L'Hôpital's rule: Difference between revisions

Latest revision as of 18:53, 31 December 2014

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He presents many compelling case histories, but at times he seems to lean too heavily on insider stories, as if showing off his contacts and knowledge of the industry. As Cooper tries to remind us, it's everyday usersnot the power users, not really the "computer literate"who are the core audience.

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L'Hôpital's rule: Difference between revisions

Latest revision as of 18:53, 31 December 2014

Contents

Beats By Dre Nz or into a new application

Tiffany And Co Canada or wikis like Wikipedia.

Ray Ban Australia among other areas.

Oakley Frogskins .

Vibram Five Fingers Canada

Navigation menu

@@ Line 1: / Line 1: @@
-<div class="thumb tright">
+== Beats By Dre Nz or into a new application ==
-<div class="thumbinner" style="width:220px;">
-{| class="wikitable" style="width:220px;"
-! {{nobreak|p \ a}} !! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10
-|-
-! 3
-|bgcolor="#FFFF00"| 0
-|bgcolor="#FFFF00"| 1 || -1 || || || || || || || ||
-|-
-! 5
-|bgcolor="#FFFF00"| 0
-|bgcolor="#FFFF00"| 1 || -1 || -1
-|bgcolor="#FFFF00"| 1 || || || || || ||
-|-
-! 7
-|bgcolor="#FFFF00"| 0
-|bgcolor="#FFFF00"| 1
-|bgcolor="#FFFF00"| 1 || -1
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-|-
-! 11
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-|bgcolor="#FFFF00"| 1
-|bgcolor="#FFFF00"| 1 || -1 || -1 || -1
-|bgcolor="#FFFF00"| 1 || -1
-|}
-<div class="thumbcaption">
-Legendre symbol (a/p) for various ''a'' (along top) and ''p'' (along left side). Only 0 ≤ ''a'' < ''p'' are shown, since due to the first property below any other ''a'' can be reduced modulo ''p''. [[Quadratic residue]]s are highlighted in yellow, and correspond precisely to the values 0 and 1.
-</div>
-</div>
-</div>
-In [[number theory]], the '''Legendre symbol''' is a [[multiplicative function]] with values 1, &minus;1, 0 that is a quadratic character modulo a [[prime number]] ''p'': its value on a (nonzero) [[quadratic residue]] mod ''p'' is 1 and on a non-quadratic residue (''non-residue'') is &minus;1. Its value on zero is 0.
-The Legendre symbol was introduced by [[Adrien-Marie Legendre]] in 1798<ref>A. M. Legendre ''Essai sur la theorie des nombres'' Paris 1798, p 186</ref> in the course of his attempts at proving the [[law of quadratic reciprocity]]. Generalizations of the symbol include the [[Jacobi symbol]] and [[Dirichlet character]]s of higher order. The notational convenience of the Legendre symbol inspired introduction of several other "symbols" used in [[algebraic number theory]], such as the [[Hilbert symbol]] and the [[Artin symbol]].
+He presents many compelling case histories, but at times he seems to lean too heavily on insider stories, as if showing off his contacts and knowledge of the industry. As Cooper tries to remind us, it's everyday usersnot the power users, not really the "computer literate"who are the core audience. <br><br>Natasha, on her part, is suitably upset at being kidnapped and held hostage. She had been wooed assiduously by Rakhal and she had succumbed to him, enjoying [http://www.sashwindowcordsauckland.co.nz/database/update.php Beats By Dre Nz] a fling for once in her utterly responsible life. Natasha believes that Rakhal intent to shun her for all but a few days a month means she is simply a vessel of fertility. Rakhal argues that she would be most honored with the land. In accepting the myth of this story, you have to subscribe to Rakhal belief system about his own land and that he is, not ignorant of the outside world, but at the least doesn't ascribe to Western philosophies in any fashion. Importantly, Natasha continues to fully stand up for herself and she varies from not willing to be the honored vessel to fighting for her love.