Products on $MU$-modules
N.
P.
Strickland
2569-2606
Abstract: Elmendorf, Kriz, Mandell and May have used their technology of modules over highly structured ring spectra to give new constructions of $MU$-modules such as $BP$, $K(n)$ and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over $MU[{\textstyle\frac{1}{2}}]_*$ that are concentrated in degrees divisible by $4$; this guarantees that various obstruction groups are trivial. We extend these results to the cases where $2=0$ or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in Baas-Sullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising $2$-local $MU_*$-modules as $MU$-modules.
Brownian sheet images and Bessel-Riesz capacity
Davar
Khoshnevisan
2607-2622
Abstract: We show that the image of a 2-dimensional set under $d$-dimensional, 2-parameter Brownian sheet can have positive Lebesgue measure if and only if the set in question has positive ($d/2$)-dimensional Bessel-Riesz capacity. Our methods solve a problem of J.-P. Kahane.
Vaught's conjecture and the Glimm-Effros property for Polish transformation groups
Greg
Hjorth;
Slawomir
Solecki
2623-2641
Abstract: We extend the original Glimm-Effros theorem for locally compact groups to a class of Polish groups including the nilpotent ones and those with an invariant metric. For this class we thereby obtain the topological Vaught conjecture.
Ultrafilters on $\omega$-their ideals and their cardinal characteristics
Saharon
Shelah;
Jörg
Brendle;
Saharon
Shelah
2643-2674
Abstract: For a free ultrafilter $\mathcal{U}$ on $\omega$ we study several cardinal characteristics which describe part of the combinatorial structure of $\,\mathcal{U}$. We provide various consistency results; e.g. we show how to force simultaneously many characters and many $\pi$-characters. We also investigate two ideals on the Baire space $\omega ^{\omega }$ naturally related to $\mathcal{U}$ and calculate cardinal coefficients of these ideals in terms of cardinal characteristics of the underlying ultrafilter.
CH with no Ostaszewski spaces
Todd
Eisworth;
Judith
Roitman
2675-2693
Abstract: There are models of CH without Ostaszeswki spaces. If $X$ is locally compact and sub-Ostaszewski, there is a forcing $P_X$ which does not add reals and which forces ``$X$ is not sub-Ostaszewski''.
The Quantum Cohomology Ring of Flag Varieties
Ionut
Ciocan-Fontanine
2695-2729
Abstract: We describe the small quantum cohomology ring of complete flag varieties by algebro-geometric methods, as presented in our previous work Quantum cohomology of flag varieties (Internat. Math. Res. Notices, no. 6 (1995), 263-277). We also give a geometric proof of the quantum Monk formula.
A speciality theorem for Cohen-Macaulay space curves
Enrico
Schlesinger
2731-2743
Abstract: We prove a version of the Halphen Speciality Theorem for locally Cohen-Macaulay curves in $\mathbb{P}^3$. To prove the theorem, we strengthen some results of Okonek and Spindler on the spectrum of the ideal sheaf of a curve. As an application, we classify curves $C$ having index of speciality as large as possible once we fix the degree of $C$ and the minimum degree of a surface containing $C$.
Spherical functions and conformal densities on spherically symmetric $CAT(-1)$-spaces
Michel
Coornaert;
Athanase
Papadopoulos
2745-2762
Abstract: Let $X$ be a $CAT(-1)$-space which is spherically symmetric around some point $x_{0}\in X$ and whose boundary has finite positive $s-$dimensional Hausdorff measure. Let $\mu =(\mu _{x})_{x\in X}$ be a conformal density of dimension $d>s/2$ on $\partial X$. We prove that $\mu _{x_{0}}$ is a weak limit of measures supported on spheres centered at $x_{0}$. These measures are expressed in terms of the total mass function of $\mu$ and of the $d-$dimensional spherical function on $X$. In particular, this result proves that $\mu$ is entirely determined by its dimension and its total mass function. The results of this paper apply in particular for symmetric spaces of rank one and semi-homogeneous trees.
An equivariant smash spectral sequence and an unstable box product
Michele
Intermont
2763-2775
Abstract: Let $G$ be a finite group. We construct a first quadrant spectral sequence which converges to the equivariant homotopy groups of the smash product $X \wedge Y$ for suitably connected, based $G$-CW complexes $X$ and $Y$. The $E^2$ term is described in terms of a tensor product functor of equivariant $\Pi$-algebras. A homotopy version of the non-equivariant Künneth theorem and the equivariant suspension theorem of Lewis are both shown to be special cases of the corner of the spectral sequence. We also give a categorical description of this tensor product functor which is analogous to the description in equivariant stable homotopy theory of the box product of Mackey functors. For this reason, the tensor product functor deserves to be called an ``unstable box product''.
Periodic traveling waves and locating oscillating patterns in multidimensional domains
Nicholas
D.
Alikakos;
Peter
W.
Bates;
Xinfu
Chen
2777-2805
Abstract: We establish the existence and robustness of layered, time-periodic solutions to a reaction-diffusion equation in a bounded domain in $\mathbb{R}^n$, when the diffusion coefficient is sufficiently small and the reaction term is periodic in time and bistable in the state variable. Our results suggest that these patterned, oscillatory solutions are stable and locally unique. The location of the internal layers is characterized through a periodic traveling wave problem for a related one-dimensional reaction-diffusion equation. This one-dimensional problem is of independent interest and for this we establish the existence and uniqueness of a heteroclinic solution which, in constant-velocity moving coodinates, is periodic in time. Furthermore, we prove that the manifold of translates of this solution is globally exponentially asymptotically stable.
