Geometry of rank 2 distributions with nonzero Wilczynski invariants

In the famous 1910 “cinq variables” paper Cartan showed in particular that for maximally nonholonomic rank 2 distributions in ?⁵ with non-zero covariant binary biquadratic form the dimension of the pseudo-group of local symmetries does not exceed 7 and among such distributions he described the one-parametric family of distributions for which this pseudo-group is exactly 7-dimensional. Using the novel interpretation of the Cartan covariant binary biquadratic form via the classical Wilczynski invariant of curves in projective spaces associated with abnormal extremals of the distributions [4, 27, 28] one can generalize this Cartan result to rank 2 distributions in ?ⁿ satisfying certain genericity assumption, called maximality of class, for arbitrary n ≥ 5.

In the present paper for any rank 2 distribution of maximal class with at least one nonvanishing generalized Wilczynski invariants we construct the canonical frame on a (2n — 3)-dimensional bundle and describe explicitly the moduli spaces of the most symmetric models. The relation of our results to the divergence equivalence of Lagrangians of higher order is given as well.     

Off-shell superpotentials and Ooguri-Vafa invariants of type Ⅱ/F theory compactification

In this paper, we calculate the off-shell superpotential of two Calabi-Yau manifolds with three parameters by integrating the period of the subsystem. We also obtain the Ooguri-Vafa invariants with open mirror symmetry.     

Topological phases of the two-leg Kitaev ladder

We study the phase diagram of the two-leg Kitaev model. Different topological phases can be characterized by either the number of Majorana modes for a deformed chain of the open ladder, or by a winding number related to the ‘h  -loop’ in the momentum space. By adding a three-spin interaction term to break the time-reversal symmetry, two originally different phases are glued together, so that the number of Majorana modes reduce to 0 or 1, namely, the topological invariant collapses to _Z2$Z_2$ from an integer Z. These observations are consistent with a recent general study [S. Tewari, J.D. Sau, arXiv:1111.6592v2].     

Rankin-Cohen Brackets and Invariant Theory

Using maps due to Ozeki and Broué-Enguehard between graded spaces of invariants for certain finite groups and the algebra of modular forms of even weight we equip these invariants spaces with a differential operator which gives them the structure of a Rankin-Cohen algebra. A direct interpretation of the Rankin-Cohen bracket in terms of transvectant for the group SL(2, C) is given.     

Longtime Behaviour of Stochastic Hamiltonian Systems: The Multidimensional Case

Hamiltonian systems perturbed by a white noise force are discussed in several dimensions. By using an appropriate scaling of the stochastic force a convergence theorem for the invariants of the deterministic motion is proved. This corresponds to convergence of the system to a stationary distribution. Especially motion in a central force field is considered; the energy and angular momentum processes are investigated.     

Hamiltonian Oscillators in 1—1—1 Resonance: Normalization and Integrability

Received July 16, 1998; accepted August 7, 1999     

使用主值空间表示的各向同性塑性本构方程

针对各向同性材料,在内变量为标量的假定下,应用张量函数表示定理给出了其塑性应变增量的不变性表示.它的3个不可约基张量取决于应力张量、相互正交且共主轴.建立3个基张量构成的张量子空间与三维主值空间的对应关系,将共主轴的张量采用笛卡尔坐标系中的矢量描述,矢量在不同坐标系下的分量均为张量的一组不可约不变量.定义塑性应变增量对应的矢量为内变量增量,使用张量函数表示理论得到,内变量演化方程除取决于应力对应的矢量和内变量本身外,还取决于应力增量在张量子空间中的投影,该投影就是应力对应矢量的增量,因此,本构方程归结为确定主值空间中矢量之间的关系.最后表明,三维主值空间与张量子空间中的流动法则是等价的.     

Modelling of chaotic systems based on modified weighted recurrent least squares support vector machines

Positive Lyapunov exponents cause the errors in modelling of the chaotic time series to grow exponentially. In this paper, we propose the modified version of the support vector machines (SVM) to deal with this problem. Based on recurrent least squares support vector machines (RLS-SVM), we introduce a weighted term to the cost function tocompensate the prediction errors resulting from the positive global Lyapunov exponents. To demonstrate the effectiveness of our algorithm, we use the power spectrum and dynamic invariants involving the Lyapunov exponents and the correlation dimension as criterions, and then apply our method to the Santa Fe competition time series. The simulation results shows that the proposed method can capture the dynamics of the chaotic time series effectively.     

Generating Replications of Chaotic Time Series

This paper addresses the problem of resampling chaotic time series. We propose a method based on resampling distances between nearest neighbours in phase space, so that the resampled time series present the original points differently positioned along the trajectory. This approach allows one to obtain time series with the same length of the original one, so that confidence intervals for Lyapunov exponents, correlation dimension and other invariants would be determined. For its generality this kind of resampling would be applicable to chaotic time series no matter the observations concern natural or life sciences. The method has been tested with common simulated chaotic systems with both clean and noisy data. Short and noisy time series, as the ones we simulated, typically occur in medicine, biology, and social sciences.     

QUADRATIC INVARIANTS AND SYMPLECTIC STRUCTURE OF GENERAL LINEAR METHODS

1. IntroductionInvestigating whether a numerical method inherits some dynamical properties possessed bythe differential equation systems being integrated is an important field of numerical analysisand has received much attention in recent years See the review articlesof Sanz-Serna[9] and Section 11.16 of Hairer et. al.[2] for more detail concerning the symplectic methods. Most of the work on canonical iotegrators has dealt with one-step formulaesuch as Runge-Kutta methods and Runge'methods …