29 October 2004

The Return of The Theorist!

George E. Hrabovsky, President, MAST

A New Focus

In the past I have turned the Mind of a Theorist into a sort of cut-down physics course. I have decided to embrace the new changes in The Citizen Scientist; that of project-orientation, and that will be the new focus of this column. I will begin the next column by working on a research project away from any area of expertise that I now possess. As I proceed, I will assume that I know no calculus and little (perhaps high school-level) physics. In this way I hope to show how an amateur theorist can develop their own roads into a topic without resorting to classical means.

What it Takes to be a Theorist

There are several traits that make for a good theorist:

Curiosity: You must always question everything. This is the only way to make sure you really understand something.

Tenacity: You are embarking on a long and hard road.

Mathematical Facility: Mathematics is the language that theory is spoken in.

Intuition: Science is not mathematics. You need to know when an idea is correct, even if the mathematics does not.

A Course in Theoretical Physics

I have decided, as with the "Mathematics Corner," to throw aside all pretense and just come right out and tell you that I intend to teach you to become theorists. My main emphasis is theoretical physics. It is entirely possible that I will adopt a project that has wide application to other sciences. Having said that, I think I will make a project to investigate heat. My question will be, "What is heat?"

The Well Stocked Theory Library

Below I will outline a set of books and online resources. As you read through the titles, keep a few things in mind. Each branch of physics exists because it deals with a specific set of problems in specific ways. It is vitally important to understand the fundamental scope and purpose of each branch. Science, and physics specifically, is all about measuring and explaining nature. This is in sharp contrast to mathematics, where there is no requirement that anything be related to reality. For mathematics that is fine; for science it is not allowed.

Everything must begin and end with nature. You should keep a running list of all of the ideas that you encounter in your readings. Most often these will be in the form of definitions, undefined technical terms, explanations, formulas, assumptions, conjectures, and proven theorems. A good source of projects is to create examples of all of these. Another source of projects is to invent new proofs for existing ideas (perhaps using different mathematics or physical ideas than existing proofs), or to prove unproven statements and conjectures.

Always look for connections between ideas. When you encounter such, note them immediately. This is another source of projects; how are two areas of science related? How can they be combined? Another source of projects is to note equivalent forms of ideas from different branches of science. Such equivalences are clues to fundamental truths.

Having said all of that, how do you approach such a vast subject? As I wrote in the 15 October 2004 installment of "Mathematics Corner," patience and one step at a time. I have found it useful to acquire a library of science. This can be very expensive, fortunately there are many freely available books on the web that are fairly easy to locate. Use such phrases as "Free Electronic Books," or, "Physics," followed by, "Course Lecture Notes," in your favorite search engine. You will need at least one book from each category, and I recommend at least two.

General Scientific Reference: These include formularies and science encyclopedias. Test preparation guides (for GRE or prelim exams at the graduate level, not the SAT or such) are good sources for this. I also have, and recommend (though it is quite expensive,) the McGraw-Hill Encyclopedia of Science and Technology on CD-ROM.

Elementary Science: I recommend that you acquire at least one high school level book on each of the following: Algebra, geometry, intermediate algebra, trigonometry, biology, chemistry, physics, and earth science. These elementary-level books will allow at least a rudimentary ability to look things up.

General Science: I recommend that you acquire at least two calculus books. One should be Thomson and Gardner's "Calculus Made Easy," a book that makes calculus understandable in a short time. I also recommend the excellent online set of calculus books by Dan Sloughter, "Difference Equations to Differential Equations" and, "The Calculus of Functions of Several Variables." I also like the Schaum's Outline of Calculus.

I also recommend the online set of books by Benjamin Crowell entitled "Light and Matter." The set includes, "Book 1, Newtonian Physics;" "Book 2, Conservation Laws;" "Book 3, Vibrations and Waves;" "Book 4, Electricity and Magnetism;" "Book 5, Optics;" and "Book 6, The Modern Revolution in Physics." Covering a lot of the same material, from a different perspective, is Christoph Schiller's excellent (and truly impressive at 1,129 pages) "Motion Mountain." MIT has a decent basic physics course freely available in their Open Courseware (along with lots of other useful things.)

I also recommend having a copy of "The Feynman Lectures on Physics." While they are not so good at learning from, they are great for looking things up. If you have a budget and want an inexpensive general physics book, I recommend the "Schaum's Outline of Physics for Scientists and Engineers." If you have money, I recommend Young and Freedman, "University Physics with Modern Physics."

