Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Part I: Controlled hysteresis
Goong
Chen;
Sze-Bi
Hsu;
Jianxin
Zhou;
Guanrong
Chen;
Giovanni
Crosta
4265-4311
Abstract: The study of nonlinear vibrations/oscillations in mechanical and electronic systems has always been an important research area. While important progress in the development of mathematical chaos theory has been made for finite dimensional second order nonlinear ODEs arising from nonlinear springs and electronic circuits, the state of understanding of chaotic vibrations for analogous infinite dimensional systems is still very incomplete. The 1-dimensional vibrating string satisfying $w_{tt}- w_{xx}=0$ on the unit interval $x \in (0,1)$ is an infinite dimensional harmonic oscillator. Consider the boundary conditions: at the left end $x=0$, the string is fixed, while at the right end $x=1$, a nonlinear boundary condition $w_{x}= \alpha w_t - \beta w_{t}^{3}, \alpha, \beta>0$, takes effect. This nonlinear boundary condition behaves like a van der Pol oscillator, causing the total energy to rise and fall within certain bounds regularly or irregularly. We formulate the problem into an equivalent first order hyperbolic system, and use the method of characteristics to derive a nonlinear reflection relation caused by the nonlinear boundary condition. Since the solution of the first order hyperbolic system depends completely on this nonlinear relation and its iterates, the problem is reduced to a discrete iteration problem of the type $u_{n+1}=F(u_n)$, where $F$ is the nonlinear reflection relation. We say that the PDE system is chaotic if the mapping $F$ is chaotic as an interval map. Algebraic, asymptotic and numerical techniques are developed to tackle the cubic nonlinearities. We then define a rotation number, following J.P. Keener , and obtain denseness of orbits and periodic points by either directly constructing a shift sequence or by applying results of M.I. Malkin to determine the chaotic regime of $\alpha$ for the nonlinear reflection relation $F$, thereby rigorously proving chaos. Nonchaotic cases for other values of $\alpha$ are also classified. Such cases correspond to limit cycles in nonlinear second order ODEs. Numerical simulations of chaotic and nonchaotic vibrations are illustrated by computer graphics.
Projectivity, transitivity and AF-telescopes
Terry
A.
Loring;
Gert
K.
Pedersen
4313-4339
Abstract: Continuing our study of projective $C^{*}$-algebras, we establish a projective transitivity theorem generalizing the classical Glimm-Kadison result. This leads to a short proof of Glimm's theorem that every $C^{*}$-algebra not of type I contains a $C^{*}$-subalgebra which has the Fermion algebra as a quotient. Moreover, we are able to identify this subalgebra as a generalized mapping telescope over the Fermion algebra. We next prove what we call the multiplier realization theorem. This is a technical result, relating projective subalgebras of a multiplier algebra $M(A)$ to subalgebras of $M(E)$, whenever $A$ is a $C^{*}$-subalgebra of the corona algebra $C(E)=M(E)/E$. We developed this to obtain a closure theorem for projective $C^{*}$-algebras, but it has other consequences, one of which is that if $A$ is an extension of an MF (matricial field) algebra (in the sense of Blackadar and Kirchberg) by a projective $C^{*}$-algebra, then $A$ is MF. The last part of the paper contains a proof of the projectivity of the mapping telescope over any AF (inductive limit of finite-dimensional) $C^{*}$-algebra. Translated to generators, this says that in some cases it is possible to lift an infinite sequence of elements, satisfying infinitely many relations, from a quotient of any $C^{*}$-algebra.
Second-order conditions in extremal problems. The abnormal points
A.
V.
Arutyunov
4341-4365
Abstract: In this paper we study a minimization problem with constraints and obtain first- and second-order necessary conditions for a minimum. Those conditions - as opposed to the known ones - are also informative in the abnormal case. We have introduced the class of 2-normal constraints and shown that for them the ``gap" between the sufficient and the necessary conditions is as minimal as possible. It is proved that a 2-normal mapping is generic.
