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  • differential equation of circles | planetmath.org
    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 3A26A06 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 3A26 03 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 3A26 03 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 3A01A45 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

    Original URL path: http://www.planetmath.org/differentialequationofcircles (2016-04-25)
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  • differentiation of Laplace transform with respect to parameter | planetmath.org
    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 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 We use the curved arrows to point from the Laplace transformed functions to the corresponding initial functions If f t x F s x normal f t x F s x f t x curvearrowleft F s x then one can differentiate both functions with respect to the parametre x x x f x t x F x s x normal subscript superscript f normal x t x subscript superscript F normal x s x displaystyle f prime x t x curvearrowleft F prime x s x 1 1 may be written also as ℒ x f t x x ℒ f t x ℒ x f t x x ℒ f t x displaystyle mathcal L frac partial partial x f t x frac partial partial x mathcal L f t x 2 Proof We differentiate partially both sides of the defining equation F s x 0 e s t f t x d t assign F s x superscript subscript 0 superscript e s t f t x d t F s x int 0 infty e st f t x dt on the right hand side under the integration sign getting F x s x 0 e s t f x t x d x subscript superscript

    Original URL path: http://www.planetmath.org/differentiationoflaplacetransformwithrespecttoparameter (2016-04-25)
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  • dihedral angle | planetmath.org
    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 3A44A10 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 dihedral angle Two distinct half planes emanating from a same line l l l divide the space ℝ 3 superscript ℝ 3 mathbb R 3 into two regions called dihedral angles The line l l l is the edge of the dihedral angle and the bounding half planes are its sides l l l The angle which the sides of a dihedral planes separate from a normal plane of the edge is the normal section of the dihedral angle Apparently all normal sections are equal According to the size of the normal section the dihedral angle may be called acute right obtuse straight skew convex and concave Unlike the angle between two planes a dihedral angle may be over 90 degrees If two planes intersect each other and if one of the four dihedral angles formed is

    Original URL path: http://www.planetmath.org/dihedralangle (2016-04-25)
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  • dilogarithm function | planetmath.org
    fragments subscript Li s fragments normal x normal normal superscript subscript n 1 superscript x n superscript n s normal mbox Li s x sum n 1 infty frac x n n s The radius of convergence of the series 1 is 1 whence the definition 1 is valid also in the unit disc of the complex plane For 0 x 1 0 x 1 0 leq x leq 1 the equation 1 is apparently equivalent to Li 2 x 0 x ln 1 t t d t fragments subscript Li 2 fragments normal x normal normal superscript subscript 0 x 1 t t d t normal displaystyle mbox Li 2 x int 0 x frac ln 1 t t dt 2 cf logarithm series of ln 1 x 1 x ln 1 x The analytic continuation of Li 2 subscript Li 2 mbox Li 2 for z 1 z 1 z geq 1 can be made by Li 2 z 0 z log 1 t t d t fragments subscript Li 2 fragments normal z normal normal superscript subscript 0 z 1 t t d t normal displaystyle mbox Li 2 z int 0 z frac log 1 t t dt 3 Thus Li 2 z subscript Li 2 z mbox Li 2 z is a multivalued analytic function of z z z Its principal branch is single valued and is got by taking the principal branch of the complex logarithm then z ℂ 1 0 arg z 1 2 π fragments z C fragments normal 1 normal fragments normal normal 0 fragments normal z 1 normal 2 π normal z in mathbb C smallsetminus 1 infty quad 0 arg z 1 2 pi For real values of x x x we have Im Li 2 x 0 for x 1 π ln x for x 1 Im subscript Li 2 x 0 forx1 πln xforx1 displaystyle mbox Im mbox Li 2 x begin cases 0 qquad mbox for x leq 1 pi ln x mbox for x 1 end cases According to 2 the derivative of the dilogarithm is Li 2 x ln 1 x x subscript superscript Li normal 2 x 1 x x mbox Li prime 2 x frac ln 1 x x In terms of the Bernoulli numbers the dilogarithm function has a series expansion more rapidly converging than 1 Li 2 x n 1 B n 1 ln 1 x n n ln 1 x 2 π fragments subscript Li 2 fragments normal x normal superscript subscript n 1 subscript B n 1 superscript 1 x n n italic fragments normal normal fragments normal 1 x normal normal 2 π normal displaystyle mbox Li 2 x sum n 1 infty B n 1 frac ln 1 x n n qquad ln 1 x 2 pi 4 Some functional equations and values Li 2 z Li 2 z 1 2 Li 2 z 2 subscript Li 2 z subscript Li 2 z 1 2 subscript

    Original URL path: http://www.planetmath.org/dilogarithmfunction (2016-04-25)
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  • direction cosines | 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 3A33B15 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 3A33B15 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 direction cosines If the non zero vector r x i y j z k normal r x normal i y normal j z normal k vec r x vec i y vec j z vec k of ℝ 3 superscript ℝ 3 mathbb R 3 forms the angles α α alpha β β beta and γ γ gamma with the positive directions of x x x axis y y y axis and z z z axis respectively then

