An alternative form of the theorem states that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the "pea and the Sun paradox".
the pea and the sun a mathematical paradox pdf free
The theorem is called a paradox because it contradicts basic geometric intuition. "Doubling the ball" by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems to be impossible, since all these operations ought, intuitively speaking, to preserve the volume. The intuition that such operations preserve volumes is not mathematically absurd and it is even included in the formal definition of volumes. However, this is not applicable here because in this case it is impossible to define the volumes of the considered subsets. Reassembling them reproduces a set that has a volume, which happens to be different from the volume at the start.
Unlike most theorems in geometry, the mathematical proof of this result depends on the choice of axioms for set theory in a critical way. It can be proven using the axiom of choice, which allows for the construction of non-measurable sets, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices.[2]
then it can be said that A and B are G-equidecomposable using k pieces. If a set E has two disjoint subsets A and B such that A and E, as well as B and E, are G-equidecomposable, then E is called paradoxical.
The heart of the proof of the "doubling the ball" form of the paradox presented below is the remarkable fact that by a Euclidean isometry (and renaming of elements), one can divide a certain set (essentially, the surface of a unit sphere) into four parts, then rotate one of them to become itself plus two of the other parts. This follows rather easily from a F2-paradoxical decomposition of F2, the free group with two generators. Banach and Tarski's proof relied on an analogous fact discovered by Hausdorff some years earlier: the surface of a unit sphere in space is a disjoint union of three sets B, C, D and a countable set E such that, on the one hand, B, C, D are pairwise congruent, and on the other hand, B is congruent with the union of C and D. This is often called the Hausdorff paradox.
They point out that while the second result fully agrees with geometric intuition, its proof uses AC in an even more substantial way than the proof of the paradox. Thus Banach and Tarski imply that AC should not be rejected solely because it produces a paradoxical decomposition, for such an argument also undermines proofs of geometrically intuitive statements.
The unit sphere S2 is partitioned into orbits by the action of our group H: two points belong to the same orbit if and only if there is a rotation in H which moves the first point into the second. (Note that the orbit of a point is a dense set in S2.) The axiom of choice can be used to pick exactly one point from every orbit; collect these points into a set M. The action of H on a given orbit is free and transitive and so each orbit can be identified with H. In other words, every point in S2 can be reached in exactly one way by applying the proper rotation from H to the proper element from M. Because of this, the paradoxical decomposition of H yields a paradoxical decomposition of S2 into four pieces A1, A2, A3, A4 as follows:
Finally, connect every point on S2 with a half-open segment to the origin; the paradoxical decomposition of S2 then yields a paradoxical decomposition of the solid unit ball minus the point at the ball's center. (This center point needs a bit more care; see below.)
There are student versions of Mathematica, Maple and MATLAB available,which sell for the cost of a typical mathematical physics textbook.If you do not wish to invest any money at this time,you can use Mathematica and MATLAB for free at computer labs on campus.For further information click here.A complete list of all the software available in the LearningTechnologies computer labs, along with the file path to find the programs on each workstation can be found at thislink. Back to the Top
1. A very good source for free mathematical textbooks can be found onFreeSciencewebpage. I can also provide another useful linkto a list of freemathematics textbooks. 2. Sean Mauch (of Caltech) provides a free massive textbook (2321 pages) entitledIntroductionto Methods of Applied Mathematics. However, many of thetopics of Physics 116C, if treated by Mauch, are in a preliminary status.3. James Nearing also provides a free textbook entitled Mathematical Tools for Physics. This book treats some of the topics of Physics 116C at a similar level of difficulty. 4. Russel L. Herman has provided notes for a secondcourse in differential equations, covering linear and nonlinear systems andboundary value problems. His monograph is entitledASecond Course in Ordinary Differential Equations: Dynamical Systems andBoundary Value Problems. Of particular interest to Physics 116Care Chapter 6 on Sturm Liouville problems, Chapter 7 on specialfunctions (including Bessel functions and a variety of orthogonalpolynomials) and Chapter 8 on Green functions.5. Kenneth Howell provides links to the chapters of his book entitledApplied DifferentialEquations, Parts 1--4. Of particular interest to Physics116C students are the additional chapters that compriseParts 5--7,which include six chapters on series solutionsto differential equations, and four chapters on boundaryvalue problems and Sturm Liouville problems. A consensed versionof his chapter on Sturm Liouville problems can be found in hislecture notesfor a course in mathematical physics. 6. R.S. Johnson, a professor of applied mathematics at the Universityof Newcastle upon Tyne has provided, free of charge, a series ofmini-books (from the Notebook Series) that cover some of the topics of Physics 116C. These includeThe series solution of second order, ordinary differential equations and special functions
