Structure and dimension of global branches of solutions to multiparameter nonlinear equations
J.
Ize;
I.
Massabò;
J.
Pejsachowicz;
A.
Vignoli
383-435
Abstract: This paper is concerned with the topological dimension of global branches of solutions appearing in different problems of Nonlinear Analysis, in particular multiparameter (including infinite dimensional) continuation and bifurcation problems. By considering an extension of the notion of essential maps defined on sets and using elementary point set topology, we are able to unify and extend, in a selfcontained fashion, most of the recent results on such problems. Our theory applies whenever any generalized degree theory with the boundary dependence property may be used, but with no need of algebraic structures. Our applications to continuation and bifurcation follow from the nontriviality of a local invariant, in the stable homotopy group of a sphere, and give information on the local dimension and behavior of the sets of solutions, of bifurcation points and of continuation points.
Volumes of tubes about complex submanifolds of complex projective space
Alfred
Gray
437-449
Abstract: Simple formulas in terms of Chern classes are given for the volume of a tube about a Kähler submanifold of a space of constant holomorphic sectional curvature. A comparison theorem which generalizes these formulas is also given. Tubes about complete intersections in complex projective space are studied in detail.
Propagation of singularities for nonstrictly hyperbolic semilinear systems in one space dimension
Lucio
Micheli
451-485
Abstract: We consider the creation and propagation of singularities in the solutions of semilinear nonstrictly hyperbolic systems in one space dimension when the initial data has jump discontinuities. We show that singularities travelling along characteristics can branch at points of degeneracy of the vector fields on all other forward characteristics. We prove a lower bound for the strength of these new singularities, and we give an example showing that our result cannot be improved in general.
On the existence and classification of extensions of actions on submanifolds of disks and spheres
Amir
Assadi;
William
Browder
487-502
Abstract: Given a $ G$-action $\psi :G \times W \to W$ and an embedding $W \subset {D^n}$, when is it possible to find a $ G$-action $\phi :G \times {D^n} \to D$ such that $ {D^n} - W$ is $ G$-free? Sufficient conditions of cohomological nature for the existence of such extensions are given and the extensions are classified. This leads to the characterization of the stationary point sets and classification of semifree actions on disks up to $G$-diffeomorphism under suitable dimension hypotheses.
A note on automorphic forms of weight one and weight three
Peter F.
Stiller
503-518
Abstract: In this paper the author develops an interesting relationship between classical automorphic forms of weights one and three, and the solutions of certain second order differential equations related to elliptic (modular) surfaces. In particular for a cusp form of weight three, it is shown that the special values of the associated Dirichlet series can be determined from the periods of an inhomogeneous differential equation, or what is the same thing, the monodromy of an associated third order differential equation. Explicit examples are provided for principal congruence subgroups $\Gamma (N)$ with $N \equiv 0\,\operatorname{mod} \,4$.
The cuspidal group and special values of $L$-functions
Glenn
Stevens
519-550
Abstract: The structure of the cuspidal divisor class group is investigated by relating this structure to arithmetic properties of special values of $L$-functions of weight two Eisenstein series. A new proof of a theorem of Kubert (Proposition 3.1) concerning the group of modular units is derived as a consequence of the method. The key lemma is a nonvanishing result (Theorem 2.1) for values of the ``$L$-function'' attached to a one-dimensional cohomology class over the relevant-congruence subgroup. Proposition 4.7 provides data regarding Eisenstein series and associated subgroups of the cuspidal divisor class group which the author hopes will simplify future calculations in the cuspidal group.
Subgraphs of random graphs
D. H.
Fremlin;
M.
Talagrand
551-582
Abstract: Let $\Delta \subseteq {[\omega ]^2}$ be an undirected graph on $\omega$, and let $u \in [0,\,1]$. Following P. Erdös and A. Hajnal, we write $(\omega ,\,2,\,u) \Rightarrow \Delta$ to mean: whenever ${E_1} \subseteq [0,\,1]$ is a measurable set of Lebesgue measure at least $u$ for every $I \in {[\omega ]^2}$, then there is some $t \in [0,\,1]$ such that $ \Delta$ appears in the graph $ {\Gamma _t} = \{ I:\,t \in {E_I}\}$ in the sense that there is a strictly increasing function $ f:\,\omega \to \omega$ such that $\{ f(i),\,f(j)\} \in {\Gamma _t}$ whenever $\{ i,\,j\} \in \Delta$. We give an algorithm for determining when $(\omega ,\,2,\,u) \Rightarrow \Delta$ for finite $\Delta$, and we show that for infinite $ \Delta ,\,(\omega ,\,2,\,u) \Rightarrow \Delta$ if there is a $\upsilon < u$ such that $ (\omega ,\,2,\,\upsilon ) \Rightarrow {\Delta ^\prime }$ for every finite $ \Delta^{\prime} \subseteq \Delta$. Our results depend on a new condition, expressed in terms of measures on $ \beta\omega$, sufficient to imply that $\Delta$ appears in $\Gamma$ (Theorem 2F), and enable us to identify the extreme points of some convex sets of measures (Theorem 5H).
