Hyperbolic invariants of knots and links
Colin
Adams;
Martin
Hildebrand;
Jeffrey
Weeks
1-56
Abstract: Tables of values for the hyperbolic volume, number of symmetries, cusp volume and conformal invariants of the cusps are given for hyperbolic knots through ten crossings and hyperbolic links of $2, 3$ and $4$ components through $9$ crossings. The horoball patterns and the canonical triangulations are displayed for knots through eight crossings and for particularly interesting additional examples of knots and links.
Crossed simplicial groups and their associated homology
Zbigniew
Fiedorowicz;
Jean-Louis
Loday
57-87
Abstract: We introduce a notion of crossed simplicial group, which generalizes Connes' notion of the cyclic category. We show that this concept has several equivalent descriptions and give a complete classification of these structures. We also show how many of Connes' results can be generalized and simplified in this framework.
Separating points from closed convex sets over ordered fields and a metric for $\tilde{R}^n$
Robert O.
Robson
89-99
Abstract: Let $R$ be an arbitrary ordered field, let $ \bar R$ be a real closure, and let $\tilde R$ and $ {\tilde R^n}$ denote the real spectra of $\bar R[X]$ and $\bar R[{X_1}, \ldots,{X_n}]$. We prove that a closed convex subset in ${R^n}$ may be separated from a point not in it via a continuous "linear" functional taking values in $ \tilde R$ and that there is a $\tilde R$-valued metric on ${\tilde R^n}$. The methods rely on the ultrafilter interpretation of points in ${\tilde R^n}$ and on the existence of suprema and infima of sets in $\tilde R$.
Quadratic models for generic local $3$-parameter bifurcations on the plane
Freddy
Dumortier;
Peter
Fiddelaers
101-126
Abstract: The first chapter deals with singularities occurring in quadratic planar vector fields. We make distinction between singularities which as a general system are of finite codimension and singularities which are of infinite codimension in the sense that they are nonisolated, or Hamiltonian, or integrable, or that they have an axis of symmetry after a linear coordinate change or that they can be approximated by centers. In the second chapter we provide quadratic models for all the known versal $ k$-parameter unfoldings with $k = 1,2,3$, except for the nilpotent focus which cannot occur as a quadratic system. We finally show that a certain type of elliptic points of codimension $ 4$ does not have a quadratic versal unfolding.
Action on Grassmannians associated with a field extension
Patrick
Rabau
127-155
Abstract: We examine the action of the general linear group ${\text{GL}}_L(V)$ on the set of all $K$-subspaces of $V$, where $L/K$ is a finite field extension and $V$ is a finite-dimensional vector space over $L$. The orbits are completely classified in the case of quadratic and cubic extensions; for infinite fields, the number of orbits is shown to be infinite if the degree of the extension is at least four. As an application we obtain $q$-analogues of tranformation and evaluation formulas for hypergeometric functions due to Gessel and Stanton.
Action on Grassmannians associated with commutative semisimple algebras
Dae San
Kim;
Patrick
Rabau
157-178
Abstract: Let $A$ be a finite-dimensional commutative semisimple algebra over a field $k$ and let $V$ be a finitely generated faithful $ A$-module. We study the action of the general linear group ${\text{GL}}_A(V)$ on the set of all $k$-subspaces of $V$ and show that, if the field $k$ is infinite, there are infinitely many orbits as soon as $A$ has dimension at least four. If $ A$ has dimension two or three, the number of orbits is finite and independent of the field; in each such case we completely classify the orbits by means of a certain number of integer parameters and determine the structure of the quotient poset obtained from the action of $ {\text{GL}}_A(V)$ on the poset of $k$-subspaces of $V$.
Unit groups and class numbers of real cyclic octic fields
Yuan Yuan
Shen
179-209
Abstract: The generating polynomials of D. Shanks' simplest quadratic and cubic fields and M.-N. Gras' simplest quartic and sextic fields can be obtained by working in the group $ {\mathbf{PG}}{{\mathbf{L}}_2}({\mathbf{Q}})$. Following this procedure and working in the group $ {\mathbf{PG}}{{\mathbf{L}}_2}({\mathbf{Q}}(\sqrt 2))$, we obtain a family of octic polynomials and hence a family of real cyclic octic fields. We find a system of independent units which is close to being a system of fundamental units in the sense that the index has a uniform upper bound. To do this, we use a group theoretic argument along with a method similar to one used by T. W. Cusick to find a lower bound for the regulator and hence an upper bound for the index. Via Brauer-Siegel's theorem, we can estimate how large the class numbers of our octic fields are. After working out the first three examples in $ \S5$, we make a conjecture that the index is $8$. We succeed in getting a system of fundamental units for the quartic subfield. For the octic field we obtain a set of units which we conjecture to be fundamental. Finally, there is a very natural way to generalize the octic polynomials to get a family of real $ {2^n}$-tic number fields. However, to select a subfamily so that the fields become Galois over $ {\mathbf{Q}}$ is not easy and still a lot of work on these remains to be done.
