Phys-197: Analytical Methods in Physics

Course Assistants

  • Regular
    • Mehmet Batu Bayındırlı
    • Şebnem Karaçubuk
    • Dilara Kosva
    • Oğuz Öner
    • Ali Pazarcı
  • Part time
    • Mert Akdenizli
    • Ege Aktener
    • Ege Çoban
    • Mehmet Dede
    • Mustafa Mullahasanoğlu

Projects

Projects will involve studying a paper of interest, writing a 3 – (infinity) page report and making a brief oral online presentation (10-15 minutes). Below are project suggestions, each student supposed to prepare one project. If you have other ideas regarding topics, you can contact me. Please, try to decide on your project topic by November 8, 2020.

Distribution of projects

  1. P.Levrie, Lost and Found: An Unpublished  ζ(2) Proof, The Mathematical Intelligencer volume 33, pages 29–32 (2011) [link]
  2. Daniele Ritelli, Another Proof of ζ(2) Using Double Integrals, The American Mathematical Monthly, 120:7, 642-645 (2013), [link]
  3. Thomas J. Osler, Finding ζ(2p) from a Product of Sines, The American Mathematical Monthly, Vol. 111, No. 1, pp. 52-54 (2004) [pdf]
  4. J. Starrett, Solving Differential Equations by Symmetry Groups, The American Mathematical Monthly , Nov., 2007, Vol. 114, No. 9, pp. 778-792 [pdf]
  5. Peter D. Lax, The Cauchy Integral Theorem, The American Mathematical Monthly , Oct., 2007, Vol. 114, No. 8, pp. 725-727 [pdf]
  6. Q. van der Hoff and A. Harding, Review of a predator-prey model with two limit cycles, International Journal of Mathematical Education in Science and Technology, 50:6, 921-933 (2019) [link]
  7. Frank R. Pfaff, A Commutative Multiplication of Number Triplets, The American Mathematical Monthly , Feb., 2000, Vol. 107, No. 2, pp. 156-162 [pdf]
  8. Dov Aharonov, Alan Beardon and Kathy Driver, Fibonacci, Chebyshev, and Orthogonal Polynomials, The American Mathematical Monthly , Aug. – Sep., 2005, Vol. 112, No. 7, pp. 612-630 [pdf] (project for two students)
  9. Alan Edelman and Gilbert Strang, Pascal Matrices, The American Mathematical Monthly , Mar., 2004, Vol. 111, No. 3, pp. 189-197 [pdf]
  10. Paul Monsky, Kepler’s First Law: A Remark, The American Mathematical Monthly, Vol. 111, No. 4, pp. 355-356 [pdf]
  11. Grzegorz Rzadkowski, A Short Proof of the Explicit Formula for Bernoulli Numbers, The American Mathematical Monthly , May, 2004, Vol. 111, No. 5, pp. 432-434 [pdf]
  12. Darko Veljan, Geometry and Convexity of cos √x, The American Mathematical Monthly , Aug. – Sep., 2004, Vol. 111, No. 7, pp. 592-595 [pdf]
  13. Jonathan M. Borwein, Hilbert’s Inequality and Witten’s Zeta-Function, The American Mathematical Monthly, Feb., 2008, Vol. 115, No. 2,pp. 125-137 [pdf] (project for two students)
  14. Grzegorz Rzadkowski, Derivatives and Eulerian Numbers, The American Mathematical Monthly , May, 2008, Vol. 115, No. 5, pp. 458-460 [pdf]
  15. Peter Hilton and Jean Pedersen, The Euler Characteristic and Pólya’s Dream, The American Mathematical Monthly , Feb., 1996, Vol. 103, No. 2, pp. 121-131[pdf]
  16. K. E. Bisshopp, The Triple Vector Product A × (B × C), The American Mathematical Monthly , Feb., 1965, Vol. 72, No. 2, pp. 163-165 [pdf]
  17. Louis Zulli, Charting the 3-Sphere–An Exposition for Undergraduates, The American Mathematical Monthly , Mar., 1996, Vol. 103, No. 3, pp. 221-229 [pdf]
  18. Walter B. Carver, Complex Numbers and their Graphical Representation, The American Mathematical Monthly , Nov., 1956, Vol. 63, No. 9, Part 2, pp. 1-7 [pdf]
  19. Shigeru Haruki, A Property of Quadratic Polynomials, The American Mathematical Monthly , Aug. – Sep., 1979, Vol. 86, No. 7, pp. 577-579 [pdf]
  20. Tyre A. Newton, A Simple Algorithm for Finding Eigenvalues and Eigenvectors for 2 × 2 Matrices, The American Mathematical Monthly , Jan., 1990, Vol. 97, No. 1, pp. 57-60 [pdf]
  21. W. R. Utz, The Matric Equation X2 = A, The American Mathematical Monthly , Dec., 1979, Vol. 86, No. 10, pp. 855-856 [pdf]
