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Arakelov Geometry and Diophantine Applications
Bridging the gap between novice and expert, the aim of this book is to present in a self-contained way a number of striking examples of current diophantine problems to which Arakelov geometry has been or may be applied. Arakelov geometry can be seen as a link between algebraic geometry and diophantine geometry.The first chapters provide some background and introduction to the subject. These are followed by a presentation of different applications to arithmetic geometry. The final part describes the recent application of Arakelov geometry to Shimura varieties and the proof of an averaged version of Colmez's conjecture. This book thus blends initiation to fundamental tools of Arakelov geometry with original material corresponding to current research.
Mathematical Formulas for Economists
This collection of formulas constitutes a compendium of mathematics for eco nomics and business. It contains the most important formulas, statements and algorithms in this significant subfield of modern mathematics and addresses primarily students of economics or business at universities, colleges and trade schools. But people dealing with practical or applied problems will also find this collection to be an efiicient and easy-to-use work of reference. First the book treats mathematical symbols and constants, sets and state ments, number systems and their arithmetic as well as fundamentals of com binatorics. The chapter on sequences and series is followed by mathematics of finance, the representation of functions of one and several independent vari ables, their differential and integral calculus and by differential and difference equations. In each case special emphasis is placed on applications and models in economics. The chapter on linear algebra deals with matrices, vectors, determinants and systems of linear equations. This is followed by the representation of struc tures and algorithms of linear programming. Finally, the reader finds formu las on descriptive statistics (data analysis, ratios, inventory and time series analysis), on probability theory (events, probabilities, random variables and distributions) and on inductive statistics (point and interval estimates, tests). Some important tables complete the work.
A Structural Framework for the Pricing of Corporate Securities : Economic and Empirical Issues
This book is the first comprehensive treatment, of structural credit risk models for the simultaneous and consistent pricing of corporate securities. Through the development of a flexible economic framework based on the firm's EBIT, the reader is taken from the economic principles of firm value models to the empirical implementation. Analytical solutions are provided, if EBIT follows an arithmetic or geometric Brownian motion.
Compactifications of Symmetric and Locally Symmetric Spaces
Noncompact symmetric and locally symmetric spaces naturally appear in many mathematical theories, including analysis (representation theory, nonabelian harmonic analysis), number theory (automorphic forms), algebraic geometry (modulae) and algebraic topology (cohomology of discrete groups). In most applications it is necessary to form an appropriate compactification of the space. The literature dealing with such compactifications is vast. The main purpose of this book is to introduce uniform constructions of most of the known compactifications with emphasis on their geometric and topological structures. The book is divided into three parts. Part I studies compactifications of Riemannian symmetric spaces and their arithmetic quotients. Part II is a study of compact smooth manifolds. Part III studies the compactification of locally symmetric spaces.
Circuits and Systems Based on Delta Modulation : Linear, Nonlinear and Mixed Mode Processing
This book is intended for students and professionals who are interested in the field of digital signal processing of delta-sigma modulated sequences. The overall focus is on the development of algorithms and circuits for linear, non-linear, and mixed mode processing of delta-sigma modulated pulse streams. The material presented here is directly relevant to applications in digital communication, DSP, instrumentation, and control.
Building the Foundation : Whole Numbers in the Primary Grades : The 23rd ICMI Study
This twenty-third ICMI Study addresses for the first time mathematics teaching and learning in the primary school (and pre-school) setting, while also taking international perspectives, socio-cultural diversity and institutional constraints into account. One of the main challenges of designing the first ICMI primary school study of this kind is the complex nature of mathematics at the early level. Accordingly, a focus area that is central to the discussion was chosen, together with a number of related questions. The broad area of Whole Number Arithmetic (WNA), including operations and relations and arithmetic word problems, forms the core content of all primary mathematics curricula. this study presents a meta-level analysis and synthesis of what is currently known about WNA, providing a useful base from which to gauge gaps and shortcomings, as well as an opportunity to learn from the practices of different countries and contexts.
Brouwer meets Husserl : On the Phenomenology of Choice Sequences
Can the straight line be analysed mathematically such that it does not fall apart into a set of discrete points, as is usually done but through which its fundamental continuity is lost? And are there objects of pure mathematics that can change through time? The mathematician and philosopher L.E.J. Brouwer argued that the two questions are closely related and that the answer to both is "yes''. To this end he introduced a new kind of object into mathematics, the choice sequence. But other mathematicians and philosophers have been voicing objections to choice sequences from the start. This book aims to provide a sound philosophical basis for Brouwer's choice sequences by subjecting them to a phenomenological critique in the style of the later Husserl.