Eigenfunctions of the Laplacian on rotationally symmetric manifolds
Michel
Marias
4367-4375
Abstract: Eigenfunctions of the Laplacian on a negatively curved, rotationally symmetric manifold $M=(\mathbf{R}^n,ds^2),$ $ds^2=dr^2+f(r)^2d\theta ^2,$ are constructed explicitly under the assumption that an integral of $f(r)$ converges. This integral is the same one which gives the existence of nonconstant harmonic functions on $M.$
The real field with convergent generalized power series
Lou
van den Dries;
Patrick
Speissegger
4377-4421
Abstract: We construct a model complete and o-minimal expansion of the field of real numbers in which each real function given on $[0,1]$ by a series $\sum c_{n} x^{\alpha _{n}}$ with $0 \leq \alpha _{n} \rightarrow \infty$ and $\sum |c_{n}| r^{\alpha _{n}} < \infty$ for some $r>1$ is definable. This expansion is polynomially bounded.
Bordism of spin 4-manifolds with local action of tori
Piotr
Mikrut
4423-4444
Abstract: We prove that bordism group of spin $4$-manifolds with singular $T$-structure, the notion introduced by Cheeger and Gromov, is an infinite cyclic group and is detected by singnature. In particular we obtain that the theorem of Atiyah and Hirzebruch of vanishing of Â-genus in case of $S^{1}$ action on spin $4n$-manifolds is not valid in case of $T$-structures on spin $4$-manifolds.
A $q$-deformation of a trivial symmetric group action
Phil
Hanlon;
Richard
P.
Stanley
4445-4459
Abstract: Let $\mathcal{A}$ be the arrangement of hyperplanes consisting of the reflecting hyperplanes for the root system $A_{n-1}$. Let $B=B(q)$ be the Varchenko matrix for this arrangement with all hyperplane parameters equal to $q$. We show that $B$ is the matrix with rows and columns indexed by permutations with $\sigma, \tau$ entry equal to $q^{i(\sigma \tau^{-1})}$ where $i(\pi)$ is the number of inversions of $\pi$. Equivalently $B$ is the matrix for left multiplication on $\mathbb{C}\mathfrak{S}_n$ by \begin{displaymath}\Gamma _n(q)=\sum _{\pi \in \mathfrak{S}_n} q^{i(\pi)} \pi . \end{displaymath} Clearly $B$ commutes with the right-regular action of $\mathfrak{S}_n$ on $\mathbb{C}\mathfrak{S}_n$. A general theorem of Varchenko applied in this special case shows that $B$ is singular exactly when $q$ is a $j(j-1)^{st}$ root of $1$ for some $j$ between $2$ and $n$. In this paper we prove two results which partially solve the problem (originally posed by Varchenko) of describing the $\mathfrak{S}_n$-module structure of the nullspace of $B$ in the case that $B$ is singular. Our first result is that \begin{displaymath}\ker(B) = \mathrm{ind}^{\mathfrak{S}_n}_{\mathfrak{S}_{n-1}} (\mathrm{Lie}_{n-1}) /\mathrm{Lie}_n\end{displaymath} in the case that $q = e^{2\pi i/n(n-1)}$ where Lie$_n$ denotes the multilinear part of the free Lie algebra with $n$ generators. Our second result gives an elegant formula for the determinant of $B$ restricted to the virtual $\mathfrak{S}_n$-module $P^\eta$ with characteristic the power sum symmetric function $p_\eta(x)$.
On the zeros of a polynomial and its derivatives
Piotr
Pawlowski
4461-4472
Abstract: If $p(z)$ is univariate polynomial with complex coefficients having all its zeros inside the closed unit disk, then the Gauss-Lucas theorem states that all zeros of $p'(z)$ lie in the same disk. We study the following question: what is the maximum distance from the arithmetic mean of all zeros of $p(z)$ to a nearest zero of $\displaystyle{p'(z)}$? We obtain bounds for this distance depending on degree. We also show that this distance is equal to $\frac{1}{3}$ for polynomials of degree 3 and polynomials with real zeros.
