Mathematical models can be used to meet many of the challenges and opportunities offered by modern biology. The description of biological phenomena requires a range of mathematical theories. This is the case particularly for the emerging field of systems biology. Mathematical Methods in Biology and Neurobiology introduces and develops these mathematical structures and methods in a systematic manner. It studies:

• discrete structures and graph theory • stochastic processes• dynamical systems and partial differential equations • optimization and the calculus of variations.

The biological applications range from molecular to evolutionary and ecological levels, for example:

• cellular reaction kinetics and gene regulation• biological pattern formation and chemotaxis• the biophysics and dynamics of neurons • the coding of information in neuronal systems• phylogenetic tree reconstruction• branching processes and population genetics• optimal resource allocation• sexual recombination • the interaction of species.

Written by one of the most experienced and successful authors of advanced mathematical textbooks, this book stands apart for the wide range of mathematical tools that are featured. It will be useful for graduate students and researchers in mathematics and physics that want a comprehensive overview and a working knowledge of the mathematical tools that can be applied in biology. It will also be useful for biologists with some mathematical background that want to learn more about the mathematical methods available to deal with biological structures and data.

This book offers an ideal graduate-level introduction to the theory of partial differential equations. The first part of the book describes the basic mathematical problems and structures associated with elliptic, parabolic, and hyperbolic partial differential equations, and explores the connections between these fundamental types. Aspects of  Brownian motion or pattern formation processes are also presented. The second part focuses on existence schemes and develops estimates for solutions of elliptic equations, such as Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. In particular, the reader will learn the basic techniques underlying current research in elliptic partial differential equations.

This revised and expanded third edition is enhanced with many additional examples that will help motivate the reader.  New features include a reorganized and extended chapter on hyperbolic equations, as well as a new chapter on the relations between different types of partial differential equations, including first-order hyperbolic systems, Langevin and Fokker-Planck equations, viscosity solutions for elliptic PDEs, and much more. Also, the new edition contains additional material on systems of elliptic partial differential equations, and it explains in more detail how the Harnack inequality can be used for the regularity of solutions.

After over a century of research, we still do not understand the principle by which a stimulus, such as an odour or a sound, is represented by distributed neural ensembles within the brain; by which an individual is developed by the control of gene networks. While past studies have made detailed analysis of response profiles of single cells and the properties of a single gene in isolation, such techniques and approaches cannot easily address holistic issues of how large ensembles of neurons and genes can integrate information both spatially and temporally. There is little doubt that much of the info processing power of the brain or a gene network resides in the activities of co-operating and competing networks of neurons and genes and that if we unlock the principles whereby info is encoded and processed within these networks as a whole, rather than within single neurones or genes in isolation, we may be able to understand how the brain or gene network works. Our book will contribute to the development of such a study.

Chemistry shapes and creates the disposition of the world''s resources and provides novel substances for the welfare and hazard of our civilisation at an exponential rate. Can we model the evolution of chemical knowledge? This book not only provides a positive answer to the question, it provides the formal models and available data to model chemical knowledge as a complex dynamical system based on the mutual interaction of the social, semiotic and material systems of chemistry. These systems, which have evolved over the history, include the scientists and institutions supporting chemical knowledge (social system); theories, concepts and forms of communication (semiotic system) and the substances, reactions and technologies (material system) central for the chemical practice. These three systems, which have traditionally been mostly studied in isolation, are brought together in this book in a grand historical narrative, on the basis of comprehensive data sets and supplemented by appropriate tools for their formal analysis. We thereby develop a comprehensive picture of the evolution of chemistry, needed for better understanding the past, present and future of chemistry as a discipline. The interdisciplinary character of this book and its non-technical language make it an ideal complement to more traditional material in undergraduate and graduate courses in chemistry, history of science and digital humanities.

This book explores and demonstrates how geometric tools can be used in data analysis. Beginning with a systematic exposition of the mathematical prerequisites, covering topics ranging from category theory to algebraic topology, Riemannian geometry, operator theory and network analysis, it goes on to describe and analyze some of the most important machine learning techniques for dimension reduction, including the different types of manifold learning and kernel methods. It also develops a new notion of curvature of generalized metric spaces, based on the notion of hyperconvexity, which can be used for the topological representation of geometric information.

