Growth Tight Actions

We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, non-elementary group $G$ acting on a $G$--space $\mathcal{X}$, we prove that if $G$ contains a strongly contracting element and if $G$ is not too badly distorted in $\mathcal{X}$, then the action of $G$ on $\mathcal{X}$ is a growth tight action. It follows that if $\mathcal{X}$ is a cocompact, relatively hyperbolic $G$--space, then the action of $G$ on $\mathcal{X}$ is a growth tight action. This generalizes all previously known results for growth tightness of cocompact actions: every already known example of a group that admits a growth tight action and has some infinite, infinite index normal subgroups is relatively hyperbolic, and, conversely, relatively hyperbolic groups admit growth tight actions. This also allows us to prove that many CAT(0) groups, including flip-graph-manifold groups and many Right Angled Artin Groups, and snowflake groups admit cocompact, growth tight actions. These provide first examples of non-relatively hyperbolic groups admitting interesting growth tight actions. Our main result applies as well to cusp uniform actions on hyperbolic spaces and to the action of the mapping class group on Teichmueller space with the Teichmueller metric. Towards the proof of our main result, we give equivalent characterizations of strongly contracting elements and produce new examples of group actions with strongly contracting elements.Comment: 29 pages, 4 figures v2 added references v3 40 pages, 6 figures, expanded preliminary sections to make paper more self-contained, other minor improvements v4 updated bibliography, to appear in Pacific Journal of Mathematic

A Conjecture about Raising Operators for Macdonald Polynomials

A multivariable hypergeometric-type formula for raising operators of the Macdonald polynomials is conjectured. It is proved that this agrees with Jing and Jozefiak's expression for the two-row Macdonald polynomials, and also with Lassalle and Schlosser's formula for partitions with length three.Comment: 13 page

Accurate determination of the Lagrangian bias for the dark matter halos

We use a new method, the cross power spectrum between the linear density field and the halo number density field, to measure the Lagrangian bias for dark matter halos. The method has several important advantages over the conventional correlation function analysis. By applying this method to a set of high-resolution simulations of 256^3 particles, we have accurately determined the Lagrangian bias, over 4 magnitudes in halo mass, for four scale-free models with the index n=-0.5, -1.0, -1.5 and -2.0 and three typical CDM models. Our result for massive halos with $M \ge M_*$ ($M_*$ is a characteristic non-linear mass) is in very good agreement with the analytical formula of Mo & White for the Lagrangian bias, but the analytical formula significantly underestimates the Lagrangian clustering for the less massive halos $M < M_*. Our simulation result however can be satisfactorily described, with an accuracy better than 15%, by the fitting formula of Jing for Eulerian bias under the assumption that the Lagrangian clustering and the Eulerian clustering are related with a linear mapping. It implies that it is the failure of the Press-Schechter theories for describing the formation of small halos that leads to the inaccuracy of the Mo & White formula for the Eulerian bias. The non-linear mapping between the Lagrangian clustering and the Eulerian clustering, which was speculated as another possible cause for the inaccuracy of the Mo & White formula, must at most have a second-order effect. Our result indicates that the halo formation model adopted by the Press-Schechter theories must be improved.Comment: Minor changes; accepted for publication in ApJ (Letters) ; 11 pages with 2 figures include

Monitoring capabilities of a mobile mapping system based on navigation qualities

Mobile mapping systems are becoming increasingly popular as they can build 3D models of the environment rapidly by using a laser scanner that is integrated with a navigation system. 3D mobile mapping has been widely used for applications such as 3D city modelling and mapping of the scanned environments. However, accurate mapping relies on not only the scanner’s performance but also on the quality of the navigation results (accuracy and robustness) . This paper discusses the potentials of using 3D mobile mapping systems for landscape change detection, that is traditionally carried out by terrestrial laser scanners that can be accurately geo-referenced at a static location to produce highly accurate dense point clouds. Yet compared to conventional surveying using terrestrial laser scanners, several advantages of mobile mapping systems can be identified. A large area can be monitored in a relatively short period, which enables high repeat frequency monitoring without having to set-up dedicated stations. However, current mobile mapping applications are limited by the quality of navigation results, especially in different environments. The change detection ability of mobile mapping systems is therefore significantly affected by the quality of the navigation results. This paper presents some data collected for the purpose of monitoring from a mobile platform. The datasets are analysed to address current potentials and difficulties. The change detection results are also presented based on the collected dataset. Results indicate the potentials of change detection using a mobile mapping system and suggestions to enhance quality and robustness

Quantum entropy of the Kerr black hole arising from gravitational perturbation

The quantum entropy of the Kerr black hole arising from gravitational perturbation is investigated by using Null tetrad and \'t Hooft\'s brick-wall model. It is shown that effect of the graviton\'s spins on the subleading correction is dependent of the square of the spins and the angular momentum per unit mass of the black hole, and contribution of the logarithmic term to the entropy will be positive, zero, and negative for different value of $a/r_+$.Comment: 8 pages, 1 figure, Latex. to appear in Phys. Rev.

