Companionship of knots and the Smith conjecture
Robert
Myers
1-32
Abstract: This paper studies the Smith Conjecture in terms of H. Schubert's theory of companionship of knots. Suppose J is a counterexample to the Smith Conjecture, i.e. is the fixed point set of an action of ${{\textbf{Z}}_p}$ on ${S^3}$. Theorem. Every essential torus in an invariant knot space $C(J)$ of J is either invariant or disjoint from its translates. Since the companions of J correspond to the essential tori in $C(J)$, this often allows one to split the action among the companions and satellites of J. In particular: Theorem. If J is composite, then each prime factor of J is a counterexample, and conversely. Theorem. The Smith Conjecture is true for all cabled knots. Theorem. The Smith Conjecture is true for all doubled knots. Theorem. The Smith Conjecture is true for all cable braids. Theorem. The Smith Conjecture is true for all nonsimple knots with bridge number less than five. In addition we show: Theorem. If the Smith Conjecture is true for all simple fibered knots, then it is true for all fibered knots. Theorem. The Smith Conjecture is true for all nonfibered knots having a unique isotopy type of incompressible spanning surface.
The group of rational solutions of $y\sp{2}=x(x-1)(x-t\sp{2}-c)$
Charles F.
Schwartz
33-46
Abstract: In this paper, we show that the Mordell-Weil group of the Weierstrass equation ${y^2}\, = \,x(x\, - \,1)(x\, - \,{t^2}\, - \,c),\,c \ne \,0,\,1$ (i.e., the group of solutions (x,y), with $x,\,y\, \in \,{\textbf{C}}(t)$) is generated by its elements of order 2, together with one element of infinite order, which is exhibited.
Twisted Lubin-Tate formal group laws, ramified Witt vectors and (ramified) Artin-Hasse exponentials
Michiel
Hazewinkel
47-63
Abstract: For any ring R let $\Lambda (R)$ denote the multiplicative group of power series of the form $1\, + \,{a_1}t\, + \, \cdots$ with coefficients in R. The Artin-Hasse exponential mappings are homomorphisms $ {W_{p,\,\infty }}(k)\, \to \,\Lambda ({W_{p,\,\infty }}(k))$, which satisfy certain additional properties. Somewhat reformulated, the Artin-Hasse exponentials turn out to be special cases of a functorial ring homomorphism $E:\,{W_{p,\,\infty }}( - )\, \to \,{W_{p,\,\infty }}({W_{p,\,\infty }}( - ))$, where ${W_{p,\,\infty }}$ is the functor of infinite-length Witt vectors associated to the prime p. In this paper we present ramified versions of both ${W_{p,\,\infty }}( - )$ and E, with ${W_{p,\,\infty }}( - )$ replaced by a functor $ W_{q,\,\infty }^F( - )$, which is essentially the functor of q-typical curves in a (twisted) Lubin-Tate formal group law over A, where A is a discrete valuation ring that admits a Frobenius-like endomorphism $ \sigma$ (we require $\sigma (a)\, \equiv \,{a^q}\,\bmod \,{\mathcal{m}}$ for all $a\, \in \,A$, where ${\mathcal{m}}$ is the maximal idea of A). These ramified-Witt-vector functors $W_{q,\,\infty }^F( - )$ do indeed have the property that, if $ k\, = \,A/{\mathcal{m}}$ is perfect, A is complete, and $l/k$ is a finite extension of k, then $ W_{q,\,\infty }^F(l)$ is the ring of integers of the unique unramified extension $ L/K$ covering $ l/k$.
Some undecidability results concerning Radon measures
R. J.
Gardner;
W. F.
Pfeffer
65-74
Abstract: We show that in metalindelöf spaces certain questions about Radon measures cannot be decided within the Zermelo-Fraenkel set theory, including the axiom of choice.
On generalized harmonic analysis
Ka Sing
Lau;
Jonathan K.
