Lie group representations and harmonic polynomials of a matrix variable
Tuong
Ton-That
1-46
Abstract: The first part of this paper deals with problems concerning the symmetric algebra of complex-valued polynomial functions on the complex vector space of n by k matrices. In this context, a generalization of the so-called ``classical separation of variables theorem'' for the symmetric algebra is obtained. The second part is devoted to the study of certain linear representations, on the above linear space (the symmetric algebra) and its subspaces, of the complex general linear group of order k and of its subgroups, namely, the unitary group, and the real and complex special orthogonal groups. The results of the first part lead to generalizations of several well-known theorems in the theory of group representations. The above representation, of the real special orthogonal group, which arises from the right action of this group on the underlying vector space (of the symmetric algebra) of matrices, possesses interesting properties when restricted to the Stiefel manifold. The latter is defined as the orbit (under the action of the real special orthogonal group) of the n by k matrix formed by the first n row vectors of the canonical basis of the k-dimensional real Euclidean space. Thus the last part of this paper is involved with questions in harmonic analysis on this Stiefel manifold. In particular, an interesting orthogonal decomposition of the complex Hilbert space consisting of all square-integrable functions on the Stiefel manifold is also obtained.
Inseparable finite solvable groups
Homer
Bechtell
47-60
Abstract: A finite group is called inseparable if the only proper normal subgroup over which it splits is the identity element. The E-residual, for the formation E of groups in which all Sylow subgroups are elementary abelian, appears to control the action of splitting. In this article, inseparable solvable groups are identified that have a metacyclic Fitting subgroup and the E-residual a p-group.
Generalized and classical solutions of the nonlinear stationary Navier-Stokes equations
Victor L.
Shapiro
61-79
Abstract: New regularity results in domains of Euclidean 3-space are established for the generalized solutions of the nonlinear stationary Navier-Stokes equations in terms of Dini criteria on the external force.
Units and one-sided units in regular rings
Gertrude
Ehrlich
81-90
Abstract: A ring R is unit regular if for every $a \in R$, there is a unit $x \in R$ such that $axa = a$, and one-sided unit regular if for every $a \in R$, there is a right or left invertible element $x \in R$ such that $axa = a$. In this paper, unit regularity and one-sided unit regularity are characterized within the lattice of principal right ideals of a regular ring R (Theorem 3). If M is an A-module and $R = {\text{End}_A}$ M is a regular ring, then R is unit regular if and only if complements of isomorphic summands of M are isomorphic, and R is one-sided unit regular if and only if complements of isomorphic summands of M are comparable with respect to the relation ``is isomorphic to a submodule of'' (Theorem 2). A class of modules is given for whose endomorphism rings it is the case that regularity in conjunction with von Neumann finiteness is equivalent to unit regularity. This class includes all abelian torsion groups and all nonreduced abelian groups with regular endomorphism rings.
Some inequalities for polynomials with a prescribed zero
Q. I.
Rahman;
G.
Schmeisser
91-103
Abstract: Let $f(z)$ be a polynomial of degree n. Given that $f(z)$ has a zero on the circle $\vert z\vert = \rho \;(0 < \rho < \infty )$, we estimate $\vert f(0)\vert$ and $\displaystyle {({(2\pi )^{ - 1}}\int _0^{2\pi }\vert f({e^{i\theta }}){\vert^2}d\theta )^{1/2}}$ in terms of $ {\max _{\vert z\vert = 1}}\vert f(z)\vert$. We also consider some other related problems.
The quasi-orbit space of continuous $C\sp{\ast} $-dynamical systems
Hiroshi
Takai
105-113
Abstract: Let $(A,G,\alpha )$ be a separable continuous $ {C^\ast}$-dynamical system. Suppose G is amenable and $\alpha$ is free on the dual  of A. Then the quasi-orbit space $ {({\text{Prim}}\;A/\alpha )^ \sim }$ of the primitive ideal space Prim A of A by $\alpha$ is homeomorphic to the induced primitive ideal space which is dense in the primitive ideal space Prim $ {C^\ast}(A;\alpha )$ of the ${C^\ast}$-crossed product ${C^\ast}(A;\alpha )$ of A by $\alpha$.
Some special decompositions of $E\sp{3}$
Charles D.
Bass
115-130
Abstract: A great deal of attention has been given to the question: which upper semicontinuous decompositions of ${E^3}$ into pointlike continua give ${E^3}$. It has recently been determined that some decompositions of ${E^3}$ into points and straight line segments give decomposition spaces which are topologically distinct from ${E^3}$. In this paper we apply a new condition to the set of nondegenerate elements of a decomposition which enables one to conclude that the resulting decomposition space is homeomorphic to $ {E^3}$.
An order topology in ordered topological vector spaces
Lyne H.
