Heteroclinic cycles for reaction diffusion systems by forced symmetry-breaking
Stanislaus
Maier-Paape;
Reiner
Lauterbach
2937-2991
Abstract: Recently it has been observed, that perturbations of symmetric ODE's can lead to highly nontrivial dynamics. In this paper we want to establish a similar result for certain nonlinear partial differential systems. Our results are applied to equations which are motivated from chemical reactions. In fact we show that the theory applies to the Brusselator on a sphere. To be more precise, we consider solutions of a semi-linear parabolic equation on the 2-sphere. When this equation has an axisymmetric equilibrium $u_\alpha$, the group orbit of $u_\alpha$ (under rotations) gives a whole (invariant) manifold $M$ of equilibria. Under generic conditions we have that, after perturbing our equation by a (small) $L\subset {{{\bf O}(3)}}$-equivariant perturbation, $M$ persists as an invariant manifold $\widetilde M$. However, the flow on $\widetilde M$ is in general no longer trivial. Indeed, we find slow dynamics on $\widetilde M$ and, in the case $L=\mathbb{T}$ (the tetrahedral subgroup of ${{{\bf O}(3)}}$), we observe heteroclinic cycles. In the application to chemical systems we would expect intermittent behaviour. However, for the Brusselator equations this phenomenon is not stable. In order to see it in a physically relevant situation we need to introduce further terms to get a higher codimension bifurcation.
An electromagnetic inverse problem in chiral media
Stephen
R.
McDowall
2993-3013
Abstract: We consider the inverse boundary value problem for Maxwell's equations that takes into account the chirality of a body in ${\mathbb R}^3$. More precisely, we show that knowledge of a boundary map for the electromagnetic fields determines the electromagnetic parameters, namely the conductivity, electric permittivity, magnetic permeability and chirality, in the interior. We rewrite Maxwell's equations as a first order perturbation of the Laplacian and construct exponentially growing solutions, and obtain the result in the spirit of complex geometrical optics.
A criterion for reduction of variables in the Willmore-Chen variational problem and its applications
Manuel
Barros;
Angel
Ferrández;
Pascual
Lucas;
Miguel
A.
Meroño
3015-3027
Abstract: We exhibit a criterion for a reduction of variables for Willmore-Chen submanifolds in conformal classes associated with generalized Kaluza-Klein metrics on flat principal fibre bundles. Our method relates the variational problem of Willmore-Chen with an elasticity functional defined for closed curves in the base space. The main ideas involve the extrinsic conformal invariance of the Willmore-Chen functional, the large symmetry group of generalized Kaluza-Klein metrics and the principle of symmetric criticality. We also obtain interesting families of elasticae in both lens spaces and surfaces of revolution (Riemannian and Lorentzian). We give applications to the construction of explicit examples of isolated Willmore-Chen submanifolds, discrete families of Willmore-Chen submanifolds and foliations whose leaves are Willmore-Chen submanifolds.
Inflection points and topology of surfaces in 4-space
Ronaldo
Alves
Garcia;
Dirce
Kiyomi Hayashida
Mochida;
Maria
del Carmen
Romero Fuster;
Maria
Aparecida
Soares Ruas
3029-3043
Abstract: We consider asymptotic line fields on generic surfaces in 4-space and show that they are globally defined on locally convex surfaces, and their singularities are the inflection points of the surface. As a consequence of the generalized Poincaré-Hopf formula, we obtain some relations between the number of inflection points in a generic surface and its Euler number. In particular, it follows that any 2-sphere, generically embedded as a locally convex surface in 4-space, has at least 4 inflection points.
Projective manifolds with small pluridegrees
Mauro
C.
Beltrametti;
Andrew
J.
