Knot modules. I
Jerome
Levine
1-50
Abstract: For a differentiable knot, i.e. an imbedding ${S^n} \subset {S^{n + 2}}$, one can associate a sequence of modules $ \{ {A_q}\}$ over the ring $Z[t,{t^{ - 1}}]$, which are the source of many classical knot invariants. If X is the complement of the knot, and $\tilde X \to X$ the canonical infinite cyclic covering, then ${A_q} = {H_q}(\tilde X)$. In this work a complete algebraic characterization of these modules is given, except for the Z-torsion submodule of ${A_1}$.
Parametrizations of Titchmarsh's $m(\lambda )$-functions in the limit circle case
Charles T.
Fulton
51-63
Abstract: For limit-circle eigenvalue problems the so-called $'m(\lambda )'$-functions of Titchmarsh [15] are introduced in such a fashion that their parametrization is built into the definition.
Unimodality and dominance for symmetric random vectors
Marek
Kanter
65-85
Abstract: In this paper a notion of unimodality for symmetric random vectors in $ {R^N}$ is developed which is closed under convolution as well as weak convergence. A related notion of stochastic dominance for symmetric random vectors is also studied.
Alternator and associator ideal algebras
Irvin Roy
Hentzel;
Giulia Maria
Piacentini Cattaneo;
Denis
Floyd
87-109
Abstract: If I is the ideal generated by all associators, $(a,b,c) = (ab)c - a(bc)$, it is well known that in any nonassociative algebra $R,I \subseteq (R,R,R) + R(R,R,R)$. We examine nonassociative algebras where $I \subseteq (R,R,R)$. Such algebras include $( - 1,1)$ algebras, Lie algebras, and, as we show, a large number of associator dependent algebras. An alternator is an associator of the type (a, a, b), (a, b, a), (b, a, a). We next study algebras where the additive span of all alternators is an ideal. These include all algebras where $I = (R,R,R)$ as well as alternative algebras. The last section deals with prime, right alternative, alternator ideal algebras satisfying an identity of the form $ [x,(x,x,a)] = \gamma (x,x,[x,a])$ for fixed $\gamma$. With two exceptions, if this algebra has an idempotent e such that $(e,e,R) = 0$, then the algebra is alternative. All our work deals with algebras with an identity element over a field of characteristic prime to 6. All our containment relations are given by identities.
Distribution of eigenvalues in the presence of higher order turning points
Anthony
Leung
111-135
Abstract: This article is concerned with the eigenvalue problem $u''(x) - {\lambda ^2}p(x)u(x) = 0,u(x) \in {L_2}( - \infty ,\infty )$, where $p(x)$ is real, analytic and possesses zeroes of arbitrary orders. Under certain conditions on $ p(x)$, approximate formulas for the eigenvalues are found. The problem considered is of interest in the study of particle scattering and wave mechanics. The formula is analogous to the quantum rule of Bohr-Sommerfeld.
Composition series and intertwining operators for the spherical principal series. I
Kenneth D.
Johnson;
Nolan R.
Wallach
137-173
Abstract: Let G be a connected semisimple Lie group with finite center and let K be a maximal compact subgroup. Let $ \pi$ be a not necessarily unitary principal series representation of G on the Hilbert space ${H^\pi }$. If ${X^\pi }$ denotes the space of K-finite vectors of ${H^\pi },\pi$ induces a representation $ {\pi _0}$ of $ U(g)$, the enveloping algebra of the Lie algebra of G, on ${X^\pi }$. In this paper, we determine when ${\pi _0}$ is irreducible, and if $ {\pi _0}$ is not irreducible we determine the composition series of $ {X^\pi }$ and the structure of the induced representations on the subquotients. Explicit computation of the intertwining operators for the different principal series representations are obtained and their relationship to polynomials defined by B. Kostant are obtained.
