The theory of scale relativity [14] is an attempt to study the consequences of giving up the hypothesis of space-time differentiability. One can show [14] [15] that a continuous but nondifferentiable space-time is necessarily fractal. Here the word fractal [12] is taken in a general meaning, as defining a set, object or space that shows structures at all scales, or on a wide range of scales. Mo...

We consider the periodic weighted operator T y = ? ?2 (2 y 0) 0 in L 2 (R; (x) 2 dx), where is a 1-periodic real positive function with (0) = 1. We assume q = 0 == 2 L 2 (0; 1). The spectrum of T consists of intervals n = + n?1 ; ? n ] separated by gaps (? n ; + n); n > 1. Using essentially the square of the gap lengths, the centre of the gap and the Dirichlet eigenvalues on the unit interval w...

Let S = {S 1, S 2} ⊂ Rd×d have a common, but not necessarily strict, Lyapunov matrix (i.e. there exists a symmetric positive-definite matrix P such that P − S T k PS k ≥ 0 for k = 1, 2). Based on a splitting theorem of the state space Rd (Dai, Huang and Xiao, arXiv:1107.0132v1[math.PR]), we establish several stability criteria for the discrete-time linear switched dynamics xn = S σn · · · S σ1 ...

For Hölder continuous cocycles over an invertible, Lipschitz base, we establish the Hölder continuity of Oseledets subspaces on compact sets of arbitrarily large measure. This extends a result of Araújo et al [On Hölder-continuity of Oseledets subspaces J. Lond. Math. Soc. 93 (2016) 194–218] by considering possibly non-invertible cocycles, which, in addition, may take values in the space of com...

The concepts of fractal tiling and hyperbolic tiling are combined to create novel fractal surfaces in Euclidean threespace. Paper folding and computer modeling are used to create these constructs. We show examples using triangular, trapezoidal, and dart-shaped prototiles. Smaller tiles deflect out of the plane of adjacent larger tiles, resulting in nonplanar surfaces, the shapes of which are de...

The 2D Wavelet-Transform Modulus Maxima (WTMM) method was used to detect microcalcifications (MC) in human breast tissue seen in mammograms and to characterize the fractal geometry of benign and malignant MC clusters. This was done in the context of a preliminary analysis of a small dataset, via a novel way to partition the wavelet-transform space-scale skeleton. For the first time, the estimat...

In our recent papers we applied fractal methods to extract the earthquake precursory signatures from scaling characteristics of the ULF geomagnetic data, obtained in a seismic active region of Guam Island during the large earthquake of 8 August 1993. We found specific dynamics of their fractal characteristics (spectral exponents and fractal dimensions) before the earthquake: appearance of the f...

State-space stability of linear shift-invariant discrete two-dimensional (2-D) dynamics is considered. An approach to the making of Lyapunov functions for 2-D dynamics is presented. It produces quadratic forms involving finite cross-terms among local states. The scope of the stability analysis based upon the notion of parallel stability expands, and the accuracy can be improved by this approach.

The robust stability problem for uncertain, linear, state-space models is considered. When a fixed Lyapunov function is used to provide an admissible perturbation set, the obtained variation bounds can be too conservative. The main purpose of this investigation is to define the conditions, under which it is always possible to construct a parameter-dependent Lyapunov function for a class of unce...