Hello!
I have been insanely busy the last week. Had a great Thanksgiving!
Just completed Chapter 9 in the book project, this is a survey of linear algebra. I am about to start in on chapter 10, Leonard Susskind's classical mechanics Lecture 5.
I am also developing the program for the Institute for Scientific Computing Summer School program. I am also working on developing courses for teaching Mathematica 8 online.
That's all for now.
Tuesday, November 30, 2010
Tuesday, November 16, 2010
Mathematica 8 Is HERE!!!
The new version of Mathematica has arrived! I have been playing with it for a day now. It is a substantial upgrade from version 7. I will write more about the new features as I play with them. One new feature is the integration with Wolfram Alpha allowing the user to use free-form commands to produce results without the use of syntax; this is limited and for people who know Mathematica, it will not be much of an addition, but it does allow people who do not know the system to begin exploring. The results also include the correct Mathematica syntax for what it thinks you want.
More later.
More later.
Recent Papers of Interest 16 November 2010
Here are some ArXiv papers I found interesting of late.
1) Takami Kuroda and Hideyuki Umeda, Three Dimensional Magneto Hydrodynamical Simulations of Gravitational Collapse of a 15Msun Star
This is an important paper for those of us who study compact objects. It begins with a nice overview of the theory of core-collapse supernovae (CCSN) and the efforts to model these. The next two sections details the code used, both MHD and GR. This is followed by the adaptive mesh scheme used. Finally they discuss the tests of the code. This accounts for half of the paper and makes a very nice overview of modeling MHD in a GR environment. The results are very nice, great paper!
2) Manjit Bhatia, P. Narayana Swamy, Elementary Quantum Mechanics in a Space-time Lattice
This paper explores the particle-in-a-box problem under the assumption of a fundamental length scale forming a spacetime lattice. This is an interesting approach.
3) Oleg Korobkin, Ernazar B. Abdikamalov, et al, Stability of general-relativistic accretion disks,
This paper is a thorough treatment of a model of unstable accretion disks near a black hole.
4) T. Fischer, I. Sagert, et al, The revival of an explosion mechanism of massive stars - the quark hadron phase transition during the early post bounce phase of core collapse supernovae,
This is a long paper (30 pages) explores the bounce during a supernova core collapse caused by the so-called quark hadron phase transition. The introduction is a nicce overview of the topic and its history. The second section, no pun intended, forms the theoretical core of the paper, describing the hybrid equation of state and other issues. The third section describes the simulation scheme. The fourth section explores the results of the model runs. The conclusion is that the quark matter description leads to early onset of deconfinement, a quark-gluon plasma. A very interesting paper!
1) Takami Kuroda and Hideyuki Umeda, Three Dimensional Magneto Hydrodynamical Simulations of Gravitational Collapse of a 15Msun Star
This is an important paper for those of us who study compact objects. It begins with a nice overview of the theory of core-collapse supernovae (CCSN) and the efforts to model these. The next two sections details the code used, both MHD and GR. This is followed by the adaptive mesh scheme used. Finally they discuss the tests of the code. This accounts for half of the paper and makes a very nice overview of modeling MHD in a GR environment. The results are very nice, great paper!
2) Manjit Bhatia, P. Narayana Swamy, Elementary Quantum Mechanics in a Space-time Lattice
This paper explores the particle-in-a-box problem under the assumption of a fundamental length scale forming a spacetime lattice. This is an interesting approach.
3) Oleg Korobkin, Ernazar B. Abdikamalov, et al, Stability of general-relativistic accretion disks,
This paper is a thorough treatment of a model of unstable accretion disks near a black hole.
4) T. Fischer, I. Sagert, et al, The revival of an explosion mechanism of massive stars - the quark hadron phase transition during the early post bounce phase of core collapse supernovae,
This is a long paper (30 pages) explores the bounce during a supernova core collapse caused by the so-called quark hadron phase transition. The introduction is a nicce overview of the topic and its history. The second section, no pun intended, forms the theoretical core of the paper, describing the hybrid equation of state and other issues. The third section describes the simulation scheme. The fourth section explores the results of the model runs. The conclusion is that the quark matter description leads to early onset of deconfinement, a quark-gluon plasma. A very interesting paper!
Friday, November 5, 2010
Project Updates
I am about to finish chapter six of the book project.
We (MAST) are planning a Summer School on Computational Physics; this will tentaively be held at the University of Wisconsin, Madison, Department of Physics and will be held from 15 June 2011 to 15 August 2011. Check out the MAST website for details.
We are exploring the possibility of Mathematica training over the web. This will include beginner training all the way up to advanced applications.
We (MAST) are planning a Summer School on Computational Physics; this will tentaively be held at the University of Wisconsin, Madison, Department of Physics and will be held from 15 June 2011 to 15 August 2011. Check out the MAST website for details.
We are exploring the possibility of Mathematica training over the web. This will include beginner training all the way up to advanced applications.
Friday, October 29, 2010
The Week of 24-30 October in Astrophysics
Here are four interesting papers from arXiv astrophysics:
1) Francesca Valsecchi, Evert Glebbeek, et al, "Formation of the black-hole binary M33 X-7 via mass-exchange in a tight massive system",
M-33 X-7 is a mystery that this paper proports to solve, maybe. This binary object seems to be a ~16 solar mass black hole orbiting a ~70 solar mass hydrogen-rich O-type companion star. This binary system has a period of ~3.5 days. Up until now, no model explained all of the weird things about this system. This paper contrives a set of circumstances to explain it all.
2) Shigehiro Nagataki, "Rotating BHs as Central Engines of Long GRBs: Faster is Better,"
This paper relates the rotational rate of a black hole to the strength of any relativistic jet produced by it. This is based on numerical relativity calculations using a magnetohydrodynamic code. Four different models are run.
3) Joan M. Centrella, John G. Baker, "Black-hole binaries, gravitational waves, and numerical relativity,"
This is a very nice review of black hole modeling and numerical relativity. Of course, one of the primary applications is the support of gravitational wave detectors.
4) Rachid Ouyed, Mathew Kostka, et al, "Quark nova imprint in the extreme supernova explosion SN 2006gy: the advent of the Quark Star,"
A fascinating possibility is that at the center of neutron stars a special form of matter is produced, one where there is a region of free quarks surrounded by a screen of gluons. This is called a quark-gluon plasma. Such a state of matter was speculative for many decades, but recent results from RHIC (Relativistic Heavy Ion Collider) have given indications that this has been achieved. The authors of this paper suggest that the supernova SN2006gy was in fact a supernova that included a boost to the explosion through the release of energy by the conversion of neutons at the center of the developing neutron star, converting the center into more stable strange quark matter. If true, this is a significant event!