Carter
131-144
Abstract: An order topology $ \Omega$ that can be defined on any partially-ordered space has as its closed sets those that contain the (o)-limits of all their (o)-convergent nets. In this paper we study the situation in which a topological vector space with a Schauder basis is ordered by the basis cone. In a Fréchet space $(E,\tau )$, we obtain necessary and sufficient conditions both for $ \tau \subset \Omega$ and for $\tau = \Omega $. Characterizations of (o)- and $\Omega$-convergence and of $\Omega$-closed sets are obtained. The equality of the order topology with the strong topology in certain dual Banach spaces is related to weak sequential completeness through the concept of a shrinking basis.
Convolution, differential equations, and entire functions of exponential type
Dale H.
Mugler
145-187
Abstract: The Whittaker-Shannon interpolation formula, or ``cardinal series", is a special case of the more general linear integro-differential equation with constant complex coefficients $\Sigma _{n = 0}^\infty {a_n}{f^{(n)}}(z) = \smallint f(z - t)d\mu (t)$ where the integral is taken over the whole real line with respect to the measure $\mu$. In this study, I show that many of these equations provide representations for particular classes of entire functions of exponential type. That is, every function in the class satisfies the equation and conversely every solution of the equation is a member of the class of functions. When the measure in the convolution integral above is chosen to be discrete, a particular form of the above type of equation is an equation of periodicity $f(z) = f(z + \tau )$. Following an extensive treatment of the general equation written above, the study concludes by offering a generalization in terms of these convolution equations of a classical theorem in complex analysis concerning periodic entire functions.
Cylindric algebras of first-order languages
Dale
Myers
189-202
Abstract: We show when two countable first-order languages have isomorphic cylindric algebras.
On the topological extension to the boundary of biholomorphic maps in $C\sp{n}$
R. Michael
Range
203-216
Abstract: Let $F:{D_1} \to {D_2}$ be a biholomorphic map between bounded domains in $ {{\mathbf{C}}^n}$ with piecewise smooth strictly pseudoconvex boundaries. It is shown that F is Hölder continuous of some positive order, and hence F extends to a homeomorphism of the closures of the domains. This generalizes recent results of G. M. Henkin and N. Vormoor for domains with smooth strictly pseudoconvex boundary.
Cartan subspaces of symmetric Lie algebras
J.
Lepowsky;
G. W.
McCollum
217-228
Abstract: A symmetric Lie algebra is defined, following J. Dixmier, to be a Lie algebra $\mathfrak{g}$ with a decomposition $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ such that $ \mathfrak{k}$ is a subalgebra of $ \mathfrak{g},[\mathfrak{k},\mathfrak{p}] \subset \mathfrak{p}$ and $[\mathfrak{p},\mathfrak{p}] \subset \mathfrak{k}$. A definition of Cartan subspace of a symmetric Lie algebra is given, and a theory is presented which parallels the standard theory of Cartan subalgebras of Lie algebras, and which generalizes the classical results for real and complex semisimple symmetric Lie algebras.
A generalization of H. Weyl's ``unitary trick''
J.
Lepowsky
229-236
Abstract: H. Weyl's ``unitary trick'' is generalized to the context of semisimple symmetric Lie algebras with Cartan subspaces, over fields of characteristic zero. As an illustration of its usefulness, the result is used to transfer to characteristic zero an important theorem in the representation theory of real semisimple Lie algebras.
Linear factorization of conical polynomials over certain nonassociative algebras
J.
Lepowsky
237-248
Abstract: Conical polynomials are defined as certain polynomials in quadratic elements of the universal enveloping algebra of a semisimple symmetric Lie algebra over a field of characteristic zero. These polynomials were used in an earlier paper to describe the conical vectors in certain induced modules. Here it is shown that when the base field is extended to a certain type of nonassociative algebra, the conical polynomials can be factored ``linearly". One such nonassociative algebra is discussed in detail--an (alternative) composition algebra intimately related to the structure of the Lie algebra and studied earlier by B. Kostant in the context of real semisimple Lie algebras. The linear factorization leads in a later paper to an extension of the earlier work on conical vectors in induced modules.
General position maps for topological manifolds in the ${2\over 3}$rds range
Jerome
Dancis
249-266
Abstract: For each proper map f of a topological m-manifold M into a topological q-manifold Q, $m \leqslant (2/3)q - 1/3$, we build an approximating map g such that the set of singularities S of g is a locally finite simplicial $(2m - q)$-complex locally tamely embedded in M, $ g(S)$ is another locally finite complex $g\vert:S \twoheadrightarrow g(S)$ is a piecewise linear map and g is a locally flat embedding on the complement of S. Furthermore if $f\vert\partial M$ is a locally flat embedding then we construct g so that it agrees with f on $\partial M$ even when $f(\partial M)$ meets $\operatorname{Int} Q \cap f({\operatorname{Int}}\;M)$. In addition we present two other general position lemmas. Also, we show that given two codimension $\geqslant 3$ locally flat topological submanifolds M and V of a topological manifold Q, $\dim \;M + \dim \;V - \dim \;Q \leqslant 3$, then we can move M so that M and V are transverse in Q.