Lee
75-97
Abstract: Motivated by Wiener's work on generalized harmonic analysis, we consider the Marcinkiewicz space $ {{\mathcal{M}}^p}({\textbf{R}})$ of functions of bounded upper average p power and the space $ {{\mathcal{V}}^p}({\textbf{R}})$ of functions of bounded upper p variation. By identifying functions whose difference has norm zero, we show that $ {{\mathcal{V}}^p}({\textbf{R}})$, $ 1\, < \,p\, < \infty$, is a Banach space. The proof depends on the result that each equivalence class in $ {{\mathcal{V}}^p}({\textbf{R}})$ contains a representative in ${L^p}({\textbf{R}})$. This result, in turn, is based on Masani's work on helixes in Banach spaces. Wiener defined an integrated Fourier transformation and proved that this transformation is an isometry from the nonlinear subspace $ {{\mathcal{W}}^2}({\textbf{R}})$ of $ {{\mathcal{M}}^2}({\textbf{R}})$ consisting of functions of bounded average quadratic power, into the nonlinear subspace $ {{\mathcal{u}}^2}({\textbf{R}})$ of $ {{\mathcal{V}}^2}({\textbf{R}})$ consisting of functions of bounded quadratic variation. By using two generalized Tauberian theorems, we prove that Wiener's transformation W is actually an isomorphism from $ {{\mathcal{M}}^2}({\textbf{R}})$ onto $ {{\mathcal{V}}^2}({\textbf{R}})$. We also show by counterexamples that W is not an isometry on the closed subspace generated by $ {{\mathcal{W}}^2}({\textbf{R}})$.
On the Hardy-Littlewood maximal function and some applications
C. J.
Neugebauer
99-105
Abstract: With a monotone family $ F\, = \,\{ {S_\alpha }\} ,\,{S_\alpha }\, \subset \,{{\textbf{R}}^n}$, we associate the Hardy-Littlewood maximal function ${M_F}f(x)\, = \,{\sup _\alpha }(1/\left\vert {{S_\alpha }} \right\vert)\int_{{S_\alpha }\, + \,x} {\left\vert f \right\vert}$. In general, ${M_F}$ is not weak type (1.1). However, if we replace in the denominator $ {S_\alpha }$ by $S_F^ {\ast} \, = \,\{ x\, - \,y:\,x,\,y\, \in \,{S_\alpha }\}$, and denote the resulting maximal function by $M_F^ {\ast}$, then $M_F^ {\ast}$ is weak type (1, 1) with weak type constant 1.
Generic deformations of varieties
Yieh Hei
Wan
107-119
Abstract: Typical families of varieties, which are defined by families of smooth maps, are studied through the method from the singularity theory of differentiable maps. It is proved that generic families of varieties (of certain types) are stable in an appropriate sense.
Subspaces of basically disconnected spaces or quotients of countably complete Boolean algebras
Eric K.
van Douwen;
Jan
van Mill
121-127
Abstract: Under ${\text{MA}}\,{\text{ + }}\,{{\text{2}}^\omega }\, = \,{\omega _2}$ there is a (compact) strongly zero-dimensional F-space of weight ${2^\omega }$ which cannot be embedded in any basically disconnected space. Dually, under $ {\text{MA}}\, + \,{2^\omega }\, = \,{\omega _2}$ there is a weakly countably complete (or almost $\sigma$-complete, or countable separation property) Boolean algebra of cardinality ${2^\omega }$ which is not a homomorphic image of any countably complete Boolean algebra. The key to our construction is the observation that if X is a subspace of a basically disconnected space and $ \beta \omega \, \subseteq \,X$ then $\beta \omega $ is a retract of X. Dually, if B is a homomorphic image of a countably complete Boolean algebra, and if h is a homomorphism from B onto $\mathcal{P}(\omega )$, the field of subsets of w, then there is an embedding $ e:\,\mathcal{P}(\omega ) \to \,B$ such that $h\, \circ \,e\, = \,{\text{i}}{{\text{d}}_{\mathcal{P}(\omega )}}$.
Permutation-partition pairs: a combinatorial generalization of graph embeddings
Saul
Stahl
129-145
Abstract: Permutation-partition pairs are a purely combinatorial generalization of graph embeddings. Some parameters are defined here for these pairs and several theorems are proved. These results are strong enough to prove virtually all the known theoretical informaton about the genus parameter as well as a new theorem regarding the genus of the amalgamation of two graphs over three points.
