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  • conjugate hyperbola | planetmath.org
    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 3A11R04 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 3A11R04 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 conjugate hyperbola The simplest form of the equation presenting a hyperbola without the mixed x y x y xy term in a rectangular coordinate system is got when the coordinate axes coincide with the principal axes of the hyperbola and it has the form x 2 a 2 y 2 b 2 1 superscript x 2 superscript a 2 superscript y 2 superscript b 2 1 displaystyle frac x 2 a 2 frac y 2 b 2 1 1 Here a 0 annotated a absent 0 a 0 is the length of the transverse semiaxis and b 0 annotated b absent 0 b 0 the length of the conjugate semiaxis of the hyperbola The equation y 2 b 2 x 2 a 2 1 superscript y 2 superscript b 2 superscript x 2 superscript a 2 1 displaystyle frac y 2 b 2 frac x 2 a 2 1 2 or x 2 a 2 y 2 b 2 1 superscript x 2 superscript a 2 superscript y 2 superscript b 2 1 frac x 2 a 2 frac y 2 b 2 1 presents the conjugate hyperbola of 1 Its transverse axis is the conjugate axis of 1 and its conjugate axis the transverse axis of 1 Both hyperbolas are conjugate hyperbolas of each other They have the common asymptotes x 2 a 2 y 2 b 2 0 superscript x 2 superscript a 2 superscript y 2 superscript b 2 0 frac x 2 a 2 frac y 2 b 2

    Original URL path: http://www.planetmath.org/conjugatehyperbola (2016-04-25)
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  • conjugated roots of equation | planetmath.org
    Edit conjugated roots of equation The rules w 1 w 2 w 1 w 2 and w 1 w 2 w 1 w 2 formulae sequence normal subscript w 1 subscript w 2 normal subscript w 1 normal subscript w 2 and normal subscript w 1 subscript w 2 normal subscript w 1 normal subscript w 2 overline w 1 w 2 overline w 1 overline w 2 quad mbox and quad overline w 1 w 2 overline w 1 overline w 2 concerning the complex conjugates of the sum and product of two complex numbers may be by induction generalised for arbitrary number of complex numbers w k subscript w k w k Since the complex conjugate of a real number is the same real number we may write a k z k a k z k normal subscript a k superscript z k subscript a k superscript normal z k overline a k z k a k overline z k for real numbers a k k 0 1 2 fragments subscript a k fragments normal k 0 normal 1 normal 2 normal normal normal a k k 0 1 2 ldots Thus for a polynomial P x a 0 x n a 1 x n 1 a n assign P x subscript a 0 superscript x n subscript a 1 superscript x n 1 normal subscript a n P x a 0 x n a 1 x n 1 ldots a n we obtain P z a 0 z n a 1 z n 1 a n a 0 z n a 1 z n 1 a n P z normal P z normal subscript a 0 superscript z n subscript a 1 superscript z n 1 normal subscript a n subscript a 0 superscript normal z n

    Original URL path: http://www.planetmath.org/conjugatedrootsofequation (2016-04-25)
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  • construction of central proportional | planetmath.org
    3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A30 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 3A30 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 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 in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit construction of central proportional Task Given two line segments p p p and q q q Using compass and straightedge construct the central proportional the geometric mean of the line segments Solution Set the line segments A D

    Original URL path: http://www.planetmath.org/constructionofcentralproportional (2016-04-25)
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  • construction of contraharmonic mean of two segments | planetmath.org
    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 3A51M15 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 construction of contraharmonic mean of two segments Let a a a and b b b mean two line segments and their lengths The contraharmonic mean x a 2 b 2 a b a 2 b 2 2 a b assign x superscript a 2 superscript b 2 a b superscript superscript a 2 superscript b 2 2 a b x frac a 2 b 2 a b frac left sqrt a 2 b 2 right 2 a b satisfying the proportion equation a b a 2 b 2 a 2 b 2 x a b superscript a 2 superscript b 2 superscript a 2 superscript b 2 x frac a b sqrt a 2 b 2 frac sqrt a 2 b 2 x can be constructed geometrically as the third proportional of the segments a b a b a b and a 2 b 2 superscript a 2 superscript b 2 sqrt a 2 b 2 the latter of which is gotten as the hypotenuse of the right triangle with catheti a a

