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Elliptic Functions (London Mathematical Society Student Texts)
Elliptic Functions (London Mathematical Society Student Texts)
by J. V. Armitage
Edition: Paperback
Price: $64.99
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3 of 3 people found the following review helpful
4.0 out of 5 stars how to solve any quintic equation in a large number of difficult steps, June 15, 2013
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I purchased this book as a follow-on to John Howie's "Fields and Galois Theory", in which at the end of the tenth chapter, after proving that there is no general solution by radicals of the general quintic equation, he states that the general solution of the quintic equation can be expressed in terms of elliptic modular functions. These are functions of the upper half-complex plane which are doubly periodic and obey certain transformation laws. The tenth chapter of "Elliptic Functions" covers this very nicely. However, I "cheated" in that I first read various entries in Wikipedia on the elliptic modular functions and related topics, in order to get a concise overview of the topic. I don't think I would have been able to achieve such a clear overview using this book, because I would probably have lost sight of the forest for the trees, or perhaps more aptly, for slogging through the dense underbrush, sweat burning my eyes.

In any event, my original plan was to go through the entire book in sequence, but after spending several weeks just going through the first two chapters, I decided to skip straight to chapter ten, which covers the general solution of the quintic. The first two chapters give an introduction to how elliptic functions were first discovered (by Niels Abel) as inversions of elliptic integrals, and how they might have been further developed along these lines, rather than as they were actually further developed, namely as theta functions by Jacobi after Abel's early death. They're interesting but not particularly essential to the core of the topic. If you want a standard exposition of the topic without this historical detour, you can start with the third chapter.

The tenth chapter clearly expounds the use of elliptic modular functions to solve the general quintic equation. It covers two different methods for doing so, of which I only studied the first. I stand in awe of how anyone could be smart enough to have figured this out. It is reasonably straightforward and concise, with references supplied to more detailed treatments. There are some minor typos, annoying but not seriously interfering with the comprehensibility of the text. (The subject matter itself provides sufficient such interference!)

There is a lot of additional theory and application of elliptic modular functions in this book, which look fascinating but which my brain is currently unable to muster the energy to attack. Perhaps some day.

I can recommend this book to anyone who wants to learn more about the theory and applications of elliptic modular functions, including the solution of the general quintic equation. The exposition is reasonably clear, there are good examples, and each chapter section is followed by exercises, the answers to which are unfortunately not provided. A background in complex analysis and linear algebra is required.

Good Luck!

Notes: (reader assumes all responsibility for relying on these notes!)

chapter 1: none

chapter 2:

p. 48 first integral (phi) should be from 0 to x, not 0 to 1

p. 48 right hand side of second equation should read
cos lemn (omega/2 - phi), not cos lemn (1/2 - phi). Omega is defined in (2.46)

Chapter 10

p. 281 (10.14): equation holds for each y sub k and x sub k where the y sub k are the roots of q(y) and the x sub k are the roots of

p. 283 line 11 should be (10.13) not (10.12). Each of the q's (I'll call them q sub i's for convenience) are functions of the respective eta sub i's and therefore of sigma sub i; the sigma sub i's are functions of the set {alpha, beta, gamma, delta, epsilon} and the s sub i's; the s sub i's are functions of the p sub i's; the p sub i's are functions of the xi sub i's; therefore the q sub i's are functions of the set {alpha beta gamma delta epsilon} and the xi sub i's.

p. 286 first line of section 10.4 refers to (10.22), not (10.12)

p. 287 third line refers to (10.26) not (10.32)

p. 287 (10.40) see p. 153

p. 288 line 2 should be (10.26) not (10.32)

p. 289 after exercise 10.4 we have section 10.5 not section 10.4

p. 291 title of section 10.6 should be transformation singular not transformations plural

p. 298 table second row should be tau to -1/tau not tau to tau minus 1/tau

p. 301 first table same mistake

p. 304 line 7 should be u^24 + 2^12 u^-24, not u^24 + 12 u^-24

A First Course in Loop Quantum Gravity
A First Course in Loop Quantum Gravity
by Rodolfo Gambini
Edition: Hardcover
Price: $62.95
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5 of 6 people found the following review helpful
5.0 out of 5 stars the first step on a thousand mile journey, April 28, 2013
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Many non-experts who are interested in physics know that the current theory of gravity, Einstein's general theory of relativity, stands apart from the theories of the other three fundamental forces, the electromagnetic, weak nuclear, and strong nuclear forces, in that the former is a classical theory whereas the latter are quantum theories. Physicists like to unify things, so they would like to have the theory of gravity be a quantum theory as well, so they can then have a "theory of everything". In addition to being satisfying on a philosophical and esthetic level, this would also potentially be able to explain what's going on at the center of black holes and at the "beginning" of the Big Bang, for which general relativity gives a singularity, an unphysical point of infinite energy density indicating that general relativity is no longer properly describing the physical reality in these situations.

