Generalized hypercomplex function theory
Robert P.
Gilbert;
Gerald
Hile
1-29
Abstract: Lipman Bers and Ilya Vekua extended the concept of an analytic function by considering the distributional solutions of elliptic systems of two equations with two unknowns and two independent variables. These solutions have come to be known as generalized (or pseudo) analytic functions. Subsequently, Avron Douglis introduced an algebra and a class of functions which satisfy (classically) the principal part of an elliptic system of 2r equations with 2r unknowns and two independent variables. In Douglis' algebra these systems of equations can be represented by a single ``hypercomplex'' equation. Solutions of such equations are termed hyperanalytic functions. In this work, the class of functions studied by Douglis is extended in a distributional sense much in the same way as Bers and Vekua extended the analytic functions. We refer to this extended class of functions as the class of generalized hyperanalytic functions.
Separable topological algebras. I
Michael J.
Liddell
31-59
Abstract: Let A be a complete topological algebra with identity and B a subalgebra of the center of A. A notion of relative topological tensor product ${\hat \otimes _B}$ for topological A modules and the resultant relative homology theory are introduced. Algebras of bidimension zero in this sense are called separable relative to B. Structure theorems are proved for such algebras under various topological assumptions on the algebra and its maximal ideal space.
$T$-faithful subcategories and localization
John A.
Beachy
61-79
Abstract: For any additive functor from a category of modules into an abelian category there is a largest Giraud subcategory for which the functor acts faithfully on homomorphisms into the subcategory. It is the largest Giraud subcategory into which the functor reflects exact sequences, and under certain conditions it is just the largest Giraud subcategory on which the functor acts faithfully. If the functor is exact and has a right adjoint, then the subcategory is equivalent to the quotient category determined by the kernel of the functor. In certain cases, the construction can be applied to a Morita context in order to obtain a recent theorem of Mueller. Similarly, the functor defines a certain reflective subcategory and an associated radical, which is a torsion radical in case the functor preserves monomorphisms. Certain results concerning this radical, when defined by an adjoint functor, can be applied to obtain two theorems of Morita on balanced modules.
An asymptotic formula in adele Diophantine approximations
Melvin M.
Sweet
81-96
Abstract: In this paper an asymptotic formula is found for the number of solutions of a system of linear Diophantine inequalities defined over the ring of adeles of an algebraic number field. The theorem proved is a generalization of results of S. Lang and W. Adams.
First order differential closures of certain partially ordered fields
Joseph E.
Turcheck
97-114
Abstract: First order algebraic differential equations (a.d.e.'s) are considered in the setting of an abstract differential field with an abstract order relation, whose properties mirror those of the usual asymptotic dominance relations of analysis. An abstract existence theorem, for such equations, is proved by constructing an extension of both the differential field and the abstract order relation. As a consequence, a first order differential closure theorem, for those differential fields with order relations which we consider, is obtained. The closure theorem has corollaries which are important to the asymptotic theory of a.d.e.'s and have application to a.d.e.'s with coefficients meromorphic in a sector of the complex plane.
Linear ordinary differential equations with Laplace-Stieltjes transforms as coefficients
James
D’Archangelo
115-145
Abstract: The n-dimensional differential system $n \times n$ complex matrix and $ A(t)$ is an $n \times n$ matrix whose entries $a(t)$ are complex valued functions which are representable as absolutely convergent Laplace-Stieltjes transforms, $\smallint _0^\infty {e^{ - st}}d\alpha (s)$, for $t > 0$. The determining functions, $\alpha (s)$, are C valued, locally of bounded variation on $ [0,\infty )$, continuous from the right, and $\alpha ( + 0) = \alpha (0) = 0$. Sufficient conditions on the determining functions are found which assure the existence of solutions of certain specified forms involving absolutely convergent Laplace-Stieltjes transforms for $t > 0$ and which behave asymptotically like certain solutions of the nonperturbed equation $\Pi _{i = 1}^m{(D - {r_i})^{e(i)}}z + \Sigma _{j = 0}^{n - 1}{a_j}(t){D^j}z = 0$, where ${r_i} \in {\mathbf{C}}$ and the ${a_j}(t)$ are like $a(t)$ above for $t > 0$.
Harmonic analysis and centers of group algebras
J.
Liukkonen;
R.
