Characteristic matrices and spectral properties of evolutionary systems
M. A.
Kaashoek;
S. M.
Verduyn Lunel
479-517
Abstract: In this paper we introduce the notion of a characteristic matrix for a large class of unbounded operators and study the precise connection between characteristic matrices and spectral properties of evolutionary systems. In particular, we study so-called multiplicity theorems. Several examples will illustrate our results.
Generalized group presentation and formal deformations of CW complexes
Richard A.
Brown
519-549
Abstract: A Peiffer-Whitehead word system $ \mathcal{W}$, or generalized group presentation, consists of generators, relators (words of order $2$), and words of higher order $n$ that represent elements of a free crossed module $(n = 3)$ or a free module $(n > 3)$. The ${P_n}$-equivalence relation on word systems generalizes the extended Nielsen equivalence relation on ordinary group presentations. Word systems, called homotopy readings, can be associated with any connected $ {\text{CW}}$ complex $ K$ by removing a maximal tree and selecting one generator or word per cell, via relative homotopy. Given homotopy readings ${\mathcal{W}_1}$ and ${\mathcal{W}_2}$ of finite ${\text{CW}}$ complexes ${K_1}$ and ${K_2}$ respectively, we show that ${\mathcal{W}_1}$ is ${P_n}$-equivalent to ${\mathcal{W}_2}$ if and only if ${K_1}$ formally $(n + 1)$-deforms to $ {K_2}$. This extends results of P. Wright (1975) and W. Metzler (1982) for the case $n = 2$. For $n \geq 3$, it follows that ${\mathcal{W}_1}$ is ${P_n}$-equivalent to ${\mathcal{W}_2}$ if and only if ${K_1}$ and ${K_2}$ have the same simple homotopy type.
Estimates of the Caccioppoli-Schauder type in weighted function spaces
Giovanni Maria
Troianiello
551-573
Abstract: We deal with imbeddings of certain weighted function spaces as well as with the corresponding norm estimates for solutions to second order elliptic problems. We redemonstrate some results of Gilbarg and Hörmander by a technique, entirely different from theirs, which enables us to cover a range of parameters excluded by them.
Cohomology of the symplectic group ${\rm Sp}\sb 4({\bf Z})$. I. The odd torsion case
Alan
Brownstein;
Ronnie
Lee
575-596
Abstract: Let ${h_2}$ be the degree two Siegel space and $ Sp(4,\mathbb{Z})$ the symplectic group. The quotient $Sp(4,\mathbb{Z})\backslash {h_2}$ can be interpreted as the moduli space of stable Riemann surfaces of genus $2$. This moduli space can be decomposed into two pieces corresponding to the moduli of degenerate and nondegenerate surfaces of genus $2$. The decomposition leads to a Mayer-Vietoris sequence in cohomology relating the cohomology of $ Sp(4,\mathbb{Z})$ to the cohomology of the genus two mapping class group $\Gamma _2^0$. Using this tool, the $3$- and $5$-primary pieces of the integral cohomology of $Sp(4,\mathbb{Z})$ are computed.
Hausdorff dimension of wild fractals
T. B.
Rushing
597-613
Abstract: We show that for every $s \in [n - 2,n]$ there exists a homogeneously embedded wild Cantor set ${C^s}$ in $\mathbb{R}^n, n \geq 3$, of (local) Hausdorff dimension $s$. Also, it is shown that for every $s \in [n - 2,n]$ and for any integer $k \ne n$ such that $1 \leq k \leq s$, there exist everywhere wild $ k$-spheres and $ k$-cells, in $\mathbb{R}^n, n \geq 3$, of (local) Hausdorff dimension $s$.
Hypoellipticity on Cauchy-Riemann manifolds
Johannes A.
Petersen
615-639
Abstract: Using a recent homotopy formula by Trèves, it is shown that the existence of $(q + 1)$-dimensional holomorphic supporting manifolds is a sufficient condition for the hypoellipticity on level $q$ and $n - q$ of a tangential Cauchy-Riemann complex of ${\text{CR}}$-dimension $n$. In the hypersurface case, this result is given microlocally.
The Mizohata structure on the sphere
Jorge
Hounie
641-649
Abstract: We prove that a compact surface that admits a Mizohata structure is (homeomorphic to) a sphere and that there exists exactly one such structure $ \mathcal{L}$ up to conjugation by diffeomorphisms. We also characterize the range and the kernel of the operator ${\delta _0}$ induced by $ \mathcal{L}$ , i.e., obtained from the exterior derivative on functions by passing to the quotient modulo ${\mathcal{L}^\bot }$ .
The monotonicity of the entropy for a family of degree one circle maps
Lluís
Alsedà;
Francesc
Mañosas
651-684
Abstract: For the natural biparametric family of piecewise linear circle maps with two pieces we show that the entropy increases when any of the two parameters increases. We also describe the regions of the parameter space where the monotonicity is strict.
On the structure of twisted group $C\sp *$-algebras
Judith A.
