The embeddings of the discrete series in the principal series for semisimple Lie groups of real rank one
M. Welleda
Baldoni Silva
303-368
Abstract: We consider the problem of finding all the ``embeddings'' of a discrete series representation in the principal series in the case of a simple real Lie group G of real rank one. More precisely, we solve the problem when G is $ \operatorname{Spin} (2n,\,1),{\text{SU}}(n,\,1),\,{\text{SP}}(n,\,1)\,{\text{or}}\,{F_4}\,(n\, \geqslant \,2)$. The problem is reduced to considering only discrete series representations with trivial infinitesimal character, by means of tensoring with finite dimensional representations. Various other techniques are employed.
A stable converse to the Vietoris-Smale theorem with applications to shape theory
Steve
Ferry
369-386
Abstract: Our main result says that if $f:\,X\, \to \,Y$ is a map between finite polyhedra which has k-connected homotopy fiber, then there is an n such that $f\, \times \,{\text{id:}}\,X\, \times \,{I^n} \to Y$ is homotopic to a map with k-connected point-inverses. This result is applied to give an algebraic characterization of compacta shape equivalent to locally n-connected compacta. We also show that a $U{V^1}$ compactum can be ``improved'' within its shape class until its homotopy theory and strong shape theory are the same with respect to finite dimensional polyhedra.
Second-order equations of fixed type in regions with corners. I
Leonard
Sarason
387-416
Abstract: A class of well-posed boundary value problems for second order equations in regions with corners and edges is studied. The boundary condition may involve oblique derivatives, and edge values may enter the graph of the associated Hilbert space operator. Uniqueness of weak solutions and existence of strong solutions is shown.
Octonion planes over local rings
Robert
Bix
417-438
Abstract: Let $\mathcal{D}$ be an octonion algebra which is a free module over a local ring R and let $J = H({\mathcal{D}_3},\gamma )$ be the quadratic Jordan algebra of Hermitian 3-by-3 matrices over R. We define the octonion plane determined by J and prove that every collineation is induced by a norm semisimilarity of J. We classify the subgroups of the collineation group normalized by the little projective group.
Invariant solutions to the oriented Plateau problem of maximal codimension
David
Bindschadler
439-462
Abstract: The principal result gives conditions which imply that a solution to the Plateau problem inherits the symmetries of its boundary. Specifically, let G be a compact connected Lie subgroup of $ {\text{SO}}(n)$. Assume the principal orbits have dimension m, there are no exceptional orbits and the distribution of $(n\, - \,m)$-planes orthogonal to the principal orbits is involutive. We show that if B is a finite sum of oriented principal orbits, then every absolutely area minimizing current with boundary B is invariant. As a consequence of the methods used, the above Plateau problems are shown to be equivalent to 1-dimensional variational problems in the orbit space. Some results concerning invariant area minimizing currents in Riemannian manifolds are also obtained.
Topological spaces with prescribed nonconstant continuous mappings
Věra
Trnková
463-482
Abstract: Given a $ {T_1}$-space Y and a ${T_3}$-space V, consider ${T_3}$-spaces X such that X has a closed covering by spaces homeomorphic to V and any continuous mapping $f:\,X \to Y$ is constant. All such spaces and all their continuous mappings are shown to form a very comprehensive category, containing, e.g., a proper class of spaces without nonconstant, nonidentical mappings or containing a space X, for every monoid M, such that all the nonconstant continuous mappings of X into itself are closed under composition and form a monoid isomorphic to M. The category of paracompact connected spaces, having a closed covering by a given totally disconnected paracompact space, has, e.g., analogous properties. Categories of metrizable spaces are also investigated.
In-between theorems in uniform spaces
D.
Preiss;
J.
Vilímovský
483-501
Abstract: Necessary and sufficient conditions for the existence of a uniformly continuous function in-between given functions $f\, \geqslant \,g$ on a uniform space are studied. It appears that the investigation of this problem is closely related to some combinatorial properties of covers and leads to the concept of perfect refinability, the latter being used, e.g., to obtain an intrinsic description of uniform real extensors. Several interesting classes of uniform spaces are characterized by special types of in-between theorems. As examples of applications we show that the usual in-between theorems in topology and their generalizations, as well as some important methods of construction of derivatives of real functions, follow easily from the general results.
A strong Stieltjes moment problem
William B.
