* ( Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation ).

To give a proof of a " yes " instance, one must specify a non-empty subset which does sum to zero.

" He added that the public as well as proper organizations were entitled to such proof and that he was ready and willing to give such proof in detail.

The theorem that bears his name may not have been his discovery, but he was probably one of the first to give a deductive proof of it.

Hilbert produced an innovative proof by contradiction using mathematical induction ; his method does not give an algorithm to produce the finitely many basis polynomials for a given ideal: it only shows that they must exist.

Such reactors often give off radioactive waste that is especially hard to safely dispose, since it must be stored in radioactive proof locations, such as underground.

Together, these first two examples give a different proof that n-dimensional Euclidean space is separable.

These papers are together believed to give a complete proof, and there are several ongoing projects to shorten and simplify this proof.

We now give an operator theoretic proof for the Cauchy – Schwarz inequality which passes to the C *- algebra setting.

Therein he attacked some often-used surrealist techniques by proclaiming, " Surrealism will at least have served to give experimental proof that total sterility and attempts at automatizations have gone too far and have led to a totalitarian system.

If, however, the subject lives a life of great extravagance, tell him he, too, can afford to give largely, the proof of his opulence being evident in his expenditure.

They must give us proof, strong as holy writ, that they have washed their hands and are worthy again to be trusted.

Proof: We only give the proof in the simplified case ; the general case is similar.

We will now give the precise meaning of this statement as well as its proof.

The MPAA's R rating is definitive proof that the organization either will never give the NC-17 rating for violence alone, or was intimidated by the subject matter.

The course of studies lasted ten years, during which time their burses continued ; but if, at the end of ten years, they had not given proof of their ability, either as teachers or as preachers, they had to give up their burse.

In fact, it is possible to give a proof that is a Noetherian ring without appealing to its order structure and this proof applies more generally to principal ideal rings ( i. e., rings in which every ideal is generated by a single element ).

Thereafter, to protect themselves, they style themselves " white men from the stars " – sorcerer-gods – and are required to give regular proof of their divinity, considerably straining both their nerves and their ingenuity.

The proof of the pudding is he didn't give a damn about those poor horses or show any respect to Mr. Scott.

" Concerning the author, the critic said, " If ever on earth there lived a man who kept the law of Christ, and could give proof of it, and be absolutely unconscious that he was giving it to them, it is this man whom the Mongols called ' our Gilmour.

She claimed to have proof that " 69 of the injured were given no medical care " and that " 80 percent of the surviving hostages are potential future invalids, including future ( occurrence of ) cancers, ( and there is a possibility that ) women who were subjected to the gas attack ( could ) give birth to defective babies ".

He had no idea how much time Budd would give him.

And you wanted no part of me when I had so much to give.

This, no doubt, is part of what Gilbert Seldes implies when he says of the arts, `` They give form and meaning to life which might otherwise seem shapeless and without sense ''.

Yet no leader had come to the fore who seemed likely to give the puissant T. R. a semblance of a race.

His very honest act called up the recent talk I had with another minister, a modest Methodist, who said: `` I feel so deeply blessed by God when I can give a message of love and comfort to other men, and I would have it no other way: and it is unworthy to think of self.

Why should Congress even consider allowing such a sum for that which can give no protection??

In the absence of a reservoir of political consensus each organized political group hopes that the elections will give them new prominence, but in a system where there is as yet no place for the less prominent.

there is no phrase or image that sounds like Hardy or that is striking enough to give individuality to the poem.

No one should wish to deny these purists the obvious pleasure they derive from all this, and to give fair warning where warning is due, no one who becomes fond of wines ever avoids acquiring some degree of purism!!

There seems to be no purpose in life that is sure -- no certain guiding principles to give stability.

This group pleads with the administration to `` give no further support for the invasion of Cuba by exile groups ''.

If one asks about this play, what it is that comes upon this community and works within it with such terrible power, there is no better answer to give than `` spirit ''.

Recognizing that there could have been no effective negotiated peace with Hitler, he points out the shocking failure to give support to the anti-Nazi underground, which very nearly eliminated Hitler in 1944.

You wonder about the Christmas card with no name on it, and it comes to you that maybe it would have been better to have made somebody else happy if you couldn't be happy yourself, to give somebody else the one they wanted -- to give them you.

The first Abstract Expressionists were considered madmen to give up their brushes and rely on the sheer force of energy to leave an image, but then the import of atomic bombs, all atomic energy, became realized, and art found no better way of expressing its power.

Perhaps the word may be included among those mysterious expressions discussed by Adolf von Harnack, “ which belong to no known speech, and by their singular collocation of vowels and consonants give evidence that they belong to some mystic dialect, or take their origin from some supposed divine inspiration .”

Furthermore, they claim that in the Bible there's no evidence showing that the office must be conveyed by laying on of hands and no Biblical command that it must be by a special class of bishops ( the laying on of hands is repeatedly used to give a commission to some person in scripture.

They had planned to sell this engine to motor manufacturers, but having heard that the Aston Martin car was no longer in production they realised that they could capitalise on the reputation of the Aston Martin name ( what we would now call the brand ) to give themselves a head start in the production of a completely new car.

Because there is no selection pressure for sight in this environment, any number of genetic abnormalities that give rise to the damage or loss of eyes could proliferate among the population with no effect on the fitness of the population.

Rowe had ordered that there should be no firing from his men until he struck his sword upon the palisades, but as he stepped forward to give the signal, he fell mortally wounded.

Pausing for a moment he scribbled on the back of an old tavern bill a note addressed to his wife, Sarah: " I have no time to say more but to beg you will give my duty to the Queen, and let her know her army has had a glorious victory.