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| In [[mathematics]], the '''multiplicity''' of a member of a [[multiset]] is the number of times it appears in the multiset. For example, the number of times a given [[polynomial equation]] has a [[Root_of_a_function|root]] at a given point.
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| The notion of multiplicity is important to be able to count correctly without specifying exceptions (for example, ''double roots'' counted twice). Hence the expression, "counted with multiplicity".
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| If multiplicity is ignored, this may be emphasized by counting the number of '''distinct''' elements, as in "the number of distinct roots". However, whenever a set (as opposed to multiset) is formed, multiplicity is automatically ignored, without requiring use of the term "distinct".
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| ==Multiplicity of a prime factor==
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| {{main|p-adic order}}
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| In the [[Integer factorization|prime factorization]], for example,
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| : 60 = 2 × 2 × 3 × 5
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| the multiplicity of the prime factor 2 is 2, while the multiplicity of each of the prime factors 3 and 5 is 1. Thus, 60 has 4 prime factors, but only 3 distinct prime factors. | |
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| ==Multiplicity of a root of a polynomial==
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| <!-- [[Eigenvalue, eigenvector and eigenspace#Definitions]] links to this section. Change the link there if you change this header -->
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| Let ''F'' be a [[field (mathematics)|field]] and ''p''(''x'') be a [[polynomial]] in one variable and coefficients in ''F''. An element ''a'' ∈ ''F'' is called a [[root of a function|root]] of multiplicity ''k'' of ''p''(''x'') if there is a polynomial ''s''(''x'') such that ''s''(''a'') ≠ 0 and ''p''(''x'') = (''x'' − ''a'')<sup>''k''</sup>''s''(''x''). If ''k'' = 1, then ''a'' is called a ''simple root''.
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| For instance, the polynomial ''p''(''x'') = ''x''<sup>3</sup> + 2''x''<sup>2</sup> − 7''x'' + 4 has 1 and −4 as [[Root_of_a_function|roots]], and can be written as ''p''(''x'') = (''x'' + 4)(''x'' − 1)<sup>2</sup>. This means that 1 is a root of multiplicity 2, and −4 is a 'simple' root (of multiplicity 1). Multiplicity can be thought of as "How many times does the [[Root_of_a_function|solution]] appear in the original equation?".
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| The [[Discriminant#Discriminant_of_a_polynomial|discriminant of a polynomial]] is zero only if the polynomial has a multiple root.
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| === Behavior of a polynomial function near a root in relation to its multiplicity ===
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| Let ''f''(''x'') be a [[polynomial]] function. Then, if ''f'' is graphed on a [[Cartesian coordinate system]], its graph will cross the ''x''-axis at real [[Root_of_a_function|zeros]] of odd multiplicity and will bounce off (not go through) the ''x''-axis at real zeros of even multiplicity. In addition, if ''f''(''x'') has a zero with a multiplicity greater than 1, the graph will be tangent to the ''x''-axis, in other words it will have slope 0 there.
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| In general, a polynomial with an ''n''-fold root will have a [[derivative]] with an (''n''−1)-fold root at that point.
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| ==Intersection multiplicity==
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| {{main|Intersection theory}}
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| In [[algebraic geometry]], the intersection of two sub-varieties of an algebraic variety is a finite union of [[irreducible variety|irreducible varieties]]. To each component of such an intersection is attached an '''intersection multiplicity'''. This notion is [[local property|local]] in the sense that it may be defined by looking what occurs in a neighborhood of any [[generic point]] of this component. It follows that without loss of generality, we may consider, for defining the intersection multiplicity, the intersection of two [[affine variety|affines varieties]] (sub-varieties of an affine space).
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| Thus, given two affine varieties ''V''<sub>1</sub> and ''V''<sub>2</sub>, let us consider an [[irreducible component]] ''W'' of the intersection of ''V''<sub>1</sub> and ''V''<sub>2</sub>. Let ''d'' be the [[dimension of an algebraic variety|dimension]] of ''W'', and ''P'' be any generic point of ''W''. The intersection of ''W'' with ''d'' [[hyperplane]]s in [[general position]] passing through ''P'' has an irreducible component that is reduced to the single point ''P''. Therefore, the [[local ring]] at this component of the [[coordinate ring]] of the intersection has only one [[prime ideal]], and is therefore an [[Artinian ring]]. This ring is thus a [[finite dimensional]] vector space over the ground field. Its dimension is the '''intersection multiplicity''' of ''V''<sub>1</sub> and ''V''<sub>2</sub> at ''W''.
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| This definition allows to state precisely [[Bézout's theorem]] and its generalizations.