Two-parameter quantum general linear supergroups

The universal R-matrix of two-parameter quantum general linear supergroups is computed explicitly based on the RTT realization of Faddeev--Reshetikhin--Takhtajan.Comment: v1: 14 pages. v2: published version, 9 pages, title changed and the section on central extension remove

Quantum Phase Diffusion in a Small Underdamped Josephson Junction

Quantum phase diffusion in a small underdamped Nb/AlO$_x$/Nb junction ($\sim$ 0.4 $\mu$m$^2$) is demonstrated in a wide temperature range of 25-140 mK where macroscopic quantum tunneling (MQT) is the dominant escape mechanism. We propose a two-step transition model to describe the switching process in which the escape rate out of the potential well and the transition rate from phase diffusion to the running state are considered. The transition rate extracted from the experimental switching current distribution follows the predicted Arrhenius law in the thermal regime but is greatly enhanced when MQT becomes dominant.Comment: 4 pages, 4 figures, 1 tabl

Jack vertex operators and realization of Jack functions

We give an iterative method to realize general Jack functions from Jack functions of rectangular shapes. We first show some cases of Stanley's conjecture on positivity of the Littlewood-Richardson coefficients, and then use this method to give a new realization of Jack functions. We also show in general that vectors of products of Jack vertex operators form a basis of symmetric functions. In particular this gives a new proof of linear independence for the rectangular and marked rectangular Jack vertex operators. Thirdly a generalized Frobenius formula for Jack functions was given and was used to give new evaluation of Dyson integrals and even powers of Vandermonde determinant.Comment: Expanded versio

Scaling properties of the redshift power spectrum: theoretical models

We report the results of an analysis of the redshift power spectrum $P^S(k,\mu)$ in three typical Cold Dark Matter (CDM) cosmological models, where $\mu$ is the cosine of the angle between the wave vector and the line-of-sight. Two distinct biased tracers derived from the primordial density peaks of Bardeen et al. and the cluster-underweight model of Jing, Mo, & B\"orner are considered in addition to the pure dark matter models. Based on a large set of high resolution simulations, we have measured the redshift power spectrum for the three tracers from the linear to the nonlinear regime. We investigate the validity of the relation - guessed from linear theory - in the nonlinear regime $$P^S(k,\mu)=P^R(k)[1+\beta\mu^2]^2D(k,\mu,\sigma_{12}(k)),$$ where $P^R(k)$ is the real space power spectrum, and $\beta$ equals $\Omega_0^{0.6}/b_l$. The damping function $D$ which should generally depend on $k$, $\mu$, and $\sigma_{12}(k)$, is found to be a function of only one variable $k\mu\sigma_{12}(k)$. This scaling behavior extends into the nonlinear regime, while $D$ can be accurately expressed as a Lorentz function - well known from linear theory - for values $D > 0.1$. The difference between $\sigma_{12}(k)$ and the pairwise velocity dispersion defined by the 3-D peculiar velocity of the simulations (taking $r=1/k$) is about 15%. Therefore $\sigma_{12}(k)$ is a good indicator of the pairwise velocity dispersion. The exact functional form of $D$ depends on the cosmological model and on the bias scheme. We have given an accurate fitting formula for the functional form of $D$ for the models studied.Comment: accepted for publication in ApJ;24 pages with 7 figures include

Dense-coding quantum key distribution based on continuous-variable entanglement

We proposed a scheme of continuous-variable quantum key distribution, in which the bright Einstein-Podolsky-Rosen entangled optical beams are utilized. The source of the entangled beams is placed inside the receiving station, where half of the entangled beams are transmitted with round trip and the other half are retained by the receiver. The amplitude and phase signals modulated on the signal beam by the sender are simultaneously extracted by the authorized receiver with the scheme of the dense-coding correlation measurement for continuous quantum variables, thus the channel capacity is significantly improved. Two kinds of possible eavesdropping are discussed. The mutual information and the secret key rates are calculated and compared with those of unidirectional transmission schemes