Beyond the apparent Banality of the mathematics classroom
New research in mathematics education deals with the complexity of the mathematics’ classroom. The classroom teaching situation constitutes a pertinent unit of analysis for research into the ternary didactic relationship which binds teachers, students and mathematical knowledge. The classroom is considered as a complex didactic system, which offers the researcher an opportunity to gauge the boundaries of the freedom that is left with regard to choices about the knowledge to be taught and the ways of organizing the students’ learning, while giveing rise to the study of interrelations between three main elements of the teaching process the: mathematical content to be taught and learned, management of the various time dimensions, and activity of the teacher who prepares and manages the class, to the benefit of the students' knowledge and the teachers' own experience.
Axiom of Choice
AC, the axiom of choice, because of its non-constructive character, is the most controversial mathematical axiom, shunned by some, used indiscriminately by others. This treatise shows paradigmatically that:Disasters happen without AC: Many fundamental mathematical results fail (being equivalent in ZF to AC or to some weak form of AC).Disasters happen with AC: Many undesirable mathematical monsters are being created (e.g., non measurable sets and undeterminate games).Illuminating examples are drawn from diverse areas of mathematics, particularly from general topology, but also from algebra, order theory, elementary analysis, measure theory, game theory, and graph theory.
Aritmetica, crittografia e codici = Arithmetic, cryptography and codes
The basic techniques of algebra and number theory useful in recent applications to cryptography and codes are developed, with the aim of being elementary and self-sufficient. The emphasis is on computational problems. This part of the volume can be useful as a textbook for a first course in algebra for mathematicians, computer scientists or engineers. Important applications of algebra and geometry to cryptography and codes are then illustrated. Both, cryptography and codes have significant applications in daily life which are illustrated here. Cryptography is developed in detail in much of its classic and current aspects, and both private and public key cryptography are developed. Cryptography with the use of elliptic curves on finite fields is also illustrated. A chapter introducing the subject is dedicated to linear codes.
Aritmetica : Un approccio computazionale = Arithmetic : A computational approach
Intended to be a contribution to the algorithmic re-reading of some classic topics of elementary number theory and an invitation to more demanding reading, according to the indications provided by the bibliography annexed to it.
Arithmetical investigations : Representation theory, orthogonal polynomials, and quantum interpolations
In this volume the author further develops his philosophy of quantum interpolation between the real numbers and the p-adic numbers. The p-adic numbers contain the p-adic integers Zp which are the inverse limit of the finite rings Z/pn. This gives rise to a tree, and probability measures w on Zp correspond to Markov chains on this tree. From the tree structure one obtains special basis for the Hilbert space L2(Zp,w). The real analogue of the p-adic integers is the interval [-1,1], and a probability measure w on it gives rise to a special basis for L2([-1,1],w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For special (gamma and beta) measures there is a "quantum" or "q-analogue" Markov chain, and a special basis, that within certain limits yield the real and the p-adic theories. This idea can be generalized variously. In representation theory, it is the quantum general linear group GLn(q)that interpolates between the p-adic group GLn(Zp), and between its real (and complex) analogue -the orthogonal On (and unitary Un )groups. There is a similar quantum interpolation between the real and p-adic Fourier transform and between the real and p-adic (local unramified part of) Tate thesis, and Weil explicit sums.
Applied Proof Theory : Proof Interpretations and Their Use in Mathematics
Ulrich Kohlenbach presents an applied form of proof theory that has led in recent years to new results in number theory, approximation theory, nonlinear analysis, geodesic geometry and ergodic theory (among others). This applied approach is based on logical transformations (so-called proof interpretations) and concerns the extraction of effective data (such as bounds) from prima facie ineffective proofs as well as new qualitative results such as independence of solutions from certain parameters, generalizations of proofs by elimination of premises. The book first develops the necessary logical machinery emphasizing novel forms of Gödel's famous functional ('Dialectica') interpretation. It then establishes general logical metatheorems that connect these techniques with concrete mathematics. Finally, two extended case studies (one in approximation theory and one in fixed point theory) show in detail how this machinery can be applied to concrete proofs in different areas of mathematics.