1) Francesca Valsecchi, Evert Glebbeek, et al, "Formation of the black-hole binary M33 X-7 via mass-exchange in a tight massive system",
M-33 X-7 is a mystery that this paper proports to solve, maybe. This binary object seems to be a ~16 solar mass black hole orbiting a ~70 solar mass hydrogen-rich O-type companion star. This binary system has a period of ~3.5 days. Up until now, no model explained all of the weird things about this system. This paper contrives a set of circumstances to explain it all.
2) Shigehiro Nagataki, "Rotating BHs as Central Engines of Long GRBs: Faster is Better,"
This paper relates the rotational rate of a black hole to the strength of any relativistic jet produced by it. This is based on numerical relativity calculations using a magnetohydrodynamic code. Four different models are run.
3) Joan M. Centrella, John G. Baker, "Black-hole binaries, gravitational waves, and numerical relativity,"
This is a very nice review of black hole modeling and numerical relativity. Of course, one of the primary applications is the support of gravitational wave detectors.
4) Rachid Ouyed, Mathew Kostka, et al, "Quark nova imprint in the extreme supernova explosion SN 2006gy: the advent of the Quark Star,"
A fascinating possibility is that at the center of neutron stars a special form of matter is produced, one where there is a region of free quarks surrounded by a screen of gluons. This is called a quark-gluon plasma. Such a state of matter was speculative for many decades, but recent results from RHIC (Relativistic Heavy Ion Collider) have given indications that this has been achieved. The authors of this paper suggest that the supernova SN2006gy was in fact a supernova that included a boost to the explosion through the release of energy by the conversion of neutons at the center of the developing neutron star, converting the center into more stable strange quark matter. If true, this is a significant event!
Wednesday, October 27, 2010
Update
Well, we just turned in our NSF proposal to develop the MAST Institute for Scientific Computing. We have letters of interest from Jack Horner, a software engineer from Los Alamos, and from the Unversity of Wisconsin, Madison, Department of Physics.
This is a very cool project!
Now that the propsal is in, I can get back to work on the book!
This is a very cool project!
Now that the propsal is in, I can get back to work on the book!
Monday, October 18, 2010
Theoretical Physics Papers #2
Only two really jumped out at me today:
1) Pablo Arrighi, Jonathan Grattage, A Quantum Game of Life,
I think most of us are familiar with the Game of Life, an example of cellular automata (CA) evolving into structures, some of which seem to be coherent. The hope is that such systems will both demonstrate evolving complexity, and reveal some of the rules for such evolution. This paper describes a universal quantum CA (QCA) in three dimensions that can simulate all other three-dimensional QCAs. One application of this is to study the flow of qubits, another is to simulate quantum gates. This is a very important paper to anyone interested in simulating a quantum computer.
2) Jonathan Hackett, Louis H. Kauffman, Octonions,
This paper is a review of the idea of the octonion, a quaternion with the proprty that a lateral rotation brings about a change in the forward orientation of the quaternion and that this is non-associative. Turning spin up to spin down would be an example of the change in orientation. This is sometimes called the Dirac belt trick.
1) Pablo Arrighi, Jonathan Grattage, A Quantum Game of Life,
I think most of us are familiar with the Game of Life, an example of cellular automata (CA) evolving into structures, some of which seem to be coherent. The hope is that such systems will both demonstrate evolving complexity, and reveal some of the rules for such evolution. This paper describes a universal quantum CA (QCA) in three dimensions that can simulate all other three-dimensional QCAs. One application of this is to study the flow of qubits, another is to simulate quantum gates. This is a very important paper to anyone interested in simulating a quantum computer.
2) Jonathan Hackett, Louis H. Kauffman, Octonions,
This paper is a review of the idea of the octonion, a quaternion with the proprty that a lateral rotation brings about a change in the forward orientation of the quaternion and that this is non-associative. Turning spin up to spin down would be an example of the change in orientation. This is sometimes called the Dirac belt trick.
Friday, October 15, 2010
Some Recent Papers on Theoretical Physics and Mathematical Physics
1) Gary Horowitz, Surprising Connections Between General Relativity and Condensed Matter,
My only objection to this paper is that it seems to be tied to anti-deSitter space (AdS). Proving things in AdS may be easier than proving them in general relativity (AdS is essentially general relativity with a negative cosmological constant), but it cannot gaurantee applicability to the real world (since we seem to have a positive cosmological constant). The paper seems to rely on an observation that black holes in AdS behave like thermal systems of one dimensional lower, and that the string theory/holographic principle of black holes allows you to extract features from general relativity that look a lot like superconductivity.
2) Geoffrey Lovelace, Mark. A. Scheel, Bela Szilagyi, Simulating merging binary black holes with nearly extremal spins,
This fascinating paper describes the merger of two rapidly rotating black holes.
My only objection to this paper is that it seems to be tied to anti-deSitter space (AdS). Proving things in AdS may be easier than proving them in general relativity (AdS is essentially general relativity with a negative cosmological constant), but it cannot gaurantee applicability to the real world (since we seem to have a positive cosmological constant). The paper seems to rely on an observation that black holes in AdS behave like thermal systems of one dimensional lower, and that the string theory/holographic principle of black holes allows you to extract features from general relativity that look a lot like superconductivity.
2) Geoffrey Lovelace, Mark. A. Scheel, Bela Szilagyi, Simulating merging binary black holes with nearly extremal spins,
This fascinating paper describes the merger of two rapidly rotating black holes.
Book Review: Quantum Mechanics by Landau and Lifshitz
Time for another book review, this time volume three of the Course in Theoretical Physics by Landau and Lifshitz. This book is available for sale in the bookstore below. This volume is titled, "Quantum Mechanics (Non-relativistic theory)," and covers, in one volume, standard non-relativistic quantum mechanics, atomic and molecular physics, group theory, and nuclear physics.
Wednesday, October 13, 2010
New developments at MAST
MAST (Madison Area Science and Technology) will be seeking funding from the NSF to open a computational science center and, if funded, it will allow us to offer internships for High School Seniors, Citizen Scientists, Undergraduate Students, Graduate Students, and even Post-docs who are interested in computational science, including, but not limited to: astrophysical fluid dynamics, compact objects, accretion disks, relativity, cosmology, mathematical physics, numerical methods, fluid dynamics, atmospheric modeling, shock wave phenomena, chaotic dynamical systems, scattering, quantum information theory, solution methods for ODEs and PDEs, and bioinformatics. We will handle all necessary educational requirements to bring the students up to speed (including training in Mathematica).
