### 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

- Arbitrary coordinate transforms computed via Jacobian ratios
- Banach contractive map theorem
- Canonical isomorphisms
- Cauchy Integral Theorem for complex integrals
- Change of variables in an ODE
- Chebyshev spaces
- Computing Laplace transforms using the Residue Theorem
- Computing flux through the surface of a volume
- Computing mass moments of inertia
- Computing the eigenvalues and eigenvectors of an operator, matrix or otherwise
- Concept of norm-equivalence and equivalence of resulting topologies
- Conformal mapping and solution of Laplace/Poisson equations
- Construction of weak variational form given strong form, plus equivalence proof
- Continuity, uniform continuity, Lipschitz continuity
- Contraction mapping theorem used for existence/uniqueness of an IVP
- Convergence, uniform convergence, and weak convergence
- Conversion of Del & Laplacian to cylindrical and spherical coordinates
- Conversion of integral limits and differentials to alternative coordinates
- Definition of metrics, norms, and inner products
- Determining a radius of convergence for a power series solution
- Determining if an operator is self-adjoint
- Direct sum of two vector spaces
- Eigenfrequencies and mode shapes for the wave equation
- Equivalence of topology given difference metrics
- Equivalence of ways to define a topology, including stronger/weaker comparison
- Euler-Bernoulli equation
- Euler-Lagrange equation in a Variation setting, especially the derivation
- Exact sequence definitions and implications
- Fixed point problem concepts, especially for integral equations
- Fourier transform and inverse transform
- Fredholm Integral Equations
- Gateaux differentials
- Gauss' Divergence Theorem
- Green's Theorem
- Hamel basis, and basis in infinite dimensional vector spaces
- Identify complex poles and classify their order
- Inductive use of Holder and Minkowski inequalities
- Inner product space definition
- Integral of a nonnegative function over a measurable set is zero iff f = 0 a.e.
- Irrotational fields and corresponding scalar potentials
- L^2 and \ell^2
- L^2 definition and how it relates to quotient spaces
- Lagrange multiplier methods for constrained minimization
- Laplace transform and inverse Laplace transform
- Laplace transform of the delta distribution
- Lebesgue Dominated Convergence Theorem
- Line integral calculation of the first and second kind
- Linear subspace definition
- Method of images?
- Minimization problem settings
- Notions of being "well-defined"
- Perturbation approaches
- Picard method
- Proof of Cauchy-Schwarz
- Proof of Holder and Minkowski Inequalities, applications to L^p spaces
- Proof that interior of A is the complement-closure-complement
- Properties that hold almost everywhere
- Quotient spaces
- Real eigenvalues and orthogonal eigenvectors for self-adjoint operators
- Separation of variables and use of Sturm-Liouville Theory
- Series convergence for matrices T like x
*{i+1} = T x*i + b, spectral radius - Show that the distance function is always continuous
- Showing that a vector field is divergence free
- Solenoidal fields
- Stokes' Theorem on a particular domain, including verifying its correctness
- Surface integral calculation
- Telegrapher's equation
- Topology on a set, including interior and closure definitions and properties
- Topology on discrete sets, e.g. neighborhoods, closed sets, separability
- Use of Fourier Transform in boundary value problems
- Use of Holder and Minkowski to demonstrate elements in L^p
- Vector space definition
- Weierstrass Theorem

### Area B: Numerical Analysis and Scientific Computation

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

### Area C: Mathematical Modeling and Applications

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

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