Trigonometric moment problems for arbitrary finite subsets of $\mathbb Z^n$
Jean-Pierre
Gabardo
4473-4498
Abstract: We consider finite subsets $\Lambda \subset \mathbf{Z}^{n}$ satisfying the extension property, i.e. the property that every collection $\{c_{\mathbf{k}}\}_{\mathbf{k} \in \Lambda - \Lambda }$ of complex numbers which is positive-definite on $\Lambda$ is the restriction to $\Lambda - \Lambda$ of the Fourier coefficients of some positive measure on $\mathbf{T}^{n}$. A simple algebraic condition on the set of trigonometric polynomials with non-zero coefficients restricted to $\Lambda$ is shown to imply the failure of the extension property for $\Lambda$. This condition is used to characterize the one-dimensional sets satisfying the extension property and to provide many examples of sets failing to satisfy it in higher dimensions. Another condition, in terms of unitary matrices, is investigated and is shown to be equivalent to the extension property. New two-dimensional examples of sets satisfying the extension property are given as well as explicit examples of collections for which the extension property fails.
Trace on the boundary for solutions of nonlinear differential equations
E.
B.
Dynkin;
S.
E.
Kuznetsov
4499-4519
Abstract: Let $L$ be a second order elliptic differential operator in $\mathbb{R}^{d}$ with no zero order terms and let $E$ be a bounded domain in $\mathbb{R}^{d}$ with smooth boundary $\partial E$. We say that a function $h$ is $L$-harmonic if $Lh=0$ in $E$. Every positive $L$-harmonic function has a unique representation \begin{equation*}h(x)=\int _{\partial E} k(x,y) \nu (dy), \end{equation*} where $k$ is the Poisson kernel for $L$ and $\nu$ is a finite measure on $\partial E$. We call $\nu$ the trace of $h$ on $\partial E$. Our objective is to investigate positive solutions of a nonlinear equation \begin{equation*}L u=u^{\alpha }\quad \text{in } E \end{equation*} for $1<\alpha \le 2$ [the restriction $\alpha \le 2$ is imposed because our main tool is the $\alpha$-superdiffusion which is not defined for $\alpha >2$]. We associate with every solution $u$ a pair $(\Gamma ,\nu )$, where $\Gamma$ is a closed subset of $\partial E$ and $\nu$ is a Radon measure on $O=\partial E\setminus \Gamma$. We call $(\Gamma ,\nu )$ the trace of $u$ on $\partial E$. $\Gamma$ is empty if and only if $u$ is dominated by an $L$-harmonic function. We call such solutions moderate. A moderate solution is determined uniquely by its trace. In general, many solutions can have the same trace. We establish necessary and sufficient conditions for a pair $(\Gamma ,\nu )$ to be a trace, and we give a probabilistic formula for the maximal solution with a given trace.
Solutions of nonlinear differential equations on a Riemannian manifold and their trace on the Martin boundary
E.
B.
Dynkin;
S.
E.