We will even be accepting a number of foreign internships.
The proposal will be submitted to the NSF by 27 October of this year.
We will even be accepting a number of foreign internships.
The proposal will be submitted to the NSF by 27 October of this year.
Monday, October 11, 2010
Book Project Milestone!
The classical mechanics book I am writing with Leonard Susskind has reached a milestone. As of now the page count stands at just over 100 pages! That is four chapters completed, the first two lectures, an introduction to calculus, and an introduction to Newtonian mechanics.
Saturday, October 9, 2010
The Inner Horizon of a Black Hole
Everyone has, by now, heard of black holes. Astronomically, they are likely to have been supermassive stars that exploded, whose remnant gravitationally collapses to a size the is smaller than a function of its mass called the Schwarzschild radius,
where M is the mass of the star. Once collapse passes this point the star is gone and it becomes a new thing, a black hole. Within the Schwarschild radius nothing can escape from the black hole, the escape velocity grows to be more than the speed of light. There is a boundary that we call the event horizon at the Schwarzschild radius, this is what separates the normal spacetime from the weirdness that happens near the black hole.
There are circumstances that can form another surface inside the event horizon. A charged black hole (Riessner-Nordstrøm) can form an inner horizon caused by a charge barrier. A spinning black hole (Kerr) can form an inner horizon due to transfer of angular momentum.
All horizons are places where, in effect, time stops with respect to an outside observer. The inner horizon must also be traversed by anything that falls into the black hole on its way to the singularity predicted to be at its center. The inner horizon gives the possibility of avoiding the singularity, but there seems no way to gaurantee a trajectory through the black hole to avoid the inner horizon prior to crossing the event horizon (you could still cross the inner horizon).
Even if you avoided the inner horizon, it is unlikely that anything other than subatomic particles will survive the experience.
where M is the mass of the star. Once collapse passes this point the star is gone and it becomes a new thing, a black hole. Within the Schwarschild radius nothing can escape from the black hole, the escape velocity grows to be more than the speed of light. There is a boundary that we call the event horizon at the Schwarzschild radius, this is what separates the normal spacetime from the weirdness that happens near the black hole.
There are circumstances that can form another surface inside the event horizon. A charged black hole (Riessner-Nordstrøm) can form an inner horizon caused by a charge barrier. A spinning black hole (Kerr) can form an inner horizon due to transfer of angular momentum.
All horizons are places where, in effect, time stops with respect to an outside observer. The inner horizon must also be traversed by anything that falls into the black hole on its way to the singularity predicted to be at its center. The inner horizon gives the possibility of avoiding the singularity, but there seems no way to gaurantee a trajectory through the black hole to avoid the inner horizon prior to crossing the event horizon (you could still cross the inner horizon).
Even if you avoided the inner horizon, it is unlikely that anything other than subatomic particles will survive the experience.
Thursday, October 7, 2010
Book Project
I have just finished lecture 2 of Classical Mechanics, that completes three chapters total. This chapter develops the ideas of conservation of energy, conservation of momentum, and the principle of least action.
Tuesday, October 5, 2010
Musings on the Initial Singularity
For those of you who do not know, the so-called initial singularity is an artifact of Big Bang cosmology implied by the expansion of the universe. If all points in spacetime are moving away from each other (not the matter within spacetime, just the spacetime envirnment-sort of like a river flowing, things in the river flow with it, but are not pulled apart), then it seems reasonable that everything had to start at a point. This time reversal leading to something that looks like a black hole was the subject of Stephen Hawking's thesis around 40 years ago. Other people have studied this time reversal process and have speculated about the dynamics of this singularity. The only problem is that they treat it as a spacetime singularity, like that of a static black hole.
The problem is, it could not have been a spacetime singularity! Spacetime did not yet exist before the Big Bang happened, so there could not have been a spacetime singularity.
This is, of course, one of the reasons why we need a truly quantum theory of gravity. This realization destroys any real chance for unification of the forces, since without spacetime curvature there can be no gravitation; and radiation pressure becomes dominant during the inflationary period. At the moment of the Big Bang, the strong, weak, and electromagnetic forces cause a rapid expansion. Spacetime starts to expand, but without a quantum theory of gravity we cannot understand this process since it is entirely in the regime of quantum mechanical distances, even if the energy densities are huge.
A very deep problem, and one that can only be solved by quantum gravity.
The problem is, it could not have been a spacetime singularity! Spacetime did not yet exist before the Big Bang happened, so there could not have been a spacetime singularity.
This is, of course, one of the reasons why we need a truly quantum theory of gravity. This realization destroys any real chance for unification of the forces, since without spacetime curvature there can be no gravitation; and radiation pressure becomes dominant during the inflationary period. At the moment of the Big Bang, the strong, weak, and electromagnetic forces cause a rapid expansion. Spacetime starts to expand, but without a quantum theory of gravity we cannot understand this process since it is entirely in the regime of quantum mechanical distances, even if the energy densities are huge.
A very deep problem, and one that can only be solved by quantum gravity.
Monday, October 4, 2010
Progress Report for the Last Week
Things I am Working On:
1. Classical Mechanics book: Almost done with lecture 2, should finish today. Thinking about writing a small chapter on dimensional analysis to derive force laws. After lecture 2 I will start on a chapter dealing with vector analysis, and with lecture 3.
2. Book reviews: Quantum Mechanics (Landau and Lifshitz), Advanced University Physics (Rogalski and Palmer).
3. Infinity Computing: Not doing much, though will start to write some code this week.
4. Quantum Computing: Beginning work on a Quantum Computer Simulator.
That's all I have been working on from a theoretical point of view.
1. Classical Mechanics book: Almost done with lecture 2, should finish today. Thinking about writing a small chapter on dimensional analysis to derive force laws. After lecture 2 I will start on a chapter dealing with vector analysis, and with lecture 3.
2. Book reviews: Quantum Mechanics (Landau and Lifshitz), Advanced University Physics (Rogalski and Palmer).
3. Infinity Computing: Not doing much, though will start to write some code this week.
4. Quantum Computing: Beginning work on a Quantum Computer Simulator.
That's all I have been working on from a theoretical point of view.
