Properties of center manifolds
Jan
Sijbrand
431-469
Abstract: The center manifold has a number of puzzling properties associated with the basic questions of existence, uniqueness, differentiability and analyticity which may cloud its profitable application in e.g. bifurcation theory. This paper aims to deal with some of these subtle properties. Regarding existence and uniqueness, it is shown that the cut-off function appearing in the usual existence proof is responsible for the selection of a single center manifold, thereby hiding the inherent nonuniqueness. Conditions are given for different center manifolds at an equilibrium point to have a nonempty intersection. This intersection will include at least the families of stationary and periodic solutions crossing through the equilibrium. In the case of nonuniqueness the difference between any two center manifolds will be less than $ {c_1}\exp ({c_2}{x^{ - 1}})$ with ${c_1}$ and ${c_2}$ constants, which explains why the formal Taylor expansions of different center manifolds are the same, while the expansions do not converge. The differentiability of a center manifold will in certain cases decrease when moving out of the origin and a simple example shows how the differentiability may be lost. Center manifolds are usually not analytic; however, an analytic manifold may exist which contains all periodic solutions of a certain type but which may otherwise not be invariant. Using this manifold, a new and very simple proof of the Lyapunov subcenter theorem is given.
The typical structure of the sets $\{x\colon\;f(x)=h(x)\}$ for $f$ continuous and $h$ Lipschitz
Zygmunt
Wójtowicz
471-484
Abstract: Let $R$ be the space of real numbers and $ C$ the space of continuous functions $ f:[0,1] \to R$ with the uniform norm. Bruckner and Garg prove that there exists a residual set $B$ in $C$ such that for every function $f \in B$ there exists a countable dense set ${\Lambda _f}$ in $R$ such that: for $\lambda \notin {\Lambda _f}$ the top and bottom levels in the direction $\lambda$ of $f$ are singletons, in between these levels there are countably many levels in the direction $\lambda$ of $f$ that consist of a nonempty perfect set together with a single isolated point, and the remaining levels in the direction $\lambda$ of $f$ are all perfect; for $\lambda \in {\Lambda _f}$ the level structure in the direction $\lambda$ of $f$ is the same except that one (and only one) of the levels has two isolated points instead of one. In this paper we show that the analogue of the above theorem holds: if we replace the family of straight lines $\{ \lambda x + c\}$ by a $2$-parameter family $H$ that is almost uniformly Lipschitz; and if we replace $ \{ \lambda x + c\}$ by a homeomorphical image of a certain $2$-parameter family $H$ that is almost uniformly Lipschitz.
Fixed points and conjugacy classes of regular elliptic elements in ${\rm Sp}(3,{\bf Z})$
Min King
Eie;
Chung Yuan
Lin
485-496
Abstract: In this paper, we obtain $13$ isolated fixed points (up to a $ \operatorname{Sp}(3,{\mathbf{Z}})$-equivalence) and $86$ conjugacy classes of regular elliptic elements in $ \operatorname{Sp}(3,{\mathbf{Z}})$. Hence the contributions from regular elliptic conjugacy classes in $ \operatorname{Sp}(3,{\mathbf{Z}})$ to the dimension formula computed via the Selberg trace formula can be computed explictly by the main theorem of [ $ {\mathbf{4}}$ or ${\mathbf{5}}$].
The Cauchy integral, Calder\'on commutators, and conjugations of singular integrals in ${\bf R}\sp n$
Margaret A. M.
Murray
497-518
Abstract: We consider the Cauchy integral and Hilbert transform for Lipschitz domains in the Clifford algebra based on ${R^n}$. The Hilbert transform is shown to be the generating function for the Calderón commutators in ${R^n}$. We make use of an intrinsic characterization of these commutators to obtain ${L^2}$ estimates. These estimates are used to show the analyticity of the Hilbert transform and of the conjugation of singular integral operators by bi-Lipschitz changes of variable in ${R^n}$.
Quotients by ${\bf C}\sp\ast \times{\bf C}\sp\ast$ actions
Andrzej
Białynicki-Birula;
Andrew John
Sommese
519-543
Abstract: Let $T \approx {{\mathbf{C}}^\ast} \times {{\mathbf{C}}^\ast}$ act meromorphically on a compact Kähler manifold $X$, e.g. algebraically on a projective manifold. The following is a basic question from geometric invariant theory whose answer is unknown even if $X$ is projective. PROBLEM. Classify all $T$-invariant open subsets $U$ of $X$ such that the geometric quotient $ U \to U/T$ exists with $U/T$ a compact complex space (necessarily algebraic if $X$ is). In this paper a simple to state and use solution to this problem is given. The classification of $U$ is reduced to finite combinatorics. Associated to the $T$ action on $X$ is a certain finite $2$-complex $ \mathcal{C}(X)$. Certain $ \{ 0,1\}$ valued functions, called moment measures, are defined in the set of $ 2$-cells of $\mathcal{C}(X)$. There is a natural one-to-one correspondence between the $U$ with compact quotients, $U/T$, and the moment measures.
