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 in 1952 the Barbados Advocate newspaper polled several prominent Barbadian politicians, lawyers, businessmen, the Speaker of the Barbados House of Assembly and later as first President of the Senate, Sir Theodore Branker, Q. C.

Then, we say Q is a generator of a one parameter symmetry Lie group.

Then the signal is either sampled or processed by the four-quadrant multiplier which produces the complex envelope components ( I, Q ) through multiplying it by the heterodyne frequency.

Then D ( Q ) contains P but, of course, not Q.

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

Then for each t the measure Q restricted to the unaugmented sigma fields is equivalent to P restricted to

Then a representation of Q is just a covariant functor from this category to the category of finite dimensional vector spaces.

For example, let P be an irreducible polynomial with integer coefficients and p be a prime number which does not divides the leading coefficient of P. Let Q be the polynomial over the finite field with p elements, which is obtained by reducing modulo p the coefficients of P. Then, if Q is separable ( which is the case for every p but a finite number ) then the degrees of the irreducible factors of Q are the lengths of the cycles of some permutation of the Galois group of P.

Then Q (√ d ) is a finite extension of Q, called a quadratic extension.

Let V a representation of G, and form the vector bundle V = Q ×< sub > G </ sub > V over M. Then the principal G-connection α on Q induces a covariant derivative on V, which is a first order linear differential operator

Then the Radon – Nikodym theorem provides the function g, equal to the density of μ with respect to Q.

Then xR is a direct summand of Q considered as R-modules.

Then Q is not Artinian as a left Z module, but it is Artinian a right Q module.

* Top Cow: Then and Now 2: Q & A with Publisher Filip Sablik by Chris Arrant, Newsarama, June 25, 2008

* Top Cow: Then and Now 1: Q & A with President Matt Hawkins by Chris Arrant, Newsarama, June 24, 2008

Then the sum S < sub > t </ sub > of the Q < sub > a, t </ sub >, where a runs over all the roots of P, can be taken as a particular Q < sub > t </ sub >.

Then there exist integers x and y such that

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 >< 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 >.

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.

My own stern hand has rent the ancient bond, And thereof shall the ending not have end: But not for me, that loved her, to be fond Lightly to please me with a newer friend Then hold it more than bravest-feathered song, That I affirm to thee, with heart of pride, I knew not what did to a friend belong Till I stood up, true friend, by thy true side ; ;

Then in 2 we show that any line involution with the properties that ( A ) It has no complex of invariant lines, and ( B ) Its singular lines form a complex consisting exclusively of the lines which meet a twisted curve, is necessarily of the type discussed in 1.

`` Then see that your youngster has a routine checkup once a year ''.

Then, following the programme he outlined in his talk at the 1958 International Congress of Mathematicians, he introduced the theory of schemes, developing it in detail in his Éléments de géométrie algébrique ( EGA ) and providing the new more flexible and general foundations for algebraic geometry that has been adopted in the field since that time.

Then he proves the antithesis, that time has no beginning, by showing that if time had a beginning, then there must have been " empty time " out of which time arose.

Then it turns out that the child has constructed this elaborate ruse in order to steal cookies and watch late-night TV without notice, giving him ur-manipulator status.

Then on July 20, 2007 guitarist Jack Owen ( ex-Cannibal Corpse ) announced that Deicide is " on hiatus " and he has joined Ohio based death / thrash combo Estuary for touring purposes.

Then, dividing the units of energy ( such as eV ) by a fundamental constant that has units of velocity ( M < sup > 0 </ sup > L < sup > 1 </ sup > T < sup >-1 </ sup >), facilitates the required conversion of using energy units to describe momentum.

Then he announced to the world: " The independence of Austria, for which he has fallen, is a principle that has been defended and will be defended by Italy even more strenuously ", and then replaced in the main square of Bolzano the statue of Walther von der Vogelweide, a Germanic troubadour, with that of Drusus, a Roman general who conquered part of Germany.

" Then examine it and test it by these rules you have, and firstly, and chiefly, by this: whether the impression has to do with the things that are up to us, or those that are not ; and if it has to do with the things that are not up to us, be ready to reply, " It is nothing to me.

Then, after the synthesis engine has mapped the design to a netlist, the netlist is translated to a gate level description where simulation is repeated to confirm the synthesis proceeded without errors.

Then he said " I'm fining everyone in this courtroom fifty cents for living in a city where a person has to steal bread in order to eat!

Then Scarlett meets Rhett Butler, a man who has a reputation as a rogue.

Then, the intended recipient ( B ), who also has been informed by A about the password ( 2a ), now approaches M and tells him the agreed password ( 3a ).

Then a general definition of isomorphism that covers the previous and many other cases is: an isomorphism is a morphism that has an inverse, i. e. there exists a morphism with and.

“ And that Christ being Lord, and God the Son of God, and appearing formerly in power as Man, and Angel, and in the glory of fire as at the bush, so also was manifested at the judgment executed on Sodom, has been demonstrated fully by what has been said .” Then I repeated once more all that I had previously quoted from Exodus, about the vision in the bush, and the naming of Joshua ( Jesus ), and continued: “ And do not suppose, sirs, that I am speaking superfluously when I repeat these words frequently: but it is because I know that some wish to anticipate these remarks, and to say that the power sent from the Father of all which appeared to Moses, or to Abraham, or to Jacob, is called an Angel because He came to men ( for by Him the commands of the Father have been proclaimed to men ); is called Glory, because He appears in a vision sometimes that cannot be borne ; is called a Man, and a human being, because He appears arrayed in such forms as the Father pleases ; and they call Him the Word, because He carries tidings from the Father to men: but maintain that this power is indivisible and inseparable from the Father, just as they say that the light of the sun on earth is indivisible and inseparable from the sun in the heavens ; as when it sinks, the light sinks along with it ; so the Father, when He chooses, say they, causes His power to spring forth, and when He chooses, He makes it return to Himself.

Then, after the pump energy stored in the laser medium has approached the maximum possible level, the introduced loss mechanism ( often an electro-or acousto-optical element ) is rapidly removed ( or that occurs by itself in a passive device ), allowing lasing to begin which rapidly obtains the stored energy in the gain medium.

Then the null set automatically has measure zero because of countable additivity, because and is finite if and only if the empty set has measure zero.

Then, five pages later, she quotes the same passage again, but this time in full, straw and all, to show how witches had hallucinations of flight ( Murray 1921, 105-6 ); she does not realise that she has thereby wrecked her previous rationalistic interpretation of the passage.

Then the scene shifts: Odysseus has been a captive of the beautiful nymph Calypso, with whom he has spent seven of his ten lost years.