In recent years there has been a fascinating development: concepts and methods originally created in the context of research in pure mathematics, and in particular in geometry, have become powerful tools in machine learning for the analysis of data. The underlying reason for this is that data are typically equipped with somekind of notion of distance, quantifying the differences between data points. Of course, to be successfully applied, the geometric tools usually need to be redefined, generalized, or extended appropriately.

Primarily aimed at mathematicians seeking an overview of the geometric concepts and methods that are useful for data analysis, the book will also be of interest to researchers in machine learning and data analysis who want to see a systematic mathematical foundation of the methods that they use.

The main intention of this book is to describe and develop the conceptual, structural and abstract thinking of mathematics. Specific mathematical structures are used to illustrate the conceptual approach; providing a deeper insight into mutual relationships and abstract common features. These ideas are carefully motivated, explained and illustrated by examples so that many of the more technical proofs can be omitted. The book can therefore be used:

·        simply as an overview of the panorama of mathematical structures and the relations between them, to be supplemented by more detailed texts whenever you want to acquire a working knowledge of some structure

·        by itself as a first introduction to abstract mathematics

·        together with existing textbooks, to put their results into a more general perspective

·        to gain a new and hopefully deeper perspective after having studied such textbooks

Mathematical Concepts has a broader scope and is less detailed than standard mathematical textbooks so that the reader can readily grasp the essential concepts and ideas for individual needs. It will be suitable for advanced mathematicians, postgraduate students and for scientists from other fields with some background in formal reasoning.

This book presents William Clifford’s English translation of Bernhard Riemann’s classic text together with detailed mathematical, historical and philosophical commentary. The basic concepts and ideas, as well as their mathematical background, are provided, putting Riemann’s reasoning into the more general and systematic perspective achieved by later mathematicians and physicists (including Helmholtz, Ricci, Weyl, and Einstein) on the basis of his seminal ideas. Following a historical introduction that positions Riemann’s work in the context of his times, the history of the concept of space in philosophy, physics and mathematics is systematically presented. A subsequent chapter on the reception and influence of the text accompanies the reader from Riemann’s times to contemporary research. 

Mathematical models can be used to meet many of the challenges and opportunities offered by modern biology. The description of biological phenomena requires a range of mathematical theories. This is the case particularly for the emerging field of systems biology. Mathematical Methods in Biology and Neurobiology introduces and develops these mathematical structures and methods in a systematic manner. It studies:

• discrete structures and graph theory • stochastic processes• dynamical systems and partial differential equations • optimization and the calculus of variations.

The biological applications range from molecular to evolutionary and ecological levels, for example:

• cellular reaction kinetics and gene regulation• biological pattern formation and chemotaxis• the biophysics and dynamics of neurons • the coding of information in neuronal systems• phylogenetic tree reconstruction• branching processes and population genetics• optimal resource allocation• sexual recombination • the interaction of species.

Written by one of the most experienced and successful authors of advanced mathematical textbooks, this book stands apart for the wide range of mathematical tools that are featured. It will be useful for graduate students and researchers in mathematics and physics that want a comprehensive overview and a working knowledge of the mathematical tools that can be applied in biology. It will also be useful for biologists with some mathematical background that want to learn more about the mathematical methods available to deal with biological structures and data.

This book offers an ideal graduate-level introduction to the theory of partial differential equations. The first part of the book describes the basic mathematical problems and structures associated with elliptic, parabolic, and hyperbolic partial differential equations, and explores the connections between these fundamental types. Aspects of  Brownian motion or pattern formation processes are also presented. The second part focuses on existence schemes and develops estimates for solutions of elliptic equations, such as Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. In particular, the reader will learn the basic techniques underlying current research in elliptic partial differential equations.

This revised and expanded third edition is enhanced with many additional examples that will help motivate the reader.  New features include a reorganized and extended chapter on hyperbolic equations, as well as a new chapter on the relations between different types of partial differential equations, including first-order hyperbolic systems, Langevin and Fokker-Planck equations, viscosity solutions for elliptic PDEs, and much more. Also, the new edition contains additional material on systems of elliptic partial differential equations, and it explains in more detail how the Harnack inequality can be used for the regularity of solutions.

