Showing posts with label differential equations. Show all posts
Showing posts with label differential equations. Show all posts

6/30/2012

Dynamical Symmetry Review

Dynamical Symmetry
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Amazing read! Five stars! This book will assuredly stimulate new levels of thinking for the reader in various aspects of dynamical symmetry and it points to discoveries yet to be made in better understanding the some principals of physics therein. ...And makes important discoveries of its own in the process. Each reader should see if they can infer and verify for themselves the import of powerful new implications based upon Maxwell's original equations set forth towards the end of the book.Algebraic Theory of Molecules (Topics in Physical Chemistry Series)

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Whenever systems are governed by continuous chains of causes and effects, their behavior exhibits the consequences of dynamical symmetries, many of them far from obvious. Dynamical Symmetry introduces the reader to Sophus Lie's discoveries of the connections between differential equations and continuous groups that underlie this observation. It develops and applies the mathematical relations between dynamics and geometry that result. Systematic methods for uncovering dynamical symmetries are described, and put to use. Much material in the book is new and some has only recently appeared in research journals.
Though Lie groups play a key role in elementary particle physics, their connection with differential equations is more often exploited in applied mathematics and engineering. Dynamical Symmetry bridges this gap in a novel manner designed to help readers establish new connections in their own areas of interest. Emphasis is placed on applications to physics and chemistry. Applications to many of the other sciences illustrate both general principles and the ubiquitousness of dynamical symmetries.

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6/22/2012

Dynamical Systems with Applications using MATLAB Review

Dynamical Systems with Applications using MATLAB
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I have been using the text for the last year or so. My familiarity with dynamical systems/non linear dynamics is over eight years now. I use this text as a reference for quick look-up on some of the more elementary techniques. Explanations are scarce and insufficient. Some minor inaccuracies are present, but can be easily disregarded if you are not particularly credulous/naive. You will have to consider Strogatz or J M T Thompson/H B Stewart texts if you want to understand what's happening physically. Or Nayfeh's body of texts on nonlinear physical systems for numerical techniques in simulating such systems or Phillip Holmes' and others for mathematical theory or geometric topology. But I love the book for its easily accessible presentation format for students and the succinctness of its prose. This book is certainly not for people from the mathematical side of dynamical systems, but great for undergrad or beginning grad level students in engineering or physics. This is mostly a cookbook, so don't expect brilliant flavors, just that you can put a meal on the table everyday.

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This introduction to dynamical systems theory guides readers through theory via example and the graphical MATLAB interface; the SIMULINK accessory is used to simulate real-world dynamical processes. Examples included are from mechanics, electrical circuits, economics, population dynamics, epidemiology, nonlinear optics, materials science and neural networks. The book contains over 330 illustrations, 300 examples, and exercises with solutions.

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5/01/2012

The Finite Element Method and Applications in Engineering Using ANSYS® Review

The Finite Element Method and Applications in Engineering Using ANSYS®
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This book is far superior to any other ANSYS FE book. It has something like 40 examples and the cd includes the batch input files. Other books on the subject (see Moaeveni) lack the # of example problems or batch file processing tutorials. Great for beginers and intermediate users who want to get the most out of ANSYS.

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This user-friendly book provides the reader with a theoretical and practical knowledge of the finite element method (FEM) and with the skills required to analyze engineering problems with ANSYS. A self-contained, introductory text, it minimizes the need for additional reference material, covering the fundamental topics in FEM as well as advanced topics concerning modeling and analysis with ANSYS. Extensive examples from various engineering disciplines are presented in a step-by-step fashion, focusing on the use of ANSYS through both the Graphics User Interface (GUI) and the ANSYS Parametric Design Language (APDL). It includes a CD-ROM with the "input" files for the example problems.

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11/12/2011

Introduction to Modeling for Biosciences Review

Introduction to Modeling for Biosciences
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This book is an ideal starting point for undergraduates, postgraduates and even researchers who want to learn the mathematical and computational techniques needed for the modelling of biological systems. The authors cover a wide range of techniques, from analytic approaches (deterministic equations, Markov Chains, master equation) to simulation based ones (agent based models and stochastic simulation algorithms). In particular, I found this book very useful in reviewing various stochastic algorithms needed to simulate biological systems (such as agent based models and Gillespie algorithms), but also in providing Java implementation for the algorithms. The authors' style is clear and this is very helpful for beginners.

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Mathematical modeling can be a useful tool for researchers in the biological scientists.Yet in biological modeling there is no one modeling technique that is suitable for all problems. Instead, different problems call for different approaches. Furthermore, it can be helpful to analyze the same system using a variety of approaches, to be able to exploit the advantages and drawbacks of each. In practice, it is often unclear which modeling approaches will be most suitable for a particular biological question, a problem which requires researchers to know a reasonable amount about a number of techniques, rather than become experts on a single one."Introduction to Modeling for Biosciences" addresses this issue by presenting a broad overview of the most important techniques used to model biological systems.In addition to providing an introduction into the use of a wide range of software tools and modeling environments, this helpful text/reference describes the constraints and difficulties that each modeling technique presents in practice, enabling the researcher to quickly determine which software package would be most useful for their particular problem.Topics and features: introduces a basic array of techniques to formulate models of biological systems, and to solve them; intersperses the text with exercises throughout the book; includes practical introductions to the Maxima computer algebra system, the PRISM model checker, and the Repast Simphony agent modeling environment; discusses agent-based models, stochastic modeling techniques, differential equations and Gillespie's stochastic simulation algorithm; contains appendices on Repast batch running, rules of differentiation and integration, Maxima and PRISM notation, and some additional mathematical concepts; supplies source code for many of the example models discussed, at the associated website http://www.cs.kent.ac.uk/imb/.This unique and practical guide leads the novice modeler through realistic and concrete modeling projects, highlighting and commenting on the process of abstracting the real system into a model.Students and active researchers in the biosciences will also benefit from the discussions of the high-quality, tried-and-tested modeling tools described in the book.Dr. David J. Barnes is a lecturer in computer science at the University of Kent, UK, with a strong background in the teaching of programming.Dr. Dominique Chu is a lecturer in computer science at the University of Kent, UK.He is an internationally recognized expert in agent-based modeling, and has also in-depth research experience in stochastic and differential equation based modeling.

