# Composition operator ( also called the substitution operator ): Given an m-ary function and m k-ary functions:

# Primitive recursion operator: Given the k-ary function and k + 2-ary function:

# Minimisation operator: Given a ( k + 1 )- ary total function:

Given any energy eigenstate, we can act on it with the lowering operator, a, to produce another eigenstate with-less energy.

Given the time reversal operator T, it does nothing to the x-operator, TxT < sup >− 1 </ sup > =

Given a densely defined linear operator A on H, its adjoint A * is defined as follows:

Given the representation of T as a multiplication operator, it is easy to characterize the Borel functional calculus: If h is a bounded real-valued Borel function on R, then h ( T ) is the operator of multiplication by the composition.

Given a linear differential operator T

Given the Reynolds operator, Hilbert's theorem is proved as follows.

Given a modular form f ( z ) of weight k, the mth Hecke operator acts by the formula

: Given a matrix-valued function A ( x ) which is symmetric and positive definite for every x, having components a < sup > ij </ sup >, the operator

) Given a fixed density, one may construct a nilpotent BV Δ operator as

Given an operator T, the range of the continuous functional calculus h → h ( T ) is the ( abelian ) C *- algebra C ( T ) generated by T. The Borel functional calculus has a larger range, that is the closure of C ( T ) in the weak operator topology, a ( still abelian ) von Neumann algebra.

Given an operator, it is sometimes useful to break up the spectrum into various parts.

Given a fixed knowledge base and a revision operator, one can define a non-monotonic inference relation using the following definition: if and only if.

Given the inverse of the metric tensor above, the explicit form of the kinetic energy operator in terms of Euler angles follows by simple substitution.

* Given a normed vector space and its continuous dual then-topology on is identical to the topology induced by the operator norm.

Given a linear operator, not necessarily closed, if the closure of its graph in happens to be the graph of some operator, that operator is called the closure of, and we say that is closable.

* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

Given the complexity of the calculations involved and the convoluted structure that a convertible bond can have, an arbitrageur often relies on sophisticated quantitative models in order to identify bonds that are trading cheap versus their theoretical value.

Given this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.

Given the federal nature of the holiday, celebrating Canada Day can be a cause of friction in the province of Quebec, where the holiday is overshadowed by Quebec's National Holiday, on June 24.

Given the rugged terrain of Ngazidja and Nzwani, and the dedication of extensive tracts to agriculture on all three islands, population pressures on Comoros are becoming increasingly critical.

Given an asymmetric subunit on a triangular face of a regular isosahedron, with three subunits per face 60 such subunits can be placed in an equivalent manner.

Given this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education.

* Frederick Douglass lecture on Haiti – Given at the World's Fair in Chicago, January 1893.

Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold ( usually defined by giving the metric in specific coordinates ), and specific matter fields defined on that manifold.

Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions.

Given an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.

Given these, a new term Anthropocene, is specifically proposed and used informally for the latest part of this epoch since approximately synchronous lithospheric evidence, or more recently atmospheric evidence, of human impacts have been found on the Earth and its ecosystems ; these impacts may be considered of global significance for future evolution of living species.

Given that this etymology does not appear in the usual scholarly works on etymology, this claim is usually dismissed as a false etymology.

Given the dual status of science as objective knowledge and as a human construct, good historiography of science draws on the historical methods of both intellectual history and social history.

Given a binary operation ★ on a set S, an element x is said to be idempotent ( with respect to ★) if

Given its heavy dependence on imported energy, Japan has aimed to diversify its sources.

Along with a hand-picked escort led by Given Campbell, Davis and his wife were captured on May 10, 1865, at Irwinville in Irwin County, Georgia.

Given the repeated failures and frustrations of workers ' revolutions and movements, Marx also sought to understand capitalism, and spent a great deal of time in the reading room of the British Museum studying and reflecting on the works of political economists and on economic data.

Given that estimation is undertaken on the basis of a least squares analysis, estimates of the unknown parameters β < sub > j </ sub > are determined by minimising a sum of squares function

Given that spinning a Dyson Sphere would result in the atmosphere pooling around the equator, the Ringworld removes all the extraneous parts of the structure, leaving a spinning band landscaped on the sun-facing side, with the atmosphere and inhabitants kept in place through centrifugal force and 1000 mile high perimeter walls ( rim walls ).

Given a Hilbert space L < sup > 2 </ sup >( m ), m being a finite measure, the inner product < ·, · > gives rise to a positive functional φ by

Given an isolated physical system, the allowed states of this system ( i. e. the states the system could occupy without violating the laws of physics ) are part of a Hilbert space H. Some properties of such a space are

Given a Hilbert space ( either finite or infinite dimensional ), its complex conjugate is the same vector space as its continuous dual space.

Given three Hilbert spaces,,