Mathematics

Statistical Thought: A Perspective and History

Shoutir Kishore Chatterjee

The book examines the distinguishing features of the various contending approaches to statistical inference (including decision-making) that are currently available in statistical ... More

The book examines the distinguishing features of the various contending approaches to statistical inference (including decision-making) that are currently available in statistical literature, and traces the historical evolution of the concepts underlying these approaches and their applications. The first part entitled, Perspective, shows that statistical inference is really a prolongation of the philosophical problem of induction, and in it, probability is involved both in the input (in the form of model) and the output (for quantifying uncertainty). Four different approaches (behavioural, instantial, pro-subjective Bayesian, and purely subjective) to such statistical induction arise due to the invocation of different conceptions of probability (objective and subjective) at the two stages of the process. The comparative characteristics, advantages, and disadvantages of the different approaches are considered, and it is concluded that each is appropriate in its natural setting. The second part entitled, History, discusses how the different types of probability originated and evolved, and how their application to statistical induction gave rise to the variety of concepts and principles associated with the different approaches. After some reference to pre-history, the developments made by the principal contributors to probability and statistics during 17th-20th centuries (from Cardano, Pascal, Fermat, Huygens, and James Bernoulli through to Daniel Bernoulli, Bayes, Laplace, Gauss, to Galton, Karl Pearon, Fisher, Jeffreys, de Finetti, Neyman, E. S. Pearson, Wald, and their successors) are delineated. Less

Statistics and Informatics in Molecular Cancer Research

Carsten Wiuf, Claus L. Andersen (eds)

This book discusses novel advances in informatics and statistics in molecular cancer research. Through eight chapters it discusses specific topics in cancer research, talks about how the ... More

This book discusses novel advances in informatics and statistics in molecular cancer research. Through eight chapters it discusses specific topics in cancer research, talks about how the topics give rise to development of new informatics and statistics tools, and explains how the tools can be applied. The focus of the book is to provide an understanding of key concepts and tools, rather than focusing on technical issues. A main theme is the extensive use of array technologies in modern cancer research — gene expression and exon arrays, SNP and copy number arrays and methylation arrays — to derive quantitative and qualitative statements about cancer, its progression and aetiology, and to understand how these technologies at one hand allow us learn about cancer tissue as a complex system and at the other hand allow us to pinpoint key genes and events as crucial for the development of the disease. Cancer is characterized by genetic and genomic alterations that influence all levels of the cell's machinery and function. Less

Stochastic Methods in Neuroscience

Carlo Laing, Gabriel J Lord (eds)

We give a brief introduction to modelling in mathematical neuroscience, to stochastic processes, and stochastic differential equations as well as an overview of the book.

Stochastic Population Processes: Analysis, Approximations, Simulations

Eric Renshaw

The vast majority of random processes in the real world have no memory — the next step in their development depends purely on their current state. Stochastic realizations are therefore ... More

The vast majority of random processes in the real world have no memory — the next step in their development depends purely on their current state. Stochastic realizations are therefore defined purely in terms of successive event-time pairs, and such systems are easy to simulate irrespective of their degree of complexity. However, whilst the associated probability equations are straightforward to write down, their solution usually requires the use of approximation and perturbation procedures. Traditional books, heavy in mathematical theory, often ignore such methods and attempt to force problems into a rigid framework of closed-form solutions. Less

The Structure of Models of Peano Arithmetic

Roman Kossak, James Schmerl

This book gives an account of the present state of research on lattices of elementary substructures and automorphisms of nonstandard models of arithmetic. Major representation theorems ... More

This book gives an account of the present state of research on lattices of elementary substructures and automorphisms of nonstandard models of arithmetic. Major representation theorems are proved, and the important particular case of countable recursively saturated models is discussed in detail. All necessary technical tools are developed. The list includes: constructions of elementary simple extensions; a partial classification of arithmetic types, in particular Gaifman's theory of definable types; forcing in arithmetic; elements of the Kirby-Paris combinatorial theory of cuts; Lascar's generic automorphisms; and applications of Abramson and Harrington's generalization of Ramsey's theorem. There are also chapters discussing ω1-like models with interesting second order properties, and a chapter on order types of nonstandard models. Less

System Control and Rough Paths

Terry Lyons, Zhongmin Qian

Systems evolve and interact, and when they do this in a continuous way, differential equations can be used to provide accurate mathematical models for the interaction or reaction of the ... More

Systems evolve and interact, and when they do this in a continuous way, differential equations can be used to provide accurate mathematical models for the interaction or reaction of the controlled system to external stimuli or forcing. If the forcing is not smooth on normal scales, one cannot use classical calculus. The theory of rough paths provides a rigorous mathematical extension of Newtonian calculus, allowing one to model the responses of systems subject to more wildly oscillating or ‘rough’ stimuli. This book defines rough paths as the completion of the piecewise smooth paths under a p-variation rough path metric. Building on the earlier work of K.T. Chen on the iterated integrals of paths, and of Young on the integration of paths with p-variation < 2, the book proves that the Itô functional, taking stimuli to responses, is uniformly continuous in the p-rough path metric; from which it is elementary to make sense of these differential equations when the external stimuli are rough paths. One important initial application of these results is to stochastic differential equations. Almost every Brownian path is a p-rough path for every p > 2. In addition, the theory allows one to consider stimuli outside of the classes traditionally treated by the Itô calculus, for example the book explains how fractional Brownian motion can often be regarded as a rough path. The basic estimates are uniform without regard to dimension, and apply to infinite dimensional noise sources. Less

