Math Flashcards
Infimum
the infimum (plural infima) of a subset S of a partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound (abbreviated as GLB) is also commonly used.
ASAP
Adjoint Sensitivity Analysis Procedure
BLUE
Best Linear Unbiased Estimator
CDF
Cumulative Distribution Function
CG
Conjugate-gradient
CONMIN
Conjugate-minimization
DASAP
Discrete Adjoint Sensitivity Analysis Procedure
DASE
Discrete Adjoint Sensitivity Equations
DDR
Discrete Response Sensitivity
DFSAP
Discrete Forward Sensitivity Analysis Procedure
DFSE
Discretized Forward Sensitivity Analysis Procedure
FAST
Fourier Amplitude Sensitivity Test
FFT
Fast Fourier Transform
FSAP
Forward Sensitivity Analysis Procedure
FSE
Forward Sensitivity Equations
HOT
Higher-Order-Term
LMQN
Limited-Memory Quasi-Newton
ME
Model Error
MGF
Moment Generating Function
MLE
Maximum Likelihood Estimator
OI
Optimal Interpolation
PDE
Partial Differential Equation
Probability Density Function
PSAS
Physical Space Statistical Analysis
QN
Quasi-Newton
SOR
Successive Over Relaxation
SQP
Sequential Quadratic Programming
SVD
Singular Value Decomposition
SWE
Shallow Water Equations
T-N
Truncated Newton
VDA
Variational Data Assimilation
IFF
If and Only If
3-D VAR
Three-Dimensional Variational Data Assimilation
4-D VAR
Four-Dimensional Variational Data Assimilation
Normal Linear Operator
A linear operator “A” is called normal if it commutes with its adjoint: AA^+ = A^+A
Hermitian Linear Operator
A linear operator “H” is called hermitian if it is equal to its adjoint: H^+ = H
Unitary Linear Operator
A linear operator “U” is called unitary if it is inverse to its adjoint: UU^+ = I
δik
Kronecker symbol
Identifiability
An unknown parameter is “identifiable” if it can be determined uniquely in all points of its domain by using the input-output relation of the system and the input-output data.
Homeomorphism
An instance of topological equivalence to another space or figure
Concatenate
link (things) together in a chain or series
EKF
Extended Kalman Filter
TV
Total Variation
Blocky Profile
“Blocky Profile” refers to functions that are piecewise constant and hence have sharply defined edges.
GLB
Greatest lower bound
Number Field “F”
An arbitrary collection of numbers within which the four operations of addition, subtraction, multiplication, and division by a nonzero number can always be carried out.
Null (Zero) Vector
the vector with all components zero
Scalar
a real vector having just one component
Linear Vector Space “V”
a set or a collection, V, of real vectors of size m is called a (linear) vector space
Neighborhood
A neighborhood of a point is any set that contains an open set that itself contains the point
Inner Product
denoted . Must be positive definite, commutative, additive, and homogeneous.
Norm of a vector x
IIxII, is a nonnegative real scalar that indicates the size or the length of the vector x, and is positive definite, homogeneous, and triangle inequality
Normed Vector Space
A vector space “V” endowed with a norm
FLOP
Floating Point Operations
Orthogonality
Two vectors x and y are called orthogonal if their inner product is zero
Span
denotes the set of all linear combination of vectors in S
Basis
If every vector in “V” can be uniquely expressed as a linear combination of those in S, then S is called a basis for V.
Dim(V)
Dimension of V
Parseval’s Identity
Parseval’s identity indicates that the total energy in any representation remains unchanged when “S” is a complete orthonormal basis.
Rectangular Matrix
A rectangular array of numbers of the field “F,” composed of m rows and n columns.
Null (Zero) Matrix
Contains all elements as 0
Principal (Main) Diagonal of A
The set of elements (a11, a22,…,amm)
Super Diagonals
Diagonals parallel to the principal diagonal and above
Sub Diagonals
Diagonals parallel to the principal diagonal and below
Column Matrix
a rectangular matrix consisting of a single column
Row Matrix
A rectangular matrix consisting of a single row
Diagonal Matrix
A square matrix with zero off-diagonal elements
Unit (Identity) Matrix
A diagonal matrix with aii=1, for all i.
Symmetrix Matrix
A square matrix “S” that coincides with its transpose
Skew-symmetric Matrix
A matrix with elements aij=-aji, for all i,j.
Hermitian Matrix
A matrix with elements aij=aji(bar), for all i,j; here, the overbear denotes complex conjugation.
Basic Operations on Matricies
- Addition (Summation)
- Scalar multiplication of a matrix by a number
- Multiplication of matrices
nilpotent
A square matrix “A” is called nilpotent iff there is an integer “p” such that A^p=0.
detA
Determinant of “A” or IAI
Rank(A)
Rank of a matrix “A.” The number of linearly independent column (rows) of “A” is called the column (row) rank of “A.”
Singular Matrix
detA=0
tr(A)
The trace of “A” is a scalar defined by the sum of the diagonal elements of “A”
R(A)
Range space of “A”
N(A)
Null Space of “A”
Reflexivity
A matrix “A” is always similar to itself
Symmetry
If A”A is similar to “B,” then “B” is similar to “A”
Transitivity
If “A” is similar to “B,” and “B” is similar to “C,” then “A” is similar to “C.”
Orthogonal Matrix
if Q^-1=Q^T
Is a permutation matrix singular or non-singular?
non-singular
Is the identity matrix a permutation matrix?
yes
Are permutation matrices orthogonal?
yes
Are Grammian matrices symmetric?
yes
Define the Spectrum of “A”
The set of all eigenvalues of “A”
Define “Spectra Radius”
The magnitude of the largest eigenvalue
rho(A)
spectral radius
characteristic vector, proper vector, latent vector
eigenvector
characteristic value, proper value, latent value, latent root, latent number, characteristic number
eigenvalue
Define MGF
the expectation of e^(tx)
The ___ distribution is widely used in radioactivity applications and in equipment failure rate analysis.
The exponential distribution
The ___ distribution is often used for weighting probabilities along the unit interval.
The beta distribution
The ___ distribution is particularly useful for modeling nonnegative phenomena, such as analysis of incomes, classroom sizes, masses or sizes of biological organisms, evaluation of neutron cross sections, scattering of subatomic particles, etc.
The log-normal distribution
Krylov Subspace
Krylov subspaces are intimately related to matrix-polynomials, and are important in reduced-space computations and error estimation.
II”A”II denotes
Norm of a square matrix “A”
Banach Space
A complete normed vector space, V
pre-Hilbert Space
A space, “V,” equipped with an inner product
Hilbert Space
A pre-Hilbert space that is complete WRT the norm. Usually denoted “H”
nondegenerate
A sequence of vectors x1, x2,… is called non degenerate if, for every p, the vectors x1, x2,…,xp are linearly independent.
Orthogonal
A sequence of vectors is called orthogonal if any two vectors of the sequence are orthogonal.
Orthogonalization
Orthogonalization of a sequence of vectors is the process of replacing the sequence boy an equivalent orthogonal sequence.
