Equivariant Novikov conjecture for groups acting on Euclidean buildings
Donggeng
Gong
2141-2183
Abstract: We prove the equivariant Novikov conjecture for groups acting on Euclidean buildings by using an equivariant Hilsum-Skandalis method. We also obtain an equivariant version of the Connes-Gromov-Moscovici theorem for almost flat $C^{*}$-algebra bundles.
Carathéodory-Toeplitz and Nehari problems for matrix valued almost periodic functions
Leiba
Rodman;
Ilya
M.
Spitkovsky;
Hugo
J.
Woerdeman
2185-2227
Abstract: In this paper the positive and strictly contractive extension problems for almost periodic matrix functions are treated. We present necessary and sufficient conditions for the existence of extensions in terms of Toeplitz and Hankel operators on Besicovitch spaces and Lebesgue spaces. Furthermore, when a solution exists a special extension (the band extension) is constructed which enjoys a maximum entropy property. A linear fractional parameterization of the set of all extensions is also provided. The techniques used in the proofs include factorizations of matrix valued almost periodic functions and a general algebraic scheme called the band method.
Topological conditions for the existence of absorbing Cantor sets
Henk
Bruin
2229-2263
Abstract: This paper deals with strange attractors of S-unimodal maps $f$. It generalizes earlier results in the sense that very general topological conditions are given that either i) guarantee the existence of an absorbing Cantor set provided the critical point of $f$ is sufficiently degenerate, or ii) prohibit the existence of an absorbing Cantor set altogether. As a by-product we obtain very weak topological conditions that imply the existence of an absolutely continuous invariant probability measure for $f$.
Periodic orbits of the restricted three-body problem
Salem
Mathlouthi
2265-2276
Abstract: We prove, using a variational formulation, the existence of an infinity of periodic solutions of the restricted three-body problem. When the problem has some additional symmetry (in particular, in the autonomous case), we prove the existence of at least two periodic solutions of minimal period $T$, for every $T>0$. We also study the bifurcation problem in a neighborhood of each closed orbit of the autonomous restricted three-body problem.
Weak*-closedness of subspaces of Fourier-Stieltjes algebras and weak*-continuity of the restriction map
M.
B.
Bekka;
E.
Kaniuth;
A.
T.
Lau;
G.
Schlichting
2277-2296
Abstract: Let $G$ be a locally compact group and $B(G)$ the Fourier-Stieltjes algebra of $G$. We study the problem of how weak*-closedness of some translation invariant subspaces of $B(G)$ is related to the structure of $G$. Moreover, we prove that for a closed subgroup $H$ of $G$, the restriction map from $B(G)$ to $B(H)$ is weak*-continuous only when $H$ is open in $G$.
Singularity of self-similar measures with respect to Hausdorff measures
Manuel
Morán;
José-Manuel
Rey
2297-2310
Abstract: Besicovitch (1934) and Eggleston (1949) analyzed subsets of points of the unit interval with given frequencies in the figures of their base-$p$ expansions. We extend this analysis to self-similar sets, by replacing the frequencies of figures with the frequencies of the generating similitudes. We focus on the interplay among such sets, self-similar measures, and Hausdorff measures. We give a fine-tuned classification of the Hausdorff measures according to the singularity of the self-similar measures with respect to those measures. We show that the self-similar measures are concentrated on sets whose frequencies of similitudes obey the Law of the Iterated Logarithm.
The irrationality of $\log(1+1/q) \log(1-1/q)$
Masayoshi
Hata
2311-2327
Abstract: We shall show that the numbers $1, \log (1+ 1/q), \log (1-1/q)$ and $\log (1+1/q)\log (1-1/q)$ are linearly independent over $\mathbf{Q}$ for any natural number $q \ge 54$. The key is to construct explicit Padé-type approximations using Legendre-type polynomials.
Approximation of the equilibrium distribution by distributions of equal point charges with minimal energy
J.
Korevaar;
M.
A.