Mosak
147-163
Abstract: The purpose of this paper is to present some results of harmonic analysis on the center of the group algebra $Z({L^1}(G))$ where G is a locally compact group. We prove that $ Z({L^1}(G))$ is a regular, Tauberian, symmetric Banach $^\ast$-algebra and contains a bounded approximate identity. Wiener's generalized Tauberian theorem is therefore applicable to $ Z({L^1}(G))$. These results complement those of I. E. Segal relating to the group algebra of locally compact abelian and compact groups. We also prove that if G contains a compact normal subgroup K such that G/K is abelian, then $ Z({L^1}(G))$ satisfies the condition of Wiener-Ditkin, so that any closed set in its maximal ideal space whose boundary contains no perfect subset is a set of spectral synthesis. We give an example of a general locally compact group for which $Z({L^1}(G))$ does not satisfy the condition of Wiener-Ditkin.
Almost equicontinuous transformation groups
Ping Fun
Lam
165-200
Abstract: A class of transformation groups called strictly almost equicontinuous transformation groups is studied. Manifolds which carry such transformation groups are determined. Applications to related classes are obtained.
Pairs of domains where all intermediate domains are Noetherian
Adrian R.
Wadsworth
201-211
Abstract: For Noetherian integral domains R and T with $R \subseteq T,(R,T)$ is called a Noetherian pair (NP) if every domain A, $R \subseteq A \subseteq T$, is Noetherian. When $\dim R = 1$ (Krull dimension) it is shown that the only NP's are those given by the Krull-Akizuki Theorem. For $\dim R \geq 2$, there is another type of NP besides the finite integral extension, namely $(R,\tilde R)$ where $\tilde R = \bigcap {\{ {R_P}\vert{\text{rk}}\;P \geq 2\} }$. Further, for every NP (R, T) with $\dim R \geq 2$ there is an integral NP extension B of R with $T \subseteq \tilde B$. In all known examples B can be chosen to be a finite integral extension of R. For such NP's it is shown that the NP relation is transitive. T may itself be an infinite integral extension R, though, and an example of this is given. It is unknown exactly which infinite integral extensions are NP's.
Henselian valued fields with prescribed value group and residue field
Linda
Hill
213-222
Abstract: A class of fields supplementary to the inertia field of a given henselian valued field is used to construct extensions of that field having prescribed value group and residue field. The extensions so-constructed are characterized, and their number investigated.
On the functional equation $f\sp{2}=e\sp{2\phi\sb{1}}+e\sp{2\phi\sb{2}}+e\sp{2\phi\sb{3}} $ and a new Picard theorem
Mark
Green
223-230
Abstract: By analogy with E. Borel's reduction of the classical Picard theorem to an analytic statement about linear relations among exponentials of entire functions, a new Picard theorem is proved by considering the functional relation ${f^2} = {e^{2{\phi _1}}} + {e^{2{\phi _2}}} + {e^{2{\phi _3}}}$ for entire functions. The analytic techniques used are those of Nevanlinna theory.
On a problem of Gronwall for Bazilevi\v c functions
John L.
Lewis
231-242
Abstract: Let $B(\alpha ,\beta ),\alpha $ positive, $\beta$ real, denote the class of normalized univalent Bazilevič functions in $K = \{ z:\vert z\vert < 1\}$ of type $\alpha ,\beta$. Let $B = { \cup _{\alpha ,\beta }}B(\alpha ,\beta )$. Let $\alpha ,0 \leq \alpha \leq 2$, and $\alpha ,0 < \alpha < \infty$, be fixed and suppose that $ f(z) = z + a{z^2} + \cdots$ is in $B(\alpha ,0)$. In this paper for given ${z_0} \in K$, the author finds a sharp upper bound for $\vert f({z_0})\vert$. Also, a sharp asymptotic bound is obtained for ${(1 - r)^2}{\max _{\vert z\vert = r}}\vert f(z)\vert$. Finally, a sharp asymptotic bound is found for ${(1 - r)^2}{\max _{\vert z\vert = r}}\vert f(z)\vert$ when f is in B with second coefficient a.
The convertibility of ${\rm Ext}\sp{n}\sb{R}(-,\,A)$
James L.
