Asymptotic formulae with remainder estimates for eigenvalue branches of the Schrödinger operator $H - \lambda W$ in a gap of $H$
S.
Z.
Levendorskii
857-899
Abstract: The Floquet theory provides a decomposition of a periodic Schrödinger operator into a direct integral, over a torus, of operators on a basic period cell. In this paper, it is proved that the same transform establishes a unitary equivalence between a multiplier by a decaying potential and a pseudo-differential operator on the torus, with an operator-valued symbol. A formula for the symbol is given. As applications, precise remainder estimates and two-term asymptotic formulas for spectral problems for a perturbed periodic Schrödinger operator are obtained.
Erratic solutions of simple delay equations
Bernhard
Lani-Wayda
901-945
Abstract: We give an example of a smooth function $g:\mathbb{R}\longrightarrow \mathbb{R}$ with only one extremum, with $\mathrm{ sign } g(x) = - \mathrm{ sign } g(-x)$ for $x \neq 0$, and the following properties: The delay equation $\dot x (t) = g(x(t-1))$ has an unstable periodic solution and a solution with phase curve transversally homoclinic to the orbit of the periodic solution. The complicated motion arising from this structure, and its robustness under perturbation of $g$, are described in terms of a Poincaré map. The example is minimal in the sense that the condition $g' < 0$ (under which there would be no extremum) excludes complex solution behavior. Based on numerical observations, we discuss the role of the erratic solutions in the set of all solutions.
Discrete threshold growth dynamics are omnivorous for box neighborhoods
Tom
Bohman
947-983
Abstract: In the discrete threshold model for crystal growth in the plane we begin with some set $A_{0} \subset {\mathbf Z}^{2}$ of seed crystals and observe crystal growth over time by generating a sequence of subsets $A_{0} \subset A_{1} \subset A_{2} \subset \dotsb$ of ${\mathbf Z}^{2}$ by a deterministic rule. This rule is as follows: a site crystallizes when a threshold number of crystallized points appear in the site's prescribed neighborhood. The growth dynamics generated by this model are said to be omnivorous if $A_{0}$ finite and $A_{i+1} \neq A_{i} \; \forall i$ imply $\bigcup _{i=0}^{\infty} A_{i} = {\mathbf Z}^{2}$. In this paper we prove that the dynamics are omnivorous when the neighborhood is a box (i.e. when, for some fixed $\rho$, the neighborhood of $z$ is $\{ x \in {\mathbf Z}^{2} : \|x-z\|_{\infty} \le \rho\})$. This result has important implications in the study of the first passage time when $A_{0}$ is chosen randomly with a sparse Bernoulli density and in the study of the limiting shape to which $n^{-1}A_{n}$ converges.
Dual kinematic formulas
Gaoyong
Zhang
985-995
Abstract: We establish kinematic formulas for dual quermassintegrals of star bodies and for chord power integrals of convex bodies by using dual mixed volumes. These formulas are extensions of the fundamental kinematic formula involving quermassintegrals to the cases of dual quermassintegrals and chord power integrals. Applications to geometric probability are considered.
Compatible complex structures on almost quaternionic manifolds
D.
V.
Alekseevsky;
S.
Marchiafava;
M.
Pontecorvo
997-1014
Abstract: On an almost quaternionic manifold $(M^{4n},Q)\;$we study the integrability of almost complex structures which are compatible with the almost quaternionic structure $Q$. If $n\geq 2$, we prove that the existence of two compatible complex structures $I_{1}, I_{2}\neq \pm I_{1}$ forces $(M^{4n},Q)\;$to be quaternionic. If $n=1$, that is $(M^{4},Q)=(M^{4},[g],or)$ is an oriented conformal 4-manifold, we prove a maximum principle for the angle function $\langle I_{1},I_{2}\rangle$ of two compatible complex structures and deduce an application to anti-self-dual manifolds. By considering the special class of Oproiu connections we prove the existence of a well defined almost complex structure $\mathbb J$ on the twistor space $Z$ of an almost quaternionic manifold $(M^{4n},Q)\;$and show that $\mathbb J$ is a complex structure if and only if $Q$ is quaternionic. This is a natural generalization of the Penrose twistor constructions.
A remarkable formula for counting nonintersecting lattice paths in a ladder with respect to turns
C.
Krattenthaler;
M.
Prohaska
1015 - 1042
On graphs with a metric end space
Kerstin
Waas
1043-1062
Abstract: R. Diestel conjectured that an infinite graph contains a topologically end-faithful forest if and only if its end space is metrizable. We prove this conjecture for uniform end spaces.
On the diophantine equation $(x^3-1)/(x-1)=(y^n-1)/(y-1)$
Maohua
Le
1063-1074
Abstract: In this paper we prove that the equation $(x^3-1)/(x-1)=$ $(y^n-1)/(y-1)$, $x,y,n\in\mathbb{N}$, $x>1$, $y>1$, $n>3$, has only the solutions $(x,y,n)=(5,2,5)$ and $(90,2,13)$ with $y$ is a prime power. The proof depends on some new results concerning the upper bounds for the number of solutions of the generalized Ramanujan-Nagell equations.
On the non-vanishing of cubic twists of automorphic $L$-series
Xiaotie
She
1075-1094
Abstract: Let $f$ be a normalised new form of weight $2$ for $\Gamma _{0} (N)$ over ${\mathbb{Q}}$ and $F$, its base change lift to $\mathbb{Q}(\sqrt {-3})$. A sufficient condition is given for the nonvanishing at the center of the critical strip of infinitely many cubic twists of the $L$-function of $F$. There is an algorithm to check the condition for any given form. The new form of level $11$ is used to illustrate our method.