Monterie
2329-2348
Abstract: Let $\omega$ denote the classical equilibrium distribution (of total charge $1$) on a convex or $C^{1,\alpha }$-smooth conductor $K$ in $\mathbb{R}^{q}$ with nonempty interior. Also, let $\omega _{N}$ be any $N$th order ``Fekete equilibrium distribution'' on $K$, defined by $N$ point charges $1/N$ at $N$th order ``Fekete points''. (By definition such a distribution minimizes the energy for $N$-tuples of point charges $1/N$ on $K$.) We measure the approximation to $\omega$ by $\omega _{N}$ for $N \to \infty$ by estimating the differences in potentials and fields, \begin{equation*}U^{\omega }-U^{\omega _{N}}\quad \text{\rm and}\quad {\mathcal{E}}^{\omega }-{\mathcal{E}}^{\omega _{N}},\end{equation*} both inside and outside the conductor $K$. For dimension $q \geq 3$ we obtain uniform estimates ${\mathcal{O}}(1/N^{1/(q-1)})$ at distance $\geq \varepsilon >0$ from the outer boundary $\Sigma$ of $K$. Observe that ${\mathcal{E}}^{\omega }=0$ throughout the interior $\Omega$ of $\Sigma$ (Faraday cage phenomenon of electrostatics), hence ${\mathcal{E}}^{\omega _{N}}={\mathcal{O}}(1/N^{1/(q-1)})$ on the compact subsets of $\Omega$. For the exterior $\Omega ^{\infty }$ of $\Sigma$ the precise results are obtained by comparison of potentials and energies. Admissible sets $K$ have to be regular relative to capacity and their boundaries must allow good Harnack inequalities. For the passage to interior estimates we develop additional machinery, including integral representations for potentials of measures on Lipschitz boundaries $\Sigma$ and bounds on normal derivatives of interior and exterior Green functions. Earlier, one of us had considered approximations to the equilibrium distribution by arbitrary distributions $\mu _{N}$ of equal point charges on $\Sigma$. In that context there is an important open problem for the sphere which is discussed at the end of the paper.
Strongly indefinite systems with critical Sobolev exponents
Josephus
Hulshof;
Enzo
Mitidieri;
Robertus
vanderVorst
2349-2365
Abstract: We consider an elliptic system of Hamiltonian type on a bounded domain. In the superlinear case with critical growth rates we obtain existence and positivity results for solutions under suitable conditions on the linear terms. Our proof is based on an adaptation of the dual variational method as applied before to the scalar case.
Approximation by polynomials with nonnegative coefficients and the spectral theory of positive operators
Roger
D.
Nussbaum;
Bertram
Walsh
2367-2391
Abstract: For $\Sigma$ a compact subset of $\mathbf{C}$ symmetric with respect to conjugation and $f: \Sigma \to \mathbf{C}$ a continuous function, we obtain sharp conditions on $f$ and $\Sigma$ that insure that $f$ can be approximated uniformly on $\Sigma$ by polynomials with nonnegative coefficients. For $X$ a real Banach space, $K \subseteq X$ a closed but not necessarily normal cone with $\overline{K - K} = X$, and $A: X \to X$ a bounded linear operator with $A[K] \subseteq K$, we use these approximation theorems to investigate when the spectral radius $\text{\rm r}(A)$ of $A$ belongs to its spectrum $\sigma (A)$. A special case of our results is that if $X$ is a Hilbert space, $A$ is normal and the 1-dimensional Lebesgue measure of $\sigma (i(A - A^{*}))$ is zero, then $\text{\rm r}(A) \in \sigma (A)$. However, we also give an example of a normal operator $A = - U -\alpha I$ (where $U$ is unitary and $\alpha > 0$) for which $A[K] \subseteq K$ and $\text{\rm r}(A) \notin \sigma (A)$.
Eigenfunctions of the Weil representation of unitary groups of one variable
Tonghai
Yang
2393-2407
Abstract: In this paper, we construct explicit eigenfunctions of the local Weil representation on unitary groups of one variable in the $p$-adic case when $p$ is odd. The idea is to use the lattice model, and the results will be used to compute special values of certain Hecke $L$-functions in separate papers. We also recover Moen's results on when a local theta lifting from $U(1)$ to itself does not vanish.
Necessary conditions for constrained optimization problems with semicontinuous and continuous data
Jonathan
M.
Borwein;
Jay
S.
Treiman;
Qiji
J.
Zhu
2409-2429
Abstract: We consider nonsmooth constrained optimization problems with semicontinuous and continuous data in Banach space and derive necessary conditions without constraint qualification in terms of smooth subderivatives and normal cones. These results, in different versions, are set in reflexive and smooth Banach spaces.
On homological properties of singular braids
Vladimir
V.
