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

for, using the fact that N and N' commute Af and so when R is sufficiently large every term in this expression for Af will be 0.

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

It is difficult to measure the direction and magnitude of R directly.

This is not a constant value like Af, but varies with the thickness of the coating and the direction and magnitude of the resultants R and Af of Fig. 6.

If the rake angle **yc of the knife is high enough and the friction angle **yt between the front of the knife and the back of the chip is low enough to give a positive value for Af, the resultant vector R will lie above the plane of the substrate.

But even more important than this is the fact that the direct search by simultaneously varying all operating conditions has produced only one optimal policy, namely, that for the given feed state and R stages.

But Af is vastly larger than R.

These will be numbered in the direction opposite to the flow of the process stream, so that stage R is the T stage from the end.

In the third column is given the optimal policy for stage R, and in the fourth, the resulting state of the stream when this policy is used.

In this case the stage R operating with conditions Af transforms the state of the stream from Af to Af, but only the probability distribution of Af is known.

that is, it is known empirically that names beginning with R are more common than those beginning with Z ; ;

The study of altruism was the initial impetus behind George R. Price's development of the Price equation, which is a mathematical equation used to study genetic evolution.

Kempthorne and his students make an assumption of unit treatment additivity, which is discussed in the books of Kempthorne and David R. Cox.

An alkyl group, generally abbreviated with the symbol R, is a functional group or side-chain that, like an alkane, consists solely of single-bonded carbon and hydrogen atoms, for example a methyl or ethyl group.

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 )).

USACC is co-chaired by Tim Cejka, President of Exxon Mobil Corporation and Reza Vaziri, President of R. V.

then there are elements x and y in R such that ax + by = d. The reason: the ideal Ra + Rb is principal and indeed is equal to Rd.

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.

The property ( EF1 ) can be restated as follows: for any principal ideal I of R with nonzero generator b, all nonzero classes of the quotient ring R / I have a representative r with.

It was around this time that J. R. R. Tolkien was employed by the OED, researching etymologies of the Waggle to Warlock range ; he parodied the principal editors as " The Four Wise Clerks of Oxenford " in the story Farmer Giles of Ham.

The key result is the structure theorem: If R is a principal ideal domain, and M is a finitely

If M is a free module over a principal ideal domain R, then every submodule of M is again free.

United States Senate | Sen. John Sherman ( Ohio ) | John Sherman ( Republican Party ( United States ) | R — List of United States Senators from Ohio | OH ), the principal author of the Sherman Antitrust Act.

L. Zangwill, ' Binet, Alfred ', in R. Gregory, The Oxford Companion to the Mind ( 1987 ) p. 88 His principal goal was to identify students who needed special help in coping with the school curriculum.

* Ariadne is a principal character in the monumental series of music dramas conceived by the Canadian composer and writer R. Murray Schafer, in his " Patria Cycle ", including " The Princess of the Stars ," " Requiems for the Party Girl "

The principal event of the golden jubilee of Indian independence was President K. R. Narayanan's midnight address to the nation during the special session of Parliament convened on the night of 14 August ; in this address, he identified the establishment of a democratic system of government and politics to be the greatest achievement of India since independence.

* E. R. Minchew, former Louisiana Tech speech professor, began his educational career as an assistant principal and coach at Bienville High School in the 1929-1930 academic year.

Born in nearby Taylor was the Louisiana Tech speech professor and long-time principal of Castor High School, E. R. Minchew.

Still another Louisiana educator, E. R. Minchew, was an assistant principal and coach at the former Lisbon High School from 1930 to 1932, when he began a 32-year stint as principal of Castor High School in Castor, a village in Bienville Parish.

* Helen R. Donaghue School – named for a former principal of the school

* The Revd R. T. France-New Testament scholar, principal 1989-1995

W. J. R. Thoenssen was the first principal, succeeded by Charles F. Tansill.

In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.

* a left principal ideal of R is a subset of R of the form Ra :=

If the ideals A and B of R are coprime, then AB = A ∩ B ; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about coprime ideals.

The main concept, however, common to all idealist epistemologies is the centrality of Reason: ( i. e.: ' Reason ' with a capital ' R '): a priori Reason: Knowledge can only be, ultimately, a product of the mind and is therefore, by definition, ' ideal '.

In fact, if I is a nonzero ideal of R then any element a of I

Universalist theories are generally forms of moral realism, though exceptions exists, such as the subjectivist ideal observer and divine command theories, and the non-cognitivist universal prescriptivism of R. M.

For instance, the ring ( in fact field ) of complex numbers, which can be constructed from the polynomial ring R over the real numbers by factoring out the ideal of multiples of the polynomial.

: R is the ideal gas constant.

An ideal P of a commutative ring R is prime if it has the following two properties:

* If R denotes the ring CY of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y < sup > 2 </ sup > − X < sup > 3 </ sup > − X − 1 is a prime ideal ( see elliptic curve ).

* In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i. e. M is contained in exactly 2 ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime.