Half De Rham complexes and line fields on odd-dimensional manifolds
Houhong
Fan
2947-2982
Abstract: In this paper we introduce a new elliptic complex on an odd-dimensional manifold with a self-dual line field. The notion of a self-dual line field is a generalization of the notion of a conformal line field. Ellipticity, Fredholm properties and Hodge decompositions of these new complexes are proved both in the case of a closed manifold and in the case of a manifold with boundary. The cohomology groups of these elliptic complexes are computed in some cases. In addition, in this paper, we generalize the notion of an anti-self-dual connection on a smooth 4-manifold to a 3-manifold with a line field and a smooth 5-manifold with a line field. The above new elliptic complexes can be twisted by anti-self-dual connections in dimensions 3 and 5, but only by flat connections in dimensions above 5. This reveals a special feature of dimensions 3 and 5.
An algebraic approach to multiparameter spectral theory
Luzius
Grunenfelder;
Tomaz
Kosir
2983-2998
Abstract: Root subspaces for multiprameter eigenvalue problems are described using coalgebraic techniques. An algorithm is given to construct bases for the root subspaces.
Topological entropy of standard type monotone twist maps
Oliver
Knill
2999-3013
Abstract: We study invariant measures of families of monotone twist maps $S_{\gamma }(q,p)$ $=$ $(2q-p+ \gamma \cdot V'(q),q)$ with periodic Morse potential $V$. We prove that there exist a constant $C=C(V)$ such that the topological entropy satisfies $h_{top}(S_{\gamma }) \geq \log (C \cdot \gamma )/3$. In particular, $h_{top}(S_{\gamma }) \to \infty$ for $|\gamma | \to \infty$. We show also that there exist arbitrary large $\gamma$ such that $S_{\gamma }$ has nonuniformly hyperbolic invariant measures $\mu _{\gamma }$ with positive metric entropy. For large $\gamma$, the measures $\mu _{\gamma }$ are hyperbolic and, for a class of potentials which includes $V(q)=\sin (q)$, the Lyapunov exponent of the map $S$ with invariant measure $\mu _{\gamma }$ grows monotonically with $\gamma$.
The Schwarzian derivative for maps between manifolds with complex projective connections
Robert
Molzon;
Karen
Pinney
Mortensen
3015-3036
Abstract: In this paper we define, in two equivalent ways, the Schwarzian derivative of a map between complex manifolds equipped with complex projective connections. Also, a new, coordinate-free definition of complex projective connections is given. We show how the Schwarzian derivative is related to the projective structure of the manifolds, to projective linear transformations, and to complex geodesics.
Cohomology of the complement of a free divisor
Francisco
J.
Castro-Jiménez;
Luis
Narváez-Macarro;
David
Mond
3037-3049
Abstract: We prove that if $D$ is a ``strongly quasihomogeneous" free divisor in the Stein manifold $X$, and $U$ is its complement, then the de Rham cohomology of $U$ can be computed as the cohomology of the complex of meromorphic differential forms on $X$ with logarithmic poles along $D$, with exterior derivative. The class of strongly quasihomogeneous free divisors, introduced here, includes free hyperplane arrangements and the discriminants of stable mappings in Mather's nice dimensions (and in particular the discriminants of Coxeter groups).
Compact self-dual Hermitian surfaces
Vestislav
Apostolov;
Johann
Davidov;
Oleg
Muskarov
3051-3063
Abstract: In this paper, we obtain a classification (up to conformal equivalence) of the compact self-dual Hermitian surfaces. As an application, we prove that every compact Hermitian surface of pointwise constant holomorphic sectional curvature with respect to either the Riemannian or the Hermitian connection is Kähler.
Extensions of codimension one immersions
Christian
Pappas
3065-3083
Abstract: We give a method for constructing all of the extensions of an immersion, and determine the CW structure and diffeomorphism type of each.
