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Then and Zariski

For f ∈ R, define D < sub > f </ sub > to be the set of ideals of R not containing f. Then each D < sub > f </ sub > is an open subset of Spec ( R ), and is a basis for the Zariski topology.

Then and tangent

Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as f. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T. Then ( 1 ) the distance from F to T is 2f, and ( 2 ) a tangent to the parabola at point T intersects the line of symmetry at a 45 ° angle.

Then rotate the tangent screw back and forth so that the reflected image passes alternately above and below the direct view.

Then consider a vector tangent to:

Let v ∈ T < sub > p </ sub > M be a tangent vector to the manifold at p. Then there is a unique geodesic γ < sub > v </ sub > satisfying γ < sub > v </ sub >( 0 )

Then each tangent space is just R. On each copy of R at the point y, we introduce the modified inner product

Then a connection on the tangent bundle of M, called an affine connection, distinguishes a class of curves called ( affine ) geodesics.

Then, multiply the tangent by the front wheel braking effort percentage and divide by the ratio of the center of gravity height to the wheelbase.

Then, if is finite-dimensional, then so is the submanifold ; likewise, the tangent space has the same dimension as.

Then the differential of φ, φ < sub >*</ sub > = dφ ( or Dφ ), is a vector bundle morphism ( over M ) from the tangent bundle TM of M to the pullback bundle φ * TN.

Then the inner product of two tangent vectors is

Then, hit the inverse tangent key ( or tan - 1 ) and the answer just calculated.

Then the unit tangent vector T may be written as

Let C ′ be the curve obtained by shrinking C by a factor of 2 toward P. Then the point R ′ corresponding to R is the center of the rectangle PXRY, and the tangent to C ′ at R ′ bisects this rectangle parallel to PY and XR.

Then all the lines which are common tangents to both P = 0 and Q = 0 are tangent to C. So, by the AF + BG theorem, the tangential equation of C has the form HP + KQ = 0.

Then, integration of the tangent field ( done numerically, if not analytically ) yields the curve.

Then the tangent cone to X at x is the spectrum of the associated graded ring of O < sub > X, x </ sub > with respect to the m-adic filtration:

Then a vector field H on TM ( that is, a section of the double tangent bundle TTM ) is a semispray on M, if any of the three following equivalent conditions holds:

Fix a point p ∈ M, and an orthonormal basis X < sub > 1 </ sub >, X < sub > 2 </ sub > of tangent vectors at p. Then the principal curvatures are the eigenvalues of the symmetric matrix

Then at time the particle acquires the velocity, as if it underwent an elastic push from the infinitely-heavy plane, which is tangent to at the point, and at time moves along the normal to at with the velocity.

The key tool is the curve X given by the set of pairs ( p, ℓ ) where p is on the conic C and ℓ is tangent to the conic D. Then X is smooth ; more specifically X is an elliptic curve.

Then construct two additional circles, each tangent to AM, BC, and to the circumcircle.

Then and space

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

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

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 and at

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 ; ;

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