Chemistry is important at this level, and the best online stuff is from Michael Blaber at http://wine1.sb.fsu.edu/chm1045/chm1045.htm for a first semester and http://wine1.sb.fsu.edu/chm1046/chm1046.htm for the second semester. For those on a budget, I recommend Linus Pauling's, "General Chemistry," published by Dover.

Mathematical Methods: There are a vast number of good books on the mathematics you need to do theory. The free online books I like are Gallier's "Algebra and Analysis for Computer Science" and Sean Mauch's (at an impressive 2,321 pages to date) "Introduction to Methods of Applied Mathematics or Advanced Mathematical Methods for Scientists and Engineers." The latter book covers a lot of topics in great detail, and each year it grows! James Nearing has written, "Mathematical Tools for Physics."

If you want to pay for a book and do not have a lot of money Dover has a good selection. The best (in my opinion) is Dennery and Krzywicki's "Mathematics for Physicists." I also like Collins' "Mathematical Methods for Physicists and Engineers."

If you want to spend some money, I like Riley, Hobson, and Bence's "Mathematical Methods for Physicists and Engineers," Cantrel's "Modern Mathematical Methods for Physicists and Engineers," and Hassani's "Mathematical Methods for Students of Physics and Related Fields, and "Mathematical Physics." I also like Matthews and Walker's "Mathematical Methods of Physics," Geroch's "Mathematical Physics," Morse and Feshbach's two volume set, "Methods of Theoretical Physics," and my favorite (though at a very advanced level) is Prakash's "Mathematical Perspectives on Theoretical Physics."

Computational Methods: You should learn to use a computer to help solve problems that are otherwise intractable. I use Mathematica as my choice of programming environments, and, while I recommend it, Mathematica is quite expensive. There are some free alternatives that do not have the same level of utility or power.

The free online books are Luttermoser's "Computational Physics" (a bit opinionated, and you don't get to the real meat until the fourth chapter) and Fitzpatrick's "Introduction to Computational Physics." I do not know of any inexpensive books on the subject. For those who want to spend money, I like Gould and Tobochnik's "An Introduction to Computer Simulation Methods: Applications to Physical Systems" (my favorite), Woolfson and Pert's "An Introduction to Computer Simulation," Koonin's "Computational Physics" and MacKeown and Newman's "Computational Techniques in Physics."

Mechanics: Mechanics is an interesting subject in its own right, particularly in light of chaos and dynamical systems theory. It is also a good place to learn about conservation laws. For free online books, Fitzpatrick has a set of notes, "Classical Mechanics," that covers Newtonian mechanics and Rosu's "Classical Mechanics" is at the graduate level. Also online are Whelan's "Dynamics," a short set of notes at an intermediate level and Morii's "Mechanics," a set of slide presentations for intermediate mechanics. Morin has produced an online intermediate level set of notes at Harvard entitled "There once was a classical theory." Sussman, Wisdom, and Mayer have produced the online graduate text, "Structure and Interpretation of Classical Mechanics."

For those on a budget, I recommend "Schaum's Outline of Theoretical Mechanics." For those who want to spend money, consider Hand and Finch's "Analytical Mechanics;" Kibble and Berkshire's "Classical Mechanics;" and Talman's "Geometric Mechanics." Other good selections are McCauley's "Classical Mechanics" and Arnold's "Mathematical Methods of Classical Mechanics." My favorite mechanics book is "Classical Dynamics" by José and Saletan.

Electrodynamics: Electrodynamics is important because almost everything we can measure is electromagnetic in one way or another. Even light is a form of electromagnetism. Free online books include (at a very high level) Shapirpov's "Classical Electrodynamics and the Theory of Relativity" and, at the graduate level, Fitzpatrick's "Advanced Classical Electromagnetism." Also consider Fitzpatrick's course in undergraduate electrodynamics, "Classical Electromagnetism," and Thildé's "Electromagnetic Field Theory."

Dover has several good books on electrodynamics, though I do not seem to own any of the low level ones. I do own, and like, Post's "Formal Structure of Electromagnetics." For those of you with money, I recommend Griffiths's "Classical Electrodynamics," Westgarde's "Electrodynamics: A Concise Introduction," Low's "Classical Field Theory," and Schwinger, et al., "Classical Electrodynamics;" as an interesting side note, Shawn Carlson has told me many stories about his personal experiences with Julian Schwinger; Shawn actually took the course that this book is based on.

