The Cauchy problem in ${\bf C}\sp N$ for linear second order partial differential equations with data on a quadric surface
Gunnar
Johnsson
1-48
Abstract: By means of a method developed essentially by Leray some global existence results are obtained for the problem referred to in the title. The partial differential equations are required to have constant principal part and the initial surface to be irreducible and not everywhere characteristic. The Cauchy data are assumed to be given by entire functions. Under these conditions the location of all possible singularities of solutions are determined. The sets of singularities can be divided into two types, K- and L-singularities. K, the set of K-singularities, is the global version of the characteristic tangent defined by Leray. The L-sets are here quadric surfaces which, in contrast to the Ksets, allow unbounded singularities. The L-sets are in turn divided into three types: initial, asymptotic, and latent singularities. The initial singularities appear when the characteristic points of the initial surface are exceptional according to Leray's local theory. These sets of singularity intersect the initial surface at characteristic points. The asymptotic case, where the set of singularities does not cut the initial surface, can be viewed as projectively equivalent to the initial case, the intersection taking place at infinite characteristic points. Finally the latent singularities are sets which intersect the initial surface, but where the solutions do not develop singularities initially. In the case of the Laplace equation with data on a real quadric surface it is shown that the K-singularities and the asymptotic singularities occur on the classical focal sets defined by Poncelet, Plücker, Darboux et al. There are also latent singularities appearing in coordinate subspaces of $ {\mathbb{R}^N}$. As a corollary a new proof is given of the fact that ellipsoids have the Pompeiu property.
The Inverse Stable Range Functor
Robert S. Y.
Young
49-56
Abstract: We give an inverse construction of the stable range for general flows which may or may not admit an invariant measure. The inverse map is then shown to be a right inverse functor of the stable range functor.
Finitely generated Kleinian groups in $3$-space and $3$-manifolds of infinite homotopy type
L.
Potyagaĭlo
57-77
Abstract: We prove the existence of a finitely generated Kleinian group $N \subset S{O_ + }(1,4)$ acting freely on an invariant component $ \Omega \subset {S^3}$ without parabolic elements such that the fundamental group ${\pi _1}(\Omega /N)$ is not finitely generated. Moreover, N is a finite index subgroup of a Kleinian group ${N_0}$ which has infinitely many conjugacy classes of elliptic elements.
Integer points on curves of genus two and their Jacobians
David
Grant
79-100
Abstract: Let C be a curve of genus 2 defined over a number field, and $ \Theta$ the image of C embedded into its Jacobian J. We show that the heights of points of J which are integral with respect to $ {[2]_\ast}\Theta$ can be effectively bounded. As a result, if P is a point on C, and $\bar P$ its image under the hyperelliptic involution, then the heights of points on C which are integral with respect to P and $\bar P$ can be effectively bounded, in such a way that we can isolate the dependence on P, and show that if the height of P is bigger than some bound, then there are no points which are S-integral with respect to P and $\bar P$. We relate points on C which are integral with respect to P to points on J which are integral with respect to $ \Theta$, and discuss approaches toward bounding the heights of the latter.
Homoclinic loop and multiple limit cycle bifurcation surfaces
L. M.
Perko
101-130
Abstract: This paper establishes the existence and analyticity of homoclinic loop bifurcation surfaces $ \mathcal{H}$ and multiplicity-two, limit cycle bifurcation surfaces $\mathcal{C}$ for planar systems depending on two or more parameters; it determines the side of $\mathcal{H}$ or $ \mathcal{C}$ on which limit cycles occur; and it shows that if $\mathcal{H}$ and $ \mathcal{C}$ intersect, then typically they do so at a flat contact.
