Copyright 2024
Hardback
ISBN 9780367481827
402 Pages 36 B/W Illustrations
April 5, 2024 by Chapman & Hall
Mixture models are a powerful tool for analyzing complex and heterogeneous datasets across many scientific fields, from finance to genomics. Mixture Models: Parametric, Semiparametric, and New Directions provides an up-to-date introduction to these models, their recent developments, and their implementation using R. It fills a gap in the literature by covering not only the basics of finite mixture models, but also recent developments such as semiparametric extensions, robust modeling, label switching, and high-dimensional modeling.
Comprehensive overview of the methods and applications of mixture models
Key topics include hypothesis testing, model selection, estimation methods, and Bayesian approaches
Recent developments, such as semiparametric extensions, robust modeling, label switching, and high-dimensional modeling
Examples and case studies from such fields as astronomy, biology, genomics, economics, finance, medicine, engineering, and sociology
Integrated R code for many of the models, with code and data available in the R Package MixSemiRob
Mixture Models: Parametric, Semiparametric, and New Directions is a valuable resource for researchers and postgraduate students from statistics, biostatistics, and other fields. It could be used as a textbook for a course on model-based clustering methods, and as a supplementary text for courses on data mining, semiparametric modeling, and high-dimensional data analysis.
1 Introduction to Mixture Models
2 Mixture models for discrete data
3 Mixture regression models
4 Bayesian mixture models
5 Label switching for mixture models
6 Hypothesis testing and model selection for mixture models
7 Robust mixture regression models
8 Mixture models for high dimensional data
9 Semiparametric mixture models
10 Semiparametric mixture regression models
Two Volume Set
Hardback
ISBN 9781032504896
750 Pages
Published September 24, 2023 by Chapman & Hall
Multiplicative Differential Equations: Volume I is the first part of a comprehensive approach to the subject. It continues a series of books written by the authors on multiplicative, geometric approaches to key mathematical topics. This volume begins with a basic introduction to multiplicative differential equations and then moves on to first and second order equations, as well as the question of existence and unique of solutions. Each chapter ends with a section of practical problems. The book is accessible to graduate students and researchers in mathematics, physics, engineering and biology.
Multiplicative Differential Equations: Volume 2 is the second part of a comprehensive approach to the subject. It continues a series of books written by the authors on multiplicative, geometric approaches to key mathematical topics.@ This volume is devoted to the theory of multiplicative differential systems. The asymptotic behavior@of the solutions of such systems is studied. Stability theory for multiplicative linear and nonlinear systems is introduced and boundary value problems for second order multiplicative linear and nonlinear equations are explored. @The authors also present first order multiplicative partial differential equations. Each chapter ends with a section of practical problems. The book is accessible to graduate students and researchers in mathematics, physics, engineering and biology.
Volume 1: 1.Introduction. 2. Elementary First Order MDEs. 3. First Order Multiplicative Linear Differential Equations. 4. Second Order Linear MDEs. 5. Existence and Uniqueness of Solutions.
Volume 2: 1. Systems Multiplicative Differential Equations. 2. Qualitative Analysis of Multiplicative Differential Systems. 3. Stability Theory. 4. Multiplicative Linear Boundary Value Problems. 5. Multiplicative Nonlinear MDEs. 6. First Order MPDE.
In his PhD thesis, Einstein derived an explicit first-order expansion for the effective viscosity of a Stokes fluid with a suspension of small rigid particles at low density. His formal derivation relied on two implicit assumptions: (i) there is a scale separation between the size of the particles and the observation scale; and (ii) at first order, dilute particles do not interact with one another. In mathematical terms, the first assumption amounts to the validity of a homogenization result defining the effective viscosity tensor, which is now well understood. Next, the second assumption allowed Einstein to approximate this effective viscosity at low density by considering particles as being isolated. The rigorous justification is, in fact, quite subtle as the effective viscosity is a nonlinear nonlocal function of the ensemble of particles and as hydrodynamic interactions have borderline integrability.