Jones;
W. J.
Thron;
Haakon
Waadeland
503-528
Abstract: This paper is concerned with double sequences of complex numbers $ C\, = \,\{ {c_n}\} _{ - \infty }^\infty$ and with formal Laurent series ${L_0}(C)\, = \,\Sigma _1^\infty \, - \,{c_{ - m}}{z^m}$ and ${L_\infty }(C)\, = \,\Sigma _0^\infty \,{c_m}{z^{ - m}}$ generated by them. We investigate the following related problems: (1) Does there exist a holomorphic function having ${L_0}(C)$ and $ {L_\infty }(C)$ as asymptotic expansions at $z\, = \,0$ and $ z\, = \,\infty$, respectively? (2) Does there exist a real-valued bounded, monotonically increasing function $\psi (t)$ with infinitely many points of increase on $[0,\,\infty )$ such that, for every integer n, ${c_n}\, = \,\int_0^\infty {{{( - t)}^n}\,d\psi (t)}$? The latter problem is called the strong Stieltjes moment problem. We also consider a modified moment problem in which the function $\psi (t)$ has at most a finite number of points of increase. Our approach is made through the study of a special class of continued fractions (called positive T-fractions) which correspond to ${L_0}(C)$ at $z\, = \,0$ and $ {L_\infty }(C)$ at $z\, = \,\infty$. Necessary and sufficient conditions are given for the existence of these corresponding continued fractions. It is further shown that the even and odd parts of these continued fractions always converge to holomorphic functions which have $ {L_0}(C)$ and ${L_\infty }(C)$ as asymptotic expansions. Moreover, these holomorphic functions are shown to be represented by Stieltjes integral transforms whose distributions $ {\psi ^{(0)}}(t)$ and ${\psi ^{(1)}}(t)$ solve the strong Stieltjes moment problem. Necessary and sufficient conditions are given for the existence of a solution to the strong Stieltjes moment problem. This moment problem is shown to have a unique solution if and only if the related continued fraction is convergent. Finally it is shown that the modified moment problem has a unique solution if and only if there exists a terminating positive T-fraction that corresponds to both ${L_0}(C)$ and ${L_\infty }(C)$. References are given to other moment problems and to investigations in which negative, as well as positive, moments have been used.
On the Wall finiteness obstruction for the total space of certain fibrations
Hans J.
Munkholm;
Erik Kjaer
Pedersen
529-545
Abstract: The problem of computing the Wall finiteness obstruction for the total space of a fibration $ p:\,E\, \to \,B$ in terms of that for the base and homological data of the fiber has been considered by D. R. Anderson and by E. K. Pedersen and L. R. Taylor. We generalize their results and show how the problem is related to the algebraically defined transfer map ${\varphi ^{\ast}}:\,{K_0}({\textbf{Z}}{\pi _1}(B))\, \to \,{K_0}({\textbf{Z}}{\pi _1}(E))$, $\varphi \, = \,{p_{\ast}}:\,{\pi _1}(E)\, \to \,{\pi _1}(B)$, whenever the latter is defined.
Canonical embeddings
J.
Morrow;
H.
Rossi
547-565
Abstract: In this paper the authors compare the embedding of a compact Riemann surface in its tangent bundle to the embedding as the diagonal in the product. These embeddings are proved to be first, but not second, order equivalent. The embedding of a hyperelliptic curve in its tangent bundle is described in an explicit way. Although it is not possible to be so explicit in the other cases, it is shown that in all cases, if the Riemann surface R has genus greater than two, then the blowdown of the zero section of the tangent bundle and the blowdown of the diagonal in the product have the same Hilbert polynomial.
Derivations on algebras of unbounded operators
Atsushi
Inoue;
Shôichi
Ota
567-577
Abstract: This paper is a study of derivations on unbounded operator algebras in connection with those in operator algebras. In particular we study spatiality of derivations in several situations. We give the characterization of derivations on general *-algebras by using positive linear functionals. We also show that a derivation with some range-property on a left $E{W^\char93 }$-algebra induced by an unbounded Hilbert algebra is strongly implemented by an operator which belongs to an algebra of measurable operators.
The Rayleigh-Schr\"odinger expansion of the Gibbs state of a classical Heisenberg ferromagnet
William G.
Faris
579-587
Abstract: The equilibrium Gibbs state of a classical Heisenberg ferromagnet is a probability measure on an infinite product of spheres. The Kirkwood-Salsburg equations may be iterated to produce a convergent high temperature expansion of this measure about a product measure. Here we show that this expansion may also be obtained as the Rayleigh-Schrödinger expansion of the ground state eigenvector of a differential operator. The operator describes a Markovian time evolution of the ferromagnet.
Chaotic behavior in piecewise continuous difference equations
James P.
Keener
589-604
Abstract: A class of piecewise continuous mappings with positive slope, mapping the unit interval into itself is studied. Families of 1-1 mappings depending on some parameter have periodic orbits for most parameter values, but have an infinite invariant set which is a Cantor set for a Cantor set of parameter values. Mappings which are not 1-1 exhibit chaotic behavior in that the asymptotic behavior as measured by the rotation number covers an interval of values. The asymptotic behavior depends sensitively on initial data in that the rotation number is either a nowhere continuous function of initial data, or else it is a constant on all but a Cantor set of the unit interval.