Sommese
3045-3064
Abstract: Let $\mathcal{L}$ be a very ample line bundle on a connected complex projective manifold $\mathcal{M}$ of dimension $n\ge 3$. Except for a short list of degenerate pairs $(\mathcal{M},\mathcal{L})$, $\kappa(K_\mathcal{M}+(n-2)\mathcal{L})=n$ and there exists a morphism $\pi: \mathcal{M} \to M$ expressing $\mathcal{M}$ as the blowup of a projective manifold $M$ at a finite set $B$, with $\mathcal{K}_M:=K_M+(n-2)L$ nef and big for the ample line bundle $L:= (\pi _*\mathcal{L})^{**}$. The projective geometry of $(\mathcal{M},\mathcal{L})$ is largely controlled by the pluridegrees $d_j:=L^{n-j}\cdot (K_M+(n-2)L)^j$ for $j=0,\ldots,n$, of $(\mathcal{M},\mathcal{L})$. For example, $d_0+d_1=2g-2$, where $g$ is the genus of a curve section of $(\mathcal{M},\mathcal{L})$, and $d_2$ is equal to the self-intersection of the canonical divisor of the minimal model of a surface section of $(\mathcal{M},\mathcal{L})$. In this article, a detailed analysis is made of the pluridegrees of $(\mathcal{M},\mathcal{L})$. The restrictions found are used to give a new lower bound for the dimension of the space of sections of $\mathcal{K}_M$. The inequalities for the pluridegrees, that are presented in this article, will be used in a sequel to study the sheet number of the morphism associated to $|2(K_\mathcal{M}+ (n-2)\mathcal{L})|$.
A finitely axiomatizable undecidable equational theory with recursively solvable word problems
Dejan
Delic
3065-3101
Abstract: In this paper we construct a finitely based variety, whose equational theory is undecidable, yet whose word problems are recursively solvable, which solves a problem stated by G. McNulty (1992). The construction produces a discriminator variety with the aforementioned properties starting from a class of structures in some multisorted language (which may include relations), axiomatized by a finite set of universal sentences in the given multisorted signature. This result also presents a common generalization of the earlier results obtained by B. Wells (1982) and A. Mekler, E. Nelson, and S. Shelah (1993).
Adams operations, localized Chern characters, and the positivity of Dutta multiplicity in characteristic $0$
Kazuhiko
Kurano;
Paul
C.
Roberts
3103-3116
Abstract: The positivity of the Dutta multiplicity of a perfect complex of $A$-modules of length equal to the dimension of $A$ and with homology of finite length is proven for homomorphic images of regular local rings containing a field of characteristic zero. The proof uses relations between localized Chern characters and Adams operations.
Fundamental groups of moduli and the Grothendieck-Teichmüller group
David
Harbater;
Leila
Schneps
3117-3148
Abstract: Let ${\mathcal{M}}_{0,n}$ denote the moduli space of Riemann spheres with $n$ ordered marked points. In this article we define the group $\operatorname{Out}^{\sharp }_{n}$ of quasi-special symmetric outer automorphisms of the algebraic fundamental group $\widehat \pi _{1}({\mathcal{M}}_{0,n})$ for all $n\ge 4$ to be the group of outer automorphisms respecting the conjugacy classes of the inertia subgroups of $\widehat \pi _{1}({\mathcal{M}}_{0,n})$ and commuting with the group of outer automorphisms of $\widehat \pi _{1}({\mathcal{M}}_{0,n})$ obtained by permuting the marked points. Our main result states that $\operatorname{Out}^{\sharp }_{n}$ is isomorphic to the Grothendieck-Teichmüller group $\widehat {\operatorname{GT}}$for all $n\ge 5$.
Universal Formulae for SU$(n)$ Casson Invariants of Knots
Hans
U.
Boden;
Andrew
Nicas
3149-3187
Abstract: An $\operatorname{SU}(n)$ Casson invariant of a knot is an integer which can be thought of as an algebraic-topological count of the number of characters of $\operatorname{SU}(n)$ representations of the knot group which take a longitude into a given conjugacy class. For fibered knots, these invariants can be characterized as Lefschetz numbers which, for generic conjugacy classes, can be computed using a recursive algorithm of Atiyah and Bott, as adapted by Frohman. Using a new idea to solve the Atiyah-Bott recursion (as simplified by Zagier), we derive universal formulae which explicitly compute the invariants for all $n$. Our technique is based on our discovery that the generating functions associated to the relevant Lefschetz numbers (and polynomials) satisfy certain integral equations.
$K$-theory of projective Stiefel manifolds
Nelza
E.
Barufatti;
Derek
Hacon
3189-3209
Abstract: Using the Hodgkin spectral sequence we calculate $K^{*}(X_{m,k})$, the complex $K$-theory of the projective Stiefel manifold $X_{m,k}$, for $mk$even. For $mk$ odd, we are only able to calculate $K^{0}(X_{m,k})$, but this is sufficient to determine the order of the complexified Hopf bundle over $X_{m,k}$.