Moduli of continuity for exponential Lipschitz classes
Paul
De Land
175-189
Abstract: Let $\Psi$ be a convex function, and let f be a real-valued function on [0, 1]. Let a modulus of continuity associated to $\Psi$ be given as $\displaystyle {Q_\Psi }(\delta ,f) = \inf \left\{ {\lambda :\frac{1}{\delta }\i... ... f(x) - f(y)\vert}}{\lambda }} \right)}\;dx\;dy\; \leqslant \Psi (1)} \right\}.$ It is shown that $ \smallint _0^1{Q_\Psi }(\delta ,f)\;d\;( - {\Psi ^{ - 1}}(c/\delta )) < \infty$ guarantees the essential continuity of f, and, in fact, a uniform Lipschitz estimate is given. In the case that $\Psi (u) = \exp \;{u^2}$ the above condition reduces to $\displaystyle \int_0^1 {{Q_{\exp \;{u^2}}}\;(\delta ,f)\frac{{d\delta }}{{\delta \sqrt {\log (c/\delta )} }}\; < \infty .} $ This exponential square condition is satisfied almost surely by the random Fourier series $ {f_t}(x) = \Sigma _{n = 1}^\infty {a_n}{R_n}(t){e^{inx}}$, where $\{ {R_n}\}$ is the Rademacher system, as long as $\displaystyle \int_0^1 {\sqrt {a_n^2{{\sin }^2}(n\delta /2)} \frac{{d\delta }}{{\delta \sqrt {\log (1/\delta )} }}\; < \infty .} $ Hence, the random essential continuity of ${f_t}(x)$ is guaranteed by each of the above conditions.
On a notion of smallness for subsets of the Baire space
Alexander S.
Kechris
191-207
Abstract: Let us call a set $ A \subseteq {\omega ^\omega }$ of functions from $\omega$ into $ \omega \;\sigma$-bounded if there is a countable sequence of functions $\{ {\alpha _n}:n \in \omega \} \subseteq {\omega ^\omega }$ such that every member of A is pointwise dominated by an element of that sequence. We study in this paper definability questions concerning this notion of smallness for subsets of ${\omega ^\omega }$. We show that most of the usual definability results about the structure of countable subsets of $ {\omega ^\omega }$ have corresponding versions which hold about $\sigma $-bounded subsets of ${\omega ^\omega }$. For example, we show that every $ \Sigma _{2n + 1}^1\;\sigma$-bounded subset of $ {\omega ^\omega }$ has a $\Delta _{2n + 1}^1$ ``bound'' $\{ {\alpha _m}:m \in \omega \}$ and also that for any $ n \geqslant 0$ there are largest $\sigma$-bounded $ \Pi _{2n + 1}^1$ and $\Sigma _{2n + 2}^1$ sets. We need here the axiom of projective determinacy if $n \geqslant 1$. In order to study the notion of $ \sigma$-boundedness a simple game is devised which plays here a role similar to that of the standard $^\ast$-games (see [My]) in the theory of countable sets. In the last part of the paper a class of games is defined which generalizes the $^\ast$- and $ ^{ \ast \ast }$- (or Banach-Mazur) games (see [My]) as well as the game mentioned above. Each of these games defines naturally a notion of smallness for subsets of ${\omega ^\omega }$ whose special cases include countability, being of the first category and $\sigma $-boundedness and for which one can generalize all the main results of the present paper.
Integral representations of invariant measures
Ashok
Maitra
209-225
Abstract: In this paper we prove, under suitable conditions, several representation theorems for invariant measures arising out of the action of a family of measurable transformations $\mathcal{J}$ on a measurable space $(X,\mathcal{A})$. Our results unify and extend results of Farrell and Varadarajan on the representation of invariant measures.
The equivalence of complete reductions
R.
Hindley
227-248
Abstract: This paper is about two properties of the $ \lambda \beta$-calculus and combinatory reduction, namely (E): all complete reductions $\rho$ and $\sigma$ of the residuals of a set of redexes in a term X have the same end; and $({{\text{E}}^ + }):\rho$ and $\sigma$ leave the same residuals of any other redex in X. Property (E) is deduced from abstract assumptions which do not imply $({{\text{E}}^ + })$. Also $({{\text{E}}^ + })$ is proved for the usual extensions of combinatory and $ \lambda \beta$-reduction, and a weak but natural form of $({{\text{E}}^ + })$ is proved for $\lambda \beta \eta $-reduction.
Fields generated by linear combinations of roots of unity
R. J.
Evans;
I. M.
Isaacs
249-258
Abstract: It is shown that a linear combination of roots of unity with rational coefficients generates a large subfield of the field generated by the set of roots of unity involved, except when certain partial sums vanish. Some related results about polygons with all sides and angles rational are also proved.
Nonsmoothable, unstable group actions
Dennis
Pixton
259-268
Abstract: For $k > 1$ there is a nonempty open set of ${C^1}$ actions of $ {{\mathbf{Z}}^k}$ on $ {S^1}$, no element of which is either topologically conjugate to a $ {C^2}$ action or structurally stable. The ${C^1}$ closure of this set contains all $ {C^2}$ actions which have compact orbits, so no such action is structurally stable in the space of ${C^1}$ actions.
