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  • continuity of natural power | planetmath.org
    3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A11R99 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 3A11R99 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 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 3A13F30 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 continuity of natural power Theorem Let n n n be arbitrary positive integer The power function x x n maps to x superscript x n x mapsto x n from ℝ ℝ mathbb R to ℝ ℝ

    Original URL path: http://www.planetmath.org/continuityofnaturalpower (2016-04-25)
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  • continuity of sine and cosine | 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 3A26A15 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 3A26A15 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 continuity of sine and cosine Theorem The real functions x sin x maps to x x x mapsto sin x and x cos x maps to x x x mapsto cos x are continuous at every real number x x x Proof Let ε ε varepsilon be an arbitrary positive number Denote Δ sin x sin z sin x fragments Δ x

    Original URL path: http://www.planetmath.org/continuityofsineandcosine (2016-04-25)
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  • continuous derivative implies bounded variation | planetmath.org
    continuous function f superscript f normal f prime has its greatest value M M M on the closed interval a b a b a b i e f x M x a b formulae sequence superscript f normal x M for all x a b f prime x leqq M quad forall x in a b Let D D D be an arbitrary partition of a b a b a b with the points x 0 a x 1 x 2 x n 1 b x n subscript x 0 a subscript x 1 subscript x 2 normal subscript x n 1 b subscript x n x 0 a x 1 x 2 ldots x n 1 b x n Consider f f f on a subinterval x i 1 x i subscript x i 1 subscript x i x i 1 x i By the mean value theorem there exists on this subinterval a point ξ i subscript ξ i xi i such that f x i f x i 1 f ξ i x i x i 1 f subscript x i f subscript x i 1 superscript f normal subscript ξ i subscript x i subscript x i 1 f x i f x i 1 f prime xi i x i x i 1 Then we get S D i 1 n f x i f x i 1 i 1 n f ξ i x i x i 1 M i 1 n x i x i 1 M b a assign subscript S D superscript subscript i 1 n f subscript x i f subscript x i 1 superscript subscript i 1 n superscript f normal subscript ξ i subscript x i subscript x i 1 M superscript subscript i 1 n subscript x i subscript x i 1 M b a S D sum i 1 n f x i f x i 1 sum i 1 n f prime xi i x i x i 1 leqq M sum i 1 n x i x i 1 M b a Thus the total variation satisfies sup D all S D s M b a subscript supremum D all subscript S D s M b a sup D mbox all S D mbox s leqq M b a infty whence f f f is of bounded variation on the interval a b a b a b 2 o superscript 2 normal o 2 underline o Define the functions G G G and H H H by setting G f f 2 H f f 2 formulae sequence assign G superscript f normal superscript f normal 2 assign H superscript f normal superscript f normal 2 G frac f prime f prime 2 quad H frac f prime f prime 2 We see that these are non negative and that f G H superscript f normal G H f prime G H Define then the functions g g g and h h h on

    Original URL path: http://www.planetmath.org/continuousderivativeimpliesboundedvariation (2016-04-25)
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  • contractive sequence | planetmath.org
    subscript a m subscript a m 1 d subscript a m 1 subscript a m δ displaystyle leqq d a m a m 1 d a m 1 a m delta d a m a m 1 d a m 1 a m 2 d a m 2 a m δ absent d subscript a m subscript a m 1 d subscript a m 1 subscript a m 2 d subscript a m 2 subscript a m δ displaystyle leqq d a m a m 1 d a m 1 a m 2 d a m 2 a m delta normal displaystyle ldots d a m a m 1 d a m 1 a m 2 d a m 2 a m 3 d a n 1 a n absent d subscript a m subscript a m 1 d subscript a m 1 subscript a m 2 d subscript a m 2 subscript a m 3 normal d subscript a n 1 subscript a n displaystyle leqq d a m a m 1 d a m 1 a m 2 d a m 2 a m 3 ldots d a n 1 a n par Now the contractiveness gives the inequalities d a 1 a 2 r d a 0 a 1 d subscript a 1 subscript a 2 r d subscript a 0 subscript a 1 d a 1 a 2 leqq rd a 0 a 1 d a 2 a 3 r d a 1 a 2 r 2 d a 0 a 1 d subscript a 2 subscript a 3 r d subscript a 1 subscript a 2 superscript r 2 d subscript a 0 subscript a 1 d a 2 a 3 leqq rd a 1 a 2 leqq r 2 d a 0 a 1 d a 3 a 4 r d a 2 a 3 r 3 d a 0 a 1 d subscript a 3 subscript a 4 r d subscript a 2 subscript a 3 superscript r 3 d subscript a 0 subscript a 1 d a 3 a 4 leqq rd a 2 a 3 leqq r 3 d a 0 a 1 normal ldots d a m a m 1 r m d a 0 a 1 d subscript a m subscript a m 1 superscript r m d subscript a 0 subscript a 1 d a m a m 1 leqq r m d a 0 a 1 normal ldots d a n 1 a n r n 1 d a 0 a 1 d subscript a n 1 subscript a n superscript r n 1 d subscript a 0 subscript a 1 d a n 1 a n leqq r n 1 d a 0 a 1 by which we obtain the estimation d a m a n d subscript a m subscript a n displaystyle d a m a n d a 0 a 1 r m r m 1 r m δ 1 absent d subscript

