The structure of Galois groups of ${\rm CM}$-fields
B.
Dodson
1-32
Abstract: A $CM$-field $K$ defines a triple $ (G,H,\rho )$, where $ G$ is the Galois group of the Galois closure of $K$, $H$ is the subgroup of $G$ fixing $K$, and $\rho \in G$ is induced by complex conjugation. A "$\rho$-structure" identifies $CM$-fields when their triples are identified under the action of the group of automorphisms of $G$. A classification of the $\rho $-structures is given, and a general formula for the degree of the reflex field is obtained. Complete lists of $\rho$-structues and reflex fields are provided for $[K:\mathbb{Q}] = 2n$, with $n = 3,4,5$ and $7$. In addition, simple degenerate Abelian varieties of $CM$-type are constructed in every composite dimension. The collection of reflex fields is also determined for the dihedral group $ G = {D_{2n}}$, with $ n$ odd and $H$ of order $2$, and a relative class number formula is found.
Projective normal flatness and Hilbert functions
U.
Orbanz;
L.
Robbiano
33-47
Abstract: Projective normal flatness of a local ring $R$ along an ideal $I$ is defined to be the flatness of the morphism on the exceptional divisor induced by blowing up $ R$ with center $ I$. It is shown that most of the important properties of normal flatness have an analogue for projective normal flatness. In particular, we study the local Hilbert function in connection with projective normal flatness. If $R/I$ is regular and $R$ projectively normally flat along $I$, then we obtain the same inequality for the local Hilbert functions under blowing up as in the permissible case.
Aposyndetic continua as bundle spaces
James T.
Rogers
49-55
Abstract: Let $\mathcal{S}$ be the $P$-adic solenoid bundle, and let $\eta :X \to {S^1}$ be a map of the continuum $ X$ onto ${S^1}$. The bundle space $B$ of the induced bundle ${\eta ^{ - 1}}\mathcal{S}$ is investigated. Sufficient conditions are obtained for $ B$ to be connected, to be aposyndetic, and to be homogeneous. Uncountably many aposyndetic, homogeneous, one-dimensional, nonlocally connected continua are constructed. Other classes of continua are placed into this framework.
Global analysis of two-parameter elliptic eigenvalue problems
H.-O.
Peitgen;
K.
Schmitt
57-95
Abstract: We consider the nonlinear boundary value problem $({\ast})Lu + \lambda f(u) = 0$, $x \in \Omega ,\,u = \sigma \phi ,\,x \in \partial \Omega $, where $L$ is a second order elliptic operator and $ \lambda$ and $ \sigma$ are parameters. We analyze global properties of solution continua of these problems as $\lambda$ and $\sigma$ vary. This is done by investigating particular sections, and special interest is devoted to questions of how solutions of the $ \sigma = 0$ problem are embedded in the two-parameter family of solutions of $ ({\ast})$. As a natural biproduct of these results we obtain (a) a new abstract method to analyze bifurcation from infinity, (b) an unfolding of the bifurcations from zero and from infinity, and (c) a new framework for the numerical computations, via numerical continuation techniques, of solutions by computing particular one-dimensional sections.
Path derivatives: a unified view of certain generalized derivatives
A. M.
Bruckner;
R. J.
O’Malley;
B. S.
Thomson
97-125
Abstract: A collection $E = \{ {E_x}:x \in R\}$ is a system of paths if each set ${E_x}$ has $x$ as a point of accumulation. For such a system $E$ the derivative $F_E'(x)$ of a function $F$ at a point $x$ is just the usual derivative at $x$ relative to the set ${E_x}$. The goal of this paper is the investigation of properties that $F$ and its derivative $F_E'$ must have under certain natural assumptions about the collection $E$. In particular, it is shown that most of the familiar properties of approximate derivatives and approximately differentiable functions follow in this setting from three conditions on the collection $E$ relating to the "thickness" of the sets $ {E_x}$ and the way in which the sets intersect.
