The book is rooted in familiar high school mathematics ?
finding patterns, polynomial functions, trigonometric identities,
the complex numbers, and counting problems ? but delves much
deeper to reveal many of the connections that make these topics
all part of the same fabric. Special topics include solving
difference equations, the Mahler polynomials, algebraic
differentiation, the Chebyshev polynomials, Cardano's formula for
the roots of a cubic polynomial, symmetric functions, complex
dynamics, the golden ratio and Fibonacci numbers, Bernoulli
polynomials, Stirling numbers of the first and second kind and
much more.
Scaffolding never looked this good! It may be true, as Gauss once
wrote, that "a cathedral is not a cathedral until the last
piece of scaffolding has been removed," but it is no less
true that appreciation of mathematics is inseparable from the art
of making mathematics. The mathematics of this book is both rich
and engaging. More than 400 exercises amplify and illustrate the
main ideas, sometimes suggesting other paths that might lead the
reader to discover the mathematics for oneself. ?Glenn Stevens,
Boston University
These beautiful problem sets allow readers to discover
mathematical ideas for themselves. The book emphasizes and
explores those ideas and their connections to the mathematics
taught in the high school classroom. I have used Cuoco's problem
sets as the foundation for several courses that I have taught to
other teachers. ?Benjamin Sinwell, Chelsea High School and PCMI
Mathematical Connections focuses on a closely-knit collection of
ideas that are at the intersection of algebra, arithmetic,
combinatorics, geometry, and calculus. Some of these ideas,
previously considered quite advanced, have become tractable
because of advances in computational technology. Others are just
beautiful classical mathematics, topics that have fallen out of
fashion and that deserve to be resurrected. While the book will
appeal to many audiences, one of its primary audiences is high
school teachers, both practicing and prospective. It can be used
as a text for undergraduate or professional courses, and the
design lends itself to self study. Of course, good mathematics
for teaching is also good for many other uses, so readers of all
persuasions can enjoy exploring some of the beautiful ideas
presented in the pages of this book.
ISBN:0-88385-739-1
260 pp., Hardbound 2005
This highly commendable work brings fresh perspective and
astonishing new insight to its venerable subject. In Professor
Kendig's skillful hands, the reader is brought to view the conic
sections within the broader framework of algebraic curves in
complex projective space. The resulting interplay is both
instructive and pleasurable. ?Basil Gordon, UCLA
This book engages the reader in a journey of discovery through a
spirited discussion among three characters: Philosopher, Teacher
and Student. Throughout the book, Philosopher pursues his dream
of a unified theory of conics, where exceptions are banished.
With a helpful teacher and example-hungry student, the trio soon
finds that conics reveal much of their beauty when viewed over
the complex numbers.
In their odyssey, they uncover a goldmine of unsuspected results.
They experience a series of "Aha!" moments as they
stumble upon living brothers to familiar conics objects like foci
and directrices. They also discover a normally-unseen ellipse
spanning the gap between the branches of any hyperbola. On the
applied side, they learn how two interfering wave sources create
systems of hyperbolas; these are used in making astonishingly
precise astronomical observations.
All these discoveries are profusely illustrated with pictures,
worked-out examples, and a CD containing 36 applets.
If you've ever needed a conics formula for area, eccentricity,
curvature and the like, look in the formula appendix. Here are
dozens of useful formulas?a set for each of eight different ways
of looking at a conic:as a cone slice; as the path of a planet
moving under the influence of a fixed sun; constructed using two
stakes and string; plus five other sets.
Conics is written in an easy, conversational style, and many
historical tidbits and other points of interest are scattered
throughout the text. Many students can self-study the book
without outside help. This book is ideal for anyone having a
little exposure to linear algebra and complex numbers.
ISBN:0-88385-335-3
432 pp., Hardbound, 2005
Packaged with a CD containing 35 applets
(Hardback) 0-19-877521-0
(Paperback) 0-19-877520-2
Publication date: 23 December 2004
412 pages, numerous figures, 234mm x 156mm
Series: Advanced Texts in Econometrics
Description
The first book to provide an intuitive introduction to GMM
combined with a unified treatment of GMM statistical theory and a
survey of recent important developments in the field.
All the main statistical results are discussed intuitively as
well as being proved formally. All the inference techniques are
illustrated using empirical examples in macroeconomics and
finance.
In addition to being an excellent graduate text, it is designed
as a resource for both theoreticians and practitioners.
Generalized Method of Moments (GMM) has become one of the main
statistical tools for the analysis of economic and financial data.
This book is the first to provide an intuitive introduction to
the method combined with a unified treatment of GMM statistical
theory and a survey of recent important developments in the field.
Providing a comprehensive treatment of GMM estimation and
inference, it is designed as a resource for both the theory and
practice of GMM: it discusses and proves formally all the main
statistical results, and illustrates all inference techniques
using empirical examples in macroeconomics and finance.
Building from the instrumental variables estimator in static
linear models, it presents the asymptotic statistical theory of
GMM in nonlinear dynamic models. Within this framework it covers
classical results on estimation and inference techniques, such as
the overidentifying restrictions test and tests of structural
stability, and reviews the finite sample performance of these
inference methods. And it discusses in detail recent developments
on covariance matrix estimation, the impact of model
misspecification, moment selection, the use of the bootstrap, and
weak instrument asymptotics.
Readership: Graduate students in econometrics and researchers in
academics, industry, and government.