<br><br>The gender wage gap also played a significant role in reported stress levels, with almost twice as many women (18 percent) as men (10 percent) citing low pay as their primary jobrelated stressor. The survey  the discharge of which coincided with Equal Payday  echoed recent findings from the National Partnership For ladies And Families, which showed that the pay gap still exists in each of the 50 states. Based on the NPWF study, the median yearly pay for fulltime female workers is more than $11,000 less than their male counterparts.<br><br>The point of these services is to make blogging a whole lot easier. However, if you use a service, you'll always have a limited feature set because you don't own the code. Another major drawback is your weblog content resides on another person's server. Although some services have the ability to export weblog content, the general approach doesn't let you move your weblog to a different location (or into a new application) at another time.<br><br>Chemistry, according to Yazit, was the key factor that held them together once the other bands that [http://www.icscancer.com/test/test/flash/scripts/user.asp Michael Kors Handbags] emerged at the same time [http://www.heritagepress.co.nz/config/links.php Oakley Sunglasses] were eventually mothballed into history's footnotes. There is also the spirit of collectivism associated to the hard graft of playing within the club/pub circuit and this was another powerful adhesive that fastened their friendship.<br><br>They setup shop in a twobedroom house, with extension cords running to the garage. Jeff [http://www.kassorg.org/photo/Award/menu.asp Tiffany Singapore] setup three Sun microstations on tables he'd made out of doors from Home Depot for under $60 each. When the test site was ready to go, Jeff asked 300 friends and acquaintances to test it. The code worked seamlessly across different computer platforms. On July 16, 1995, Bezos opened his site to the world, and told his 300 beta testers to get the word out. In 30 days, with no press, Amazon had sold books in all 50 states and 45 foreign countries. By September, it had sales of $20,000 per week. Bezos and his team continued improving the site, introducing such unheardof features as oneclick shopping, testimonials, and email order verification.<ul>
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-== Definition ==
+== Tiffany And Co Canada  or wikis like Wikipedia. ==
-Let ''p'' be an odd [[prime number]]. An integer ''a'' is a [[quadratic residue]] modulo ''p'' if it is [[modular arithmetic|congruent]] to a [[square number|perfect square]] modulo ''p'' and is a quadratic nonresidue modulo ''p'' otherwise. The '''Legendre symbol''' is a function of ''a'' and ''p'' defined as follows:
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-:<math>
+== Ray Ban Australia  among other areas. ==
-\left(\frac{a}{p}\right) =
-\begin{cases}
-\;\;\,1 \text{ if } a \text{ is a quadratic residue modulo}\ p
-\text{ and } a \not\equiv 0\pmod{p} \\
--1 \text{ if } a \text{ is a quadratic non-residue modulo}\ p\\
-\;\;\,0 \text{ if } a \equiv 0 \pmod{p}.
-\end{cases}
-</math>
-Legendre's original definition was by means of an explicit formula:
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-:<math> \left(\frac{a}{p}\right) \equiv a^{(p-1)/2}\ \pmod{ p}\;\;\text{  and } \left(\frac{a}{p}\right) \in \{-1,0,1\}.</math>