Quartic surfaces of elliptic ruled type
Yumiko
Umezu
127-143
Abstract: Let $X$ be a normal quartic surface whose resolutions are birationally equivalent to elliptic ruled surfaces. We classify the singularities on $ X$ and then describe the global structure of $X$.
A nonlinear integral equation occurring in a singular free boundary problem
Klaus
Höllig;
John A.
Nohel
145-155
Abstract: We study the Cauchy problem $\displaystyle \left\{ \begin{gathered}{u_t} = \phi {({u_x})_x},\qquad (x,t) \in... ...{{\mathbf{R}}_ + }, u( \cdot ,0) = f \end{gathered} \right.$ with the piecewise linear constitutive function $\phi (\xi ) = {\xi _ + } = \max (0,\xi )$ and with smooth initial data $f$ which satisfy $x \in {\mathbf{R}}$, and $ f''(0) > 0$. We prove that free boundary $s$, given by ${u_x}(s{(t)^ + },t) = 0$, is of the form $\displaystyle s(t) = - \kappa \sqrt t + o\left( {\sqrt t } \right),\qquad t \to {0^ + },$ where the constant $\kappa = 0.9034 \ldots$ is the (numerical) solution of a particular nonlinear equation. Moreover, we show that for any $ \alpha \in (0,1/2)$, $\displaystyle \left\vert {\frac{{{d^2}}} {{d{t^2}}}f(s(t))} \right\vert = O({t^{\alpha - 1}}),\qquad t \to {0^ + }.$ The proof involves the analysis of a nonlinear singular integral equation.
An integral inequality with applications
M. A.
Leckband
157-168
Abstract: Using a technical integral inequality, J. Moser proved a sharp result on exponential integrability of a certain space of Sobolev functions. In this paper, we show that the integral inequality holds in a general setting using nonincreasing functions and a certain class of convex functions. We then apply the integral inequality to extend the above result by J. Moser to other spaces of Sobolev functions. A second application is given generalizing some different results by M. Jodeit.
Where the continuous functions without unilateral derivatives are typical
Jan
Malý
169-175
Abstract: An alternative proof of the existence of a Besicovitch function (i.e. a continuous function which has nowhere a unilateral derivative) is presented. The method consists in showing the residuality of Besicovitch functions in special subspaces of the Banach space of all continuous functions on $[0,1]$ and yields Besicovitch functions with additional properties of Morse or Hölder type. A way how to obtain functions with a similar behavior on normed linear spaces is briefly mentioned.
Conjugacy classes of hyperbolic matrices in ${\rm Sl}(n,\,{\bf Z})$ and ideal classes in an order
D. I.
Wallace
177-184
Abstract: A bijection is proved between $\operatorname{Sl} (n,{\mathbf{Z}})$-conjugacy classes of hyperbolic matrices with eigenvalues $ \{ {\lambda _1}, \ldots ,{\lambda _n}\}$ which are units in an $n$-degree number field, and narrow ideal classes of the ring ${R_k} = {\mathbf{Z}}[{\lambda _i}]$. A bijection between $\operatorname{Gl} (n,{\mathbf{Z}})$-conjugacy classes and the wide ideal classes, which had been known, is repeated with a different proof.
A Cayley-Dickson process for a class of structurable algebras
B. N.
Allison;
J. R.
Faulkner
185-210
Abstract: In this paper, we study the class of all simple structurable algebras with the property that the space of skew-hermitian elements has dimension $1$. These algebras with involution have arisen in the study of Lie algebra constructions. The reduced algebras are isotopic to $2 \times 2$ matrix algebras. We study a Cayley-Dickson process for rationally constructing some algebras in the class including division algebras and nonreduced nondivision algebras. An important special case of the process endows the direct sum of two copies of a $28$-dimensional degree $4$ central simple Jordan algebra $\mathcal{B}$ with the structure of an algebra with involution. In preparatory work, we obtain a procedure for giving the space $ {\mathcal{B}_0}$ of trace zero elements of any such Jordan algebra $\mathcal{B}$ the structure of a $ 27$-dimensional exceptional Jordan algebra.
Abelian subgroups of topological groups
Siegfried K.