Application of the extremum principle to investigating certain extremal problems
L.
Mikołajczyk;
S.
Walczak
147-155
Abstract: Denote by C, K, X, respectively, a complex plane, the disc $\{ z\, \in \,{\textbf{C:}}\,\left\vert z \right\vert\, < \,1\} $ and any compact Hausdorff space. Denote by P a set of probabilistic measures defined on Borel subsets of the space X. For $\mu \, \in \,P$, let $ f(z)\, = \,\int_X {q(z,\,t)\,d\mu } ,\,z\, \in \,K$, and ${\mathcal{F}}\, = \,\{ f:\,\mu \, \in \,P\}$. Consider a finite sequence of real functions $ {F_0},\,{F_{1,}}\, \ldots ,\,{F_m}$ defined in the space ${R^{2n}}$. Let ${\zeta _1},\, \ldots ,\,{\zeta _k}$ be fixed points of the disc K and $\eta (f)\, = \,[\operatorname{re} \,{f^{(0)}}(\zeta ),\,\operatorname{im} \,{f... ...e} \,{f^{({n_k})}}({\zeta _k}),\,\operatorname{im} \,{f^{({n_k})}}({\zeta _k})]$, where $ f\, \in \,{\mathcal{F}},\,n\, = \,{n_1}\, + \, \cdots \, + \,{n_k}\, + \,k$. Let ${F_j}(f)\, = \,{F_j}(\eta (f)),\,j\, = \,0,\,1,\, \ldots ,\,m$. We consider the following extremal problem. Determine a minimum of the functional ${F_0}(f)$ under the conditions $ {F_j}(f)\, \leqslant \,0,\,j\, = \,1,\,2,\, \ldots ,\,m,\,f\, \in \,{\mathcal{F}}$. We apply the extremum principle to solve this problem. In the linear case this problem was investigated in [11].
Lifting surgeries to branched covering spaces
Hugh M.
Hilden;
José María
Montesinos
157-165
Abstract: It is proved that if ${M^n}$ is a branched covering of a sphere, branched over a manifold, so is ${M^n}\, \times \,{S^m}$, but the number of sheets is one more. In particular, the n-dimensional torus is an n-fold simple covering of ${S^n}$ branched over an orientable manifold. The proof involves the development of a new technique to perform equivariant handle addition. Other consequences of this technique are given.
Some examples of sequence entropy as an isomorphism invariant
F. M.
Dekking
167-183
Abstract: With certain geometrically diverging sequences A and the shift T on dynamical systems arising from substitutions we associate a Markov shift S such that the A-entropy of T equals the usual entropy of S. We present examples to demonstrate the following results. Sequence entropy can distinguish between an invertible ergodic transformation and its inverse. A-entropy does not depend monotonically on A. The variational principle for topological sequence entropy need not hold.
Equivariant dynamical systems
M. J.
Field
185-205
Abstract: The basic properties of vector fields and diffeomorphisms invariant under the action of a compact Lie group are presented. A Kupka-Smale density theorem for equivariant dynamical systems and an existence theorem for equivariant Morse-Smale systems on an arbitrary compact G-manifold are proved.
