Problem of the Week, #121

A former colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his students, giving a prize of a cookie if they could solve it. Here is one of them.

Problem 121

The Maryland “Big Game” lottery is played by selecting 5 different numbers in \{ 1,2,3,\dots, 50\} and then selecting one of the numbers in \{ 1,2,3,\dots, 36\}. The first section is an unordered selection without replacement (so, arrange them in increasing order if you like) but the second selection can repeat one of the 5 numbers initially picked.

How many ways can this be done?

Problem of the week, 137

A former colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his US Naval Academy students, giving a prize of a cookie if they could solve it. One of them is given below.

Chain addition is a technique employed in cryptography for extending a short sequence of digits, called the seed to a longer sequence of pseudorandom digits. Quoting David Kahn (in Kahn on Codes, MacMillan, New York, 1983, p. 154), “the first two digits of the [seed] are added together modulo 10 [which means they are added and the carry is neglected] and the result placed at the end of the [sequence], then the second and third digits are added and the sum placed at the end, and so forth, using also the newly generated digits when the [seed] is exhausted, until the desired length is obtained”. Thus, the seed 3964 yields the sequence 3964250675632195… .

Periodic pattern

Periodic pattern

a. Show that this sequence eventually repeats itself.
b. Show that the sequence begins repeating itself with “3964”.
c. EXTRA CREDIT: How many digits are there before the first repetition of “3964”?

Problem of the week, 148

A former colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his US Naval Academy students, giving a prize of a cookie if they could solve it. One of them is given below.

 

Suppose p and q are each monic polynomials of degree 4 with real coefficients and the intersection of their graphs is {(1, 3), (5, 21)}. If p(3) – q(3) = 20, what is the area enclosed by their graphs?

Problem of the week, 150

A former colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his US Naval Academy students, giving a prize of a cookie if they could solve it. One of them is given below.
 

 

Let a, b, and c be real numbers and let f and g be real valued functions of a real variable such that \lim_{x\to a} g(x) = b and \lim_{x\to b} f(x) = c.
a. Give an example in which \lim_{x\to a} f(g(x)) \not= c.
b. Give an additional condition on f alone and show that it
guarantees \lim_{x\to a} f(g(x)) = c.
c. Give an additional condition on g alone and show that it
guarantees \lim_{x\to a} f(g(x)) = c.

Real world applications of representation theory

(Subtitle: Representation theorists will rule the world one day just you wait)

 

This post describes some applications of representation theory of non-abelian groups to various fields and gives some references.

  • Engineering.
    • Tensegrity – the design of “strut-and-cable” constructions.Want to build a building with cables and struts but don’t know representation theory? Check out these references:
      • R. Connelly and A. Back, “Mathematics and tensegrity”, Amer Scientist, April-May 1998, pages 142-151
      • symmetric tensegrities
    • Telephone network designs.This is the information age with more and more telephone lines needed every day. Want to reach out and touch someone? You need representation theory.
      • F. Bien, “Construction of telephone networks by group representations”, Notices A. M. S. 36(1989)5-22
    • Nonlinear network problems.This is cheating a little since the works in the reference below really use the theory of Lie groups instead of representation theory itself. Still, there is a tangential relation at least between representation theory of Lie groups and the solution to certain nonlinear network problems.
      • C. Desoer, R. Brockett, J. Wood, R. Hirshorn,
        A. Willsky, G. Blankenship, Applications of Lie group theory to nonlinear network problems, (Supplement to IEEE Symposium on Circuit Theory, 1974), Western Periodicals Co., N. Hollywood, CA, 1974
    • Control theory.
      • R. W. Brockett, “Lie theory and control systems defined on spheres”, SIAM J on Applied Math 25(1973) 213-225
    • Robotics.The future is not in plastics (see the movie “The Graduate“) but in robotics.
      How do you figure out their movements before building them? You guessed it, using representation theory.

      • G. Chirikjian, “Determination and synthesis of discretely actuated manipulator workspaces using harmonic analysis”, in Advances in Robotic Kinematics, 5, 1996, Springer-Verlag
      • G. Chirikjian and I. Ebert-Uphoff, “Discretely actuated manipulator workspace generation by closed-form convolution”, in ASME Design Engineering Technical Conference, August 18-22 1996
    • Radar design.W. Schempp, Harmonic analysis on the Heisenberg nilpotent Lie group, with
      applications to signal theory
      , Longman Scientific & Technical, New York (Copublished in the U.S. with Wiley), 1986.
    • Antenna design.B. Hassibi, B. Hochwald, A. Shokrollahi, W. Sweldens, “Representation theory for high-rate multiple antenna code design,” 2000 preprint (see A. Shokrollahi’s site for similar works).
    • Design of stereo systems.We’re talkin’ quadrophonic state-of-the-art.
      • K. Hannabus, “Sound and symmetry”, Math. Intelligencer, 19, Fall 1997, pages 16-20
    • Coding theory. Interesting progress in coding theory has been made using group theory and representation theory. Here are a few selected references.
      • F. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes,
        North-Holland/Elsevier, 1993 (8th printing)
      • I. Blake and R. Mullin, Mathematical Theory of Coding, Academic Press, 1975
        49(1995)215-223
      • J.-P. Tillich and G. Zemor,
        “Optimal cycle codes constructed from Ramanujan graphs,” SIAM J on Disc. Math. 10(1997)447-459
      • H. Ward and J. Wood, “Characters and the equivalence of codes,” J. Combin. Theory A 73348-352
      • J. Lafferty and D. Rockmore, “Spectral Techniques for Expander Codes” , (Extended Abstract) 1997 Symposium on Theory of Computation (available
        at  Dan Rockmore’s web page)
  • Mathematical physics.
    Any complete list of books and papers in this field which use representation theory would be much too long for the limited goal we have here (which is simply
    to list some real-world applications). A small selection is given below.

