New Elements of Analysis of a Degenerate Chenciner Bifurcation

A new transformation of parameters for generic discrete-time dynamical systems with two independent parameters is defined, for when the degeneracy occurs. Here the classical transformation of parameters (α1,α2)→(β1,β2) is not longer regular at (0,0); therefore, implicit function theorem (IFT) cannot be applied around the origin, and a new transformation is necessary. The approach in this article to a case of Chenciner bifurcation is theoretical, but it can provide an answer for a number of applications of dynamical systems. We studied the bifurcation scenario and found out that, by this transformation, four different bifurcation diagrams are obtained, and the non-degenerate Chenciner bifurcation can be described by two bifurcation diagrams.

Microbial mediation of textures and minerals – terrestrial or parent body processes?

Evolution of chondritic parent body is influenced by thermal, impact metamorphism and aqueous alteration, studied in Mező-Madaras, Knyahinya, Mócs and Nyírábrány in aspect of high resolution in situ textural, mineralogical and organic geochemical characteristics, using optical microscopy, FTIR-ATR and Raman spectroscopy. Our observations focused on Fe-containing opaque grains, glass, olivines and pyroxenes, which were well populated by micrometer-sized microbial filamentous elements in their boundary region within matrix and inside the minerals resembling mineralized microbially produced textures (MMPT), affecting 70-80 vol% of samples. In MMPT iron oxides (ferrihydrite, goethite), olivine, montmorillonite, kandite minerals and various hydrocarbon compounds were identified. (1) Data confirmed dense and invasive terrestrial microbially mediated contamination in the chondrites, supported by microtexture, micromineralogy and embedded organic compounds. As the classical transformation processes are supposed nowadays to have been happened on the parent bodies, a contradiction arose: how could it be that these classical products are manifested in microbially mediated texture? (2) Based on terrestrial analogies, microbial mediation is a sudden process comparing to geological times, very ancient, widespread and occur in various environments under determined conditions. It can consume previous and also produce new minerals. After formation, MMPT can survive billions of years proposing occurrence on parent bodies.

The connection of an old geodetic datum with a new one using Least Squares Collocation: The Greek case

In the present paper we are dealing with the problem of the connection of two local geodetic datums in which the classical transformation procedures fail to give sufficient solution. Thereafter, we additionally implemented Least Squares Collocation in order to model the remaining residuals of the initial transformation. We tested this methodology in an area located in Northern Greece and we found significant improvement of the transformation results. The implementation of Least Squares Collocation refines the statistical behaviour of the inconsistencies regarding their extreme values (minimum and maximum) from almost 70 to 23 cm, while the standard deviation is reduced from the level of 27 to 10 cm.

Image Encryption Algorithm Based on 3D Arnold Cat and Logistic Map

Arnold cat map is a classical transformation of image encryption, but it has some shortcomings such as short key quantities, small period and so on. On the basis of Arnold cat map, it presented an algorithm of image encryption based on 3D Arnold cat and chaotic map. Simulation experiments show that the encryption algorithm has characters of strong keys, better effect and fast.

Non-Terminating Basic Hypergeometric Series and the q-Zeilberger Algorithm

We present a systematic method for proving non-terminating basic hypergeometric identities. Assume that k is the summation index. By setting a parameter x to xqn, we may find a recurrence relation of the summation by using the q-Zeilberger algorithm. This method applies to almost all non-terminating basic hypergeometric summation formulae in the work of Gasper and Rahman. Furthermore, by comparing the recursions and the limit values, we may verify many classical transformation formulae, including the Sears–Carlitz transformation, transformations of the very well-poised 8φ7 series, the Rogers–Fine identity and the limiting case of Watson's formula that implies the Rogers–Ramanujan identities.