Kuznetsov
4521-4552
Abstract: Let $L$ be a second order elliptic differential operator on a Riemannian manifold $E$ with no zero order terms. We say that a function $h$ is $L$-harmonic if $Lh=0$. Every positive $L$-harmonic function has a unique representation \begin{equation*}h(x)=\int _{E'} k(x,y) \nu (dy), \end{equation*} where $k$ is the Martin kernel, $E'$ is the Martin boundary and $\nu$ is a finite measure on $E'$ concentrated on the minimal part $E^{*}$ of $E'$. We call $\nu$ the trace of $h$ on $E'$. Our objective is to investigate positive solutions of a nonlinear equation \begin{equation*}L u=u^{\alpha }\quad \text{on } E \tag{*} \end{equation*} for $1<\alpha \le 2$ [the restriction $\alpha \le 2$ is imposed because our main tool is the $(L,\alpha )$-superdiffusion, which is not defined for $\alpha >2$]. We associate with every solution $u$ of (*) a pair $(\Gamma ,\nu )$, where $\Gamma$ is a closed subset of $E'$ and $\nu$ is a Radon measure on $O=E'\setminus \Gamma$. We call $(\Gamma ,\nu )$ the trace of $u$ on $E'$. $\Gamma$ is empty if and only if $u$ is dominated by an $L$-harmonic function. We call such solutions moderate. A moderate solution is determined uniquely by its trace. In general, many solutions can have the same trace. In an earlier paper, we investigated the case when $L$ is a second order elliptic differential operator in $\mathbb{R}^{d}$ and $E$ is a bounded smooth domain in $\mathbb{R}^{d}$. We obtained necessary and sufficient conditions for a pair $(\Gamma ,\nu )$ to be a trace, and we gave a probabilistic formula for the maximal solution with a given trace. The general theory developed in the present paper is applicable, in particular, to elliptic operators $L$ with bounded coefficients in an arbitrary bounded domain of $\mathbb{R}^{d}$, assuming only that the Martin boundary and the geometric boundary coincide.
Consequences of contractible geodesics on surfaces
J.
Denvir;
R.
S.
Mackay
4553-4568
Abstract: The geodesic flow of any Riemannian metric on a geodesically convex surface of negative Euler characteristic is shown to be semi-equivalent to that of any hyperbolic metric on a homeomorphic surface for which the boundary (if any) is geodesic. This has interesting corollaries. For example, it implies chaotic dynamics for geodesic flows on a torus with a simple contractible closed geodesic, and for geodesic flows on a sphere with three simple closed geodesics bounding disjoint discs.
The Santaló-regions of a convex body
Mathieu
Meyer;
Elisabeth
Werner
4569-4591
Abstract: Motivated by the Blaschke-Santaló inequality, we define for a convex body $K$ in $\mathbf{R}^n$ and for $t \in \mathbf{R}$ the Santaló-regions $S(K,t)$ of $K$. We investigate the properties of these sets and relate them to a concept of affine differential geometry, the affine surface area of $K$.
The Dynkin-Lamperti arc-sine laws for measure preserving transformations
Maximilian
Thaler
4593-4607
Abstract: Arc-sine laws in the sense of renewal theory are proved for return time processes generated by transformations with infinite invariant measure on sets satisfying a type of Darling-Kac condition, and an application to real transformations with indifferent fixed points is discussed.
Lower bounds for dimensions of representation varieties
Andy
R.
Magid
4609-4621
Abstract: The set of $n$-dimensional complex representations of a finitely generated group $\Gamma$ form a complex affine variety $R_{n}(\Gamma )$. Suppose that $\rho $ is such a representation and consider the associated representation $Ad \circ \rho$ on $n \times n$ complex matrices obtained by following $\rho$ with conjugation of matrices. Then it is shown that the dimension of $R_{n}(\Gamma )$ at $\rho$ is at least the difference of the complex dimensions of $Z^{1}(\Gamma , Ad \circ \rho )$ and $H^{2}(\Gamma , Ad \circ \rho )$. It is further shown that in the latter cohomology $\Gamma$ may be replaced by various proalgebraic groups associated to $\Gamma$ and $\rho$.
Atomic maps and the Chogoshvili-Pontrjagin claim
M.
Levin;
Y.
Sternfeld
4623-4632
Abstract: It is proved that all spaces of dimension three or more disobey the Chogoshvili-Pontrjagin claim. This is of particular interest in view of the recent proof (in Certain 2-stable embeddings, by Dobrowolski, Levin, and Rubin, Topology Appl. 80 (1997), 81-90) that two-dimensional ANRs obey the claim. The construction utilizes the properties of atomic maps which are maps whose fibers ($=$point inverses) are atoms ($=$hereditarily indecomposable continua). A construction of M. Brown is applied to prove that every finite dimensional compact space admits an atomic map with a one-dimensional range.
