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