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-By [[Euler's criterion]], which had been discovered earlier and was known to Legendre, these two definitions are equivalent.<ref>Hardy & Wright, Thm. 83.</ref> Thus Legendre's contribution lay in introducing a convenient ''notation'' that recorded quadratic residuosity of ''a'' mod ''p''. For the sake of comparison, [[Carl Friedrich Gauss|Gauss]] used the notation <math>a\mathrm{R}p</math>, <math>a\mathrm{N}p</math> according to whether ''a'' is a residue or a non-residue modulo ''p''.
+== Oakley Frogskins . ==
-For typographical convenience, the Legendre symbol is sometimes written as (''a''|''p'') or (''a''/''p''). The sequence (''a''|''p'') for ''a'' equal to 0,1,2,... is [[periodic sequence|periodic]] with period ''p'' and is sometimes called the '''Legendre sequence''', with {0,1,−1} values occasionally replaced by {1,0,1} or {0,1,0}.<ref name=KimSong01>Jeong-Heon Kim and Hong-Yeop Song, "Trace Representation of Legendre Sequences," ''Designs, Codes, and Cryptography'' '''24''', p. 343–348 (2001).</ref>
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-== Properties of the Legendre symbol ==
+== Vibram Five Fingers Canada ==
-There are a number of useful properties of the Legendre symbol which, together with the law of [[quadratic reciprocity]], can be used to compute it efficiently.
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-* The Legendre symbol is periodic in its first (or top) argument: if ''a'' ≡ ''b'' (mod ''p''), then
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-::<math>
-\left(\frac{a}{p}\right) = \left(\frac{b}{p}\right).
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-</math>
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-* The Legendre symbol is a [[completely multiplicative function]] of its top argument:
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-::<math>
-\left(\frac{ab}{p}\right) = \left(\frac{a}{p}\right)\left(\frac{b}{p}\right).
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-</math>
+ </ul>
-: In particular, the product of two numbers that are both quadratic residues or quadratic non-residues modulo ''p'' is a residue, whereas the product of a residue with a non-residue is a non-residue. A special case is the Legendre symbol of a square:
-::<math>
-\left(\frac{x^2}{p}\right) =
-\begin{cases} 1&\mbox{if }p\nmid x\\0&\mbox{if }p\mid x.
-\end{cases}
-</math>
-* When viewed as a function of ''a'', the Legendre symbol <math>\left(\frac{a}{p}\right)</math> is the unique quadratic (or order 2) [[Dirichlet character]] modulo ''p''.
-* The first supplement to the law of quadratic reciprocity:
-::<math>
-\left(\frac{-1}{p}\right)
-= (-1)^\tfrac{p-1}{2}
-=\begin{cases}
-\;\;\,1\mbox{ if }p \equiv 1\pmod{4} \\
--1\mbox{ if }p \equiv 3\pmod{4}.  \end{cases}</math>
-* The second supplement to the law of quadratic reciprocity:
-::<math>
-\left(\frac{2}{p}\right)
-= (-1)^\tfrac{p^2-1}{8}
-=\begin{cases}
-\;\;\,1\mbox{ if }p \equiv 1\mbox{ or }7 \pmod{8} \\
--1\mbox{ if }p \equiv 3\mbox{ or }5 \pmod{8}.  \end{cases}</math>
-* Special formulas for the Legendre symbol <math>\left(\frac{a}{p}\right)</math> for small values of ''a'':
-:* For an odd prime ''p'' ≠ 3,
-::<math>
-\left(\frac{3}{p}\right)
-= (-1)^{\big\lfloor \frac{p+1}{6}\big\rfloor}
-=\begin{cases}
-\;\;\,1\mbox{ if }p \equiv 1\mbox{ or }11 \pmod{12} \\
--1\mbox{ if }p \equiv 5\mbox{ or }7 \pmod{12}.  \end{cases}</math>
-:* For an odd prime ''p'' ≠ 5,
-::<math>
-\left(\frac{5}{p}\right)
-=(-1)^{\big\lfloor \frac{p+2}{5}\big \rfloor}
-=\begin{cases}
-\;\;\,1\mbox{ if }p \equiv 1\mbox{ or }4 \pmod5 \\