Graphs with the circuit cover property
Brian
Alspach;
Luis
Goddyn;
Cun Quan
Zhang
131-154
Abstract: A circuit cover of an edge-weighted graph (G, p) is a multiset of circuits in G such that every edge e is contained in exactly $p(e)$ circuits in the multiset. A nonnegative integer valued weight vector p is admissible if the total weight of any edge-cut is even, and no edge has more than half the total weight of any edge-cut containing it. A graph G has the circuit cover property if (G, p) has a circuit cover for every admissible weight vector p. We prove that a graph has the circuit cover property if and only if it contains no subgraph homeomorphic to Petersen's graph. In particular, every 2-edge-connected graph with no subgraph homeomorphic to Petersen's graph has a cycle double cover.
Spectral analysis for the generalized Hermite polynomials
Allan M.
Krall
155-172
Abstract: The operator theory associated with the Hermite polynomials does not extend to the generalized Hermite polynomials because the even and odd polynomials satisfy different differential equations. We show that this leads to two problems, each of interest on its own. We then weld them together to form a united spectral expansion. In addition, the exponent $\mu$ in the weight $\vert x{\vert^{2\mu }}{e^{ - {x^2}}}$ has traditionally always been greater than $- \frac{1}{2}$. We show what happens if $\mu \leq - \frac{1}{2}$. Finally, we examine the differential equations in left-definite spaces.
Representable $K$-theory of smooth crossed products by ${\bf R}$ and ${\bf Z}$
N. Christopher
Phillips;
Larry B.
Schweitzer
173-201
Abstract: We show that the Thorn isomorphism and the Pimsner-Voiculescu exact sequence both hold for smooth crossed products of Fréchet algebras by $ \mathbb{R}$ and $\mathbb{Z}$ respectively. We also obtain the same results for ${L^1}$-crossed products of Banach algebras by $\mathbb{R}$ and $ \mathbb{Z}$.
Minimal torsion in isogeny classes of elliptic curves
Raymond
Ross
203-215
Abstract: Let K be a field finitely generated over its prime field, and let $ w(K)$ denote the number of roots of unity in K. If K is of characteristic 0, then there is an integer D, divisible only by those primes dividing $w(K)$, such that for any elliptic curve $ E/K$ without complex multiplication over K, there is an elliptic curve $ E\prime/K$ isogenous to E such that $ E\prime{(K)_{{\text{tors}}}}$ is of order dividing D. In case K admits a real embedding, we show $D = 2$, and a nonuniform result is proved in positive characteristic.
Generalised Castelnuovo inequalities
Liam A.
Donohoe
217-260
Abstract: Given a Riemann surface of genus p, denoted by ${X_p}$, admitting j linear series of dimension r and degree n Accola derived a polynomial function $f(j,n,r)$ so that $p \leq f(j,n,r)$ and exhibited plane models of Riemann surfaces attaining equality in the inequality. In this paper we provide a classification of all such ${X_p}$ when $r \geq 6$. In addition we classify curves, $ {X_p}$, of maximal genus when ${X_p}$ admits two linear series which have a common dimension but different degrees.
The Brown-Peterson homology of Mahowald's $X\sb k$ spectra
Dung Yung
Yan
261-289
Abstract: We compute the Brown-Peterson homology of Mahowald's ${X_k}$ spectrum which is the Thom spectrum induced from $ \Omega {J_{{2^k} - 1}}{S^2} \to {\Omega ^2}{S^3} - {\text{BO}}$, and the edge homomorphism of the Adams-Novikov spectral sequence for $ {\pi _\ast}({X_k})$. We then compute the nonnilpotent elements of ${\pi _\ast}({X_k})$.
$3$-primary $v\sb 1$-periodic homotopy groups of $F\sb 4$ and $E\sb 6$
Martin
Bendersky;
Donald M.
Davis
291-306
Abstract: We compute the 3-primary ${v_1}$-periodic homotopy groups of the exceptional Lie groups ${F_4}$ and ${E_6}$. The unstable Novikov spectral sequence is used for the most delicate part of the analysis.
Mean value inequalities in Hilbert space
F. H.
Clarke;
Yu. S.
