Search Results for ""

The Pauli matrices, also called the Pauli spin matrices, are complex matrices that arise in Pauli's treatment of spin in quantum mechanics. They are defined by sigma_1 = ...

A quadratic form involving n real variables x_1, x_2, ..., x_n associated with the n×n matrix A=a_(ij) is given by Q(x_1,x_2,...,x_n)=a_(ij)x_ix_j, (1) where Einstein ...

The scalar triple product of three vectors A, B, and C is denoted [A,B,C] and defined by [A,B,C] = A·(BxC) (1) = B·(CxA) (2) = C·(AxB) (3) = det(ABC) (4) = |A_1 A_2 A_3; B_1 ...

Roman (1984, p. 2) describes umbral calculus as the study of the class of Sheffer sequences. Umbral calculus provides a formalism for the systematic derivation and ...

The simple process of voting leads to surprisingly counterintuitive paradoxes. For example, if three people vote for three candidates, giving the rankings A, B, C; B, C, A; ...

The generalized hypergeometric function is given by a hypergeometric series, i.e., a series for which the ratio of successive terms can be written ...

A harmonic number is a number of the form H_n=sum_(k=1)^n1/k (1) arising from truncation of the harmonic series. A harmonic number can be expressed analytically as ...

Consider decomposition the factorial n! into multiplicative factors p_k^(b_k) arranged in nondecreasing order. For example, 4! = 3·2^3 (1) = 2·3·4 (2) = 2·2·2·3 (3) and 5! = ...

An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set. For ...

The conjugate transpose of an m×n matrix A is the n×m matrix defined by A^(H)=A^_^(T), (1) where A^(T) denotes the transpose of the matrix A and A^_ denotes the conjugate ...