10 April 2008

Qualifying exam concepts index

Some concepts appearing in Areas A, B, and C qualifying exam packet, poorly categorized and in no particular order. May will be a fun month.

Area A: Applicable Mathematics

  1. Arbitrary coordinate transforms computed via Jacobian ratios
  2. Banach contractive map theorem
  3. Canonical isomorphisms
  4. Cauchy Integral Theorem for complex integrals
  5. Change of variables in an ODE
  6. Chebyshev spaces
  7. Computing Laplace transforms using the Residue Theorem
  8. Computing flux through the surface of a volume
  9. Computing mass moments of inertia
  10. Computing the eigenvalues and eigenvectors of an operator, matrix or otherwise
  11. Concept of norm-equivalence and equivalence of resulting topologies
  12. Conformal mapping and solution of Laplace/Poisson equations
  13. Construction of weak variational form given strong form, plus equivalence proof
  14. Continuity, uniform continuity, Lipschitz continuity
  15. Contraction mapping theorem used for existence/uniqueness of an IVP
  16. Convergence, uniform convergence, and weak convergence
  17. Conversion of Del & Laplacian to cylindrical and spherical coordinates
  18. Conversion of integral limits and differentials to alternative coordinates
  19. Definition of metrics, norms, and inner products
  20. Determining a radius of convergence for a power series solution
  21. Determining if an operator is self-adjoint
  22. Direct sum of two vector spaces
  23. Eigenfrequencies and mode shapes for the wave equation
  24. Equivalence of topology given difference metrics
  25. Equivalence of ways to define a topology, including stronger/weaker comparison
  26. Euler-Bernoulli equation
  27. Euler-Lagrange equation in a Variation setting, especially the derivation
  28. Exact sequence definitions and implications
  29. Fixed point problem concepts, especially for integral equations
  30. Fourier transform and inverse transform
  31. Fredholm Integral Equations
  32. Gateaux differentials
  33. Gauss' Divergence Theorem
  34. Green's Theorem
  35. Hamel basis, and basis in infinite dimensional vector spaces
  36. Identify complex poles and classify their order
  37. Inductive use of Holder and Minkowski inequalities
  38. Inner product space definition
  39. Integral of a nonnegative function over a measurable set is zero iff f = 0 a.e.
  40. Irrotational fields and corresponding scalar potentials
  41. L^2 and \ell^2
  42. L^2 definition and how it relates to quotient spaces
  43. Lagrange multiplier methods for constrained minimization
  44. Laplace transform and inverse Laplace transform
  45. Laplace transform of the delta distribution
  46. Lebesgue Dominated Convergence Theorem
  47. Line integral calculation of the first and second kind
  48. Linear subspace definition
  49. Method of images?
  50. Minimization problem settings
  51. Notions of being "well-defined"
  52. Perturbation approaches
  53. Picard method
  54. Proof of Cauchy-Schwarz
  55. Proof of Holder and Minkowski Inequalities, applications to L^p spaces
  56. Proof that interior of A is the complement-closure-complement
  57. Properties that hold almost everywhere
  58. Quotient spaces
  59. Real eigenvalues and orthogonal eigenvectors for self-adjoint operators
  60. Separation of variables and use of Sturm-Liouville Theory
  61. Series convergence for matrices T like x{i+1} = T xi + b, spectral radius
  62. Show that the distance function is always continuous
  63. Showing that a vector field is divergence free
  64. Solenoidal fields
  65. Stokes' Theorem on a particular domain, including verifying its correctness
  66. Surface integral calculation
  67. Telegrapher's equation
  68. Topology on a set, including interior and closure definitions and properties
  69. Topology on discrete sets, e.g. neighborhoods, closed sets, separability
  70. Use of Fourier Transform in boundary value problems
  71. Use of Holder and Minkowski to demonstrate elements in L^p
  72. Vector space definition
  73. Weierstrass Theorem

