Laplace operators and the $\mathfrak{h}$ module structure of certain cohomology groups
Floyd L.
Williams
1-57
Abstract: Let $\mathfrak{n}$ be the maximal nilpotent ideal of a Borel subalgebra of a complex semisimple Lie algebra $\mathfrak{g}$. Under the adjoint action $ \mathfrak{n},\mathfrak{g}/\mathfrak{n}$, and $ \mathfrak{n}$) are $\mathfrak{n}$ modules. Laplace operators for these three modules are computed by techniques which extend those introduced by B. Kostant in [6]. The kernels of these operators are then determined and, in view of the existence of a Hodge decomposition, the detailed structure of the first degree cohomology groups of $\mathfrak{n}$ with coefficients in $ \mathfrak{n},\mathfrak{g}/\mathfrak{n}$, and $\mathfrak{h}$ of $\mathfrak{g}$.
Periodic solutions of $x'' + g(x) + \mu h(x) = 0$
G. J.
Butler;
H. I.
Freedman
59-74
Abstract: Necessary and sufficient conditions for $ x'' + f(x) = 0$ to admit at least one nontrivial periodic solution are given. The results are applied to
Modules over quadratic and quaternion rings and transformations of quadratic forms
Bart
Rice
75-86
Abstract: A study is made of transformations carrying certain quadratic and quaternary quadratic forms into multiples of themselves, and it is shown how these are related to the study of modules over quadratic and quaternion rings. Special automorphic transformations of n-ary quadratic forms may also exhibit a structure like those in the quadratic and quaternary cases.
Finite groups with a proper $2$-generated core
Michael
Aschbacher
87-112
Abstract: H. Bender's classification of finite groups with a strongly embedded subgroup is an important tool in the study of finite simple groups. This paper proves two theorems which classify finite groups containing subgroups with similar but somewhat weaker embedding properties. The first theorem, classifying the groups of the title, is useful in connection with signalizer functor theory. The second theorem classifies a certain subclass of the class of finite groups possessing a permutation representation in which some involution fixes a unique point.
Formal groups and Hopf algebras over discrete rings
Robert A.
Morris;
Bodo
Pareigis
113-129
Abstract: A theory of formal schemes and groups over abitrary rings is presented. The flat formal schemes in this theory have coalgebras of distributions which behave in the usual way. Frobenius and Verschiebung maps are studied.
Subspaces of the nonstandard hull of a normed space
C. Ward
Henson;
L. C.
Moore
131-143
Abstract: Normed spaces which are isomorphic to subspaces of the nonstandard hull of a given normed space are characterized. As a consequence it is shown that a normed space is B-convex if and only if the nonstandard hull contains no subspace isomorphic to ${l_1}$ and a Banach space is super-reflexive if and only if the nonstandard hull is reflexive. Also, embeddings of second dual spaces into the nonstandard hull are studied. In particular, it is shown that the second dual space of a normed space E is isometric to a complemented subspace of the nonstandard hull of E.
Symplectic homogeneous spaces
Bon Yao
Chu
145-159
Abstract: It is proved in this paper that for a given simply connected Lie group G with Lie algebra $ \mathfrak{g}$, every left-invariant closed 2-form induces a symplectic homogeneous space. This fact generalizes the results in [7] and [12] that if ${H^1}(\mathfrak{g}) = {H^2}(\mathfrak{g}) = 0$, then the most general symplectic homogeneous space covers an orbit in the dual of the Lie algebra $\mathfrak{g}$. A one-to-one correspondence can be established between the orbit space of equivalent classes of 2-cocycles of a given Lie algebra and the set of equivalent classes of simply connected symplectic homogeneous spaces of the Lie group. Lie groups with left-invariant symplectic structure cannot be semisimple; hence such groups of dimension four have to be solvable, and connected unimodular groups with left-invariant symplectic structure are solvable [4].
Sufficient sets for some spaces of entire functions
Dennis M.
