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  • classical isoperimetric problem | planetmath.org
    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 classical isoperimetric problem The points a a a and b b b on the x x x axis have to be connected by an arc with a given length l l l such that the area between the x x x axis and the arc is as great as possible Denote the equation of the searched arc by y y x y y x y y x The task which belongs to the isoperimetric problems can be formulated as to maximise a b y d x to maximise superscript subscript a b y d x displaystyle mbox to maximise quad int a b y dx 1 under the constraint condition a b 1 y 2 d x l superscript subscript a b 1 superscript y normal 2 d x l displaystyle int a b sqrt 1 y prime 2 dx l 2 We have the integrands f x y y y g x y y 1 y 2 formulae sequence f x y superscript y normal y g x y superscript y normal 1 superscript y normal 2 f x y y prime equiv y quad g x y y prime equiv sqrt 1 y prime 2 The conditional variation problem for the functional in 1 may be considered as a free variation problem without conditions for the functional a b f λ g d x superscript subscript a b f λ g d x int a b f lambda g dx where λ λ lambda is a Lagrange multiplier For this end we need the Euler Lagrange differential equation y f λ g d d x y f λ g 0 y f λ g d d x superscript y normal f λ g 0 displaystyle frac partial partial y f lambda g frac d dx frac partial partial y prime f lambda g 0 3 Since the expression f λ g f λ g f lambda g does not depend explicitly on x x x the differential equation 3 has by the Beltrami identity a first integral of the form f

    Original URL path: http://www.planetmath.org/classicalisoperimetricproblem (2016-04-25)
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  • classification of complex numbers | planetmath.org
    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 3A49K22 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 3A49K05 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 3A49K05 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 classification of complex numbers The set ℂ ℂ mathbb C of all complex numbers and many of its subsets may be partitioned classified into two subsets by certain criterion of the numbers A F i r s t c l a s s i f i c a t i o n Complex numbers contain 1 algebraic numbers 2 transcendental numbers Algebraic numbers contain 1 algebraic integers entire algebraic numbers 2 algebraic fractions fractional algebraic numbers Algebraic integers contain 1 rational integers 2 non rational integers Algebraic fractions contain 1 rational fractions 2 non rational fractions Transcendental

    Original URL path: http://www.planetmath.org/classificationofcomplexnumbers (2016-04-25)
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  • closed complex plane | planetmath.org
    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 closed complex plane The complex plane ℂ ℂ mathbb C i e the set of the complex numbers z z z satisfying z z z infty is open but not closed since it doesn t contain the accumulation points of all sets of complex numbers for example of the set 1 2 3 1 2 3 normal 1 2 3 ldots One can supplement ℂ ℂ mathbb C to the closed complex plane ℂ ℂ mathbb C cup infty by adding to ℂ ℂ mathbb C the infinite point infty which represents the lacking accumulation points One settles that infty infty where the latter infty means the real infinity The resulting space is the one point compactification of ℂ ℂ mathbb C The open sets are the open sets in ℂ ℂ mathbb C together with sets containing infty whose complement is compact in ℂ ℂ mathbb C Conceptually one thinks of the additional open sets

    Original URL path: http://www.planetmath.org/closedcomplexplane (2016-04-25)
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  • coefficients of Laurent series | planetmath.org
    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 Primary tabs View active tab Coauthors PDF Source Edit coefficients of Laurent series Suppose that f f f is analytic in the annulus z ℂ R 1 z a R 2 z ℂ normal subscript R 1 z a subscript R 2 z in mathbb C vdots R 1 z a R 2 where R 1 subscript R 1 R 1 may be 0 and R 2 subscript R 2 R 2 may be infty Then the coefficients of the Laurent series expansion n c n z a n superscript subscript n subscript c n superscript z a n sum n infty infty c n z a n of f f f can be obtained from c n 1 2 π i γ f t t a n 1 d t n 0 1 2 fragments subscript c n 1 2 π i subscript contour integral γ f t superscript t a n 1 d t fragments normal n 0 normal plus or minus 1 normal plus or minus 2 normal normal normal normal displaystyle c n frac 1 2 pi i oint gamma frac f t t a n 1 dt quad n 0 pm 1 pm 2 ldots 1 where the path γ γ gamma goes anticlockwise once around the point z a z a z a within the annulus Especially the residue of f f f in the point a a a is c 1 1 2 π i γ f t d t subscript c 1 1 2 π i subscript contour integral γ f t d t displaystyle c 1 frac 1 2 pi i oint gamma f t dt 2 Remark Usually the Laurent series of a function i e the coefficients c n subscript c n c n are not determined by using the integral formula 1 but directly from known series expansions Often it is sufficient to know the value of c 1 subscript c 1 c 1 or the residue which is used to compute integrals see the Cauchy residue theorem cf 2 There is also the usable Rule In the case that the limit lim z a z a f z subscript normal z a z a f z

