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Given the space X = Spec ( R ) with the Zariski topology, the structure sheaf O < sub > X </ sub > is defined on the D < sub > f </ sub > by setting Γ ( D < sub > f </ sub >, O < sub > X </ sub >) = R < sub > f </ sub >, the localization of R at the multiplicative system
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Given and space
* 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 any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).
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 these two assumptions, the coordinates of the same event ( a point in space and time ) described in two inertial reference frames are related by a Galilean transformation.
Given a basis of a vector space, every element of the vector space can be expressed uniquely as a finite linear combination of basis vectors.
Given infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the universe.
Given a set of training examples of the form, a learning algorithm seeks a function, where is the input space and
Given a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.
Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows.
Given the date of his publication and the widespread, permanent distribution of his work, it appears that he should be regarded as the originator of the concept of space sailing by light pressure, although he did not develop the concept further.
Given an arbitrary topological space ( X, τ ) there is a universal way of associating a completely regular space with ( X, τ ).
Given any embedding of a Tychonoff space X in a compact Hausdorff space K the closure of the image of X in K is a compactification of X.
Given a completely regular space X there is usually more than one uniformity on X that is compatible with the topology of X.
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 and X
: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.
Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).
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 a ( random ) sample the relation between the observations Y < sub > i </ sub > and the independent variables X < sub > ij </ sub > is formulated as
# Given any point x in X, and any sequence in X converging to x, the composition of f with this sequence converges to f ( x )
Given and =
* 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 the state at some initial time ( t = 0 ), we can solve it to obtain the state at any subsequent time.
Given Ω microstates at a particular energy, the probability of finding the system in a particular microstate is p = 1 / Ω.
* Given a recursively enumerable set A of positive integers that has insoluble membership problem, ⟨ a, b, c, d | a < sup > n </ sup > ba < sup > n </ sup > = c < sup > n </ sup > dc < sup > n </ sup >: n ∈ A ⟩ is a finitely generated group with a recursively enumerable presentation whose word problem is insoluble
The quantum circuits used for this algorithm are custom designed for each choice of N and the random a used in f ( x ) = a < sup > x </ sup > mod N. Given N, find Q = 2 < sup > q </ sup > such that < math > N ^ 2
# Given u and v in W, then they can be expressed as u = ( u < sub > 1 </ sub >, u < sub > 2 </ sub >, 0 ) and v = ( v < sub > 1 </ sub >, v < sub > 2 </ sub >, 0 ).
# Given u in W and a scalar c in R, if u = ( u < sub > 1 </ sub >, u < sub > 2 </ sub >, 0 ) again, then cu = ( cu < sub > 1 </ sub >, cu < sub > 2 </ sub >, c0 ) = ( cu < sub > 1 </ sub >, cu < sub > 2 </ sub >, 0 ).
Given a ring R and a unit u in R, the map ƒ ( x ) = u < sup >− 1 </ sup > xu is a ring automorphism of R. The ring automorphisms of this form are called inner automorphisms of R. They form a normal subgroup of the automorphism group of R.
Given ω = ( xθ, yθ, zθ ), with v = ( x, y, z ) a unit vector, the correct skew-symmetric matrix form of ω is
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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