Coxeter functors without diagrams
Maurice
Auslander;
María Inés
Platzeck;
Idun
Reiten
1-46
Abstract: The first part of this paper is devoted to a generalization of the notion of partial Coexeter functor from diagrams to certain types of artin rings and artin algebras. The rest of the paper is devoted to a discussion of the connection between the various Coxeter functors which exist for diagrams as well as for artin rings and artin algebras.
Modeloids. I
Miroslav
Benda
47-90
Abstract: If A is a set and  is the collection of finite nonrepeating sequences of its elements then a modeloid E on A is an equivalence relation on  which preserves length, is hereditary, and is invariant under the action of permutations. The pivotal operation on modeloids is the derivative. The theory of this operation turns out to be very rich with connections leading to diverse branches of mathematics. For example, in §3 we associate an action space with a modeloid and in §5 we characterize the action spaces which are associated with the basic modeloids, i.e., those which are derivatives of themselves. What emerges is a kind of stability for the action space. We then show that action spaces with this stability can be approximated by finite actions and, subject to certain requirements, this approximation is unique (see Proposition 5.7). Algebraically, the countable basic modeloids correspond to closed subgroups of the symmetric groups. This and the study of automorphisms of modeloids let us show, without any algebra, that the only nontrivial normal subgroups of the finite ( $ \left( { \geqslant \,\,5} \right)$) symmetric groups are the alternating groups. The last section gives, hopefully, credence to the thesis that the essence of model theory is the study of modeloids.
Ultrafiltres \`a la fa\c con de Ramsey
Maryvonne
Daguenet-Teissier
91-120
Abstract: Let $\beta {\text{N}}$ be the set of ultrafilters on N; $ {\mathcal{u}}\, \in \,\beta {\text{N}}$ is ``absolu'' [6] (Ramsey [4]) if all its free images by continuous maps $ \beta {\text{N}}\, \to \,\beta {\text{N}}$ are isomorphic. We study here a weaker Ramsey-like property, which implies the existence of fiber products $\mathcal{D}\,\, \otimes {\,_E}\,\mathcal{D}\,\left( { \otimes _\textbf{E}^k\,\mathcal{D}} \right)$ extending the usual product ${\mathcal{D}}\, \otimes \,{\mathcal{D}}\,\left( {{ \otimes ^{k\,}}{\mathcal{D}}} \right)$. This can be translated in the language of model-theory on the one hand as the existence of repeated almagamated sums and on the other hand by some properties of sets of indiscernibles associated with ultrafilters having this property (§5). We show that the class of ultrafilters we study strictly contains the class of Ramsey ultrafilters (§1) and is (§2) strictly (§3) contained in the class of p-point ultrafilters [9] ("$\delta$-stables'' [6]) and contains the free images of its elements (§4). In §2 we also give a characterization of p-point ultrafilters in terms of the product ${ \otimes ^k}\,{\mathcal{D}}$. In §3 we show the link with weakly Ramsey ultrafilters of Blass [3] and more generally we study ultrafilters $ {\mathcal{D}}$ on N having only a finite number $i\left( {\mathcal{D}} \right)$ of free images up to isomorphism and such that $\char93 \,\tau {\,^{ - \,1}}\left( {{\mathcal{D}},\,{\mathcal{D}}} \right)\, = \,2i\left( {\mathcal{D}} \right)\, + \,1$, where $\char93 {\tau ^{ - 1}}\left( {{\mathcal{D}},{\mathcal{D}}} \right)$ is the number of ultrafilters on $ {{\text{N}}^2}$ finer than the filter generated by $\left( {D\, \times \,D} \right)$ with $D\, \in \,{\mathcal{D}}$.
Isotoping mappings to open mappings
John J.
Walsh
121-145
Abstract: Let f be a quasi-monotone mapping from a compact, connected manifold $ {M^m}\,(m\, \geqslant \,3)$ onto a space Y; then there is an open mapping g from M onto Y such that, for each $ y\, \in \,Y,\,{g^{ - 1}}(y)$ is not a point and $ {g^{ - 1}}(y)$ and ${f^{ - 1}}(y)$ are equivalently embedded in M (in particular, $ {g^{ - 1}}(y)$ and ${f^{ - 1}}(y)$ have the same shape). Applying the result with f equal to the identity mapping on M yields a continuous decomposition of M into cellular sets each of which is not a point.
