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  • cyclometric functions | planetmath.org
    tan y x y x tan y x and π 2 y π 2 π 2 y π 2 frac pi 2 y frac pi 2 defined in the whole ℝ ℝ mathbb R arccot x arccot x arccot x is the angle y y y satisfying cot y x y x cot y x and 0 y π 0 y π 0 y pi defined in the whole ℝ ℝ mathbb R Those functions are denoted also sin 1 x superscript 1 x sin 1 x cos 1 x superscript 1 x cos 1 x tan 1 x superscript 1 x tan 1 x and cot 1 x superscript 1 x cot 1 x We here use these notations temporarily for giving the corresponding multivalued functions n 0 1 2 n 0 plus or minus 1 plus or minus 2 normal n 0 pm 1 pm 2 sin 1 x n π 1 n arcsin x superscript 1 x n π superscript 1 n x sin 1 x n pi 1 n arcsin x cos 1 x 2 n π arccos x superscript 1 x plus or minus 2 n π x cos 1 x 2n pi pm arccos x tan 1 x n π arctan x superscript 1 x n π x tan 1 x n pi arctan x cot 1 x n π arccot x superscript 1 x n π arccot x cot 1 x n pi arccot x Some formulae arcsin x arccos x π 2 x x π 2 arcsin x arccos x frac pi 2 arctan x arccot x π 2 x arccot x π 2 arctan x arccot x frac pi 2 arcsin x 0 x d t 1 t 2 d t x superscript subscript 0 x d t 1 superscript t 2 d t arcsin x int 0 x frac dt sqrt 1 t 2 dt arctan x 0 x d t 1 t 2 d t x superscript subscript 0 x d t 1 superscript t 2 d t arctan x int 0 x frac dt 1 t 2 dt arcsin x x 1 2 x 3 3 1 3 2 4 x 5 5 1 3 5 2 4 6 x 7 7 x 1 fragments x x 1 2 normal superscript x 3 3 normal 1 3 normal 2 4 normal superscript x 5 5 normal 1 3 5 normal 2 4 6 normal superscript x 7 7 normal fragments normal normal x normal 1 normal arcsin x x frac 1 2 cdot frac x 3 3 frac 1 cdot 3 2 cdot 4 cdot frac x 5 5 frac 1 cdot 3 cdot 5 2 cdot 4 cdot 6 cdot frac x 7 7 ldots quad x leqq 1 arctan x x x 3 3 x 5 5 x 7 7 x 1 fragments x x superscript x 3 3 superscript x 5 5 superscript x 7 7 normal fragments normal normal x normal 1 normal arctan

    Original URL path: http://www.planetmath.org/cyclometricfunctions (2016-04-25)
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  • d'Alembert and D. Bernoulli solutions of wave equation | planetmath.org
    s assign u x t 1 2 f x c t f x c t 1 2 c superscript subscript x c t x c t g s d s displaystyle u x t frac 1 2 f x ct f x ct frac 1 2c int x ct x ct g s ds 1 of the wave equation in one dimension in the special case when the other initial condition is u t x 0 g x 0 assign subscript superscript u normal t x 0 g x 0 displaystyle u prime t x 0 g x equiv 0 2 We shall see that the solution is equivalent with the solution of D Bernoulli We expand the given function f f f to the Fourier sine series on the interval 0 p 0 p 0 p f y n 1 A n sin n π y p with A n 2 p 0 p f x sin n π x p d x n 1 2 fragments f fragments normal y normal superscript subscript n 1 subscript A n n π y p with subscript A n 2 p superscript subscript 0 p f fragments normal x normal n π x p d x fragments normal n 1 normal 2 normal normal normal f y sum n 1 infty A n sin frac n pi y p quad mbox with A n frac 2 p int 0 p f x sin frac n pi x p dx quad n 1 2 ldots Thus we may write f x c t n 1 A n sin n π x p n π c t p n 1 A n sin n π x p cos n π c t p cos n π x p sin n π c t p f x c t n 1 A n sin n π x p n π c t p n 1 A n sin n π x p cos n π c t p cos n π x p sin n π c t p f x ct n1 Ansin nπxp nπctp n1 An sin nπxpcos nπctp cos nπxpsin nπctp f x ct n1 Ansin nπxp nπctp n1 An sin nπxpcos nπctp cos nπxpsin nπctp displaystyle begin cases f x ct sum n 1 infty A n sin left frac n pi x p frac n pi ct p right sum n 1 infty A n left sin frac n pi x p cos frac n pi ct p cos frac n pi x p sin frac n pi ct p right f x ct sum n 1 infty A n sin left frac n pi x p frac n pi ct p right sum n 1 infty A n left sin frac n pi x p cos frac n pi ct p cos frac n pi x p sin frac n pi ct p right end cases Adding these equations and dividing by 2 yield u x t 1

