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87.10.61.143: /* Vorticity */

2015-01-08T12:51:18Z

Vorticity

← Older revision		Revision as of 14:51, 8 January 2015
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en>Wavelength: inserting 11 hyphens: —> "two-dimensional" [10 instances] and "Two-dimensional"—wikt:two-dimensional

2014-02-21T18:43:40Z

inserting 11 hyphens: —> "two-dimensional" [10 instances] and "Two-dimensional"—wikt:two-dimensional

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en>Crowsnest: /* Vorticity */ vector calculus is independent of the used specific coordinate system; unify vector notation in article

2014-01-30T15:38:32Z

Vorticity: vector calculus is independent of the used specific coordinate system; unify vector notation in article

← Older revision		Revision as of 17:38, 30 January 2014
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	The ~~writer~~'s ~~name~~ is ~~Andera~~ and ~~she thinks it seems quite great~~. ~~Credit authorising~~ is ~~how she makes~~ a ~~living~~. ~~Her family lives~~ in ~~Ohio~~. ~~To climb~~ is ~~something she would~~ by ~~no means give up~~.<br><br>~~Take~~ a ~~look at my website~~ :: ~~psychics online~~ ([~~http~~://~~mergernexus~~.~~com~~/~~index~~.~~php?do~~=/~~profile~~-~~145700~~/~~info~~/ ~~learn more here~~])		In [[mathematics]], a '''Sheffer sequence''' or '''poweroid''' is a [[polynomial sequence]], i.e., a sequence { ''p''<sub>''n''</sub>(''x'') : ''n'' = 0, 1, 2, 3, . . . } of [[polynomial]]s in which the index of each polynomial equals its degree, satisfying conditions related to the [[umbral calculus]] in combinatorics. They are named for [[Isador M. Sheffer]].

			==Definition==
			Fix a polynomial sequence ''p''<sub>''n''</sub>. Define a linear operator ''Q'' on polynomials in ''x'' by

			:<math>Qp_n(x) = np_{n-1}(x)\, .</math>

			This determines ''Q'' on all polynomials. The polynomial sequence ''p''<sub>''n''</sub> is a ''Sheffer sequence'' if the linear operator ''Q'' just defined is ''shift-equivariant''. Here, we define a linear operator ''Q'' on polynomials to be ''shift-equivariant'' if, whenever ''f''(''x'') = ''g''(''x'' + ''a'') = ''T''<sub>''a''</sub> ''g''(''x'') is a "shift" of ''g''(''x''), then (''Qf'')(''x'') = (''Qg'')(''x'' + ''a''); i.e., ''Q'' commutes with every [[shift operator]]: ''T''<sub>''a''</sub>''Q'' = ''QT''<sub>''a''</sub>. Such a ''Q'' is a [[delta operator]].

			==Properties==

			The set of all Sheffer sequences is a [[group (mathematics)\|group]] under the operation of '''umbral composition''' of polynomial sequences, defined as follows. Suppose { ''p''<sub>''n''</sub>(x) : ''n'' = 0, 1, 2, 3, ... } and { ''q''<sub>''n''</sub>(x) : ''n'' = 0, 1, 2, 3, ... } are polynomial sequences, given by

			:<math>p_n(x)=\sum_{k=0}^n a_{n,k}x^k\ \mbox{and}\ q_n(x)=\sum_{k=0}^n b_{n,k}x^k.</math>

			Then the umbral composition <math>p \circ q</math> is the polynomial sequence whose ''n''th term is

			:<math>(p_n\circ q)(x)=\sum_{k=0}^n a_{n,k}q_k(x)=\sum_{0\le k \le \ell \le n} a_{n,k}b_{k,\ell}x^\ell</math>

			(the subscript ''n'' appears in ''p''<sub>''n''</sub>, since this is the ''n'' term of that sequence, but not in ''q'', since this refers to the sequence as a whole rather than one of its terms).

			The neutral element of this group is the standard monomial basis

			:<math>e_n(x) = x^n = \sum_{k=0}^n \delta_{n,k} x^k.</math>

			Two important subgroups are the group of [[Appell sequence]]s, which are those sequences for which the operator ''Q'' is mere differentiation, and the group of sequences of [[binomial type]], which are those that satisfy the identity
			:<math>p_n(x+y)=\sum_{k=0}^n{n \choose k}p_k(x)p_{n-k}(y).</math>
			A Sheffer sequence { ''p''<sub>''n''</sub>(''x'') : ''n'' = 0, 1, 2, . . . } is of binomial type if and only if both

