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Additional Materials

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All Additional Materials 2004-10-17
Download one notebook with all of the individual problems and solutions listed below. The evaluated version of the notebook contains about 600 bitmap graphics and its printed version is about 380 pages.
Evaluated (.nb file, 60471 kB) | GZ (.gz file, 16146 kB) | Unevaluated (.nb file, 1318 kB)

Graphics

Chapter 1

  1. Stick Breaking 2004-10-07
    We consider the recursive breaking of a stick into n (randomly chosen) pieces. The resulting distributions of the stick lengths are visualized.
    Evaluated (.nb file, 119 kB) | Unevaluated (.nb file, 73 kB)
  2. Aperiodic Tiling Animation 2004-10-07
    We take part of an aperiodic tiling and rotate the polygons of the tiling around their vertices.
    Evaluated (.nb file, 198 kB) | Unevaluated (.nb file, 74 kB)
  3. Circles on Curves 2004-10-07
    We take two discretized Lissajou figures, interpolate between them and make an animation how circles connecting neighboring points change.
    Evaluated (.nb file, 1207 kB) | Unevaluated (.nb file, 71 kB)

Chapter 2

  1. Rotating Möbius Bands 2004-10-07
    An animation of two interlocked, rotating Möbius bands is constructed.
    Evaluated (.nb file, 151 kB) | Unevaluated (.nb file, 73 kB)
  2. Rotating Tori with Tilings 2004-10-07
    An animation of two interlocked, rotating tori with tilings and their projections is constructed.
    Evaluated (.nb file, 1014 kB) | Unevaluated (.nb file, 85 kB)
  3. Klein Bottle With Hexagonal Wireframe 2004-10-07
    We construct a Klein bottle whose surface is made from hexagons.
    Evaluated (.nb file, 456 kB) | Unevaluated (.nb file, 79 kB)
  4. Lattice Sites Visited by Random Walkers 2004-10-07
    We consider 1000 random walkers on a 3D cubic lattice, each making 1000 steps. We visualize the lattice sites visited.
    Evaluated (.nb file, 274 kB) | Unevaluated (.nb file, 78 kB)
  5. 2D Minkowski Function 2004-10-07
    The first few orders of a possible 2D generalization of the classical Minkowski function inside the unit triangle is calculated and visualized.
    Evaluated (.nb file, 801 kB) | Unevaluated (.nb file, 87 kB)

Chapter 3

  1. 2D Chladny Figures Animation 2004-10-07
    For a eightfold degenerate eigenfunctions, the time-dependent vibrations of a square plate are visualized.
    Evaluated (.nb file, 569 kB) | Unevaluated (.nb file, 74 kB)
  2. Charged ngon Animation 2004-10-07
    The equi-contour lines of a set of charged moving triangles, quadrilaterals, pentagons, ... are visualized.
    Evaluated (.nb file, 1113 kB) | Unevaluated (.nb file, 76 kB)
  3. Charged Truchet Pattern 2004-10-07
    The equi-contour lines of a charged Truchet pattern is visualized.
    Evaluated (.nb file, 964 kB) | Unevaluated (.nb file, 76 kB)