Sturm-Liouville Theory
Partial differential equations: classification and canonical forms
7. A textbook entitled Partial Differential Equations for Scientists and Engineers, by Geoffrey Stephenson is available from the following PDF link.8. A textbookentitled Introduction to Probability, by Charles M. Grinstead and J. Laurie Snell has been generously made available free of charge from thefollowing PDF link, along with PDF solutions to all odd-numbered problems.I especially recommend Chapter 4 for its superbtreatment of conditional probability, with applications to the MontyHall problem, the two-child paradox, and other purported paradoxes thatarise in probability theory.9. Another free textbook on probability entitled Introduction to Probability byDimitri P. Bertsekas and John N. Tsitsiklis is based on a coursetaught at MIT. The level of difficulty is appropriate for Physics116C students. [PDF] Back to the Top
1. All the examples of the indicial equations given in Boas possess real roots. But a quadratic equation with real coefficientscan also have complex roots. Applying the Frobenius method in thecase of complex indicial roots is treated pedagogically in a paper entitledThe Frobenius method for complex roots of the indicial equation by Joseph L. Neuringer, International Journal of Mathematical Educationin Science and Technology, Volume 9, Issue 1, 1978, pages 71--77.    [PDF]2. The most readable account that I have come across on the Sturm Liouville problem, suitable for Physics 116C students, is Chapter 11of R. Kent Nagle, Edward B. Saff and Arthur David Snider,Fundamentals of Differential Equations and Boundary Value Problems, 6th Edition (Pearson Education, Inc., Boston, MA, 2012). Unfortunately, this book is too expensive, and there are no previews available either from google or amazon. You may be able to find the most recent or previous editions of this book in the Science and Engineering Library.3. If you were wondering whether the Laguerre differential equationcan also be solved by factorizing the equation into a product offirst order differential operators and constructing ladder (raising andlowering) operators, check out the article by H. Fakhri and A. Chenaghlouentitled Ladderoperators for the associated Laguerre functions inJ. Phys. A: Math. Gen. 37(2004) pp. 7499--7507.     [PDF]4. The factorization method for solving differential equations wasmade popular by physicists. The method was described in a classicreview paper by L. Infeld and T.E. Hullentitled The Factorization Method in Reviews of Modern Physics 23 (1951) pp. 21--68.     [PDF]5. One of the best elementary treatments of partial differentialequations suitable for physicists can be found in a book by Gabriel Barton,Elements of Green's Functions and Propagation---Potentials,Diffusion and Waves, published byOxford Science Publications in 1989.Among other things, there is a very clear discussion of which boundary conditions constitute a well-posed problem.6. Spherical harmonics play a critical role in computer graphics.Colorful visual representations of the spherical harmonics provideadded insight into their properties and significance. See forexample, the following two articles:Volker Schönefeld, Spherical Harmionics     [PDF]
Peter-Pike Sloan, Stupid Spherical Harmonics Tricks     [PDF]
7. In the Monty Hall problem, there are three doors. Behind one ofthem is a brand new car, whereas the other two doors conceal goats.Monty Hall asks you to choose a door without opening it. Then, heopens one of the other doors to reveal a goat. At this point,Monty asks you whether youwish to switch your choice of doors or to stand pat. Indeed, what is the probabilty of winning the car if you switch your choice of doors?When Marylin vos Savant answered this question in the September 9, 1990issue of Parade magazine, the reader response was overwhelming, manyof whom could not believe her proposed answer that switching resultedin a probability of 2/3 for winning the car. Wikipediaprovides an excellent introduction to this problem. For a moredetailed account, read about the historyof the Monty Hall problem and its many variants in this enertainingbook by Jason Rosenhouse entitledThe Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brain Teaser (Oxford University Press, Oxford, UK, 2009).8. Apparent paradoxes often arise in probability theory. One suchparadox is the called the two-child paradox, which asks for theprobability that a two-child family has two boys given that one of the children is known to be a boy. Once again,Wikipediaprovides an excellent introduction to this problem. I also highly recommend Chapter 4 of the free textbook entitled Introduction to Probability, by Charles M. Grinstead and J. LaurieSnell previously cited in Section XIII of this website. In addition,check out a fascinating article by Peter Lynch,The Two-Child Paradox: Dichotomy and Ambiguity in Irish Mathematical Society Bulletin 67(2011) pp. 67--73 [PDF].9. The Monty Hall problem and the two-child paradox can be analyzedby using conditional probabilites. Once again, I highly recommend Chapter 4 ofthe free textbook entitled Introduction to Probability, by Charles M. Grinstead and J. LaurieSnell previously cited in Section XIII of this website.Another excellent treatmentof these and other well-known puzzles in probability can be foundin an article by Maya Bar-Hillel and RumaFalk, Some teasers concerning conditional probabilities in Cognition 11 (1982) pp. 109--122 [ PDF]. 