Thursday, September 30, 2010
Blog Update
I have not posted for a couple of days because I am hard at work on transcribing Lecture 4 of Classical Mechanics. This is about symmetry and conservation laws.
I am also working on a book review for a book I received for review from CRC Press, Advanced University Physics.
Lots on my plate this week.
I am also working on a book review for a book I received for review from CRC Press, Advanced University Physics.
Lots on my plate this week.
Friday, September 24, 2010
Book project update
I have just completed a 15 page chapter on calculus that covers limits, derivatives, integrals, differential equations, infinite series, and partial derivatives. This chapter is for the proposed classical mechanics book I am working on with Leonard Susskind.
I took a very minimalist approach to the subject. If you want to see how I crammed so much into so little, wait for the book... :-)
Actually, some of this material will likely find its way into my column (and I took some of it from my existing columns).
I took a very minimalist approach to the subject. If you want to see how I crammed so much into so little, wait for the book... :-)
Actually, some of this material will likely find its way into my column (and I took some of it from my existing columns).
Wednesday, September 22, 2010
The Grand Design - The Straight Story
I have just completed reading, "The Grand Design," by Stephen Hawking and Leonard Mlodinow. It was a good book. I do not agree with everything they write, but then it is for a lay audience and not for specialists. This book is available for sale from the bookstore below.
Let me state one thing right up front and make it completely clear: Nowhere in the book do the authors state that God does not exist! Period! As I suspected (see a previous post) all the authors state is that the current physical theories are such that no appeal to God is necessary to explain the existence of the universe. In other words, the physical laws are rich enough to explain how the universe came into being out of nothing.
The one objection that I have to the book is the reliance on M-theory, this is a theory that I do not understand; so it is difficult for me to critique it. I know just enough to get myself into trouble, so I will refrain.
I enjoyed the book a lot! if you are interested in physics, this is worth getting.
Ignore all the hype, it is nothing but lies! No one who has read the book can truthfully state that the authors said that God does not exist. It is a very nioe history of the development of theories of the universe of all types.
Let me state one thing right up front and make it completely clear: Nowhere in the book do the authors state that God does not exist! Period! As I suspected (see a previous post) all the authors state is that the current physical theories are such that no appeal to God is necessary to explain the existence of the universe. In other words, the physical laws are rich enough to explain how the universe came into being out of nothing.
The one objection that I have to the book is the reliance on M-theory, this is a theory that I do not understand; so it is difficult for me to critique it. I know just enough to get myself into trouble, so I will refrain.
I enjoyed the book a lot! if you are interested in physics, this is worth getting.
Ignore all the hype, it is nothing but lies! No one who has read the book can truthfully state that the authors said that God does not exist. It is a very nioe history of the development of theories of the universe of all types.
Monday, September 20, 2010
Secret Weapons of The Theoretical Physicist
Tonight I am thinking about a set of the most powerful techniques of the theoretist, classically called back-of-the-envelope calculations. These consitute two broad categories of methods.
The first is the method of estimation, where you use some basic reasoning abilities to determine a rough estimate of the quantities you are considering. For example, you want to estimate the number of hairs on someone's head. You think about it and decide that if you knew the area of the part of the head covered by hair, then knew the average area of a hair, you could figure it out. Then you make the necessary assumptions about the quantities (either by looking up the information, or by figuring it out for yourself).
The second is the ability to check the correctness of equations and to derive new equations by studying the relevant units of the quantities. All terms of any equation must have the same units. When you chaeck the terms, any that do not have the correct units are just wrong. By including powers of each quantity in each term and making sure all of the units come out the same, you invent a system of algebraic equations that can be solved. This gives you the necessary powers of each variable and shows you the structure of each term within a factor of a constant of proportionality. In this way you can learn what each term must look like. This allows you to invent new equations if you know the units involved. This is called dimensional analysis.
I will be writing about these ideas in my column in the coming weeks. I will also be discussing this in the Classical Mechanics book I am writing with Leonard Susskind.
The first is the method of estimation, where you use some basic reasoning abilities to determine a rough estimate of the quantities you are considering. For example, you want to estimate the number of hairs on someone's head. You think about it and decide that if you knew the area of the part of the head covered by hair, then knew the average area of a hair, you could figure it out. Then you make the necessary assumptions about the quantities (either by looking up the information, or by figuring it out for yourself).
The second is the ability to check the correctness of equations and to derive new equations by studying the relevant units of the quantities. All terms of any equation must have the same units. When you chaeck the terms, any that do not have the correct units are just wrong. By including powers of each quantity in each term and making sure all of the units come out the same, you invent a system of algebraic equations that can be solved. This gives you the necessary powers of each variable and shows you the structure of each term within a factor of a constant of proportionality. In this way you can learn what each term must look like. This allows you to invent new equations if you know the units involved. This is called dimensional analysis.
I will be writing about these ideas in my column in the coming weeks. I will also be discussing this in the Classical Mechanics book I am writing with Leonard Susskind.
Sunday, September 19, 2010
An Apology and a Reminder
I just deleted a comment on a previous post, I will not tell who sent it, or exactly what it was about. It had the broad topic of mysticism and religion. While I am happy to discuss such things, this is not the forum for such discussions.
Please do not send comments about such topics to me. I will not publish them.
If you want to discuss physics, that is fine. If you think you have proved Newton, Galileo, Einstein, Bohr, Planck, Feynman, Hawking, etc. wrong, tell the New York Times, not me! I sincerely doubt that your theory will meet my standards...
If you have interesting ideas about physics, that is fine. I have an open mind, but it is not so open that I am liable to fall in... I also am completely skeptical of wild claims (ask anyone that has been a speaker at a physics colloqium or seminar that I have attended...).
This blog is about theoretical physics education and research. Let's limit it to that.
George
Please do not send comments about such topics to me. I will not publish them.
If you want to discuss physics, that is fine. If you think you have proved Newton, Galileo, Einstein, Bohr, Planck, Feynman, Hawking, etc. wrong, tell the New York Times, not me! I sincerely doubt that your theory will meet my standards...
If you have interesting ideas about physics, that is fine. I have an open mind, but it is not so open that I am liable to fall in... I also am completely skeptical of wild claims (ask anyone that has been a speaker at a physics colloqium or seminar that I have attended...).
This blog is about theoretical physics education and research. Let's limit it to that.
