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  • generating function of Laguerre polynomials | planetmath.org
    planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A09 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A09 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A12D99 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A12D99 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader

    Original URL path: http://www.planetmath.org/generatingfunctionoflaguerrepolynomials (2016-04-25)
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  • generating function of Legendre polynomials | planetmath.org
    line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A09 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A33E30 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A33E30 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit For finding the generating function F t n 0 P n z t n F t superscript subscript n 0 subscript P n z superscript t n F t sum n 0 infty P n z t n of the sequence of the Legendre polynomials P 0 z 1 subscript P 0 z 1 P 0 z 1 P 1 z z subscript P 1 z z P 1 z z P 2 z 1 2 3 z 2 1 subscript P 2 z 1 2 3 superscript z 2 1 P 2 z frac 1 2 3z 2 1 P

    Original URL path: http://www.planetmath.org/generatingfunctionoflegendrepolynomials (2016-04-25)
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  • generators of inverse ideal | planetmath.org
    sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A30B10 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A30B10 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit generators of inverse ideal Theorem Let R R R be a commutative ring with non zero unity and let T T T be the total ring of fractions of R R R If ๐”ž a 1 a n ๐”ž subscript a 1 normal subscript a n mathfrak a a 1 ldots a n is an invertible fractional ideal of R R R with ๐”ž ๐”Ÿ R ๐”ž ๐”Ÿ R mathfrak ab R then also the inverse ideal ๐”Ÿ ๐”Ÿ mathfrak b can be generated by n n n elements of T T T Proof The equation ๐”ž ๐”Ÿ 1 ๐”ž ๐”Ÿ 1 mathfrak ab 1 implies the existence of the elements a i superscript subscript a i normal a i prime of ๐”ž ๐”ž mathfrak a and b i superscript subscript b i normal b i prime of ๐”Ÿ ๐”Ÿ mathfrak b i 1 m i 1 normal m i 1 ldots m such that a 1 b 1 a m b m 1 superscript subscript a 1 normal superscript subscript b 1 normal normal superscript subscript a m normal superscript subscript b m normal 1 a 1 prime b 1 prime cdots a m prime b m prime 1 Because the a i superscript subscript a i normal a i prime s are in ๐”ž ๐”ž mathfrak a they may be expressed as a i j 1 n r i j a j i 1 m fragments superscript subscript a

    Original URL path: http://www.planetmath.org/generatorsofinverseideal (2016-04-25)
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  • generatrices of hyperbolic paraboloid | planetmath.org
    all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit generatrices of hyperbolic paraboloid Since the equation x 2 a 2 y 2 b 2 2 z superscript x 2 superscript a 2 superscript y 2 superscript b 2 2 z frac x 2 a 2 frac y 2 b 2 2z of hyperbolic paraboloid can be gotten by multiplying the equations in the pair x a y b 2 z h x a y b h xayb 2zh xaybh displaystyle begin cases displaystyle frac x a frac y b frac 2z h displaystyle frac x a frac y b h end cases 1 of equations of planes the intersection line of these planes is contained in the surface of the hyperbolic paraboloid for each value of the parameter h h h So the surface has the family of generatrices rulings given by all real values of h h h The same concerns the other family x a y b 2 z k x a y b k xayb 2zk xaybk displaystyle begin cases displaystyle frac x a frac y b frac 2z k displaystyle frac x a frac y b k end cases 2 of lines It is easily seen that any point of the hyperbolic paraboloid is passed through by exactly two generatrices one from the family 1 and the other from family 2 Thus the surface is a doubly ruled surface The latter of the equations 1 tells that all generatrices the first family are parallel to the vertical plane x a y b 0 x a y b 0 frac x a frac y b 0 the so called director plane of the hyperbolic paraboloid this plane is also a plane of symmetry of the surface According to the latter equation 2