Schneider
161-180
Abstract: B. A. Taylor [13] has shown that the lattice points in the plane form a sufficient set for the space of entire functions of order less than two. We obtain a generalization of this result to functions of several variables and to more general spaces of entire functions. For example, we prove that if $S \subset {{\mathbf{C}}^n}$ such that $d(z,S) \leq \operatorname{const}\vert z{\vert^{1 - \rho /2}}$ for all $z \in {{\mathbf{C}}^n}$, then S is a sufficient set for the space of entire functions on ${{\mathbf{C}}^n}$ of order less than $ \rho$. The proof involves estimating the growth rate of an entire function from its growth rate on S. We also introduce the concept of a weakly sufficient set and obtain sufficient conditions for a set to be weakly sufficient. We prove that sufficient sets are weakly sufficient and that certain types of effective sets [8] are weakly sufficient.
Polyanalytic functions with exceptional values
P.
Krajkiewicz
181-210
Abstract: Let $ f(z) = \sum\nolimits_{k = 0}^n {{{\bar z}^k}{f_k}(z)} $ where the functions ${f_0},{f_1}, \cdots ,{f_n}$ are analytic on some annular neighborhood A of the point $\infty$ and $ {f_n} \equiv 1$ on A and z denotes the complex conjugate of z. If f does not vanish on A it is shown that the functions ${f_0},{f_1}, \cdots ,{f_{n - 1}}$ have a nonessential isolated singularity at the point infinity.
Spectrum and direct integral
Edward A.
Azoff
211-223
Abstract: Let $T = \smallint _Z^ \oplus T(\mathcal{E})$ be a direct integral of Hilbert space operators, and equip the collection $\mathcal{C}$ of compact subsets of C with the Hausdorff metric topology. Consider the [set-valued] function sp which associates with each $\mathcal{E} \in Z$ the spectrum of $T(\mathcal{E})$. The main theorem of this paper states that sp is measurable. The relationship between $\sigma (T)$ and $\{ \sigma (T(\mathcal{E}))\}$ is also examined, and the results applied to the hyperinvariant subspace problem. In particular, it is proved that if $ \sigma (T(\mathcal{E}))$ consists entirely of point spectrum for each $\mathcal{E} \in Z$, then either T is a scalar multiple of the identity or T has a hyperinvariant subspace; this generalizes a theorem due to T. Hoover.
Compact sets definable by free $3$-manifolds
W. H.
Row
225-244
Abstract: Shape conditions are given that force a compactum (i.e., a compact metric space) embedded in the interior of a nonclosed, piecewise-linear 3-manifold to have arbitrarily close, compact, polyhedral neighborhoods each component of which is a 3-manifold with free fundamental group (i.e., to be definable by free 3-manifolds). For compact, connected ANR's these conditions reduce to the criterion of having a free fundamental group. Additional conditions are given that insure definability by handlebodies or cubes-with-handles. An embedding of Menger's universal 1-dimensional curve in Euclidean 3-space is shown to have the property that all tame surfaces, separating in 3-space a fixed pair of points, cannot be adjusted (by a small space homeomorphism) to intersect the embedded curve in a 0-dimensional set.
Nonattainability of a set by a diffusion process
Avner
Friedman
245-271
Abstract: Consider a system of n stochastic differential equations $ d\xi = b(\xi )dt + \sigma (\xi )dw$. Let M be a k-dimensional submanifold in $ {R^n},k \leq n - 1$. For $x \in M$, denote by $ d(x)$ the rank of $\sigma {\sigma ^ \ast }$ restricted to the linear space of all normals to M at x. It is proved that if $ d(x) \geq 2$ for all $x \in M$, then $ \xi (t)$ does not hit M at finite time, given $\xi (0) \notin M$, i.e., M is nonattainable. The cases $ d(x) \geq 1,d(x) \geq 0$ are also studied.
Constructions in algebra
A.
Seidenberg
273-313
Abstract: It is shown how to construct a primary decomposition and to find the associated prime ideals of a given ideal in a polynomial ring. This is first done from a classical, and then from a strictly constructivist, point of view.
Commutative twisted group algebras
Harvey A.
