Cosmoi of internal categories
Ross
Street
271-318
Abstract: An internal full subcategory of a cartesian closed category $\mathcal{A}$, is shown to give rise to a structure on the 2-category $ Cat(\mathcal{A})$ of categories in $ \mathcal{A}$ which introduces the notion of size into the analysis of categories in $\mathcal{A}$ and allows proofs by transcendental arguments. The relationship to the currently popular study of locally internal categories is examined. Internal full subcategories of locally presentable categories (in the sense of Gabriel-Ulmer) are studied in detail. An algorithm is developed for their construction and this is applied to the categories of double categories, triple categories, and so on.
Matrix-valued special functions and representation theory of the conformal group. I. The generalized gamma function
Kenneth I.
Gross;
Wayne J.
Holman
319-350
Abstract: This article examines in detail the matrix-valued gamma function $\displaystyle {\Gamma ^{{\lambda ^0}}}\,(\alpha )\, = \,\int_P {{e^{ - {\text{tr}}(r)}}{\lambda ^0}(r,\,\bar r){{(\det \,r)}^{\alpha - 2}}\,} dr$ associated to the conformal group $ G\, = \,U(2,\,2)$. Here, $ \alpha$ is a continuous complex parameter, $ {\lambda ^0}$ runs through a family of ``weights'' of $K\, = \,U(2)\, \times \,U(2)$, P is the cone of $2\, \times \,2$ positive-definite Hermitian matrices, and the integral is well known to converge absolutely for ${\text{Re}}(\alpha )\, > \,1$. However, until now very little has been known about the analytic continuation for the general weight $ {\lambda ^0}$. The results of this paper include the following: The complete analytic continuation of ${\Gamma ^{{\lambda ^0}}}$ is determined for all weights $ {\lambda ^0}$. In analogy to the case of the classical gamma function it is proved that for any weight $ {\lambda ^0}$ the mapping $\alpha \, \to \,{\Gamma ^{{\lambda ^0}}}\,{(\alpha )^{ - 1}}$ is entire. A new integral formula is given for the inverse of the gamma function. An explicit calculation is given for the normalized variant of the gamma matrix that arises in the reproducing kernel for the spaces in which the holomorphic discrete series of G is realized, and one observes that the behavior of the analytic continuation for weights ``in general position'' is markedly different from the special cases in which the gamma function ``is scalar". The full analytic continuation of the holomorphic discrete series for G is determined. The gamma function for the forward light cone (the boundary orbit) is found, and the associated Hardy space of vector-valued holomorphic functions is described. Analogs are given for some of the well-known formulas for the classical gamma function. As an epilogue, applications of the matrix-valued gamma function, such as generalizations to $2\, \times \,2$ matrix space of the classical binomial theorem, are announced. These applications require a detailed understanding of the (generalized) Bessel functions associated to the conformal group that will be treated in the sequel to this paper.
Equivariant homotopy theory and Milnor's theorem
Stefan
Waner
351-368
Abstract: The foundations of equivariant homotopy and cellular theory are examined; an equivariant Whitehead theorem is proved, and the classical results by Milnor about spaces with the homotopy-type of a CW complex are generalized to the equivariant case. The ambient group G is assumed compact Lie. Further results include equivariant cellular approximation and the procedure for replacement of an arbitrary G-space by a G-CW complex.
Equivariant fibrations and transfer
Stefan
Waner
369-384
Abstract: The basic properties of equivariant fibrations are described, including an equivariant version of the Ďold Theorem. The foundations of equivariant stable homotopy theory are described, and the theory of equivariant transfer is developed.
Equivariant classifying spaces and fibrations
Stefan
Waner
385-405
Abstract: Explicit classifying spaces for equivariant fibrations are constructed using the geometric two-sided bar construction. The constructions are then extended to classify stable equivariant spherical fibrations and equivariant K-theory. The ambient groups is assumed compact Lie.
On the existence of eigenvalues of differential operators dependent on a parameter
Sh.
Strelitz;
S.
Abramovich
407-429
Abstract: In this paper we obtain results about the existence of eigenvalues for a system which depends polynomially on $ \lambda$, $ k\, = \,1,...,\,N$. In order to get these results we prove that this system can be reduced to a standard system of the form $ k\, = \,1,...,\,n$.
On meromorphic solutions of algebraic differential equations
Sh.
Strelitz
431-440
Abstract: The Malmquist Theorem is generalized for equations of the type
Cyclic extensions of parafree groups
Peng Choon
Wong
441-456
Abstract: Let $ 1\, \to \,F\, \to \,G\, \to \,T\, \to \,1$ be a short exact sequence where F is parafree and T is infinite cyclic. We examine some properties of G when $F/F'$ is a free ZT-module. Here $ F'$ is the commutator subgroup of F and ZT is the integral group ring of T. In particular, we show G is parafree and ${\gamma _n}F/{\gamma _{n + 1}}F$ is a free ZT-module for every $n \geqslant 1$ (where ${\gamma _n}F$ is the nth term of the lower central series of F).
Parametrizations of $G\sb{\delta }$-valued multifunctions
H.
