The functional determinant of a four-dimensional boundary value problem
Thomas P.
Branson;
Peter B.
Gilkey
479-531
Abstract: Working on four-dimensional manifolds with boundary, we consider, elliptic boundary value problems (A, B), A being the interior and B the boundary operator. These problems (A, B) should be valued in a tensorspinor bundle; should depend in a universal way on a Riemannian metric g and be formally selfadjoint; should behave in an appropriate way under conformal change $g \to {\Omega ^2}g$, $\Omega$ a smooth positive function; and the leading symbol of A should be positive definite. We view the functional determinant det $ {A_B}$ of such a problem as a functional on a conformal class $\{ {\Omega ^2}g\}$, and develop a formula for the quotient of the determinant at $ {\Omega ^2}g$ by that at g. (Analogous formulas are known to be intimately related to physical string theories in dimension two, and to sharp inequalities of borderline Sobolev embedding and Moser-Trudinger types for the boundariless case in even dimensions.) When the determinant in a background metric ${g_0}$ is explicitly computable, the result is a formula for the determinant at each metric ${\Omega ^2}{g_0}$ (not Just a quotient of determinants). For example, we compute the functional determinants of the Dirichlet and Robin (conformally covariant Neumann) problems for the Laplacian in the ball $ {B^4}$, using our general quotient formulas in the case of the conformal Laplacian, together with an explicit computation on the hemisphere ${H^4}$.
Densely defined selections of multivalued mappings
M. M.
Čoban;
P. S.
Kenderov;
J. P.
Revalski
533-552
Abstract: Rather general suficient conditions are found for a given multivalued mapping $F:X \to Y$ to possess an upper semicontinuous and compact-valued selection G which is defined on a dense $ {G_\delta }$-subset of the domain of F. The case when the selection G is single-valued (and continuous) is also investigated. The results are applied to prove some known as well as new results concerning generic differentiability of convex functions, Lavrentieff type theorem, generic well-posedness of optimization problems and generic non-multivaluedness of metric projections and antiprojections.
On $U$-rank $2$ types
Ludomir
Newelski
553-581
Abstract: Let T be a superstable theory with $< {2^{{\aleph _0}}}$ countable models. We study some special types $p \in S(\emptyset )$ of U-rank 2 called skeletal (cf. [Bu4]). We reduce an eventual version of the problem of counting isomorphism types of sets $p(M)$ for countable M to a problem from linear algebra.
Subanalytic functions
Adam
Parusiński
583-595
Abstract: We prove a strong version of rectilinearization theorem for subanalytic functions. Then we use this theorem to study the properties of arc-analytic functions.
Partition identities and labels for some modular characters
G. E.
Andrews;
C.
Bessenrodt;
J. B.
Olsson
597-615
Abstract: In this paper we prove two conjectures on partitions with certain conditions. A motivation for this is given by a problem in the modular representation theory of the covering groups ${\hat S_n}$ of the finite symmetric groups $ {S_n}$ in characteristic 5. One of the conjectures (Conjecture B below) has been open since 1974, when it was stated by the first author in his memoir [A3]. Recently the second and third author (jointly with A. O. Morris) arrived at essentially the same conjecture from a completely different direction. Their paper [BMO] was concerned with decomposition matrices of $ {\hat S_n}$ in characteristic 3. A basic difficulty for obtaining similar results in characteristic 5 (or larger) was the lack of a class of partitions which would be "natural" character labels for the modular characters of these groups. In this connection two conjectures were stated (Conjectures A and ${B^\ast}$ below), whose solutions would be helpful in the characteristic 5 case. One of them, Conjecture $ {{\text{B}}^\ast}$, is equivalent to the old Conjecture B mentioned above. Conjecture A is concerned with a possible inductive definition of the set of partitions which should serve as the required labels.
Best comonotone approximation
Frank
Deutsch;
Jun
Zhong
617-627
Abstract: A general theory of best comonotone approximation in $ C[a,b]$ by elements of an n-dimensional extended Chebyshev subspace is described. In particular, theorems on the existence, (in general) nonuniqueness, and characterization of best comonotone approximations are established.
The algorithmic theory of finitely generated metabelian groups
Gilbert
Baumslag;
Frank B.
Cannonito;
Derek J. S.