On differential equations associated with Euler product expressions
Ian
Knowles
545-573
Abstract: A method is given by which one may associate (uniquely) certain differential equations with analytic functions defined by certain Euler product expressions. Some of the consequences of this construction include results relating the location of the zeros of the analytic functions to asymptotic properties of the solutions of the differential equations, as well as a differential equation characterization of those Dirichlet series with multiplicative coefficients.
Fractional integrals on weighted $H\sp p$ spaces
Angel E.
Gatto;
Cristian E.
Gutiérrez;
Richard L.
Wheeden
575-589
Abstract: We characterize the pairs of doubling weights $(u,v)$ on ${R^n}$ such that $\displaystyle \parallel {I_\alpha }f{\parallel _{H_u^q}} \leq c\parallel f{\parallel _{H_v^p}}, \quad 0 < p < q < \infty $ , where ${I_\alpha },\alpha > 0$, is the fractional integral operator. We also consider the behavior of an associated maximal function. Applications of the results to Sobolev inequalities in weighted $ {L^p}$ spaces are given.
Invariant regions for systems of conservation laws
David
Hoff
591-610
Abstract: We describe necessary and sufficient conditions for a region in ${{\mathbf{R}}^n}$ to be invariant for (Glimm) solutions of the system of $n$ conservation laws ${u_t} + f{(u)_x} = 0$. We also make some observations about the invariance of such regions for certain finite difference approximations of solutions of systems of conservation laws.
Error bounds for Glimm difference approximations for scalar conservation laws
David
Hoff;
Joel
Smoller
611-642
Abstract: We derive error bounds for the Glimm difference approximation to the solution of a genuinely nonlinear scalar conservation law with BV initial data. We show that the ${L^1}$ error is bounded by $O(\Delta {x^{1/6}}\vert\log \Delta x\vert)$ in the general case, and by $O(\Delta {x^{1/2}}\vert\log \Delta x\vert)$ for a generic class of piecewise constant data.
A kernel approach to the local solvability of the tangential Cauchy Riemann equations
A.
Boggess;
M.-C.
Shaw
643-658
Abstract: An integral kernel approach is given for the proof of the theorem of Andreotti and Hill which states that the $Y(q)$ condition of Kohn is a sufficient condition for local solvability of the tangential Cauchy Riemann equations on a real hypersurface in $ {{\mathbf{C}}^n}$. In addition, we provide an integral kernel approach to nonsolvability for a certain class of real hypersurfaces in the case when $Y(q)$ is not satisfied.
Convexity conditions and intersections with smooth functions
S.
Agronsky;
A. M.
Bruckner;
M.
Laczkovich;
D.
Preiss
659-677
Abstract: A continuous function that agrees with each member of a family $\mathcal{F}$ of smooth functions in a small set must itself possess certain desirable properties. We study situations that arise when $\mathcal{F}$ consists of the family of polynomials of degree at most $n$, as well as certain larger families and when the small sets of agreement are finite. The conclusions of our theorems involve convexity conditions. For example, if a continuous function $f$ agrees with each polynomial of degree at most $n$ in only a finite set, then $f$ is $(n + 1)$-convex or $(n + 1)$-concave on some interval. We consider also certain variants of this theorem, provide examples to show that certain improvements are not possible and present some applications of our results.
Six standard deviations suffice
Joel
Spencer
679-706
Abstract: Given $ n$ sets on $n$ elements it is shown that there exists a two-coloring such that all sets have discrepancy at most $K{n^{1/2}}$, $K$ an absolute constant. This improves the basic probabilistic method with which $K = c{(\ln n)^{1/2}}$. The result is extended to $ n$ finite sets of arbitrary size. Probabilistic techniques are melded with the pigeonhole principle. An alternate proof of the existence of Rudin-Shapiro functions is given, showing that they are exponential in number. Given $n$ linear forms in $n$ variables with all coefficients in $[ - 1, + 1]$ it is shown that initial values $ {p_1}, \ldots ,{p_n} \in \{ 0,1\}$ may be approximated by ${\varepsilon _1}, \ldots ,{\varepsilon _n} \in \{ 0,1\}$ so that the forms have small error.
Convolution equations in spaces of distributions with one-sided bounded support
R.
Shambayati;
Z.