Packer;
Iain
Raeburn
685-718
Abstract: We first give general structural results for the twisted group algebras $ {C^{\ast} }(G,\sigma )$ of a locally compact group $G$ with large abelian subgroups. In particular, we use a theorem of Williams to realise ${C^{\ast}}(G,\sigma )$ as the sections of a ${C^{\ast}}$-bundle whose fibres are twisted group algebras of smaller groups and then give criteria for the simplicity of these algebras. Next we use a device of Rosenberg to show that, when $\Gamma$ is a discrete subgroup of a solvable Lie group $G$, the $K$-groups ${K_ {\ast} }({C^{\ast} }(\Gamma ,\sigma ))$ are isomorphic to certain twisted $K$-groups ${K^{\ast} }(G/\Gamma ,\delta (\sigma ))$ of the homogeneous space $G/\Gamma$, and we discuss how the twisting class $\delta (\sigma ) \in {H^3}(G/\Gamma ,\mathbb{Z})$ depends on the cocycle $\sigma$. For many particular groups, such as $ {\mathbb{Z}^n}$ or the integer Heisenberg group, $ \delta (\sigma )$ always vanishes, so that ${K_ {\ast} }({C^{\ast} }(\Gamma ,\sigma ))$ is independent of $\sigma$, but a detailed analysis of examples of the form ${\mathbb{Z}^n} \rtimes \mathbb{Z}$ shows this is not in general the case.
$H\sp p$- and $L\sp p$-variants of multiparameter Calder\'on-Zygmund theory
Anthony
Carbery;
Andreas
Seeger
719-747
Abstract: We consider Calderón-Zygmund operators on product domains. Under certain weak conditions on the kernel a singular integral operator can be proved to be bounded on ${H^p}(\mathbb{R} \times \mathbb{R} \times \cdots \times \mathbb{R}), 0 < p \leq 1$, if its behaviour on ${L^2}$ and on certain scalar-valued and vector-valued rectangle atoms is known. Another result concerns an extension of the authors' results on $ {L^p}$-variants of Calderón-Zygmund theory [1,23] to the product-domain-setting. As an application, one obtains estimates for Fourier multipliers and pseudo-differential operators.
Anomalies associated to the polar decomposition of ${\rm GL}(n,{\bf C})$
Steven
Rosenberg
749-760
Abstract: Let $D$ be a selfadjoint elliptic differential operator on a hermitian bundle over a compact manifold. For positive $D$, the variation of the functional determinant of $ D$ under positive definite hermitian gauge transformations is calculated. This corresponds to computing a gauge anomaly in the nonunitary directions of the polar decomposition of the frame bundle $ {\text{GL}}(E)$. The variation of the eta invariant for general $D$ is also calculated. If $ D$ is not selfadjoint, the integrand in the heat equation proof of the Atiyah-Singer Index Theorem is interpreted as an anomaly for ${D^{\ast} }D$ . In particular, the gauge anomaly for semiclassical Yang-Mills theory is computed.
Eta invariants of Dirac operators on foliated manifolds
Goran
Perić
761-782
Abstract: We define the eta function of Dirac operators on foliated manifolds. We show that the eta functions are regular at the origin thus defining corresponding eta invariants of foliated manifolds. Under the hypothesis of invertibility of the operator in question, we prove the Atiyah-Singer relative index theorem for Dirac operators on foliated manifolds. Then we discuss the homotopy invariance of the index with respect to secondary data.
A PL geometric study of algebraic $K$ theory
Bi Zhong
Hu
783-808
Abstract: This paper manages to apply the Farrell-Jones theory on algebraic $ K$-groups of closed negatively curved riemannian manifolds to Gromov's hyperbolic group theory. The paper reaches the conclusion that for any finite polyhedron $K$ with negative curvature, $ \operatorname{Wh}({\pi _1}K) = 0$ .
On the generalized Ramanujan-Nagell equation $x\sp 2-D=2\sp {n+2}$
Mao Hua
Le
809-825
Abstract: Let $D$ be a positive integer which is odd. In this paper we prove that the equation ${x^2} - D = {2^{n + 2}}$ has at most three positive integer solutions $(x,n)$ except when $D = {2^{2m}} - 3 \cdot {2^{m + 1}} + 1$ , where $m$ is a positive integer with $m \geq 3$ .
Entropy for canonical shifts
Marie
Choda
827-849
Abstract: For a $ ^{\ast}$-endomorphism $ \sigma$ of an injective finite von Neumann algebra $A$ , we investigate the relations among the entropy $H(\sigma )$ for $\sigma$ , the relative entropy $H(A\vert\sigma (A))$ of $ \sigma (A)$ for $ A$ , the generalized index $ \lambda (A,\sigma (A))$, and the index for subfactors. As an application, we have the following relations for the canonical shift $ \Gamma$ for the inclusion $N \subset M$ of type II$_{1}$ factors with the finite index $[M:N]$, $\displaystyle H(A\vert\Gamma (A)) \leq 2H(\Gamma ) \leq \log \lambda {(A,\Gamma (A))^{ - 1}} = 2\log [M:N],$ where $A$ is the von Neumann algebra generated by the two of the relative commutants of $M$. In the case of that $ N \subset M$ has finite depth, then all of them coincide.
