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Generalizing from a straight line (i.e., first degree polynomial) to a kth degree polynomial y=a_0+a_1x+...+a_kx^k, (1) the residual is given by ...

Levenberg-Marquardt is a popular alternative to the Gauss-Newton method of finding the minimum of a function F(x) that is a sum of squares of nonlinear functions, ...

The direct sum of modules A and B is the module A direct sum B={a direct sum b|a in A,b in B}, (1) where all algebraic operations are defined componentwise. In particular, ...

An operator A:f^((n))(I)|->f(I) assigns to every function f in f^((n))(I) a function A(f) in f(I). It is therefore a mapping between two function spaces. If the range is on ...

The pair group of a group G is the group that acts on the 2-subsets of {1,...,p} whose permutations are induced by G. Pair groups can be calculated using PairGroup[g] in the ...

The representation, beloved of engineers and physicists, of a complex number in terms of a complex exponential x+iy=|z|e^(iphi), (1) where i (called j by engineers) is the ...

Let f:R×R->R be a one-parameter family of C^3 maps satisfying f(-x,mu)=-f(x,mu) (1) (partialf)/(partialx)|_(mu=0, x=0)=0 (2) (partial^2f)/(partialxpartialmu)|_(mu=0, x=0)>0 ...

A second-order partial differential equation arising in physics, del ^2psi=-4pirho. If rho=0, it reduces to Laplace's equation. It is also related to the Helmholtz ...

For a polynomial P=sum_(k=0)^na_kz^k, (1) several classes of norms are commonly defined. The l_p-norm is defined as ||P||_p=(sum_(k=0)^n|a_k|^p)^(1/p) (2) for p>=1, giving ...

A short exact sequence of groups A, B, and C is given by two maps alpha:A->B and beta:B->C and is written 0->A->B->C->0. (1) Because it is an exact sequence, alpha is ...