V. I. Arnold reveals some unexpected connections between such apparently unrelated theories as Galois fields, dynamical systems, ergodic theory, statistics, chaos and the geometry of projective structures on finite sets. The author blends experimental results with examples and geometrical explorations to make these findings accessible to a broad range of mathematicians, from undergraduate students to experienced researchers.
One of the traditional ways mathematical ideas and even new areas of mathematics are created is from experiments. One of the best-known examples is that of the Fermat hypothesis, which was conjectured by Fermat in his attempts to find integer solutions for the famous Fermat equation. This hypothesis led to the creation of a whole field of knowledge, but it was proved only after several hundred years. This book, based on the author's lectures, presents several new directions of mathematical research. All of these directions are based on numerical experiments conducted by the author, which led to new hypotheses that currently remain open, i.e., are neither proved nor disproved. The hypotheses range from geometry and topology (statistics of plane curves and smooth functions) to combinatorics (combinatorial complexity and random permutations) to algebra and number theory (continuous fractions and Galois groups). For each subject, the author describes the problem and presents numerical results that led him to a particular conjecture. In the majority of cases there is an indication of how the readers can approach the formulated conjectures (at least by conducting more numerical experiments). Written in Arnold's unique style, the book is intended for a wide range of mathematicians, from high school students interested in exploring unusual areas of mathematics on their own, to college and graduate students, to researchers interested in gaining a new, somewhat nontraditional perspective on doing mathematics. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession. Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).
This book exposes and comments on the consequences of Reg NMS and MiFID on market microstructure. It covers changes in market design, electronic trading, and investor and trader behaviors. The emergence of high frequency trading and critical events like the'Flash Crash' of 2010 are also analyzed in depth.Using a quantitative viewpoint, this book explains how an attrition of liquidity and regulatory changes can impact the whole microstructure of financial markets. A mathematical Appendix details the quantitative tools and indicators used through the book, allowing the reader to go further independently.This book is written by practitioners and theoretical experts and covers practical aspects (like the optimal infrastructure needed to trade electronically in modern markets) and abstract analyses (like the use on entropy measurements to understand the progress of market fragmentation).As market microstructure is a recent academic field, students will benefit from the book's overview of the current state of microstructure and will use the Appendix to understand important methodologies. Policy makers and regulators will use this book to access theoretical analyses on real cases. For readers who are practitioners, this book delivers data analysis and basic processes like the designs of Smart Order Routing and trade scheduling algorithms.In this second edition, the authors have added a large section on orderbook dynamics, showing how liquidity can predict future price moves, and how High Frequency Traders can profit from it. The section on market impact has also been updated to show how buying or selling pressure moves prices not only for a few hours, but even for days, and how prices relax (or not) after a period of intense pressure.Further, this edition includes pages on Dark Pools, Circuit Breakers and added information outside of Equity Trading, because MiFID 2 is likely to push fixed income markets towards more electronification. The authors explore what is to be expected from this change in microstructure. The appendix has also been augmented to include the propagator models (for intraday price impact), a simple version of Kyle's model (1985) for daily market impact, and a more sophisticated optimal trading framework, to support the design of trading algorithms.
This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fine subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.