On vanishing of characteristic numbers in Poincaré complexes
Yanghyun
Byun
3085-3095
Abstract: Let $G_r(X)\subset \pi _r(X)$ be the evaluation subgroup as defined by Gottlieb. Assume the Hurewicz map $G_r(X)\rightarrow H_r(X; R)$ is non-trivial and $R$ is a field. We will prove: if $X$ is a Poincaré complex oriented in $R$-coefficient, all the characteristic numbers of $X$ in $R$-coefficient vanish. Similarly, if $R=Z_p$ and $X$ is a $Z_p$-Poincaré complex, then all the mod $p$ Wu numbers vanish. We will also show that the existence of a non-trivial derivation on $H^*(X; Z_p)$ with some suitable conditions implies vanishing of mod $p$ Wu numbers.
Presentation and central extensions of mapping class groups
Sylvain
Gervais
3097-3132
Abstract: We give a presentation of the mapping class group $\mathcal {M}$ of a (possibly bounded) surface, considering either all twists or just non-separating twists as generators. We also study certain central extensions of $\mathcal {M}$. One of them plays a key role in studying TQFT functors, namely the mapping class group of a $p_1$-structure surface. We give a presentation of this extension.
Conjugate points and shocks in nonlinear optimal control
N.
Caroff;
H.
Frankowska
3133-3153
Abstract: We investigate characteristics of the Hamilton-Jacobi-Bellman equation arising in nonlinear optimal control and their relationship with weak and strong local minima. This leads to an extension of the Jacobi conjugate points theory to the Bolza control problem. Necessary and sufficient optimality conditions for weak and strong local minima are stated in terms of the existence of a solution to a corresponding matrix Riccati differential equation.
A group of paths in $\mathbb{R}^2$
Richard
Kenyon
3155-3172
Abstract: We define a group structure on the set of compact ``minimal'' paths in $\mathbb {R} ^2$. We classify all finitely generated subgroups of this group $G$: they are free products of free abelian groups and surface groups. Moreover, each such group occurs in $G$. The subgroups of $G$ isomorphic to surface groups arise from certain topological $1$-forms on the corresponding surfaces. We construct examples of such $1$-forms for cohomology classes corresponding to certain eigenvectors for the action on cohomology of a pseudo-Anosov diffeomorphism. Using $G$ we construct a non-polygonal tiling problem in $\mathbb {R} ^2$, that is, a finite set of tiles whose corresponding tilings are not equivalent to those of any set of polygonal tiles. The group $G$ has applications to combinatorial tiling problems of the type: given a set of tiles $P$ and a region $R$, can $R$ be tiled by translated copies of tiles in $P$?
The automorphism group of a coded system
Doris
Fiebig;
Ulf-Rainer
Fiebig
3173-3191
Abstract: We give a general construction of coded systems with an automorphism group isomorphic to $\mathbf Z\oplus G$ where $G$ is any preassigned group which has a ``continuous block presentation'' (the isomorphism will map the shift to $(1,e_G))$. Several applications are given. In particular, we obtain automorphism groups of coded systems which are abelian, which are finitely generated and one which contains $\mathbf Z[1/2]$. We show that any group which occurs as a subgroup of the automorphism group of some subshift with periodic points dense already occurs for some synchronized system.
Packing measure of the sample paths of fractional Brownian motion
Yimin
Xiao
3193-3213
Abstract: Let $X(t) (t \in % \mathbf {R})$ be a fractional Brownian motion of index $% \alpha$ in $% \mathbf {R}^d.$ If $1 < % \alpha d$, then there exists a positive finite constant $K$ such that with probability 1, \begin{displaymath}\hbox {$\phi$-$p(X([0,t]))$} = Kt \ \hbox {for any } t > 0 ,\end{displaymath} where $% \phi (s) = s^{\frac 1 {% \alpha }}/ (\log \log \frac 1 s)^{\frac 1 {2 % \alpha }}$ and $\phi$-$p (X([0,t]))$ is the $\phi$-packing measure of $X([0,t])$.