Zeeman's filtration of homology
Clint
McCrory
147-166
Abstract: Geometric interpretations of Zeeman's filtrations of the homology and cohomology of a triangulable space are given, using an analysis of his spectral sequence for Poincaré duality.
A nonunitary pairing of polarizations for the Kepler problem
J. H.
Rawnsley
167-180
Abstract: The half-form pairing of two polarizations of the Kepler manifold is found and shown to define a bounded linear isomorphism of the two Hilbert spaces, but is not unitary.
A $3$-local characterization of $L\sb{7}(2)$
Larry
Finkelstein;
Daniel
Frohardt
181-194
Abstract: Recent work of Gorenstein and Lyons on finite simple groups has led to standard form problems for odd primes. The present paper classifies certain simple groups which have a standard 3-component of type ${L_5}\left( 2 \right)$.
Starlike, convex, close-to-convex, spiral-like, and $\Phi $-like maps in a commutative Banach algebra with identity
L. F.
Heath;
T. J.
Suffridge
195-212
Abstract: Let C(X) be the space of continuous functions on a compact $ {T_2}$-space X where each point of X is a ${G_\delta }$. If $F:\,B\, \to \,C\,(X)$ is a biholomorphic (in the sense that F and $ {F^{ - 1}}$ are Fréchet differentiable) map of $ B\, = \,\{ \,f\left\vert {\,\left\Vert f \right\Vert} \right.\, < \,1\}$ onto a convex domain with $ DF(0)\, = \,I$, then F is Lorch analytic (i.e., $DF\,(f)(g)\, = \,{a_f}g $ for some ${a_{f}} \, \in \,C\,(X))$). Let R be a commutative Banach algebra with identity such that the Gelfand homomorphism of R into $C(\mathcal{m})$ is an isometry. Starlike, convex, close-to-convex, spirallike and $\Phi$-like functions are defined in $B\, = \,\{ x\, \in \,R\,\left\vert {\,\left\Vert x \right\Vert} \right.\, < \,1\}$ for L-analytic functions in B and they are related to associated complex-valued holomorphic functions in $\Delta \, = \,\{ z\, \in \,\left. {\textbf{C}} \right\vert\,\,\left\vert z \right\vert\, < \,1\}$.
Defining Lagrangian immersions by phase functions
J. Alexander
Lees
213-222
Abstract: In order to analyze the singularities of the solutions of certain partial differential equations, Hörmander, in his paper on Fourier integral operators, extends the method of stationary phase by introducing the class of nondegenerate phase functions. Each phase function, in turn, defines a lagrangian submanifold of the cotangent bundle of the manifold which is the domain of the corresponding differential operator. Given a lagrangian submanifold of a cotangent bundle, when is it globally defined by a nondegenerate phase function? A necessary and sufficient condition is here found to be the vanishing of two topological obstructions; one in the cohomology and the other in the k-theory of the given lagrangian submanifold.
Borel parametrizations
R. Daniel
Mauldin
223-234
Abstract: Let X and Y be uncountable Polish spaces and B a Borel subset oi $ X\, \times \,Y$ such that for each x, ${B_x}$ is uncountable. A Borel parametrization of B is a Borel isomorphism, g, of $X\, \times \,E$ onto B where E is a Borel subset of Y such that for each x, $ g\left( {x,\, \cdot } \right)$ maps E onto $ {B_x}\, = \,\left\{ {y:\,\left( {x,\,y} \right)\, \in \,B} \right\}$. It is shown that B has a Borel parametrization if and only if B contains a Borel set M such that for each x, ${M_x}$ is a nonempty compact perfect set, or, equivalently, there is an atomless conditional probability distribution, $\mu$, so that for each x, $\mu \left( {x,\,{B_x}} \right)\, > \, 0$. It is also shown that if Y is dense-in-itself and ${B_x}$ is not meager, for each x, then B has a Borel parametrization.
Control problems governed by a pseudo-parabolic partial differential equation
Luther W.