On the self-intersections of foliation cycles
Yoshihiko
Mitsumatsu
851-860
Abstract: The existence of a transverse invariant measure imposes a strong restriction on the transverse complexity of a foliated manifold. The homological self-intersection of the corresponding foliation cycle measures the complexity around its support. In the present paper, the vanishing of the self-intersection is proven under some regularity condition on the measure.
An inverse problem for circle packing and conformal mapping
Ithiel
Carter;
Burt
Rodin
861-875
Abstract: Figure 1.1 illustrates the fact that if a region is almost packed with circles of radius $ \varepsilon$ in the hexagonal pattern and if the unit disk is packed in an isomorphic pattern with circles of varying radii then, after suitable normalization, the correspondence of circles converges to the Riemann mapping function as $\varepsilon \to 0$ (see [15]). In the present paper an inverse of this result is obtained as illustrated by Figure 1.2; namely, if the unit disk is almost packed with $\varepsilon $-circles there is an isomorphic circle packing almost filling the region such that, after suitable normalization, the circle correspondence converges to the conformal map of the disk onto the region as $ \varepsilon \to 0$. Note that this set up yields an approximate triangulation of the region by joining the centers of triples of mutually tangent circles. Since this triangulation is intimately related to the Riemann mapping it may be useful for grid generation [18].
Generation and propagation of interfaces in reaction-diffusion systems
Xinfu
Chen
877-913
Abstract: This paper is concerned with the asymptotic behavior, as $\varepsilon \searrow 0$, of the solution $ ({u^\varepsilon },{v^\varepsilon })$ of the second initial-boundary value problem of the reaction-diffusion system: $\displaystyle \left\{ {\begin{array}{*{20}{c}} {u_t^\varepsilon - \varepsilon \... ...varepsilon } - \gamma {\upsilon ^\varepsilon }} \end{array} } \right.$ where $\gamma > 0$ is a constant. When $v \in ( - 2\sqrt 3 /9,2\sqrt 3 /9)$, $f$ is bistable in the sense that the ordinary differential equation $ {u_t} = f(u,v)$ has two stable solutions $ u = {h_ - }(v)$ and $u = {h_ + }(v)$ and one unstable solution $u = {h_0}(v)$, where ${h_ - }(v), {h_0}(v)$, and ${h_ + }(v)$ are the three solutions of the algebraic equation $ f(u,v) = 0$. We show that, when the initial data of $v$ is in the interval $( - 2\sqrt 3 /9,2\sqrt 3 /9)$, the solution $ ({u^\varepsilon },{v^\varepsilon })$ of the system tends to a limit $ (u,v)$ which is a solution of a free boundary problem, as long as the free boundary problem has a unique classical solution. The function $u$ is a "phase" function in the sense that it coincides with ${h_ + }(v)$ in one region ${\Omega _ + }$ and with $ {h_ - }(v)$ in another region ${\Omega _ - }$. The common boundary (free boundary or interface) of the two regions ${\Omega _ - }$ and $ {\Omega _ + }$ moves with a normal velocity equal to $\mathcal{V}(v)$, where $\mathcal{V}( \bullet )$ is a function that can be calculated. The local (in time) existence of a unique classical solution to the free boundary problem is also established. Further we show that if initially $ u( \bullet, 0) - {h_0}(v( \bullet, 0))$ takes both positive and negative values, then an interface will develop in a short time $ O(\varepsilon \vert\ln \varepsilon \vert)$ near the hypersurface where $u(x,0) - {h_0}(v(x,0)) = 0$.
Brauer-Hilbertian fields
Burton
Fein;
David J.
Saltman;
Murray
Schacher
915-928
Abstract: Let $F$ be a field of characteristic $ p$ ($p = 0$ allowed), and let $ F(t)$ be the rational function field in one variable over $F$. We say $F$ is Brauer-Hilbertian if the following holds. For every $\alpha$ in the Brauer group $\operatorname{Br}(F(t))$ of exponent prime to $ p$, there are infinitely many specializations $ t \to a \in F$ such that the specialization $\bar \alpha \in \operatorname{Br}(F)$ is defined and has exponent equal to that of $ \alpha$. We show every global field is Brauer-Hilbertian, and if $ K$ is Hilbertian and $ F$ is finite separable over $K(t)$, $F$ is Brauer-Hilbertian.
Single loop space decompositions
David J.
Anick
929-940
Abstract: The method of single loop space decompositions, in which $\Omega X$ is factored into minimal factors, is an important one for understanding the unstable homotopy of many simply-connected spaces $ X$. This paper begins with a survey of the major known theorems along these lines. We then give a necessary and sufficient condition for $ \Omega X$ to be decomposable as a product of spaces belonging to a certain list. We conclude with a nontrivial instance of an application of this condition.