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The expectation value of a function f(x) in a variable x is denoted <f(x)> or E{f(x)}. For a single discrete variable, it is defined by <f(x)>=sum_(x)f(x)P(x), (1) where P(x) ...

On a Lie group, exp is a map from the Lie algebra to its Lie group. If you think of the Lie algebra as the tangent space to the identity of the Lie group, exp(v) is defined ...

A differential k-form can be integrated on an n-dimensional manifold. The basic example is an n-form alpha in the open unit ball in R^n. Since alpha is a top-dimensional ...

The fractional derivative of f(t) of order mu>0 (if it exists) can be defined in terms of the fractional integral D^(-nu)f(t) as D^muf(t)=D^m[D^(-(m-mu))f(t)], (1) where m is ...

A two-dimensional piecewise linear map defined by x_(n+1) = 1-y_n+|x_n| (1) y_(n+1) = x_n. (2) The map is chaotic in the filled region above and stable in the six hexagonal ...

A global field is either a number field, a function field on an algebraic curve, or an extension of transcendence degree one over a finite field. From a modern point of view, ...

The energy of a graph is defined as the sum of the absolute values of its graph eigenvalues (i.e., the sum of its graph spectrum terms). Other varieties of graph energy are ...

The inhomogeneous Helmholtz differential equation is del ^2psi(r)+k^2psi(r)=rho(r), (1) where the Helmholtz operator is defined as L^~=del ^2+k^2. The Green's function is ...

The group algebra K[G], where K is a field and G a group with the operation *, is the set of all linear combinations of finitely many elements of G with coefficients in K, ...

Let f be an entire function of finite order lambda and {a_j} the zeros of f, listed with multiplicity, then the rank p of f is defined as the least positive integer such that ...