White
235-246
Abstract: Let G be a bounded domain in ${R^n}$ and $Q\, = \,G\, \times \,\left( {0,\,T} \right)$. We consider the solution $ y\left( u \right)$ of the pseudo-parabolic initial-value problem \begin{multline}\left( {1\, + \,M\left( x \right)} \right)\,{y_t}\,\left( u \rig... ... \right)\, = \,0\,{\text{in}}\,{L^2}\,\left( G \right), \end{multline} , to be the state corresponding to the control u. Here $M\left( x \right)$ and $L\left( x \right)$ are symmetric uniformly strongly elliptic second-order partial differential operators. The control problem is to find a control $ {u_0}$ in a fixed ball in ${L^2}\left( Q \right)$ such that (i) the endpoint of the corresponding state $ y\left( { \cdot ,\,T;\,{u_0}} \right)$ lies in a given neighborhood of a target Z in $ {L^2}\left( G \right)$ and (ii) ${u_0}$ minimizes a certain energy functional. In this paper we establish results concerning the controllability of the states and the compatibility of the constraints, existence and uniqueness of the optimal control, existence and properties of Lagrange multipliers associated with the constraints, and regularity properties of the optimal control.
Weak cuts of combinatorial geometries
Hien Q.
Nguyen
247-262
Abstract: A weak cut of a Combinatorial Geometry G is a generalization of a modular cut, corresponding to the family of the new dependent sets in a weak map image of G. The use of weak cuts allows the construction of all weak images of G, an important result being that, to any family $ {\mathcal{M}}$ of independent sets of G, is associated a unique weak cut ${\mathcal{C}}$ containing ${\mathcal{M}}$. In practice, the flats of the weak image defined by $ {\mathcal{C}}$ can be constructed directly. The weak cuts corresponding to known weak maps, such as truncation, projection, elementary quotient, are determined. The notion of weak cut is particularly useful in the study of erections. Given a geometry F and a weak image G, an F-erection of G is an erection of G which is a weak image of F. The main results are that the set of all F-erections of G is a lattice with the weak map order, and that the free F-erection can be constructed explicitly. Finally, a problem involving higher order erection is solved.
Space curves that point almost everywhere
J. B.
Wilker
263-274
Abstract: We construct a simple, closed, continuously differentiable curve $r:\,[0,\,1]\, \to \,{E^d}\,(d\, \geqslant \,3)$ whose tangent vector never points twice in the same direction of $ {S^{d\, - \,1}}$ yet sweeps out a set of directions equal to almost all of ${S^{d\, - \,1}}$.
A converse of the Borel formula
Ronald M.
Dotzel
275-287
Abstract: When an elementary Abelian p-group acts on a $ {Z_p}$-homology sphere (p a prime), it is known that the Borel formula must hold. Here we ask that the Borel formula hold and determine how this restricts, homologically, the type of space which can occur, assuming spherical fixed sets and connectedness. This is done by constructing a linear model of the action and an equivariant map to the model, the mapping cone of which yields certain homological information.
The first order theory of $N$-colorable graphs
William H.
Wheeler
289-310
Abstract: Every N-colorable graph without loops or multiple edges is a substructure of a direct power of a particular, finite, N-coloarable graph. Consequently, the class of N-colorable graphs without loops or endpoints can be recursively axiomatized by a first order, universal Horn theory. This theory has a model-companion which has a primitive recursive elimination of quantifiers and is decidable, complete, ${\aleph _0}$-categorical, and independent. The N-colorable graphs without loops or multiple edges which have a proper, prime model extension for the model-companion are precisely the finite, amalgamation bases.
Deforming twist-spun knots
R. A.
Litherland
311-331
Abstract: In [15] Zeeman introduced the process of twist-spinning an n-knot to obtain an (n + l)-knot, and proved the remarkable theorem that a twist-spun knot is fibred. In [2] Fox described another deformation which can be applied during the spinning process, and which he called rolling. We show that, provided one combines the rolling with a twist, the resulting knot is again fibred. In fact, this result holds for a larger class of deformations, defined below.
CR submanifolds of a Kaehler manifold. II
Aurel
Bejancu
333-345
Abstract: The differential geometry of CR submanifolds of a Kaehler manifold is studied. Theorems on parallel normal sections and on a special type of flatness of the normal connection on a CR submanifold are obtained. Also, the nonexistence of totally umbilical proper CR submanifolds in an elliptic or hyperbolic complex space is proven.