Almost isolated spectral parts and invariant subspaces
C. R.
Putnam
267-277
Abstract: Let T be an operator with spectrum $ \sigma (T)$ on a Hilbert space. A compact subset E of $\sigma (T)$ is called a disconnected part of $\sigma (T)$ if, for some bounded open set A, E is the closure of $ \sigma (T) \cap A$ and $\sigma (T) - E$ is the union of the isolated parts of $\sigma (T)$ lying completely outside the closure of A. The set E is called an almost isolated part of $ \sigma (T)$ if, in addition, the set A can be chosen so as to have a rectifiable boundary $ \partial A$ on which the subset $ \sigma (T) \cap \partial A$ has arc length measure 0. The following results are obtained. If T is subnormal and if E is a disconnected part of $ \sigma (T)$ then there exists a reducing subspace $ \mathfrak{M}$ of T for which $ \sigma (T\vert\mathfrak{M}) = E$. If ${T^\ast}$ is hyponormal and if E is an almost isolated part of $ \sigma (T)$ then there exists an invariant subspace $ \mathfrak{M}$ of T for which $ \sigma (T\vert\mathfrak{M}) = E$. An example is given showing that if T is arbitrary then an almost isolated part of $\sigma (T)$ need not be the spectrum of the restriction of T to any invariant subspace.
An index theorem for $p$-adic differential operators
A.
Adolphson
279-293
Abstract: A system of first order linear differential operators satisfying conditions arising naturally in geometry (rational function coefficients, regular singularities, non-Liouville exponents) is considered. It is shown that the index of the system on certain spaces of holomorphic functions can be calculated by restricting to a subspace of rational functions. This is applied to obtain an explicit formula for the index of a single kth order linear differential operator.
A product formula for generalizations of the Kervaire invariant
Edgar H.
Brown
295-311
Abstract: Formulas are developed for the Arf invariant of the product of two manifolds in terms of invariants of the factors. If the Wu orientations are carefully chosen the formula is $ \sigma (M \times N) = \sigma (M)\sigma (N)$.
The generalized Fredholm operators
Kung Wei
Yang
313-326
Abstract: Let X, Y be Banach spaces over either the real field or the complex field. A continuous linear operator will be called a generalized Fredholm operator if $ T(X)$ is closed in Y, and Ker T and Coker T are reflexive Banach spaces. A theory similar to the classical Fredholm theory exists for the generalized Fredholm operators; and the similarity brings out the correspondence: Reflexive Banach spaces $\leftrightarrow$ finite-dimensional spaces, weakly compact operators $\leftrightarrow$ compact operators, generalized Fredholm operators $ \leftrightarrow$ Fredholm operators, Tauberian operators with closed range $\leftrightarrow $ semi-Fredholm operators.
On the uniqueness of solutions to hyperbolic boundary value problems
C. C.
Travis
327-336
Abstract: The paper is concerned with the uniqueness of solutions to non-well-posed hyperbolic boundary value problems. Both regular and singular boundary value problems are considered. For the singular problem a class of boundary conditions is considered that has not appeared in the literature before in connection with this problem.
Piecewise linear bundles in the metastable range
Kenneth C.
Millett
337-350
Abstract: For numerable vector bundles a nonzero section determines a unique trivial line subbundle containing the section and this subbundle is a direct summand of the bundle. The main result, a consequence of concordance-isotopy theory, states that in the metastable range a nonzero section to a piecewise linear ${{\mathbf{R}}^n}$ bundle determines a unique trivial line subbundle and that this is the best possible result. This fact is then compared with the known failure of the summand property below the stable range.
Signature of links
Louis H.
Kauffman;
Laurence R.
Taylor
351-365
Abstract: Let L be an oriented tame link in the three sphere ${S^3}$. We study the Murasugi signature, $\sigma (L)$, and the nullity, $\eta (L)$. It is shown that $\sigma (L)$ is a locally flat topological concordance invariant and that $\eta (L)$ is a topological concordance invariant (no local flatness assumption here). Known results about the signature are re-proved (in some cases generalized) using branched coverings.
Some locally convex spaces of continuous vector-valued functions over a completely regular space and their duals
A.
Katsaras
367-387
Abstract: The strict, superstrict and the $ {\beta _F}$ topologies are defined on a space A of continuous functions from a completely regular space into a Banach space E. Properties of these topologies are discussed and the corresponding dual spaces are identified with certain spaces of operator-valued measures. In case E is a Banach lattice, A becomes a lattice under the pointwise ordering and the strict and superstrict duals of A coincide with the spaces of all $\tau$-additive and all $\sigma$-additive functionals on A respectively.
Finite groups as isometry groups
D.
Asimov
388-390
Abstract: We show that given any finite group G of cardinality $k + 1$, there is a Riemannian sphere ${S^{k - 1}}$ (imbeddable isometrically as a hypersurface in $ {{\mathbf{R}}^k}$) such that its full isometry group is isomorphic to G. We also show the existence of a finite metric space of cardinality $k(k + 1)$ whose full isometry group is isomorphic to G.
Erratum to: ``A constructive ergodic theorem'' (Trans. Amer. Math. Soc. {\bf 164} (1972), 115--137)
J. A.
Nuber
393