    Original URL path: http://www.planetmath.org/dalembertanddbernoullisolutionsofwaveequation (2016-04-25)
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  • d'Alembert's equation | planetmath.org
    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 3A35L05 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 3A35L05 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 d Alembert s equation The first order differential equation y φ d y d x x ψ d y d x y normal φ d y d x x ψ d y d x y varphi frac dy dx cdot x psi frac dy dx is called d Alembert s differential equation here φ φ varphi and ψ ψ psi mean some known differentiable real functions If we denote d y d x p assign d y d x p frac dy dx p the equation is y φ p x ψ p y normal φ p x ψ p y varphi p cdot x psi p We take p p p as a new variable and derive the equation with respect to p p p getting p φ p x φ p ψ p d p d x p φ p x superscript φ normal p superscript ψ normal p d p d x p varphi p x varphi prime p psi prime p frac dp dx If the equation p φ p 0 p φ p 0 p varphi p 0 has the roots p p 1 p subscript p 1 p p 1 p 2 subscript p 2 p 2 p k subscript p k p k then we have d p ν d x 0 d subscript p ν d x 0 frac dp nu dx 0 for all ν ν nu s and therefore there are the special solutions y p ν

    Original URL path: http://www.planetmath.org/dalembertsequation (2016-04-25)
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  • decimal expansion | planetmath.org
    3 ν 10 1 ν 1 10 2 ν 2 10 3 ν 3 formulae sequence ν subscript ν 1 subscript ν 2 subscript ν 3 normal ν superscript 10 1 subscript ν 1 superscript 10 2 subscript ν 2 superscript 10 3 subscript ν 3 normal displaystyle nu nu 1 nu 2 nu 3 ldots nu 10 1 nu 1 10 2 nu 2 10 3 nu 3 ldots 1 where ν m n ν m n nu lfloor frac m n rfloor is the integer part of m n m n frac m n and the integers ν j subscript ν j nu j are the remainders of 10 j m n normal superscript 10 j m n lfloor 10 j cdot frac m n rfloor when divided by 10 thus 0 ν j 10 0 subscript ν j 10 0 leqq nu j 10 We may suppose that m m m and n n n are coprime if necessary reduce the fraction Then the length l l l of the period depends only on the denominator n n n In the case that gcd n 10 1 n 10 1 gcd n 10 1 the period length is the least positive integer l l l such that 10 l 1 mod n superscript 10 l annotated 1 pmod n 10 l equiv 1 mathop rm mod n the period length does not change if we multiply the fraction by a suitable power of 10 and then reduce all prime factors of 10 from the denominator In every case the period length is a factor of the number φ n φ n varphi n where φ φ varphi is Euler s totient function Examples 1 8 0 125000 0 124999 1 8 0 125000 normal 0 124999 normal frac 1 8 0 125000 ldots 0 124999 ldots one digit periods N B two possibilities 1 12 0 08333 1 12 0 08333 normal frac 1 12 0 08333 ldots one digit per 1 37 0 027 027 027 1 37 superscript 0 normal superscript 027 normal superscript 027 normal superscript 027 normal normal frac 1 37 0 prime 027 prime 027 prime 027 prime ldots three digit per 1 82 0 0 12195 12195 12195 1 82 superscript 0 0 normal superscript 12195 normal superscript 12195 normal superscript 12195 normal normal frac 1 82 0 0 prime 12195 prime 12195 prime 12195 prime ldots five digit per 1 25351 0 000039446 1 25351 0 000039446 normal frac 1 25351 0 000039446 ldots hundred digit per The tail of infinitely many 0 s as in 0 125000 is of course usually not written out Such a tail is possible only when n n n has no other prime factors except prime factors of the base of the digit system in question If the tails of 0 s are not accepted then the digital expansion of every positive rational is unique then e g 0 124999 is the only expansion

