However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

) were also backward compatible, which is often the case, then, by induction, version N will also accept input that worked under any prior version after, and including, the latest one that was not backward compatible.

Cilia of eukaryotic cells can also produce chemotaxis ; in this case it is mainly a Ca < sup > 2 +</ sup > dependent induction of the microtubular system of the basal body and the beat of the 9 + 2 microtubules within cilia.

Braid's original description of his induction is as follows: Braid himself later acknowledged that the hypnotic induction technique was not necessary in every case and subsequent researchers have generally found that on average it contributes less than previously expected to the effect of hypnotic suggestions ( q. v., Barber, Spanos & Chaves, 1974 ).

They are sometimes useful for constructing the base case of an induction argument.

In this case the sequence is called a composition series for M. In order to prove a statement inductively using composition series, the statement is first proved for simple modules, which form the base case of the induction, and then the statement is proved to remain true under an extension of a module by a simple module.

In case No. 3 the motor generator can be synchronous / synchronous or induction / synchronous.

Engines must be based on a Type 1 engine case, with no electronic fuel injection or ignition and no forced induction, with an unlimited capacity.

In the case of an electrical generator, a time-varying magnetic field inside the generator creates an electric field via electromagnetic induction, which in turn creates an energy difference between generator terminals.

By reasoning using " induction " Bacon meant the ability to gradually generalize a finding based on accumulating data-he advised proceeding by this method ( building a case from the ground up ).

The next day Lucille Hester released a letter she claimed he had drafted three years before he died, on October 29, 1999, in case he did not live to see his induction.

In proof by mathematical induction, a single " base case " is proved, and an " induction rule " is proved, which establishes that if a certain case is true, then another case is true.

Applying the induction rule repeatedly, starting from the independently proved base case, proves many, often infinitely many, other cases.

It can be shown that additivity implies the homogeneity in all cases where α is rational ; this is done by proving the case where α is a natural number by mathematical induction and then extending the result to arbitrary rational numbers.

First we prove the theorem for the 2-colour case, by induction on r + s.

In the 2-color case, if R ( r − 1, s ) and R ( r, s − 1 ) are both even, the induction inequality can be strengthened to < u > R ( r, s ) ≤ R ( r − 1, s ) + R ( r, s − 1 ) − 1 .</ u >

For instance, proofs by mathematical induction have two parts: the " base case " that shows that the theorem is true for a particular initial value such as n

Don't like to bother no one unless we have to, which I figger we do, in your case.

But in our case -- and neither my wife nor I have extreme views on integration, nor are we given to emotional outbursts -- the situation has ruined one or two valued friendships and come close to wrecking several more.

Now we can argue that the irresistible fate of Oedipus Rex was nothing more than the irresistible unconscious longings of Oedipus projected outward, but this externalization of unconscious conflict makes all the difference between a story and a clinical case history.

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

There may be a case of this sort, but it is not one we wish to argue, here.

Just as in the case of every prodigy child, we must watch for the efficacy of my teaching to show up in the future -- if he should master all the strenuous exercises I inflicted on him.

We were given a job and we carried it out, and later, his case was taken up by the Disciplinary Committee.

We can be virtuous only if we control our lower natures, the passions in this case, and strengthen our rational side ; ;

But, here again, comparative benchmarks are lacking, and we do not know, in any case, what measure of profoundity and intensity to expect from healthy, young, secure and relatively inexperienced persons ; ;

In such a case, however, we would encourage the recipient country to get on with its programing task, supply it with substantial technical assistance in performing that task, and make it plain that an expansion or even a continuation of our assistance to the country's development was conditional upon programing progress being made.

Here, for the case of squares inscribed in plane curves, we remove the restriction to convexity and give certain other results.

Continuing with the case in which **zg is a Af curve on a quadric Q, we first observe that the second regulus of Q consists precisely of the lines which join the two free intersections of **zg and the planes through any one of the multiple secants.

Are we prepared to say that in that case nothing bad occurred in the sense in which we said it did??