Robinson
629-648
Abstract: Algorithms are constructed which, when an explicit presentation of a finitely generated metabelian group G in the variety $ {\mathcal{A}^2}$ is given, produce finitary presentations for the derived subgroup $G\prime$, the centre $Z(G)$, the Fitting subgroup $\operatorname{Fit}(G)$, and the Frattini subgroup $\varphi (G)$. Additional algorithms of independent interest are developed for commutative algebra which construct the associated set of primes $\operatorname{Ass}(M)$ of a finitely generated module M over a finitely generated commutative ring R, and the intersection ${\varphi _R}(M)$ of the maximal submodules of M.
Nonorientable $4$-manifolds with fundamental group of order $2$
Ian
Hambleton;
Matthias
Kreck;
Peter
Teichner
649-665
Abstract: In this paper we classify nonorientable topological closed 4-manifolds with fundamental group $ \mathbb{Z}/2$ up to homeomorphism. Our results give a complete list of such manifolds, and show how they can be distinguished by explicit invariants including characteristic numbers and the $\eta$-invariant associated to a normal $ Pin^c$-structure by the spectral asymmetry of a certain Dirac operator. In contrast to the oriented case, there exist homotopy equivalent nonorientable topological 4-manifolds which are stably homeomorphic (after connected sum with ${S^2} \times {S^2}$) but not homeomorphic.
Approximation properties for group $C\sp *$-algebras and group von Neumann algebras
Uffe
Haagerup;
Jon
Kraus
667-699
Abstract: Let G be a locally compact group, let $ C_r^\ast(G)$ (resp. ${\text{VN}}(G)$) be the ${C^\ast}$-algebra (resp. the von Neumann algebra) associated with the left regular representation l of G, let $A(G)$ be the Fourier algebra of G, and let $ {M_0}A(G)$ be the set of completely bounded multipliers of $ A(G)$. With the completely bounded norm, ${M_0}A(G)$ is a dual space, and we say that G has the approximation property (AP) if there is a net $\{ {u_\alpha }\}$ of functions in $ A(G)$ (with compact support) such that $ {u_\alpha } \to 1$ in the associated weak $^\ast$-topology. In particular, G has the AP if G is weakly amenable ( $\Leftrightarrow A(G)$ has an approximate identity that is bounded in the completely bounded norm). For a discrete group $\Gamma$, we show that $\Gamma$ has the ${\text{AP}} \Leftrightarrow C_r^\ast(\Gamma )$ has the slice map property for subspaces of any ${C^\ast}$-algebra $\Leftrightarrow {\text{VN}}(\Gamma )$ has the slice map property for $\sigma$-weakly closed subspaces of any von Neumann algebra (Property $ {S_\sigma }$). The semidirect product of weakly amenable groups need not be weakly amenable. We show that the larger class of groups with the AP is stable with respect to semidirect products, and more generally, this class is stable with respect to group extensions. We also obtain some results concerning crossed products. For example, we show that the crossed product $ M{ \otimes _\alpha }G$ of a von Neumann algebra M with Property ${S_\sigma }$ by a group G with the AP also has Property $ {S_\sigma }$.
Homotopy groups in Lie foliations
Enrique
Macias-Virgós
701-711
Abstract: According to the results of Fédida and Molino [9], the structure of a G-Lie foliation F on a compact manifold M can be described by means of four locally trivial fibre bundles. In this paper we study the relations that those fibrations imply among the (rational) homotopy groups of: the manifold M, the generic leaf L, its closure $N = \bar L$, the basic manifold W, the Lie group G, and the structural Lie group H. Also, we prove that those relations are a particular case of an algebraic result concerning generalized homology theories.
Dirichlet problem at infinity for harmonic maps: rank one symmetric spaces
Harold
Donnelly
713-735
Abstract: Given a symmetric space M, of rank one and noncompact type, one compactifies M by adding a sphere at infinity, to obtain a manifold $M\prime$ with boundary. If $\bar M$ is another rank one symmetric space, suppose that $f:\partial M\prime \to \partial \bar M\prime$ is a continuous map. The Dirichlet problem at infinity is to construct a proper harmonic map $u:M \to \bar M$ with boundary values f. This paper concerns existence, uniqueness, and boundary regularity for this Dirichlet problem.
Measures of chaos and a spectral decomposition of dynamical systems on the interval
B.
Schweizer;
J.