In the present memoir, we establish Einstein's effective viscosity formula in the most general setting. In addition, we pursue the low-density expansion to arbitrary order in form of a cluster expansion, where the summation of hydrodynamic interactions crucially requires suitable renormalizations. In particular, we justify a celebrated result by Batchelor and Green on the second-order correction and we explicitly describe all higher-order renormalizations for the first time. In some specific settings, we further address the summability of the whole cluster expansion. Our approach relies on a combination of combinatorial arguments, variational analysis, elliptic regularity, probability theory, and diagrammatic integration methods.
Published: 28 June 2023
Hardcover
574 pages
This two-volume set presents forty-eight papers on applied mathematics by Fields Medal winner Shing-Tung Yau.
The papers are organized topically into thirteen sections: circuit theory, MRI data analysis, matrix eigenvalues, scientific computing, evolutionary dynamics (in mathematical biology), quantum information (mathematical physics), nonlinear filtering, graph theory, medical imaging, computer vision, computer graphics and geometric modeling, artificial intelligence, and computational geometry and numerical analysis.
Also included are commentaries on Yaufs work in the various topics: S.T. Chui on Yaufs work in circuit theory; Xudong Huang on his work in MRI Data analysis; Mei-Heng Yueh on his work in scientific computing; Gabor Lippner on his work in quantum state transfer; Xiuqiong Chen on his work in nonlinear filtering; Yong Lin on his works on graphs; and Xianfeng David Gu on his work in computer graphics, computer vision, digital geometry processing, medical imaging and artificial intelligence.
The collection begins with prefaces by Bjorn Engquist (University of Texas) and Tony Chang (president, KAUST).
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Shing-Tung Yau is a Professor Emeritus of Harvard University, where he was the William Caspar Graustein Professor of Mathematics and Professor of Physics at Harvard University. He is now Chair Professor of Mathematics at Tsinghua University in Beijing. He is the winner of the Fields Medal, the National Medal of Science, the Crafoord Prize, the Veblen Prize, the Wolf Prize, and a MacArthur Fellowship. A member of the National Academy of Sciences, Yau has received honorary degrees from ten universities. He is also the director of several mathematical institutes in China. Yau has written and edited more than twenty books and is the editor of numerous mathematics journals.
Published: 30 November 2023
Paperback
178 pages
This book is based upon the first in a series of lecture notes written by the author for his courses in mathematical physics at Qiuzhen College of Tsinghua University, Beijing. Its purpose is to explain the key physical elements of Lagrangian and Hamiltonian mechanics, and their relationship to modern developments in geometry. Heavy emphasis is placed on different faces of concrete examples that illustrate the bridge between mathematics and physics.
Si Li received his PhD in mathematics at Harvard University, and is currently professor in the Department of Mathematical Sciences at Tsinghua University. His specialty is mathematical physics, especially the interplay between quantum field theory and geometry/topology. He is the winner of the 2016 Morningside Gold Medal, for his construction of the quantum B-twisted string field theory, and for his solution to the Landau?Ginzburg Mirror Symmetry Conjecture. He was an invited plenary speaker at the String-Math conference in 2018 (Japan) and in 2021 (Brazil).
Published: 30 November 2023
Paperback
180 pages
This book is based upon a series of lecture notes written by the author for his courses in mathematical physics at Qiuzhen College of Tsinghua University. The essence of this book is to explain both the physical and the geometric aspects of modern electromagnetism, as well as the bridge between these two viewpoints. The first part of this book explains key ingredients of electromagnetic fields and their classical dynamics. The second part explains Maxwell theory in terms of the geometry of
fiber bundle. We use the language of differential forms throughout this book to illustrate the geometric and topological natures of Maxwellfs equations.
Si Li received his PhD in mathematics at Harvard University, and is currently professor in the Department of Mathematical Sciences at Tsinghua University. His specialty is mathematical physics, especially the interplay between quantum field theory and geometry/topology. He is the winner of the 2016 Morningside Gold Medal, for his construction of the quantum B-twisted string field theory, and for his solution to the Landau?Ginzburg Mirror Symmetry Conjecture. He was an invited plenary speaker at the String-Math conference in 2018 (Japan) and in 2021 (Brazil).