Then I < sub > x </ sub > and I < sub > x </ sub >< sup > 2 </ sup > are real vector spaces and the cotangent space is defined as the quotient space T < sub > x </ sub >< sup >*</ sup > M = I < sub > x </ sub > / I < sub > x </ sub >< sup > 2 </ sup >.

Then the quotient space X /~ can be naturally identified with a torus: take a square piece of paper, bend and glue together the upper and lower edge to form a cylinder, then bend the resulting cylinder so as to glue together its two open ends, resulting in a torus.

Then divide 18 by 12 to get a quotient of 1 and a remainder of 6.

Then, this quotient algebra is converted into a Poisson algebra by introducing a Poisson bracket derivable from the action, called the Peierls bracket.

Then I and I < sup > 2 </ sup > are real vector spaces, and T < sub > x </ sub > M may be defined as the dual space of the quotient space I / I < sup > 2 </ sup >.

Then R / I is a ring with unity, ( respectively, R / A is a finitely generated module ), and so the above theorems can be applied to the quotient to conclude that there is a maximal ideal ( respectively maximal right ideal ) of R containing I ( respectively, A ).

Then the quotient, i. e. the remaining part of p ( x ), can be factored in the usual way with one of the other root-finding algorithms.

Then the latest entry to the quotient, 2, is multiplied by the divisor 4 to get 8, which is the largest multiple of 4 that does not exceed 10 ; so 8 is written below 10, and the subtraction 10 minus 8 is performed to get the remainder 2, which is placed below the 8.

Then this new quotient digit 5 is multiplied by the divisor 4 to get 20, which is written at the bottom below the existing 20.

Then, for some quotient polynomial Q ( x ) and remainder polynomial R ( x ) with degree ( R ) < degree ( D ),

Then, the order parameter space can be written as the Lie group quotient

Suppose that H is a locally compact Hausdorff group with a compact subgroup K. Then H acts on the quotient space X = H / K.

Then PSL ( 2, 7 ) is defined to be the quotient group

Then, the quotient of by the nullspace of its bilinear form is naturally isomorphic ( as a G-module with an invariant bilinear form ) to if r ≠ 0, and to if r

Then ( I: J ) is itself an ideal in R. The ideal quotient is viewed as a quotient because if and only if.

Then the quotient and remainder of the division are obtained as usual using div.

Then a presheaf on X is a contravariant functor from O ( X ) to the category of sets, and a sheaf is a presheaf which satisfies the gluing axiom.

Then the sheaf axioms can be expressed as the exactness of the sequence

) Then a S-scheme X is projective if and only if it is proper and there exists a very ample sheaf on X relative to S. Indeed, if X is proper, then an immersion corresponding to the very ample line bundle is necessarily closed.

Let X be a smooth projective variety where all of its irreducible components have dimension n. Then one has the following version of the Serre duality: for any locally free sheaf on X,

Then H < sup > 2 </ sup > and higher vanish for dimensional reason and the space of the global sections of the structure sheaf is one-dimensional.

Then sheaf cohomology enables us to consider a similar extension problem while " continuously varying " the Abelian group.

Then K < sub > X </ sub > is the sheaf associated to the presheaf K < sub > X </ sub >< sup > pre </ sup >.

Then the map that associates to a sheaf its global sections is a covariant functor to.

Then ΠE is a supermanifold given by the sheaf Γ ( ΛE < sup >*</ sup >).

Given a simply connected open subset D of C < sup > n </ sup >, there is an associated sheaf O < sub > D </ sub > of holomorphic functions on D. Throughout, U is any open subset of D. Then the set O < sub > D </ sub >( U ) of holomorphic functions from U to C has a natural ( componentwise ) C-algebra structure and one can collate sections that agree on intersections to create larger sections ; this is outlined in more detail at sheaf.

Then, in 1905, to explain the photoelectric effect ( 1839 ), i. e., that shining light on certain materials can function to eject electrons from the material, Albert Einstein postulated, based on Planck ’ s quantum hypothesis, that light itself consists of individual quantum particles, which later came to be called photons ( 1926 ).

Then the Netherlands were finally reunited with the origins of the " alluvial deposits of the French rivers ," of which the country in the view of Napoleon consists.

Then the Alexandroff extension of X is a certain compact space X * together with an open embedding c: X → X * such that the complement of X in X * consists of a single point, typically denoted ∞.

Then an enriched category C ( alternatively, in situations where the choice of monoidal category needs to be explicit, a category enriched over M, or M-category ), consists of

His idea of being close to his daughter consists in his nocturnal visits to her bedroom when she is fast asleep: Then he gropes around under the sheets — in a harmless way, imagining what it would be like to have a normal child and envisaging the day when he will actually come face to face with her.

Life Is Real Only Then, When " I Am " ( a book first privately printed in 1974 ) consists of the incomplete text of the third Series of All and Everything by G. I. Gurdjieff.

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 is totally unimodular if and only if every simple arbitrarily-oriented cycle in consists of alternating forwards and backwards arcs.

# Let be a scheme of finite type over C. Then there is a topological space X < sup > an </ sup > which as a set consists of the closed points of X with a continuous inclusion map λ < sub > X </ sub >: X < sup > an </ sup > → X.

Then H consists of functions of the form

Then, we use the so-called keyhole contour, which consists of a small circle about the origin of radius ε say, extending to a line segment parallel and close to the positive real axis but not touching it, to an almost full circle, returning to a line segment parallel, close, and below the positive real axis in the negative sense, returning to the small circle in the middle.

: Then the kernel of L consists of all functions f ∈ C for which f ( 0. 3 ) = 0.

: Then the kernel of D consists of all functions in C < sup >∞</ sup >( R ) whose derivatives are zero, i. e. the set of all constant functions.

Let A and B be sets of vertices in a ( possibly infinite ) digraph G. Then there exists a family P of disjoint A-B-paths and a separating set which consists of exactly one vertex from each path in P.

Then consists of integers between and.

) Then the ideal consists of all function germs which vanish to order k at p. We may now define the jet space at p by

Then the Cauchy problem consists of finding the solution u of the differential equation which satisfies