Extensions of measures invariant under countable groups of transformations
Adam
Krawczyk;
Piotr
Zakrzewski
211-226
Abstract: We consider countably additive, nonnegative, extended real-valued measures vanishing on singletons. Given a group $G$ of bijections of a set $ X$ and a $G$-invariant measure $m$ on $X$ we ask whether there exists a proper $ G$-invariant extension of $ m$. We prove, among others, that if $ \mathbb{Q}$ is the group of rational translations of the reals, then there is no maximal $ \mathbb{Q}$-invariant extension of the Lebesgue measure on $\mathbb{R}$. On the other hand, if ${2^\omega }$ is real-valued measurable, then there exists a maximal $\sigma$-finite $ \mathbb{Q}$-invariant measure defined on a proper $\sigma$-algebra of subsets of $\mathbb{R}$.
Brownian motion in a wedge with variable skew reflection
L. C. G.
Rogers
227-236
Abstract: Does planar Brownian motion confined to a wedge by skew reflection on the sides approach the vertex of the wedge? This question has been answered by Varadhan and Williams in the case where the direction of reflection is constant on each of the sides, but here we address the question when the direction reflected is allowed to vary. A necessary condition, and a sufficient condition, are obtained for the vertex to be reached. The conditions are of a geometric nature, and the gap between them is quite small.
On the braid index of alternating links
Kunio
Murasugi
237-260
Abstract: We show that, at least for an alternating fibered link or $ 2$-bridge link $ L$, there is an exact formula which expresses the braid index ${\mathbf{b}}(L)$ of $L$ as a function of the $2$-variable generalization ${P_L}(l,m)$ of the Jones polynomial.
Characterizations of turbulent one-dimensional mappings via $\omega$-limit sets
Michael J.
Evans;
Paul D.
Humke;
Cheng Ming
Lee;
Richard J.
O’Malley
261-280
Abstract: The structure of $ \omega$-limit sets for nonturbulent functions is studied, and various characterizations for turbulent and chaotic functions are obtained. In particular, it is proved that a continuous function mapping a compact interval into itself is turbulent if and only if there exists an $\omega$-limit set which is a unilaterally convergent sequence
Parity and generalized multiplicity
P. M.
Fitzpatrick;
Jacobo
Pejsachowicz
281-305
Abstract: Assuming that $ X$ and $Y$ are Banach spaces and $ \alpha :[a,b] \to \mathcal{L}(X,Y)$ is a path of linear Fredholm operators with invertible endpoints, in $[{\text{F}} - \text{P}1]$ we defined a homotopy invariant of $\alpha,\sigma (\alpha,I) \in {{\mathbf{Z}}_2}$, the parity of $\alpha$ on $I$. The parity plays a fundamental role in bifurcation problems, and in degree theory for nonlinear Fredholm-type mappings. Here we prove (a) that, generically, the parity is a $\bmod\, 2$ count of the number of transversal intersections of $\alpha (I)$ with the set of singular operators, (b) that if $ {\lambda _0}$ is an isolated singular point of $\alpha$, then the local parity $\displaystyle \sigma (\alpha,{\lambda _0}) \equiv \mathop {\lim }\limits_{\vare... ...\to 0} \sigma (\alpha,[{\lambda _0} - \varepsilon,{\lambda _0} + \varepsilon ])$ remains invariant under Lyapunov-Schmidt reduction, and (c) that $\sigma (\alpha,{\lambda _0}) = {(- 1)^{{M_G}({\lambda _0})}}$, where ${M_G}({\lambda _0})$ is any one of the various concepts of generalized multiplicity which have been defined in the context of linearized bifurcation data.
Cyclic Galois extensions and normal bases
C.
Greither
307-343
Abstract: A Kummer theory is presented which does not need roots of unity in the ground ring. For $R$ commutative with $ {p^{ - 1}} \in R$ we study the group of cyclic Galois extensions of fixed degree $ {p^n}$ in detail. Our theory is well suited for dealing with cyclic $ {p^n}$-extensions of a number field $K$ which are unramified outside $p$. We then consider the group $ \operatorname{Gal}({\mathcal{O}_K}[{p^{ - 1}}],{C_{{p^n}}})$ of all such extensions, and its subgroup $ {\text{NB}}({\mathcal{O}_K}[{p^{ - 1}}],{C_{{p^n}}})$ of extensions with integral normal basis outside $p$. For the size of the latter we get a simple asymptotic formula $ (n \to \infty)$, and the discrepancy between the two groups is in some way measured by the defect $\delta$ in Leopoldt's conjecture.