Applied Fuzzy Arithmetic : An Introduction with Engineering Applications
Applied Fuzzy Arithmetic provides a well-structured compendium that offers both a deeper knowledge about the theory of fuzzy arithmetic and an extensive view on its applications in the engineering sciences, making it a resource for students, researchers, and practical engineers. The first part of the book gives an introduction to the theory of fuzzy arithmetic, which aims to present the subject in a well-organized and comprehensible form. The derivation of fuzzy arithmetic from the original fuzzy set theory and its evolution towards a successful implementation is presented with existing formulations of fuzzy arithmetic included and integrated in the overall context. The second part of the book presents a diversified exposition of the application of fuzzy arithmetic, addressing different areas of the engineering sciences, such as mechanical, geotechnical, biomedical, and control engineering.
An Introduction to the Mathematics of Money : Saving and Investing
This is an undergraduate textbook on the basic aspects of personal savings and investing with a balanced mix of mathematical rigor and economic intuition. It uses routine financial calculations as the motivation and basis for tools of elementary real analysis rather than taking the latter as given. Proofs using induction, recurrence relations and proofs by contradiction are covered. Inequalities such as the Arithmetic-Geometric Mean Inequality and the Cauchy-Schwarz Inequality are used. Basic topics in probability and statistics are presented.
An Introduction to Number Theory
An Introduction to Number Theory provides an introduction to the main streams of number theory. Starting with the unique factorization property of the integers, the theme of factorization is revisited several times throughout the book to illustrate how the ideas handed down from Euclid continue to reverberate through the subject. In particular, the book shows how the Fundamental Theorem of Arithmetic, handed down from antiquity, informs much of the teaching of modern number theory. The result is that number theory will be understood, not as a collection of tricks and isolated results, but as a coherent and interconnected theory. A number of different approaches to number theory are presented, and the different streams in the book are brought together in a chapter that describes the class number formula for quadratic fields and the famous conjectures of Birch and Swinnerton-Dyer. The final chapter introduces some of the main ideas behind modern computational number theory and its applications in cryptography.
Algèbre, Chapitres 1 à 3 = Algebra, Chapters 1 to 3
To do algebra is essentially to calculate, that is to say to perform, on elements of a set, (<algebraic operations n, the best-known example of which is provided by the (<four rules)) of elementary arithmetic. This is not the place to retrace the slow process of progressive abstraction by which the notion of algebraic operation, initially restricted to natural integers and to measurable quantities, gradually widened its field, as it grew. at the same time generalized the notion of ((number O, until, going beyond the latter, it came to apply to elements which no longer had any character ((numeric)>, for example to permutations of a - seems (see Historical Note in chap. 1).
Algebraic Geometry : An Introduction
The book starts with easily-formulated problems with non-trivial solutions – for example, Bézout’s theorem and the problem of rational curves – and uses these problems to introduce the fundamental tools of modern algebraic geometry: dimension; singularities; sheaves; varieties; and cohomology. The treatment uses as little commutative algebra as possible by quoting without proof (or proving only in special cases) theorems whose proof is not necessary in practice, the priority being to develop an understanding of the phenomena rather than a mastery of the technique. A range of exercises is provided for each topic discussed, and a selection of problems and exam papers are collected in an appendix to provide material for further study.
A Remarkable Collection of Babylonian Mathematical Texts : Manuscripts in the Schøyen Collection: Cuneiform Texts I
This new text from Jöran Friberg, the leading expert on Babylonian mathematics, presents 130 previously unpublished mathematical clay tablets from the Norwegian Schøyen collection, and provides a synthesis of the author's most important work. Through a close study of these tablets, Friberg has made numerous amazing discoveries, including the first known examples of pre-Classical labyrinths and mazes, a new understanding of the famous table text Plimpton 322, and new evidence of Babylonian familiarity with sophisticated mathematical ideas and objects, such as the three-dimensional Pythagorean equation and the icosahedron.
A First Course in Modular Forms
This book introduces the theory of modular forms with an eye toward the Modularity Theorem: All rational elliptic curves arise from modular forms. The topics covered include • elliptic curves as complex tori and as algebraic curves, • modular curves as Riemann surfaces and as algebraic curves, • Hecke operators and Atkin–Lehner theory, • Hecke eigenforms and their arithmetic properties, • the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms, • elliptic and modular curves modulo p and the Eichler–Shimura Relation, • the Galois representations associated to elliptic curves and to Hecke eigenforms. As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their applications to number theory.
