One obtains the value f ( r ) by substitution of the value r for the symbol X in P. One reason to distinguish between polynomials and polynomial functions is that over some rings different polynomials may give rise to the same polynomial function ( see Fermat's little theorem for an example where R is the integers modulo p ).

One can show that C < sub > τ </ sub >( X ) = C < sub > ρ </ sub >( X ) in the above construction so that the rings C ( X ) and C *( X ) are typically only studied for completely regular spaces X.

One of the earliest patterns is for a crocheted afghan with tatted rings forming a raised design.

One reason for the increased popularity of expensive engagement rings is its relationship to human sexuality and the woman's marriage prospects.

It may also have been associated with the Holy Trinity, as in the three hares motif, representing the " One in Three and Three in One " of which the triangle or three interlocking shapes such as rings are common symbols.

It was Sauron's design to control all these rings and their bearers through the One Ring, forged in secret for this purpose, but only the Nine succumbed completely to its power and its seduction:

Sauron intended the rings to subvert these races of Middle-earth to his power, since the One Ring was the Ruling Ring that controlled the others.

Sauron's plan was not completely successful ; for the Elves hid their rings and did not use them while Sauron held the One, and the Dwarves ' rings did not respond to the One's control as Sauron expected.

Though the other rings could be destroyed in dragon-fire, the One could be unmade only in the Cracks of Doom where it was forged.

Another source was some of the Continental currency issued earlier in 1776 ; these were designed by Franklin and featured the motto We Are One surrounded by thirteen rings, each with the name of a colony.

* The collection of Posie rings that supposedly inspired the One Ring in J. R. R. Tolkien's The Lord of the Rings

* " One ringy-dingy ... two ringy-dingies ..." ( Ernestine's mimicking of the rings while she was waiting for someone to pick up the receiver on the other end of the telephone lines )

One technique is a " fan chart ", which features a half circle chart with concentric rings: the person of interest is the inner circle, the second circle is divided in two ( each side is one parent ), the third circle is divided in four, and so forth.

One started from the most general axioms, and then specialized, for example, by introducing modules over commutative rings, and limiting to vector spaces over the real numbers only when absolutely necessary.

One of the most widely recognized skills performed on the rings is the Iron Cross, which is executed by extending both arms straight out from the sides of the body while suspended mid air for at least two seconds.

One way that leads to generalisations is to allow reducible algebraic sets ( and fields k that aren't algebraically closed ), so the rings R may not be integral domains.

One can check that U is a functor from the category of rings to the category of groups: every ring homomorphism f: R → S induces a group homomorphism U ( f ): U ( R ) → U ( S ), since f maps units to units.

One such side quest is ring collection ; rings provide Link with various bonuses and abilities, such as improved defense.

One such method, based on principles of dendroclimatology, uses the width and other characteristics of tree rings to infer temperature.

One more players than the number of rings are designated as " weasels ", all but one standing in the rings.

One thing was certain -- his method was effective, so effective that after a time even the warning notices were often unnecessary.

One of Greg's bombs hung up, and he was miles from the target before he could get rid of it.

One swallow was all he would have ; ;

One girl expressed what was obviously in their minds.

One is tempted to say that, on the difference between the concepts of sovereignty in these two preambles, the worst war of the Nineteenth century was fought.

One beatnik got the woman he was living with so involved in drugs and self-analysis and all-night sessions of sex that she was beginning to crack up.

One evening, while a volley-ball game was being played in the yard among the prisoners remaining there, a simulated melee was staged -- just as the gates were opened to admit other prisoners returning from work.

One of the pictures was of a man with hat drawn over his face ceremoniously lighting a cigarette ; ;

One thing Papa had not taught Henrietta was how to handle a young man as high-spirited and opinionated as herself.

One cause of Schopenhauer's pessimism was the fact that he failed to learn the guitar.

One was an article on the U.N. by Alice Widener from the Cincinnati Enquirer.

One day he assigned me to lay bare a `` plot '' by the Duponts to supply munitions to a wholly fictitious revolution he said was about to occur in Cuba.

One shawl was so tremendous that she could not wear it, so she draped it over the banister on the second floor, and it hung over the stairway.

One White House dog was immortalized in a painting.

One person she helped was my brother.

One of Sherman's most serious shortcomings, however, was his mistrust of his cavalry.

One can imagine that with her and Gorton there it was no place for anyone with weak nerves.

One historical authority presents laborious and circuitous testimony tending to arouse suspicion that Massachusetts was behind the clouds settling down on the embattled Gorton.

One of the people who was afraid of Alfred was his own brother, Lew.

One example of this was his assertion that `` all servile revolts must be dealt with by physical force ''.

One day he was visited by a delegation of would-be imitators who wanted to know his secret.