A filtration of spectra arising from families of subgroups of symmetric groups
Kathryn
Lesh
3211-3237
Abstract: Let ${\mathcal F}_{n}$ be a family of subgroups of $\Sigma_{n}$ which is closed under taking subgroups and conjugates. Such a family has a classifying space, $B{\mathcal F}_{n}$, and we showed in an earlier paper that a compatible choice of ${\mathcal F}_{n}$ for each $n$ gives a simplicial monoid $\coprod_{n} B{\mathcal F}_{n}$, which group completes to an infinite loop space. In this paper we define a filtration of the associated spectrum whose filtration quotients, given an extra condition on the families, can be identified in terms of the classifying spaces of the families of subgroups that were chosen. This gives a way to go from group theoretic data about the families to homotopy theoretic information about the associated spectrum. We calculate two examples. The first is related to elementary abelian $p$-groups, and the second gives a new expression for the desuspension of $Sp^{m}(S^{0})/Sp^{m-1}(S^{0})$ as a suspension spectrum.
Steiner type formulae and weighted measures of singularities for semi-convex functions
Andrea
Colesanti;
Daniel
Hug
3239-3263
Abstract: For a given convex (semi-convex) function $u$, defined on a nonempty open convex set $\Omega\subset\mathbf{R}^n$, we establish a local Steiner type formula, the coefficients of which are nonnegative (signed) Borel measures. We also determine explicit integral representations for these coefficient measures, which are similar to the integral representations for the curvature measures of convex bodies (and, more generally, of sets with positive reach). We prove that, for $r\in \{0,\ldots,n\}$, the $r$-th coefficient measure of the local Steiner formula for $u$, restricted to the set of $r$-singular points of $u$, is absolutely continuous with respect to the $r$-dimensional Hausdorff measure, and that its density is the $(n-r)$-dimensional Hausdorff measure of the subgradient of $u$. As an application, under the assumptions that $u$ is convex and Lipschitz, and $\Omega$ is bounded, we get sharp estimates for certain weighted Hausdorff measures of the sets of $r$-singular points of $u$. Such estimates depend on the Lipschitz constant of $u$ and on the quermassintegrals of the topological closure of $\Omega$.
Sharp weighted inequalities for the vector-valued maximal function
Carlos
Pérez
3265-3288
Abstract: We prove in this paper some sharp weighted inequalities for the vector-valued maximal function $\overline M_q$ of Fefferman and Stein defined by \begin{displaymath}\overline M_qf(x)=\left(\sum _{i=1}^{\infty}(Mf_i(x))^{q}\right)^{1/q},\end{displaymath} where $M$ is the Hardy-Littlewood maximal function. As a consequence we derive the main result establishing that in the range $1<q<p<\infty$ there exists a constant $C$ such that \begin{displaymath}\int _{\mathbf{R}^{n}}\overline M_qf(x)^p\, w(x)dx\le C\, \int _{\mathbf{R}^n}|f(x)|^{p}_{q}\, M^{[\frac pq]+1}w(x) dx.\end{displaymath} Furthermore the result is sharp since $M^{[\frac pq]+1}$ cannot be replaced by $M^{[\frac pq]}$. We also show the following endpoint estimate \begin{displaymath}w(\{x\in \mathbf{R}^n:\overline M_qf(x)>\lambda\})\,\le \frac C\lambda \int _{\mathbf{R}^n} |f(x)|_q\, Mw(x)dx,\end{displaymath} where $C$ is a constant independent of $\lambda$.
Absolutely continuous S.R.B. measures for random Lasota-Yorke maps
Jérôme
Buzzi
3289-3303
Abstract: A. Lasota and J. A. Yorke proved that a piecewise expanding interval map admits finitely many ergodic absolutely continuous invariant probability measures. We generalize this to the random composition of such maps under conditions which are natural and less restrictive than those previously studied by Morita and Pelikan. For instance our conditions are satisfied in the case of arbitrary random $\beta$-transformations, i.e., $x\mapsto \beta x\mod 1$ on $[0,1]$ where $\beta$ is chosen according to any stationary stochastic process (in particular, not necessarily i.i.d.) with values in $]1,\infty [$. RSESUM´E. A. Lasota et J. A. Yorke ont montré qu'une application de l'intervalle dilatante par morceaux admet un nombre fini de mesures de probabilité invariantes et ergodiques absolument continues. Nous généralisons ce résultat à la composition aléatoire de telles applications sous des conditions naturelles, moins restrictives que celles précédemment envisagées par Morita et Pelikan. Par exemple, nos conditions sont satisfaites par toute $\beta$-transformation aléatoire, i.e., $x\mapsto \beta x\mod 1$ sur $[0,1]$ avec $\beta$ choisi selon un processus stochastique stationnaire quelconque (en particulier, non-nécessairement i.i.d.) à valeurs dans $]1,\infty [$.