George
Friday, September 17, 2010
Downplaying the Hawking-Mlodnow "Grand Design" Controversy
People, mostly reporters, have been driving me crazy again. Specifically, they are making statements about the new book, "Grand Design," by Stephen Hawking and Leonard Mlodnow without quoting the book directly. Now, let me say up front that I have not yet read the book. I have heard the statements that Hawking has made in the past along the lines of, "God is not necessary to the creation of the universe." At no point, including in this book, has either author made that statement that God does not exist, nor that God did not create the universe.
Despite this, reports are that they have said these things in their book. Having seen statements made by the authors and listening to interviews with Mlodnow, I cannot believe that people are ignoring the authors!
Come on people! Stop making an issue out of nothing!
Despite this, reports are that they have said these things in their book. Having seen statements made by the authors and listening to interviews with Mlodnow, I cannot believe that people are ignoring the authors!
Come on people! Stop making an issue out of nothing!
Thursday, September 16, 2010
Pythagorean Triples
Pythagorean triples are groupings of three integers that each form the sides of a right triangle. I have been playing with their distributions, and have made some plots in Mathematica. I think these are pretty cool.
Wednesday, September 15, 2010
The Mind of a Theorist - The Relaunch of the Column
You can find the rewritten and reformated column at www.madscitech.org
I will be placing a new column up every week.
I will be placing a new column up every week.
Tuesday, September 14, 2010
New Version of Mathematica Announced
I was just informed by Wolfram Research that a new version of Mathematica will be coming out soon. As I understand it, this version is going to be a big step forward. This version will have built-in GPU support for CUDA, built-in control systems and wavelets, advanced data visualization tools, and a huge raft of other things I have heard about, but have not seen officially so will keep my mouth (fingers?) shut about.
This is exciting news since there is a personal version of Mathematica for only about $300, with a $500 upgrade to the most current professional version. There is no difference between the personal and profession versions.
This is exciting news since there is a personal version of Mathematica for only about $300, with a $500 upgrade to the most current professional version. There is no difference between the personal and profession versions.
Monday, September 13, 2010
Book Review: The Classical Theory of Fields
Time for another book review. You can purchase this book at my book store below.
This time I am going for volume 2 of the Course of Theoretical Physics by Landau and Lifshitz. This volume is titled, "The Classical Theory of Fields," and covers, in one volume, special and general relativity, electrodynamics in a vacuum, and optics in a vacuum. The authors assume you are familiar with vector analysis and the electromagnetic phenomena equivalent to that covered in a general physics course (charges, electric and magnetic fields, and induction).
This time I am going for volume 2 of the Course of Theoretical Physics by Landau and Lifshitz. This volume is titled, "The Classical Theory of Fields," and covers, in one volume, special and general relativity, electrodynamics in a vacuum, and optics in a vacuum. The authors assume you are familiar with vector analysis and the electromagnetic phenomena equivalent to that covered in a general physics course (charges, electric and magnetic fields, and induction).
Sunday, September 12, 2010
The Wonders of Physics
Earlier this year, I participated in a public demonstration lecture called the Wonders of Physics. In this year's show I demonstrated the physics of board-breaking.
I had a lot of fun, and you can find a video from a friend of mine, showing my participation in one of the earlier shows: http://www.youtube.com/watch?v=4eQWRLZG1rU
The official web site for the Wonders of Physics, where you can see all of the shows over the years, is here: http://sprott.physics.wisc.edu/wop.htm
I had a lot of fun, and it is a nice to give such presentations to the public, who hopefully will become interested in physics.
I had a lot of fun, and you can find a video from a friend of mine, showing my participation in one of the earlier shows: http://www.youtube.com/watch?v=4eQWRLZG1rU
The official web site for the Wonders of Physics, where you can see all of the shows over the years, is here: http://sprott.physics.wisc.edu/wop.htm
I had a lot of fun, and it is a nice to give such presentations to the public, who hopefully will become interested in physics.
Saturday, September 11, 2010
Book Review: Mechanics, by Landau and Lifshitz
It is my intent to review every book in my store, below, so I will start with the first book in the store. You can purchase this book at my book store below.
This is the first volume of the classic Course of Theoretical Physics due to Lev Landau and the Landau Institiute of Theoretical Physics in Russia. These ten volumes were the subject of what Landau belived to be the required preparation in physics for any theoretical physicist.
This volume begins with a touching introduction to Lev Landau and his philosophy. It also lays out the prerequisites for this volume. To quote: "...ability to solve any indefinite integral that can be expressed in terms of elementary functions and to solve any ordinary differential equation of the standard type, knowledge of vector analysis and tensor algebra as well as the principles of the theory of functions of a complex variable (theory of residues, Laplace method). If you need this ackground I recommend Hassani, Mathematical Methods for Students of Physics and Related Fields, Second Edition.
The first chapter describes Lagrange's equations of motion from the starting point of generalized coordinates. This lays the ground work for a very nice presentation of the least action principle. Then there is a nice section of the Galilean transformations; it is important to realize that there is no discussion of special relativity in the book. The book then describes the Lagrangian of a free particle in Cartesian, spherical, and cylindrical coordinates. The final section of the first chapter not only discusses the Lagrangian of a system of particles, but nicely links Lagrange's equations to Newtonian theory.
The second chapter presents one of the most important aspects of theoretical physics, the conservation laws. This begins with a unified treatment of integrals of the motion, the conservation of energy, how conservation of energy implies the homogeneity of time, and conservative systems. This leads into the conservation of momentum, how the conservation of momentum implies the homogeneity of space, generalized momenta, and generalized forces. There is a brief discussion of systems of particles, the center of mass, and internal energy. This leads to a section on the conservation of angular momentum, how this implies the isotropy of space, and the idea of the central field. The final section deals with a couple of different topics such as the behavior of the equations of motion under transformations and the virial theorem.
The third chapter is a very practical one about applying the equations of motion to specific situations. This begins with one dimensional motion and a discussion of energy diagrams and oscillations. This leads into a discussion of how to interpret energy diagrams and derive the potential from a specific period of oscillation. The next topic covered is the beginning of the two-body problem in the form of the reduced mass of a system. The two-body problem is then reduced to a one-body problem in a central field in a very clear presentation. This leads to a nice discussion of the Kepler problem for bound and unbounded orbits.
Chapter four is an important discussion of the classical theory of particle scattering. This begins by the idea that particles can break up, this also introduces the lab and center-of-mass frames of reference. Then the authors describe elastic collisions in quite intuitive way. Then the notions of impact parameters and scattering-cross sections are covered. As an example of scattering due to fields, there is a section on Rutherford scattering. The chapter ends with small-angle scattering.
