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Given a functor F: J → C ( thought of as an object in C < sup > J </ sup >), the limit of F, if it exists, is nothing but a terminal morphism from Δ to F. Dually, the colimit of F is an initial morphism from F to Δ.
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Given and functor
Given a functor U and an object X as above, there may or may not exist an initial morphism from X to U. If, however, an initial morphism ( A, φ ) does exist then it is essentially unique.
Given an arbitrary contravariant functor G from C to Set, Yoneda's lemma asserts that
Given an object X, a functor G ( taking for simplicity the first functor to be the identity ) and an isomorphism proof of unnaturality is most easily shown by giving an automorphism that does not commute with this isomorphism ( so ).
Given a concrete category ( C, U ) and a cardinal number N, let U < sup > N </ sup > be the functor C → Set determined by U < sup > N </ sup >( c ) = ( U ( c ))< sup > N </ sup >.
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Given any finite flat commutative group scheme G over S, its Cartier dual is the group of characters, defined as the functor that takes any S-scheme T to the abelian group of group scheme homomorphisms from the base change G < sub > T </ sub > to G < sub > m, T </ sub > and any map of S-schemes to the canonical map of character groups.
Given a continuous map there is a map defined by This makes into a functor from the category of topological spaces into itself.
Given and F
Given a field F, the assertion “ F is algebraically closed ” is equivalent to other assertions:
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* Shay Given – former goalkeeper for both the Republic of Ireland and Newcastle United F. C ..
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Given any subset F =
Given and J
Given two fractional ideals I and J, one defines their product IJ as the set of all finite sums: the product IJ is again a fractional ideal.
Given a set S with three subsets, J, K, and L, the following holds:
Given a hermitian structure on a vector space, J and Ω are related via
Given that the degree specializes in international law, and is not teaching a first degree in U. S. law ( the J. D.
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Given the partition function Z in terms of the source field J, the energy functional is its logarithm.
* Kim T. J. A Combined Land Use-Transportation Model When Zonal Travel Demand is Endogenously Given, Transportation Research, 17B, pp. 449 – 462.
Given a symplectic form ω and a linear complex structure J, one may define an associated symmetric bilinear form g < sub > J </ sub > on V < sub > J </ sub >
* Given, Brian J.
Given the coefficient sequence for some M < J and all the difference sequences, k = M ,..., J-1, one computes recursively
Given Armstrong's concerns, the administration employed general counsel J. Patrick Abell to file a friendly test case to determine the constitutionality of the incentive package.
Given a high-profile spot on the all-star compilation Roll Wit tha Flava as their first recording opportunity, Zhane lived up to the pressure and came away with one of the hip-hop party anthems of all time, " Hey, Mr. D. J.
Given a simply connected and open subset D of R < sup > 2 </ sup > and two functions I and J which are continuous on D then an implicit first-order ordinary differential equation of the form
Given an arbitrary Riemannian metric g on an almost complex manifold M one can construct a new metric g ′ compatible with the almost complex structure J in an obvious manner:
Given and →
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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 ).
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Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).
Given metric spaces ( X, d < sub > 1 </ sub >) and ( Y, d < sub > 2 </ sub >), a function f: X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d < sub > 1 </ sub >( x, y ) < δ, we have that d < sub > 2 </ sub >( f ( x ), f ( y )) < ε.
Given a morphism f: B → A the associated natural transformation is denoted Hom ( f ,–).
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: Given two sets, A and T, of equal size, together with a weight function C: A × T → R. Find a bijection f: A → T such that the cost function:
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