A ``Tits-alternative'' for subgroups of surface mapping class groups
John
McCarthy
583-612
Abstract: It has been observed that surface mapping class groups share various properties in common with the class of linear groups (e.g., $ [\mathbf{BLM},\,\mathbf{H}]$). In this paper, the known list of such properties is extended to the ``Tits-Alternative'', a property of linear groups established by J. Tits $[\mathbf{T}]$. In fact, we establish that every subgroup of a surface mapping class group is either virtually abelian or contains a nonabelian free group. In addition, in order to establish this result, we develop a theory of attractors and repellers for the action of surface mapping classes on Thurston's projective lamination spaces $ [\mathbf{Th1}]$. This theory generalizes results known for pseudo-Anosov mapping classes $ [\mathbf{FLP}]$.
Convergence of conditional expectations and strong laws of large numbers for multivalued random variables
Fumio
Hiai
613-627
Abstract: Fatou's lemmas and Lebesgue's convergence theorems are established for multivalued conditional expectations of random variables having values in the closed subsets of a separable Banach space. Strong laws of large numbers are also given for such multivalued random variables.
Elements of finite order for finite monadic Church-Rosser Thue systems
Friedrich
Otto
629-637
Abstract: A Thue system $ T$ over $\Sigma$ is said to allow nontrivial elements of finite order, if there exist a word $u \in {\Sigma ^ \ast }$ and integers $n \ge 0$ and $k \ge 1$ such that $u \nleftrightarrow \,_T^ \ast \lambda$ and ${u^{n + k}} \leftrightarrow \,_T^ \ast {u^n}$. Here the following decision problem is shown to be decidable: Instance. A finite, monadic, Church-Rosser Thue system $T$ over $\Sigma$. Question. Does $ T$ allow nontrivial elements of finite order? By a result of Muller and Schupp this implies in particular that given a finite monadic Church-Rosser Thue system $T$ it is decidable whether the monoid presented by $T$ is a free group or not.
Completely unstable dynamical systems
Sudhir K.
Goel;
Dean A.
Neumann
639-668
Abstract: We associate with the $ {C^r}\,(r\, \ge \,1)$ dynamical system $\phi$ on an $m$-manifold $M$, the orbit space $M/\phi$, defined to be the set of orbits of $\phi$ with the quotient topology. If $\phi$ is completely unstable, $ M/\phi$ turns out to be a ${C^r}\,(m\, - \,1)$-nonseparated manifold. It is known that for a completely unstable flow $ \phi$ on a contractible manifold $ M,\,M/\phi$ is Hausdorff if and only if $\phi$ is parallelizable. In general, we place an order on the non-Hausdorff points of $M/\phi$ (essentially) by setting $\bar p < \bar q$ if and only if ${\pi ^{ - 1}}(\bar q) \subseteq {J^ + }({\pi ^{ - 1}}(\bar p))$. Our result is that $(M,\,\phi)$ is topologically equivalent to $(M^{\prime},\,\phi ^{\prime})$ if and only if $ M/\phi$ is order isomorphic to $M^{\prime}/\phi ^{\prime}$.
Maps between surfaces
Richard
Skora
669-679
Abstract: The Uniqueness Conjecture states if $\phi ,\,\psi :\,M \to N$ are $d$-fold, simple, primitive, branched coverings between closed, connected surfaces, then $ \phi$ and $\psi$ are equivalent. The Uniqueness Conjecture is proved in the case that $M$ and $N$ are nonorientable and $N = \mathbf{R}{P^2}$ or Klein bottle. It is also proved in the case that $M$ and $N$ are nonorientable and $d/2 < d\chi (N) - \chi (M)$. As an application it is shown that two $d$-fold, branched coverings $\phi :{M_1} \to N,\,\psi :{M_2} \to N$ between closed, connected surfaces are branched cobordant.