Chapter Five is on the theory of oscillations. Naturally, this begins with free oscillations in one dimension. The book then turns its attention to forced oscillations and resonance. Then the authors treat the idea of oscillations in more than one degree of freedom, including eigenfrequences and normal modes. This is then applied to the classical theory of molecules. Then damped oscillators are covered along with a discussion of dissipative functions in Lagrangian dynamics. The next section introduces dissipation and damped and driven oscillations. At this point the book leaves the realm of simple oscillating systems and goes into parametric oscillations and the Mathieu equation. There is a discussion of nonlinear oscillators, laying the ground-work for the study of chaos, this also includes a section of the resonance of nonlinear oscillators—introducing the Duffing oscillator without calling it that. Finally, there is an interesting section on the motion of a particle in a rapidly oscillating field.
Chapter six is a thorough treatment of rigid body dynamics. This begins with a discussion of the kinematics of rigid bodies. Then the authors introduce the inertia tensor. Indeed, this is one of the few sections in the book with more than a few problems (there are nine here). The section on angular momentum uses the axes of inertia to define angular momentum. Then the equations of motion for a rigid body are derived. This leads to the notion of Euler angles, which in turn leads to the Euler equations. These notions are then applied to the motion of a top. Then they shift gears a little and discuss rigid bodies in contact. Finally, and I think this might have done better at the beginning of the chapter, is a section on motion in noninertial frames, specifically rotating frames. This chapter seems very conventional, though it must be remembered that the first version of it came out in 1960, so a lot of other books are based on material found in here.
The final chapter is an overview of Hamiltonian theory. This begins, reasonably enough, with Hamilton's canonical equations of motion and relates the Hamiltonian and the Lagrangian in the normal way. The Routhian has its own section. Then they address the extremely important topic of the Poisson brackets. The next section establishes the idea of a functional without using those words. Then the authors discuss the principle of Maupertuis. Then they discuss canonical transformations and link them to Poisson brackets, and discuss conjugate variables, laying some of the groundwork for quantum mechanics. The authors next introduct the idea of phase space through a discussion of Liouville's theorem. The next section discusses the Hamilton-Jacobi equation and the idea of general and complete integrals. This is followed by a treatment of the method of separation of variables. Then the authors have an excellent and clear discussion of adiabatic invariants, a subject with applications in plasma physics. Then they discuss action-angle variables. Then the authors turn to the validity of the adiabatic invariant. The chapter, and the book, ends with a section on conditionally periodic motion.
The book, with all problems included, is just 167 pages long and comprises a total of 52 sections. This is something you could easily cover in two months of dedicated study. I think the book could use an update, but it's a classic as it is and its choice of topics are pretty good. I like it a lot, even if it doesn't really cover chaos in any meaningful way.
This is the first volume of the classic Course of Theoretical Physics due to Lev Landau and the Landau Institiute of Theoretical Physics in Russia. These ten volumes were the subject of what Landau belived to be the required preparation in physics for any theoretical physicist.
This volume begins with a touching introduction to Lev Landau and his philosophy. It also lays out the prerequisites for this volume. To quote: "...ability to solve any indefinite integral that can be expressed in terms of elementary functions and to solve any ordinary differential equation of the standard type, knowledge of vector analysis and tensor algebra as well as the principles of the theory of functions of a complex variable (theory of residues, Laplace method). If you need this ackground I recommend Hassani, Mathematical Methods for Students of Physics and Related Fields, Second Edition.
The first chapter describes Lagrange's equations of motion from the starting point of generalized coordinates. This lays the ground work for a very nice presentation of the least action principle. Then there is a nice section of the Galilean transformations; it is important to realize that there is no discussion of special relativity in the book. The book then describes the Lagrangian of a free particle in Cartesian, spherical, and cylindrical coordinates. The final section of the first chapter not only discusses the Lagrangian of a system of particles, but nicely links Lagrange's equations to Newtonian theory.
The second chapter presents one of the most important aspects of theoretical physics, the conservation laws. This begins with a unified treatment of integrals of the motion, the conservation of energy, how conservation of energy implies the homogeneity of time, and conservative systems. This leads into the conservation of momentum, how the conservation of momentum implies the homogeneity of space, generalized momenta, and generalized forces. There is a brief discussion of systems of particles, the center of mass, and internal energy. This leads to a section on the conservation of angular momentum, how this implies the isotropy of space, and the idea of the central field. The final section deals with a couple of different topics such as the behavior of the equations of motion under transformations and the virial theorem.
The third chapter is a very practical one about applying the equations of motion to specific situations. This begins with one dimensional motion and a discussion of energy diagrams and oscillations. This leads into a discussion of how to interpret energy diagrams and derive the potential from a specific period of oscillation. The next topic covered is the beginning of the two-body problem in the form of the reduced mass of a system. The two-body problem is then reduced to a one-body problem in a central field in a very clear presentation. This leads to a nice discussion of the Kepler problem for bound and unbounded orbits.
Chapter four is an important discussion of the classical theory of particle scattering. This begins by the idea that particles can break up, this also introduces the lab and center-of-mass frames of reference. Then the authors describe elastic collisions in quite intuitive way. Then the notions of impact parameters and scattering-cross sections are covered. As an example of scattering due to fields, there is a section on Rutherford scattering. The chapter ends with small-angle scattering.
Chapter Five is on the theory of oscillations. Naturally, this begins with free oscillations in one dimension. The book then turns its attention to forced oscillations and resonance. Then the authors treat the idea of oscillations in more than one degree of freedom, including eigenfrequences and normal modes. This is then applied to the classical theory of molecules. Then damped oscillators are covered along with a discussion of dissipative functions in Lagrangian dynamics. The next section introduces dissipation and damped and driven oscillations. At this point the book leaves the realm of simple oscillating systems and goes into parametric oscillations and the Mathieu equation. There is a discussion of nonlinear oscillators, laying the ground-work for the study of chaos, this also includes a section of the resonance of nonlinear oscillators—introducing the Duffing oscillator without calling it that. Finally, there is an interesting section on the motion of a particle in a rapidly oscillating field.
