DATE PUBLISHED: May 2024 AVAILABILITY: Available FORMAT: Hardback ISBN: 9781009282277
Using diverse real-world examples, this text examines what models used for data analysis mean in a specific
research context. What assumptions underlie analyses, and how can you check them? Building on the successful '
Data Analysis and Graphics Using R,' 3rd edition (Cambridge, 2010), it expands upon topics including cluster analysis,
exponential time series, matching, seasonality, and resampling approaches. An extended look at p-values leads to an
exploration of replicability issues and of contexts where numerous p-values exist, including gene expression.
Developing practical intuition, this book assists scientists in the analysis of their own data, and familiarizes students
in statistical theory with practical data analysis. The worked examples and accompanying commentary teach readers
to recognize when a method works and, more importantly, when it doesn't. Each chapter contains copious exercises.
Selected solutions, notes, slides, and R code are available online, with extensive references pointing to detailed guides
to R.
Encourages readers to learn by working through examples, building practical intuition
Revisits the same data set several times throughout the book, teaching readers to recognize when a method works and,
more importantly, when it doesn't
Emphasizes diagnostic checks and offers strategies for moving ahead when checks fail
Advocates for a judicious use of automation to leave the user free to think about, and attend to, the really important
issues of model choice and checks on assumptions
Contrasts p-values with Bayes factors for the different perspective offered in a classical hypothesis testing context.
Simulation is used extensively, both to build an intuitive understanding of statistical variation and as a complement
to statistical theory
1. Learning from data, and tools for the task
2. Generalizing from models
3. Multiple linear regression
4. Exploiting the linear model framework
5. Generalized linear models and survival analysis
6. Time series models
7. Multilevel models, and repeated measures
8. Tree-based classification and regression
9. Multivariate data exploration and discrimination
Epilogue
A. The R system a brief overview
References
References to R packages
Index of R functions
Subject index.
Not yet published - available from July 2024
FORMAT: Hardback
ISBN: 9781009430913
'Functional Analysis Revisited' is not a first course in functional analysis although it covers the basic notions of functional
analysis, it assumes the reader is somewhat acquainted with them. I
t is by no means a second course either: there are too many deep subjects that are not within scope here.
Instead, having the basics under his belt, the author takes the time to carefully think through their fundamental
consequences. In particular, the focus is on the notion of completeness and its implications, yet without venturing
too far from areas where the description 'elementary' is still valid. The author also looks at some applications, perhaps
just outside the core of functional analysis, that are not completely trivial. The aim is to show how functional analysis
influences and is influenced by other branches of contemporary mathematics. This is what we mean by
'Functional Analysis Revisited.'
Helps the reader gain a deeper understanding of functional analysis and its applications with accessible step-by-step
explanations and examples
Demonstrates how the fundamental notion of completeness permeates contemporary mathematics
Reinforces learning with short chapter summaries and a variety of thoughtfully designed standard and nonstandard
exercises
Introduction
1. Complete metric spaces
2. Banach's principle
3. Picard's theorem
4. Banach spaces
5. Renewal equation in the McKendrickon Foerster model
6. Riemann integral for vector-valued functions
7. The Stoneeierstrass theorem
8. Norms do differ
9. Hilbert spaces
10. Complete orthonormal sequences
11. Heat equation
12. Completeness of the space of operators
13. Working in L(X)
14. The Banachteinhaus theorem and strong convergence
15. We go deeper, deeper we go (into the structure of complete spaces)
16. Semigroups of operators
Appendix. Two consequences of the Hahn anach theorem
References
Index.
Not yet published - available from October 2024
FORMAT: Hardback
ISBN: 9781009200585
The N-body problem has been investigated since Isaac Newton, however vast tracts of the problem remain open.
Showcasing the vibrancy of the problem, this book describes four open questions and explores progress made over
the last 20 years. After a comprehensive introduction, each chapter focuses on a different open question, highlighting
how the stance taken and tools used vary greatly depending on the question. Progress on question one,
'Are the central configurations finite?', uses tools from algebraic geometry. Two, 'Are there any stable periodic orbits?',
is dynamical and requires some understanding of the KAM theorem. The third, 'Is every braid realised?', requires
topology and variational methods. The final question, 'Does a scattered beam have a dense image?', is quite new and
formulating it precisely takes some effort. An excellent resource for students and researchers of mathematics,
astronomy, and physics interested in exploring state-of-the-art techniques and perspectives on this classical problem.
Emphasizes the vitality of the field through open problems, summarizing the methods used and providing readers
with ideas for new research directions
Accessible to those working in any area of mathematics, with the necessary physics covered in an introductory
chapter and dedicated appendix
Takes an intrinsic geometric approach to formulating the problems, allowing readers to draw pictures of
higher-dimensional problems and build intuition not available through standard treatments
Part I. Tour, Problem, and Structures: -1. A tour of solutions
0. The problem and its structure
Part II. The Questions:
1. Are the central configurations finite?
2. Are there any stable periodic orbits?
3. Is every braid realized?
4. Does a scattered beam have a dense image?
Appendices: A. Geometric mechanics
B. Reduction and Poisson brackets
C. The three-body problem and the shape sphere
D. The orthogonal group and its Lie algebra
E. Braids, homotopy and homology
F. The Jacobi aupertuis metric
G. Regularizing binary collisions
H. One-degree of freedom and central scattering
References
Index.
