Asymptotic completeness for classes of two, three, and four particle Schr\"odinger operators
George A.
Hagedorn
1-75
Abstract: Formulas for the resolvent $ {(z\, - \,H)^{ - 1}}$ are derived, where $ H\, = \,{H_0}\, + \,{\Sigma _{i < j}}{\lambda _{ij}}{V_{ij}}$ is an N particle Schrödinger operator with the center of mass motion removed. For a large class of two-body potentials and generic couplings $\{ {\lambda _{ij}}\}$, these formulas are used to prove asymptotic completeness in the $N\, \leqslant \,4$ body problem when the space dimension is $ m\, \geqslant \,3$. The allowed potentials belong to a space of dilation analytic multiplication operators which fall off more rapidly than $ {r^{ - 2 - \varepsilon }}$ at $\infty$. In particular, Yukawa potentials, generalized Yukawa potentials, and potentials of the form $ {(1\, + \,r)^{ - 2 - \varepsilon }}$ are allowed.
On the excursion process of Brownian motion
Frank B.
Knight
77-86
Abstract: Let $W_0^ + \,(t)$ denote the scaled excursion process of Brownian motion, and let $l_0^ + \,(a),\,0\, \leqslant \,a,$ be its local time at a. The joint distribution of $l_0^ + \,(a),\,\beta (a),$ and $\gamma (a)$ is obtained, where $\beta (a)$ and $ \gamma (a)$ are the last exit time and the first passage time of a by $W_0^{ + }\,(t)$.
Approximate torus fibrations of high dimensional manifolds can be approximated by torus bundle projections
R. E.
Goad
87-97
Abstract: In this paper, we prove that approximate torus fibrations of high dimensional manifolds can be approximated by torus bundle projections. The principal tools are the torus trick developed by Kirby and Siebenmann, a surgery theorem concerning homotopy structures on torii due to Hsiang and Wall, a theorem on the space of homeomorphisms of the torus due to Hamstrom and a generalization of hereditary homotopy equivalence developed by the author.
The interior operator logic and product topologies
Joseph
Sgro
99-112
Abstract: In this paper we present a model theory of the interior operator on product topologies with continuous functions. The main results are a completeness theorem, an axiomatization of topological groups, and a proof of an interpolation and definability theorem.
Topological equivalence of gradient vectorfields
Douglas S.
Shafer
113-126
Abstract: This paper is a study of the behavior of the topological equivalence class of the planar gradient vectorfield $X\, = \,{\operatorname{grad} _g}\,V$, in a neighborhood of a degenerate singularity of V, as g varies over all Riemannian metrics. It is shown that under simple restrictions on V the topological equivalence class of X is determined by its first nonvanishing jet, and that only finitely many equivalence classes occur (for fixed V). In this case, when the degree of the first nonvanishing jet of V is less than five, necessary and sufficient conditions for change in equivalence class are given, both in terms of the coefficients of the homogeneous part of V and geometrically in terms of its level curves. A catalogue of possible phase portraits, up to topological equivalence, is included. Necessary conditions are given for change in higher degree.
Fibrewise localization and completion
J. P.
May
127-146
Abstract: The behavior of fibrewise localization and completion on the classifying space level is analyzed. The relationship of these constructions to fibrewise joins and smash products and to orientations of spherical fibrations is also analyzed. This theory is essential to validate Sullivan's proof of the Adams conjecture.
Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms
Aroldo
Kaplan
147-153
Abstract: We introduce a class of nilpotent Lie groups which arise naturally from the notion of composition of quadratic forms, and show that their standard sublaplacians admit fundamental solutions analogous to that known for the Heisenberg group.
Fractional differentiation and Lipschitz spaces on local fields
C. W.
Onneweer
155-165
Abstract: In this paper we continue our study of differentiation on a local field K. We define strong derivatives of fractional order $\alpha \, > \,0$ for functions in $ {L_r}(\textbf{K})$, $ 1\, \leqslant \,r\, < \,\infty$. After establishing a number of basic properties for such derivatives we prove that the spaces of Bessel potentials on K are equal to the spaces of strongly $ {L_r}(\textbf{K})$-differentiable functions of order $\alpha \, > \,0$ when $1\, \leqslant \,r\, \leqslant \,2$. We then focus our attention on the relationship between these spaces and the generalized Lipschitz spaces over K. Among others, we prove an inclusion theorem similar to a wellknown result of Taibleson for such spaces over $ {\textbf{R}^n}$.