Lifting idempotents and exchange rings
W. K.
Nicholson
269-278
Abstract: Idempotents can be lifted modulo a one-sided ideal L of a ring R if, given $x \in R$ with $x - {x^2} \in L$, there exists an idempotent $e \in R$ such that $e - x \in L$. Rings in which idempotents can be lifted modulo every left (equivalently right) ideal are studied and are shown to coincide with the exchange rings of Warfield. Some results of Warfield are deduced and it is shown that a projective module P has the finite exchange property if and only if, whenever $P = N + M$ where N and M are submodules, there is a decomposition $P = A \oplus B$ with $A \subseteq N$ and $B \subseteq M$.
On the zeros of Stieltjes and Van Vleck polynomials
Neyamat
Zaheer;
Mahfooz
Alam
279-288
Abstract: Stieltjes and Van Vleck polynomials arise in the study of the polynomial solutions of the generalized Lamé differential equation. Our object is to generalize a theorem due to Marden on the location of the zeros of Stieltjes and Van Vleck polynomials. In fact, our generalization is two-fold: Firstly, we employ sets which are more general than the ones used by Marden for prescribing the location of the complex constants occurring in the Lamé differential equation; secondly, Marden deals only with the standard form of the said differential equation, whereas our result is equally valid for yet another form of the same differential equation. The part of our main theorem concerning Stieltjes polynomials may also be regarded as a generalization of Lucas' theorem to systems of partial fraction sums.
Zeroes of holomorphic vector fields and Grothendieck duality theory
N. R.
O’Brian
289-306
Abstract: The holomorphic fixed point formula of Atiyah and Bott is discussed in terms of Grothendieck's theory of duality. The treatment is valid for an endomorphism of a compact complex-analytic manifold with arbitrary isolated fixed points. An expression for the fixed point indices is then derived for the case where the endomorphism belongs to the additive group generated by a holomorphic vector field with isolated zeroes. An application and some examples are given. Two generalisations of these results are also proved. The first deals with holomorphic vector bundles having sufficient homogeneity properties with respect to the action of the additive group on the base manifold, and the second with additive group actions on algebraic varieties.
Existence of integrals and the solution of integral equations
Jon C.
Helton
307-327
Abstract: Functions are from R to N or $R \times R$ to N, where R denotes the real numbers and N denotes a normed complete ring. If S, T and G are functions from $R \times R$ to N, each of $ S({p^ - },p),S({p^ - },{p^ - }),T({p^ - },p)$ and $T({p^ - },{p^ - })$ exists for $a < p \leqslant b$, each of $ S(p,{p^ + }),S({p^ + },{p^ + }),T(p,{p^ + })$ and $T({p^ + },{p^ + })$ exists for $a \leqslant p < b$, G has bounded variation on [a, b] and $ \smallint _a^bG$ exists, then each of $\displaystyle \int_a^b S \left[ {G - \int G } \right]T\quad {\text{and}}\quad \int_a^b {S\left[ {1 + G - \prod {(1 + G)} } \right]} \;T$ exists and is zero. These results can be used to solve integral equations without the existence of integrals of the form $\displaystyle \int_a^b {\left\vert {G - \int G } \right\vert = 0} \quad {\text{and}}\quad \int_a^b {\left\vert {1 + G - \prod {(1 + G)} } \right\vert} = 0.$ This is demonstrated by solving the linear integral equation $\displaystyle f(x) = h(x) + (LR)\int_a^x {(fG + fH)}$ and the Riccati integral equations $\displaystyle f(x) = w(x) + (LRLR)\int_a^x {(fH + Gf + fKf)}$ without the existence of the previously mentioned integrals.
Knots with infinitely many minimal spanning surfaces
Julian R.
Eisner
329-349
Abstract: We show that if $ {k_1}$ and ${k_2}$ are nonfibered knots, then the composite knot $K = {k_1}\char93 {k_2}$ has an infinite collection of minimal spanning surfaces, no two of which are isotopic by an isotopy which leaves the knot K fixed. This result is then applied to show that whether or not a knot has a unique minimal spanning surface can depend on what definition of spanning surface equivalence is used.
Quasi-Anosov diffeomorphisms and hyperbolic manifolds
Ricardo
Mañé
351-370
Abstract: Let f be a diffeomorphism of a smooth manifold N and $M \subset N$ a compact boundaryless submanifold such that it is a hyperbolic set for f. The diffeomorphism f/M is characterized and it is proved that it is Anosov if and only if M is an invariant isolated set of f (i.e. the maximal invariant subset of some compact neighborhood). Isomorphisms of vector bundles with the property that the zero section is an isolated subset are studied proving that they can be embedded in hyperbolic vector bundle isomorphisms.