Smítal
737-754
Abstract: Let $f:[0,1] \to [0,1]$ be continuous. For $x,y \in [0,1]$, the upper and lower (distance) distribution functions, $ F_{xy}^\ast$ and $ {F_{xy}}$, are defined for any $t \geq 0$ as the lim sup and lim inf as $n \to \infty$ of the average number of times that the distance $ \vert{f^i}(x) - {f^i}(y)\vert$ between the trajectories of x and y is less than t during the first n iterations. The spectrum of f is the system $\Sigma (f)$ of lower distribution functions which is characterized by the following properties: (1) The elements of $ \Sigma (f)$ are mutually incomparable; (2) for any $F \in \Sigma (f)$, there is a perfect set ${P_F} \ne \emptyset$ such that ${F_{uv}} = F$ and $F_{uv}^\ast \equiv 1$ for any distinct u, $v \in {P_F}$; (3) if S is a scrambled set for f, then there are F, G in $\Sigma (f)$ and a decomposition $S = {S_F} \cup {S_G}$ (${S_G}$ may be empty) such that ${F_{uv}} \geq F$ if u, $v \in {S_F}$ and $ {F_{uv}} \geq G$ if u, $v \in {S_G}$. Our principal results are: (1) If f has positive topological entropy, then $\Sigma (f)$ is nonempty and finite, and any $F \in \Sigma (f)$ is zero on an interval $[0,\varepsilon]$, where $\varepsilon > 0$ (and hence any ${P_F}$ is a scrambled set in the sense of Li and Yorke). (2) If f has zero topological entropy, then $ \Sigma (f) = \{ F\}$ where $F \equiv 1$. It follows that the spectrum of f provides a measure of the degree of chaos of f. In addition, a useful numerical measure is the largest of the numbers $\int_0^1 {(1 - F(t))dt} $, where $F \in \Sigma (f)$.
On an integral representation for the genus series for $2$-cell embeddings
D. M.
Jackson
755-772
Abstract: An integral representation for the genus series for maps on oriented surfaces is derived from the combinatorial axiomatisation of 2-cell embeddings in orientable surfaces. It is used to derive an explicit expression for the genus series for dipoles. The approach can be extended to vertex-regular maps in general and, in this way, may shed light on the genus series for quadrangulations. The integral representation is used in conjunction with an approach through the group algebra, $ \mathbb{C}{\mathfrak{G}_n}$, of the symmetric group [11] to obtain a factorisation of certain Gaussian integrals.
The dynamics of continuous maps of finite graphs through inverse limits
Marcy
Barge;
Beverly
Diamond
773-790
Abstract: Suppose that $ f:G \to G$ is a continuous piecewise monotone function on a finite graph G. Then the following are equivalent: (i) f has positive topological entropy; (ii) there are disjoint intervals ${I_1}$, and ${I_2}$ and a positive integer n with $\displaystyle {I_1} \cup {I_2} \subseteq {f^n}({I_1}) \cap {f^n}({I_2});$ (iii) the inverse limit space constructed by using f on G as a single bonding map contains an indecomposable subcontinuum. This result generalizes known results for the interval and circle.
Composition operators with closed range
Nina
Zorboska
791-801
Abstract: We characterize the closed-range composition operators on weighted Bergman spaces in terms of the ranges of the inducing maps on the unit disc. The method uses Nevanlinna's counting function and Luecking's results on inequalities on Bergman spaces.
Amenable actions of groups
Scot
Adams;
George A.
Elliott;
Thierry
Giordano
803-822
Abstract: The equivalence between different characterizations of amenable actions of a locally compact group is proved. In particular, this answers a question raised by R. J. Zimmer in 1977.
Infinitesimally stable endomorphisms
Hiroshi
Ikeda
823-833
Abstract: It is well known that infinitesimal stability of diffeomorphisms is an open property. However, infinitesimal stability of endomorphisms is not an open property. So we consider the interior of the set of all infinitesimally stable endomorphisms. We prove that if f belongs to the interior of the set of all infinitesimally stable endomorphisms, then f is $\Omega$-stable. This means a generalization of Smale's $\Omega$-stability theorem for diffeomorphisms. Moreover, it is proved that for Anosov endomorphisms structural stability is equivalent to lying in the interior of the set of infinitesimally stable endomorphisms.
There is just one rational cone-length
Octavian
Cornea
835-848
Abstract: We show that the homotopic nilpotency of the algebra of piecewise polynomial forms on a simply-connected, finite type, CW-complex coincides with the strong L.S. category of the rationalization of that space. This is used to prove that, in the rational, simply-connected context all reasonable notions of cone-length agree. Both these two results are obtained as parts of a more general and functorial picture.