    Original URL path: http://www.planetmath.org/coefficientsoflaurentseries (2016-04-25)
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  • commensurable numbers | planetmath.org
    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 commensurable numbers Two positive real numbers a a a and b b b are commensurable iff there exists a positive real number u u u such that a m u b n u formulae sequence a m u b n u displaystyle a mu quad b nu 1 with some positive integers m m m and n n n If the positive numbers a a a and b b b are not commensurable they are incommensurable Theorem The positive numbers a a a and b b b are commensurable if and only if their ratio is a rational number m n m n displaystyle frac m n m n ℤ m n ℤ m n in mathbb Z Proof The equations 1 imply the proportion a b m n a b m n displaystyle frac a b frac m n 2 Conversely if 2 is valid with m n ℤ m n ℤ m n in mathbb Z then we can write a m b n b n b n formulae sequence a normal m b n b normal n b n a m cdot frac b n quad b n cdot frac b n which means that a a a and b b b are multiples of b n b n displaystyle frac b n and thus commensurable Q E D Example The lengths of the side and the diagonal of square are always incommensurable 0 1 Commensurability as relation The commensurability is an equivalence relation in the set ℝ subscript ℝ mathbb R of the positive reals the reflexivity and the symmetry are trivial if a b

    Original URL path: http://www.planetmath.org/commensurablenumbers (2016-04-25)
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  • common point of triangle medians | planetmath.org
    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 3A03E02 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 common point of triangle medians Theorem The three medians of a triangle intersect one another in one point which divides each median in the ratio 2 1 normal 2 1 2 1 A A A B B B C C C D D D E E E F F F Proof Let the medians of a triangle A B C A B C ABC be A D A D AD B E B E BE and C F C F CF Any median vector is the arithmetic mean of the side vectors emanating from the same vertex Using vectors let us form three ways all beginning from the vertex A A A the first going simply 2 3 2 3 2 3 of the median vector A D normal A D overrightarrow AD blue in the picture 2 3 A D 2 3 1 2 A B A C 1 3 A B A C 2 3 normal A D normal 2 3 1 2 normal A B normal A C 1 3 normal A B normal A C displaystyle frac 2 3 overrightarrow AD frac 2 3 cdot frac 1 2 overrightarrow AB overrightarrow AC frac 1 3 overrightarrow AB overrightarrow AC 1 The second way goes first the side vector A B normal A B overrightarrow AB and then 2 3 2 3 2 3 of the median vector B E normal B E overrightarrow BE green in the picture A B 2 3 B E A B 2 3 1 2 A B A C A B 1 3 A B A C normal A B 2 3 normal B E normal A B normal 2 3 1 2 normal A B normal A C normal A B 1 3 normal A B normal A C displaystyle overrightarrow AB frac 2 3 overrightarrow BE overrightarrow AB frac 2 3 cdot

    Original URL path: http://www.planetmath.org/commonpointoftrianglemedians (2016-04-25)
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  • comparison of Pythagorean means | planetmath.org
    the geometric mean g u v g u v g u v the arithmetic mean a u v a u v a u v and the contraharmonic mean c u v c u v c u v obey the order u h u v g u v a u v c u v v u h u v g u v a u v c u v v displaystyle u leq h u v leq g u v leq a u v leq c u v leq v 1 The part u h u v g u v a u v v u h u v g u v a u v v displaystyle u leq h u v leq g u v leq a u v leq v 2 of 1 was known already by the ancient Babylonians Therefore it may be called the Babylonian inequality chain Horst Hischer The below diagram plots the means h x 1 h x 1 h x 1 in black g x 1 g x 1 g x 1 in blue a x 1 a x 1 a x 1 in cyan and c x 1 c x 1 c x 1 in green for 0 x 1 0 x 1 0 leq x leq 1 psaxes Dx 10 Dy 10 0 0 0 5 0 5 5 7 5 7 0 0 0 1 1 1 1 1 1 1 2 1 2 frac 1 2 x x x y y y psplot linecolor green 05x x mul 25 add x 5 add div psplot linecolor red 05x x mul 25 add 2 div sqrt psplot linecolor cyan 05x 5 add 2 div psplot linecolor blue 055 x mul sqrt psplot linecolor black 0510 x mul x 5 add div psplot linecolor

    Original URL path: http://www.planetmath.org/comparisonofpythagoreanmeans (2016-04-25)
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  • compass and straightedge construction of parallel line | planetmath.org
    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 3A01A20 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 3A01A20 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 3A01A17 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 3A01A17 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 compass and straightedge construction of parallel line Task Construct the line parallel to a given

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