Hein
243-264
Abstract: Let R be a commutative ring and $\operatorname{Mod} (R)$ the category of R-modules. Call a contravariant functor $ F:\operatorname{Mod} (R) \to \operatorname{Mod} (R)$ convertible if for every direct system $ \{ {X_\alpha }\}$ in $ \operatorname{Mod} (R)$ there is a natural isomorphism $ \gamma :F(\mathop {\lim }\limits_ \to {X_\alpha }) \to \mathop {\lim }\limits_ \leftarrow F({X_\alpha })$. If A is in $\operatorname{Mod} (R)$ and n is a positive integer then ${\text{Ext}}_R^n( - ,A)$ is not in general convertible. The purpose of this paper is to study the convertibility of Ext, and in so doing to find out more about Ext as well as the modules A that make $ {\text{Ext}}_R^n( - ,A)$ convertible for all n. It is shown that $ {\text{Ext}}_R^n( - ,A)$ is convertible for all A having finite length and all n. If R is Noetherian then A can be Artinian, and if R is semilocal Noetherian then A can be linearly compact in the discrete topology. Characterizations are studied and it is shown that if A is a finitely generated module over the semilocal Noetherian ring R, then ${\text{Ext}}_R^1( - ,A)$ is convertible if and only if A is complete in the J-adic topology where J is the Jacobson radical of R. Morita-duality is characterized by the convertibility of ${\text{Ext}}_R^1( - ,R)$ when R is a Noetherian ring, a reflexive ring or an almost maximal valuation ring. Applications to the vanishing of Ext are studied.
Homogeneity and extension properties of embeddings of $S\sp{1}$ in $E\sp{3}$
Arnold C.
Shilepsky
265-276
Abstract: Two properties of embeddings of simple closed curves in ${E^3}$ are explored in this paper. Let $ {S^1}$ be a simple closed curve and $ f({S^1}) = S$ an embedding of ${S^1}$ in ${E^3}$. The simple closed curve S is homogeneously embedded or alternatively f is homogeneous if for any points p and q of S, there is an automorphism h of $ {E^3}$ such that $ h(S) = S$ and $ h(p) = q$. The embedding f or the simple closed curve S is extendible if any automorphism of S extends to an automorphism of ${E^3}$. Two classes of wild simple closed curves are constructed and are shown to be homogeneously embedded. A new example of an extendible simple closed curve is constructed. A theorem of H. G. Bothe about extending orientation-preserving automorphisms of a simple closed curve is generalized.
Fields of constants of integral derivations on a $p$-adic field
Henry W.
Thwing;
Nickolas
Heerema
277-290
Abstract: Let ${K_0}$ be a p-adic subfield of a p-adic field K with residue fields ${k_0} \subset k$. If ${K_0}$ is algebraically closed in K and k is finitely generated over ${k_0}$ then ${K_0}$ is the subfield of constants of an analytic derivation on K or equivalently, ${K_0}$ is the invariant subfield of an inertial automorphism of K. If (1) $ {k_0}$ is separably algebraically closed in k, (2) $[k_0^{{p^{ - 1}}} \cap k:{k_0}] < \infty$, and (3) k is not algebraic over $ {k_0}$ then there exists a p-adic subfield ${K_0}$ over ${k_0}$ which is algebraically closed in K. All subfields over ${k_0}$ are algebraically closed in K if and only if ${k_0}$ is algebraically closed in k. Every derivation on k trivial on ${k_0}$ lifts to a derivation on K trivial on ${K_0}$ if k is separable over $ {k_0}$. If k is finitely generated over ${k_0}$ the separability condition is necessary. Applications are made to invariant fields of groups of inertial automorphisms on p-adic fields and of their ramification groups.
Some mapping theorems
R. C.
Lacher
291-303
Abstract: Various mapping theorems are proved, culminating in the following result for mappings f from a closed $ (2k + 1)$-manifold M to another, N: If ``almost all'' point-inverses of f are strongly acyclic in dimensions less than k and if ``almost all'' point-inverses of f have Euler characteristic equal to one, then all but finitely many point-inverses are totally acyclic. (Here ``almost all'' means ``except on a zero-dimensional set in N".) More can be said when $k = 1$: If f is a monotone map between closed 3-manifolds and if the Euler characteristic of almost-all point-inverses is one, then all but finitely many point-inverses of f are cellular in M; consequently M is the connected sum of N and some other closed 3-manifold and f is homotopic to a spine map. Other results include an acyclicity criterion using the idea of ``nonalternating'' mapping and the following result for PL maps $\phi$ between finite polyhedra X and Y: If the Euler characteristic of each point-inverse of $\phi$ is the integer c then $\chi (X) = c\chi (Y)$.
Cohomology of nilradicals of Borel subalgebras
George F.
Leger;
Eugene M.
Luks
305-316
Abstract: Let $\mathfrak{N}$ be the maximal nilpotent ideal in a Borel subalgebra of a complex simple Lie algebra. The cohomology groups $ {H^1}(\mathfrak{N},\mathfrak{N}),{H^1}(\mathfrak{N},{\mathfrak{N}^\ast})$ and the $\mathfrak{N}$-invariant symmetric bilinear forms on $\mathfrak{N}$ are determined. The main result is the computation of $ {H^2}(\mathfrak{N},\mathfrak{N})$.
