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Page "Symmetry group" ¶ 45
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Then and equivalence
Then all elements of X equivalent to each other are also elements of the same equivalence class.
Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows.
Then the equivalence class of the pair can be identified with the rational number, and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of rational numbers.
Suppose Then, by the equivalence of the permutation representation and the group action.
Then the quotient sheaf consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms.
: Let X be a projective scheme over C. Then the functor associating the coherent sheaves on X to the coherent sheaves on the corresponding complex analytic space X < sup > an </ sup > is an equivalence of categories.
Then one imposes a set of equivalence relations upon the words, where the relations are the defining relations of the algebraic object at hand.
Let S a multiplicatively closed subset of R, i. e. for any s and t ∈ S, the product st is also in S. Then the localization of M with respect to S, denoted S < sup >− 1 </ sup > M, is defined to be the following module: as a set, it consists of equivalence classes of pairs ( m, s ), where m ∈ M and s ∈ S. Two such pairs ( m, s ) and ( n, t ) are considered equivalent if there is a third element u of S such that
Then the growth rate of the group G can be defined as the corresponding equivalence class of the function
Then an Eilenberg – MacLane space exists, as a CW-complex, and is unique up to a weak homotopy equivalence.

Then and class
Then epistolatory me was a foreign correspondent dispatching exciting cables and communiques, full of dash and wit and glamor, quoting from the books I read, imitating the grand styles of the authors recommended by a teacher in whose special, after-school class I was enrolled.
Then, once this claim ( expressed in the previous sentence ) is proved, it will suffice to prove " φ is either refutable or satisfiable " only for φ's belonging to the class C. Note also that if φ is provably equivalent to ψ ( i. e., ( φ ≡ ψ ) is provable ), then it is indeed the case that " ψ is either refutable or satisfiable " → " φ is either refutable or satisfiable " ( the soundness theorem is needed to show this ).
⟨ H ⟩, be the group generated by H. Then the word problem in H < sup >*</ sup > is solvable: given two words h, k in the generators H of H < sup >*</ sup >, write them as words in X and compare them using the solution to the word problem in G. It is easy to think that this demonstrates a uniform solution the word problem for the class K ( say ) of finitely generated groups that can be embedded in G. If this were the case the non-existence of a universal solvable word problem group would follow easily from Boone-Rogers.
Then, in 1960, Pope John XXIII issued the Code of Rubrics, completely ending the ranking of feast days by doubles etc., and replacing it by a ranking, applied not only to feast days but to all liturgical days, as I, II, III, and IV class days.
Let be the mean of the values in associated with class c, and let be the variance of the values in associated with class c. Then, the probability of some value given a class,, can be computed by plugging into the equation for a Normal distribution parameterized by and.
The other class of Dedekind rings which is arguably of equal importance comes from geometry: let C be a nonsingular geometrically integral affine algebraic curve over a field k. Then the coordinate ring k of regular functions on C is a Dedekind domain.
Then, viral glycoprotein gp42 interacts with cellular MHC class II molecules.
Then a new datum is associated with the class such that it's best sparsely represented by the corresponding dictionary.
Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.
Then every Hodge class on X is algebraic.
Then every cohomology class in H < sup > 2k </ sup >( X, Z ) ∩ H < sup > k, k </ sup >( X ) is the cohomology class of an algebraic cycle with integral coefficients on X.
Then every non-torsion cohomology class in H < sup > 2k </ sup >( X, Z ) ∩ H < sup > k, k </ sup >( X ) is the cohomology class of an algebraic cycle with integral coefficients on X.
Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.
Then every Hodge class on X is a linear combination with rational coefficients of Chern classes of vector bundles on X.
Then every Hodge class on X is a linear combination with rational coefficients of Chern classes of coherent sheaves on X.
Then, there was feudal society, in which lord and serf played the roles of class war.
Then he went to the Chandragiri government high school, till class 9.

Then and identity
Then, about halfway through, or sometimes even during the final act, one of the suspects usually dies, often because they have inadvertently deduced the killer's identity and need silencing.
Then is a derivation and is linear, i. e., and, and a Lie algebra homomorphism, i. e.,, but it is not always an algebra homomorphism, i. e. the identity does not hold in general.
Then is a group whose identity element is The group inverse of an arbitrary group element is the function inverse
Then an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.
Then,, and the constant function turns into the identity by substitution.
Then whose identity do they reveal?
Then the trace of the identity matrix is the dimension of the space ; this leads to generalizations of dimension using trace.
Then use this identity to calculate the other root.
Then, after proving that the Draques are members of the crime ring, Nick reveals the identity of the leader of the gang: Dr. Bruce Clayworth.
Then, as before in the English Reformation, this project was driven by a need for a national identity and history that addressed present concerns, hence analysis and commentary typically reflected the critic's political views.
Then he created a Native identity, telling people that he was the child of a Scottish father and Apache mother.
Then one day I met Dr. Lee and through a miracle of surgery he restored my voice enabling me to resume my real identity as Philip Gault, Lawyer.
Then, where is the vector ( 1, 0 ,..., 0 )< sup > T </ sup >, ||·|| is the Euclidean norm and is an m-by-m identity matrix, set
Then he killed George Odekirk, the forger that created his " Erik Lehnsherr " alias to prevent his true identity from being discovered by Sabra and Gabrielle Haller.
Then the disguised Letitia enters and reveals her true identity to the overjoyed Doricourt, who also learns that Hardy was not ill after all.
for x, y in A and r, s in K. Then * is an associative operation with identity element ( 0, 1 ).
Then again, Dasmariñas was annihilated and identity was lost in a heap of rubble.
Then this polynomial sequence is of binomial type, i. e. it satisfies the binomial identity
Then one can define the connection form, an antisymmetric matrix of 1-forms which satisfy from the following identity

1.282 seconds.