Ergodic Matrix Research Articles

An Efficient Adaptive Ergodic Matrix and Chaotic System for Image Encryption

Secrete permutations are widely used in multimedia encryption algorithms, and many permutation-only ciphers have been proposed in recent years for protection of digital images and videos. Recently, ergodic matrices are used to realize the pureposition permutation algorithms. An effective decryption algorithm is developed for all pureposition permutation algorithms. Thereby, it is pointed out that all permutation only image ciphers are insecure against known/chosen-plaintext attacks. This paper presents an image encryption scheme that uses an adaptive ergodic matrix for each block encryption (by adapting the ergodic matrix from one block to another), and combines shuffling the positions, by adaptive ergodic matrix, and changing the grey values of image pixels based on preprocessing the discrete output signal of the Chen’s chaotic system. Thereby, significantly increasing the resistance to known/chosen-plaintext and statistical attacks. Moreover, visually and computationally, experimental results demonstrate the high security and significantly superior encryption quality of the proposed scheme.

A Rapid-Mutation Approximation for Cell Population Dynamics

Carcinogenesis and cancer progression are often modeled using population dynamics equations for a diverse somatic cell population undergoing mutations or other alterations that alter the fitness of a cell and its progeny. Usually it is then assumed, paralleling standard mathematical approaches to evolution, that such alterations are slow compared to selection, i.e., compared to subpopulation frequency changes induced by unequal subpopulation proliferation rates. However, the alterations can be rapid in some cases. For example, results in our lab on in vitro analogues of transformation and progression in carcinogenesis suggest there could be periods where rapid alterations triggered by horizontal intercellular transfer of genetic material occur and quickly result in marked changes of cell population structure.We here initiate a mathematical study of situations where alterations are rapid compared to selection. A classic selection-mutation formalism is generalized to obtain a "proliferation-alteration" system of ordinary differential equations, which we analyze using a rapid-alteration approximation. A system-theoretical estimate of the total-population net growth rate emerges. This rate characterizes the diverse, interacting cell population acting as a single system; it is a weighted average of subpopulation rates, the weights being components of the Perron-Frobenius eigenvector for an ergodic Markov-process matrix that describes alterations by themselves. We give a detailed numerical example to illustrate the rapid-alteration approximation, suggest a possible interpretation of the fact that average aneuploidy during cancer progression often appears to be comparatively stable in time, and briefly discuss possible generalizations as well as weaknesses of our approach.

The Methods of Hilbert Spaces and Structure of the Fixed-Point Set of Lipschitzian Mapping

The purpose of this paper is to prove, by asymptotic center techniques and the methods of Hilbert spaces, the following theorem. Let be a Hilbert space, let be a nonempty bounded closed convex subset of and let be a strongly ergodic matrix. If is a lipschitzian mapping such that , then the set of fixed points is a retract of . This result extends and improves the corresponding results of [7, Corollary 9] and [8, Corollary 1].

The Analysis of Longitudinal Ordinal Response Data in Continuous Time

Abstract A simple Markov model is developed for assessing the predictive effect of time-dependent covariates on an intermittently observed ordinal response in continuous time. This is accomplished by reparameterizing an ergodic intensity matrix in terms of its equilibrium distribution and a parametrically independent component that assesses the rate of movement between ordinal categories. The effect of covariates on the equilibrium distribution can then be modeled using any link appropriate for ordinal data. A robust maximum likelihood estimator based on this model that is consistent and asymptotically normal is constructed. Practical data analysis issues are discussed, and a simple diagnostic tool for assessing model adequacy is developed. The utility of these methods is demonstrated with several analyses of visual acuity data, including a comparison analysis based on generalized estimating equation (GEE) methods.

A Remark on Fixed Point Theorems for Lipschitzian Mappings

The aim of this note is to give the following theorem: Let p > 1 and let E be a p-uniformly convex Banach space, C a nonempty bounded closed convex subset of E, and A = [an,k]n,k ≥ 1 a strongly ergodic matrix. If T: C → C is a mapping such that g = lim infn → ∞infm = 0,1,2… ∑∞k = 1an,k · ||Tk + m||p < 1 + c, where c > 0 is some constant, then T has a fixed point in C.

Sensitivity Analysis of the Fixed Point Vector of Transition Matrices

Abstract The fixed point vector of a regular or ergodic transition matrix is given for a transition matrix, P, when an entire row is changed. The computation of the fixed point vector involves only vector-matrix multiplication and addition, and is based on the initial computation of the pseudo-inverse of (P – I). Sensitivity analysis of the Markov Chain is easily performed utilizing this result.