The Theory of Infinite Soluble Groups

John C. Lennox, Derek J. S. Robinson

This book provides a comprehensive account of the theory of infinite soluble groups, from its foundations up to research level. Topics covered include: polycyclic groups, Cernikov ... More

This book provides a comprehensive account of the theory of infinite soluble groups, from its foundations up to research level. Topics covered include: polycyclic groups, Cernikov groups, Mal’cev completions, soluble linear groups, P. Hall’s theory of finitely generated soluble groups, soluble groups with finite rank, soluble groups whose abelian subgroups satisfy finiteness conditions, simple modules over polycyclic groups, the Jategaonkar-Roseblade theorem, centrality in finitely generated soluble groups and the Lennox-Roseblade theorem, algorithmic problems for polycyclic and metabelian groups, cohomological topics including groups with finite (co)homological dimension and vanishing theorems, finitely presented soluble groups, constructible soluble groups, the Bieri-Strebel invariant, subnormality, and soluble groups. Less

Thermoelasticity with Finite Wave Speeds

Józef Ignaczak, Martin Ostoja-Starzewski

Generalized dynamic thermoelasticity is a vital area of research in continuum mechanics, free of the classical paradox of infinite propagation speeds of thermal signals in Fourier‐type ... More

Generalized dynamic thermoelasticity is a vital area of research in continuum mechanics, free of the classical paradox of infinite propagation speeds of thermal signals in Fourier‐type heat conduction. Besides that paradox, the classical dynamic thermoelasticity theory offers either unsatisfactory or poor descriptions of a solid's response to a fast transient loading (say, due to short laser pulses) or at low temperatures. Several models were developed and intensively studied over the past four decades, and this book is the first monograph on the subject since the 1970s, aiming to provide a point of reference in the field. It focuses on dynamic thermoelasticity governed by hyperbolic equations, and, in particular, on the two leading theories: that of Lord‐Shulman (with one relaxation time), and that of Green‐Lindsay (with two relaxation times). While the resulting field equations are linear partial differential ones, the complexity of theories is due to the coupling of mechanical with thermal fields. The book is concerned with the mathematical aspects of both theories — existence and uniqueness theorems, domain of influence theorems, convolutional variational principles — as well as with the methods for various initial/boundary value problems. In the latter respect, following the establishment of the central equation of thermoelasticity with finite wave speeds, there are extensive presentations of: the exact, aperiodic‐in‐time solutions of Green‐Lindsay theory; Kirchhoff‐type formulas and integral equations in Green‐Lindsay theory; thermoelastic polynomials; moving discontinuity surfaces; and time‐periodic solutions. This is followed by a chapter on physical aspects of generalized thermoelasticity, with a review of several applications. The book closes with a chapter on a nonlinear hyperbolic theory of a rigid heat conductor for which a number of asymptotic solutions are obtained using a method of weakly nonlinear geometric optics. The book is augmented by an extensive bibliography. Less

Thiele: Pioneer in Statistics

Steffen L. Lauritzen

Thorvald Nicolai Thiele was a brilliant Danish researcher of the 19th century. He was a professor of Astronomy at the University of Copenhagen and the founder of Hafnia, the first Danish ... More

Thorvald Nicolai Thiele was a brilliant Danish researcher of the 19th century. He was a professor of Astronomy at the University of Copenhagen and the founder of Hafnia, the first Danish private insurance company. Thiele worked in astronomy, mathematics, actuarial science, and statistics, his most spectacular contributions were in the latter two areas, where his published work was far ahead of his time. This book is concerned with his statistical work. It evolves around his three main statistical masterpieces, which are now translated into English for the first time: 1) his article from 1880 where he derives the Kalman filter; 2) his book from 1889, where he lays out the subject of statistics in a highly original way, derives the half-invariants (today known as cumulants), the notion of likelihood in the case of binomial experiments, the canonical form of the linear normal model, and develops model criticism via analysis of residuals; and 3) an article from 1899 where he completes the theory of the half-invariants. This book also contains three chapters, written by A. Hald and S. L. Lauritzen, which describe Thiele's statistical work in modern terms and puts it into an historical perspective. Less

Traces and Determinants of Pseudodifferential Operators

Simon Scott

This text provides a broad account of the theory of traces and determinants on geometric algebras of differential and pseudodifferential operators over compact manifolds. Trace and ... More

This text provides a broad account of the theory of traces and determinants on geometric algebras of differential and pseudodifferential operators over compact manifolds. Trace and determinant functionals on geometric operator algebras provide a means of constructing refined invariants in analysis, topology, differential geometry, analytic number theory and QFT. The consequent interactions around such invariants have led to significant advances both in pure mathematics and theoretical physics. As the fundamental tools of trace theory have become well understood and clear general structures have emerged, so the need for specialist texts which explain the basic theoretical principles and the computational techniques has become increasingly exigent. This text is the first to deal with the general theory of traces and determinants of operators on manifolds in a broad context, encompassing a number of the principle applications and backed up by specific computations which set out in detail to newcomers the nuts-and-bolts of the basic theory. Both the microanalytic approach to traces and determinants via pseudodifferential operator theory and the more computational approach directed by applications in geometric analysis, are developed in a general framework that will be of interest to mathematicians and physicists in a number of different fields. Less