Which families of $l$-modal maps are full?
R.
Galeeva;
S.
van Strien
3215-3221
Abstract: In this paper we shall show that certain conditions which are sufficient for a family of one-dimensional maps to be full cannot be dispensed with.
Foxby duality and Gorenstein injective and projective modules
Edgar
E.
Enochs;
Overtoun
M. G.
Jenda;
Jinzhong
Xu
3223-3234
Abstract: In 1966, Auslander introduced the notion of the $G$-dimension of a finitely generated module over a Cohen-Macaulay noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander and Bridger (Mem. Amer. Math. Soc., vol. 94, 1969)). Enochs and Jenda attempted to dualize the notion of $G$-dimensions. It seemed appropriate to call the modules with $G$-dimension 0 Gorenstein projective, so the basic problem was to define Gorenstein injective modules. These were defined in Math. Z. 220 (1995), 611--633 and were shown to have properties predicted by Auslander's results. The way we define Gorenstein injective modules can be dualized, and so we can define Gorenstein projective modules (i.e. modules of $G$-dimension 0) whether the modules are finitely generated or not. The investigation of these modules and also Gorenstein flat modules was continued by Enochs, Jenda, Xu and Torrecillas. However, to get good results it was necessary to take the base ring Gorenstein. H.-B. Foxby introduced a duality between two full subcategories in the category of modules over a local Cohen-Macaulay ring admitting a dualizing module. He proved that the finitely generated modules in one category are precisely those of finite $G$-dimension. We extend this result to modules which are not necessarily finitely generated and also prove the dual result, i.e. we characterize the modules in the other class defined by Foxby. The basic result of this paper is that the two classes involved in Foxby's duality coincide with the classes of those modules having finite Gorenstein projective and those having finite Gorenstein injective dimensions. We note that this duality then allows us to extend many of our results to the original Auslander setting.
An $\omega_2$-minimal Boolean algebra
Mariusz
Rabus
3235-3244
Abstract: For every linear order $L$ we define a notion of $L$-minimal Boolean algebra and then give a consistent example of an $\omega _{2}$-minimal algebra. The Stone space $X$ of our algebra contains a point $\{*\}$ such that $X-\{*\}$ is an example of a countably tight, initially $\aleph _{1}$-compact, non-compact space. This answers a question of Dow and van Douwen.
Computation of Nielsen numbers for maps of closed surfaces
O.
Davey;
E.
Hart;
K.
Trapp
3245-3266
Abstract: Let $X$ be a closed surface, and let $f: X \rightarrow X$ be a map. We would like to determine $\text {Min}(f):= \mathrm {min} \{ | \mathrm {Fix}g| : g \simeq f\}.$ Nielsen fixed point theory provides a lower bound $N(f)$ for $\text {Min}(f)$, called the Nielsen number, which is easy to define geometrically and is difficult to compute. We improve upon an algebraic method of calculating $N(f)$ developed by Fadell and Husseini, so that the method becomes algorithmic for orientable closed surfaces up to the distinguishing of Reidemeister orbits. Our improvement makes tractable calculations of Nielsen numbers for many maps on surfaces of negative Euler characteristic. We apply the improved method to self-maps on the connected sum of two tori including classes of examples for which no other method is known. We also include the application of this algebraic method to maps on the Klein bottle $K$. Nielsen numbers for maps on $K$ were first calculated (geometrically) by Halpern. We include a sketch of Halpern's never published proof that $N(f)= \text {Min}(f)$ for all maps $f$ on $K$.
On quadratic forms of height two and a theorem of Wadsworth
Detlev
W.