    Original URL path: http://www.planetmath.org/decimalexpansion (2016-04-25)
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  • decomposable curve | planetmath.org
    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 3A11A99 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 decomposable curve An algebraic curve f x y 0 f x y 0 f x y 0 is decomposable if the polynomial f x y f x y f x y is reducible in ℝ x y ℝ x y mathbb R x y that is if there are polynomials g x y g x y g x y and h x y h x y h x y with positive degree in ℝ x y ℝ x y mathbb R x y such that f x y g x y h x y f x y g x y h x y f x y g x y h x y Example The quadratic curve x 2 a 2 y 2 b 2 0 superscript x 2 superscript a 2 superscript y 2 superscript b 2 0 displaystyle frac x 2 a 2 frac y 2 b 2 0 1 is decomposable since the equation may be written x a y b x a y b 0 x a y b x a y b 0 left frac x a frac y b right left frac x a frac y b right 0 or equivalently x a y b 0 x a y b

    Original URL path: http://www.planetmath.org/decomposablecurve (2016-04-25)
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  • definition of prime ideal by Artin | planetmath.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 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 definition of prime ideal by Artin Lemma Let R R R be a commutative ring and S S S a multiplicative semigroup consisting of a subset of R R R If there exist ideals of R R R which are disjoint with S S S then the set mathfrak S of

    Original URL path: http://www.planetmath.org/definitionofprimeidealbyartin (2016-04-25)
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  • degree of algebraic number | planetmath.org
    2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A06A06 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 degree of algebraic number Theorem The degree of any algebraic number α α alpha in the number field ℚ ϑ ℚ ϑ mathbb Q vartheta divides the degree of ϑ ϑ vartheta The zeroes of the characteristic polynomial g x g x g x of α α alpha consist of the algebraic conjugates of α α alpha each of which having equal multiplicity as zero of g x g x g x Proof Let the minimal polynomial of α α alpha be a x x k a 1 x k 1 a k assign a x superscript x k subscript a 1 superscript x k 1 normal subscript a k a x x k a 1 x k 1 ldots a k and all zeroes of this be α 1 α α 2 α k subscript α 1 α subscript α 2 normal subscript α k alpha 1 alpha alpha 2 ldots alpha k Denote the canonical polynomial of α α alpha with respect to the primitive element ϑ ϑ vartheta by r x r x r x then a r ϑ a α 0 a r ϑ a α 0 a r vartheta a alpha 0 If a r x φ x assign a r x φ x a r x varphi x then the equation φ x 0 φ x 0 varphi x 0 has rational coefficients and is satisfied by ϑ ϑ vartheta Since the minimal polynomial f x f x f x of ϑ ϑ vartheta is irreducible it must divide φ x φ x varphi x and all algebraic conjugates 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 of ϑ ϑ vartheta make φ x φ x varphi x zero Hence we have a α i a r ϑ i 0 for i 1 2 n formulae sequence a superscript α i a r subscript ϑ i 0 for i 1 2 normal n a alpha i a r vartheta i 0 quad mbox for quad i 1 2 ldots n where the numbers α i superscript α i alpha i are the ℚ ϑ ℚ ϑ mathbb Q vartheta conjugates of α α alpha Thus these ℚ ϑ ℚ ϑ mathbb Q vartheta conjugates are roots of the irreducible equation a x 0 a x 0 a x 0 whence a x a x a x must divide the characteristic polynomial g x g x g x Let the power a x m superscript a x m a x m exactly divide g x g x g x when g

    Original URL path: http://www.planetmath.org/degreeofalgebraicnumber (2016-04-25)
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  • delay theorem | 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 3A12F05 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 3A12F05 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 3A12E05 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

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