Chapter six is a thorough treatment of rigid body dynamics. This begins with a discussion of the kinematics of rigid bodies. Then the authors introduce the inertia tensor. Indeed, this is one of the few sections in the book with more than a few problems (there are nine here). The section on angular momentum uses the axes of inertia to define angular momentum. Then the equations of motion for a rigid body are derived. This leads to the notion of Euler angles, which in turn leads to the Euler equations. These notions are then applied to the motion of a top. Then they shift gears a little and discuss rigid bodies in contact. Finally, and I think this might have done better at the beginning of the chapter, is a section on motion in noninertial frames, specifically rotating frames. This chapter seems very conventional, though it must be remembered that the first version of it came out in 1960, so a lot of other books are based on material found in here.
The final chapter is an overview of Hamiltonian theory. This begins, reasonably enough, with Hamilton's canonical equations of motion and relates the Hamiltonian and the Lagrangian in the normal way. The Routhian has its own section. Then they address the extremely important topic of the Poisson brackets. The next section establishes the idea of a functional without using those words. Then the authors discuss the principle of Maupertuis. Then they discuss canonical transformations and link them to Poisson brackets, and discuss conjugate variables, laying some of the groundwork for quantum mechanics. The authors next introduct the idea of phase space through a discussion of Liouville's theorem. The next section discusses the Hamilton-Jacobi equation and the idea of general and complete integrals. This is followed by a treatment of the method of separation of variables. Then the authors have an excellent and clear discussion of adiabatic invariants, a subject with applications in plasma physics. Then they discuss action-angle variables. Then the authors turn to the validity of the adiabatic invariant. The chapter, and the book, ends with a section on conditionally periodic motion.
The book, with all problems included, is just 167 pages long and comprises a total of 52 sections. This is something you could easily cover in two months of dedicated study. I think the book could use an update, but it's a classic as it is and its choice of topics are pretty good. I like it a lot, even if it doesn't really cover chaos in any meaningful way.
Friday, September 10, 2010
Conservation of Energy and The Nature of Forces
Today I have been toying with a bunch of things:
1) Conservation of energy, I have been writing this section for the book project. Nice approach, if I do say so myself. Did you know that we really don't know what energy is other than a calculated number that remains the same in a closed system? We have no better understanding of energy than that. Despite being able to calculate it, predict it, and use it in other kinds of calculations; we have no idea what energy IS!
2) In Newtonian mechanics the entire game is chasing the forces. We have F=ma, and people think this is the definition of force. No! This states that the numerical quantity of force is equal to the product of mass and acceleration. If you do not know what F is prior to doing a calculation you will not be likely to figure it out.
3) I have been messing around, playing with trigonometric and hyperbolic functions.
Been having a lot of fun with these things.
1) Conservation of energy, I have been writing this section for the book project. Nice approach, if I do say so myself. Did you know that we really don't know what energy is other than a calculated number that remains the same in a closed system? We have no better understanding of energy than that. Despite being able to calculate it, predict it, and use it in other kinds of calculations; we have no idea what energy IS!
2) In Newtonian mechanics the entire game is chasing the forces. We have F=ma, and people think this is the definition of force. No! This states that the numerical quantity of force is equal to the product of mass and acceleration. If you do not know what F is prior to doing a calculation you will not be likely to figure it out.
3) I have been messing around, playing with trigonometric and hyperbolic functions.
Been having a lot of fun with these things.
Thursday, September 9, 2010
Brief on Calculus and Appreciation to Leonard Susskind for his Lectures
Hello everyone!
I have been working on the book project with Leonard Susskind. I just completed an 11-page overview of calculus with detailed material on limits, derivatives (including a short table of derivatives), integrals (with another short table of integrals), differential equations (covering mostly separation of variables), and Taylor series.
I am thinking of adding sections on partial derivatives, maxima and minima, and maybe curvature.
Of course, Leonard has not committed to anything yet; I certainly have not produced enough material to do more than whet his appetite for more. It is possible that something could happen to prevent me from finishing, and he probably does not want to stick his neck, or his reputation, out for someone who is not too well known.
I am having great fun writing this material. It is extremely challenging to develop short and clear explanations for these topics. My olf Mind of a Theorist column is good for this, the philosophy that Iadopted for the column is the same the Leonard has in the lectures.
To quote from the introduction, "What is this course about? Who am I teaching to? While undergraduate or graduate students will be able to read this, it is mostly designed for people who are interested in getting into the meat of physics right away. This is not a standard physics course. This is the real deal, theoretical physics at full scale! We use equations, and sometimes hard equations, but we try to use the simplest equations that will do the job. Basically, we try to keep it minimal. The goal here is to get to the basic ideas fast. So we will be telling you what you really, really, need to know to get to the next level. Sometimes the basics can be hard; we will do them anyway, but will only spend the minimum time required to get them right. And here we mean getting them really right, not by metaphors or analogies, but equations when necessary."
I am dazzled by the structure of the video lectures. In lecture one he introduces the ideas of dynamical systems, phase space, and conservation laws with no mathematics; just some diagrams and very clear explanations. Wonderful!
I have been working on the book project with Leonard Susskind. I just completed an 11-page overview of calculus with detailed material on limits, derivatives (including a short table of derivatives), integrals (with another short table of integrals), differential equations (covering mostly separation of variables), and Taylor series.
I am thinking of adding sections on partial derivatives, maxima and minima, and maybe curvature.
Of course, Leonard has not committed to anything yet; I certainly have not produced enough material to do more than whet his appetite for more. It is possible that something could happen to prevent me from finishing, and he probably does not want to stick his neck, or his reputation, out for someone who is not too well known.
I am having great fun writing this material. It is extremely challenging to develop short and clear explanations for these topics. My olf Mind of a Theorist column is good for this, the philosophy that Iadopted for the column is the same the Leonard has in the lectures.
To quote from the introduction, "What is this course about? Who am I teaching to? While undergraduate or graduate students will be able to read this, it is mostly designed for people who are interested in getting into the meat of physics right away. This is not a standard physics course. This is the real deal, theoretical physics at full scale! We use equations, and sometimes hard equations, but we try to use the simplest equations that will do the job. Basically, we try to keep it minimal. The goal here is to get to the basic ideas fast. So we will be telling you what you really, really, need to know to get to the next level. Sometimes the basics can be hard; we will do them anyway, but will only spend the minimum time required to get them right. And here we mean getting them really right, not by metaphors or analogies, but equations when necessary."
I am dazzled by the structure of the video lectures. In lecture one he introduces the ideas of dynamical systems, phase space, and conservation laws with no mathematics; just some diagrams and very clear explanations. Wonderful!