Grosser;
Wolfgang N.
Herfort
211-223
Abstract: In [1] Šmidt's conjecture on the existence of an infinite abelian subgroup in any infinite group is settled by counterexample. The well-known Hall-Kulatilaka Theorem asserts the existence of an infinite abelian subgroup in any infinite locally finite group. This paper discusses a topological analogue of the problem. The simultaneous consideration of a stronger condition--that centralizers of nontrivial elements be compact--turns out to be useful and, in essence, inevitable. Thus two compactness conditions that give rise to a profinite arithmetization of topological groups are added to the classical list (see, e.g., [13 or 4]).
Examples of unoriented area-minimizing surfaces
Frank
Morgan
225-237
Abstract: A comprehensive study is made of constructions of area-minimizing flat chains modulo two. Many have singularities. For instance, any bounded, area-minimizing submanifold of $ {{\mathbf{R}}^n}$ occurs as the singular set of some area-minimizing flat chain modulo two in some $ {{\mathbf{R}}^N}$.
On the diagonal of an operator
Peng
Fan
239-251
Abstract: Characterizations of zero-diagonal operators (i.e., operators that have a diagonal whose entries consist entirely of zeros) and the norm-closure of these operators are obtained. Also included are new characterizations of trace class operators, self-commutators of bounded operators, and others.
Banach spaces which are $M$-ideals in their biduals
Peter
Harmand;
Åsvald
Lima
253-264
Abstract: We investigate Banach spaces $X$ such that $X$ is an $M$-ideal in $ {X^{{\ast}{\ast}}}$. Subspaces, quotients and ${c_0}$-sums of spaces which are $M$-ideals in their biduals are again of this type. A nonreflexive space $X$ which is an $M$-ideal in $ {X^{{\ast}{\ast}}}$ contains a copy of ${c_0}$. Recently Lima has shown that if $ K(X)$ is an $M$-ideal in $L(X)$ then $X$ is an $M$-ideal in $ {X^{{\ast}{\ast}}}$. Here we show that if $X$ is reflexive and $K(X)$ is an $M$-ideal in $L(X)$, then $ K{(X)^{{\ast}{\ast}}}$ is isometric to $L(X)$, i.e. $K(X)$ is an $M$-ideal in its bidual. Moreover, for real such spaces, we show that $K(X)$ contains a proper $M$-ideal if and only if $X$ or $ {X^{\ast}}$ contains a proper $M$-ideal.
Applications of uniform convexity of noncommutative $L\sp{p}$-spaces
Hideki
Kosaki
265-282
Abstract: We consider noncommutative ${L^p}$-spaces, $ 1 < p < \infty$, associated with a von Neumann algebra, which is not necessarily semifinite, and obtain some consequences of their uniform convexity. Among other results, we obtain (i) the norm continuity of the "absolute value part" map from each ${L^p}$-space onto its positive part; (ii) a certain continuity result on Radon-Nikodym derivatives in the context of positive cones introduced by H. Araki; and (iii) the necessary and sufficient condition for certain ${L^p}$-norm inequalities to become equalities. Some dominated convergence theorems for a probability gage are also considered.
Decomposability of Radon measures
R. J.
Gardner;
W. F.
Pfeffer
283-293
Abstract: A topological space is called metacompact or metalindelöf if each open cover has a point-finite or point-countable refinement, respectively. It is well known that each Radon measure is expressible as a sum of Radon measures supported on a disjoint family of compact sets, called a concassage. If the unions of measurable subsets of the members of a concassage are also measurable, the Radon measure is called decomposable. We show that Radon measures in a metacompact space are always saturated, and therefore decomposable whenever they are complete. The previous statement is undecidable in ZFC if "metacompact" is replaced by "metalindelöf". The proofs are based on structure theorems for a concassage of a Radon measure. These theorems also show that the union of a concassage of a Radon measure in a metacompact space is a Borel set, which is paracompact in the subspace topology whenever the main space is regular.
