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 ) 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.

The b value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like a, reflects the function in the y-axis when it is negative.

:: Rule 10. 05 ( b ) Comment: Rule 10. 05 ( b ) shall not apply if the fielder merely looks toward or feints toward another base before attempting to make the putout at first base.

The order of the elements in each pair of G is important: if a ≠ b, then aRb and bRa can be true or false, independently of each other.

An ideal I of A is called prime if I ≠ A and if a ∧ b in I always implies a in I or b in I.

In an " extended model " which includes hot dark matter in the form of neutrinos, then if the " physical baryon density " Ω < sub > b </ sub > h < sup > 2 </ sup > is estimated at about 0. 023 ( this is different from the ' baryon density ' Ω < sub > b </ sub > expressed as a fraction of the total matter / energy density, which as noted above is about 0. 046 ), and the corresponding cold dark matter density Ω < sub > c </ sub > h < sup > 2 </ sup > is about 0. 11, the corresponding neutrino density Ω < sub > v </ sub > h < sup > 2 </ sup > is estimated to be less than 0. 0062.

In calculus, this picture also gives a geometric proof of the derivative if one sets and interpreting b as an infinitesimal change in a, then this picture shows the infinitesimal change in the volume of an n-dimensional hypercube, where the coefficient of the linear term ( in ) is the area of the n faces, each of dimension

For given nonzero integers a and b there is a nonzero integer of minimal absolute value among all those of the form ax + by with x and y integers ; one can assume d > 0 by changing the signs of both s and t if necessary.

The etymology is uncertain, but a strong candidate has long been some word related to the Biblical פוך ( pūk ), " paint " ( if not that word itself ), a cosmetic eye-shadow used by the ancient Egyptians and other inhabitants of the eastern Mediterranean.

Syllabaries typically contain 50 to 400 glyphs ( though the Múra-Pirahã language of Brazil would require only 24 if it did not denote tone, and Rotokas would require only 30 ), and the glyphs of logographic systems typically number from the many hundreds into the thousands.

Moseley, after discussions with Bohr who was at the same lab ( and who had used Van den Broek's hypothesis in his Bohr model of the atom ), decided to test Van den Broek and Bohr's hypothesis directly, by seeing if spectral lines emitted from excited atoms fit the Bohr theory's demand that the frequency of the spectral lines be proportional to a measure of the square of Z.

* Any large snake that " crushes " its prey ( see Constriction ), if applied loosely, could be called anaconda.

For example, the open interval ( 0, 1 ) does not have a least element: if x is in ( 0, 1 ), then so is x / 2, and x / 2 is always strictly smaller than x.

For example, if one defines categories in terms of sets, that is, as sets of objects and morphisms ( usually called a small category ), or even locally small categories, whose hom-objects are sets, then there is no category of all sets, and so it is difficult for a category-theoretic formulation to apply to all sets.

Astatine is sometimes described as being a black solid ( assuming that it follows this trend ), or as having a metallic appearance ( if it is a metalloid or a metal ).

Flowers across the Apiaceae are fairly uniform and are usually perfect ( hermaphroditic ) and actinomorphic, but some are andromonoecious, polygamomonoecious, or even dioecious ( as in Acronema ), with a distinct calyx and corolla, but the calyx if often highly reduced, to the point of being undetectable in many species, while the corolla can be white, yellow, pink or purple.

Any type of focal lesion of the central nervous system ( such as stroke, brain tumour, multiple sclerosis ) will cause the type of ataxia corresponding to the site of the lesion: cerebellar if in the cerebellum, sensory if in the dorsal spinal cord ( and rarely in the thalamus or parietal lobe ), vestibular if in the vestibular system ( including the vestibular areas of the cerebral cortex ).

Fischer insisted that the match be the first to ten wins ( draws not counting ), but that the champion would retain the crown if the score was tied 9 – 9.

The term became popular again in Australia first, when George Giffen, in his memoirs ( With Bat and Ball, 1899 ), used the term as if it were well known.

The automorphism group of an object X in a category C is denoted Aut < sub > C </ sub >( X ), or simply Aut ( X ) if the category is clear from context.

When questioned by the Sadducees about the resurrection of the dead ( in a context relating to who one's spouse would be if one had been married several times in life ), Jesus said that marriage will be irrelevant after the resurrection as the resurrected will be ( at least in this respect ) like the angels in heaven.

The first division into major and minor arts dates back to Leon Battista Alberti's works ( De re aedificatoria, De statua, De pictura ), focusing the importance of intellectual skills of the artist rather than the manual skills ( even if in other forms of art there was a project behind ).

