Strengthening Quality of Chaotic Bit Sequences

We discuss chaos and its quality as measured through the 0-1 test for chaos. When the 0-1 test indicates deteriorating quality of chaos, because of the finite precision representations of real numbers in digital implementations, then the process may eventually lead to a periodic sequence. A simple method for improving the quality of a chaotic signal is to mix the signal with another signal by using the XOR operation. In this paper, such mixing of weak chaotic signals is considered, yielding new signals with improved quality (with K values from the 0-1 test close to 1). In some sense, such a mixing of signals could be considered as a two-layer prevention strategy to maintain chaos. That fact may be important in those applications when the hardware resources are limited. The 0-1 test is used to show the improved chaotic behavior in the case when a continuous signal (for example, from the Chua, Rössler or Lorenz system) intermingles with a discrete signal (for example, from the logistic, Tinkerbell or Henon map). The analysis is presented for chaotic bit sequences. Our approach can further lead to hardware applications, and possibly, to improvements in the design of chaotic bit generators. Several illustrative examples are included.

A study of defect-based error estimates for the Krylov approximation of φ-functions

Prior recent work, devoted to the study of polynomial Krylov techniques for the approximation of the action of the matrix exponential etAv, is extended to the case of associated φ-functions (which occur within the class of exponential integrators). In particular, a posteriori error bounds and estimates, based on the notion of the defect (residual) of the Krylov approximation are considered. Computable error bounds and estimates are discussed and analyzed. This includes a new error bound which favorably compares to existing error bounds in specific cases. The accuracy of various error bounds is characterized in relation to corresponding Ritz values of A. Ritz values yield properties of the spectrum of A (specific properties are known a priori, e.g., for Hermitian or skew-Hermitian matrices) in relation to the actual starting vector v and can be computed. This gives theoretical results together with criteria to quantify the achieved accuracy on the fly. For other existing error estimates, the reliability and performance are studied by similar techniques. Effects of finite precision (floating point arithmetic) are also taken into account.

Exact Analysis of the Finite Precision Error Generated in Important Chaotic Maps and Complete Numerical Remedy of These Schemes

A first aim of the present work is the determination of the actual sources of the “finite precision error” generation and accumulation in two important algorithms: Bernoulli’s map and the folded Baker’s map. These two computational schemes attract the attention of a growing number of researchers, in connection with a wide range of applications. However, both Bernoulli’s and Baker’s maps, when implemented in a contemporary computing machine, suffer from a very serious numerical error due to the finite word length. This error, causally, causes a failure of these two algorithms after a relatively very small number of iterations. In the present manuscript, novel methods for eliminating this numerical error are presented. In fact, the introduced approach succeeds in executing the Bernoulli’s map and the folded Baker’s map in a computing machine for many hundreds of thousands of iterations, offering results practically free of finite precision error. These successful techniques are based on the determination and understanding of the substantial sources of finite precision (round-off) error, which is generated and accumulated in these two important chaotic maps.

Effective moduli of rocks predicted by the Kuster–Toksöz and Mori–Tanaka models

Natural rocks are polymineral composites with complex microstructures. Such strong heterogeneities significantly affect the estimation of effective moduli by some theoretical models. First, we have compared the effective moduli of isotropic rocks predicted by the Kuster–Toksöz (KT) model and the Mori–Tanaka (MT) model. The widely used KT model only has finite precision in many cases because of its assumption that is restricted to the first-order scattering approximation. However, the MT model based on the Eshelby tensor in mesomechanics has the advantage of predicting effective moduli of rocks, especially when the volume fraction of embedded inclusions is sufficiently large. In addition, the MT model can be used to predict the effective modulus of anisotropic rocks, but the KT model cannot. For a certain kind of shale or tight sandstones, which are viewed as isotropic composites, both the models work well. For the medium containing spherical pores, both the models produce the same results, whereas for ellipsoidal pores the MT model is more accurate than the KT model, validated by the finite element simulations. In what follows, the applicable ranges of simplified formulas for pores with needle, coin and disk shapes, widely used in engineering, are quantitatively given based on the comparison with the results according to the reduced ellipsoidal formulas of the MT and KT models. These findings provide a comprehensive understanding of the two models in calculating the effective modulus of rocks, which are beneficial to such areas as petroleum exploration and exploitation, civil engineering, and geophysics.

Analysis, Evaluation and Exact Tracking of the Finite Precision Error Generated in Arbitrary Number of Multiplications

In the present paper, a novel approach is introduced for the study, estimation and exact tracking of the finite precision error generated and accumulated during any number of multiplications. It is shown that, as a rule, this operation is very “toxic”, in the sense that it may force the finite precision error accumulation to grow arbitrarily large, under specific conditions that are fully described here. First, an ensemble of definitions of general applicability is given for the rigorous determination of the number of erroneous digits accumulated in any quantity of an arbitrary algorithm. Next, the exact number of erroneous digits produced in a single multiplication is given as a function of the involved operands, together with formulae offering the corresponding probabilities. In case the statistical properties of these operands are known, exact evaluation of the aforementioned probabilities takes place. Subsequently, the statistical properties of the accumulated finite precision error during any number of successive multiplications are explicitly analyzed. A method for exact tracking of this accumulated error is presented, together with associated theorems. Moreover, numerous dedicated experiments are developed and the corresponding results that fully support the theoretical analysis are given. Eventually, a number of important, probable and possible applications is proposed, where all of them are based on the methodology and the results introduced in the present work. The proposed methodology is expandable, so as to tackle the round-off error analysis in all arithmetic operations.

Exact and approximate similarities of non-necessarily rational planar, parametrized curves, using centers of gravity and inertia tensors

We provide an algorithm to detect whether two bounded, planar parametrized curves are similar, i.e. whether there exists a similarity transforming one of the curves onto the other. The algorithm is valid for completely general parametrizations, and can be adapted to the case when the input is given with finite precision, using the notion of approximate [Formula: see text]. The algorithm is based on the computation of centers of gravity and inertia tensors of the considered curves or of the planar regions enclosed by the curves, which have nice properties when a similarity transformation is applied. In more detail, the centers of gravity are mapped onto each other, and the matrices representing the inertia tensors satisfy a simple relationship: when the similarity is a congruence (i.e. distances are preserved) the matrices are congruent, and in the more general case the relationship is analogous, but involves the square of the scaling constant. Using both properties, and except for certain pathological cases, the similarities can be found. Additional ideas are presented for the case of closed, i.e. compact, curves.