A review of dynamic recrystallization phenomena in metallic materials

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

Every organisation from the scale of whole countries down to small companies has a list of system developments which have ended in various forms of disaster. The nature of the failures varies but typical examples are: cost overruns; timescale overruns and sometimes, loss of life. The post-mortems to these systems reveal a wide range of reasons all the way from hardware failures, through software errors right to major system level mistakes. More importantly a large number of these systems share one attribute: complexity. This paper presents a fresh look at the nature of complexity in the building of computer based systems.

http://ieeexplore.ieee.org/iel3/4458/12639/00581896.pdf

Complex systems

What do you do to start reading dimension theory? Searching the book that you love to read first or find an interesting book that will make you want to read? Everybody has difference with their reason of reading a book. Actuary, reading habit must be from earlier. Many people may be love to read, but not a book. It's not fault. Someone will be bored to open the thick book with small words to read. In more, this is the real condition. So do happen probably with this dimension theory.

Aspects of topology: On dimension theory

In this chapter we will embark upon the task of systematically identifying important specific phenomena associated with the asymptotic behavior of smooth dynamical systems We will build upon the results of our survey of specific examples in Chapter 1 as well as on the insights gained from the general structural approach outlined and illustrated in Chapter 2 Most of the properties discussed in the present chapter are in fact topological invariants and can be defined for broad classes of topological dynamical systems, including symbolic ones The predominance of topological invariants fits well with the picture that emerges from the considerations of Sections 21, 23, 24, and 26 The considerations of the previous chapter make it very plausible that smooth dynamical systems are virtually never differentiably stable and can only rarely be classified locally up to smooth conjugacy In contrast, structural and the related topological stability seem to be fairly widespread phenomena We will consider three broad classes of asymptotic invariants: (i) growth of the numbers of orbits of various kinds and of the complexity of orbit families, (ii) types of recurrence, and (iii) asymptotic distribution and statistical behavior of orbits The first two classes are of a purely topological nature; they are discussed in the present chapter The last class is naturally related to ergodic theory and hence we will provide an introduction to key aspects of that subject This will require some space so we put that material into a separate chapter The two chapters are intimately connected

Introduction to the Modern Theory of Dynamical Systems: PRINCIPAL CLASSES OF ASYMPTOTIC TOPOLOGICAL INVARIANTS

We are concerned with families of dynamical systems which are related to generalized radix representations. The properties of these dynamical systems lead to new results on the characterization of bases of Pisot number systems as well as canonical number systems.

/pdf/generalized-radix-representations-and-dynamical-systems-i-2e4s7l4y0k.pdf

Generalized radix representations and dynamical systems I

Le but de ce survol est d'aborder definitions et proprietes concernant la numeration d'un point de vue dynamique: nous nous concentrons sur les systemes de numeration, leur compactification, et. les systemes dynamiques qui peuvent etre definis dessus. La notion de systeme de numeration fibre unifie la presentation. De nombreux exemples sont etudies. Plusieurs numerations sur les entiers naturels, relatifs, les nombres reels ou les nombres complexes sont presentees. Nous portons une attention speciale a la β-numeration ainsi qu'a ses generalisations, aux systemes de numeration abstraits, aux systemes dits "shift radix", de meme qu'aux G-echelles et aux odometres. Un paragraphe d'applications conclut ce survol.

/pdf/dynamical-directions-in-numeration-2m636zpu4b.pdf

Dynamical directions in numeration

Substitutions are combinatorial objects (one replaces a letter by a word) which produce sequences by iteration. They occur in many mathematical fields, roughly as soon as a repetitive process appears. In the present monograph we deal with topological and geometric properties of substitutions, in particular, we study properties of the Rauzy fractals associated to substitutions. To be more precise, let be a substitution over the finite alphabet A. We assume that the incidence matrix of is primitive and that its dominant eigenvalue is a unit Pisot number (i.e., an algebraic integer greater than one whose norm is equal to one and all of whose Galois conjugates are of modulus strictly smaller than one). It is well-known that one can attach to a set which is called central tile or Rauzy fractal of . Such a central tile is a compact set that is the closure of its interior and decomposes in a natural way in n=|A| subtiles (1),,(n). The central tile as well as its subtiles are graph directed self-affine sets that often have fractal boundary. Pisot substitutions and central tiles are of high relevance in several branches of mathematics like tiling theory, spectral theory, Diophantine approximation, the construction of discrete planes and quasicrystals as well as in connection with numeration like generalized continued fractions and radix representations. The questions coming up in all these domains can often be reformulated in terms of questions related to the topology and the geometry of the underlying central tile. After a thorough survey of important properties of unit Pisot substitutions and their associated Rauzy fractals the present monograph is devoted to the investigation of a variety of topological properties of and its subtiles. Our approach is an algorithmic one. In particular, we dwell upon the question whether and its subtiles induce a tiling, calculate the Hausdorff dimension of their boundary, give criteria for their connectivity and homeomorphy to a closed disk and derive properties of their fundamental group. The basic tools for our criteria are several classes of graphs built from the description of the tiles (i) (1in) as the solution of a graph directed iterated function system and from the structure of the tilings induced by these tiles. These graphs are of interest in their own right. For instance, they can be used to construct the boundaries as well as (i) (1in) and all points where two, three or four different tiles of the induced tilings meet. When working with central tiles in one of the above mentioned contexts it is often useful to know such intersection properties of tiles. In this sense the present monograph also aims at providing tools for ``everyday's life'' when dealing with topological and geometric properties of substitutions. Many examples are given throughout the text in order to illustrate our results. Moreover, we give perspectives for further directions of research related to the topics discussed in this monograph.

/pdf/topological-properties-of-rauzy-fractals-4kal7ay29b.pdf

Topological Properties of Rauzy Fractals

In the present paper we give an overview of topological properties of self-affine tiles. After reviewing some basic results on self-affine tiles and their boundary we give criteria for their local connectivity and connectivity. Furthermore, we study the connectivity of the interior of a family of tiles associated to quadratic number systems and give results on their fundamental group. If a self-affine tile tessellates the space the structure of the set of its ‘neighbors’ is discussed.

A Survey on Topological Properties of Tiles Related to Number Systems

/pdf/generalized-radix-representations-and-dynamical-systems-ii-i77jy4rsda.pdf

Jörg M. Thuswaldner

Papers

Generalized radix representations and dynamical systems I

Dynamical directions in numeration

Topological Properties of Rauzy Fractals

A Survey on Topological Properties of Tiles Related to Number Systems

Generalized radix representations and dynamical systems II