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Page "Involution (mathematics)" ¶ 20
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Given and module
Given two tensor bundles E → M and F → M, a map A: Γ ( E ) → Γ ( F ) from the space of sections of E to sections of F can be considered itself as a tensor section of if and only if it satisfies A ( fs ,...) = fA ( s ,...) in each argument, where f is a smooth function on M. Thus a tensor is not only a linear map on the vector space of sections, but a C < sup >∞</ sup >( M )- linear map on the module of sections.
Given a set, a free module on is a free module with basis.
Given a module, M, a projective resolution of M is an infinite exact sequence of modules
Given a module A over a ring R, and a submodule B of A, the quotient space A / B is defined by the equivalence relation
* Given a field K, the multiplicative group ( K < sup > s </ sup >)< sup >×</ sup > of a separable closure of K is a Galois module for the absolute Galois group.
Given a finite separable field extension L / K and a torus T over L, we have a Galois module isomorphism
Let M be a ( left or right ) module over some ring R. Given a chain of submodules of M of the form
Given a coherent such I, the quotient sheaf O < sub > D </ sub > / I is such that / I ( U ) is always a module over O < sub > D </ sub >( U );

Given and M
* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).
Given two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both
Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).
Given an evaluation e of variables by elements of M < sub > w </ sub >, we
Given a system of n-dimensional variables ( physical variables ), in k ( physical ) dimensions, write the dimensional matrix M, whose rows are the dimensions and whose columns are the variables: the ( i, j ) th entry is the power of the ith unit in the jth variable.
Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping
Given two complexes M < sub >*</ sub > and N < sub >*</ sub >, a chain map between the two is a series of homomorphisms from M < sub > i </ sub > to N < sub > i </ sub > such that the entire diagram involving the boundary maps of M and N commutes.
Given an SVD of M, as described above, the following two relations hold:
Given a set M of molecules, chemical reactions can be roughly defined as pairs r =( A, B ) of subsets from M.
Given a Hermitian form Ψ on a complex vector space V, the unitary group U ( Ψ ) is the group of transforms that preserve the form: the transform M such that Ψ ( Mv, Mw ) = Ψ ( v, w ) for all v, w ∈ V. In terms of matrices, representing the form by a matrix denoted, this says that.
Given the morphological distinctness of the Cape Verde birds and the fact that the Cape Verde population was isolated from other populations of Red Kites, it cannot be conclusively resolved at this time whether the Cape Verde population was not a distinct subspecies ( as M. migrans fasciicauda ) or even species that frequently absorbed stragglers from the migrating European populations into its gene pool.
Given an orientable Haken manifold M, by definition it contains an orientable, incompressible surface S. Take the regular neighborhood of S and delete its interior from M. In effect, we've cut M along the surface S. ( This is analogous, in one less dimension, to cutting a surface along a circle or arc.
Given such a G-module M, it is natural to consider the subgroup of G-invariant elements:
Given a smooth curve γ on ( M, g ) and a vector field V along γ its derivative is defined by

Given and over
" Given its general release in January 1931, City Lights proved to be a popular and financial success — eventually grossing over $ 5 million.
Given the large number of U. S. military personnel and their dependents residing in Europe, it was expected that over 7 % of donors would be deferred due to the policy.
Given that the ColecoVision could produce near arcade-quality ports, industry magazines like Electronic Games were unanimous in their enthusiasm over the console.
Given any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).
Given that the dispute with Pizarro over Cuzco had kept intensifying, Almagro spent a great deal of time and money equipping a company of 500 men for a new exploration south of Peru.
* Given any set X, there is an equivalence relation over the set of all possible functions X → X.
* Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G.
Given these origins, there have been various suggestions over the years to rename the town ( for example, to " Invernevis ").
Given a field K, the corresponding general linear groupoid GL < sub >*</ sub >( K ) consists of all invertible matrices whose entries range over K. Matrix multiplication interprets composition.
Given the differences of opinion within the Lord Aberdeen cabinet over the direction of foreign policy with regard to relations between Britain and the French under Napoleon III, it is not surprising that debate raged within the government as Louis Bonaparte, now assuming the title of Emperor Napoleon III of France.
On poverty, Hoover said that " Given the chance to go forward with the policies of the last eight years, we shall soon with the help of God, be in sight of the day when poverty will be banished from this nation ", and promised, " We in America today are nearer to the final triumph over poverty than ever before in the history of any land ," but within months, the Stock Market Crash of 1929 occurred, and the world's economy spiraled downward into the Great Depression.
Given a vector space V over the field R of real numbers, a function is called sublinear if
Given the extraordinary competition and technological craftsmanship required for top search engine placement, the implication of the term " search engine optimization " has deteriorated over the last decade.
Given that the cost of replacing an executive can run over 100 % of his or her annual salary, any investment of time and energy in re-recruitment will likely pay for itself many times over if it helps a business retain just a handful of key players that would have otherwise left.
Given a function ƒ defined over the reals x, and its derivative ƒ < nowiki > '</ nowiki >, we begin with a first guess x < sub > 0 </ sub > for a root of the function f. Provided the function is reasonably well-behaved a better approximation x < sub > 1 </ sub > is
The Fascist government had direct rule only over the Benadir territory Given the defeat of the Dervish movement in the early 1920s and the rise of fascism in Europe, on 10 July 1925, Mussolini gave the green light to De Vecchi to start the takeover of the north-eastern sultanates.
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 particular wing and its velocity through the air, debates over which mathematical approach is the most convenient to use can be misperceived by novices as differences of opinion about the basic principles of flight.
" Given their 3: 00 a. m. start time and non-inclusion ( at frontman John Fogerty's insistence ) in the Woodstock film, members have expressed bitterness over their experiences at the famed festival.
Given two affine spaces and, over the same field, a function is an affine map if and only if for every family of weighted points in such that
Given that radio communication channels are allocated by agencies such as the Federal Communication Commission giving a prescribed ( maximum ) bandwidth, the advantage of QPSK over BPSK becomes evident: QPSK transmits twice the data rate in a given bandwidth compared to BPSK-at the same BER.
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

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