Additionally, if we assume that there are no viscous forces in the medium ( the bulk and shear viscosities are zero ), the momentum equation takes the form

However, there are certain limitations: they may not administer the sacraments and related functions whose celebration is reserved to bishops, priests, deacons, or seminarians ( the male clergy ), namely, Holy Orders ( they may make provision for an ordained cleric to help train and to admit some of their members, if needed, as altar servers, Eucharistic ministers, or lectors-the minor ministries which are now open to the non-ordained ).

They may not administer Penance ( Reconciliation ), Anointing of the Sick ( Extreme Unction ), or function as an ordained celebrant or concelebrant of the Mass ( by virtue of their office and their training and institution, they may act, if the need arises, as altar servers, lectors, ushers, porters, or Eucharistic ministers of the Cup, and if need be, the Host ).

There is considerable speculation as to why the observable universe is apparently composed almost entirely of matter ( as opposed to a mixture of matter and antimatter ), whether there exist other places that are almost entirely composed of antimatter instead, and what sorts of technology might be possible if antimatter could be harnessed.

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.

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.

But what you could not know, of course, was how smoothly the Victorian Fitzgerald was to lead into an American Fitzgerald of my own vintage under whose banner we adolescents were to come, if not of age, then into a bright, taut semblance of it.

William Wimsatt and Cleanth Brooks, it seems to me, have a penetrating insight into the way in which this control is effected: `` For if we say poetry is to talk of beauty and love ( and yet not aim at exciting erotic emotion or even an emotion of Platonic esteem ) and if it is to talk of anger and murder ( and yet not aim at arousing anger and indignation ) -- then it may be that the poetic way of dealing with these emotions will not be any kind of intensification, compounding, or magnification, or any direct assault upon the affections at all.

What I want is to have this evidence come before Congress and if the Attorney General does not report it, as I am very sure he won't, as he has refused to do anything of the kind, I then wish that a committee of seven Representatives be appointed with power to take the evidence.

The Acropolis was unique in the world and if that imcomparable work flooded by moonlight wasn't enough for both natives and tourists, then they were quite simply barbarians and the hell with them.

Sturley quoted Quiney as having written on November 1 that if he had `` more monei presente much might be done to obtaine our Charter enlargd, ij faires more, with tole of corne, bestes, and sheepe, and a matter of more valewe then all that ''.

If man is actually the product of his environment and if science can discover the laws of human nature and the ways in which environment determines what people do, then someone -- a someone probably standing outside traditional systems of values -- can turn around and develop completely efficient means for controlling people.

And if you bore each other then, heaven help you.

He invited Mr. Case to stop by to say hello if he ever visited the academy and then added that he was on the managerial staff of the freshman football team

The prevailing official attitude then seemed to be that fallout, if not exactly good for you, might not be much worse than a bad cold.

And if the foreigners fighting in the Katanga Army are mercenaries then Lafayette and Von Steuben were mercenaries too, as were also the members of the Lafayette Escadrille in the early part of World War 1, and of Chennault's Flying Tigers in the early days of World War 2.

Rachel came close to the bed, bent as if she would kiss him, then moved away.

and then I was adding my own voice to the crescendo of sound, hurling more vile language than I ever thought I knew, sobbing and shouting, and aware that if I had passed water before, it was not enough, for my pants were soaking wet.

`` I'll have a drink, then, if you'll have one with me ''.

And, if we follow the Rayburn pattern, as consciously or by an instinctual political sense I like to think I have followed it, then the very nature of our loyalty to our own immediate areas must necessarily be reflected in the devotion of our services to our country.

During nighttime hours, because of the intense skywave propagation then prevailing, no large number of stations can be permitted to operate on one of these channels, if the wide area service for which these frequencies are assigned is to be rendered satisfactorily by the dominant stations which must be relied upon to render it.

My husband enters the motel office, signs up for a room, and then solemnly asks the proprieter if he accepts pets.

`` We, the Subscribers, do agree, that as soon as a convenient Number of Persons have subscribed to this, or a similar Writing, We will present a petition to the Hon'ble General Court of the Commonwealth of Massachusetts, praying for an Act incorporating into a Body politic the subscribers to such Writing with Liberty to build such a Bridge, and a Right to demand a Toll equal to that received at Malden Bridge, and on like Terms, and if such an Act shall be obtained, then we severally agree each with the others, that we will hold in the said Bridge the several shares set against our respective Names, the whole into two hundred shares being divided, and that we will pay such sums of Money at such Times and in such Manners, as by the said proposed Corporation, shall be directed and required ''.

You can save all that shimming time if you set your windows in one, two, three order -- first the stud on one side, then the window, then the stud on the other side.

( C ) if Af is the operator induced on Af by T, then the minimal polynomial for Af is Af.

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.