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9/01/2011

Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems (Computational Neuroscience) Review

Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems (Computational Neuroscience)
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This book is a detailed overview of the computational modeling of nervous systems from the molecular and cellular level and from the standpoint of human psychophysics and psychology. They divide their conception of modeling into descriptive, mechanistic, and interpretive models. My sole interest was in Part 3, which covers the mathematical modeling of adaptation and learning, so my review will be confined to these chapters. The virtue of this book, and others like it, is the insistence on empirical validation of the models, and not their justification by "thought experiments" and arm-chair reasoning, as is typically done in philosophy.
Part 3 begins with a discussion of synaptic plasticity and to what degree it explains learning and memory. The goal here is to develop mathematical models to understand how experience and training modify the neuronal synapses and how these changes effect the neuronal patterns and the eventual behavior. The Hebb model of neuronal firing is ubiquitous in this area of research, and the authors discuss it as a rule that synapses change in proportion to the correlation of the activities of pre- and postsynaptic neurons. Experimental data is immediately given that illustrates long-term potentiation (LTP) and long-term depression (LTD). The authors concentrate mostly on models based on unsupervised learning in this chapter. The rules for synaptic modification are given as differential equations and describe the rate of change of the synaptic weights with respect to the pre- and postsynaptic activity. The covariance and BCM rules are discussed, the first separately requiring postsynaptic and presynaptic activity, the second requiring both simultaneously. The authors consider ocular dominance in the context of unsupervised learning and study the effect of plasticity on multiple neurons. The last section of the chapter covers supervised learning, in which a set of inputs and the desired outputs are imposed during training.
In the next chapter, the authors consider the area of reinforcement learning, beginning with a discussion of the mathematical models for classical conditioning, and introducing the temporal difference learning algorithm. The authors discuss the Rescorla-Wagner rule , which is a trial-by-trial learning rule for the weight adjustments, in terms of the reward, the prediction, and the learning rate. They then discuss more realistic policies such as static action choice, where the reward/punishment immediately follows the action taken, and sequential action choice, where rewards may be delayed. The authors discuss foraging behavior of bees as an example of static action choice, reducing it to a stochastic two-armed bandit problem. The maze task for rats is discussed as an example of sequential action choice, and the authors reduce it to the "actor-critic algorithm." A generalized reinforcement learning algorithm is then discussed, with the rat water maze problem given as an example.
Chapter 10 is an overview of what the authors call "representational learning", which, as they explain, is a study of neural representations from a computational point of view. The goal is to begin with sensory input and find out how representations are generated on the basis of these inputs. That such representations are necessary is based on for example the consideration of the visual system, since, argue the authors, what is presented at the retina is too crude for an accurate representation of the visual world. The main strategy in the chapter is to begin with a deterministic or probabilistic input and construct a recognition algorithm that gives an estimate of the input. The algorithms constructed are all based on unsupervised learning, and hence the existence and nature of the causes must be computed using heuristics and the statistics of the input data. These two requirements are met via the construction of first a generative model and then a recognition model in the chapter. The familiar 'expectation maximization' is discussed as a method of optimization between real and synthetic data in generative models. A detailed overview of expectation maximization is given in the context of 'density estimation'. The authors then move on to discuss causal models for density estimation, such as Gaussian mixtures, the K-means algorithm, factor analysis, and principal components analysis. They then discuss sparse coding, as a technique to deal with the fact that the cortical activity is not Gaussian. They illustrate an experimental sample, showing the activity follows an exponential distribution in a neuron in the inferotemporal area of the macaque brain. The reader will recognize 'sparse' probability distributions as being 'heavy-tailed', i.e. having values close to zero usually, but ones far from zero sometimes. The authors emphasize the difficulties in the computation of the recognition distribution explicitly. The Olshausen/Field model is used to give a deterministic approximate recognition model for this purpose. The authors then give a fairly detailed overview of a two-layer, nonlinear 'Helmholtz machine' with binary inputs. They illustrate how to obtain the expectation maximization in terms of the Kullback-Leibler divergence. The learning in this model takes place via stochastic sampling and occurs in two phases, the so-called "wake and sleep" algorithm. The last section of the chapter gives a general discussion of how recent interest in coding, transmitting, and decoding images has led to much more research into representational learning algorithms. They discuss multi-resolution decomposition and its relationship to the coding algorithms available.

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Theoretical neuroscience provides a quantitative basis for describingwhat nervous systems do, determining how they function, and uncovering the generalprinciples by which they operate. This text introduces the basic mathematical andcomputational methods of theoretical neuroscience and presents applications in avariety of areas including vision, sensory-motor integration, development, learning,and memory.The book is divided into three parts. Part I discusses the relationshipbetween sensory stimuli and neural responses, focusing on the representation ofinformation by the spiking activity of neurons. Part II discusses the modeling ofneurons and neural circuits on the basis of cellular and synaptic biophysics. PartIII analyzes the role of plasticity in development and learning. An appendix coversthe mathematical methods used, and exercises are available on the book's Website.

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