Zielezny
707-713
Abstract: Let $ \mathcal{D}\prime(0,\infty )$ be the space of distributions on $ R$ with support in $[0,\infty )$ and $\mathcal{S}\prime(0,\infty )$ its subspace consisting of tempered distributions. We characterize the distributions $S \in \mathcal{D}\prime(0,\infty )$ for which $S\, \ast \mathcal{D}\prime(0,\infty ) = \mathcal{D}\prime(0,\infty )$, where $\ast$ is the convolution. We also characterize the distributions $S \in \mathcal{S}\prime(0,\infty )$ for which $S \ast \mathcal{S}\prime(0,\infty ) = \mathcal{S}\prime(0,\infty )$.
Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces; the case of finite volume
Peter D.
Lax;
Ralph S.
Phillips
715-735
Abstract: Let $\Gamma$ be a discrete subgroup of automorphisms of $ {{\mathbf{H}}^n}$, with fundamental polyhedron of finite volume, finite number of sides, and $N$ cusps. Denote by $ {\Delta _\Gamma }$ the Laplace-Beltrami operator acting on functions automorphic with respect to $\Gamma$. We give a new short proof of the fact that ${\Delta _\Gamma }$ has absolutely continuous spectrum of uniform multiplicity $N$ on $( - \infty ,{((n - 1)/2)^2})$, plus a standard discrete spectrum. We show that this property of the spectrum is unchanged under arbitrary perturbation of the metric on a compact set. Our method avoids Eisenstein series entirely and proceeds instead by constructing explicitly a translation representation for the associated wave equation.
On the proper Steenrod homotopy groups, and proper embeddings of planes into $3$-manifolds
Matthew G.
Brin;
T. L.
Thickstun
737-755
Abstract: Standard algebraic invariants of proper homotopy type are discussed. These do not naturally fit into long exact sequences. Groups of proper homotopy classes of proper maps of Euclidean spaces and open annuli which do naturally form a long exact sequence are defined, and a diagram relating these groups to the standard algebraic invariants of proper homotopy type is given. The structures defined are used to compare several notions of essentiality for proper maps. Some results and examples are given for proper maps of spaces into manifolds of dimension $ 2$ and $3$. These results are used to add information to a theorem of Brown and Feustel about properly embedding planes in $3$-manifolds.
Obstruction theory and multiparameter Hopf bifurcation
Jorge
Ize
757-792
Abstract: The Hopf bifurcation problem is treated as an example of an equivariant bifurcation. The existence of a local bifurcating solution is given by the nonvanishing of an obstruction to extending a map defined on a complex projective space and is computed using the complex Bott periodicity theorem. In the case of the classical Hopf bifurcation the results of Chow, Mallet-Paret and Yorke are recovered without using any special index as the Fuller degree: There is bifurcation if the number of exchanges of stability is nonzero. A global theorem asserts that the sum of the local invariants on a bounded component of solutions must be zero.
Equivariant diffeomorphisms with simple recurrences on two-manifolds
W.
de Melo;
G. L.
dos Reis;
P.
Mendes
793-807
Abstract: We consider the class of diffeomorphisms, on compact two-dimensional manifolds, which are invariant under the action of a compact Lie group $G$ and whose nonwandering set consists of a finite number of $G$-orbits. We describe the modulus of stability of almost all diffeomorphisms in this class.
Directed graphs and traveling waves
David
Terman
809-847
Abstract: The existence of traveling wave solutions for equations of the form $ {u_t} = {u_{xx}} + F\prime(u)$ is considered. All that is assumed about $ F$ is that it is sufficiently smooth, ${\lim _{\vert u\vert \to \infty }}F(u) = - \infty$, $F$ has only a finite number of critical points, each of which is nondegenerate, and if $ A$ and $B$ are distinct critical points of $ F$, then $F(A) \ne F(B)$. The results demonstrate that, for a given function $F$, there may exist zero, exactly one, a finite number, or an infinite number of waves which connect two fixed, stable rest points. The main technique is to identify the phase planes, which arise naturally from the problem, with an array of integers. While the phase planes may be very complicated, the arrays of integers are always quite simple to analyze. Using the arrays of integers one is able to construct a directed graph; each path in the directed graph indicates a possible ordering, starting with the fastest, of which waves must exist. For a large class of functions $ F$ one is then able to use the directed graphs in order to determine how many waves connect two stable rest points.
Characteristic classes of transversely homogeneous foliations
Chal
Benson;
David B.
Ellis
849-859
Abstract: The foliations studied in this paper have transverse geometry modeled on a homogeneous space $G/H$ with transition functions given by the left action of $G$. It is shown that the characteristic classes for such a foliation are determined by invariants of a certain flat bundle. This is used to prove that when $ G$ is semisimple, the characteristic classes are rigid under smooth deformations, extending work of Brooks, Goldman and Heitsch.