--1\mbox{ if }p \equiv 2\mbox{ or }3 \pmod5.  \end{cases}</math>
-* The [[Fibonacci numbers]] 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...  are defined by the recurrence {{nowrap|F<sub>1</sub> {{=}} F<sub>2</sub> {{=}} 1,}} {{nowrap|F<sub>n+1</sub> {{=}} F<sub>n</sub> + F<sub>n&minus;1</sub>.}} If ''p'' is a prime number then
-:<math>
-F_{p-\left(\frac{p}{5}\right)} \equiv 0 \pmod p,\;\;\;
-F_{p} \equiv \left(\frac{p}{5}\right) \pmod p.
-</math>
-<blockquote>
-For example,
-:<math>
-\begin{align}
-&(\tfrac{2}{5}) &= &-1, &F_3  &= 2, \;\;\;\;&F_2 &=1,\\
-&(\tfrac{3}{5}) &= &-1,  &F_4  &= 3,&F_3&=2,\\
-&(\tfrac{5}{5}) &= &\;\;\;\;0,  &F_5  &= 5,\\
-&(\tfrac{7}{5}) &= &-1,  &F_8  &= 21,&F_7&=13,\\
-&(\tfrac{11}{5}) &= &\;\;\,1,  &F_{10}  &= 55, &F_{11}&=89.
-\end{align}
-</math>
-This result comes from the theory of [[Lucas sequence]]s, which are used in [[primality testing]].<ref>Ribenboim, p. 64; Lemmermeyer, ex 2.25-2.28, pp. 73–74.</ref> See [[Wall–Sun–Sun prime]].
-</blockquote>
-== Legendre symbol and quadratic reciprocity ==
-Let ''p'' and ''q'' be odd primes. Using the Legendre symbol, the [[quadratic reciprocity]] law can be stated concisely:
-: <math>
-\left(\frac{q}{p}\right)
-= \left(\frac{p}{q}\right)(-1)^{\tfrac{p-1}{2}\tfrac{q-1}{2}}.
-</math>
-Many [[proofs of quadratic reciprocity]] are based on Legendre's formula
-:<math>
-\left(\frac{a}{p}\right) \equiv a^{\tfrac{p-1}{2}}\pmod p.
-</math>
-In addition, several alternative expressions for the Legendre symbol were devised in order to produce various proofs of the quadratic reciprocity law.
-* Gauss introduced the [[quadratic Gauss sum]] and used the formula
-::<math>
-\sum_{k=0}^{p-1}\zeta^{ak^2}=
-\left(\frac{a}{p}\right)\sum_{k=0}^{p-1}\zeta^{k^2},
-\quad \zeta = e^{2\pi i/p}
-</math>
-:in his fourth<ref>Gauss, "Summierung gewisser Reihen von besonderer Art" (1811), reprinted in ''Untersuchungen ...'' pp. 463–495</ref> and sixth<ref>Gauss, "Neue Beweise und Erweiterungen des Fundamentalsatzes in der Lehre von den quadratischen Resten" (1818) reprinted in ''Untersuchungen ...'' pp. 501–505</ref> proofs of quadratic reciprocity.
-* [[Leopold Kronecker|Kronecker's]] proof<ref>Lemmermeyer, ex. p. 31, 1.34</ref>  first establishes that
-::<math>
-\left(\frac{p}{q}\right)
-=\sgn\left(\prod_{i=1}^{\frac{q-1}{2}}\prod_{k=1}^{\frac{p-1}{2}}\left(\frac{k}{p}-\frac{i}{q}\right)\right).</math>
-: Reversing the roles of ''p'' and ''q'', he obtains the relation between <math>\left(\frac{p}{q}\right)</math> and <math>\left(\frac{q}{p}\right).</math>
-* One of [[Gotthold Eisenstein|Eisenstein]]'s proofs<ref>Lemmermeyer, pp. 236 ff.</ref> begins by showing that
-::<math>
-\left(\frac{q}{p}\right)
-=\prod_{n=1}^{\frac{p-1}{2}} \frac{\sin\left(\frac{2\pi qn}{p}\right)}{\sin\left(\frac{2\pi n}{p}\right)}.
-</math>
-: Using certain [[elliptic function]]s instead of the [[sine function]], Eisenstein was able to prove [[cubic reciprocity|cubic]] and [[quartic reciprocity|quartic]] reciprocity as well.
-== Related functions ==
-* The [[Jacobi symbol]] <math>\left(\tfrac{a}{n}\right)</math> is a generalization of the Legendre symbol that allows for a composite second (bottom) argument ''n'', although ''n'' must still be odd and positive.  This generalization provides an efficient way to compute all Legendre symbols without performing factorization along the way.
-* A further extension is the [[Kronecker symbol]], in which the bottom argument may be any integer.
-* The [[power residue symbol]] &nbsp; <math>\left(\tfrac{a}{p}\right)_n</math> generalizes the Legendre symbol to higher power ''n''. The Legendre symbol represents the [[power residue symbol]] for ''n'' = 2.