One approach to a quantum theory of gravity is the famous (perhaps infamous) "string theory". Another approach, the approach discussed in this book, is loop quantum gravity, in which a more traditional approach to quantizing the gravitational field is employed. In a nutshell, loop quantum gravity is a canonical quantization of a modification of the Hamiltonian formulation of general relativity originally developed by Arnowitt, Deser, and Misner in the early 1960s ("ADM formalism"), said modification being the replacement of the position and momentum variables used by ADM with a new set of variables introduced by Ashtekar in the 1980s. Canonical quantization refers to the quantization procedure based on the Hamiltonian originally developed by Dirac in the 1920s and further developed by him in the 1950s. Loop quantum gravity uses a "loop representation" which has as a key result that there is a minimum allowed length, area, and volume in the universe, i.e, these quantities are quantized. This in turn leads to the result that the universe has a minimum size, and if we play the movie of cosmic expansion backwards, we reach this minimum size before we reach the singularity, and the universe starts to expand again, the so-called "Big Bounce". Also, the relation between the surface area and entropy of a black hole reduces to a relationship between the number of area quanta which make up the surface area and the entropy.

This book is directed at the advanced undergraduate level, and aims to fill a gap in the pedagogy of this subject, which requires much more math and physics background to really understand than any undergraduate would have. As a result, it takes a very simplistic approach and should be considered no more than a very preliminary introduction to the topic.

The first part of the book covers all the basic background concepts needed to understand the program of loop quantum gravity, including general relativity, the generalized Hamiltonian with constraints, and canonical quantization. It then goes on to discuss the ADM formalism, Ashtekar variables, the loop representation, and applications of loop quantum gravity to cosmology and black hole thermodynamics. It ends with a very honest evaluation of the limitations and incompleteness of the theory at the present time. Loop quantum gravity is still a work in progress, and although there have been many important advances in terms of formulating a theory, there is as yet no validation experimentally, nor any obvious path to such a validation.

You will not really understand loop quantum gravity when you finish this book, only understand a very superficial version of it "dumbed down" for the intended readership of this book, but still extremely formidable in terms of its demands on the reader mathematically speaking. If you haven't already had at least an introductory exposure to general relativity, quantum mechanics, Lagrangian and Hamiltonian mechanics, and quantum field theory, you're in for some very tough sledding indeed. If you have this background, after you finish this book you will be prepared to delve more deeply into the topic if you have the time and inclination. Included are references to more advanced texts as well as important review papers to guide this further exploration.

I give this book 5 stars for identifying a pedagogical need, clearly defining the intended readership and the goals, and carefully describing the inevitable limitations and gaps in trying to present such a sophisticated, difficult topic to a relatively unprepared audience.

A Concise History of Solar and Stellar Physics
A Concise History of Solar and Stellar Physics
by Jean Louis Tassoul
Edition: Hardcover
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3 of 4 people found the following review helpful
5.0 out of 5 stars nice review of the history of solar and stellar physics, October 29, 2012
This is an excellent review of the history of solar and stellar physics. It describes the process by which our understanding of the Sun and stars progressed from ignorance to its current state of knowledge. It covers all aspects of the topic and includes some fairly basic mathematical discussions which can be skipped without seriously impairing the reader's understanding of the material. Some relevant math is covered in a series of Appendices. I found this book to be a fascinating overview of solar and stellar physics from the historical vantagepoint, which would be a very useful complement to more standard ahistorical treatments. Brief biographical sketches of many of the important figures in solar and stellar physics are presented. I recommend this book to anyone who is interested in the history of astronomy and physics.

Paleoclimatology, Volume 68, Second Edition: Reconstructing Climates of the Quaternary (International Geophysics)
Paleoclimatology, Volume 68, Second Edition: Reconstructing Climates of the Quaternary (International Geophysics)
by Raymond S. Bradley
Edition: Hardcover
Price: $113.00
41 used & new from $3.02

4 of 5 people found the following review helpful
5.0 out of 5 stars detailed, authoritative review, October 9, 2012
This is an authoritative review of the methods of paleoclimatology research and their application to the reconstruction of climates during the Quaternary Era (the last 2.5 million years). The first two chapters provide an excellent overview of climatology in general and paleoclimatology in particular. The next two chapters are a detailed discussion of the theory, practice, and limitations of the various dating methods used in paleoclimatology, including radioisotope, chemical, paleomagnetic, and biological methods. There follow chapters on the evidence concerning paleoclimates provided by ice cores, marine sediments and corals, non-marine geological and biological evidence, pollen, tree rings, and historical documents. The final chapter discusses climate modelling as applied to paleoclimatology. This book is written at a very advanced level, but I believe many interested laypeople would also find it rewarding, as the author avoids jargon and explains everything from scratch without assuming prior knowledge on the part of the reader. I would recommend this book to anyone interested in geophysics, climate, and climate change. Five stars.

