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We formulate this proof directly for the case of polynomials over a UFD R, which is hardly different from its special case for R = Z.
Some Related Sentences
We and formulate
We are seeking to formulate an Islamic society which in the final analysis will produce an Islamic state.
This is true for all sciences: the goal is to connect a “ natural phenomenon ” with its “ immediate cause .” We formulate hypotheses elucidating, as we see it, the relation of cause and effect for particular phenomena.
*" We hear from time to time that the Irish people are determined to formulate their own politics, and not to take them from Rome ; but events constantly demonstrate that not only the religion but the politics of Ireland are those of the Church of Rome, and that the Irish people are still being exploited in the interest of clericalism and for the proselytising of England.
< BLOCKQUOTE > We are seeking to formulate an Islamic society which in the final analysis will produce an Islamic state.
" We are living in times of unparalleled threat and hostility from unseen and often unknown enemies ", Woodson states on the MilitaryWeek. com website ; " The tendency of governments facing this situation is to turn inward, to sit on information as they try to formulate a strategy to meet this daunting challenge.
We need to think about what contribution the Celtic peoples and their cultures can make to the development of European unity, to formulate proposals and suggestions, disseminating them, instead of adopting an aloof sort of attitude.
We and proof
We shall show that the polynomials Af behave in the manner described in the first paragraph of the proof.
Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.
We approach the proof of Theorem 2 by successively restricting the class of all formulas φ for which we need to prove " φ is either refutable or satisfiable ".
We have proved that φ is either satisfiable or refutable, and this concludes the proof of the Lemma.
We will now follow the strategy of David Hilbert ( 1862 – 1943 ) who gave a simplification of the original proof of Charles Hermite.
We now give an operator theoretic proof for the Cauchy – Schwarz inequality which passes to the C *- algebra setting.
We will go over a typical application of Zorn's lemma: the proof that every nontrivial ring R with unity contains a maximal ideal.
Dan Olinger, a professor at the fundamentalist Bob Jones University in Greenville said, “ We want to be good citizens and participants, but we ’ re not really interested in using the iron fist of the law to compel people to everything Christians should do .” Bob Marcaurelle, interim pastor at Mountain Springs Baptist Church in Piedmont, said the Middle Ages were proof enough that Christian ruling groups are almost always corrupted by power.
We provide a sketch of a proof for the case where the underlying field of scalars is the complex numbers.
" We " in this sense often refers to " the reader and the author ," since the author often assumes that the reader knows certain principles or previous theorems for the sake of brevity ( or, if not, the reader is prompted to look them up ), for example, so that the author does not need to explicitly write out every step of a mathematical proof.
We include here a proof that DTIME ( f ( n )) is a strict subset of DTIME ( f ( 2n + 1 )< sup > 3 </ sup >) as it is simpler.
We call a a witness for the compositeness of n ( sometimes misleadingly called a strong witness, although it is a certain proof of this fact ).
* Example: " We cannot wait for the final proof – the smoking gun – that could come in the form of a mushroom cloud.
" We " in this sense often refers to " the reader and the author ", since the author often assumes that the reader knows certain principles or previous theorems for the sake of brevity ( or, if not, the reader is prompted to look them up ), for example, so that the author does not need to explicitly write out every step of a mathematical proof.
We and directly
We propose a method for selecting only dictionary information required by the text being translated and a means for passing the information directly to the occurrences in text.
* The song No Joy in Mudville from Death Cab for Cutie's album We Have the Facts and We're Voting Yes directly references the poem.
Frank Capra's seven-film series Why We Fight is said to have been directly inspired by and America's response to Triumph of the Will.
::“ We certainly cannot hope directly to compare their effects except within a limited future ; and all the arguments, which have ever been used in Ethics, and upon which we commonly act in common life, directed to shewing that one course is superior to another, are ( apart from theological dogmas ) confined to pointing out such probable immediate advantages …
* Post-rock band, Deadhorse, refer their 2010 album release We Can Create Our Own World to be directly influenced by the book and Ray Bradbury's vision in evoke imagination in his readers.
We found evidence that the Corporation for Public Broadcasting ( CPB ) former Chairman violated statutory provisions and the Director ’ s Code of Ethics by dealing directly with one of the creators of a new public affairs program during negotiations with the Public Broadcasting Service ( PBS ) and the CPB over creating the show.
The manifestos directly state: " We speak unto you by parables, but would willingly bring you to the right, simple, easy, and ingenuous exposition, understanding, declaration, and knowledge of all secrets.
In a letter to Autoweek Magazine, Donald Petersen, then Ford Chairman, said, " We are moving toward building a quality culture at Ford and the many changes that have been taking place here have their roots directly in Deming's teachings.
Of particular importance to the development of folk rock were the subtle folk influences evident in such Beatles ' compositions as " I'll Be Back ", " Things We Said Today ", and " I'm a Loser ", with the latter song being directly inspired by folk singer-songwriter Bob Dylan.
We shall therefore now consider only arguments that prove the theorem directly for any matrix using algebraic manipulations only ; these also have the benefit of working for matrices with entries in any commutative ring.
Tottenham Court Road Tube Station is replicated as part of the set for the Queen musical, We Will Rock You, which plays at the Dominion Theatre, directly above the actual tube station.
He proposed the " Radical Creed ," which, " as I understand it, is this: We have not abandoned our old belief in liberty, justice, and Self-help, but we say that under certain conditions the people cannot help themselves, and that then they should be helped by the State representing directly the whole people.
We thus see that, not only directly but also through the replies that it called forth, the system of Hutcheson, or at least the system of Hutcheson combined with that of Shaftesbury, contributed, in large measure, to the formation and development of some of the most important of the modern schools of ethics.
However, Clark was determined that VI Corps should strike directly for Rome as evidenced in his later writing: " We not only wanted the honour of capturing Rome, but felt that we deserved it ... Not only did we intend to become the first army to seize Rome from the south, but we intended to see that people at home knew that it was the Fifth Army that did the job, and knew the price that had been paid for it .".
We pursue our vision through direct services, education, advocacy, and outreach, in conjunction with our partners, to directly benefit these communities.
We do not see the frame directly, but infer its presence by its characteristic expressions and language.
Washington and Lee maintains a rigorous honor system that traces directly to Robert E. Lee, who said, " We have but one rule here, and it is that every student must be a gentleman.