BASIC IDEAS OF AMMANN BAR GRIDS

1993 ◽  
Vol 07 (06n07) ◽  
pp. 1437-1453 ◽  
Author(s):  
REINHARD LÜCK

The formalism of the Ammann bar grid with two spacings is systematically analyzed. Simple two-by-two matrices describe the self-similar deflation. The deflation and the inflation factors are the two eigenvalues of the matrix. The ratio of the two spacings is derived as another characteristic quantity of the matrices. The maximum number of spacing ratios and the ratios themselves have been derived for a given inflation factor by the matrix formalism. Applications to non-periodic patterns with eightfold, tenfold and twelvefold symmetry are demonstrated. The strong relationship to Fibonacci and Fibonacci related series is explained. The significance of the Ammann bars for matching rules, inflation/deflation procedures and phason movement is outlined.

1998 ◽  
Vol 13 (26) ◽  
pp. 2085-2094 ◽  
Author(s):  
B. SATHIAPALAN

We use the matrix formalism to investigate what happens to strings above the Hagedorn temperature. We show that is not a limiting temperature but a temperature at which the continuum string picture breaks down. We study a collection of N D-0-branes arranged to form a string having N units of light-cone momentum. We find that at high temperatures the favored phase is one where the string worldsheet has disappeared and the low-energy degrees of freedom consists of N2 massless particles ("gluons"). The nature of the transition is very similar to the deconfinement transition in large-N Yang–Mills theories.


Minerals ◽  
2021 ◽  
Vol 11 (10) ◽  
pp. 1060
Author(s):  
Dominic Papineau ◽  
Jiayu Yin ◽  
Kevin Devine ◽  
Deng Liu ◽  
Zhenbing She

Chemically oscillating reactions are abiotic reactions that produce characteristic, periodic patterns during the oxidation of carboxylic acids. They have been proposed to occur during the early diagenesis of sediments that contain organic matter and to partly explain the patterns of some enigmatic spheroids in malachite, phosphorite, jasper chert, and stromatolitic chert from the rock record. In this work, circularly concentric self-similar patterns are shown to form in new chemically oscillating reaction experiments with variable mixtures of carboxylic acids and colloidal silica. This is carried out to best simulate in vitro the diagenetic formation of botryoidal quartz and carbonate in two Ediacaran-age geological formations deposited after the Marinoan–Nantuo snowball Earth event in South China. Experiments performed with alkaline colloidal silica (pH of 12) show that this compound directly participates in pattern formation, whereas those with humic acid particles did not. These experiments are particularly noteworthy since they show that pattern formation is not inhibited by strong pH gradients, since the classical Belousov–Zhabotinsky reaction occurs in solution with a pH around 2. Our documentation of hundreds of classical Belousov–Zhabotinsky experiments yields a number of self-similar patterns akin to those in concretionary structures after the Marinoan–Nantuo snowball Earth event. Morphological, compositional, and size dimensional comparisons are thus established between patterns from these experiments and in botryoidal quartz and carbonate from the Doushantuo and Denying formations. Selected specimens exhibit circularly concentric layers and disseminations of organic matter in quartz and carbonate, which also occurs in association with sub-micron-size pyrite and sub-millimetre iron oxides within these patterns. X-ray absorption near edge structure (XANES) analyses of organic matter extracted from dolomite concretions in slightly younger, early Cambrian Niutitang Formation reveal the presence of carboxylic and N-bearing molecular functional groups. Such mineral assemblages, patterns, and compositions collectively suggest that diagenetic redox reactions take place during the abiotic decay of biomass, and that they involve Fe, sulphate, and organic matter, similarly to the pattern-forming experiments. It is concluded that chemically oscillating reactions are at least partly responsible for the formation of diagenetic siliceous spheroids and concretionary carbonate, which can relate to various other persistent problems in Earth and planetary sciences.


Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2948
Author(s):  
Sergey Korchagin ◽  
Ekaterina Pleshakova ◽  
Irina Alexandrova ◽  
Vitaliy Dolgov ◽  
Elena Dogadina ◽  
...  

Composite materials consisting of a dielectric matrix with conductive inclusions are promising in the field of micro- and optoelectronics. The properties of a nanocomposite material are strongly influenced by the characteristics of the substances included in its composition, as well as the shape and size of inclusions and the orientation of particles in the matrix. The use of nanocomposite material has significantly expanded and covers various systems. The anisotropic form of inclusions is the main reason for the appearance of optical anisotropy. In this article, models and methods describing the electrical conductivity of a layered nanocomposite of a self-similar structure are proposed. The method of modeling the electrical conductivity of individual blocks, layers, and composite as a whole is carried out similarly to the method of determining the dielectric constant. The advantage of the method proposed in this paper is the removal of restrictions imposed on the theory of generalized conductivity associated with the need to set the dielectric constant.


2005 ◽  
Vol 94 (2) ◽  
pp. 163-179 ◽  
Author(s):  
Rosalba Saija ◽  
Maria Antonia Iatı` ◽  
Arianna Giusto ◽  
Paolo Denti ◽  
Ferdinando Borghese

2011 ◽  
Vol 130-134 ◽  
pp. 3023-3026
Author(s):  
Yi Min Tian ◽  
Ao Zhang

Matrix exponential computstion is a difficulty thing when the order of the matrix get big and big after discretion. When we use Lie group method to get numeric solution of a differential equation, we often face this problem.Li group method is a kind of prosperous method, its basic ideas is to keep the numeric solution in a manifold which is less than the Euclid space while bigger than the analytic solution manifold, so we can get more exact numeric solution than other method. So we discussed the generalized polar decompositions method for matrix exponential.


2009 ◽  
Vol 24 (05) ◽  
pp. 799-815
Author(s):  
SERGE N. ANDRIANOV ◽  
NIKOLAY S. EDAMENKO ◽  
YURY V. TERESHONKOV

We treat here the process of simulation of ion micro- and nanoprobes in detail using the matrix formalism for Lie algebraic tools. Similar approach allows realizing necessary analytical and numerical modeling procedures. Nowadays ion micro- and nanoprobes are extensively applied in different branches of science and industry. It is known that similar facilities are very sensitive to certain of steering parameters of the systems. In other words, similar beam lines are high precision systems, requiring preliminary modeling for thorough analysis of possible optimal working regimes. In this paper we consider analytical and numerical models, which allow one to study effect of various aberrations on basic beam characteristics. Research process performs from linear to nonlinear model with step by step including nonlinear effects of different nature. Previous papers of the authors consider some aspects of nonlinear models. The present paper deals with full conception of modeling process, generalizing most essential aberrations and providing adequate solution methods.


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