Zeros of successive derivatives of entire functions of the form $h(z){\rm exp}(-e\sp{z})$
Albert
Edrei
207-226
Abstract: Consider $f(z)\, = \,h(z)\exp ( - {e^z})\,(z\, = \,x\, + \,iy)$ where $h(z)$ is a real entire function of finite order having no zeros in some strip $\{ x\, + \,iy:\,\left\vert {y - \pi } \right\vert\, < \,{\eta _1},\,x\, > \,{x_0}\} \,(0\, < \,{\eta _1})$. The author studies the power series (1) $f(\tau \, + \,z)\, = \,\Sigma _{n = 0}^\infty \,{a_n}{z^n}$ ($\tau$ real) and the number $N({\tau _1},\,{\tau _2};\,n)$ of real zeros of ${f^{(n)}}(z)$ which lie in the interval $[{\tau _1},\,{\tau _2}]$. He proves (2) $N({\tau _1},{\tau _2};\,n)\, \sim \,({\tau _2} - {\tau _1})n/{(\log \,n)^2}\,(n \to \infty )$. With regard to the expansion (1) he determines a positive, strictly increasing, unbounded sequence $ \{ {v_k}\} _{k = 1}^\infty$ such that $({v_{k + 1}} - {v_k})/\log \,{v_k} \to 1\,(k \to \infty )$ and having the following properties: (i) if $ {v_k}\, < \,n\, < \,{v_{k + 1}}$, then $ {a_n}\, \ne \,0$ and all the ${a_n}$ have the same sign; (ii) if in addition $ {v_{k + 1}}\, < \,m\, < \,{v_{k + 2}}$, tnen ${a_m}{a_n}\, < 0$. It is possible to deduce from (2) the complete characterization of the final set (in the sense of Pólya) of $\exp ( - {e^z})$.
$p$-subgroups of compact Lie groups and torsion of infinite height in $H\sp{\ast} (BG)$
Mark
Feshbach
227-233
Abstract: The relation between elementary abelian p-subgroups of a connected compact Lie group G and the existence of p-torsion in $ {H^ {\ast} }(G)$ has been known for some time [B-S]. In this paper we prove that if G is any compact Lie group then ${H^ {\ast} }(BG)$ contains p-torsion of infinite height iff G contains an elementary abelian p-group not contained in a maximal torus. The hard direction is proven using the double coset theorem for the transfer. A third equivalent condition is also given.
Littlewood-Paley and multiplier theorems on weighted $L\sp{p}$ spaces
Douglas S.
Kurtz
235-254
Abstract: The Littlewood-Paley operator $\gamma (f)$, for functions f defined on $ {{\textbf{R}}^n}$, is shown to be a bounded operator on certain weighted $ {L^p}$ spaces. The weights satisfy an ${A_p}$ condition over the class of all n-dimensional rectangles with sides parallel to the coordinate axes. The necessity of this class of weights demonstrates the 1-dimensional nature of the operator. Results for multipliers are derived, including weighted versions of the Marcinkiewicz Multiplier Theorem and Hörmander's Multiplier Theorem.
Statically tame periodic homeomorphisms of compact connected $3$-manifolds. II. Statically tame implies tame
Edwin E.
Moise
255-280
Abstract: Let f be a periodic homeomorphism $M\, \leftrightarrow \,M$, where M is a compact connected 3-manifold (without boundary). Suppose that for each i, the fixed-point set of $ {f^i}$ is a tame set. Then f is simplicial, relative to some triangulation of M.
On the boundary values of Riemann's mapping function
R. J. V.
Jackson
281-297
Abstract: Classically, the calculus of variations is required to prove the existence of a biholomorphism from the unit disk to a given simply-connected, smooth domain in the complex plane. Here, the problem is reduced to the solution of an ordinary differential equation along the boundary of the domain. The sole coefficient in this equation is identified with the bounded term in the asymptotic expansion of the Bergman kernel function. It is shown that this coefficient can not depend upon any differential expression involving only the curvature function of the boundary.
Generation of analytic semigroups by strongly elliptic operators under general boundary conditions
H. Bruce
Stewart
299-310
Abstract: Strongly elliptic operators are shown to generate analytic semigroups of evolution operators in the topology of uniform convergence, when realized under general boundary conditions on (possibly) unbounded domains. An application to the existence and regularity of solutions to parabolic initial-boundary value problems is indicated.
The fundamental theorem on torsion classes of lattice-ordered groups
Jorge
Martinez
311-317
Abstract: This paper generalizes the earlier notion of a torsion class to a setting where its significance can be fully realized. The dual notion of a torsion-free class is herein defined and the fundamental Connection Theorem is proved. In addition, a few restrictions are considered, in particular, how to view the application of the main theorem to the hereditary classes.
Erratum to: ``A construction of uncountably many weak von Neumann transformations'' [Trans. Amer. Math. Soc. {\bf 257} (1980), no. 2, 397--410]
Karl
David
318-318