Fractal Activity Time Models for Risky Asset with Dependence and Generalized Hyperbolic Distributions

Risky asset models with the dependence through fractal activity time are described. The construction of the fractal activity time is implemented via superpositions of Ornstein-Uhlenbeck type processes driven by Lévy noise. The model features both tractable dependence structure and desired marginal distributions of the returns from the generalized hyperbolic class: the Variance Gamma and normal inverse Gaussian. These distributions provide good fit to real financial data. Pricing formulae for the proposed models are derived.     

On a numerical technique for finding multiple zeros and its dynamic

An optimal method is developed for approximating the multiple zeros of a nonlinear function, when the multiplicity is known. Analysis of convergence for the proposed technique is studied to reveal the fourth-order convergence. We further investigate the dynamics of such multiple zero finders by using basins of attraction and their corresponding fractals in the complex plane. A fourth-order method will also be presented, when the multiplicity m is not known. Numerical comparisons will be made to support the underlying theory of this paper.     

Spatial and Spatiotemporal Karhunen-Loève-Type Representations on Fractal Domains

Abstract

We study the spectral properties of spatial and spatiotemporal Gaussian random fields defined as the solutions to stochastic elliptic, parabolic, and hyperbolic fractional pseudodifferential equations on compact fractal domains. The fractal dimension of the domain modifies the asymptotic properties of the eigenvalues that define the pure point spectra of the covariance functions of the solutions and their Karhunen-Loève-type expansions. The eigenfunction systems involved constitute orthogonal bases of the corresponding trace spaces on fractal sets. The Hölder exponent of the sample paths of the random fields is computed in terms of the fractional order of mean-quadratic variation on their increments. Such an exponent also depends on the Hausdorff dimension of the domain.     

Measurements of flame-front curvature based on Fourier transformation

An optimized image processing method for calculating the local curvature of two-dimensional digitized flame-front contours is proposed. This contour-smoothing method is based on Fourier transformation of the entire contour and is different from conventional methods in which a chosen number of neighbouring pixel points is locally fitted to an ad hoc smooth function along the contour on a point-by-point basis. The maximum curvature that can be measured with the proposed method is determined by the inner cut-off scale from a fractal length measure of the flame-front contour and is limited by image spatial resolution only. Curvature measurements are also made in highly stretched turbulent premixed methane–air jet flames. The measured probability density functions of local flame curvature are consistent with the generalized form of a symmetric Gaussian distribution suggested by Bradley et al. (Combustion and Flame, 135, 503–523 (2003)). Advantages and limitations of the proposed method are also discussed.     

Precise (m,k)-Zipf diagram analysis of mathematical and financial time series when m=6, k=2

For studying short-range time correlations in financial signals, we have envisaged to combine the Zipf method and the i-variability diagrams (VD) as useful tools. The 2-VD describes the local curvature short-range correlations. We have resulted into ranking the 2-VD data according to their frequency of occurrence. After having tested the ideas and estimated the error bars on a Brownian motion signal, we have examined two stocks, i.e. SGP and OXHP closing price and volume of transaction long series. A precise (m,k)-Zipf diagram analysis when m=6, k=2 has been shown to lead to a non-immediate information on the signal behavior, even taking into account error bars. The set of curvatures (translated into “words”) indicates a Brownian motion-like set for the closing price local curvature of such signals over a 6 day span. Moreover, it has been shown that the conjecture about a simple relationship between the Hurst exponent H and the ζ exponent of Zipf plots does not seem to be substantiated here.     

THE NAGUMO EQUATION ON SELF－SIMILAR FRACTAL SETS

The Nagumo equationut ut=△u+bu(u-a)(1-u),t＞0 is investigated with initial data and zero Neumann boundary conditions on post-critically finite (p.c.f.) self-similar fractals that have regular harmonic structures and satisfy the separation condition. Such a nonlinear diffusion equation has no travelling wave solutions because of the“pathological” property of the fractal. However, it is shown that a global Hoelder continuous solution in spatial variables exists on the fractal considered. The Sobolev-type inequality plays a crucial role, which holds on such a class of p.c.f self-similar fractals. The heat kernel has an eigenfunction expansion and is well-defined due to a Weyl‘s formula. The large time asymptotic behavior of the solution is discussed, and the solution tends exponentially to the equilibrium state of the Nagumo equation as time tends to infinity if b is small.     

表面分形球、柱的光散射

分形和分维是近些年才兴起的一门处理复杂事物的新方法￣［１］。本文主要研究具有表面分形的球、柱的光散射特征。我们首先利用几何光学近似研究了一种表面满足自仿射原理的分形均匀球的散射，其散射特性与分维存在一定的关系。我们还利用微扰法讨论了表面具有沟槽的分形柱的光散射的分形特征。     

小波级数的Bouligand维数(英)

本文讨论小波级数的上、下Buoligangd维数，在小波级数满足一定的条件下，给出了与Weierstrass函数相应的结果．     

黎曼流形上不变集的分形维数估计

本文利用整体Lyapunov指数给出了光滑黎曼流形上C1-映射不变集的分形维数的上界.