    Original URL path: http://www.planetmath.org/contractivesequence (2016-04-25)
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  • contradictory statement | planetmath.org
    because of the meaning of the terms employed In propositional logic a contradictory statement a k a contradiction is a statement which is false regardless of the truth values of the substatements which form it According to G Peano one may generally denote a contradiction with the symbol curlywedge For a simple example the statement P P P P P wedge lnot P is a contradiction for any statement P P P The negation Q Q lnot Q of every contradiction Q Q Q is a tautology and vice versa fragments normal lnot curlywedge curlyvee lnot curlyvee curlywedge To test a given statement or form to see if it is a contradiction one may construct its truth table If it turns out that every value of the last column is F then the statement is a contradiction Cf the entry contradiction Keywords false Related Tautology LogicalConnective Contradiction Synonym contradiction Type of Math Object Definition Major Section Reference Groups audience World Writable Buddy List of pahio Mathematics Subject Classification 03B05 no label found Add a correction Attach a problem Ask a question Search form Search Home Articles Questions Forums Planetary Bugs HS Secondary University Tertiary Graduate Advanced Industry Practice Research Topics

    Original URL path: http://www.planetmath.org/contradictorystatement (2016-04-25)
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  • contrageometric proportion | planetmath.org
    m x y m x y frac m x y m frac x y defining the harmonic mean of x x x and y y y into the proportion equation m x y m y x m x y m y x frac m x y m frac y x defining their contraharmonic mean one also may convert the proportion equation m x y m m y m x y m m y frac m x y m frac m y defining the geometric mean into a new equation m x y m y m m x y m y m displaystyle frac m x y m frac y m 1 defining the contrageometric mean m m m of x x x and y y y Thus the three positive numbers x x x m m m y y y satisfying 1 are in contrageometric proportion One integer example is 1 4 6 1 4 6 1 4 6 Solving m m m from 1 one gets the expression m x y x y 2 4 y 2 2 f x y fragments m x y superscript x y 2 4 superscript y 2 2 normal f fragments normal x normal y normal normal displaystyle m frac x y sqrt x y 2 4y 2 2 f x y 2 Suppose now that 0 x y 0 x y 0 leq x leq y Using 2 we see that m x y 0 2 4 y 2 2 x y 2 x m x y superscript 0 2 4 superscript y 2 2 x y 2 x m geq frac x y sqrt 0 2 4y 2 2 frac x y 2 geq x y 2 m 2 y x 2 y x x y 2 4 y 2 2 y x y x 2 4 y 2 y x 2 0 superscript y 2 superscript m 2 superscript y x 2 y x superscript x y 2 4 superscript y 2 2 y x superscript y x 2 4 superscript y 2 y x 2 0 y 2 m 2 frac y x 2 y x sqrt x y 2 4y 2 2 frac y x sqrt y x 2 4y 2 y x 2 geq 0 accordingly x f x y y x f x y y displaystyle x leq f x y leq y 3 Thus the contrageometric mean of x x x and y y y also is at least equal to their arithmetic mean We can also compare m m m with their quadratic mean by watching the difference x 2 y 2 2 2 m 2 y x 1 2 y x 2 4 y 2 y 0 superscript superscript x 2 superscript y 2 2 2 superscript m 2 y x 1 2 superscript y x 2 4 superscript y 2 y 0 left sqrt frac x 2 y 2 2 right 2 m 2 y x left frac 1 2 sqrt y

    Original URL path: http://www.planetmath.org/contrageometricproportion (2016-04-25)
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  • contraharmonic Diophantine equation | planetmath.org
    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 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 contraharmonic

    Original URL path: http://www.planetmath.org/contraharmonicdiophantineequation (2016-04-25)
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  • contraharmonic means and Pythagorean hypotenuses | planetmath.org
    contraharmonic mean c u 2 v 2 u v c superscript u 2 superscript v 2 u v c frac u 2 v 2 u v of the positive integers u u u and v v v with u v u v u v Then u v u 2 v 2 u v 2 2 u v fragments u v normal superscript u 2 superscript v 2 superscript fragments normal u v normal 2 2 u v u v mid u 2 v 2 u v 2 2uv whence u v 2 u v fragments u v normal 2 u v normal u v mid 2uv and we have the positive integers a u v u 2 v 2 u v b 2 u v u v fragments a normal u v superscript u 2 superscript v 2 u v normal b normal 2 u v u v a u v frac u 2 v 2 u v quad b frac 2uv u v satisfying a 2 b 2 u 2 v 2 2 2 u v 2 u v 2 u 4 2 u 2 v 2 v 4 4 u 2 v 2 u v 2 u 4 2 u 2 v 2 v 4 u v 2 u 2 v 2 2 u v 2 c 2 superscript a 2 superscript b 2 superscript superscript u 2 superscript v 2 2 superscript 2 u v 2 superscript u v 2 superscript u 4 2 superscript u 2 superscript v 2 superscript v 4 4 superscript u 2 superscript v 2 superscript u v 2 superscript u 4 2 superscript u 2 superscript v 2 superscript v 4 superscript u v 2 superscript superscript u 2 superscript v 2 2 superscript u v 2 superscript c 2 a 2 b 2 frac u 2 v 2 2 2uv 2 u v 2 frac u 4 2u 2 v 2 v 4 4u 2 v 2 u v 2 frac u 4 2u 2 v 2 v 4 u v 2 frac u 2 v 2 2 u v 2 c 2 2 superscript 2 2 circ Suppose that c c c is the hypotenuse of the Pythagorean triple a b c a b c a b c whence c 2 a 2 b 2 superscript c 2 superscript a 2 superscript b 2 c 2 a 2 b 2 Let us consider the rational numbers u c b a 2 v c b a 2 fragments u normal c b a 2 normal v normal c b a 2 normal displaystyle u frac c b a 2 quad v frac c b a 2 1 If the triple is primitive then two of the integers a b c a b c a b c are odd and one of them is even if not then similarly or all of a b c a b c a b c are even Therefore c b a plus or minus c

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