Elementary Climate Physics
Elementary Climate Physics
by F. W. Taylor
Edition: Paperback
Price: $59.50
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1 of 2 people found the following review helpful
5.0 out of 5 stars excellent balanced overview, October 3, 2012
This is an excellent overview of the physics of Earth's climate, including solar radiation, astronomical influences, the physics of the atmosphere and ocean, remote sensing of climate variables, and climate modelling. There is also a chapter on climates of other planets. Portions of it are at the advanced undergraduate level, but someone without an extensive background in math or physics could find much of it understandable and rewarding. Those interested in learning more about the science behind global warming and climate change will find the relevant material presented in a very balanced manner with detailed descriptions of the limitations of our current understanding of climate and its prediction.

Ordinary Geniuses: Max Delbruck, George Gamow, and the Origins of Genomics and Big Bang Cosmology
Ordinary Geniuses: Max Delbruck, George Gamow, and the Origins of Genomics and Big Bang Cosmology
by Gino Segrè
Edition: Hardcover
63 used & new from $0.01

2 of 2 people found the following review helpful
3.0 out of 5 stars Good personal but not scientific histories, September 23, 2012
This is a double biography of two important scientists of the twentieth century, George Gamow and Max Delbruck. Although both were trained as physicists, Delbruck made his most important discoveries in the field of genetics studying bacteriophages, viruses which infect bacteria. Gamow did important work in nuclear physics and was one of the pioneers of the Big Bang theory. I read this book at the same time I was reading the masterly biography of Hans Bethe by Silvan Schweber, and it suffered greatly in comparison. Its author, Gino Segre, is a professor of physics at Penn, and has done a good job of telling his subjects' personal histories, which are interesting in their own right. However the description of their scientific work is much less effective. The author appears to have concluded that his readers would not understand the science, because he treats it in a superficial way, almost as an afterthought. Someone who didn't know anything about Gamow's and Delbruck's work prior to reading this book would have a great deal of trouble understanding what they discovered and why it was important. It is of course no easy task to explain complicated science to laypersons, but that's the challenge any biographer of science faces, and in my opinion the author of this book did not succeed in meeting this challenge. It doesn't seem as though he even tried.

Nuclear Forces: The Making of the Physicist Hans Bethe
Nuclear Forces: The Making of the Physicist Hans Bethe
by Silvan S. Schweber
Edition: Hardcover
Price: $35.00
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5 of 6 people found the following review helpful
5.0 out of 5 stars superlative in every way, September 23, 2012
This is a superlative biography of one of the twentieth century's most important physicists. It is the very model of how to write an authoritative scientific and personal life. The author is a professor of physics and of the history of ideas at Brandeis, and thus possesses the essential expertise to write such a life. He had unparalleled access to primary sources for Bethe's life, and has produced a masterpiece.

The work focuses mainly on the first half of Bethe's long and distinguished career, up until about 1940, which includes the elucidation of the nuclear reactions which power stars for which he won the Nobel Prize. There is a brief concluding chapter which covers Bethe's role in the coming of age of quantum electrodynamics after the war, beginning with the Shelter Island Conference in June 1947, after which Bethe made his celebrated calculation of the Lamb shift which demonstrated the success of the technique of mass renormalization in avoiding the divergences which had plagued the theory.

This biography has everything one could possibly want. It is both remarkably detailed and beautifully organized. The author does an excellent job of partially separating the discussions of his subject's scientific and personal lives so that the two threads can be followed more easily. The scientific portion requires some background in physics and math to be fully understandable, but many of the more technical details are put into footnotes which greatly aids the flow of the narrative. After reading so many biographies of scientists by non-scientists in which the science is treated superficially, it gave me great pleasure to see Bethe's scientific contributions discussed with the depth and authority they deserve. The author is not just a scientist, however. He also handles the details of Bethe's personal life with great skill, insight, and humanity. He has brought his subject fully to life as a person as well as a physicist. Through his personal life we experience the high culture of Germany and Europe before the Nazis, and we become acquainted with many other famous and not-so-famous scientists of the time.