  22. A. J. B. Ward, A link between the matrix equation AX –XB = C and the matrix quadratic, International Journal of Mathematical Education in Science and Technology, 25:3, 407-411 (1994) [link]
  23. Munir Mahmood, Sali Hammad and Ibtihal Mahmood, An efficient algorithm for computing the roots of general quadratic, cubic and quartic equations, International Journal of Mathematical Education in Science and Technology, 45:7, 1095-1103 (2014) [link]
  24. N. Karjanto, Properties of Chebyshev polynomials, arXiv:2002.01342
  25. M. Z. Rafat, D. Dobie, Throwing π at a wall, arXiv:1901.06260
  26. A. Barnes, Geometrical meaning of the curl operation when A⋅curlA≠0, American Journal of Physics 45, 371 (1977) [link] (see also link)
  27. J. P. McKelvey, The case of the curious curl, American Journal of Physics 58, 306 (1990) [link]
  28. Franco Battaglia, and Thomas F. George, Tensors: A guide for undergraduate students, American Journal of Physics 81, 498 (2013) [link]
  29. Roger B. Nelsen, Elementary Proofs of the Trigonometric Inequalities of Huygens, Cusa, and Wilker, Mathematics Magazine, 93:4, 276-283 (2020) [link]
  30. Alan F. Beardon, Complex Exponents and Group Theory, Mathematics Magazine, 93:3, 186-192 (2020) [link]
  31. John Engbers & Adam Hammett, Silos, Snow Cones, Railroad Spikes, and Spherical Coordinates, Mathematics Magazine, 93:2, 143-147 (2020) [link]
  32. Syed Abbas, Proving the Extended Binomial Theorem Using Ordinary Differential Equations, Mathematics Magazine, 93:1, 33-35 (2020) [link]
  33. David Treeby, Randomly Generated Identities, The College Mathematics Journal, 51:2, 90-94 (2020) [link]
  34. Russell Euler, Maclaurin Expansion of Arctan x via L’Hôspital’s Rule, The College Mathematics Journal, 24:4, 347-350 (1993) [link]
  35. E. John Hornsby Jr., Geometrical and Graphical Solutions of Quadratic Equations, The College Mathematics Journal, 21:5, 362-378 (1990) [link]
  36. Floyd Vest, Graphing the Complex Roots of a Quadratic Equation, The College Mathematics Journal, 16:4, 257-261(1985) [link]
  37. Elena Anne Marchisotto & Gholam-Ali Zakeri, An Invitation to Integration in Finite Terms, The College Mathematics Journal, 25:4, 295-308 (1994) [link]
  38. Arthur T. Benjamin & Daniel Walton, Counting on Chebyshev Polynomials, Mathematics Magazine, 82:2, 117-126, (2009) [link]
  39. David Speyer & Bernd Sturmfels, Tropical Mathematics, Mathematics Magazine, 82:3, 163-173, (2009) [link]
  40. Raymond A. Beauregard & Vladimir A. Dobrushkin, Powers of a Class of Generating Functions, Mathematics Magazine, 89:5, 359-363 (2016) [link]
  41. Harvey Diamond, Defining Exponential and Trigonometric Functions Using Differential Equations, Mathematics Magazine, 87:1, 37-42 (2014) [link]
  42. Connie Xu, A Unique Area Property of the Quadratic Function, Mathematics Magazine, 87:1, 52-56 (2014) [link]
  43. Ernst E. Scheufens, From Fourier Series to Rapidly Convergent Series for Zeta(3), Mathematics Magazine, 84:1, 26-32 (2011) [link]
  44. Alexander Kheifets & James Propp, A Counterexample to Integration by Parts, Mathematics Magazine, 83:3, 222-225 (2010) [link]
  45. Shirley Llamado Yap, Differential Equations—Not Just a Bag of Tricks!, Mathematics Magazine, 83:1, 3-14 (2010) [link]
  46. Gary Brookfield, Yet Another Elementary Solution of the Brachistochrone Problem, Mathematics Magazine, 83:1, 59-63 (2010) [link]
  47. Peter M. Knopf, The Operator (x d/dx)^n and Its Applications to Series, Mathematics Magazine, 76:5, 364-371 (2003) [link]
  48. James P. Lesko & Wendy D. Smith, A Laplace Transform Technique for Evaluating Infinite Series, Mathematics Magazine, 76:5, 394-398 (2003) [link]
  49. G. Hossein Behforooz, Thinning Out the Harmonic Series, Mathematics Magazine, 68:4, 289-293 (1995) [link]