			:<math>p_0(x) = 1\,</math>

			and

			:<math>p_n(0) = 0\mbox{ for } n \ge 1. \,</math>

			The group of Appell sequences is [[abelian group\|abelian]]; the group of sequences of binomial type is not. The group of Appell sequences is a [[normal subgroup]]; the group of sequences of binomial type is not. The group of Sheffer sequences is a [[semidirect product]] of the group of Appell sequences and the group of sequences of binomial type. It follows that each [[coset]] of the group of Appell sequences contains exactly one sequence of binomial type. Two Sheffer sequences are in the same such coset if and only if the operator ''Q'' described above – called the "[[delta operator]]" of that sequence – is the same linear operator in both cases. (Generally, a ''delta operator'' is a shift-equivariant linear operator on polynomials that reduces degree by one. The term is due to F. Hildebrandt.)

			If ''s''<sub>''n''</sub>(''x'') is a Sheffer sequence and ''p''<sub>''n''</sub>(''x'') is the one sequence of binomial type that shares the same delta operator, then

			:<math>s_n(x+y)=\sum_{k=0}^n{n \choose k}p_k(x)s_{n-k}(y).</math>

			Sometimes the term ''Sheffer sequence'' is ''defined'' to mean a sequence that bears this relation to some sequence of binomial type. In particular, if { ''s''<sub>''n''</sub>(''x'') } is an Appell sequence, then

			:<math>s_n(x+y)=\sum_{k=0}^n{n \choose k}x^ks_{n-k}(y).</math>

			The sequence of [[Hermite polynomials]], the sequence of [[Bernoulli polynomials]], and the [[monomial]]s { ''x<sup>n</sup>'' : ''n'' = 0, 1, 2, ... } are examples of Appell sequences.

			A Sheffer sequence ''p''<sub>''n''</sub> is characterised by its [[exponential generating function]]

			:<math> \sum_{n=0}^\infty \frac{p_n(x)}{n!} t^n = A(t) \exp(x B(t)) \, </math>

			where ''A'' and ''B'' are (formal) power series in ''t''. Sheffer sequences are thus examples of [[generalized Appell polynomials]] and hence have an associated [[recurrence relation]].

			==Examples==
			Examples of polynomial sequences which are Sheffer sequences include:
			* The [[Abel polynomials]];
			* The [[Bernoulli polynomials]];
			* The central factorial polynomials;
			* The [[Hermite polynomials]];
			* The [[Laguerre polynomials]];
			* The [[Mahler polynomials]];
			* The [[monomial]]s { ''x<sup>n</sup>'' : ''n'' = 0, 1, 2, ... } ;
			* The [[Mott polynomials]];

			==References==

			* {{cite journal \| authorlink=Gian-Carlo Rota \| author=G.-C. Rota \| coauthors=D. Kahaner, and [[Andrew Odlyzko\|A. Odlyzko]] \| title=On the foundations of combinatorial theory VIII: Finite Operator Calculus \| journal=Journal of Mathematical Analysis and its Applications \| volume=42 \| issue=3 \| date=June 1973 \| pages=684–750 \| doi=10.1016/0022-247X(73)90172-8 }} Reprinted in the next reference.
			* {{cite book \| author=G.-C. Rota \| title=Finite operator calculus \| coauthors=P. Doubilet, C. Greene, D. Kahaner, A. Odlyzko and R. Stanley \| publisher=Academic Press \| year=1975 \| isbn=0-12-596650-4 }}
			* {{cite journal \| author=I. M. Sheffer \|authorlink=Isador M. Sheffer\| title=Some Properties of Polynomial Sets of Type Zero \| journal=[[Duke Mathematical Journal]] \| volume=5 \| pages=590–622 \| year=1939 \| doi=10.1215/S0012-7094-39-00549-1 \| issue=3 }}
			*{{Cite book \| last1=Roman \| first1=Steven \| title=The umbral calculus \| url=http://books.google.com/books?id=JpHjkhFLfpgC \| publisher=Academic Press Inc. [Harcourt Brace Jovanovich Publishers] \| location=London \| series=Pure and Applied Mathematics \| isbn=978-0-12-594380-2 \| mr=741185 Reprinted by Dover, 2005 \| year=1984 \| volume=111 \| postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}

			==External links==
			* {{MathWorld\|ittle=Sheffer Sequence\|id=ShefferSequence}}

			[[Category:Polynomials]]
			[[Category:Factorial and binomial topics]]

en>Crowsnest: /* Continuity: The Derivation */ lowercase, see WP:CAPITAL

2012-04-27T11:35:21Z

Continuity: The Derivation: lowercase, see WP:CAPITAL

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