If anyone asked us, after we made the remark that the suffering was a bad thing, whether we should think it relevant to what we said to learn that the incident had never occurred and no pain had been suffered at all, we should say that it made all the difference in the world, that what we were asserting to be bad was precisely the suffering we thought had occurred back there, that if this had not occurred, there was nothing left to be bad, and that our assertion was in that case mistaken.

This is not, however, the case, and development is a term which we can apply to Hardy only in a very limited sense.

This is the principle which we will invoke in every case to set up a functional equation.

Our discussion of this involves using Erik Erikson's schema of `` identity vs. identity diffusion '' as a conceptual tool in superimposing a few common denominators onto the diverse personality and family configurations of the unwed mothers from whose case histories we quoted earlier.

The American liberal may, in the world of to-day, have a strong case ; ;

It may establish the relation of the figure of the dancer to light and color, in which case changes in the light or color will set off a kaleidescope of visual designs.

And in any case, answers may clarify but they do not change anything.

In connection with any claim decided by the Commission pursuant to this Title in which an award is made, the Commission may, upon the written request of the claimant or any attorney heretofore or hereafter employed by such claimant, determine and apportion the just and reasonable attorney's fees for services rendered with respect to such claim, but the total amount of the fees so determined in any case shall not exceed 10 per centum of the total amount paid pursuant to the award.

in the case of any assignment of an award, or any part thereof, which is made in writing and duly acknowledged and filed, after such award is certified to the Secretary of the Treasury, payment may, in the discretion of the Secretary of the Treasury, be made to the assignee, as his interest may appear.

Whenever the Secretary of the Treasury, or the Comptroller General of the United States, as the case may be, shall find that any person is entitled to any such payment, after such payment shall have been received by such person, it shall be an absolute bar to recovery by any other person against the United States, its officers, agents, or employees with respect to such payment.

All payments authorized under Section 7 of this Title shall be disbursed from the proper fund, as the case may be, and all amounts covered into the Treasury to the credit of the aforesaid funds are hereby permanently appropriated for the making of the payments authorized by Section 7 of this Title.

Position may not be everything, but in the case of a pool it can certainly contribute difficulties, social and/or physical.

Westinghouse has a similar system, with two classifications representing various levels of competence on the strictly technical side: consulting engineer or scientist, as the case may be, and advisory engineer or scientist.

Such classifications may be called `` somewhat advanced '' or `` somewhat delayed '', as the case may be, reserving `` moderate '' for dots upon or close to the mean.

Another case may be given in illustration of a successful use of analysis, and also of the employment of a procedure for intensive analysis.

When this is the case, the appropriate section of this manual will indicate that `` Continuation Cards '' may be used.

Statistically this has represented a loss to the nation, although one must admit that in an individual case the decision in retrospect may have been a wise one.

We may now take up for consideration a hard case which seems to require either no action employing economic pressure or else action that would seem to violate the principles set forth above.

You may think we didn't need Nancy and Jean, but you always get what you can when you can, and we had no guarantee that a fingerprint record on them couldn't be useful before we were through with this case.

There was no directive for it -- the Security Council's resolution had not mentioned political matters, and in any case the United Nations by the terms of its charter may not interfere in the political affairs of any nation, whether to unify it, federalize it or Balkanize it.

Presumably, if the reverse is the case and the good effect is more certain than the evil result that may be forthcoming, not only must the good and the evil be prudentially weighed and found proportionate, but also calculation of the probabilities and of the degree of certainty or uncertainty in the good or evil effect must be taken into account.

The arithmetic mean may be misinterpreted as the median to imply that most values are higher or lower than is actually the case.

In tort, equity, or other civil matters either party to a previous case may file an appeal.

However, certain critical interlocutory court orders, such as the denial of a request for an interim injunction, or an order holding a person in contempt of court, can be appealed immediately although the case may otherwise not have been fully disposed of.

It may, in addition, send the case back (" remand " or " remit ") to the lower court for further proceedings to remedy the defect.

More rarely, a script may have separate letters for tones, as is the case for Hmong and Zhuang.