The solution of length four equations over groups
Martin
Edjvet;
James
Howie
345-369
Abstract: Let $G$ be a group, $F$ the free group generated by $t$ and let $r(t) \in G \ast F$. The equation $r(t) = 1$ is said to have a solution over $ G$ if it has a solution in some group that contains $G$. This is equivalent to saying that the natural map $G \to \langle G \ast F\vert r(t)\rangle$ is injective. There is a conjecture (attributed to M. Kervaire and F. Laudenbach) that injectivity fails only if the exponent sum of $t$ in $r(t)$ is zero. In this paper we verify this conjecture in the case when the sum of the absolute values of the exponent of $t$ in $r(t)$ is equal to four.
Algebraic hulls and smooth orbit equivalence
Alessandra
Iozzi
371-384
Abstract: For $i = 1,2,$ let ${\mathcal{F}_i}$ be foliations on smooth manifolds ${M_i}$ determined by the actions of connected Lie groups ${H_i}$; we describe here some results which provide an obstruction, in terms of a geometric invariant of the actions, to the existence of a diffeomorphism between the $ \mathcal{F}_i^{\prime}{\text{s}}$.
The holomorphic discrete series of an affine symmetric space and representations with reproducing kernels
G.
Ólafsson;
B.
Ørsted
385-405
Abstract: Consider a semisimple connected Lie group $G$ with an affine symmetric space $ X$. We study abstractly the intertwining operators from the discrete series of $ X$ into representations with reproducing kernel and, in particular, into the discrete series of $G$; each such is given by a convolution with an analytic function. For $X$ of Hermitian type, we consider the holomorphic discrete series of $X$ and here derive very explicit formulas for the intertwining operators. As a corollary we get a multiplicity one result for the series in question.
A measure of smoothness related to the Laplacian
Z.
Ditzian
407-422
Abstract: A $K$-functional on $f \in C\,({R^d})$ given by $\displaystyle \tilde K\,(f,{t^2})= \inf (\vert\vert f - g\vert\vert + {t^2}\vert\vert\Delta g\vert\vert;g \in {C^2}\,({R^d}))$ will be shown to be equivalent to the modulus of smoothness $\displaystyle \tilde w\,(f,t)= \mathop {\sup }\limits_{0 < h \leq t} \,\left\Ve... ...,df(x) - \sum\limits_{i = 1}^d {[f(x + h{e_i}) + f(x - h{e_i})]} } \right\Vert.$ The situation for other Banach spaces of functions on ${R^d}$ will also be resolved.
The cohomology of certain function spaces
Martin
Bendersky;
Sam
Gitler
423-440
Abstract: We study a spectral sequence converging to the cohomology of the configuration space of $n$ ordered points in a manifold. A chain complex is constructed with homology equal to the ${E_2}$ term. If the field is the rationals and the manifold is formal then the spectral sequence is shown to collapse. The results are applied to compute the Anderson spectral sequence converging to the cohomology of a function space.
Remarks on forced equations of the double pendulum type
Gabriella
Tarantello
441-452
Abstract: Motivated by the double pendulum equation we consider Lagrangian systems with potential $V = V(t,q)$ periodic in each of the variables $t,q = ({q_1}, \ldots,{q_N})$. We study periodic solutions for the corresponding equation of motion subject to a periodic force $f = f(t)$. If $f$ has mean value zero, the corresponding variational problem admits a $ {{\mathbf{Z}}^N}$ symmetry which yields $N + 1$ distinct periodic solutions (see [9]). Here we consider the case where the average of $f$, though bounded, is no longer required to be zero. We show how this situation becomes more delicate, and in general it is only possible to claim no more than two periodic solutions.
Some model theory of compact Lie groups
Ali
Nesin;
Anand
Pillay
453-463
Abstract: We consider questions of first order definability in a compact Lie group $G$. Our main result is that if such $G$ is simple (and centerless) then the Lie group structure of $G$ is first order definable from the abstract group structure. Along the way we also show (i) if $ G$ is non-Abelian and connected then a copy of the field $\mathbb{R}$ is interpretable. in $(G, \cdot)$, and (ii) any "$ 1$-dimensional" field interpretable in $ (\mathbb{R}, +, \cdot)$ is definably (i.e., semialgebraically) isomorphic to the ground field $ \mathbb{R}$.