Hoffmann
3267-3281
Abstract: Let $\varphi$ and $\psi$ be anisotropic quadratic forms over a field $F$ of characteristic not $2$. Their function fields $F(% \varphi )$ and $F(\psi )$ are said to be equivalent (over $F$) if $% \varphi \otimes F(\psi )$ and $\psi \otimes F(% \varphi )$ are isotropic. We consider the case where $\dim % \varphi =2^n$ and $% \varphi$ is divisible by an $(n-2)$-fold Pfister form. We determine those forms $\psi$ for which $% \varphi$ becomes isotropic over $F(\psi )$ if $n\leq 3$, and provide partial results for $n\geq 4$. These results imply that if $F(% \varphi )$ and $F(\psi )$ are equivalent and $\dim % \varphi =\dim \psi$, then $% \varphi$ is similar to $\psi$ over $F$. This together with already known results yields that if $% \varphi$ is of height $2$ and degree $1$ or $2$, and if $\dim % \varphi =\dim \psi$, then $F(% \varphi )$ and $F(\psi )$ are equivalent iff $F(% \varphi )$ and $F(\psi )$ are isomorphic over $F$.
On multiplicities in polynomial system solving
M.
G.
Marinari;
H.
M.
Möller;
T.
Mora
3283-3321
Abstract: This paper deals with the description of the solutions of zero dimensional systems of polynomial equations. Based on different models for describing solutions, we consider suitable representations of a multiple root, or more precisely suitable descriptions of the primary component of the system at a root. We analyse the complexity of finding the representations and of algorithms which perform transformations between the different representations.
Quadratic forms for the 1-D semilinear Schrödinger equation
Carlos
E.
Kenig;
Gustavo
Ponce;
Luis
Vega
3323-3353
Abstract: This paper is concerned with 1-D quadratic semilinearSchrödinger equations. We study local well posedness in classical Sobolev space $H^s$ of the associated initial value problem and periodic boundary value problem. Our main interest is to obtain the lowest value of $s$ which guarantees the desired local well posedness result. We prove that at least for the quadratic cases these values are negative and depend on the structure of the nonlinearity considered.
Weierstrass points on cyclic covers of the projective line
Christopher
Towse
3355-3378
Abstract: We are interested in cyclic covers of the projective line which are totally ramified at all of their branch points. We begin with curves given by an equation of the form $y^{n}=f(x)$, where $f$ is a polynomial of degree $d$. Under a mild hypothesis, it is easy to see that all of the branch points must be Weierstrass points. Our main problem is to find the total Weierstrass weight of these points, $BW$. We obtain a lower bound for $BW$, which we show is exact if $n$ and $d$ are relatively prime. As a fraction of the total Weierstrass weight of all points on the curve, we get the following particularly nice asymptotic formula (as well as an interesting exact formula): \begin{equation*}\lim _{d\to \infty }\frac {BW}{g^{3}-g}=\frac {n+1}{3(n-1)^{2}}, \end{equation*} where $g$ is the genus of the curve. In the case that $n=3$ (cyclic trigonal curves), we are able to show in most cases that for sufficiently large primes $p$, the branch points and the non-branch Weierstrass points remain distinct modulo $p$.
$S$-integral points of $\mathbb{P}^n-\{2n+1$ hyperplanes in general position over number fields and function fields\}
Julie
T.-Y.
Wang
3379-3389
Abstract: For the number field case we will give an upper bound on the number of the $S$-integral points in $\mathbb {P}^n(K)-\{ 2n+1\text { hyperplanes in general}$$\text {position}\}$. The main tool here is the explicit upper bound of the number of solutions of $S$-unit equations (Invent. Math. 102 (1990), 95--107). For the function field case we will give a bound on the height of the $S$-integral points in $\mathbb {P}^n(K)-\{ 2n+1\text { hyperplanes in general position}\}$. We will also give a bound for the number of ``generators" of those $S$-integral points. The main tool here is the $S$-unit Theorem by Brownawell and Masser (Proc. Cambridge Philos. Soc. 100 (1986), 427--434).