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 k < sub > x </ sub > := R < sub > x </ sub >/ m < sub > x </ sub > is a field and m < sub > x </ sub >/ m < sub > x </ sub >< sup > 2 </ sup > is a vector space over that field ( the cotangent space ).

If we take E to be the sum of the even exterior powers of the cotangent bundle, and F to be the sum of the odd powers, define D = d + d *, considered as a map from E to F. Then the topological index of D is the Euler characteristic of the Hodge cohomology of M, and the analytical index is the Euler class of the manifold.

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, 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 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 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 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, 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 at last the darkness began to dissolve.

Then he was on his way at a gallop.

Then, helpfully, as she merely stared at him in weary silence, `` Maybe you could write it down for me, huh??

Then, on July 2, there occurred another incident which set tongues to wagging at a furious clip.

Then the boy straight-armed himself up, twisting at the last moment so he landed sitting.

`` Then I return to the United States for engagements at the Hollywood Bowl and in Philadelphia '', he added.

Then I spent the next two days at the baseball park and at Jack Doyle's pool parlors.

Then I spoke at the ninetieth birthday party of W. E. Burghardt Du Bois, who embarked on a fictional trilogy at eighty-nine and who, with The Crisis, had created a Negro intelligentsia that had never existed in America before him.

Then, all but blind, he said there was nothing in Back to Methuselah --, -- `` G.B.S. ought to have known that '', -- and `` I look at my bookshelves despairingly, knowing that I can have nothing more to do with them ''.

Then we have surviving at least one instance of a poem prepared for another, in Naturam non Pati Senium, and perhaps also the De Idea Platonica.

Certainly, the meaning is clearer to one who is not familiar with Biblical teachings, in the New English Bible which reads: `` Then Jesus arrived at Jordan from Galilee, and he came to John to be baptized by him.

Then see what a boom in all trades, as well as slum clearance at no cost to taxpayers, will happen.

Then the audience saw a small, dim figure appear at the edge of the Presidential box.

Then she looked at the old woman again, her eyes calm.

Then, when he had it pointed down the hill, he stopped to gaze at her through the window.

Then he stood back to look at Mr. Jack, who was pulling on his pigskin gloves.

Then he looked at his finger, at the wrinkled, heavy knuckle and the thick nail he used like a knife to pry up, slit, and open.

Then you can do the finishing touches at your leisure.

Then, for the above parasites, feed continuously at these levels: Feeder cattle -- 2-5 grams of phenothiazine daily ; ;

Then, at last, his day would fall into an ordered pattern and he would be free to read, or garden or just wander through the woods in the late afternoon, accompanied by his dogs.

Then at least he would have a place to hang his tools and something to work on.

Then, my mother blushed at this small lie ; ;

Then there exist integers x and y such that

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

Then, for any given sequence of integers a < sub > 1 </ sub >, a < sub > 2 </ sub >, …, a < sub > k </ sub >, there exists an integer x solving the following system of simultaneous congruences.

Then G is a group under composition, meaning that ∀ x ∈ A ∀ g ∈ G ( = ), because G satisfies the following four conditions:

Then, in terms of P < sub > n </ sub >( x ), the remainder

Then setting x =

Then, Gödel defined essences: if x is an object in some world, then the property P is said to be an essence of x if P ( x ) is true in that world and if P entails all other properties that x has in that world.

Then b < sub > 0 </ sub > is the value of p ( x < sub > 0 </ sub >).

Then, x ( or x to some power ) is repeatedly factored out.

Then the probability density function f *( x ) of the size biased population is

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

Then N < sub > x </ sub > is a directed set, where the direction is given by reverse inclusion, so that S ≥ T if and only if S is contained in T. For S in N < sub > x </ sub >, let x < sub > S </ sub > be a point in S. Then ( x < sub > S </ sub >) is a net.