Then every linear operator T in V can be written as the sum of a diagonalizable operator D and a nilpotent operator N which commute.

Then, in a virtually unprecedented move, the Court invited William T. Coleman, Jr. to argue the government's position in an amicus curiae brief, thus ensuring that the prosecution's position would be the one the Court wished to hear.

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 >.

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 one need only check the records in each bucket T against those in buckets T where k ranges between − m and m.

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.

Line QP can be extended beyond P to some point T, and line GP can be extended beyond P to some point R. Then and are vertical, so they are equal ( congruent ).

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 according to the second isomorphism theorem S ∩ T is normal in T and ST / S ≅ T /( S ∩ T ).

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 T. W.

Then the map T admits one and only one fixed-point x < sup >*</ sup > in X ( this means T ( x < sup >*</ sup >) = x < sup >*</ sup >).

Then, thus by T, which means w R w using the definition of.

Let K be a closed subset of a compact set T in R < sup > n </ sup > and let C < sub > K </ sub > be an open cover of K. Then

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 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, 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 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.

Then, under the null hypothesis that M < sub > 2 </ sub > is the true model, the difference between the deviances for the two models follows an approximate chi-squared distribution with k-degrees of freedom.

Then the unconditional distribution of Z is non-central chi-squared with k degrees of freedom, and non-centrality parameter.

Then follows a site investigation of soil, rock, fault distribution and bedrock properties on and below an area of interest to determine their engineering properties including how they will interact with, on or in a proposed construction.

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 the random number of successes we have seen, X, will have the negative binomial ( or Pascal ) distribution:

Then Bernardo Bertolucci cast him in a small speechless role as a window upholsterer in the film La Luna which evaded American distribution due to its subject matter.

Then the joint distribution of is multivariate normal with mean vector and covariance matrix

Then, the relativistic expression for the angular distribution of the bremsstrahlung ( considering only the dominant dipole radiation contribution ), is

Let be the mean of the values in associated with class c, and let be the variance of the values in associated with class c. Then, the probability of some value given a class,, can be computed by plugging into the equation for a Normal distribution parameterized by and.

Then the cevians are concurrent if and only if a mass distribution can be assigned to the vertices such that each cevian intersects the opposite facet at its center of mass.

Then, expensive video equipment was required: Laserdisc player, PC, genlock, and recording deck for producing the master ; subsequently two or more video decks were then needed for producing distribution copies.

Three books by Gurdjieff were published in the English language in the United States after his death: Beelzebub's Tales to His Grandson published in 1950 by E. P. Dutton & Co. Inc., Meetings with Remarkable Men, published in 1963 by E. P. Dutton & Co. Inc., and Life is Real Only Then, When ' I Am ', printed privately by E. P. Dutton & Co. and published in 1978 by Triangle Editions Inc. for private distribution only.

Then they are independent, but not necessarily identically distributed, and their joint probability distribution is given by the Bapat – Beg theorem.

Then there is some probability distribution m on the interval and some random variable Y such that

Then the joint distribution

" Then, on May 14, 1861, " An act regulating the sale of prizes and the distribution thereof ," was also passed.

Then, the field radiated by the small source is a spherical wave which is modulated by the FT of the source distribution, as in eqn.

Then the phase space distribution determines the probability that the system will be found in the infinitesimal phase space volume.

Then the Wishart distribution is the probability distribution of the p × p random matrix

Then, if n ≥ p, has a Wishart distribution with n degrees of freedom if it has a probability density function given by

Then, Boltzmann's entropy formula gives for the probability distribution function, A = const.

Then the theorem says that the asymptotic distribution of these invariants is uniform over the group, so that a conjugacy class with k elements occurs with frequency asymptotic to

Then the joint posterior distribution of the probabilities p < sub > 1 </ sub >, ..., p < sub > m </ sub > is given by ;