On bordism groups of immersions
Guillermo
Pastor
295-301
Abstract: The bordism group of immersions of oriented $n$-manifolds into ${{\mathbf{R}}^{n + k}}$ is identified with the stable homotopy group $\Pi _{n + k}^s(\operatorname{MSO} (k))$. We study these groups for $n - 2 \leqslant k \leqslant n$, and discuss the behaviour of double points and their relation with the corresponding bordism groups of embeddings.
Suspension spectra and homology equivalences
Nicholas J.
Kuhn
303-313
Abstract: Let $ f:{\Sigma ^\infty }X \to {\Sigma ^\infty }Y$ be a stable map between two connected spaces, and let $ {E_{\ast}}$ be a generalized homology theory. We show that if ${E_{\ast}}(f)$ is an isomorphism then ${E_{\ast}}({\Omega ^\infty }f):{E_{\ast}}(QX) \to {E_{\ast}}(QY)$ is a monomorphism, but possibly not an epimorphism. Applications of this theorem include results of Miller and Snaith concerning the $ K$-theory of the Kahn-Priddy map.
The quotient semilattice of the recursively enumerable degrees modulo the cappable degrees
Steven
Schwarz
315-328
Abstract: In this paper, we investigate the quotient semilattice $\underline R /\underline M $ of the r.e. degrees modulo the cappable degrees. We first prove the $ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{R} /\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{M} $ counterpart of the Friedberg-Muchnik theorem. We then show that minimal elements and minimal pairs are not present in $\underline R /\underline M$. We end with a proof of the $ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{R} /\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{M} $ counterpart to Sack's splitting theorem.
Compact spaces and spaces of maximal complete subgraphs
Murray
Bell;
John
Ginsburg
329-338
Abstract: We consider the space $M(G)$ of all maximal complete subgraphs of a graph $G$ and, in particular, the space $M(P)$ of all maximal chains of an ordered set $P$. The main question considered is the following: Which compact spaces can be represented as $ M(G)$ for some graph $ G$ or as $M(P)$ for some ordered set $P$? The former are characterized as spaces which have a binary subbase for the closed sets which consists of clopen sets. We give an example to show that this does not include all zero-dimensional supercompact spaces. The following negative result is obtained concerning ordered sets: Let $D$ be an uncountable discrete space and let $ \alpha D$ denote the one-point compactification of $D$. Then there is no ordered set $P$ such that $M(P) \simeq \alpha D$.
Quadratic forms of height two
Robert W.
Fitzgerald
339-351
Abstract: Quadratic forms of height two and leading form defined over the base field are determined over several fields. Also forms of height and degree two over an arbitrary field are classified.
Extensions of tight set functions with applications in topological measure theory
Wolfgang
Adamski
353-368
Abstract: Let $ {\mathcal{K}_1},\,{\mathcal{K}_2}$ be lattices of subsets of a set $ X$ with $ {\mathcal{K}_1} \subset {\mathcal{K}_2}$. The main result of this paper states that every semifinite tight set function on ${\mathcal{K}_1}$ can be extended to a semifinite tight set function on $ {\mathcal{K}_2}$. Furthermore, conditions assuring that such an extension is uniquely determined or $\sigma$-smooth at $\phi$ are given. Since a semifinite tight set function defined on a lattice $ \mathcal{K}$ [and being $ \sigma$-smooth at $ \phi$] can be identified with a semifinite $ \mathcal{K}$-regular content [measure] on the algebra generated by $\mathcal{K}$, the general results are applied to various extension problems in abstract and topological measure theory.
Factoring compact and weakly compact operators through reflexive Banach lattices
C. D.
Aliprantis;
O.
Burkinshaw
369-381
Abstract: When does a weakly compact operator between two Banach spaces factor through a reflexive Banach lattice? This paper provides some answers to this question. One of the main results: If an operator between two Banach spaces factors through a Banach lattice with weakly compact factors, then it also factors through a reflexive Banach lattice. In particular, the square of a weakly compact operator on a Banach lattice factors through a reflexive Banach lattice. Similar results hold for compact operators. For instance, the square of a compact operator on a Banach lattice factors with compact factors through a reflexive Banach lattice.