Shift equivalence and the Conley index
John
Franks;
David
Richeson
3305-3322
Abstract: In this paper we introduce filtration pairs for an isolated invariant set of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Last, we define partially ordered Morse decompositions and prove the existence of Morse set filtrations for such decompositions.
Hyperbolic minimizing geodesics
Daniel
Offin
3323-3338
Abstract: We apply the intersection theory for Lagrangian submanifolds to obtain a Sturm type comparison theorem for linearized Hamiltonian flows. Applications to the theory of geodesics are considered, including a sufficient condition that arclength minimizing closed geodesics, for an $n$-dimensional Riemannian manifold, are hyperbolic under the geodesic flow. This partially answers a conjecture of G. D. Birkhoff.
The characters of the generalized Steinberg representations of finite general linear groups on the regular elliptic set
Allan
J.
Silberger;
Ernst-Wilhelm
Zink
3339-3356
Abstract: Let $k$ be a finite field, $k_{n}\vert k$ the degree $n$ extension of $k$, and $G=\operatorname{GL}_{n}(k)$ the general linear group with entries in $k$. This paper studies the ``generalized Steinberg" (GS) representations of $G$ and proves the equivalence of several different characterizations for this class of representations. As our main result we show that the union of the class of cuspidal and GS representations of $G$ is in natural one-one correspondence with the set of Galois orbits of characters of $k_{n}^{\times }$, the regular orbits of course corresponding to the cuspidal representations. Besides using Green's character formulas to define GS representations, we characterize GS representations by associating to them idempotents in certain commuting algebras corresponding to parabolic inductions and by showing that GS representations are the sole components of these induced representations which are ``generic" (have Whittaker vectors).
Trees and valuation rings
Hans
H.
Brungs;
Joachim
Gräter
3357-3379
Abstract: A subring $B$ of a division algebra $D$ is called a valuation ring of $D$ if $x\in B$ or $x^{-1}\in B$ holds for all nonzero $x$ in $D$. The set $\mathcal{B}$ of all valuation rings of $D$ is a partially ordered set with respect to inclusion, having $D$ as its maximal element. As a graph $\mathcal{B}$ is a rooted tree (called the valuation tree of $D$), and in contrast to the commutative case, $\mathcal{B}$ may have finitely many but more than one vertices. This paper is mainly concerned with the question of whether each finite, rooted tree can be realized as a valuation tree of a division algebra $D$, and one main result here is a positive answer to this question where $D$ can be chosen as a quaternion division algebra over a commutative field.
Hereditary crossed products
Jeremy
Haefner;
Gerald
Janusz
3381-3410
Abstract: We characterize when a crossed product order over a maximal order in a central simple algebra by a finite group is hereditary. We need only concentrate on the cases when the group acts as inner automorphisms and when the group acts as outer automorphisms. When the group acts as inner automorphisms, the classical group algebra result holds for crossed products as well; that is, the crossed product is hereditary if and only if the order of the group is a unit in the ring. When the group is acting as outer automorphisms, every crossed product order is hereditary, regardless of whether the order of the group is a unit in the ring.
A generalized Brauer construction and linear source modules
Robert
Boltje;
Burkhard
Külshammer
3411-3428
Abstract: For a complete discrete valuation ring $\mathcal{O}$ with residue field $F$, a subgroup $H$ of a finite group $G$ and a homomorphism $\varphi: H \to \mathcal{O}^\times$, we define a functor $V \mapsto \overline{\overline{V}} (H,\varphi)$ from the category of $\mathcal{O} G$-modules to the category of $FN_G(H,\varphi)$-modules and investigate its behaviour with respect to linear source modules.