The number of groups of a given genus
T. W.
Tucker
167-179
Abstract: It is shown that the number of groups with a given genus greater than one is finite. The proof depends heavily on V. K. Proulx's classification of groups of genus one. The key observation is that as the number of vertices of a graph imbedded on a given surface increases, the average face size of the imbedding approaches the average face size of a toroidal imbedding. The result appears to be related to Hurwitz's theorem bounding the order of a group of conformal automorphisms on a Riemann surface of genus g.
On special classes of entire functions whose zeros and growth are restricted
Carl L.
Prather
181-189
Abstract: The present paper is an investigation of the uniform limits on bounded sets of entire functions of genus $\leqslant \,2p$ whose zeros are real, or lie in an even number of sectors (or correspondingly on rays) of a certain size, both the number and size depending on p. A characterization of the uniform limits of entire functions of genus $\leqslant \,2p$ whose zeros lie in these sectors is given.
Convolution equations in spaces of infinite-dimensional entire functions of exponential and related types
J.-F.
Colombeau;
B.
Perrot
191-198
Abstract: We prove results of existence and approximation of the solutions of the convolution equations in spaces of entire functions of exponential type on infinite dimensional spaces. In particular we obtain: let E be a complex, quasi-complete and dual nuclear locally convex space and $ \Omega$ a convex balanced open subset of E; let $\mathcal{H} (\Omega )$ be the space of the holomorphic functions on $\Omega$, equipped with the compact open topology and $\mathcal{F}$; equip this space $\mathcal{F}$. Then, ``every nonzero convolution operator on
Basic sequences in non-Schwartz Fr\'echet spaces
Steven F.
Bellenot
199-216
Abstract: Obliquely normalized basic sequences are defined and used to characterize non-Schwartz-Fréchet spaces. It follows that each non-Schwartz-Fréchet space E has a non-Schwartz subspace with a basis and a quotient which is not Montel (which has a normalized basis if E is separable). Stronger results are given when more is known about E, for example, if E is a subspace of a Fréchet $ {l_p}$-Köthe sequence space, then E has the Banach space $ {l_p}$ as a quotient and E has a subspace isomorphic to a non-Schwartz $ {l_p}$-Köthe sequence space. Examples of Fréchet-Montel spaces which are not subspaces of any Fréchet space with an unconditional basis are given. The question of the existence of conditional basic sequences in non-Schwartz-Fréchet spaces is reduced to questions about Banach spaces with symmetric bases. Nonstandard analysis is used in some of the proofs and a new nonstandard characterization of Schwartz spaces is given.
Minimal excessive measures and functions
E. B.
Dynkin
217-244
Abstract: Let H be a class of measures or functions. An element h of H is minimal if the relation $h\, = \,{h_1}\, + \,{h_2}$, ${h_1}$, $ {h_2} \in H$ implies that $ {h_1}$, ${h_2}$ are proportional to h. We give a limit procedure for computing minimal excessive measures for an arbitrary Markov semigroup $ {T_t}$ in a standard Borel space E. Analogous results for excessive functions are obtained assuming that an excessive measure $ \gamma$ on E exists such that $ {T_t}f\, = 0$ if $ f\, = \,0$ $\gamma$-a.e. In the Appendix, we prove that each excessive element can be decomposed into minimal elements and that such a decomposition is unique.
Derivation alternator rings with idempotent
Irvin R.
Hentzel;
Harry F.
Smith
245-256
Abstract: A nonassociative ring is called a derivation alternator ring if it satisfies the identities $ (yz,\,x,\,x)\, = \,y(z,\,x,\,x)\, + \,(y,\,x,\,x)z,\,(x,\,x,\,yz)\, = \,y(x,\,x,\,z)\, + \,(x,\,x,\,y)z$ and $(x,\,x,\,x)\, = 0$. Let R be a prime derivation alternator ring with idempotent $e \ne 1$ and characteristic $ \ne 2$. If R is without nonzero nil ideals of index 2, then R is alternative.
Two approaches to supermanifolds
Marjorie
Batchelor
257-270
Abstract: The problem of supplying an analogue of a manifold whose sheaf of functions contains anticommuting elements has been approached in two ways. Either one extends the sheaf of functions formally, as in the category of graded manifolds [3], [8], or one mimicks the usual definition of a manifold, having replaced Euclidean space with a suitable product of the odd and even parts of an exterior algebra as in the category of supermanifolds [6]. This paper establishes the equivalence of the category of supermanifolds with the category of graded manifolds.