Tuesday, September 7, 2010
Wonderful Opportunity
I have begun converting Leonard Susskind's video lecture series on Modern Physics to book form. I have just completed Lecture 1 of the video course Modern Physics: Classical Mechanics and am now working on a chapter on calculus.
So far, so good. These are still early days, but I plan to convert them all in time.
Whether it becomes anything is, at this point, up in the air; we will see how it goes.
I am honored to be working with someone so passionate about providing a good resource on theoretical physics to enthusiasts. I am honored that he didn't just slam the door in my face! I am having a great time in converting these lectures, not only because I get to work with one of the giants of theoretical physics; but also because I am forced to think deeply about the meaning of the lectures and their context. It is a wonderful opportunity.
Thanks, Leonard!
So far, so good. These are still early days, but I plan to convert them all in time.
Whether it becomes anything is, at this point, up in the air; we will see how it goes.
I am honored to be working with someone so passionate about providing a good resource on theoretical physics to enthusiasts. I am honored that he didn't just slam the door in my face! I am having a great time in converting these lectures, not only because I get to work with one of the giants of theoretical physics; but also because I am forced to think deeply about the meaning of the lectures and their context. It is a wonderful opportunity.
Thanks, Leonard!
Wednesday, September 1, 2010
Great Find!
Dover Publications has produced a version of Richard Feynman and Albert Hibbs' excellent textbook, "Quantum Mechanics and Path Integrals." The text has been cleaned of typographic and grammatical errors, and the notation has been standardized, while maintaining the rough edge to the original. This is probably the best text for the path integral approach to quantum mechanics. Enjoy!
Monday, August 30, 2010
Creeping Featuritis
In the computer world there is a term used to describe a project, intended to be accomplished in finite time, where the client keeps wanting new things added before the project is completed; called creeping-featureitis. When building a computer project, this is a bad thing.
In theoretical physics, since we know we do not know all of the rules of the game, this is the norm. We are always adding new things: ideas, theorems, examples, counterexamples, contradictions, etc.
It is not surprising that theorists have computer code that is ever expanding; that is how we think!
To that extent, all theorists engage in creeping-featuritis and it is the way it should be.
In theoretical physics, since we know we do not know all of the rules of the game, this is the norm. We are always adding new things: ideas, theorems, examples, counterexamples, contradictions, etc.
It is not surprising that theorists have computer code that is ever expanding; that is how we think!
To that extent, all theorists engage in creeping-featuritis and it is the way it should be.
Wednesday, August 25, 2010
Weird Fractional Quantum Hall Effect
In a recent publication to Physical Review Letters, "Optical Probing of the Spin Polarization of the v= 5/2 Quantum Hall State" the authors: M. Stern, P. Plochocka, V. Umansky, D. K. Maude, M. Potemski, and I. Bar-Joseph, describe a phenomena resulting in a pseudoparticle having non-integer charge!
This is all based on the fractional quantum Hall effect.
Recall the classical Hall effect, discovered by Edwin Hall in 1879. This effect occurs when you apply a magnetic field perpendicular to a current in a conductor, producing a voltage difference across the conductor. This voltage difference is sometimes called the Hall Voltage.
If you have a system of electrons in a plane (two-dimensional) or on a surface that are at low temperatures, you can produce Hall voltages of only quantized values by applying a strong magnetic field. This is the integer quantum Hall effect. You can calculate the Hall Resistance,
R= (h/e^2)/v
Where h is Planck's constant, e is the fundamental charge, and v is called the filling factor.
In some cases the electrons behave as if they are a fluid. In this case they behave as a pseudoparticle with fractional charge. Very interesting! This occurs when v is fractional, it is specifically interesting at v = 5/2.
I have not yet finished the letter, but something mysterious occurs at v = 5/2. There is a lot of interesting physics here!
George
This is all based on the fractional quantum Hall effect.
Recall the classical Hall effect, discovered by Edwin Hall in 1879. This effect occurs when you apply a magnetic field perpendicular to a current in a conductor, producing a voltage difference across the conductor. This voltage difference is sometimes called the Hall Voltage.
If you have a system of electrons in a plane (two-dimensional) or on a surface that are at low temperatures, you can produce Hall voltages of only quantized values by applying a strong magnetic field. This is the integer quantum Hall effect. You can calculate the Hall Resistance,
R= (h/e^2)/v
Where h is Planck's constant, e is the fundamental charge, and v is called the filling factor.
In some cases the electrons behave as if they are a fluid. In this case they behave as a pseudoparticle with fractional charge. Very interesting! This occurs when v is fractional, it is specifically interesting at v = 5/2.
I have not yet finished the letter, but something mysterious occurs at v = 5/2. There is a lot of interesting physics here!
George
Friday, August 20, 2010
Hellow World
Hello everyone! My name is George, and I used to write a column on The Citizen Scientist (an e-publication put out by the Society for Amateur Scientists) called The Mind of a Theorist. This publication has ended (along with my column), at least temporarily, so I have relaunched it here.
The goal of the column was to provide good content about theoretical physics in an educational way.
I was forced to change the format due to editorial interference.
I want to return to the old way of doing it.
To that end, I will be going over my old columns and resurrecting them for the dead here. In some cases I will not change much, while in some I will trash the old and ring in the new right away.
With each post I will discuss a physics news item, my research, a topic of interest to readers, a comment from a post, or other such. I will also present something educational; something to help you become a theoretical physicist - even an amateur one; and then once a week I will issue a theorist challenge. At the moment, I will not be able to provide any prizes, but if I get enough support that might happen. I will always answer the challenge in the next week.
About me: I am the President of the nonprofit organization Madison Area Science and Technology (MAST). We do science without regard to credentials. You can check out our web site at http://www.madscitech.org/ .
The goal of the column was to provide good content about theoretical physics in an educational way.
I was forced to change the format due to editorial interference.
I want to return to the old way of doing it.
To that end, I will be going over my old columns and resurrecting them for the dead here. In some cases I will not change much, while in some I will trash the old and ring in the new right away.
With each post I will discuss a physics news item, my research, a topic of interest to readers, a comment from a post, or other such. I will also present something educational; something to help you become a theoretical physicist - even an amateur one; and then once a week I will issue a theorist challenge. At the moment, I will not be able to provide any prizes, but if I get enough support that might happen. I will always answer the challenge in the next week.
About me: I am the President of the nonprofit organization Madison Area Science and Technology (MAST). We do science without regard to credentials. You can check out our web site at http://www.madscitech.org/ .
That is all for now, I will now check out how to include mathematical symbols in text.
George
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