Wave Phenomena: The ability to understand oscillating systems is very important, not just in physics, but in every science. The phenomena covered here include abstract waves, optics, acoustics, and water waves. The free online books are Morii's course notes for Physics 15 C from MIT, "Wave Phenomena," Mei's "Wave Propagation," Richardson's "Waves and Optics," and Louro's "Physics 323 Notes."

For those on a budget, I recommend the Dover books Elmore and Heald's "Physics of Waves" and Towne's "Wave Phenomena." If you have money you want to spend, then these are my recommendations: First, at the graduate level, Whitham's "Linear and Nonlinear Waves." At an undergraduate level we have Hecht's classic, "Optics," One that I really like is Pedrotti and Pedrotti's "Introduction to Optics." I also like Born and Wolf's rather high level classic "Principles of Optics" and Blackstock's "Fundamentals of Acoustics."

Quantum Mechanics: So far we have been talking about books that relate to everyday phenomena. Quantum mechanics describes what happens when things get very small. It turns out that normal classical mechanics no longer works at this small scale, where the atom lives.

The free online books include Porter's intermediate level "Notes for Caltech's Physics 195." There is also a set of course notes available from MIT's Open Courseware program. Also online are Biswas's "Quantum Mechanics—Concepts and Applications;" Horgan's "Part IIA Quantum Physics" (which assumes you have already studied some quantum mechanics); and Norbury's excellent "Quantum Mechanics." More online choices include Fitzpatrick's "Quantum Mechanics" and Luttermoser's "Physics 4617/5617: Quantum Physics." At the graduate level we have Pratt's "Physics 851;" Schulten's "Notes on Quantum Mechanics;" and Grosche's "Introduction to the Feynman Path Integral."

For those of you on a budget, I recommend " Schaum's Outline of Quantum Mechanics" (it is quite good). I also recommend the following Dover books. At the intermediate level we have Chester's "Primer of Quantum Mechanics" and the more applied "Quantum Mechanics for Applied Physics and Engineering" by Fromhold, Jr. At the graduate level we have Bohm's "Quantum Theory."

For those who have money to spend, I recommend, at the undergraduate level, Griffiths' "Introduction to Quantum Mechanics." At the graduate level I recommend Shankar's "Principles of Quantum Mechanics;" Ballantine's "Quantum Mechanics;" and Dirac's "Principles of Quantum Mechanics."

Thermal Physics: This covers the topics of thermodynamics, kinetic theory, and statistical mechanics. The free online books are Sethna's "Entropy, Order Parameters, and Complexity;" Kinaret's "Statistical Physics;" Fitzpatrick's "Thermodynamics and Statistical Mechanics;" and Gould and Tobochnik's "Thermal Physics."

For those on a budget, I recommend the following Dover Books. Ferrmi's (yes, that Fermi) "Thermodynamics;" Lavenda's "Thermodynamics of Irreversible Processes;" and Reiss's "Methods of Thermodynamics." Also consider Hecht's "Statistical Thermodynamics and Kinetic Theory" and, one that I particularly like, Wannier's "Statistical Physics." Also there are Mattuck's "A Guide to Feynman Diagrams in the Many-Body Problem" and "Methods of Quantum Field Theory in Statistical Physics" by Abrikosov, et al. A true masterpiece is Goodstein's "State of Matter."

For those with money, consider Sears and Salinger's "Thermodynamics, Kinetic Theory, and Statistical Thermodynamics." I particularly like Baierlein's "Thermal Physics" and Phillies' "Elementary Lectures in Statistical Mechanics. Others I like are Chandler's "Introduction to Modern Statistical Mechanics;" Feynman's (yes, that Feynman) "Statistical Mechanics;" Goldenfield's "Lectures on Phase Transitions and the Renormalization Group;" and Nozières and Pines' "The Theory of Quantum Liquids."

Physical Chemistry: These books cover the application of physics to chemistry and can cover such topics as thermodynamics, statistical mechanics, kinetic theory, chemical dynamics, and quantum chemistry. Free online books in this category are Ulness' "Physical Chemistry;" Shurko's "Introduction to Physical Chemistry;" and Batista's "Introductory Quantum Chemistry." Also, Muskgrave's "Chem 160 Physical Chemistry: Quantum Chemistry;" Kirby's "Introduction to Quantum Chemistry;" Simons' "An Introduction to Theoretical Chemistry and his book "Quantum Mechanics in Chemistry."