    Original URL path: http://www.planetmath.org/directioncosines (2016-04-25)
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  • discontinuity of characteristic function | planetmath.org
    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 3A51N20 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 3A51N20 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 Theorem For a subset A A A of ℝ n superscript ℝ n mathbb R n the set of the discontinuity points of the characteristic function χ A subscript χ A chi A is the boundary of A A A Proof Let a a a be a discontinuity point of χ A subscript χ A chi A Then any neighborhood of a a a contains the points b b b and c c c such that χ A b 1 subscript χ A b 1 chi A b 1 and χ A c 0 subscript χ A c 0 chi A c 0 Thus b A b A b in A and c A c A c notin A whence a a a is a boundary point of A A A If on the contrary a a a is a boundary point of A A A and U a U a U a an arbitrary neighborhood of a a a it follows that U a U a U a contains both points belonging to A A A and points not belonging to A A A So we have in U a U a U a the points b b b and c c c such that χ A b 1 subscript χ A b 1 chi A

    Original URL path: http://www.planetmath.org/discontinuityofcharacteristicfunction (2016-04-25)
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  • discriminant in algebraic number field | planetmath.org
    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 Primary tabs View active tab Coauthors PDF Source Edit discriminant in algebraic number field Let us consider the elements α 1 α 2 α n subscript α 1 subscript α 2 normal subscript α n alpha 1 alpha 2 ldots alpha n of an algebraic number field ℚ ϑ ℚ ϑ mathbb Q vartheta of degree n n n Let ϑ 1 ϑ ϑ 2 ϑ n subscript ϑ 1 ϑ subscript ϑ 2 normal subscript ϑ n vartheta 1 vartheta vartheta 2 ldots vartheta n be the algebraic conjugates of the primitive element ϑ ϑ vartheta and α i r i ϑ i 1 2 n fragments subscript α i subscript r i fragments normal ϑ normal fragments normal i 1 normal 2 normal normal normal n normal alpha i r i vartheta quad i 1 2 ldots n the canonical forms of the elements α i subscript α i alpha i Then the ℚ ϑ ℚ ϑ mathbb Q vartheta conjugates of those elements are α i j r i ϑ j superscript subscript α i j subscript r i subscript ϑ j alpha i j r i vartheta j Using these one can define the discriminant Δ α 1 α 2 α n normal Δ subscript α 1 subscript α 2 normal subscript α n Delta alpha 1 alpha 2 ldots alpha n of the elenents α i subscript α i alpha i as Δ α 1 α 2 α n det r i ϑ j 2 det α i j 2 assign normal Δ subscript α 1 subscript α 2 normal subscript α n superscript subscript r i subscript ϑ j 2 superscript superscript subscript α i j 2 Delta alpha 1 alpha 2 ldots alpha n det

    Original URL path: http://www.planetmath.org/discriminantinalgebraicnumberfield (2016-04-25)
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  • discriminant of algebraic number | planetmath.org
    subscript a n f x x n a 1 x n 1 ldots a n we have f ϑ n ϑ n 1 n 1 a 1 ϑ n 2 2 a n 2 ϑ a n 1 ℚ ϑ superscript f normal ϑ n superscript ϑ n 1 n 1 subscript a 1 superscript ϑ n 2 normal 2 subscript a n 2 ϑ subscript a n 1 ℚ ϑ f prime vartheta n vartheta n 1 n 1 a 1 vartheta n 2 ldots 2a n 2 vartheta a n 1 in mathbb Q vartheta The norm of f ϑ superscript f normal ϑ f prime vartheta in ℚ ϑ ℚ ℚ ϑ ℚ mathbb Q vartheta mathbb Q is the product of all ℚ ϑ ℚ ϑ mathbb Q vartheta conjugates f ϑ i superscript superscript f normal ϑ i f prime vartheta i of f ϑ superscript f normal ϑ f prime vartheta which is N f ϑ f ϑ 1 f ϑ 2 f ϑ n f ϑ 1 f ϑ 2 f ϑ n N superscript f normal ϑ superscript superscript f normal ϑ 1 superscript superscript f normal ϑ 2 normal superscript superscript f normal ϑ n superscript f normal subscript ϑ 1 superscript f normal subscript ϑ 2 normal superscript f normal subscript ϑ n mbox N f prime vartheta f prime vartheta 1 f prime vartheta 2 cdots f prime vartheta n f prime vartheta 1 f prime vartheta 2 cdots f prime vartheta n On the other side the polynonomial f x f x f x in its linear factors is f x x ϑ 1 x ϑ 2 x ϑ n f x x subscript ϑ 1 x subscript ϑ 2 normal x subscript ϑ n f x x vartheta 1 x vartheta 2 cdots x vartheta n whence its derivative may be written f x ν 1 n x ϑ 1 x ϑ ν 1 x ϑ ν 1 x ϑ n superscript f normal x superscript subscript ν 1 n x subscript ϑ 1 normal x subscript ϑ ν 1 x subscript ϑ ν 1 normal x subscript ϑ n f prime x sum nu 1 n x vartheta 1 cdots x vartheta nu 1 x vartheta nu 1 cdots x vartheta n Substituting x ϑ ν x subscript ϑ ν x vartheta nu gives simply f ϑ ν j ν ϑ ν ϑ j for ν 1 n formulae sequence superscript f normal subscript ϑ ν subscript product j ν subscript ϑ ν subscript ϑ j for ν 1 normal n f prime vartheta nu prod j neq nu vartheta nu vartheta j quad mbox for nu 1 ldots n Multiplying these equations we obtain N f ϑ ν 1 n f ϑ ν i j ϑ i ϑ j N superscript f normal ϑ superscript subscript product ν 1 n superscript f normal subscript ϑ ν subscript product i j subscript ϑ i subscript ϑ

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