Ledyaev
307-324
Abstract: We establish a new mean value theorem applicable to lower semi-continuous functions on Hilbert space. A novel feature of the result is its "multidirectionality": it compares the value of a function at a point to its values on a set. We then discuss some refinements and consequences of the theorem, including applications to calculus, flow invariance, and generalized solutions to partial differential equations. Résumé. On établit un nouveau théorème de la valeur moyenne qui s'applique aux fonctions semicontinues inférieurement sur un espace de Hilbert. On déduit plusieurs conséquences du résultat portant, par exemple, sur les fonctions monotones et sur les solutions généralisées des équations aux dérivées partielles.
A general view of reflexivity
Don
Hadwin
325-360
Abstract: Various concepts of reflexivity for an algebra or linear space of operators have been studied by operator theorists and algebraists. This paper contains a very general version of reflexivity based on dual pairs of vector spaces over a Hausdorff field. The special cases include topological, algebraic and approximate reflexivity. In addition general versions of hyperreflexivity and direct integrals are introduced. We prove general versions of many known (and some new) theorems, often with simpler proofs.
Groups and fields interpretable in separably closed fields
Margit
Messmer
361-377
Abstract: We prove that any infinite group interpretable in a separably closed field F of finite Eršov-invariant is definably isomorphic to an F-algebraic group. Using this result we show that any infinite field K interpretable in a separably closed field F is itself separably closed; in particular, in the finite invariant case K is definably isomorphic to a finite extension of F.
Structural instability of exponential functions
Zhuan
Ye
379-389
Abstract: We first prove some equivalent statements on J-stability of families of critically finite entire functions. Then, with these in hand, a conjecture concerning stability of the family of exponential functions is affirmatively answered in some cases.
A $4$-dimensional Kleinian group
B. H.
Bowditch;
G.
Mess
391-405
Abstract: We give an example of a 4-dimensional Kleinian group which is finitely generated but not finitely presented, and is a subgroup of a cocompact Kleinian group.
A Banach space not containing $c\sb 0, l\sb 1$ or a reflexive subspace
W. T.
Gowers
407-420
Abstract: An infinite-dimensional Banach space is constructed which does not contain ${c_0}$, ${l_1}$ or an infinite-dimensional reflexive subspace. In fact, it does not even contain $ {l_1}$ or an infinite-dimensional subspace with a separable dual.
A differential operator for symmetric functions and the combinatorics of multiplying transpositions
I. P.
Goulden
421-440
Abstract: By means of irreducible characters for the symmetric group, formulas have previously been given for the number of ways of writing permutations in a given conjugacy class as products of transpositions. These formulas are alternating sums of binomial coefficients and powers of integers. Combinatorial proofs are obtained in this paper by analyzing the action of a partial differential operator for symmetric functions.
On nonlinear delay differential equations
A.
Iserles
441-477
Abstract: We examine qualitative behaviour of delay differential equations of the form $\displaystyle y\prime (t) = h(y(t),\;y(qt)),\quad y(0) = {y_0},$ where h is a given function and $q > 0$. We commence by investigating existence of periodic solutions in the case of $h(u,v) = f(u) + p(v)$, where f is an analytic function and p a polynomial. In that case we prove that, unless q is a rational number of a fairly simple form, no nonconstant periodic solutions exist. In particular, in the special case when f is a linear function, we rule out periodicity except for the case when $q = 1/\deg p$. If, in addition, p is a quadratic or a quartic, we show that this result is the best possible and that a nonconstant periodic solution exists for $q = \frac{1}{2}$ or $ \frac{1}{4}$, respectively. Provided that g is a bivariate polynomial, we investigate solutions of the delay differential equation by expanding them into Dirichlet series. Coefficients and arguments of these series are derived by means of a recurrence relation and their index set is isomorphic to a subset of planar graphs. Convergence results for these Dirichlet series rely heavily upon the derivation of generating functions of such graphs, counted with respect to certain recursively-defined functionals. We prove existence and convergence of Dirichlet series under different general conditions, thereby deducing much useful information about global behaviour of the solution.