Given the repressive nature of the regime of Napoleon III, Bartholdi took no immediate action on the idea except to discuss it with Laboulaye.

Given " mappings " of input variables into membership functions and truth values, the microcontroller then makes decisions for what action to take, based on a set of " rules ", each of the form:

Given the need for immediate action was no longer pressing, François once again demanded he be allowed to wait for his artillery supplies.

" Given that declaration, along with seven years of bombings and the wording of their communiques throughout that time that strove to present an image of a powerful organization spread secretly throughout all sectors of society, the authorities took significant action.

Given his standing, he was one of the few officers who might have persuaded Hitler to follow more rational courses of action.

: Given that the particle begins at position x < sub > 1 </ sub > at time t < sub > 1 </ sub > and ends at position x < sub > 2 </ sub > at time t < sub > 2 </ sub >, the physical trajectory that connects these two endpoints is an extremum of the action integral.

Given the option, most swimmers choose to use a dolphin kicking action, but there still is a small minority of swimmers who prefer the breaststroke kick, for recreational swimming and even for competition.

For a finite group G, the left regular representation λ ( over a field K ) is a linear representation on the K-vector space V whose basis is the elements of G. Given g ∈ G, λ ( g ) is the linear map determined by its action on the basis by left translation by g, i. e.

Given a fundamental domain for the group action ( for the full, orientation-reversing symmetry group, a ( 2, 3, 7 ) triangle ), the reflection domains ( images of this domain under the group ) give a tiling of the quartic such that the automorphism group of the tiling equals the automorphism group of the surface – reflections in the lines of the tiling correspond to the reflections in the group ( reflections in the lines of a given fundamental triangle give a set of 3 generating reflections ).

Given this closure property for CSAs, they form a monoid under tensor product, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse class to that of an algebra A is the one containing the opposite algebra A < sup > op </ sup > ( the opposite ring with the same action by K since the image of K → A is in the center of A ).

Given a set of action names, the set of CCS processes is defined by the following BNF grammar:

Given an action that must be repeated, for instance, five times, different languages ' for loops will be written differently.

Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action.

Given an action of a group G on a topological space X by homeomorphisms, a fundamental domain ( also called fundamental region ) for this action is a set D of representatives for the orbits.

Given an action of one group scheme on another by automorphisms, one can form semidirect products by following the usual set-theoretic construction.

Given the proposed manner of action of the muscle pump to increase arterial blood flow, it would seem impossible for a muscle contraction and skeletal muscle hyperemia to be uncoupled.

Given a faithful irreducible representation ρ of G, the lattice Yang-Mills action is the sum over all lattice sites of the ( real component of the ) trace over the n links e < sub > 1 </ sub >, ..., e < sub > n </ sub > in the Wilson loop,

Given the popularity of the first-person shooter genre, it was assumed that all tactical-level military gameplay necessarily involved individual combat action.

However, in his Travels With Charley, Steinbeck mentions wearing a naval officer's cap, ' Given me by a British torpedo boat captain, a very gentle gentleman and a murderer ,' which phrase suggests that Steinbeck may well have observed action at first hand.

Given the exacerbated ethnic tensions and the lack of government control in the East, Rwanda was to take action against the security threat posed by the génocidaires that had found refuge in eastern Zaire.

" Given the unprecedented decline in new construction, however, more action is necessary.

Given two Lie algebras and, their direct sum is the Lie algebra consisting of the vector space

Given a manifold and a Lie algebra valued 1-form, over it, we can define a family of p-forms:

Given a function and a vector field X defined on M, the Lie derivative of a function ƒ along a vector field is simply the application of the vector field.

* Given any manifold, there is a Lie groupoid called the pair groupoid, with as the manifold of objects, and precisely one morphism from any object to any other.

* Given a Lie group acting on a manifold, there is a Lie groupoid called the translation groupoid with one morphism for each triple with.

Let X be any Lie algebra over K. Given a unital associative K-algebra U and a Lie algebra homomorphism: h: X → U < sub > L </ sub >, ( notation as above ) we say that U is the universal enveloping algebra of X if it satisfies the following universal property: for any unital associative K-algebra A and Lie algebra homomorphism f: X → A < sub > L </ sub > there exists a unique unital algebra homomorphism g: U → A such that: f (-) = g < sub > L </ sub > ( h (-)).

Given any compact Lie group G one can take its identity component G < sub > 0 </ sub >, which is connected.

Given a basis e < sup > i </ sup > of the Lie algebra g, the matrix elements of the Killing form are given by

Given an element x of a Lie algebra, one defines the adjoint action of x on as the endomorphism with

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* 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 a finite dimensional real quadratic space with quadratic form, the geometric algebra for this quadratic space is the Clifford algebra Cℓ ( V, Q ).

Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows.

Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:

Given a list of operations and axioms in universal algebra, the corresponding algebras and homomorphisms are the objects and morphisms of a category.

A Clifford algebra Cℓ ( V, Q ) is a unital associative algebra over K together with a linear map satisfying for all defined by the following universal property: Given any associative algebra A over K and any linear map such that

So, a collection of functions with given signatures generate a free algebra, the term algebra T. Given a set of equational identities ( the axioms ), one may consider their symmetric, transitive closure E. The quotient algebra T / E is then the algebraic structure or variety.

Given his animosity to infinitesimals it is fitting that the result was couched in terms of algebra rather than analysis.

Given a bounded lattice with largest and smallest elements 1 and 0, and a binary operation, these together form a Heyting algebra if and only if the following hold:

Given a set with three binary operations and, and two distinguished elements 0 and 1, then is a Heyting algebra for these operations ( and the relation defined by the condition that when ) if and only if the following conditions hold for any elements and of:

Given an algebra, a homomorphism h thus defines two algebras homomorphic to, the image h () and The two are isomorphic, a result known as the homomorphic image theorem.