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Page "Analytic function" ¶ 11
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is and because
This is puzzling to an outsider conscious of the classic tradition of liberalism, because it is clear that these Democrats who are left-of-center are at opposite poles from the liberal Jefferson, who held that the best government was the least government.
In fact it has caused us to give serious thought to moving our residence south, because it is not easy for the most objective Southerner to sit calmly by when his host is telling a roomful of people that the only way to deal with Southerners who oppose integration is to send in troops and shoot the bastards down.
Had the situation been reversed, had, for instance, England been the enemy in 1898 because of issues of concern chiefly to New England, there is little doubt that large numbers of Southerners would have happily put on their old Confederate uniforms to fight as allies of Britain.
But it is more than irony: one of the main reasons why nationalism is no longer a tenable concept is because it has spread throughout the planet.
The `` approximate '' is important, because even after the order of the work has been established by the chance method, the result is not inviolable.
Often it is recognized that all the details of the pattern may not be essential to the outcome but, because the pattern was empirically determined and not developed through theoretical understanding, one is never quite certain which behavior elements are effective, and the whole pattern becomes ritualized.
Yet often fear persists because, even with the most rigid ritual, one is never quite free from the uneasy feeling that one might make some mistake or that in every previous execution one had been unaware of the really decisive act.
Or is it relevant because it teaches us something useful to know about ourselves??
I knew that a conversation with the author would not settle such questions, because a man is not the same as his writing: in the last analysis, the questions had to be settled by the work itself.
I suggested that one must let it in because it is the truth, but Beckett did not take to the word truth.
That is why the form itself becomes a preoccupation, because it exists as a problem separate from the material it accommodates.
It is because there is not only darkness but also light that our situation becomes inexplicable.
If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;
Jazz is good not only because it promotes wholeness but because of its decided sexual effect.
It is worth dwelling in some detail on the crisis of this story, because it brings together a number of characteristic elements and makes of them a curious, riddling compound obscurely but centrally significant for Mann's work.

is and if
`` That is, if we can be sure this is Colcord's money '' --
Let me pass over the trip to Sante Fe with something of the same speed which made Mrs. Roebuck `` wonduh if the wahtahm speed limit '' ( 35 m.p.h. ) `` is still in ee-faket ''.
It is the last of the three tests of manhood which the women impose, to discover if a male is worthy of survival there.
There was a measure of protection in its concrete walls and ceiling, but the engineers who hastily installed it were well aware that concrete is not much better than prayer, if as efficacious, when a direct hit comes along.
`` And if the dive goes OK he has the exclusive import rights to your line for this country, is that right ''??
`` All right, if you can't do your arithmetic during school hours you can do it after school is out '', Miss Langford said firmly, not smiling.
Why, in the first place, call himself a liberal if he is against laissez-faire and favors an authoritarian central government with womb-to-tomb controls over everybody??
And if he is so scornful of the rights of states, why not advocate a different sort of constitution that he could more sincerely support??
If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.
I think it is essential, however, to pinpoint here the difference between the two concepts of sovereignty that went to war in 1861 -- if only to see better how imperative is our need today to clarify completely our far worse confusion on this subject.
Unruly hair goes straight up from his forehead, standing so high that the top falls gently over, as if to show that it really is hair and not bristle.
If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.
He will not curb his instinctual desires but release the energy within him that makes him feel truly and fully alive, even if it is only for this brief moment before the apocalypse of annihilation explodes on earth.
If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.
And if I have gone into so much detail about so small a work, that is because it is also so typical a work, representing the germinal form of a conflict which remains essential in Mann's writing: the crude sketch of Piepsam contains, in its critical, destructive and self-destructive tendencies, much that is enlarged and illuminated in the figures of, for instance, Naphta and Leverkuhn.
All such imitations of negative quality have given rise to a compensatory response in the form of a heroic and highly individualistic humanism: if man can neither know nor love reality as it is, he can at least invent an artistic `` reality '' which is its own world and which can speak to man of purely personal and subjective qualities capable of being known and worthy of being loved.
But Aristotle kept the principle of levels and even augmented it by describing in the Poetics what kinds of character and action must be imitated if the play is to be a vehicle of serious and important human truths.
Operating as a one man police force in fact if not in name, he is at once more independent and more dedicated than the police themselves.

is and polynomial
The characteristic polynomial for A is Af and this is plainly also the minimal polynomial for A ( or for T ).
If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.
The theorem which we prove is more general than what we have described, since it works with the primary decomposition of the minimal polynomial, whether or not the primes which enter are all of first degree.
( C ) if Af is the operator induced on Af by T, then the minimal polynomial for Af is Af.
We shall find a polynomial Af such that Af is the identity on Af and is zero on the other Af, and so that Af, etc..
Note also that if Af, then Af is divisible by the polynomial p, because Af contains each Af as a factor.
It is clear that each vector in the range of Af is in Af for if **ya is in the range of Af, then Af and so Af because Af is divisible by the minimal polynomial P.
Thus Af is divisible by the minimal polynomial P of T, i.e., Af divides Af.
Hence the minimal polynomial for Af is Af.
If Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.
In the notation of the proof of Theorem 12, let us take a look at the special case in which the minimal polynomial for T is a product of first-degree polynomials, i.e., the case in which each Af is of the form Af.
The diagonalizable operator D and the nilpotent operator N are uniquely determined by ( A ) and ( B ) and each of them is a polynomial in T.
for since it is nilpotent, the minimal polynomial for this operator is of the form Af for some Af ; ;

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