For those on a budget, I recommend "Schaum's Outline of Physical Chemistry" and the following Dover books: Pilar's "Elementary Quantum Chemistry;" Craig and Thirunamachandran's "Molecular Quantum Electrodynamics;" Levine's "Quantum Mechanics of Molecular Rate Processes;" Bishop's "Group Theory and Chemistry;" and Child's "Molecular Collision Theory."

For those who have money to spend, I recommend Levine's "Physical Chemistry" and Haken and Wolf's two-volume set, "The Physics of Atoms and Quanta" and "Molecular Physics and Elements of Quantum Chemistry." I really like Atkins' "Molecular Quantum Mechanics." Alan Hinchliffe has two books, "Chemical Modeling" and "Modeling Molecular Structures." I also like Leach's "Molecular Modeling" and Billing and Mikkelson's "Advanced Molecular Dynamics and Chemical Kinetics."

Organic Chemistry: Organic compounds are not only interesting in their own right, they have wide applications throughout science. Several free online books are available. From Georgia Tech we have Collard and Deutsch's "Chem 2311" and Boggs' "Chem 2312." from Ohio State we have Lowary's "Chemistry 251 Lecture Notes" and "Chemistry 252 Lecture Notes." There is also Young's "Organic Chemistry Online."

The only inexpensive book that I know of is "Schaum's Outline of Organic Chemistry," and it is quite good. If you have money, I recommend Streitwieser and Heathcock's "Introduction to Organic Chemistry" and "Introduction to Organic and Biochemistry" by Hein, et al. My favorite is McDonald's "Deductive Organic Chemistry."

Inorganic Chemistry: Inorganic compounds are also interesting in many contexts, and they are very useful in many applications. The free online books are Greenblatt's "Chem 371 Notes" and MacDonald's "Inorganic Chemistry Chem 59-250" and "Main Group Chemistry Chem 59-651."

I know of no inexpensive book on this subject, though some elements of it are covered in the Dover book by Bishop, "Group Theory and Chemistry." For those with money, the only book I know is Miessler and Tarr's "Inorganic Chemistry."

Continuum Physics: This is the physics of matter in the large scale. It includes elasticity, fluid dynamics, gas dynamics, electrodynamics of matter, thermal physics of matter, plasma physics, large-scale solid state physics, large-scale condensed matter physics, and exotic matter. Free online books include Backus' "Continuum Mechanics" and, at a slightly simpler level, Kennet's "Introduction to Continuum Mechanics." There is also the course in continuum mechanics from MIT's Open Courseware program. An excellent resource is Heinbockel's "Introduction to Tensor Analysis and Continuum Mechanics." Also consider Burgess and van Elst's "Fluid Dynamics;" Prieve's "Fluid Mechanics with Vector Field Theory;" and Fitzpatrick's "Introduction to Plasma Physics."

For those on a budget, I recommend "Schaum's Outline of Continuum Mechanics" and "Schaum's Outline of Fluid Dynamics." I also like the following Dover books: Segel's "Mathematics Applied to Continuum Mechanics" and, a favorite of mine, Meyer's "Introduction to Mathematical Fluid Dynamics." Another favorite is Aris's "Vectors, Tensors, and the Basic Equations of Fluid Mechanics." A classic is Green and Zerna's "Theoretical Elasticity." A book that is generally good (though it lacks in its discussion of the dynamics of tornadoes, which is fanciful at best) is Vanyo's "Rotating Fluids in Engineering and Science." One of my favorite books is Marsden and Hughes' (though anything by Marsden is going to be good) "Mathematical Foundations of Elasticity." There is the true masterpiece, "Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena" by Zel'dovich and Raizer. Finally, there is "Foundations of Radiation Hydrodynamics" by Mihalis and Weibel-Mihalis.

If you have money, I recommend anything by Truesdall. I have his, "Elements of Continuum Mechanics." I recommend Fox's "Mechanics" and "Physical Hydrodynamics" by Guyon, et al. I also recommend "Physical Fluid Dynamics," the classic by Tritton; Pai's excellent, "Modern Fluid Mechanics;" Marsden and Chorin's "A Mathematical Introduction to Fluid Mechanics;" and Batchelor's classic "An Introduction to Fluid Dynamics. I also like Warsi's "Fluid Dynamics Theoretical and Computational Approaches."