Contents
1 Introduction
2 The Instrumental Variable Estimator in the Linear Regression
Model
3 GMM Estimation in Correctly Specified Models
4 GMM Estimation in Misspecified Models
5 Hypothesis Testing
6 Asymptotic Theory and Finite Sample Behaviour
7 Moment Selection in Theory and in Practice
8 Alternative Approximations in Finite Sample Behaviour
9 Empirical Examples
10 Related Methods of Estimation
Appendix: Mixing processes and Nonstationarity
(Hardback) 0-19-927865-2
(Paperback) 0-19-927869-5
Publication date: 7 April 2005
472 pages, 234mm x 156mm
Series: Advanced Texts in Econometrics
Description
Harvey is a well known author
Includes substantive introductions to each section
This book presents a collection of readings which give the reader
an idea of the nature and scope of unobserved components (UC)
models and the methods used to deal with them. It contains four
parts, three of which concern recent theoretical developments in
classical and Bayesian estimation of linear, nonlinear, and non
Gaussian UC models, signal extraction and testing, and one is
devoted to selected econometric applications.
The first part focuses on the linear state space model; the
readings provide insight on prediction theory, signal extraction,
and likelihood inference for non stationary and non invertible
processes, diagnostic checking, and the use of state space
methods for spline smoothing.
Part II deals with applications of linear UC models to various
estimation problems concerning economic time series, such as
trend-cycle decompositions, seasonal adjustment, and the
modelling of the serial correlation induced by survey sample
design.
The issues involved in testing in linear UC models are the theme
of part III, which considers tests concerned with whether or not
certain variance parameters are zero, with special reference to
stationarity tests.
Finally, part IV is devoted to the advances concerning classical
and Bayesian inference for non linear and non Gaussian state
space models, an area that has been evolving very rapidly during
the last decade, paralleling the advances in computational
inference using stochastic simulation techniques.
The book is intended to give a relatively self-contained
presentation of the methods and applicative issues. For this
purpose, each part comes with an introductory chapter by the
editors that provides a unified view of the literature and the
many important developments that have occurred in the last years.
Readership: Academics and graduate students in econometrics;
practioners and consultants.
Contents
Signal Extraction and Likelihood Inference for Linear UC Models
1 Introduction
2 P. Burridge and K.F. Wallis: Prediction Theory for
Autoregressive-Moving Average Processes
3 S.J. Koopman: Exact Initial Kalman Filtering and Smoothing for
Non-stationary Time Series Models
4 P. de Jong: Smoothing and Interpolation with the State Space
Model
5 A.C. Harvey and S.J. Koopman: Diagnostic Checking of Unobserved
Components in Time Series Models
6 R. Kohn, C.F. Ansley and C. Wong: Nonparametric Spline
Regression with Autoregressive Moving Average Errors
Unobserved Components in Economic Time Series
7 Introduction
8 M.W. Watson: Univariate Detrending Methods with Stochastic
Trends
9 A.C. Harvey and A. Jaeger: Detrending, Stylized Facts and the
Business Cycle
10 A. Maravall: Stochastic Linear Trends, Models and Estimators
11 D. Pfeffermann: Estimation and Seasonal Adjustment of
Population Means Using Data from Repeated Surveys
12 A.C. Harvey, S.J. Koopman and M. Riani: The Modelling and
Seasonal Adjustment of Weekly Observations
Testing in Unobserved Components Models
13 Introduction
14 J. Nyblom: Testing for Deterministic Linear Trends in a Times
Series
15 F. Canova and B.E. Hansen: Are Seasonal Patterns Stable Over
Time? A Test for Seasonal Stability
Non-Linear and Non- Gaussian Models
16 Introduction
17 A.C. Harvey and C. Fernandes: Times Series Models for Count
Data or Qualitative Observations
18 Carter and Kohn: On Gibbs Sampling for State Space Models
19 P. de Jong and N. Shephard: The Simulation Smoother
20 N. Shephard and M.K. Pitt: Likelihood Analysis of Non-Gaussian
Measurement Time Series
21 J. Durbin and S.J. Koopman: Time Series Analysis of Non-Gaussian
Observations based on State Space Models from both Classical and
Bayesian Perspectives
22 S. Kim, N. Shephard, and S. Chib: Stochastic Volatility:
Liklihood Inference and Comparison with ARCH Models
23 A. Doucet, S.J. Godsill, and C. Andrieu: On Sequential Monte
Carlo Sampling Methods for Bayesian Filtering
Paper | January 2006 | ISBN: 0-691-12488-4
Cloth | January 2006 | ISBN: 0-691-12487-6
288 pp. | 6 x 9 | 45 line illus.
This book studies the dynamics of iterated holomorphic mappings
from a Riemann surface to itself, concentrating on the classical
case of rational maps of the Riemann sphere. This subject is
large and rapidly growing and the lectures discussed here are
intended to introduce some key ideas in the field, and to form a
basis for further study.
The reader is assumed to be familiar with the rudiments of
complex variable theory and of two-dimensional differential
geometry, as well as some basic topics from topology. This third
edition contains a number of minor additions and improvements: A
historical survey has been added, and the definition of Lattes
map has been made more inclusive. The statement of Lemma 18.17
has been sharpened, and the material on two complex variables has
been expanded. Recent results on effective computability have
been added, and the references have been expanded and updated.
Written in the author's usual brilliant style, this book makes
difficult mathematics look easy. It is an accessible source for
much of what has been accomplished in the field.
John Milnor is Professor of Mathematics and Co-Director of the
Institute for Mathematical Sciences at Stony Brook University,
the State University of New York. He is the author of Topology
from the Differential Viewpoint, Singular Points of Complex
Hypersurfaces, Morse Theory, Introduction to Algebraic K-Theory,
Characteristic Classes (with James Stasheff), and Lectures on the
H-Cobordism Theorem (Princeton).
Annals of Mathematics Studies