
Vamanamurthy;
M.
Vuorinen
687-706
Abstract: Most distortion theorems for $K$-quasiconformal mappings in ${{\mathbf{R}}^n}$, $n \geqslant 2$, depend on both $n$ and $K$ in an essential way, with bounds that become infinite as $n$ tends to $\infty$. The present authors obtain dimension-free versions of four well-known distortion theorems for quasiconformal mappings--namely, bounds for the linear dilatation, the Schwarz lemma, the $ \Theta$-distortion theorem, and the $\eta$-quasisymmetry property of these mappings. They show that the upper estimates they have obtained in each of these four main results remain bounded as $ n$ tends to $\infty$ with $K$ fixed. The proofs are based on a "dimensioncancellation" property of the function $ t \mapsto {\tau ^{ - 1}}(\tau (t)/K),\,t > 0,\,K > 0$, where $\tau (t)$ is the capacity of a Teichmüller extremal ring in $ {{\mathbf{R}}^n}$. The authors also prove a dimension-free distortion theorem for the absolute (cross) ratio under $ K$-quasiconformal mappings of $ {\overline {\mathbf{R}} ^n}$, from which several other distortion theorems follow as special cases.
Fine structure of the integral exponential functions below $2\sp {2\sp x}$
Bernd I.
Dahn
707-716
Abstract: Integral exponential functions are the members of the least class of real functions containing $1$, the identity function, and closed under addition, multiplication, and binary exponentiation sending $f$ and $g$ to ${f^g}$. This class is known to be wellordered by the relation of eventual dominance. It is shown that for each natural number $n$ the order type of the integral exponential functions below $ {2^{{x^n}}}$ (below ${x^{{x^n}}}$) is exactly ${\omega ^{{\omega ^{2n - 1}}}}$ ( ${\omega ^{{\omega ^{2n}}}}$ respectively). The proof, using iterated asymptotic expansions, contains also a new proof that integral exponential functions below $ {2^{{2^x}}}$ are wellordered.
On the factorizations of ordinary linear differential operators
G. J.
Etgen;
G. D.
Jones;
W. E.
Taylor
717-728
Abstract: Relations are found between the nonvanishing of certain Wronskians and disconjugacy properties of $ {L_n}y + py = 0$, where $ {L_n}y$ is a disconjugate operator and $p$ is sign definite. The results are then used to show ways in which $ {L_n}y + py$ can be factored.
On the local behavior of $\Psi(x,y)$
Adolf
Hildebrand
729-751
Abstract: $\Psi (x,y)$ denotes the number of positive integers $\leqslant x$ and free of prime factors $> y$. In the range $y \geqslant \exp ({(\log \log x)^{5/3 + \varepsilon }})$, $\Psi (x,y)$ can be well approximated by a "smooth" function, but for $y \leqslant {(\log x)^{2 - \varepsilon }}$, this is no longer the case, since then the influence of irregularities in the distribution of primes becomes apparent. We show that $ \Psi (x,y)$ behaves "locally" more regular by giving a sharp estimate for $\Psi (cx,y)/\Psi (x,y)$, valid in the range $x \geqslant y \geqslant 4\log x$, $1 \leqslant c \leqslant y$.
A strong containment property for discrete amenable groups of automorphisms on $W\sp \ast$ algebras
Edmond E.
Granirer
753-761
Abstract: Let $G$ be a countable group of automorphisms on a $ {W^{\ast}}$ algebra $\mathcal{M}$ and let ${\phi _0}$ be a ${w^{\ast}}{G_\delta }$ point of the set of $ G$ invariant states on $\mathcal{M}$ which belong to $ {w^{\ast}}\operatorname{cl} \operatorname{Co} E$, where $E$ is a set of (possibly pure) states on $\mathcal{M}$. If $G$ is amenable, then the cyclic representation ${\pi _{{\phi _0}}}$ corresponding to $ {\phi _0}$ is contained in $ \{ \oplus {\pi _\phi };\phi \in E\}$. This property characterizes amenable groups. Related results are obtained.
On the a.e. convergence of the arithmetic means of double orthogonal series
F.
Móricz
763-776
Abstract: The extension of the coefficient test of Menšov and Kaczmarz ensuring the a.e. $(C,1,1)$-summability of double orthogonal series has been stated by two authors. Unfortunately, their proofs turned out to be deficient. Now we present a general theory, in the framework of which a complete proof of this test can also be obtained. Besides, we extend the relevant theorems of Kolmogorov and Kaczmarz from single orthogonal series to double ones, establishing the a.e. equiconvergence of the lacunary subsequences of the rectangular partial sums and of the entire sequence of the arithmetic means. The corresponding tests ensuring the a.e. $(C,1,0)$ and $(C,0,1)$-summability are also treated.
All triangles are Ramsey
Peter
Frankl;
Vojtěch
Rödl
777-779
Abstract: Given a triangle $ ABC$ and an integer $ r$, $r \geqslant 2$, it is shown that for $ n$ sufficiently large and an arbitrary $r$-coloring of ${R^n}$ one can find a monochromatic copy of $ ABC$.