Arbitrarily large continuous algebras on one generator
Jiří
Adámek;
Václav
Koubek;
Evelyn
Nelson;
Jan
Reiterman
681-699
Abstract: Generation of order-continuous algebras is investigated for various concepts of continuity. For the continuity understood as the preservation of joins of countably-directed sets, arbitrarily large infinitary continuous algebras on one generator are constructed.
Projective modules in the category ${\scr O}\sb S$: self-duality
Ronald S.
Irving
701-732
Abstract: Given a parabolic subalgebra $ {\mathfrak{p}_S}$ of a complex, semisimple Lie algebra $ \mathfrak{g}$, there is a naturally defined category ${\mathcal{O}_S}$ of $ \mathfrak{g}$-modules which includes all the $ \mathfrak{g}$-modules induced from finite-dimensional $ {\mathfrak{p}_S}$-modules. This paper treats the question of whether certain projective modules in $ {\mathcal{O}_S}$ are isomorphic to their dual modules. The projectives in question are the projective covers of those simple modules occurring in the socles of generalized Verma modules. Their self-duality is proved in a number of cases, and additional information is obtained on the generalized Verma modules.
Projective modules in the category ${\scr O}\sb S$: Loewy series
Ronald S.
Irving
733-754
Abstract: Let $\mathfrak{g}$ be a complex, semisimple Lie algebra with a parabolic subalgebra ${\mathfrak{p}_S}$. The Loewy lengths and Loewy series of generalized Verma modules and of their projective covers in $ {\mathcal{O}_S}$ are studied with primary emphasis on the case in which ${\mathfrak{p}_S}$ is a Borel subalgebra and ${\mathcal{O}_S}$ is the category $\mathcal{O}$. An examination of the change in Loewy length of modules under translation leads to the calculation of Loewy length for Verma modules and for self-dual projectives in $\mathcal{O}$, assuming the Kazhdan-Lusztig conjecture (in an equivalent formulation due to Vogan). In turn, it is shown that the Loewy length results imply Vogan's statement, and lead to the determination of Loewy length for the self-dual projectives and certain generalized Verma modules in ${\mathcal{O}_S}$. Under the stronger assumption of Jantzen's conjecture, the radical and socle series are computed for self-dual projectives in $\mathcal{O}$. An analogous result is formulated for self-dual projectives in ${\mathcal{O}_S}$ and proved in certain cases.
The Schur multiplier of a nilpotent group
Ursula Martin
Webb
755-763
Abstract: In this paper we obtain upper and lower bounds for the rank of the Schur multiplier of a nilpotent group in terms of the nilpotency class and the number of generators and rank of the derived quotient.
Differential identities in prime rings with involution
Charles
Lanski
765-787
Abstract: Let $R$ be a prime ring with involution. Using work of V. K. Kharchenko it is shown that any generalized identity for $R$ involving derivations of $R$ and the involution of $R$ is a consequence of the generalized identities with involution which $R$ satisfies. In obtaining this result, a generalization, to rings satisfying a GPI, of the classical theorem characterizing inner derivations of finite-dimensional simple algebras is required. Consequences of the main theorem are that in characteristic zero no outer derivation of $R$ can act algebraically on the set of symmetric elements of $R$, and if the images of the set of symmetric elements under the derivations of $R$ satisfy a polynomial relation, then $ R$ must satisfy a generalized polynomial identity.
Factorization of diagonally dominant operators on $L\sb 1([0,1],X)$
Kevin T.
Andrews;
Joseph D.
Ward
789-800
Abstract: Let $X$ be a separable Banach space. It is shown that every diagonally dominant invertible operator on $ {L_1}([0,\,1],\,X)$ can be factored uniquely as a product of an invertible upper triangular operator and an invertible unit lower triangular operator.
Multipliers on the space of semiperiodic sequences
Manuel Núñez
Jiménez
801-811
Abstract: Semiperiodic sequences are defined to be the uniform limit of periodic sequences. They form a space of continuous functions on a compact group $\Delta$. We study the properties of the Radon measures on $\Delta$ in order to classify the multipliers for the space of semiperiodic sequences, paying special attention to those which can be realized as transference functions of physically constructible filters.