Perturbation Methods

This is a concise and precise text. This book amplified my understanding of asymptotics (which is limited) and is an excellent complement to Bender and Orszag.

I found the discussion of the limitation of convergence of series (unless it is extremely rapid), the value of divergent series, the art of the approach by analyzing the behaviour (order) of the functions within sections of the domain helpful conceptually. The more complex middle and end of the book are subjects I hope to return to (this is a personal limitation not a criticism of the book). The first half, however, significantly improved my understanding.

Electromagnetic Vibrations, Waves, and Radiation

This book is some 40 years old. I, however, found it very instructive and enjoyable. It forms part of the Syllabus of the Lewin MIT 8.03 course. I now understand why.  The book is clearly written. The book is filled with worked examples that are interesting and flow from the text. There is consistent explicit use of units and after examples are used they are put in context of the text,

The book covers: oscillations, waves, electromagnetic radiation, transmission of electromagnetic radiation, reflection, refraction, scattering, interaction of electromagnetic fields with matter, diffraction/interference and coherence. This journey is based on classical physics. The authors, however, are quick to point out the limitations of this approach but defend t he practice based on the reasonableness of the approximation for the scale of domain under consideration and the use of expectation values.

The color of the sky, why gold is yellow and silver shiny gray and the frequency dependence of a number of other phenomena as discussed. One example illustrated the tissue penetration of microwave energy into biological tissues. These were all delightful examples.

I am certain that I am completely ignorant of a number of limitations of this book. I also cannot hope to have deep insights into the necessity of quantum mechanics for the nanoscale understanding. However, the clarity and well organized nature of this 40 year old book has certainly diminished my ignorance in a constructive manner.

 

 

Little steps

I have been walking to improve fitness (and perhaps mental health).  An iPhone app Runkeeper allows you to record your GPS position against time and calculates your average speed, pace and “caloric use” (the latter based on a lot of assumptions). The app stores your data and you can review it on web site runkeeper.com.  It allows you to export the data (you go to runkeeper.com/settings  and go to “Export Data” ). A zip file containing the GPS data in GPX format as well as some CSV files containing some of the calculated data.

The web site displays the data in am attractive format. As an exercise in Mathematica I aimed to extract the data from the GPX files and “play” with the data. The graphic that follows is an example.

Not so hidden in plain sight

I have been musing with the Image Processing features of Mathematica. A home maintenance projet provided an opportunity: painting over a repaired wall defect.

I took before and after photographs. The position was not exact and there were some shadowing realted to time of day. However, this still allowed processing. The painting with computer color matched at the hardware store and the whole region was repainted and roughly two layers were applied.

The exercise was to see if the underlying defect could still be detected despite the painting. I used ColorConvert, HistogramTransform (using Manipulate and Slider2D to modify image) and then GradientOrientationFilter.

The results are shown in the graphic below. The top row are the cropped before and after photographs and the corresponding processed images are shown directly below. Note the square defect is easily discernible and the second image of the second row.

Stability, Instability and Chaos

This was a good book. Ir was above my expertise but was instructive nonetheless. It covered the usual value of linearization. It also covered perturbation methods then bifurcation theory (saddlenode, transcritical, pitchfork and Hopf) and then discusses chaos through the horseshoe map concept. I found the discussion of uniomodal maps instructive, particularly unimodal maps without chaos.

I rushed through this book. It was above my head and my rushing reflected my mental state and time commitments. However, despite these caveats with respect to my review, it was a helpful book and motivates me to learn more. The author focuses on concepts and proofs a conceptually constructed. The bibliography is useful.

 

In a Spin

I have been contemplating the Coriolis effect. In the above diagram the blue axes are an inertial reference frame with the blue arrows representing the unit vectors. The red arrows show an instant from a rotating reference frame.

The following shows the derivation of the Coriolis and centrifugal “fictitious” forces terms. Considere.g. ( )

Mathematical Biology: I. An Introduction

This book is a classic. I basically skimmed through this (partly a reflection of a current dificulty with focus and concentration). This book covers a large number of areas: simple population models, sex determination in crocodiles, mathematical models of marriage, biological oscillators, diffusion and chemotaxis, wave phenomena in biological systems and finally a brief discussion of fractals in biology (uses and misuses).