Not yet published - available from October 2024
FORMAT: Hardback ISBN: 9781009445412
The Traveling Salesman Problem (TSP) is a central topic in discrete mathematics and theoretical computer science.
It has been one of the driving forces in combinatorial optimization. The design and analysis of better and better
approximation algorithms for the TSP has proved challenging but very fruitful. This is the first book on approximation
algorithms for the TSP, featuring a comprehensive collection of all major results and an overview of the most
intriguing open problems. Many of the presented results have been discovered only recently, and some are
published here for the first time, including better approximation algorithms for the asymmetric TSP and its
path version. This book constitutes and advances the state of the art and makes it accessible to a wider audience.
Featuring detailed proofs, over 170 exercises, and 100 color figures, this book is an excellent resource for teaching,
self-study, and further research.
Serves as a self-contained resource on approximation algorithms for the Traveling Salesman Problem, covering all major
results and putting recent developments into context
Serves as a starting point for future research with several of the authors' previously unpublished results
Guides students and self-learners through the field through pedagogical explanations, detailed proofs, and many
exercises and color figures
Preface
1. Introduction
2. Linear programming relaxations of the Symmetric TSP
3. Linear programming relaxations of the Asymmetric TSP
4. Duality, cuts, and uncrossing
5. Thin trees and random trees
6. Asymmetric Graph TSP
7. Constant-factor approximation for the Asymmetric TSP
8. Algorithms for subtour cover
9. Asymmetric Path TSP
10. Parity correction of random trees
11. Proving the main payment theorem for hierarchies
12. Removable pairings
13. Ear-Decompositions, matchings, and matroids
14. Symmetric Path TSP and T-tours
15. Best-of-Many Christofides and variants
16. Path TSP by dynamic programming
17. Further results, related problems
18. State of the art, open problems
Bibliography
Index.
The goal of this book is to present new mathematical techniques for studying the behavior of mean-field systems
with disordered interactions. We mostly focus on certain problems of statistical inference in high dimension, and on
spin glasses. The techniques we present aim to determine the free energy of these systems, in the limit of large
system size, by showing that they asymptotically satisfy a Hamilton-Jacobi equation.
The first chapter is a general introduction to statistical mechanics, with a focus on the Curie eiss model. We give a brief
introduction to convex analysis and large deviation principles in Chapter 2, and identify the limit free energy of the Curie
eiss model using these tools. In Chapter 3, we define the notion of viscosity solution to a Hamilton-Jacobi equation,
and use it to recover the limit free energy of the Curie-eiss model. We discover technical challenges to applying
the same method to generalized versions of the Curie-eiss model, and develop a new selection principle based on
convexity to overcome these. We then turn to statistical inference in Chapter 4, focusing on the problem of
recovering a large symmetric rank-one matrix from a noisy observation, and we see that the tools developed in
the previous chapter apply to this setting as well. Chapter 5 is preparatory work for a discussion of the more
challenging case of spin glasses. The first half of this chapter is a self-contained introduction to Poisson point
processes, including limit theorems on extreme values of independent and identically distributed random variables.
We finally turn to the setting of spin glasses in Chapter 6. For the Sherrington irkpatrick model, we show how to
relate the Parisi formula with the Hamilton-Jacobi approach. We conclude with a more informal discussion on the status of current research for more challenging models.
Sparse polynomial approximation is an important tool for approximating high-dimensional functions from limited samples
task commonly arising in computational science and engineering. Yet, it lacks a complete theory. There is a
well-developed theory of best s-term polynomial approximation, which asserts exponential or algebraic rates
of convergence for holomorphic functions. There are also increasingly mature methods such as (weighted)
1
-minimization for practically computing such approximations. However, whether these methods achieve the rates
of the best s-term approximation is not fully understood. Moreover, these methods are not algorithms per se,
since they involve exact minimizers of nonlinear optimization problems. This paper closes these gaps by affirmatively
answering the following question: are there robust, efficient algorithms for computing sparse polynomial approximations
to finite- or infinite-dimensional, holomorphic and Hilbert-valued functions from limited samples that achieve the
same rates as the best s-term approximation? We do so by introducing algorithms with exponential or algebraic
convergence rates that are also robust to sampling, algorithmic and physical discretization errors. Our results involve
several developments of existing techniques, including a new restarted primal-dual iteration for solving weighted
1
-minimization problems in Hilbert spaces. Our theory is supplemented by numerical experiments demonstrating
the efficacy of these algorithms.