Vershinin
2431-2455
Abstract: Homology of objects which can be considered as singular braids, or braids with crossings, is studied. Such braids were introduced in connection with Vassiliev's theory of invariants of knots and links. The corresponding algebraic objects are the braid-permutation group $BP_{n}$ of R. Fenn, R. Rimányi and C. Rourke and the Baez-Birman monoid $SB_{n}$ which embeds into the singular braid group $SG_{n}$. The following splittings are proved for the plus-constructions of the classifying spaces of the infinite braid-permutation group and the singular braid group \begin{equation*}\mathbb{Z}\times BBP_{\infty }^{+}\simeq \Omega ^{\infty }S^{\infty }\times S^{1} \times Y, \end{equation*} \begin{equation*}\mathbb{Z}\times BSG_{\infty }^{+}\simeq S^{1}\times \Omega ^{2} S^{2}\times W, \end{equation*} where $Y$ is an infinite loop space and $W$ is a double loop space.
Fixed point sets of deformations of polyhedra with local cut points
Peter
Wolfenden
2457-2471
Abstract: A locally finite simplicial complex $X$ is said to be 2-dimensionally connected if $X - \{\text{local cut points of } X \}$ is connected. Such spaces exhibit ``classical'' behavior in that they all admit deformations with one fixed point, and they admit fixed point free deformations if and only if the Euler characteristic is zero. A result of G.-H. Shi implies that, for non 2-dimensionally connected spaces, the fixed point sets of deformations are equivalent to the fixed point sets of certain combinatorial maps which he calls good displacements. U. K. Scholz combined Shi's results with a theorem of P. Hall to obtain a characterization of all finite simplicial complexes which admit fixed point free deformations. In this paper we begin by explicitly capturing the combinatorial structure of a non 2-dimensionally connected polyhedron in a bipartite graph. We then apply an extended version of Hall's theorem to this graph to get a realization theorem which gives necessary and sufficient conditions for the existence of a deformation with a prescribed finite fixed point set. Scholz's result, and a characterization of all finite simplicial complexes without fixed point free deformations that admit deformations with a single fixed point follow immediately from this realization theorem.
Abelian subgroups of pro-$p$ Galois groups
Antonio
José
Engler;
Jochen
Koenigsmann
2473-2485
Abstract: It is proved that non-trivial normal abelian subgroups of the Galois group of the maximal Galois $p$-extension of a field $F$ (where $p$ is an odd prime) arise from $p$-henselian valuations with non-$p$-divisible value group, provided $\# (\dot {F}/\dot {F}^{p})\geq p^{2}$ and $F$ contains a primitive $p$-th root of unity. Also, a generalization to arbitrary prime-closed Galois-extensions is given.
On the distribution of mass in collinear central configurations
Peter
W.
Lindstrom
2487-2523
Abstract: Moulton's Theorem says that given an ordering of masses, $m_1,m_2, \dotsc,m_n$, there exists a unique collinear central configuration with center of mass at the origin and moment of inertia equal to 1. This theorem allows us to ask the questions: What is the distribution of mass in this standardized collinear central configuration? What is the behavior of the distribution as $n\to\infty$? In this paper, we define continuous configurations, prove a continuous version of Moulton's Theorem, and then, in the spirit of limit theorems in probability theory, prove that as $n\to\infty$, under rather general conditions, the discrete mass distributions of the standardized collinear central configurations have distribution functions which converge uniformly to the distribution function of the unique continuous standardized collinear central configuration which we determine.
Geometric aspects of multiparameter spectral theory
Luzius
Grunenfelder;
Tomaz
Kosir
2525-2546
Abstract: The paper contains a geometric description of the dimension of the total root subspace of a regular multiparameter system in terms of the intersection multiplicities of its determinantal hypersurfaces. The new definition of regularity used here is proved to restrict to the familiar definition in the linear case. A decomposability problem is also considered. It is shown that the joint root subspace of a multiparameter system may be decomposable even when the root subspace of each member is indecomposable.
An intersection number for the punctual Hilbert scheme of a surface
Geir
Ellingsrud;
Stein
Arild
Strømme
2547-2552
Abstract: We compute the intersection number between two cycles $A$ and $B$ of complementary dimensions in the Hilbert scheme $H$ parameterizing subschemes of given finite length $n$ of a smooth projective surface $S$. The $(n+1)$-cycle $A$ corresponds to the set of finite closed subschemes the support of which has cardinality 1. The $(n-1)$-cycle $B$ consists of the closed subschemes the support of which is one given point of the surface. Since $B$ is contained in $A$, indirect methods are needed. The intersection number is $A.B=(-1)^{n-1}n$, answering a question by H. Nakajima.