Regular elements in rings with involution
Charles
Lanski
317-325
Abstract: The purpose of this paper is to determine when a symmetric element, regular with respect to other symmetries, is regular in the ring. This result is true for simple rings, for prime rings with either Goldie chain condition, and for semiprime Goldie rings. Examples are given to show that these results are the best that can be hoped for.
Extremal problems in classes of analytic univalent functions with quasiconformal extensions
J. Olexson
McLeavey
327-343
Abstract: This work solves many of the classical extremal problems posed in the class of functions $ {\Sigma _{K(\rho )}}$, the class of functions in $\Sigma$ with $K(\rho )$-quasiconformal extensions into the interior of the unit disk where $K(\rho )$ is a piecewise continuous function of bounded variation on $ [r,1],0 \leq r < 1$. The approach taken is a variational technique and results are obtained through a limiting procedure. In particular, sharp estimates are given for the Golusin distortion functional, the Grunsky quadratic form, the first coefficient, and the Schwarzian derivative. Some extremal problems in $ {S_{K(\rho )}}$, the subclass of functions in S with $K(\rho )$-quasiconformal extensions to the exterior of the unit disk, are also solved.
A $2$-sphere in $E\sp{3}$ with vertically connected interior is tame
J. W.
Cannon;
L. D.
Loveland
345-355
Abstract: A set X in $ {E^3}$ is said to have vertical number n if the intersection of each vertical line with X contains at most n components. The set X is said to have vertical order n if each vertical line intersects X in at most n points. A set with vertical number 1 is said to be vertically connected. We prove that a 2-sphere in ${E^3}$ with vertically connected interior is tame. This result implies as corollaries several previously known taming theorems involving vertical order and vertical number along with several more general and previously unknown results.
On a certain sum in number theory. II
Břetislav
Novák
357-364
Abstract: We derive ``exact order'' of the function $\displaystyle \sum\limits_{k \leq \sqrt x } {{k^\rho }{{\min }^\beta }\left( {\frac{{\sqrt x }}{k},\frac{1}{{{P_k}}}} \right)}.$ Here $ \rho$ and $\beta$ are nonnegative real numbers and, for given real ${\delta _1},{\delta _2}, \cdots ,{\delta _r},{P_k} = {\max _j}\langle k{\delta _j}\rangle$ where $\langle t\rangle$, for real t, denotes distance of t from the nearest integer. Using our results, we obtain the solution of the basic problem in the theory of lattice points with weight in rational many-dimensional ellipsoids.
Square integrable differentials on Riemann surfaces and quasiconformal mappings
Carl David
Minda
365-381
Abstract: If $ K{(f)^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}$ where $ K(f)$ is the maximal dilatation of f. In addition, f defines an isomorphism of the square integrable harmonic differentials and some important subspaces are preserved. It is shown that not all important subspaces are preserved. The relationship of this to other work is investigated; in particular, the connection with the work of Nakai on the isomorphism of Royden algebras induced by a quasiconformal mapping is explored. Finally, the induced isomorphisms are applied to the classification theory of Riemann surfaces to show that various types of degeneracy are quasiconformally invariant.
Semigroups over trees
M. W.
Mislove
383-400
Abstract: A semigroup over a tree is a compact semigroup S such that $\mathcal{H}$ is a congruence on S and $S/\mathcal{H}$ is an abelian tree with idempotent endpoints. Each such semigroup is characterized as being constructible from cylindrical subsemigroups of S and the tree $ S/\mathcal{H}$ in a manner similar to the construction of the hormos. Indeed, the hormos is shown to be a particular example of the construction given herein when $S/\mathcal{H}$ is an I-semigroup. Several results about semigroups whose underlying space is a tree are also established as lemmata for the main results.
The closure of the space of homeomorphisms on a manifold
William E.
Haver
401-419
Abstract: The space, $ \bar H(M)$, of all mappings of the compact manifold M onto itself which can be approximated arbitrarily closely by homeomorphisms is studied. It is shown that $\bar H(M)$ is homogeneous and weakly locally contractible. If M is a compact 2-manifold without boundary, then $\bar H(M)$ is shown to be locally contractible.
On free products of finitely generated abelian groups
Anthony M.
Gaglione
421-430
Abstract: Let the group G be a free product of a finite number of finitely generated abelian groups. Let $G'$ be its commutator subgroup. It is proven here that the ``quasi-G-simple'' commutators, defined below, are free generators of $ G'$.