Group extensions and tame pairs
Michael
L.
Mihalik
1095-1107
Abstract: Tame pairs of groups were introduced to study the missing boundary problem for covers of compact 3-manifolds. In this paper we prove that if $1\to A\to G\to B\to 1$ is an exact sequence of infinite finitely presented groups or if $G$ is an ascending HNN-extension with base $A$ and $H$ is a certain type of finitely presented subgroup of $A$, then the pair $(G,H)$ is tame. Also we develop a technique for showing certain groups cannot be the fundamental group of a compact 3-manifold. In particular, we give an elementary proof of the result of R. Bieri, W. Neumann and R. Strebel: A strictly ascending HNN-extension cannot be the fundamental group of a compact 3-manifold.
A Weakly Chainable Tree-Like Continuum without the Fixed Point Property
Piotr
Minc
1109-1121
Abstract: An example of a fixed points free map is constructed on a tree-like, weakly chainable continuum.
A generalization of Snaith-type filtration
Greg
Arone
1123-1150
Abstract: In this paper we describe the Goodwillie tower of the stable homotopy of a space of maps from a finite-dimensional complex to a highly enough connected space. One way to view it is as a partial generalization of some well-known results on stable splittings of mapping spaces in terms of configuration spaces.
Riesz transforms for $1\le p\le 2$
Thierry
Coulhon;
Xuan
Thinh
Duong
1151-1169
Abstract: It has been asked (see R. Strichartz, Analysis of the Laplacian$\dotsc$, J. Funct. Anal. 52 (1983), 48-79) whether one could extend to a reasonable class of non-compact Riemannian manifolds the $L^p$ boundedness of the Riesz transforms that holds in ${\mathbb R}^n$. Several partial answers have been given since. In the present paper, we give positive results for $1\leq p\leq 2$ under very weak assumptions, namely the doubling volume property and an optimal on-diagonal heat kernel estimate. In particular, we do not make any hypothesis on the space derivatives of the heat kernel. We also prove that the result cannot hold for $p>2$ under the same assumptions. Finally, we prove a similar result for the Riesz transforms on arbitrary domains of ${\mathbb R}^n$.
Building blocks for quadratic Julia sets
Joachim
Grispolakis;
John
C.
Mayer;
Lex
G.
Oversteegen
1171-1201
Abstract: We obtain results on the structure of the Julia set of a quadratic polynomial $P$ with an irrationally indifferent fixed point $z_0$ in the iterative dynamics of $P$. In the Cremer point case, under the assumption that the Julia set is a decomposable continuum, we obtain a building block structure theorem for the corresponding Julia set $J=J(P)$: there exists a nowhere dense subcontinuum $B\subset J$ such that $P(B)=B$, $B$ is the union of the impressions of a minimally invariant Cantor set $A$ of external rays, $B$ contains the critical point, and $B$ contains both the Cremer point $z_0$ and its preimage. In the Siegel disk case, under the assumption that no impression of an external ray contains the boundary of the Siegel disk, we obtain a similar result. In this case $B$ contains the boundary of the Siegel disk, properly if the critical point is not in the boundary, and $B$ contains no periodic points. In both cases, the Julia set $J$ is the closure of a skeleton $S$ which is the increasing union of countably many copies of the building block $B$ joined along preimages of copies of a critical continuum $C$ containing the critical point. In addition, we prove that if $P$ is any polynomial of degree $d\ge 2$ with a Siegel disk which contains no critical point on its boundary, then the Julia set $J(P)$ is not locally connected. We also observe that all quadratic polynomials which have an irrationally indifferent fixed point and a locally connected Julia set have homeomorphic Julia sets.
Invariant Measures for Set-Valued Dynamical Systems
Walter
Miller;
Ethan
Akin
1203-1225
Abstract: A continuous map on a compact metric space, regarded as a dynamical system by iteration, admits invariant measures. For a closed relation on such a space, or, equivalently, an upper semicontinuous set-valued map, there are several concepts which extend this idea of invariance for a measure. We prove that four such are equivalent. In particular, such relation invariant measures arise as projections from shift invariant measures on the space of sample paths. There is a similarly close relationship between the ideas of chain recurrence for the set-valued system and for the shift on the sample path space.
Germs of Kloosterman Integrals for $GL(3)$
Hervé
Jacquet;
Yangbo
Ye
1227-1255
Abstract: In an earlier paper we introduced the concept of Shalika germs for certain Kloosterman integrals. We compute explicitly the germs in the case of the group $GL(3)$.
On locally linearly dependent operators and derivations
Matej
Bresar;
Peter
Semrl
1257-1275
Abstract: The first section of the paper deals with linear operators $T_i:U\longrightarrow V$, $i = 1,\ldots,n$, where $U$ and $V$ are vector spaces over an infinite field, such that for every $u \in U$, the vectors $T_1 u,\ldots,T_n u$ are linearly dependent modulo a fixed finite dimensional subspace of $V$. In the second section, outer derivations of dense algebras of linear operators are discussed. The results of the first two sections of the paper are applied in the last section, where commuting pairs of continuous derivations $d,g$ of a Banach algebra ${\cal A}$ such that $(dg)(x)$ is quasi-nilpotent for every $x \in {\cal A}$ are characterized.
Composition algebras over rings of genus zero
S.
Pumplün
1277-1292
Abstract: The theory of composition algebras over locally ringed spaces and some basic results from algebraic geometry are used to characterize composition algebras over open dense subschemes of curves of genus zero.