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 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 quotient sheaf consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms.

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, advised by the Architect of the Capitol, the Joint Committee for the Library, traditionally responsible for the works of art in the building, ordered the space cleared and painted in fresco, to show `` the Peace after the Civil War '', `` the Spanish-American War '', and `` the Birth of Aviation '', to match as nearly as feasible Brumidi's technique and composition.

Then Af, the maximization being over all admissible Af and the integration over the whole of stage space.

Then we can express the variables as the sum of the ( time averaged ) mean field () that varies in space and a small fluctuating field () that varies in space and time.

Then is a compact topological space ; this follows from the Tychonoff theorem.

Then by Arzelà – Ascoli theorem the space K is compact.

Then the cotangent space at x is defined as the dual space of T < sub > x </ sub > M:

Then, considering all three colour channels, and assuming that the colour channels are expressed in a γ = 1 colour space ( that is to say, the measured values are proportional to light intensity ), we have:

Then an " estimator " is a function that maps the sample space to a set of sample estimates.

Then ( Ω, F, P ) is a probability space, with sample space Ω, event space F and probability measure P.

Then came Gaia ( Earth ), Tartarus ( the cave-like space under the earth ; the later-born Erebus is the darkness in this space ), and Eros ( Sexual Desire-the urge to reproduce, not the emotion of love as is the common misconception ).

Then the Zariski tangent space at a point p ∈ X is the collection of K-derivations D: O < sub > X, p </ sub >→ K, where K is the ground field and O < sub > X, p </ sub > is the stalk of O < sub > X </ sub > at p.

Then, when personal computers became available, gamers could simply " sit down and play " without learning masses of rules, clearing physical space, and finding and coordinating schedules with opponents.

Let X be a normed topological vector space over F, compatible with the absolute value in F. Then in X *, the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak -* topology.

Then the trace of the identity matrix is the dimension of the space ; this leads to generalizations of dimension using trace.

) Then X < sub > i </ sub > is the value ( or realization ) produced by a given run of the process at time i. Suppose that the process is further known to have defined values for mean μ < sub > i </ sub > and variance σ < sub > i </ sub >< sup > 2 </ sup > for all times i. Then the definition of the autocorrelation between times s and t is

Then X is reflexive if and only if each X < sub > j </ sub > is reflexive.

Then X is separable if and only if X ′ is separable.

Then X is compact if and only if X is a complete lattice ( i. e. all subsets have suprema and infima ).

Then all elements of X equivalent to each other are also elements of the same equivalence class.

Then the expectation of this random variable X is defined as

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 joint distribution of X and Y is completely determined by our channel and by our choice of, the marginal distribution of messages we choose to send over the channel.

Then ƒ is invertible if there exists a function g with domain Y and range X, with the property:

* Let the index set I of an inverse system ( X < sub > i </ sub >, f < sub > ij </ sub >) have a greatest element m. Then the natural projection π < sub > m </ sub >: X → X < sub > m </ sub > is an isomorphism.

Then for a specific value x of X, the function L ( θ | x )

Then the observation that X

Then the emf, E < sub > X </ sub >, of the same cell containing the solution of unknown pH is measured.

Then X has cardinality at most and cardinality at most if it is first countable.

Then A is dense in C ( X, R ) if and only if it separates points.

Then ρ will be the finest completely regular topology on X which is coarser than τ.