Smith
315-326
Abstract: A twisted group algebra $ {L^1}(A,G;T,\alpha )$ is commutative iff A and G are, T is trivial and $\alpha$ is symmetric: $\alpha (\gamma ,g) = \alpha (g,\gamma )$. The maximal ideal space ${L^1}(A,G;\alpha )\hat \emptyset$ of a commutative twisted group algebra is a principal $G\hat \emptyset$ bundle over $A\hat \emptyset$. A class of principal $G\hat \emptyset$ bundles over second countable locally compact M is defined which is in 1-1 correspondence with the (isomorphism classes of) ${C_\infty }(M)$-valued commutative twisted group algebras on G. If G is finite only locally trivial bundles can be such duals, but in general the duals need not be locally trivial.
Wreath products and existentially complete solvable groups
D.
Saracino
327-339
Abstract: It is known that the theory of abelian groups has a model companion but that the theory of groups does not. We show that for any fixed $n \geq 2$ the theory of groups solvable of length $ \leq n$ has no model companion. For the metabelian case $(n = 2)$ we prove the stronger result that the classes of finitely generic, infinitely generic, and existentially complete metabelian groups are all distinct. We also give some algebraic results on existentially complete metabelian groups.
General theory of the factorization of ordinary linear differential operators
Anton
Zettl
341-353
Abstract: The problem of factoring the general ordinary linear differential operator $Ly = {y^{(n)}} + {p_{n - 1}}{y^{(n - 1)}} + \cdots + {p_0}y$ into products of lower order factors is studied. The factors are characterized completely in terms of solutions of the equation $Ly = 0$ and its adjoint equation ${L^ \ast }y = 0$. The special case when L is formally selfadjoint of order $n = 2k$ and the factors are of order k and adjoint to each other reduces to a well-known result of Rellich and Heinz: $L = {Q^ \ast }Q$ if and only if there exist solutions ${y_1}, \cdots ,{y_k}$ of $Ly = 0$ satisfying $W({y_1}, \cdots ,{y_k}) \ne 0$ and $[{y_i};{y_j}] = 0$ for $i,j = 1, \cdots ,k$; where [ ; ] is the Lagrange bilinear form of L.
Coherent extensions and relational algebras
Marta C.
Bunge
355-390
Abstract: The notion of a lax adjoint to a 2-functor is introduced and some aspects of it are investigated, such as an equivalent definition and a corresponding theory of monads. This notion is weaker than the notion of a 2-adjoint (Gray) and may be obtained from the latter by weakening that of 2functor and replacing the adjointness equations by adding 2-cells satisfying coherence conditions. Lax monads are induced by and resolve into lax adjoint pairs, the latter via 2-categories of lax algebras. Lax algebras generalize the relational algebras of Barr in the sense that a relational algebra for a monad in Sets is precisely a lax algebra for the lax monad induced in Rel. Similar considerations allow us to recover the T-categories of Burroni as well. These are all examples of lax adjoints of the ``normalized'' sort and the universal property they satisfy can be expressed by the requirement that certain generalized Kan extensions exist and are coherent. The most important example of relational algebras, i.e., topological spaces, is analysed in this new light also with the purpose of providing a simple illustration of our somewhat involved constructions.
Prime and search computability, characterized as definability in certain sublanguages of constructible $L\sb{\omega }{}\sb{1,\omega }$
Carl E.
Gordon
391-407
Abstract: The prime computable (respectively, search computable) relations of an arbitrary mathematical structure are shown to be those relations R such that both R and its complement are definable by disjunctions of recursively enumerable sets of quantifier free (respectively, existential) formulas of the first order language for the structure. The prime and search computable functions are also characterized in terms of recursive sequences of terms and formulas of this language.
Complementation for right ideals in generalized Hilbert algebras
John
Phillips
409-417
Abstract: Let $\mathfrak{A}$ be a generalized Hilbert algebra and let $ \mathcal{J}$ be a closed right ideal of $ \mathfrak{A}$. Let ${\mathcal{J}^ \bot }$ denote the pre-Hilbert space orthogonal complement of $\mathcal{J}$ in $ \mathfrak{A}$. The problem investigated in this paper is: for which algebras $\mathfrak{A}$ is it true that $ \mathfrak{A} = \mathcal{J} \oplus {\mathcal{J}^ \bot }$ for every closed right ideal $ \mathcal{J}$ of $\mathfrak{A}$? In the case that $\mathfrak{A}$ is achieved, a slightly stronger property is characterized and these characterizations are then used to investigate some interesting examples.