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Multidimensional Continued Fractions
Mathematician Fritz Schweiger, whose academic affiliation is not provided, provides an introduction to a field of research that has seen remarkable progress in recent decades, concentrating on multidimensional continued fractions which can be described by fractional linear maps or equivalently by a set of (n + 1) x (n + 1) matrices. Addressing the question of periodicity, he refines the problem of convergence to the question of whether these algorithms give "good" simultaneous Diophantine approximations. He notes that these algorithms are not likely to provide such "good" approximations which satisfy the n-dimensional Dirichlet property. Also studied are the ergodic properties of these maps. Annotation copyrighted by Book News Inc., Portland, OR
Reciprocals
The theoretical issues addressed in the present volume are semantic and cognitive properties of reciprocal events, syntactic properties of reciprocals, and the relationship of reciprocals to other grammatical categories. Several papers discuss the history of reciprocal constructions, offering alternative hypotheses regarding the grammaticalization of reciprocals. The formal, functional, typological and historical approaches in the present volume complement each other, contributing together to the understanding of forms, and syntactic and semantic properties of reciprocal markers. Several papers in the present volume make a double contribution to the problems of reciprocal constructions: they provide new descriptive data and they address theoretical issues at the same time. The languages discussed include: English, Dutch, German, Greek, Polish, Nyulnyulan (Australia), Amharic (Ethio-Semitic), Bilin (Cushitic), Chadic languages, Bantu, Halkomelem (Salishan), Mandarin, Yukaghir and a number of Oceanic languages. The volume also includes a study of grammaticalization of reciprocals and reflexives in African languages.
New Mathematics Education Research and Practice
Mathematics education research has blossomed into many different areas which we can see in the programmes of the ICME conferences as well as in the various survey articles in the Handbooks. However, all of these lines of research are trying to grapple with a common problem, the complexity of the process of learning mathematics.
Typical Dynamics of Volume Preserving Homeomorphisms
This 2000 book provides a self-contained introduction to typical properties of homeomorphisms. Examples of properties of homeomorphisms considered include transitivity, chaos and ergodicity. A key idea here is the interrelation between typical properties of volume preserving homeomorphisms and typical properties of volume preserving bijections of the underlying measure space. The authors make the first part of this book very concrete by considering volume preserving homeomorphisms of the unit n-dimensional cube, and they go on to prove fixed point theorems (Conley–Zehnder– Franks). This is done in a number of short self-contained chapters which would be suitable for an undergraduate analysis seminar or a graduate lecture course. Much of this work describes the work of the two authors, over the last twenty years, in extending to different settings and properties, the celebrated result of Oxtoby and Ulam that for volume homeomorphisms of the unit cube, ergodicity is a typical property.
Noun Phrase Structure in the Languages of Europe
The result of over five years of close collaboration among an international group of leading typologists within the EUROTYP program, this volume is about the morphology and syntax of the noun phrase. Particular attention is being paid to nominal inflectional categories and inflectional systems and to the syntax of determination, modification, and conjunction. Its areal focus, like that of other EUROTYP volumes, is on the languages of Europe; but in order to appreciate what is peculiarly European about their noun phrases, a more comprehensive and genuinely typological view is being taken at the full range of cross-linguistic variation within this structural domain. There has been no shortage lately of contributions to the theory of noun phrase structure; the present volume is, however, unique in the extent to which its theorizing is empirically grounded.
Diophantine Approximation
This volume contains 21 research and survey papers on recent developments in the field of diophantine approximation, which are based on lectures given at a conference at the Erwin Schrödinger-Institute (Vienna, 2003). The articles are either in the spirit of more classical diophantine analysis or of a geometric or combinatorial flavor. Several articles deal with estimates for the number of solutions of diophantine equations as well as with congruences and polynomials.
Math Worlds
An international group of distinguished scholars brings a variety of resources to bear on the major issues in the study and teaching of mathematics, and on the problem of understanding mathematics as a cultural and social phenomenon. All are guided by the notion that our understanding of mathematical knowledge must be grounded in and reflect the realities of mathematical practice. Chapters on the philosophy of mathematics illustrate the growing influence of a pragmatic view in a field traditionally dominated by platonic perspectives. In a section on mathematics, politics, and pedagogy, the emphasis is on politics and values in mathematics education. Issues addressed include gender and mathematics, applied mathematics and social concerns, and the reflective and dialogical nature of mathematical knowledge. The concluding section deals with the history and sociology of mathematics, and with mathematics and social change. Contributors include Philip J. Davis, Helga Jungwirth, Nel Noddings, Yehuda Rav, Michael D. Resnik, Ole Skovsmose, and Thomas Tymoczko.