Numerics

Chapter 1

  1. Random Walk with Varying Step Size 2004-10-07
    A 1D random walk with a step-dependent step size is considered. For various step dependencies, the resulting distributions of the walkers position are visualized.
    Evaluated (.nb file, 160 kB) | Unevaluated (.nb file, 73 kB)
  2. Three-Disk Scattering 2004-10-07
    The scattering of a point particle on three disk is considered. For various disk diameters, the dependence of the exit point is calculated and visualized.
    Evaluated (.nb file, 1015 kB) | Unevaluated (.nb file, 91 kB)
  3. Vibrating Hilbert Curve Spring System 2004-10-07
    A 2D Hilbert curve is embedded in a square grid (so that the edges of the Hilbert curve are along neighboring lattice points). The eigenmodes of the spring system for various stiffnesses of the springs is investigated. The resulting vibrations are visualized.
    Evaluated (.nb file, 5975 kB) | Unevaluated (.nb file, 80 kB)
  4. Hypercube Projections 2004-10-07
    The optimal projections of some hypercubes in two and three dimensions are calculated and visualized.
    Evaluated (.nb file, 1406 kB) | Unevaluated (.nb file, 86 kB)
  5. Currents through a Penrose Tiling 2004-10-07
    The current through the finite-resistance edges of an aperiodic tiling made are calculated and visualized.
    Evaluated (.nb file, 1880 kB) | Unevaluated (.nb file, 92 kB)
  6. Various Time-dependent PDEs 2004-10-07
    Numerical solutions to a variety of time-dependent PDEs are calculated and visualized.
    Evaluated (.nb file, 5985 kB) | Unevaluated (.nb file, 164 kB)
  7. Vibrating Koch Drum 2004-10-07
    The eigenvibrations of a 2D square Koch drum are calculated and visualized. The Weyl-Berry law for the cumulative mode density is tested.
    Evaluated (.nb file, 3856 kB) | Unevaluated (.nb file, 140 kB)
  8. Wigner Function of a Duffing Oscillator 2004-10-07
    The time-dependence of the Wigner function of a Duffing oscillator is calculated and visualized. The results are compared with the evolution of the classical counterpart.
    Evaluated (.nb file, 4021 kB) | Unevaluated (.nb file, 149 kB)
  9. Circular Aperture Diffraction Integral 2004-10-07
    The diffraction integral for a circular aperture is calculated and visualized.
    Evaluated (.nb file, 249 kB) | Unevaluated (.nb file, 78 kB)
  10. Cauchy-Born Hypothesis Failure 2004-10-07
    Examples of failure of the Cauchy-Born Hypothesis for square lattices are founds and visualized.
    Evaluated (.nb file, 626 kB) | Unevaluated (.nb file, 86 kB)
  11. Schwarz-Christoffel Map 2004-10-07
    Some examples for the Schwarz-Christoffel Map for some 2D polygons are calculated and visualized.
    Evaluated (.nb file, 551 kB) | Unevaluated (.nb file, 91 kB)
  12. EU Voting Scheme 2004-10-07
    The normalized Banzhaf index for the EU is calculated.
    Evaluated (.nb file, 218 kB) | Unevaluated (.nb file, 85 kB)
  13. Wave Propagation on a Torus 2004-10-07
    An animation showing how an initially localized perturbation evolves on a torus under the wave equation is calculated.
    Evaluated (.nb file, 460 kB) | Unevaluated (.nb file, 85 kB)

Chapter 2

  1. Infinite Cosine Product Integral 2004-10-07
    The value of an integral over an infinite product of cosine functions is calculated.
    Evaluated (.nb file, 426 kB) | Unevaluated (.nb file, 89 kB)
  2. Pi-related Continued Fraction 2004-10-07
    The digits in a special pi-related continued fraction are calculated.
    Evaluated (.nb file, 91 kB) | Unevaluated (.nb file, 76 kB)
  3. Graphics of the Argument of Cyclotomic Polynomials 2004-10-07
    Graphics showing the argument of cyclotomic polynomials are constructed.
    Evaluated (.nb file, 705 kB) | Unevaluated (.nb file, 77 kB)