-== Computational example ==
-The above properties, including the law of quadratic reciprocity, can be used to evaluate any Legendre symbol. For example:
-:<math>\left ( \frac{12345}{331}\right )</math>
-:<math>=\left ( \frac{3}{331}\right ) \left ( \frac{5}{331}\right ) \left ( \frac{823}{331}\right )</math>
-:<math>=\left ( \frac{3}{331}\right ) \left ( \frac{5}{331}\right ) \left ( \frac{161}{331}\right )</math>
-:<math>=\left ( \frac{3}{331}\right ) \left ( \frac{5}{331}\right ) \left ( \frac{7}{331}\right ) \left ( \frac{23}{331}\right )</math>
-:<math>= (-1) \left ( \frac{331}{3}\right ) \left ( \frac{331}{5}\right ) (-1) \left ( \frac{331}{7}\right ) (-1) \left ( \frac{331}{23}\right )</math>
-:<math>=-\left ( \frac{1}{3}\right ) \left ( \frac{1}{5}\right ) \left ( \frac{2}{7}\right ) \left ( \frac{9}{23}\right )</math>
-:<math>=-\left ( \frac{1}{3}\right ) \left ( \frac{1}{5}\right ) \left ( \frac{2}{7}\right ) \left ( \frac{3^2}{23}\right )</math>
-:<math>= - \left (1\right ) \left (1\right ) \left (1\right ) \left (1\right ) = -1.</math>
-Or using a more efficient computation:
-:<math>\left ( \frac{12345}{331}\right )=\left ( \frac{98}{331}\right )=\left ( \frac{2 \cdot 7^2}{331}\right )=\left ( \frac{2}{331}\right )=(-1)^\tfrac{331^2-1}{8}=-1.</math>
-The article [[Jacobi symbol#Calculations_using_the_Legendre_symbol|Jacobi symbol]] has more examples of Legendre symbol manipulation.
-==Notes==
-{{reflist|2}}
-==References==
-*{{citation
-  | last1 = Gauss  | first1 = Carl Friedrich
-  | last2 = Maser | first2 = H. (translator into German)
-  | title = Untersuchungen über höhere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition)
-  | publisher = Chelsea
-  | location = New York
-  | year = 1965
-  | isbn = 0-8284-0191-8}}
-*{{citation
-  | last1 = Gauss  | first1 = Carl Friedrich
-  | last2 = Clarke | first2 = Arthur A. (translator into English)
-  | title = Disquisitiones Arithmeticae (Second, corrected edition)
-  | publisher = [[Springer Science+Business Media|Springer]]
-  | location = New York
-  | year = 1986
-  | isbn = 0-387-96254-9}}
-*{{citation
-  | last1 = Bach  | first1 = Eric
-  | last2 = Shallit | first2 = Jeffrey
-  | title = Algorithmic Number Theory (Vol I: Efficient Algorithms)
-  | publisher = [[The MIT Press]]
-  | location = Cambridge
-  | year = 1996
-  | isbn = 0-262-02405-5}}
-*{{citation
-  | last1 = Hardy  | first1 = G. H.
-  | authorlink1=G. H. Hardy
-  | last2 = Wright | first2 = E. M.
-  | title = An Introduction to the Theory of Numbers (Fifth edition)
-  | publisher = [[Oxford University Press]]
-  | location = Oxford
-  | year = 1980
-  | isbn = 978-0-19-853171-5}}
-*{{citation
-  | last1 = Ireland  | first1 = Kenneth
-  | last2 = Rosen  | first2 = Michael
-  | title = A Classical Introduction to Modern Number Theory (Second edition)
-  | publisher = [[Springer Science+Business Media|Springer]]
-   | location = New York
-  | year = 1990
-  | isbn = 0-387-97329-X}}
-*{{citation
-  | last1 = Lemmermeyer  | first1 = Franz
-  | title = Reciprocity Laws: from Euler to Eisenstein
-  | publisher = [[Springer Science+Business Media|Springer]]
-  | location = Berlin
-  | year = 2000
-  | isbn = 3-540-66957-4}}
-*{{citation
-  | last1 = Ribenboim  | first1 = Paulo
-  | title = The New Book of Prime Number Records
-  | publisher = [[Springer Science+Business Media|Springer]]
-  | location = New York
-  | year = 1996
-  | isbn = 0-387-94457-5}}
-== External links ==
-*[http://www.math.fau.edu/richman/jacobi.htm  Jacobi symbol calculator]
-[[Category:Modular arithmetic]]
-[[Category:Quadratic residue]]

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