    • Differential equations (such as the heat equation, Schrodinger wave equation, etc).M. Craddock, “The symmetry groups of linear partial differential equations
      and representation theory, I” J. Diff. Equations 116(1995)202-247
    • Mechanics.
      • D.H. Sattinger, O.L. Weaver, Lie Groups and Algebras With Applications to Physics, Geometry, and Mechanics (Applied Mathematical Sciences, Vol 61) , Springer Verlag, 1986
      • Johan Belinfante, “Lie algebras and inhomogeneous simple materials”,
        SIAM J on Applied Math 25(1973)260-268
    • Models for elementary particles.
    • Quantum mechanics.
      • Eugene Wigner, “Reduction of direct products and restriction of representations to subgroups: the everyday tasks of the quantum theorists”, SIAM J on Applied Math 25(1973) 169-185
      • V. Vladimirov, I. Volovich, and E. Zelenov, “Spectral theory in p-adic quantum mechanics and representation theory,” Soviet Math. Doklady 41(1990)40-44
    • p-adic string theory.
      • Y. Manin, “Reflections on arithmetical physics,” in Conformal invariance and string theory, Academic Press, 1989, pages 293-303
      • V. Vladimirov, I. Volovich, and E. Zelenov, p-adic analysis and mathematical physics, World Scientific, 1994
      • V. Vladimirov, “On the Freund-Witten adelic formula for Veneziano amplitudes,” Letters in Math. Physics 27(1993)123-131
  • Mathematical chemistry.
    • Spectroscopy.B. Judd, “Lie groups in Atomic and molecular spectroscopy”, SIAM J on Applied Math 25(1973) 186-192
    • Crystallography.
      • G. Ramachandran and R. Srinivasan, Fourier methods in crystallography,
        New York, Wiley-Interscience, 1970.
      • T. Janssen, Crystallographic groups, North-Holland Pub., London, 1973.
      • J. Zak, A. Casher, M. Gluck, Y. Gur, The irreducible representations of space groups, W. A. Benjamin, Inc., New York, 1969.
    • Molecular strucure of the Buckyball.
      • F. Chung and S. Sternberg, “Mathematics and the buckyball”, American Scientist 83(1993)56-71
      • F. Chung, B. Kostant, and S. Sternberg, “Groups and the buckyball”, in Lie theory and geometry, (ed. J.-L. Brylinski et al), Birkhauser, 1994
      • G. James, “The representation theory for the Buckminsterfullerene,” J. Alg. 167(1994)803-820
  • Knot theory (which, in turn, has applications to modeling DNA) uses representation theory. F. Constantinescu and F. Toppan, “On the linearized Artin braid representation,” J. Knot Theory and its Ramifications, 2(1993)
  • The Riemann hypothesis.
    Think you’re going to solve the Riemann hypothesis without using
    representation theory? Check this paper out: A. Connes, “Formule de traces en geometrie non-commutative et hypothese de Riemann”, C. R. Acad. Sci. Paris 323 (1996)1231-1236. (For those who argue that this is not a real-world application, we refer to Barry Cipra’s article, “Prime Formula Weds Number Theory and Quantum Physics,” Science, 1996 December 20, 274, no. 5295, page 2014, in Research News.)
  • Circuit design, statistics, signal processing, …
    See the survey paper
    D. Rockmore, “Some applications of generalized FFTs” in Proceedings of the DIMACS
    Workshop on Groups and Computation, June 7-10, 1995 eds. L. Finkelstein and W. Kantor, (1997) 329–369. (available at  Dan Rockmore’s web page)
  • Vision – See the survey papers by Jacek Turski:Geometric Fourier Analysis of the Conformal Camera for Active Vision, SIAM Review, Volume 46 Issue 2 pages 230-255, 2004 Society for Industrial and Applied Mathematics, and, Geometric Fourier Analysis for Computational Vision, JFAA 11, 1-23, 2005.

mathematics problem 155

A colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his students, giving a prize of a cookie if they could solve it. Here is one of them.

Mathematics Problem, #155

We can represent a triangle with sides of length a, b, c by the ordered triple (a, b, c). Changing the order of the sides doesn’t change the triangle, so (a, b, c), (b, a, c), (b, c, a), (c, b, a), (c, a, b), and (a, c, b) all represent the same triangle. To avoid confusion, let’s agree to write (a, b, c) with a < b < c. We say that a triangle <I (a, b, c) is integral if a, b, and c are integers. How many integral triangles are there with longest side less than or equal to 100 ?

Mathematics Problem 154

A colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his students, giving a prize of a cookie if they could solve it. Here is one of them.

Mathematics Problem, #154

Find the volume of the intersection of three cylinders, each of radius a, which are centered on the x-axis, the y-axis, and the z-axis. That is, find the volume of the three dimensional region


E = {(x,y,z) | x2 + y2 < a2, y2 + z2 < a2, z2 + x2 < a2}.