There is so much to enjoy in this book. The Introduction contains some thoughtful comments on the historiography of science. There are 5 appendices with fascinating material on the courses Bethe took as an undergraduate and graduate, his doctoral thesis, the announcement of his defense of his professorial thesis (Habiltationisschrift), and a marvellous brief history of the genesis of quantum mechanics which is the best I've ever seen. There are 81 pages of footnotes which superbly complement and extend the text and indicate the extraordinary depth and breadth of the author's research on his subject, followed by 38 pages of references which reflect the author's great erudition. This is a book which will repay many rereadings.

All in all, the author has done a superlative job of bringing his subject to life personally and professionally, placing him in the historical and cultural context of his times, and explaining the nature and importance of his scientific discoveries. This is as good a biography as I have ever read. It would seem to be the equivalent in its genre of the celebrated "Bethe Bible", the landmark series of authoritative reviews on nuclear physics which Bethe wrote in the mid 1930s and which became the basis for all subsequent progress in the field. If I had more than five stars to give, I would happily do so.

Great Solid State Physicists of the 20th Century
Great Solid State Physicists of the 20th Century
by Julio A. Gonzalo Gonzalez
Edition: Hardcover
Price: $138.00
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2 of 2 people found the following review helpful
3.0 out of 5 stars Interesting but not great, May 14, 2012
This book is a pastiche of things: biographical portraits of some 20th century physicists who made important contributions to solid state physics (the Braggs pere and fils, Peter Debye, John Bardeen, and Lev Landau), an extensive discussion of the scientific contributions of Debye, the Nobel Prize lectures of Bardeen, and a list of all the Nobel Prizes given in Physics up to 2002, with an explanation of the ones relating to solid state physics. It has some interesting things in it, but it's not terribly well edited, and the translation of one of the contributions about Lev Landau from the original Polish is hilariously rough. There are loads of typos, although the ones in Bardeen's lectures for the most part can be figured out without too much difficulty. On the other hand, his lectures are somewhat mystifying to someone without a solid background in the topics he's talking about, and no effort is made elsewhere in the book to clarify what he's saying.

I would recommend this book to anyone who is curious about solid state physics and some of the major contributors to its advancement. I would certainly not recommend buying it at the quoted price of $104! If you can't find it in a library, you'd best skip it. It won't be a great loss.

Turing's Cathedral: The Origins of the Digital Universe
Turing's Cathedral: The Origins of the Digital Universe
by George B. Dyson
Edition: Hardcover
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5 of 6 people found the following review helpful
3.0 out of 5 stars OK, not great, a bit disappointing., May 8, 2012
I bought this book after watching Dyson discuss it on cable TV. I thought it would be fascinating to learn more about how John von Neumann and his cadre built the first "modern" digital electronic computer, the MANIAC, the granddaddy of all subsequent computers, after WWII. The impetus was to do calculations necessary to design the H-bomb (plus the impetus to just build it, period). The book describes in detail the setting (the Institute for Advanced Studies in Princeton, NJ) and the characters (first among them von Neumann, for whom many proofs of his genius are cited). There's a fair amount of detail about the technical aspects of the computer as well. But what is missing, and what I found most disappointing, is the lack of an overall description of the theory behind digital electronic computing. Perhaps Dyson felt it would be too technical for most readers, but he could have put it in an Appendix, and he certainly could have made room, because it's not until Chapter 5 that he actually starts describing the project itself. In fact, he throws in the history of the founding and development of New Jersey, the Revolutionary War campaign which included the Battle of Princeton, and all sorts of other interesting but marginally relevant at best information.

For those readers who simply want lots of interesting historical tidbits and aren't too interested in the details of how the MANIAC actually worked, this book will be quite enjoyable. For those readers who hoped for a more in-depth discussion of the theory behind the project, they'll have to look elsewhere.

Heisenberg's War: The Secret History Of The German Bomb
Heisenberg's War: The Secret History Of The German Bomb
by Thomas Powers
Edition: Paperback
Price: $19.73
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4 of 5 people found the following review helpful
3.0 out of 5 stars spoiler alert!, April 23, 2012
Spoiler alert! Don't read the rest of the review if you want to find out what the author thinks was Heisenberg's contribution to the German nuclear bomb effort.

In this thoroughly researched and documented book, the author makes the case that Heisenberg, one of the greatest physicists of the 20th century and far and away the most brilliant German physicist who chose to remain in Germany throughout the Nazi period, effectively sabotaged the German nuclear research program before and during WWII.

I found the first few chapters, in which the author describes the process by which physicists realized that a fission bomb could be made, and the last 2 chapters, in which the author convincingly argues that Heisenberg deliberately concealed this knowledge from the Nazis and sabotaged the German nuclear program through a passive-aggressive campaign of "go slow, do little", to be worthwhile reading.

In between there are hundreds of pages of less riveting material which only someone truly fascinated by the topic would be interested in. I would guess that of the 483 pages in this book, only about 150 repay the reader's attention.
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