Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations

We introduce the notion of signature for relations in mapping class groups and show that the signature of a Lefschetz fibration over the 2-sphere is the sum of the signatures for basic relations contained in its monodromy. Combining explicit calculations of the signature cocycle with a technique of substituting positive relations, we give some new examples of non-holomorphic Lefschetz fibrations of genus and which violate slope bounds for non-hyperelliptic fibrations on algebraic surfaces of general type.

     

Fundamental Groups of Closed Manifolds with Positive Curvature and Torus Actions

We determine the fundamental group of a closed n-manifold of positive sectional curvature on which a torus T^k (k large) acts effectively and isometrically. Our results are: (A) If k>(n − 3)/4 and n ≥ 17, then the fundamental group π₁(M) is isomorphic to the fundamental group of a spherical 3-space form. (B) If k ≥ (n/6)+1 and n≠ 11, 15, 23, then any abelian subgroup of π₁(M) is cyclic. Moreover, if the T^k-fixed point set is empty, then π₁(M) is isomorphic to the fundamental group of a spherical 3-space form.Mathematics Subject Classification (2000). 53-XX*Supported partially by NSF Grant DMS 0203164 and by a reach found from Beijing normal university.**Supported partially by NSFC 10371008.     

An approximate analysis of the interaction between two tearing modes of the same helicity

The interaction between tearing modes with the mode numbersm=1,n=1 andm=2,n=2 is investigated for different initial amplitudes of the modes, using a single helicity approximation, a step-current profile and a time-independent resistivity. Also included are the results on the temporal behaviour of the amplitude of the uncoupled modem=2,n=2 in the situation with identical equilibrium parameters as ours.     

Transition from decelerated to accelerated cosmic expansion in braneworld universes

Braneworld theory provides a natural setting to treat, at a classical level, the cosmological effects of vacuum energy. Non-static extra dimensions can generally lead to a variable vacuum energy, which in turn may explain the present accelerated cosmic expansion. We concentrate our attention in models where the vacuum energy decreases as an inverse power law of the scale factor. These models agree with the observed accelerating universe, while fitting simultaneously the observational data for the density and deceleration parameter. The redshift at which the vacuum energy can start to dominate depends on the mass density of ordinary matter. For _m = 0.3, the transition from decelerated to accelerated cosmic expansion occurs at z _T ≈ 0.48 ± 0.20, which is compatible with SNe data. We set a lower bound on the deceleration parameter today, namely > − 1 + 3 _m /2, i.e., > − 0.55 for _m = 0.3. The future evolution of the universe crucially depends on the time when vacuum starts to dominate over ordinary matter. If it dominates only recently, at an epoch z < 0.64, then the universe is accelerating today and will continue that way forever. If vacuum dominates earlier, at z > 0.64, then the deceleration comes back and the universe recollapses at some point in the distant future. In the first case, quintessence and Cardassian expansion can be formally interpreted as the low energy limit of our model, although they are entirely different in philosophy. In the second case there is no correspondence between these models and ours.     

Commensurability of Hyperbolic Manifolds with Geodesic Boundary

Suppose , let M ₁, M ₂ be n-dimensional connected complete finite-volume hyperbolic manifolds with nonempty geodesic boundary, and suppose that π₁ (M ₁) is quasi-isometric to π₁ (M ₂) (with respect to the word metric). Also suppose that if n=3, then ∂M ₁ and ∂M ₂ are compact. We show that M ₁ is commensurable with M ₂. Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifolds with non-compact geodesic boundary which are not commensurable with each other. We also prove that if M is as M ₁ above and G is a finitely generated group which is quasi-isometric to π₁ (M), then there exists a hyperbolic manifold with geodesic boundary M′ with the following properties: M′ is commensurable with M, and G is a finite extension of a group which contains π₁ (M′) as a finite-index subgroupMathematics Subject Classification (2000). Primary: 20F65; secondary: 30C65, 57N16     

124 W diode-side-pumped Nd:YAG laser in dynamic fundamental mode

A high power dynamic fundamental mode Nd:YAG laser is experimentally demonstrated with a stagger-pumped laser module and a V-shaped resonator. The rod is pumped symmetrically by staggered bar modules. And dynamic fundamental mode is achieved under different pump levels. The maximal continuous wave (CW) output of 124 W (M²=1.4) is achieved with a dual rod. Average output of 112 W, pulse width of 120 ns, pulse energy of 11.2 mJ and peak power of 93 kW are obtained in Q-switched operation of 10 kHz.     

The action of the etale fundamental group scheme on the connected component of the essentially finite one

We follow the pattern in [14, Section 4] to define an action of the etale fundamental group scheme on the local component of the essentially finite fundamental group scheme of Nori. We show that the associated representation is faithful when X is a curve of genus ⩾2.     

On some hypersurfaces of the homogeneous nearly Kähler

In this paper, we study hypersurfaces of the homogeneous nearly Kähler manifold with typical properties. We first show that in the NK there exist neither totally umbilical hypersurfaces nor hypersurfaces of parallel second fundamental form. Then we investigate hypersurfaces of such that its shape operator A and induced almost contact structure ϕ satisfy the condition , and as the main result, a complete classification of this remarkable family of hypersurfaces in is presented.     

非时齐马氏过程的耦合基本定理

本文将耦合方法应用于非时齐马氏过程,推广了时齐情形的耦合基本定理,为后续研究非时齐马氏过程的耦合提供了理论基础.     

弹簧地基上大直径单桩海上风机的一阶自振频率

自振频率是海上风机前期动力设计的主要难点之一,计算精度要求极高,而桩-土相互作用对自振频率影响显著。目前,针对桩-土相互作用普遍采用弹簧地基简化模型,为对比分析不同弹簧地基简化模型的准确性和精度,本文采用回传射线矩阵法,建立固定端、单弹簧、双弹簧和三弹簧这4种地基简化模型,基于Bernoulli-Euler梁理论和Timoshenko梁理论,求解风机系统的自振频率,并与风机基频实测值作对比,进一步分析三弹簧简化地基模型中地基旋转刚度、地基耦合刚度和地基水平刚度对风机系统基频的影响。研究结果表明,桩-土相互作用不可忽略,三弹簧地基简化模型计算风机系统基频精度最高;地基刚度对风机系统基频的敏感性次序为,地基耦合刚度>地基旋转刚度>地基水平刚度;当地基旋转(水平)刚度较小时,地基耦合刚度与地基旋转(水平)刚度的耦合效应对风机系统的基频影响较大;地基耦合刚度与地基旋转刚度的耦合效应对风机系统基频的影响程度大于地基耦合刚度与地基水平刚度的耦合效应。