Area B: Numerical Analysis and Scientific Computation

  1. Adaptive integration
  2. Best approximation results for projections
  3. Cholesky factorization
  4. Construct matrix condition numbers based on perturbed matrices and vectors
  5. Cost analysis of common matrix operations
  6. Definition of a norm and norm equivalence
  7. Demonstrating that eigenvalue computation has no closed form solution
  8. Determining an orthonormal basis, the LQ decomposition
  9. Determining equispaced interpolating subdomains with some error bound
  10. Determining the rank of a matrix in finite precision arithmetic
  11. Effect of matrix rank on Gram-Schmidt
  12. Eigenvalues and eigenvectors
  13. Estimation of an interval given information at a finite number of points
  14. Existence and uniqueness of an interpolating form given particular information
  15. Finite difference construction based on Taylor truncation, including stability
  16. For Galerkin-like minimization methods, show that error is "orthogonal"
  17. Forward and backward error analysis
  18. Gaussian elimination error propagation when used without pivoting
  19. Gaussian quadrature and the use of orthogonal polynomials
  20. Gershgorin Circle Theorem
  21. Gram-Schmidt, including how floating error arises
  22. Hermite matrices and their properties
  23. Householder transforms/rotators
  24. LU Decomposition
  25. Lagrangian interpolating polynomials
  26. Linear independence, linear combinations
  27. Linear least squares, properties, the normal equations, including computing
  28. Modified Gram-Schmidt
  29. Operational identities in finite precision arithmetic
  30. Partitioning of symmetric positive definite matrices
  31. Positive and negative definiteness, and relation to eigen- and singular values
  32. Proof of Taylor-like error estimate for an interpolating polynomial error bound
  33. Proof that Gaussian n-point quadrature is exact for polynomials of order 2n-1
  34. Proofs Cline made a fuss about :)
  35. Properties of a matrix norm, including the sub-multiplicative one
  36. Properties of an induces matrix norm
  37. QR Factorization
  38. Rank and Nullity Theorem
  39. Residual matrices
  40. Richardson extrapolation, acceleration rates, and required regularity
  41. Runge-Kutta methods for solving initial value problems
  42. Separation of a 2-tensor into a symmetric and antisymmetric part
  43. Shifting of Gaussian quadrature rules through affine transformations
  44. Similarity and congruence transformations
  45. Simpson's rule
  46. Singular value decomposition and how it relates to rank, nullspace, and range
  47. Singularity to working precision, and the distance to nonsingular matrices
  48. Special matrix structures, e.g. symmetric, banded, diagonal, skewsymmetric
  49. Spline construction, including cubic Hermite splines
  50. Trapezoidal rule and relationship to Lagrange polynomials
  51. Two definitions of matrix norms and how they relate to minmag and maxmag
  52. Use of QR factorization in the least squares problem
  53. Ways to test if a matrix is positive definite, and correctness guarantees
  54. Why/when is QR preferred over Gram-Schmidt

Area C: Mathematical Modeling and Applications

  1. Balance laws
  2. Boltzmann Transport Equation
  3. Calculus of variations applied to simple problems
  4. Cauchy's principal and the Cauchy stress tensor
  5. Cauchy-Green deformation tensors
  6. Computing deformation gradient, tensors, and principal deformation directions
  7. Conservation laws and their derivations
  8. Darcy's law and its derivation
  9. Definitions of basic continuum mechanics concepts
  10. Derivation of Navier-Stokes equations, including term-by-term interpretation
  11. Determine lattice specific energy
  12. Developing model of water through a rigid trough
  13. Energy density calculation given a potential
  14. Energy per particle
  15. Formal homogenization assuming periodic small scales, including anisotropics
  16. Green strain
  17. Lattice energy calculation
  18. Main sources of non-numerical error in simulations
  19. Main steps in constructing a mathematical model
  20. Mathematical formulation of Schrödinger's equation, constraints, and solutions
  21. Multiscale/matched asymptotic approaches
  22. Nondimensionalization of the transport equation
  23. Optimization and the adjoint problem, classes of optimization/control problems
  24. PDE types (e.g. Parabolic, Hyperbolic, Elliptic) for 1st and 2nd order
  25. Permeability constants
  26. Piola transformation
  27. Stokes' equation
  28. Well-posedness of second order, linear differential equations

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