    Original URL path: http://www.planetmath.org/constructionofcontraharmonicmeanoftwosegments (2016-04-25)
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  • construction of fourth proportional | planetmath.org
    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 construction of fourth proportional Task Given three line segments a a a b b b and c c c Using compass and straightedge construct the fourth proportional of the line segments Solution Draw an angle α α alpha and denote its

    Original URL path: http://www.planetmath.org/constructionoffourthproportional (2016-04-25)
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  • construction of tangent | planetmath.org
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    Original URL path: http://www.planetmath.org/constructionoftangent (2016-04-25)
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  • content of polynomial | planetmath.org
    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 The content of a polynomial f f f may be defined in any polynomial ring R x R x R x over a commutative ring R R R as the ideal of R R R generated by the coefficients of the polynomial It is denoted by cont f cont f operatorname cont f or c f c f c f Coefficient module is a little more general concept If R R R is a unique factorisation domain and f g R x f g R x f g in R x the Gauss lemma I implies 1 1 In a UFD one can use as contents of f f f and g g g the greatest common divisors a a a and b b b of the coefficients of these polynomials when one has f x a f 1 x f x a subscript f 1 x f x af 1 x g x b g 1 x g x b subscript g 1 x g x bg 1 x with f 1 x subscript f 1 x f 1 x and g 1 x subscript g 1 x g 1 x primitive polynomials Then f x g x a b f 1 x g 1 x f x g x a b subscript f 1 x subscript g 1 x f x g x abf 1 x g 1 x and since also f 1 g 1 subscript f 1 subscript g 1 f 1 g 1 is a primitive polynomial we see that c f g a b c f c g c f g a b c f c g c fg ab c f c g that c f g c f c g c f g c f c g displaystyle c fg c f c g 1 For an arbitrary commutative ring R R R there is only the containment c f g c

    Original URL path: http://www.planetmath.org/contentofpolynomial (2016-04-25)
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  • continuation of exponent | planetmath.org
    when x 0 assign 1 subscript ν 0 x when x0 νxewhen x0 1 qquad qquad qquad nu 0 x begin cases infty quad mbox when x 0 frac nu x e mbox when x neq 0 end cases defined in the base field k k k is an exponent of k k k Proof The exponent ν ν nu of K K K attains in the set k 0 k 0 k smallsetminus 0 also non zero values otherwise k k k would be included in ν subscript ν mathcal O nu the ring of the exponent ν ν nu Since any element ξ ξ xi of K K K are integral over k k k it would then be also integral over ν subscript ν mathcal O nu which is integrally closed in its quotient field K K K see theorem 1 in ring of exponent the situation would mean that ξ ν ξ subscript ν xi in mathcal O nu and thus the whole K K K would be contained in ν subscript ν mathcal O nu This is impossible because an exponent of K K K attains also negative values So we infer that ν ν nu does not vanish in the whole k 0 k 0 k smallsetminus 0 Furthermore ν ν nu attains in k 0 k 0 k smallsetminus 0 both negative and positive values since ν a ν a 1 ν a a 1 ν 1 0 ν a ν superscript a 1 ν a superscript a 1 ν 1 0 nu a nu a 1 nu aa 1 nu 1 0 Let p p p be such an element of k k k on which ν ν nu attains as its value the least possible positive integer e e e in the field k k k and let a a a be an arbitrary non zero element of k k k If ν a m q e r q r ℤ 0 r e fragments ν fragments normal a normal m q e r fragments normal q normal r Z normal 0 r e normal normal nu a m qe r quad q r in mathbb Z 0 leqq r e then ν a p q m q e r ν a superscript p q m q e r nu ap q m qe r and thus r 0 r 0 r 0 on grounds of the choice of p p p This means that ν a ν a nu a is always divisible by e e e i e that the values of the function ν 0 subscript ν 0 nu 0 in k 0 k 0 k smallsetminus 0 are integers Because ν 0 p 1 subscript ν 0 p 1 nu 0 p 1 and ν 0 p l l subscript ν 0 superscript p l l nu 0 p l l the function attains in k k k every integer value Also the conditions ν 0 a

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