10. Continuous sample spaces can lead to apparent paradoxical behavior. I highly recommend Chapter 2 ofthe free textbook entitled Introduction to Probability, by Charles M. Grinstead and J. LaurieSnell previously cited in Section XIII of this website. This chapterincludes a discussion of Buffon's needle (and the relation of pi tothe probability of where a dropped needle lies). This chapter alsoprovides a treatment ofBertrand's paradox, which involves a randomly drawn chord in the unit circle.11. Finally, I can resist in mentioningTheBanach-Tarski paradox which exhibits some very unintuitive phenomena that can arise in dealing with sets with an uncountable infinite number of points. For a very readable account, check out the book byLeonard M. Wapner entitledThe Pea and the Sun: A Mathematical Paradox (A.K. Peters, Ltd., Wellesley, MA, 2005). For a slightly more technical treatment(which is still accessible to advanced undergraduate students), have a look at R. French, The Banach-Tarski Theorem, The Mathematical Intelligencer 10 (1988) 21--28.12. The law of large numbers and the central limit theorem are treatedat an elementary level in Chapters 8 and 9, respectively of Introduction to Probability, by Charles M. Grinstead and J. LaurieSnell, previously cited in Section XIII of this website.13. David L. Streiner provides an article entitled Maintaining Standards: Differences between the Standard Deviation and Standard Error, and When to Use Each, which should help illuminate the distinction between the standard deviation and the standard error (also known as the standard deviation of the mean). [PDF]X. Other Web Pages of Interest1. A superb resource for both the elementary functions and thespecial functions of mathematical physics is theHandbookof Mathematical Functions by Milton Abramowitz and Irene A. Stegun,which is freely available on-line. The home page for thisresource can be foundhere. There, youwill find links to a framesinterface of the book. Another scan of the book can be found here.A third independent link to the book can be foundhere.2. The NISTHandbook of Mathematical Functions (publishedby Cambridge University Press), together with its Webcounterpart, the NIST Digital Library of Mathematical Functions(DLMF), is the culmination of a project that was conceived in 1996 atthe National Institute of Standards and Technology (NIST). The projecthad two equally important goals: to develop an authoritativereplacement for the highly successful Handbook of MathematicalFunctions with Formulas, Graphs, and Mathematical Tables, published in1964 by the National Bureau of Standards (M. Abramowitz andI. A. Stegun, editors); and to disseminate essentially the sameinformation from a public Web site operated by NIST. The new Handbookand DLMF are the work of many hands: editors, associate editors,authors, validators, and numerous technical experts.The NISTHandbook coversthe properties of mathematical functions, from elementarytrigonometric functions to the multitude of special functions.All of the mathematical information contained in the Handbook is alsocontained in the DLMF, along with additional features such as moregraphics, expanded tables, and higher members of some families offormulas. A PDF copy of the handbook is provided here: [PDF]3. Another very useful reference for both the elementary functions and thespecial functions of mathematical physics is An Atlasof Functions (2nd edition) by Keith B. Oldham, Jan Myland and JeromeSpanier, published by Springer Science in 2009.This resource is freely available on-line to studentsat the University of California at thislink.4. Yet another excellent website for both the elementary functions and thespecial functions of mathematical physics is the Wolfram Functionssite. This site was created with Mathematica and is developed and maintainedby Wolfram Research with partial support from the National Science Foundation. 5. One of the classic references to special functions is a threevolume set entitled Higher Transcendental Functions (edited by A. Erdelyi), which wascompiled in 1953 and is based in part on notes left by Harry Bateman.This was the primary reference for a generation of physicists andapplied mathematicians, which is colloquially referred to as theBateman Manuscripts. Although much of the material goes considerablybeyond the level of Physics 116C, this esteemed reference work continues to be a valuable resources for students and professionals. PDF versions of the three volumes are now available free of charge. Check out the three volumes by clicking on the relevant links here: [Volume 1 Volume 2 Volume 3].6. Google is invaluable for searching for mathematical information.For example, if I need the first few Bessel function zeros, I searchwith google for "Bessel function zeros." The first hit is Besselfunction zeros on the Wolfram MathWorld site. 7. Wikipedia provides some good articles on various mathematicaltopics. For example, a simple exposition on the the law of large numbers can be foundhere. The Wikipedia treatment of central limittheorem is more technical. What I discussed in class was the classical CLT (or Lindeberg-Levy CLT), which is presented in the firstparagraph of the Wikipedia article. However, Wikipedia has anotherpage that illustrates the central limit theorem which is quiteilluminating. Check it out at thislink. Back to the Top 2ff7e9595c
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