For plasma physics I recommend Goldston and Rutherford's "Introduction to Plasma Physics;" Davidson's "An Introduction to Magnetohydrodynamics; and I like Hazeltine and Waelbrock's "The Framework of Plasma Physics;"

I recommend Turrell's "Gas Dynamics;" and the classic work by Chapman and Cowling, "The Mathematical Theory of Non-Uniform Gases." Finally, I really like "Principles of Condensed Matter Physics" by Chaikin and Lubensky.

Quantum Physics of Matter: This field includes elementary particle physics, nuclear physics, atomic physics, molecular physics, quantum many-body theory, the quantum theories of solids, fluids, gases, plasmas, superconductivity, and exotic matter (including superfluids and Bose-Einstein condensates.)

The free online books are Susan Cooper's "Particle and Nuclear Physics;" Surrow's "Introduction to Nuclear and Particle Physics;" Murayama's "Physics 129A Notes;" Wackeroth's "Elementary Particle Physics;" and Johnson's "Lectures on Atomic Physics." At a more elementary level we have Fox's "Atomic and Laser Physics;" MIT's"Atomic and Optical Physics;" and, at a very high level, there is Kleinert's "Gauge Fields in Condensed Matter."

There are also Marko's "Order and Disorder in Condensed Matter;" Neto's "Introduction to Condensed Matter Physics;" Sander's "Condensed Matter Physics;" and Altland and Simons' excellent text, "Concepts of Theoretical Solid State Physics." Finally, there are Dresselhaus' "Solid State Physics;" Galperin's "Modern Solid State Physics;" Casalbuoni's "Lecture Notes on Superconductivity;" and Simons and Altland's "Theories of Mesoscopic Physics."

For those on a budget I can recommend the following Dover books: Harrison's "Solid State Theory," and "Theoretical Solid State Physics," the high-level and excellent two-volume series by Jones and March.

For those with money to spend I recommend Coughlin and Dodd's "The Ideas of Particle Physics" and, my favorite, Rolnick's "Fundamental Particles and Their Interactions." Then there is "Quarks and Leptons: An Introductory Course in Modern Particle Physics" by Halzen and Martin (I actually know Francis Halzen, he is a great guy.) To round out the particle physics books I also recommend Ho-Kin and Pham's "Elementary Particles and Their Interactions;" and Muta's "Foundations of Quantum Chromodynamics;"

For nuclear physics there is Greenwood's "An Introduction to Nuclear Physics;" and Krane's "Introductory Nuclear Physics." I recommend Heyde's "Basic Ideas and Concepts in Nuclear Physics." For atomic and molecular physics there is Erkoç and Uzer's "Atomic and Molecular Physics;" Friedrich's "Theoretical Atomic Physics;" and Buyana's "Molecular Physics." I also recommend Bethe (yes, Hans Bethe) and Jackiw's "Intermediate Quantum Mechanics."

To finish the physics of matter I reccomend Imry's "Introduction to Mesoscopic Physics;" and Ibach and Lüth's "Solid State Physics." One of my favorite books is Ashcroft and Mermin's "Solid State Physics." A very useful book is Modinos' "Quantum Theory of Matter." Finally, there are Tinkham's "Introduction to Superconductivity;" Ketterson and Song's "Superconductivity;" and the classic work by Tilley and Tilley, "Superfluidity and Superconductivity."

General Relativity: One of the two most successful theories of physics is that of general relativity. Thus, it is important to be able to learn about it. There are many free online books about relativity. My favorite is Sean Carroll's notes, "Lecture Notes on General Relativity." There are also Aldrovandi and Pereira's "An Introduction to General Relativity" and their book "An Introduction to Gravitation Theory" and Blau's "Lecture Notes on General Relativity."

For those on a budget, I recommend the Dover book, "The Principle of Relativity." while this book is attributed only to Einstein, it contains numerous seminal papers on relativity, including Einstein's original papers. I also like Pauli's book, "The Theory of Relativity," also from Dover.

For those with money to spend I can think of no better introductory text than Ray D'Inverno's "Introducing Einstein's Relativity." The single best book on relativity, in my opinion, is Misner, Thorne, and Wheeler's "Gravitation." Another great book is Wald's, "General Relativity," and Hawking (yes, THAT Hawking,) and Ellis' "The Large-Scale Structure of Spacetime."