There is a systematic exploration of these various models and the important insights from linearization, perturbation methods for stability analysis was repeatedly illustrated. The graphics with comparisons to experimental data were well chosen and demonstrated.

The book highlighted to me deeo deficits in my knowledge and forms motivation for reducing my ignorance. I am looking forward to the Second Volume which explores spatial patterns and excitable media.

The Art of Modeling Dynamic Systems: Forecasting for Chaos, Randomness and Determinism

I was disappointed by this book. It is ambitious in its title and aims. It covers an enormous range of topics. The book is a philosophical treatise from an experienced practitioner. The scope of material though broad is more superficial than I expected and the structure more disconnected than I anticipated.

There are some broad themes in the book. The author encourages the reader to look at dynamical systems within a classification system relating to level of randomness. The other major theme is a constructivism over formality in applied Mathematics. This recurrent refrain was unnecessary. The repetition whenever an example of unsatisfactory nature of the concept of infinity in real world context of finite precision was not edifying.

I was looking forward to deeper insights. The book did provide me insight into many of my own deficiencies of education. However, it is a book that kept pointing from itself to somewhere else without providing (me) sufficient depth and connection to make the journey worthwhile. This may reflect the patchiness of my own education. However, the books title overstates its content for this reader.

The Theoretical Minimum

I read this book cover to cover.  The somewhat (deliberately) ambiguous title refers to the minimum amount of theoretical foundation necessary to start doing physics. This book essentially covers classical mechanics and is based on the successful video lecture series of Professor Susskind (http://www.newpackettech.com/Resources/Susskind/Susskind.htm).

The book is written clearly and concisely. There is a calculus and vector calculus refresher course: that necessary for the exposition. The chapter preambles are amusing entres to the content. There is a rapid progression from Newton;s law, to the Lagrangian and Hamiltonian formulations of mechanics. The relationship between symmetry and conservation laws are clearly discussed and illustrated. The utility of Poisson brackets was shown repeatedly.  Finally, gauge transformation and gauge invariance are discussed in the context of electromagnetic force: a clear derivation of the Lorentz force from the magnetic potential  in the Hamiltonian framework. The author hints at the links between this “theoretical minimum” and Quantum Mechanics and General Relativity.

I have read a number of books on these topics but found the clarity and conciseness wonderful.

I look forward to the video lectures for these area appearing in print and I will be immersing myself in these also.

Invisible in the Storm

 

I enjoyed this book. It is a systematic account of the development of weather forecasting. The writing style is engaging there is a nice balance of historical and biographical detail as well as explanation of the relevant concepts from Mathematics and Physics. It is deliberately limited on the technical aspects.  I enjoyed the journey:  from developing physical model of the various aspects of weather; the remarkable power of simpler models in large scale phenomena in )the pre-computer era) in  motivating ongoing pursuit of the ‘holy grail’ of accurate prediction;  the convergence of technologies to  increase accuracy of measurement with  increasing scope and level of detail of measurement  and the increasing power of electronic computing unmasking the problem of chaos; and the need for more sophisticated mathematics to ‘tame the chaos’: to understand the constraints, to better characterize attractors, to develop horizons of prediction and to use statistical (probabilistic) methods such as  ensemble forecasting to place bounds on the uncertainties.

The book shows how complex and amazing the weather remains. The human endeavor to understand it has involved different personalities, different viewpoints,  changing capabilities. The book describes the importance of non-linearities in explaining the complex and chaotic observed phenomena…it strikes me that although the book presents the a systematic and incremental increase in understanding, that the complex interactions of people, time and technology make progress non-linear.  The birth of meteorology and its connections to the Mathemticsm Physics and Astronomy is fascinating. The book is another example of the unreasonable effectiveness of Mathematics in explaining the world, as well as the vision, persistence and  hard work of  Mathematicians, Physicists, and Metereologists.