Determining the input dimension of a neural network for nonlinear time series prediction

Determining the input dimension of a feed-forward neural network for nonlinear time series prediction plays an important role in the modelling.The paper first summarizes the current methods for determining the input dimension of the neural network.Then inspired by the fact that the correlation dimension of a nonlinear dynamic system is the most important feature of it ,the paper pressents a new idea that the input dimension of the neural network for nonlinear time series prediction can be taken as an integer just greater than or equal to the correlation dimension.Fimally,some validation examples and results are given.     

高维非线性动力学系统降维方法的若干进展

综述近年来非线性动力系统降维理论与方法的研究现状.主要介绍非线性动力系统现有降维方法的基本思想、特点与局限性;这些方法包括: 基于中心流形理论的降维方法, Lyapunov-Schmidt (L-S)方法, 非线性Galerkin方法和本征正交分解技术(proper orthogonaldecomposition, POD)方法;并简单介绍了基于规范形理论和快慢流形动力系统的降维方法.最后提出关于高维非线性动力系统降维的一些新设想,并讨论了今后研究工作的方向.      

Fractal characteristics of defects evolution in parallel fibre reinforced composite in quasi-static process of fracture

With appropriate numerical methods of composites modelling we can observe defects growth without using invasive experimental methods. In this paper we present results of simulation of quasi-static process of fracture in composite using cellular automata and fractal characteristics of defects growth based on box-counting method. Defects evolution in composite are treated as a dynamical system depending on external and internal conditions and properties of fibres.     

n阶矩阵的可交换空间的维数

本文通过研究矩阵的若当相似形矩阵的可交换空间 ,从而得到了矩阵的可交换空间的维数公式     

湍流的分形特征

综述了湍流分维测量的最新成果,涉及湍流分维测量的传统证据、分维测量方法以及湍流分维测量成果,重点详细介绍了湍流分维测量的各种方法及其优缺点,讨论了各家的测量成果,指出了具有广泛影响的 Ｓｒｅｅｎｉｖａｓａｎ等人使用的分维测量方法存在的问题,介绍了作者最新的成果．最后,综述了Ｎ－Ｓ方程与分形、标度律、湍流的理论分形模型等与湍流分维应用相关的领域的最新进展．对湍流分形研究进行了讨论和展望      

Supervised Sparse and Functional Principal Component Analysis

Principal component analysis (PCA) is an important tool for dimension reduction in multivariate analysis. Regularized PCA methods, such as sparse PCA and functional PCA, have been developed to incorporate special features in many real applications. Sometimes additional variables (referred to as supervision) are measured on the same set of samples, which can potentially drive low-rank structures of the primary data of interest. Classical PCA methods cannot make use of such supervision data. In this article, we propose a supervised sparse and functional principal component (SupSFPC) framework that can incorporate supervision information to recover underlying structures that are more interpretable. The framework unifies and generalizes several existing methods and flexibly adapts to the practical scenarios at hand. The SupSFPC model is formulated in a hierarchical fashion using latent variables. We develop an efficient modified expectation-maximization (EM) algorithm for parameter estimation. We also implement fast data-driven procedures for tuning parameter selection. Our comprehensive simulation and real data examples demonstrate the advantages of SupSFPC. Supplementary materials for this article are available online.     

Poincaré series and an application to Weyl algebras

Let A n be the n-th Weyl algebra over a field of characteristic 0 and M a finitely generated module over A n . By further exploring the relationship between the Poincar′e series and the dimension and the multiplicity of M , we are able to prove that the tensor product of two finitely generated modules over A n has the multiplicity equal to the product of the multiplicities of both modules. It turns out that we can compute the dimensions and the multiplicities of some homogeneous subquotient modules of A n .     

Generalized framework for auxiliary extra dimensions

The theory of gravity with an auxiliary extra dimension is known to give the ghost-free cubic completion of the Fierz-Pauli mass term in the decoupling limit. Our work generalizes the boundary condition in the auxiliary dimension that avoids ghosts order-by-order, and to all orders, in the decoupling limit. Furthermore, we extend the formalism to the case of many auxiliary dimensions, and we show that the multi-dimensional extension with the rotationally invariant boundaries of the bulk, is equivalent to the model with a single auxiliary dimension. The above constructions require the appropriate adjustment of the boundary condition, which we discuss in detail. The other possible extension of the original model by the Gauss-Bonnet term is studied as well.     

On class dimension of flat association schemes in affine and affine-symplectic spaces

In this paper, we obtain upper bounds of the class dimension of flat association schemes in affine and affine-symplectic spaces and construct resolving sets for these schemes.     

Some properties of the n-dimensional plane symmetric solution

An n-dimensional static plane symmetric solution of Einstein field equation, which is judged as the source of n-dimensional Taub solution, is presented in our previous work. The properties of geodesics of this solution are studied in this Letter. The essence of the source is also investigated. A phantom with dust and photon is suggested as the substance of the source matter.