Symbolics

Chapter 1

  1. Bose Gas Virial Series Convergence Radius 2004-10-07
    The convergence radius for the Bose gas virial series in four dimensions is estimated.
    Evaluated (.nb file, 108 kB) | Unevaluated (.nb file, 82 kB)
  2. Midpoint Quadrature Formula 2004-10-07
    The optimal coefficients for a midpoint quadrature rule is calculated.
    Evaluated (.nb file, 93 kB) | Unevaluated (.nb file, 78 kB)
  3. MacMahon Master Theorem 2004-10-07
    A one-liner checking the MacMahon Master Theorem for an explicit integer is implemented.
    Evaluated (.nb file, 75 kB) | Unevaluated (.nb file, 74 kB)
  4. Adler-Moser Polynomials 2004-10-07
    Low order Adler-Moser polynomials polynomials are calculate and visualized.
    Evaluated (.nb file, 384 kB) | Unevaluated (.nb file, 81 kB)
  5. ODE for Yablonskii-Vorob'ev Polynomials 2004-10-07
    Based on an ansatz, the ODE for the ODE for Yablonskii-Vorob'ev Polynomials is conjectured.
    Evaluated (.nb file, 85 kB) | Unevaluated (.nb file, 79 kB)
  6. Implicit Hypocycloidal Torus 2004-10-07
    An implicit polynomial description of the hypocycloidal torus visualized in Chapter 2 of the Graphics volume of the GuideBooks is calculated.
    Evaluated (.nb file, 974 kB) | Unevaluated (.nb file, 81 kB)
  7. Second Feigenbaum Constant 2004-10-07
    The second Feigenbaum constant is estimated and the Feigenbaum functions is visualized along the real axis and over the complex plane.
    Evaluated (.nb file, 278 kB) | Unevaluated (.nb file, 85 kB)
  8. Sequence of Delta Function Potentials 2004-10-07
    The poles of the Green's function of the Schrödinger equation with a sequence of delta potentials are calculated and the dependence on the potential strength is visualized.
    Evaluated (.nb file, 1171 kB) | Unevaluated (.nb file, 89 kB)
  9. Implicit Poynting Vector 2004-10-07
    For a special case, an implicit description for the equi0surfaces of a accelerated charge is calculated.
    Evaluated (.nb file, 271 kB) | Unevaluated (.nb file, 84 kB)
  10. Symmetric Sphere Points 2004-10-07
    Solutions of a diophantine equation are used to distribute points on a sphere.
    Evaluated (.nb file, 345 kB) | Unevaluated (.nb file, 73 kB)
  11. Isospectral Polygons 2004-10-07
    Eigenvalues and eigenfunctions of the two isospectral polygons bilby and hawk are calculated using high-order finite elements. The resulting eigenfunctions and their gradients are visualized.
    Evaluated (.nb file, 5197 kB) | Unevaluated (.nb file, 124 kB)
  12. Probability Distribution of a Quotient 2004-10-07
    The exact probability distribution of a quotient with numerator and dominator coming from a rectangle in the complex plane are calculated and visualized.
    Evaluated (.nb file, 202 kB) | Unevaluated (.nb file, 88 kB)
  13. Gosper Spring System 2004-10-07
    The eigenvibrations for a Gosper curve with springs along its line segments are calculated and visualized.
    Evaluated (.nb file, 1417 kB) | Unevaluated (.nb file, 90 kB)
  14. Average Distance Distribution 2004-10-07
    The probability distribution of the length of the distance between two points chosen randomly within the unit square is calculated.
    Evaluated (.nb file, 164 kB) | Unevaluated (.nb file, 83 kB)
  15. Dirichlet-Neumann Boundary Condition Transition 2004-10-07
    The transition of the eigenfunctions of the Helmholtz equation from Dirichlet to Neumann boundary conditions on part of the boundary of a square are visualized.
    Evaluated (.nb file, 10994 kB) | Unevaluated (.nb file, 106 kB)
  16. Generalized WKB Approximation 2004-10-07
    The validity of a recently proposed generalized WKB approximation is investigated for the harmonic oscillator.
    Evaluated (.nb file, 404 kB) | Unevaluated (.nb file, 90 kB)
  17. Multi-Dimensional Integral 2004-10-07
    A special multi-dimensional integral is calculated.
    Evaluated (.nb file, 115 kB) | Unevaluated (.nb file, 82 kB)
  18. Kobussen-Leubner Lopez-Lagrangians for the Harmonic Oscillator 2004-10-07
    Special Kobussen-Leubner Lopez-Lagrangians are calculated for the harmonic oscillator and the resulting phase space trajectories are visualized.
    Evaluated (.nb file, 435 kB) | Unevaluated (.nb file, 89 kB)

Chapter 2

  1. Hénon-Heiles Eigenfunctions 2004-10-07
    The eigenfunctions of the 2D Schrodinger equation are calculated for the Hénon-Heiles potential and the resulting eigenfunctions are visualized.
    Evaluated (.nb file, 5158 kB) | Unevaluated (.nb file, 224 kB)

Chapter 3

  1. Rational Values of Gauss Hypergeometric Function 2004-10-07
    A search is carried out to calculate rational values of the Gauss hypergeometric function.
    Evaluated (.nb file, 75 kB) | Unevaluated (.nb file, 74 kB)
  2. Discrete Harmonic Oscillator Wave Functions 2004-10-07
    The discrete harmonic Oscillator wave functions based on Kravchuk polynomials are calculated and visualized.
    Evaluated (.nb file, 450 kB) | Unevaluated (.nb file, 81 kB)
  3. Average Length of Smallest Component of Multidimensional Unit Vectors 2004-10-07
    The average length of smallest component of multidimensional unit vectors is estimated numerically based on a recursive implementation to calculate uniformly distributed points on a hypersphere.
    Evaluated (.nb file, 179 kB) | Unevaluated (.nb file, 82 kB)
  4. ODE for Jacobi Elliptic Function sn with Respect to the Modulus 2004-10-07
    An ODE for Jacobi elliptic function sn with respect to the modulus is derived.
    Evaluated (.nb file, 124 kB) | Unevaluated (.nb file, 76 kB)
  5. A Certain Zeta Sum 2004-10-07
    A certain family of sums involving the Riemann Zeta function are calculated to high order.
    Evaluated (.nb file, 144 kB) | Unevaluated (.nb file, 81 kB)
  6. Harmonic Numbers Inversion 2004-10-07
    A series expansion for the inverse function of the harmonic numbers is derived.
    Evaluated (.nb file, 117 kB) | Unevaluated (.nb file, 74 kB)
  7. Sliding Coin 2004-10-07
    The rotational and translational movement of a sliding coin is numerically calculated and visualized.
    Evaluated (.nb file, 141 kB) | Unevaluated (.nb file, 84 kB)
  8. Theta Function Identities 2004-10-07
    Using a series expansion approach, sets of new identities for the Jacobi theta functions are calculated.
    Evaluated (.nb file, 412 kB) | Unevaluated (.nb file, 138 kB)