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| This definition generalizes the multiplicity of a root of a polynomial in the following way. The roots of a polynomial ''f'' are points on the [[affine line]], which are the components of the algebraic set defined by the polynomial. The coordinate ring of this affine set is <math>R=K[X]/\langle f\rangle, </math> where ''K'' is an [[algebraically closed field]] containing the coefficients of ''f''. If <math>f(X)=\prod_{i=1}^k (X-\alpha_i)^{m_i}</math> is the factorization of ''f'', then the local ring of ''R'' at the prime ideal <math>\langle X-\alpha_i\rangle</math> is <math>K[X]/\langle (X-\alpha)^{m_i}\rangle.</math> This is a vector space over ''K'', which has the multiplicity <math>m_i</math> of the root as a dimension.
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| This definition of intersection multiplicity, which is essentially due to [[Jean-Pierre Serre]] in his book ''Local algebra'', works only for the set theoretic components (also called ''isolated components'') of the intersection, not for the [[embedded prime|embedded components]]. Theories have been developed for handling the embedded case (see [[intersection theory]] for details).
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| ==In complex analysis==
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| Let ''z''<sub>0</sub> be a root of a [[holomorphic function]] '' ƒ '', and let ''n'' be the least positive integer such that, the ''n''<sup>th</sup> derivative of ''ƒ'' evaluated at ''z''<sub>0</sub> differs from zero. Then the power series of ''ƒ'' about ''z''<sub>0</sub> begins with the ''n''<sup>th</sup> term, and ''ƒ'' is said to have a root of multiplicity (or “order”) ''n''. If ''n'' = 1, the root is called a simple root (Krantz 1999, p. 70).
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| We can also define the multiplicity of the [[Zero (complex analysis)|zeroes]] and [[Pole (complex analysis)|poles]] of a [[meromorphic function]] thus: If we have a meromorphic function ''ƒ'' = ''g''/''h'', take the [[Taylor series|Taylor expansions]] of ''g'' and ''h'' about a point ''z''<sub>0</sub>, and find the first non-zero term in each (denote the order of the terms ''m'' and ''n'' respectively). if ''m'' = ''n'', then the point has non-zero value. If ''m'' > ''n'', then the point is a zero of multiplicity ''m'' − ''n''. If ''m'' < ''n'', then the point has a pole of multiplicity ''n'' − ''m''.
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| ==See also==
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| * [[Zero (complex analysis)]]
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| * [[Set (mathematics)]]
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| * [[Fundamental theorem of algebra]]
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| * [[Fundamental theorem of arithmetic]]
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| * Algebraic multiplicity and geometric multiplicity of an [[eigenvalue]]
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| * [[Frequency (statistics)]]
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| ==References==
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| *Krantz, S. G. ''Handbook of Complex Variables''. Boston, MA: Birkhäuser, 1999. ISBN 0-8176-4011-8.
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| [[Category:Set theory]]
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| [[Category:Mathematical analysis]]
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If an existing Word - Press code is found vulnerable, Word - Press will immediately issue an update for that. It is thus, on these grounds that compel various web service provider companies to integrate the same in their packages too. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. 2- Ask for the designs and graphics that will be provided along with the Word - Press theme. The number of options offered here is overwhelming, but once I took the time to begin to review the video training, I was amazed at how easy it was to create a squeeze page and a membership site.
Creating a website from scratch can be such a pain. When you write a new post, you'll see a small bar that goes across the text input area. This plugin is a must have for anyone who is serious about using Word - Press. You can add new functionalities and edit the existing ones to suit your changing business needs. Now a days it has since evolved into a fully capable CMS platform which make it, the best platform in the world for performing online business.
The entrepreneurs can easily captivate their readers by using these versatile themes. As of now, Pin2Press is getting ready to hit the market. This platform can be customizedaccording to the requirements of the business. If you have any issues pertaining to exactly where and how to use wordpress dropbox backup, you can get in touch with us at the page. Newer programs allow website owners and internet marketers to automatically and dynamically change words in their content to match the keywords entered by their web visitors in their search queries'a feat that they cannot easily achieve with older software. After that the developer adds the unordered list for navigations.
Word - Press has plenty of SEO benefits over Joomla and Drupal. The SEOPressor Word - Press SEO Plugin works by analysing each page and post against your chosen keyword (or keyword phrase) and giving a score, with instructions on how to improve it. Specialty about our themes are that they are easy to load, compatible with latest wordpress version and are also SEO friendly. Word - Press is the most popular open source content management system (CMS) in the world today. This includes enriching the content with proper key words, tactfully defining the tags and URL.
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