An algebraic characterization of connected sum factors of closed $3$-manifolds
W. H.
Row
347-356
Abstract: Let M and N be closed connected 3-manifolds. A knot group of M is the fundamental group of the complement of a tame simple closed curve in M. Denote the set of knot groups of M by K(M). A knot group G of M is realized in N if G is the fundamental group of a compact submanifold of N with connected boundary. Theorem. Every knot group of N is realized in M iff N is a connected sum factor of M. Corollary 1. $ K\,(M)\, = \,K\,(N)$ iff M is homeomorphic to N. Given M, there exists a knot group ${G_M}$ of M that serves to characterize M in the following sense. Corollary 2. $ {G_M}$ is realized in N and ${G_N}$, is realized in M iff M is homeomorphic to N. Our proof depends heavily on the work of Bing, Feustal, Haken, and Waldhausen in the 1960s and early 1970s. A. C. Conner announced Corollary 1 for orientable 3-manifolds in 1969 which Jaco and Myers have recently obtained using different techniques.
Nonstandard measure theory: avoiding pathological sets
Frank
Wattenberg
357-368
Abstract: The main results in this paper concern representing Lebesgue measure by nonstandard measures which avoid certain pathological sets. An (external) set E is S-thin if InfmA|A standard,* $A\, \supseteq \,E$ = 0 and Q-thin if Inf*mA|A internal, $A\, \supseteq \,E$ = 0. It is shown that any *finite sample which represents Lebesgue measure avoids every S-thin set and that given any Q-thin set E there is a *finite sample avoiding E which represents Lebesgue measure. In the last part of the paper a particular pathological set ${\mathcal{H}}\,\, \subseteq \, * \left[ {0,\,1} \right]$ is constructed which is important for the study of approximate limits, derivatives etc. It is shown that every *finite sample which represents Lebesgue measure assigns inner measure zero and outer measure one to this set and that Loeb measure does the same. Finally, it is shown that Loeb measure can be extended to a $ \sigma$-algebra including ${\mathcal{H}}$ in such a way that ${\mathcal{H}}$ is assigned zero measure.
Orthogonal polynomials defined by a recurrence relation
Paul G.
Nevai
369-384
Abstract: R. Askey has conjectured that if a system of orthogonal polynomials is defined by the three term recurrence relation $\displaystyle x{p_{n\, - \,1}}\left( x \right)\, = \,\frac{{{\gamma _{n\, - \,1... ...amma _{n\, - \,2}}}} {{{\gamma _{n\, - \,1}}}}\,{p_{n\, - \,2}}\left( x \right)$ and $\displaystyle {\alpha _n}\, = \,\frac{{{{( - 1)}^n}}} {n}\,{\text{const}}\,{\text{ + }}O\left( {\frac{1} {{{n^2}}}} \right),$ $\displaystyle \frac{{{\gamma _n}}} {{{\gamma _{n + 1}}}}\, = \,\frac{1} {2}\, +... ...1)}^n}}} {n}\,{\text{const}}\,{\text{ + }}O\left( {\frac{1} {{{n^2}}}} \right),$ then the logarithm of the absolutely continuous portion of the corresponding weight function is integrable. The purpose of this paper is to prove R. Askey's conjecture and solve related problems.
A simultaneous lifting theorem for block diagonal operators
G. D.
Allen;
J. D.
Ward
385-397
Abstract: Stampfli has shown that for a given $T\, \in \,B\left( H \right)$ there exists a $K\, \in \,C\left( H \right)$ so that $\sigma \left( {T\, + \, K} \right)\,= \,{\sigma _w}\left( T \right)$. An analogous result holds for the essential numerical range ${W_e}\left( T \right)$. A compact operator K is said to preserve the Weyl spectrum and essential numerical range of an operator $T\, \in \,B\left( H \right)$ if $\sigma \left( {T\, + \, K} \right)\, = \,{\sigma _w}\left( T \right)$ and $\overline {W\left( {T \, + \, K} \right)} \, = \,{W_e}\left( T \right)$. Theorem. For each block diagonal operator T, there exists a compact operator K which preserves the Weyl spectrum and essential numerical range of T. The perturbed operator $T \, + \, K$ is not, in general, block diagonal. An example is given of a block diagonal operator T for which there can be no block diagonal perturbation which preserves the Weyl spectrum and essential numerical range of T.
Erratum to: ``Pullback de Rham cohomology of the free path fibrations'' [Trans. Amer. Math. Soc. {\bf 242} (1978), 307--318; MR 57:17678]
Kuo Tsai
Chen
398-398