We say that N is nilpotent if there is some positive integer R such that Af.

It is very easy to establish by induction on R that if F is in Af then Af ; ;

The Republicans some weeks ago served notice through Senator Thruston B. Morton ( R ) of Kentucky, chairman of the Republican National Committee, that the Kennedy administration would be held responsible if the outcome in Laos was a coalition government susceptible of Communist domination.

Where territory is occupied in the course of hostilities by an enemy's force, even if the annexation of the occupied country is proclaimed by the enemy, there can be no change of allegiance during the progress of hostilities on the part of a citizen of the occupied country ( R v Vermaak ( 1900 ) 21 NLR 204 ( South Africa )).

A natural-born subject owes allegiance wherever they may be, so that where territory is occupied in the course of hostilities by an enemy's force, even if the annexation of the occupied country is proclaimed by the enemy, there can be no change of allegiance during the progress of hostilities on the part of a citizen of the occupied country ( R v Vermaak ( 1900 ) 21 NLR 204 ( South Africa )).

In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X

: if R ( a, b ) and R ( b, a ), then a = b,

: if R ( a, b ) with a ≠ b, then R ( b, a ) must not hold.

Conversely, a subset R defines a binary function if and only if, for any x in X and y in Y, there exists a unique z in Z such that ( x, y, z ) belongs to R.

The definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ring R. It also can be easily generalized to n-ary functions, where the proper term is multilinear.

That is, if R is a PID, and a and b are elements of R, and d is a greatest common divisor of a and b,

If is an outward pointing in-plane normal, whereas is the unit vector perpendicular to the plane ( see caption at right ), then the orientation of C is chosen so that a tangent vector to C is positively oriented if and only if forms a positively oriented basis for R < sup > 3 </ sup > ( right-hand rule ).

) The exterior derivative of a k-form in R < sup > 3 </ sup > is defined as the ( k + 1 )- form from above ( and in R < sup > n </ sup > if, e. g., < math >

Two ideals A and B in the commutative ring R are called coprime ( or comaximal ) if A + B = R. This generalizes Bézout's identity: with this definition, two principal ideals ( a ) and ( b ) in the ring of integers Z are coprime if and only if a and b are coprime.

In general, if y = f ( x ), then it can be transformed into y = af ( b ( x − k )) + h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis.

His comment on Numbers 23: 19 has a still more polemical tone: “ God is not a man that he should lie ; neither the son of man, that he should repent ; < font face =" times new roman " size = 3 > if a man says: ‘ I am a god ’ he is a liar ; if he says: ‘ I am a son of man ’ he will have cause to regret it ; and if he says, ‘ I will go up to heaven ’ he has said but will not keep his word ” last phrase is borrowed from B ' midbar 23: 19 ( Yer.

These aberrations are avoided if, according to Abbe, the sine condition, sin u ' 1 / sin u1 = sin u ' 2 / sin u2, holds for all rays reproducing the point O.

In the statement < code >< u > if </ u > token = if < u > then </ u > < u > result </ u >

What the compiler would see in the above example would be "< code >< u > if </ u > token =

The horizontal line y = c is a horizontal asymptote of the function y = ƒ ( x ) if

E. g., if n =

* completely additive if a ( mn ) = a ( m ) + a ( n ) for all natural numbers m and n ;

* completely multiplicative if a ( mn ) = a ( m ) a ( n ) for all natural numbers m and n ;

* additive if a ( mn ) = a ( m ) + a ( n ) for all coprime natural numbers m and n ;

Set-theoretically, one may represent a binary function as a subset of the Cartesian product X × Y × Z, where ( x, y, z ) belongs to the subset if and only if f ( x, y ) = z.

Furthermore, for every a ∈ A we have that a ∧-a = 0 ∈ I and then a ∈ I or-a ∈ I for every a ∈ A, if I is prime.

In particular, if α = m + 1 / 2 with m a nonnegative integer, the above relations imply directly that

Similarly, if we limit the number of literals per clause to 2 and change the OR operations to XOR operations, the result is exclusive-or 2-satisfiability, a problem complete for SL = L.

For if such were the case, either the plane of the two lines would meet **zg in more than K points or, alternatively, the order of the image regulus of the pencil determined by the two lines would be too high.

In abstract algebra, a field extension L / K is called algebraic if every element of L is algebraic over K, i. e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i. e. which contain transcendental elements, are called transcendental.

An extension L / K is algebraic if and only if every sub K-algebra of L is a field.

To see this, note that if L is any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is contained within the algebraic closure of K.

because if M is any algebraically closed field containing K, then the elements of M which are algebraic over K form an algebraic closure of K.

The algebraic closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K is finite.

The separable closure is the full algebraic closure if and only if K is a perfect field.

`` 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.