Completely continuous composition operators
Joseph A.
Cima;
Alec
Matheson
849-856
Abstract: A composition operator ${T_b}f = f \circ b$ is completely continuous on ${H^1}$ if and only if $\vert b\vert < 1$ a.e. If the adjoint operator $ T_b^\ast$ is completely continuous on VMOA, then ${T_b}$ is completely continuous on $ {H^1}$. Examples are given to show that the converse fails in general. Two results are given concerning the relationship between the complete continuity of an operator and of its adjoint in the presence of certain separability conditions on the underlying Banach space.
The Picard group, closed geodesics and zeta functions
Mark
Pollicott
857-872
Abstract: In this article we consider the Picard group $ {\text{SL}}(2,\mathbb{Z}[i])$, viewed as a discrete subgroup of the isometries of hyperbolic space. We fix a canonical choice of generators and then construct a Markov partition for the action of the group on the sphere at infinity. Our main application is to the study of the zeta function associated to the associated three-dimensional hyperbolic manifold.
Notes sur la propri\'et\'e de Namioka
Ahmed
Bouziad
873-883
Abstract: We show that the class of co-Namioka compacts is stable under the arbitrary product if and only if it is stable under the finished product. We also prove that if X is a Valdivia compact space, then for every co-Namioka compact Y the product $X \times Y$ is co-Namioka. Several examples of co-Namioka compacts are given.
Hochschild homology in a braided tensor category
John C.
Baez
885-906
Abstract: An r-algebra is an algebra A over k equipped with a Yang-Baxter operator $R:A \otimes A \to A \otimes A$ such that $R(1 \otimes a) = a \otimes 1$, $R(a \otimes 1) = 1 \otimes a$, and the quasitriangularity conditions $ R(m \otimes I) = (I \otimes m)(R \otimes I)(I \otimes R)$ and $R(I \otimes m) = (m \otimes I)(I \otimes R)(R \otimes I)$ hold, where $ m:A \otimes A \to A$ is the multiplication map and $I:A \to A$ is the identity. R-algebras arise naturally as algebra objects in a braided tensor category of k-modules (e.g., the category of representations of a quantum group). If $m = m{R^2}$, then A is both a left and right module over the braided tensor product $ {A^e} = A\hat \otimes {A^{{\text{op}}}}$, where $ {A^{{\text{op}}}}$ is simply A equipped with the "opposite" multiplication map ${m^{{\text{op}}}} = mR$. Moreover, there is an explicit chain complex computing the braided Hochschild homology ${H^R}(A) = \operatorname{Tor}^{{A^e}}(A,A)$. When $m = mR$ and ${R^2} = {\text{id}}_{A \otimes A}$, this chain complex admits a generalized shuffle product, and there is a homomorphism from the r-commutative differential forms $ {\Omega _R}(A)$ to $ {H^R}(A)$.
A distortion theorem for biholomorphic mappings in ${\bf C}\sp 2$
Roger W.
Barnard;
Carl H.
FitzGerald;
Sheng
Gong
907-924
Abstract: Let ${J_f}$ be the Jacobian of a normalized biholomorphic mapping f from the unit ball $ {B^2}$ into ${\mathbb{C}^2}$. An expression for the $\log \det {J_f}$ is determined by considering the series expansion for the renormalized mappings F obtained from f under the group of holomorphic automorphisms of ${B^2}$. This expression is used to determine a bound for $\vert\det {J_f}\vert$ and $\vert\arg \det {J_f}\vert$ for f in a compact family X of normalized biholomorphic mappings from ${B^2}$ into $ {\mathbb{C}^2}$ in terms of a bound $C(X)$ of a certain combination of second-order coefficients. Estimates are found for $C(X)$ for the specific family X of normalized convex mappings from ${B^2}$ into ${\mathbb{C}^2}$.
The Jacobson radical of a CSL algebra
Kenneth R.
Davidson;
John Lindsay
Orr
925-947
Abstract: Extrapolating from Ringrose's characterization of the Jacobson radical of a nest algebra, Hopenwasser conjectured that the radical of a CSL algebra coincides with the Ringrose ideal (the closure of the union of zero diagonal elements with respect to finite sublattices). A general interpolation theorem is proved that reduces this conjecture for completely distributive lattices to a strictly combinatorial problem. This problem is solved for all width two lattices (with no restriction of complete distributivity), verifying the conjecture in this case.