Subelliptic harmonic maps
Jürgen
Jost;
Chao-Jiang
Xu
4633-4649
Abstract: We study a nonlinear harmonic map type system of subelliptic PDE. In particular, we solve the Dirichlet problem with image contained in a convex ball.
Singular limit of solutions of the porous medium equation with absorption
Kin
Ming
Hui
4651-4667
Abstract: We prove that as $m\to \infty$ the solutions $u^{(m)}$ of $u_{t}=\Delta u^{m}-u^{p}$, $(x,t)\in R^{n}\times (0,T)$, $T>0$, $m>1$, $p>1$, $u\ge 0$, $u(x,0)=f(x)\in L^{1}(R^{n})\cap L^{\infty }(R^{n})$, converges in $L^{1}_{loc}(R^{n}\times (0,T))$ to the solution of the ODE $v_{t}=-v^{p}$, $v(x,0)=g(x)$, where $g\in L^{1}(R^{n})$, $0\le g\le 1$, satisfies $g-\Delta \widetilde {g}=f$ in $\mathcal{D}'(R^{n})$ for some function $\widetilde {g}\in L^{\infty }_{loc}(R^{n})$, $\widetilde {g}\ge 0$, satisfying $\widetilde {g}(x)=0$ whenever $g(x)<1$ for a.e. $x\in R^{n}$, $\int _{E}\widetilde {g}dx\le C|E|^{2/n}$ for $n\ge 3$ and $\int _{E}|\nabla \widetilde {g}|dx\le C|E|^{1/2}$ for $n=2$, where $C>0$ is a constant and $E$ is any measurable subset of $R^{n}$.
A class of parabolic $k$-subgroups associated with symmetric $k$-varieties
A.
G.
Helminck;
G.
F.
Helminck
4669-4691
Abstract: Let $G$ be a connected reductive algebraic group defined over a field $k$ of characteristic not 2, $\sigma$ an involution of $G$ defined over $k$, $H$ a $k$-open subgroup of the fixed point group of $\sigma$, $G_k$ (resp. $H_k$) the set of $k$-rational points of $G$ (resp. $H$) and $G_k/H_k$ the corresponding symmetric $k$-variety. A representation induced from a parabolic $k$-subgroup of $G$ generically contributes to the Plancherel decomposition of $L^2(G_k/H_k)$ if and only if the parabolic $k$-subgroup is $\sigma$-split. So for a study of these induced representations a detailed description of the $H_k$-conjucagy classes of these $\sigma$-split parabolic $k$-subgroups is needed. In this paper we give a description of these conjugacy classes for general symmetric $k$-varieties. This description can be refined to give a more detailed description in a number of cases. These results are of importance for studying representations for real and $\mathfrak p$-adic symmetric $k$-varieties.
Model aspherical manifolds with no periodic maps
Wim
Malfait
4693-4708
Abstract: A. Borel proved that, if the fundamental group $E$ of an aspherical manifold $M$ is centerless and the outer automorphism group of $E$ is torsion-free, then $M$ admits no periodic maps, or equivalently, there are no non-trivial finite groups of homeomorphisms acting effectively on $M$. In the literature, taking off from this result, several examples of (rather complex) aspherical manifolds exhibiting this total lack of periodic maps have been presented. In this paper, we investigate to what extent the converse of Borel's result holds for aspherical manifolds $M$ arising from Seifert fiber space constructions. In particular, for e.g. flat Riemannian manifolds, infra-nilmanifolds and infra-solvmanifolds of type (R), it turns out that having a centerless fundamental group with torsion-free outer automorphism group is also necessary to conclude that all finite groups of affine diffeomorphisms acting effectively on the manifold are trivial. Finally, we discuss the problem of finding (less complex) examples of such aspherical manifolds with no periodic maps.