Bourgin-Yang type theorem and its application to $Z_2$-equivariant Hamiltonian systems
Marek
Izydorek
2807-2831
Abstract: We will be concerned with the existence of multiple periodic solutions of asymptotically linear Hamiltonian systems with the presence of $Z_2$-action. To that purpose we prove a new version of the Bourgin-Yang theorem. Using the notion of the crossing number we also introduce a new definition of the Morse index for indefinite functionals.
``Best possible'' upper and lower bounds for the zeros of the Bessel function $J_\nu(x)$
C.
K.
Qu;
R.
Wong
2833-2859
Abstract: Let $j_{\nu,k}$ denote the $k$-th positive zero of the Bessel function $J_\nu(x)$. In this paper, we prove that for $\nu>0$ and $k=1$, 2, 3, $\ldots$, \begin{displaymath}\nu - \frac{a_k}{2^{1/3}} \nu^{1/3} < j_{\nu,k} < \nu - \frac{a_k}{2^{1/3}} \nu^{1/3} + \frac{3}{20} a_k^2 \frac{2^{1/3}}{\nu^{1/3}} \,. \end{displaymath} These bounds coincide with the first few terms of the well-known asymptotic expansion \begin{displaymath}j_{\nu,k} \sim \nu - \frac{a_k}{2^{1/3}} \nu^{1/3} + \frac{3}{20} a_k^2 \frac{2^{1/3}}{\nu^{1/3}} + \cdots \end{displaymath} as $\nu\to\infty$, $k$ being fixed, where $a_k$ is the $k$-th negative zero of the Airy function $\operatorname{Ai}(x)$, and so are ``best possible''.
Chaotic solutions in differential inclusions: chaos in dry friction problems
Michal
Feckan
2861-2873
Abstract: The existence of a continuum of many chaotic solutions is shown for certain differential inclusions which are small periodic multivalued perturbations of ordinary differential equations possessing homoclinic solutions to hyperbolic fixed points. Applications are given to dry friction problems. Singularly perturbed differential inclusions are investigated as well.
On the enhancement of diffusion by chaos, escape rates and stochastic instability
Pierre
Collet;
Servet
Martínez;
Bernard
Schmitt
2875-2897
Abstract: We consider stochastic perturbations of expanding maps of the interval where the noise can project the trajectory outside the interval. We estimate the escape rate as a function of the amplitude of the noise and compare it with the purely diffusive case. This is done under a technical hypothesis which corresponds to stability of the absolutely continuous invariant measure against small perturbations of the map. We also discuss in detail a case of instability and show how stability can be recovered by considering another invariant measure.
Partial subdifferentials, derivates and Rademacher's Theorem
D.
N.
Bessis;
F.
H.
Clarke
2899-2926
Abstract: In this paper, we present new partial subdifferentiation formulas in nonsmooth analysis, based upon the study of two directional derivatives. Simple applications of these formulas include a new elementary proof of Rademacher's Theorem in ${\mathbb R}^n$, as well as some results on Gâteaux and Fréchet differentiability for locally Lipschitz functions in a separable Hilbert space. RÉSUMÉ. Dans cet article, nous présentons de nouvelles formules de sousdifférentiation partielle en analyse nonlisse, basées sur l'étude de deux dérivées directionnelles. Une simple application de ces formules nous permet d'obtenir une nouvelle preuve élémentaire du théorème de Rademacher dans ${\mathbb R}^{n}$, ainsi que certains résultats sur la différentiabilité Gâteaux ou Fréchet des fonctions localement Lipschitz sur un espace de Hilbert séparable.
Rotation and entropy
William
Geller;
Michal
Misiurewicz
2927-2948
Abstract: For a given map $f: X \to X$ and an observable $\varphi : X \to \mathbb{R} ^{d},$ rotation vectors are the limits of ergodic averages of $\varphi .$ We study which part of the topological entropy of $f$ is associated to a given rotation vector and which part is associated with many rotation vectors. According to this distinction, we introduce directional and lost entropies. We discuss their properties in the general case and analyze them more closely for subshifts of finite type and circle maps.
Compressions of resolvents and maximal radius of regularity
C.
Badea;
M.
Mbekhta
2949-2960
Abstract: Suppose that $\lambda - T$ is left invertible in $L(H)$ for all $\lambda \in \Omega$, where $\Omega$ is an open subset of the complex plane. Then an operator-valued function $L(\lambda)$ is a left resolvent of $T$ in $\Omega$ if and only if $T$ has an extension $\tilde{T}$, the resolvent of which is a dilation of $L(\lambda)$ of a particular form. Generalized resolvents exist on every open set $U$, with $\overline{U}$ included in the regular domain of $T$. This implies a formula for the maximal radius of regularity of $T$ in terms of the spectral radius of its generalized inverses. A solution to an open problem raised by J. Zemánek is obtained.
Equations in the Q-completion of a torsion-free hyperbolic group
O.
Kharlampovich;
E.
Lioutikova;
A.
Myasnikov
2961-2978
Abstract: In this paper we prove the algorithmic solvability of finite systems of equations over the Q-completion of a torsion-free hyperbolic group.
Left-symmetric algebras for $\mathfrak{gl}(n)$
Oliver
Baues
2979-2996
Abstract: We study the classification problem for left-symmetric algebras with commutation Lie algebra ${\mathfrak{gl}}(n)$ in characteristic $0$. The problem is equivalent to the classification of étale affine representations of ${\mathfrak{gl}}(n)$. Algebraic invariant theory is used to characterize those modules for the algebraic group $\operatorname{SL}(n)$ which belong to affine étale representations of ${\mathfrak{gl}}(n)$. From the classification of these modules we obtain the solution of the classification problem for ${\mathfrak{gl}}(n)$. As another application of our approach, we exhibit left-symmetric algebra structures on certain reductive Lie algebras with a one-dimensional center and a non-simple semisimple ideal.