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  • determining the continuations of exponent | planetmath.org
    sink 23this 3E 7B msc 3A11D45 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 determining the continuations of exponent Task Let ν 0 subscript ν 0 nu 0 be the 3 adic triadic exponent valuation of the field ℚ ℚ mathbb Q of the rational numbers and let mathfrak o be the ring of the exponent Determine the integral closure mathfrak O of mathfrak o in the extension field ℚ 5 ℚ 5 mathbb Q sqrt 5 and the continuations of ν 0 subscript ν 0 nu 0 to this field The triadic exponent of ℚ ℚ mathbb Q at any non zero rational number 3 n u v superscript 3 n u v displaystyle frac 3 n u v where u u u and v v v are integers not divisible by 3 is defined as ν 0 3 n u v n assign subscript ν 0 superscript 3 n u v n nu 0 left frac 3 n u v right n Any number of the quadratic field ℚ 5 ℚ 5 mathbb Q sqrt 5 is of the form r s 5 r s 5 r s sqrt 5 with r r r and s s s rational numbers When α r s 5 α r s 5 alpha r s sqrt 5 belongs to mathfrak O the rational coefficients of the quadratic equation x 2 2 r x r 2 5 s 2 0 superscript x 2 2 r x superscript r 2 5 superscript s 2 0 x 2 2rx r 2 5s 2 0 satisfied by α α alpha belong to the ring mathfrak o whence one has ν 0 2 r 0 ν 0 r 2 5 c 2 0 formulae sequence subscript ν 0 2 r 0 subscript ν 0 superscript r 2 5 superscript c 2 0 nu 0 2r geqq 0 quad nu 0 r 2 5c 2 geqq 0 The first of these inequalities implies that ν 0 r 0 subscript ν 0 r 0 nu 0 r geqq 0 since 2 2 2 is a unit of mathfrak o As for s s s if one had ν 0 s 0 subscript ν 0 s 0 nu 0 s 0 then ν 0 5 s 2 2 ν 0 s 0 subscript ν 0 5 superscript s 2 2 subscript ν 0 s 0 nu 0 5s 2 2 nu 0 s 0 and therefore one had ν 0 r 2 5 s 2 min ν 0 r 2 ν 0 5 s 2 0 subscript ν 0 superscript r 2 5 superscript s 2 subscript ν 0 superscript r 2 subscript ν 0 5 superscript s 2 0 nu 0 r 2 5s 2 min nu 0 r 2 nu 0 5s

    Original URL path: http://www.planetmath.org/determiningthecontinuationsofexponent (2016-04-25)
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  • developable surface | planetmath.org
    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 3A13A18 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 3A12J20 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 3A12J20 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 3A13F30 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

    Original URL path: http://www.planetmath.org/developablesurface (2016-04-25)
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  • diametral points | planetmath.org
    3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51M20 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 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 diametral points Two points P 1 subscript P 1 P 1 and P 2 subscript P 2 P 2 on the circumference of a circle or on a sphere are diametral if the line segment P 1 P 2 subscript P 1 subscript P 2 P 1 P 2 connecting them passes through the centre of the circle resp the sphere i e is a diametre Equivalently the shortest distance of the diametral points P 1 subscript P 1 P 1 and P 2 subscript P 2 P 2 on the circle is maximal on the circle resp on the sphere namely a half of the perimetre It s easily justified that a point of a circle resp a sphere has exactly one diametral point A circle c c c is a diametral circle of a given circle c 0 subscript c 0 c 0 if c c c intersects c 0 subscript c 0 c 0 diametrically i e in two diametral points of c 0 subscript c 0 c 0 If the equation of c 0 subscript c 0 c 0 is

    Original URL path: http://www.planetmath.org/diametralpoints (2016-04-25)
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  • difference | planetmath.org
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    Original URL path: http://www.planetmath.org/difference (2016-04-25)
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  • difference of squares | planetmath.org
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    Original URL path: http://www.planetmath.org/differenceofsquares (2016-04-25)
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  • difference of vectors | planetmath.org
    2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26C99 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 3A26C99 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 3A13A99 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 3A13A99 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 difference of vectors Let a normal a vec a and b normal b vec b be two vectors in the plane or in a vector space

    Original URL path: http://www.planetmath.org/differenceofvectors (2016-04-25)
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  • difference quotient | planetmath.org
    2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A53A45 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 3A53A45 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 difference quotient The difference quotient f z Δ z f z Δ z f z normal Δ z f z normal Δ z frac f z Delta z f z Delta z of a complex or real function f f f is the ratio of the change of the function to the corresponding change of the variable The difference quotient is needed in defining the derivative of the function In the case of a real function f f f the difference quotient may be interpreted as the slope of a secant line of the graph of the function Related Derivative2 Slope SecondDerivativeAsSimpleLimit Type of

    Original URL path: http://www.planetmath.org/differencequotient (2016-04-25)
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  • differential | planetmath.org
    property we can write f x Δ x f x Δ x f x Δ x f x normal Δ x f x normal Δ x superscript f normal x normal Δ x frac f x Delta x f x Delta x f prime x langle Delta x rangle This allows us to express the increment of the function f x Δ x f x Δ f assign f x normal Δ x f x normal Δ f f x Delta x f x Delta f in the form Δ f f x Δ x Δ x f x Δ x Δ x Δ x normal Δ f superscript f normal x normal Δ x normal Δ x superscript f normal x normal Δ x normal Δ x normal Δ x displaystyle Delta f f prime x langle Delta x rangle Delta x f prime x Delta x langle Delta x rangle Delta x 1 This result may be uttered as the Theorem If the derivative f x superscript f normal x f prime x exists then the increment Δ f normal Δ f Delta f of the function corresponding to the increment of the argument from x x x to x Δ x x normal Δ x x Delta x may be divided into two essentially different parts 1 superscript 1 1 circ One part is proportional to the increment Δ x normal Δ x Delta x of the argument i e it equals this increment multiplied by a coefficient f x superscript f normal x f prime x which is independent on the increment 2 superscript 2 2 circ The ratio of the other part Δ x Δ x normal Δ x normal Δ x langle Delta x rangle Delta x to the increment Δ x normal Δ x Delta x of the argument tends to 0 along with Δ x normal Δ x Delta x As well the converse of the theorem is true By Leibniz the former part f x Δ x superscript f normal x normal Δ x f prime x Delta x is called the differential or the differential increment of the function and denoted by d f x d f x df x briefly d f d f df It is easily checked that when one has set the tangent line of the curve at the point x f x x f x x f x the differential increment d f x d f x df x geometrically means the increment of the ordinate of the tangent corresponding to the transition from the abscissa x x x to the ascissa x Δ x x normal Δ x x Delta x The differential of the identity function f x x f x x f x equiv x f x 1 superscript f normal x 1 f prime x equiv 1 is d x 1 Δ x Δ x d x normal 1 normal Δ x normal Δ x dx 1 cdot Delta x Delta

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