Isotropic Subspaces of Polylinear Forms

The dimensions of maximal isotropic subspaces of symmetric and skew-symmetric forms of degree d in general position are found.     

Dirichlet Forms and Brownian Motion Penetrating Fractals

Can a Brownian motion penetrate the two-dimensional Sierpinski gasket? This question was studied in [8], and an affirmative answer was given. In this paper, the problem is studied with a different approach, using Dirichlet forms and function space theory. The results obtained are somewhat different from, and from certain aspects more general than, the results in [8].

     

Unique Orbital Normal Form for Vector Fields of Hopf-Zero Singularity

Normal forms for vector fields of Hopf-zero singularity in R³ are studied. Multiple Lie bracket method is used to give unique normal forms under both conjugacy and orbital equivalence for such vector fields with a generic quadratic part.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.     

Analysis of periodic-quasiperiodic nonlinear systems via Lyapunov-Floquet transformation and normal forms

In this paper a general technique for the analysis of nonlinear dynamical systems with periodic-quasiperiodic coefficients is developed. For such systems the coefficients of the linear terms are periodic with frequency ω while the coefficients of the nonlinear terms contain frequencies that are incommensurate with ω. No restrictions are placed on the size of the periodic terms appearing in the linear part of system equation. Application of Lyapunov-Floquet transformation produces a dynamically equivalent system in which the linear part is time-invariant and the time varying coefficients of the nonlinear terms are quasiperiodic. Then a series of quasiperiodic near-identity transformations are applied to reduce the system equation to a normal form. In the process a quasiperiodic homological equation and the corresponding ‘solvability condition’ are obtained. Various resonance conditions are discussed and examples are included to show practical significance of the method. Results obtained from the quasiperiodic time-dependent normal form theory are compared with the numerical solutions. A close agreement is found.     

Dispersion and localisation in structured Rayleigh beams

This paper brings a comparative analysis between dynamic models of couple-stress elastic materials and structured Rayleigh beams on a Winkler foundation. Although physical phenomena have different physical origins, the underlying equations appear to be similar, and hence mathematical models have a lot in common. In the present work, our main focus is on the analysis of dispersive waves, band-gaps and localised waveforms in structured Rayleigh beams. The Rayleigh beam theory includes the effects of rotational inertia which are neglected in the Euler–Bernoulli beam theory. This makes the approach applicable to higher frequency regimes. Special attention is given to waves in pre-stressed Rayleigh beams on elastic foundations.     

Equivariant Sternberg–Chen Theorem

Smooth (C ) vector fields with linear symmetries and anti-symmetries are considered. We prove that provided symmetry group is compact then a smooth conjugacy in Sternberg–Chen Theorem can be chosen symmetric.     

Normal Forms for High Co-Dimension Bifurcations of Nonlinear Time-Periodic Systems with Nonsemisimple Eigenvalues

The normal forms for time-periodic nonlinear variational equations witharbitrary linear Jordan form undergoing bifurcations of highco-dimension are found. First, the equations are transformed via theLyapunov–Floquet (L–F) transformation into an equivalent form in whichthe linear matrix is constant with degenerate nonsemisimple lineareigenvalues while the nonlinear monomials have periodic coefficients. Byconsidering the resulting coupling of the bases of the near identitytransformation, the solvability condition for an arbitrary Jordan matrixis then derived. It is shown that time-independent and/or time-dependentnonlinear resonance terms remain in the normal form for various Jordanmatrices. Specifically, the normal forms for quadratic and cubicnonlinearities with the following linear Jordan forms are explicitlyderived: double zero eigenvalues (co-dimension two bifurcation), triplezero eigenvalues (co-dimension three bifurcation), and two repeatedpairs of purely imaginary eigenvalues (co-dimension two bifurcation). Acommutative system with cubic nonlinearities and a double inverted pendulum with a periodicfollower force are used as illustrative examples.     

Stack structures on GIT quotients parametrizing hypersurfaces

We suggest to endow Mumford's GIT quotient scheme with a stack structure, by replacing Proj(?) of the invariant ring with its stack theoretic analogue. We analyse the stacks resulting in this way from classically studied invariant rings, and in particular for binary forms of low degree. Our viewpoint is that the stack structure carries interesting geometric information that is intrinsically present in the invariant ring, but lost when passing to its Proj(?). © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

中西方传统经济思想特点的比较

中西方传统经济思想由于面临的地理环境、经济社会形态不同,导致中西方传统经济思想的特点不同。具体而言,中国古代内陆的地理环境和农耕文明的自然经济形态,导致中国传统经济思想是从治国平天下的目标出发来研究经济管理,是中国特有的大一统国家政权和小农经济条件下的产物,特点表现为强调国家本位,抑兼并,国家干预经济占据经济政策的主导地位,经济思想带有浓厚的伦理文化的属性。古希腊海洋文明的地理环境和商业文明的活跃,导致古希腊走上了城邦制分权社会,经济思想的特点是强调庄园本位,维护私有产权制度是关注的重点,从追求财富到研究分工、交换和价值实现为核心,经济思想带有浓厚的宗教文化属性。中西方传统经济思想的特点对之后中西方社会经济的发展产生深远的影响。