After over a century of research, we still do not understand the principle by which a stimulus, such as an odour or a sound, is represented by distributed neural ensembles within the brain; by which an individual is developed by the control of gene networks. While past studies have made detailed analysis of response profiles of single cells and the properties of a single gene in isolation, such techniques and approaches cannot easily address holistic issues of how large ensembles of neurons and genes can integrate information both spatially and temporally. There is little doubt that much of the info processing power of the brain or a gene network resides in the activities of co-operating and competing networks of neurons and genes and that if we unlock the principles whereby info is encoded and processed within these networks as a whole, rather than within single neurones or genes in isolation, we may be able to understand how the brain or gene network works. Our book will contribute to the development of such a study.

Chemistry shapes and creates the disposition of the world''s resources and provides novel substances for the welfare and hazard of our civilisation at an exponential rate. Can we model the evolution of chemical knowledge? This book not only provides a positive answer to the question, it provides the formal models and available data to model chemical knowledge as a complex dynamical system based on the mutual interaction of the social, semiotic and material systems of chemistry. These systems, which have evolved over the history, include the scientists and institutions supporting chemical knowledge (social system); theories, concepts and forms of communication (semiotic system) and the substances, reactions and technologies (material system) central for the chemical practice. These three systems, which have traditionally been mostly studied in isolation, are brought together in this book in a grand historical narrative, on the basis of comprehensive data sets and supplemented by appropriate tools for their formal analysis. We thereby develop a comprehensive picture of the evolution of chemistry, needed for better understanding the past, present and future of chemistry as a discipline. The interdisciplinary character of this book and its non-technical language make it an ideal complement to more traditional material in undergraduate and graduate courses in chemistry, history of science and digital humanities.

This book explores and demonstrates how geometric tools can be used in data analysis. Beginning with a systematic exposition of the mathematical prerequisites, covering topics ranging from category theory to algebraic topology, Riemannian geometry, operator theory and network analysis, it goes on to describe and analyze some of the most important machine learning techniques for dimension reduction, including the different types of manifold learning and kernel methods. It also develops a new notion of curvature of generalized metric spaces, based on the notion of hyperconvexity, which can be used for the topological representation of geometric information.

In recent years there has been a fascinating development: concepts and methods originally created in the context of research in pure mathematics, and in particular in geometry, have become powerful tools in machine learning for the analysis of data. The underlying reason for this is that data are typically equipped with somekind of notion of distance, quantifying the differences between data points. Of course, to be successfully applied, the geometric tools usually need to be redefined, generalized, or extended appropriately.

Primarily aimed at mathematicians seeking an overview of the geometric concepts and methods that are useful for data analysis, the book will also be of interest to researchers in machine learning and data analysis who want to see a systematic mathematical foundation of the methods that they use.

The main intention of this book is to describe and develop the conceptual, structural and abstract thinking of mathematics. Specific mathematical structures are used to illustrate the conceptual approach; providing a deeper insight into mutual relationships and abstract common features. These ideas are carefully motivated, explained and illustrated by examples so that many of the more technical proofs can be omitted. The book can therefore be used:

·        simply as an overview of the panorama of mathematical structures and the relations between them, to be supplemented by more detailed texts whenever you want to acquire a working knowledge of some structure

·        by itself as a first introduction to abstract mathematics

·        together with existing textbooks, to put their results into a more general perspective

·        to gain a new and hopefully deeper perspective after having studied such textbooks

Mathematical Concepts has a broader scope and is less detailed than standard mathematical textbooks so that the reader can readily grasp the essential concepts and ideas for individual needs. It will be suitable for advanced mathematicians, postgraduate students and for scientists from other fields with some background in formal reasoning.

This book presents William Clifford’s English translation of Bernhard Riemann’s classic text together with detailed mathematical, historical and philosophical commentary. The basic concepts and ideas, as well as their mathematical background, are provided, putting Riemann’s reasoning into the more general and systematic perspective achieved by later mathematicians and physicists (including Helmholtz, Ricci, Weyl, and Einstein) on the basis of his seminal ideas. Following a historical introduction that positions Riemann’s work in the context of his times, the history of the concept of space in philosophy, physics and mathematics is systematically presented. A subsequent chapter on the reception and influence of the text accompanies the reader from Riemann’s times to contemporary research.