  50. Kenneth S. Miller, Derivatives of Noninteger Order, Mathematics Magazine, 68:3, 183-1922 (1995) [link]
  51. F. M. S. Lima, A shortcut for solving the ubiquitous integral ∫ x^n e^{αx} dx using only derivatives, International Journal of Mathematical Education in Science and Technology, 51:1, 136-140 (2020) [link]
  52. Yukio Kobayashi, Geometrical meaning of arithmetic series , and in terms of the elementary combinatorics, International Journal of Mathematical Education in Science and Technology, 42:5, 657-664 (2011) [link]
  53. Miguel Casquilho & Jorge Buescu, A minimum distance: arithmetic and harmonic means in a geometric dispute, International Journal of Mathematical Education in Science and Technology, 42:3, 399-405 (2011) [link]
  54. Yukio Kobayashi, Integral representation of the pictorial proof of , International Journal of Mathematical Education in Science and Technology, 42:2, 235-239 (2011) [link]
  55. Thierry Dana-Picard, Integral presentations of Catalan numbers and Wallis formula, International Journal of Mathematical Education in Science and Technology, 42:1, 122-129 (2011) [link]
  56. Shiv K. Gupta & Waclaw Szymanski, Cubic Polynomials with Rational Roots and Critical Points, The College Mathematics Journal, 41:5, 365-369 (2010) [link]
  57. Duane W. DeTemple, Counting Squares to Sum Squares, The College Mathematics Journal, 41:3, 214-221 (2010) [link]
  58. Robert D. Poodiack & Kevin J. LeClair, Fundamental Theorems of Algebra for the Perplexes, The College Mathematics Journal, 40:5, 322-336 (2009) [link]
  59. Hongwei Chen, The Fresnel Integrals Revisited, The College Mathematics Journal, 40:4, 259-262 (2009) [link]
  60. Richard Hammack, A Geometric View of Complex Trigonometric Functions, The College Mathematics Journal, 38:3, 210-217 (2007) [link]
  61. S. A. Fulling, How to Avoid the Inverse Secant (and Even the Secant Itself), The College Mathematics Journal, 36:5, 381-387 (2005) [link]
  62. Poo-Sung Park, Ramanujan’s Continued Fraction for a Puzzle, The College Mathematics Journal, 36:5, 363-365 (2005) [link]
  63. Eugene Boman & Frank Uhlig, When 1/(a+b)=1/a+1/b, Anyway?, The College Mathematics Journal, 33:4, 296-300 (2002) [link]
  64. Herb Bailey, On Running in the Rain, The College Mathematics Journal, 33:2, 88-92 (2002) [link]
  65. Asuman Aksoy & Mario Martelli, Mixed Partial Derivatives and Fubini’s Theorem, The College Mathematics Journal, 33:2, 126-130 (2002) [link]
  66. Frank Burk, Summing Series via Integrals, The College Mathematics Journal, 31:3, 178-181 (2000) [link]
  67. A. P. Stone, Power Series Expansions for Trigonometric Functions via Solutions to Initial Value Problems, Mathematics Magazine, 64:4, 247-251 (1991) [link]
  68. T. Y. Lam, On the Diagonalization of Quadratic Forms, Mathematics Magazine, 72:3, 231-235 (1999) [link]
  69. Seyoum Getu, Evaluating Determinants via Generating Functions, Mathematics Magazine, 64:1, 45-53 (1991) [link]
  70. Dumitru Acu, Some Algorithms for the Sums of Integer Powers, Mathematics Magazine, 61:3, 189-191 (1988) [link]
  71. W. B. R. Lickorish & K. C. Millett, The New Polynomial Invariants of Knots and Links, Mathematics Magazine, 61:1, 3-23 (1988) [link] (project for two students)

Examination

The grade will be composed as follows: problem sessions 30%, homework 30%, typed notes of a project (pdf and tex-files) 10%, presentation 10%, exams 20%. The projects will be evaluated according to the clarity of exposition, effort, and calculations.

Literature

I am not going to follow any specific book. List of some books:

  • Y. B. Zeldovich, I. M. Yaglom, Higher Math for Beginning Physicists and Engineers
  • Victor V. Prasolov, Polynomials, 1999
  • Victor V. Prasolov, Intuitive Topology, 1995
  • A. Kurosh, Higher algebra, 1975
  • Van der Waerden, Modern Algebra, 1930
  • E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, 1902

Final results of students