    Original URL path: http://www.planetmath.org/generatricesofhyperbolicparaboloid (2016-04-25)
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  • generatrices of one-sheeted hyperboloid | planetmath.org
    View active tab Coauthors PDF Source Edit generatrices of one sheeted hyperboloid The one sheeted hyperboloid is a ruled surface which is seen from its equation written in the form y 2 b 2 z 2 c 2 1 x 2 a 2 superscript y 2 superscript b 2 superscript z 2 superscript c 2 1 superscript x 2 superscript a 2 displaystyle frac y 2 b 2 frac z 2 c 2 1 frac x 2 a 2 1 or y b z c y b z c 1 x a 1 x a y b z c y b z c 1 x a 1 x a displaystyle left frac y b frac z c right left frac y b frac z c right left 1 frac x a right left 1 frac x a right 2 In fact 2 may be thought to be formed by multiplying the equations in the pair y b z c h 1 x a y b z c 1 h 1 x a ybzc h 1xa ybzc 1h 1xa displaystyle begin cases displaystyle frac y b frac z c h left 1 frac x a right displaystyle frac y b frac z c frac 1 h left 1 frac x a right end cases 3 which represents a line in the space h h h is an arbitrary parameter For any h 0 h 0 h neq 0 each point x y z x y z x y z on the line 3 satisfies also 2 This means that the line 3 lies on the hyperboloid i e it s a question of a generatrix ruling of the one sheeted hyperboloid Giving distinct real values to the parameter h h h we get an infinite family of the generatrices 3 Further one of these lines passes through every point of the hyperboloid Actually if the point P 1 x 1 y 1 z 1 subscript P 1 subscript x 1 subscript y 1 subscript z 1 P 1 x 1 y 1 z 1 satisfies the equation 2 of the surface we have the proportion equation y 1 b z 1 c 1 x 1 a 1 x 1 a y 1 b z 1 c subscript y 1 b subscript z 1 c 1 subscript x 1 a 1 subscript x 1 a subscript y 1 b subscript z 1 c frac frac y 1 b frac z 1 c 1 frac x 1 a frac 1 frac x 1 a frac y 1 b frac z 1 c and if we assign in 3 to h h h the value of the left hand side of the proportion then P 1 subscript P 1 P 1 satisfies also the equations 3 But since the equation 2 may be splitted also as y b z c k 1 x a y b z c 1 k 1 x a ybzc k 1xa ybzc 1k 1xa displaystyle begin cases displaystyle frac

    Original URL path: http://www.planetmath.org/generatricesofonesheetedhyperboloid (2016-04-25)
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  • geometric constructions by Euclid | planetmath.org
    org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51M04 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51M04 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit geometric constructions by Euclid The geometric constructions using compass and straightedge consist of three simple fundamental tasks as given in Euclid s The Elements in ancient Greek ฮฃ ฯ„ o ฮน ฯ‡ ฮต ฮน ฮฑ normal ฮฃ ฯ„ o ฮน ฯ‡ ฮต normal ฮน ฮฑ Sigma tau o iota chi varepsilon acute iota alpha transliterated Stoikheia These fundamental tasks are as follows 1 Drawing a line through two given points 2 Drawing a circle having a given point as its center and passing through another given point 3 Setting a plane passing through three given non collinear points where one performs tasks based on the two preceding tasks Example The usual task of drawing a circle with a given point as its center and with a given line segment as its radius a fundamental task in many textbooks can be reduced to Euclid s fundamental tasks one needs five circles Remark It can be proven that all geometric constructions with compass and straightedge are possible using only the compass See e g compass and straightedge construction of parallel line In the text of Euclid the constructions are not listed separately but are combined with the theorems as propositions A way to tell whether a proposition is a theorem or a construction is to go to the end of the proof and see if it ends with QED in which case it is a theorem or with QEF in which case it is a construction Note that QEF is an abbreviation for the Latin phrase quod erat faciendum meaning which was to be done Here is a list of the geometric constructions to be found in The Elements I

    Original URL path: http://www.planetmath.org/geometricconstructionsbyeuclid (2016-04-25)
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  • geometric sequence | planetmath.org
    8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51 00 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51 00 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit geometric sequence A sequence of the form a a r a r 2 a r 3 a a r a superscript r 2 a superscript r 3 normal a ar ar 2 ar 3 ldots of real or

    Original URL path: http://www.planetmath.org/geometricsequence (2016-04-25)
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  • gradient theorem | planetmath.org
    2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A40 00 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit gradient theorem If u u x y z u u x y z u u x y z is continuously differentiable function in a simply connected domain D D D of โ„ 3 superscript โ„ 3 mathbb R 3 and P x 0 y 0 z 0 P subscript x 0 subscript y 0 subscript z 0 P x 0 y 0 z 0 and Q x 1 y 1 z 1 Q subscript x 1 subscript y 1 subscript z 1 Q x 1 y 1 z 1 lie in this domain then P Q u d s u x 1 y 1 z 1 u x 0 y 0 z 0 superscript subscript P Q normal normal u normal d s u subscript x 1 subscript y 1 subscript z 1 u subscript x 0 subscript y 0 subscript z 0 displaystyle int P Q nabla u cdot vec ds u x 1 y 1 z 1 u x 0 y 0 z 0 1 where the line integral of the left hand side is taken along an arbitrary path in D D D The equation 1 is illustrated by the fact that u d s u x d x u y d

    Original URL path: http://www.planetmath.org/gradienttheorem (2016-04-25)
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