Sarbadhikari;
S. M.
Srivastava
457-466
Abstract: Let T, X be Polish spaces, $ \mathcal{J}$ a countably generated sub-$\sigma$-field of $ {\mathcal{B}_T}$, the Borel $\sigma$-field of T, and $F:\,T\, \to \,X$ a multifunction such that $ F(t)$ is a ${G_\delta }$ in X for each $t\, \in \,T$. F is $\mathcal{J}$-measurable and $ {\text{Gr}}(F)\, \in \,J\, \otimes \,{\mathcal{B}_X}$, where ${\text{Gr}}(F)$ denotes the graph of F. We prove the following three results on F. (I) There is a map $f:\,T\, \times \,\Sigma \, \to \,X$ such that for each $ t\, \in \,T,\,f(t,\, \cdot )$ is a continuous, open map from $\Sigma$ onto $F(t)$ and for each $\sigma \, \in \,\Sigma ,\,f( \cdot ,\,\sigma )$ is $ \mathcal{J}$-measurable, where $\Sigma$ is the space of irrationals. (II) The multifunction F is of Souslin type. (III) If X is uncountable and $F(t),\,t\, \in \,T$, are all dense-in-itself then there is a $\mathcal{J}\, \otimes \,{\mathcal{B} _X}$-measurable map $ f:\,T\, \times \,X\, \to \,X$ such that for each $t\, \in \,T,\,f(t,\, \cdot )$ is a Borel isomorphism of X onto $F(t)$.
Scattering theory and polynomials orthogonal on the real line
J. S.
Geronimo;
K. M.
Case
467-494
Abstract: The techniques of scattering theory are used to study polynomials orthogonal on a segment of the real line. Instead of applying these techniques to the usual three-term recurrence formula, we derive a set of two two-term recurrence formulas satisfied by these polynomials. One of the advantages of these new recurrence formulas is that the Jost function is related, in the limit as $n\, \to \,\infty $, to the solution of one of the recurrence formulas with the boundary conditions given at $n\, = \,0$. In this paper we investigate the properties of the Jost function and the spectral function assuming the coefficients in the recurrence formulas converge at a particular rate.
On oscillatory elliptic equations on manifolds
A.
Baider;
E. A.
Feldman
495-504
Abstract: In this note we investigate the possibility of oscillatory behavior for a second-order selfadjoint elliptic operators on noncompact Riemannian manifolds (M, g). Let A be such an operator which is semibounded below and symmetric on $C_0^\infty (M)\, \subseteq \,{L^2}(M,\,d\mu )$ where $ d\mu$ is a volume element on M. If $\varphi$ is a $ {C^\infty }$ function such that $ A\varphi \, = \,\lambda \varphi$, we would naively say that $\varphi$ is oscillatory (and by extension $\lambda$ is oscillatory if it possesses such an eigenfunction $\varphi$) if $M\, - \,{\varphi ^{ - 1}}(0)$ has an infinite number of bounded connected components. For technical reasons this is not quite adequate for a definition. However, in §1 we give the usual definition of oscillatory which is a slight generalization of the one above. Let ${\Lambda _0}$ be the number below which this phenomenon cannot occur; $ {\Lambda _0}$ is the oscillatory constant for the operator A. In that A is semibounded and symmetric on $ C_0^\infty (M)\, \subseteq \,{L^2}(M,\,d\mu )$, A has a Friedrichs extension. Let $ {\Lambda _c}$ be the bottom of the continuous spectrum of the Friedrichs extension of A. Our main result is ${\Lambda _0}\, = \,{\Lambda _c}$.
The Witt ring of a space of orderings
Murray
Marshall
505-521
Abstract: The theory of ``space of orderings'' generalizes the reduced theory of quadratic forms over fields (or, more generally, over semilocal rings). The category of spaces of orderings is equivalent to a certain category of ``abstract Witt rings". In the particular case of the space of orderings of a formally real field K, the corresponding abstract Witt ring is just the reduced Witt ring of K. In this paper it is proved that if $X\, = \,(X,\,G)$ is any space of orderings with Witt ring W(X), and $X\, \to \,Z$ is any continuous function, then g is represented by an element of W(X) if and only if ${\Sigma _{\sigma \, \in \,V}}g(\sigma )\, \equiv \,0\,\bmod \,\left\vert V \right\vert$ holds for all finite fans $V \subseteq X$. This generalizes a recent field theoretic result of Becker and Bröcker. Following the proof of this, applications are given to the computation of the stability index of X, and to the representation of continuous functions $ g:\,X\, \to \, \pm 1$ by elements of G.
A nonlinear Volterra equation with rapidly decaying solutions
Olof J.
Staffans
523-530
Abstract: We study the asymptotic behavior of the solutions of a nonlinear integrodifferential Volterra equation with a convolution kernel. More specifically, we give conditions which imply that a solution x satisfies $ x(t)\, = \,O({t^{ - p}})\,(t \to \infty )$, where p is an arbitrary, positive real number.
The fixed point property and unbounded sets in Hilbert space
William O.
Ray
531-537
Abstract: It is shown that a closed convex subset K of a real Hilbert space H has the fixed point property for nonexpansive mappings if and only if K is bounded.
Erratum to: ``The behavior of the support of solutions of the equation of nonlinear heat conduction with absorption in one dimension''
Barry F.
Knerr
539-539