Quantum Field Theory: The second of the two most successful theories of physics is quantum field theory. The best of the free online books is Warren Siegel's magnum opus, "Fields." Also online are Varchenko's "Mathematics of Quantum Field Theory;" MIT's "Geometry of Quantum Field Theory;" van Baal's "A Course in Field Theory;" and Norbury's "Quantum Field Theory." Also, Casalbuoni's "Quantum Field Theory" and his "Advanced Quantum Field Theory." And Gunion's "Class Notes for Quantum Field Theory I, II, and III;" Oeckl's "Quantum Geometry and Quantum Field Theory;" Simons' "Concepts in Theoretical Physics;" and Hindmarsh's "Relativistic Quantum Fields I, II, and III."

For those on a budget, I recommend the Dover book by Heitler, "The Quantum Theory of Radiation," a truly excellent book, if somewhat out of date. For those who can spend money, I recommend Feynman's "Quantum Electrodynamics;" Veltman's "Diagrammatica;" Kreimer's "Knots and Feynman Diagrams;" and "Relativistic Quantum Mechanics and Field Theory" by Gross, who recently shared the Nobel prize.

My favorite on this subject is the wonderful book by Zee, "Quantum Field Theory in a Nutshell." My second favorite is the set by Weinberg (also a Nobel winner) "The Quantum Theory of Fields I, II, and III." My third favorite is by Peskin and Schroeder, "An Introduction to Quantum Field Theory."

Collections: These are collections of papers, chapters, or books written by a single author about a wide breadth of physics. My favorite free online book is "Applications of Classical Physics" by Kip Thorne and Roger Blandford. James Binney has written about many topics in mathematics and physics. For those on a budget, I recommend "The Pauli Lectures on Physics," published by Dover books.

For those who have money, I highly recommend Landau and Lifshitz's "Course in Theoretical Physics." This set, which holds a place of honor on my bookshelves, includes vol. I: Mechanics, vol. II: The Classical Theory of Fields, vol. III: Quantum Mechanics, vol. IV: Statistical Physics (part 1), vol. V: Fluid Mechanics, vol. VI: Theory of Elasticity, vol. VII: Electrodynamics of Continuous Media, vol. VIII: Statistical Physics (part 2), and vol. IX: Physical Kinetics."

Next to Landau and Lifshitz on my book shelves is the series by Sommerfeld, "Lectures on Theoretical Physics," This includes vol. I: Mechanics, vol. II: Mechanics of Deformable Bodies, vol. III: Electrodynamics, vol. IV: Optics, vol. V: Thermodynamics and Statistical Mechanics, and vol. VI: Partial Differential Equations," I also greatly prize Thirring's set of books, "A Course in Mathematical Physics," which includes vol. I: Classical Dynamics, vol. II: Classical Field Theory, vol. III: Quantum Mechanics of Atoms and Molecules, and vol. IV: Quantum Mechanics of Large Systems."

So what do you do with all of these books? It would take a lifetime to read them all. You use them as reference material. When you develop a project idea, you begin to search through them and learn about the subject of your project idea. Ask questions and find the answers. When you can no longer find the answers, you should have enough information to begin to develop your own answers. When you have completed the answer to your own question, let it sit for a day. Then assume you got the wrong answer and find ut where you made your mistake. Let it sit for another day, then assume the answer was correct and defend it from the perceived mistake of the previous day.

Continue in this fashion until you either are no longer able to find anything wrong with your answer, or you can no longer defend its difficulties. When that happens, call an expert and ask nicely if they would mind looking over your work at their convenience. Mind you, every professional scientist gets hundreds or more calls from crackpots who think they have made a major discovery, only to find that they don't have the least clue what they are talking about. Be patient, it is hard to fight against that sort of trend. If you are truly careful in your work you might surprise the professionals. More likely you have overlooked something; if so, accept it and ask for advice. In this way you will develop a working relationship with a professional scientist, who will then be more likely to accept your work in the future. Eventually you will be correct and then this pro might actually want to work with you.

I will be demonstrating how to perform such a project over the next set of columns. I hope they will be illuminating. I hope you will become involved. I look forward to seeing your contributions. Any acceptable contributions will be included as addenda to my columns, so long as the contributor gives their consent to me and the Editor.
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Created by Mathematica (October 17, 2004)