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  • fractional ideal of commutative ring | planetmath.org
    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 fractional ideal of commutative ring Definition Let R R R be a commutative ring having a regular element and let T T T be the total ring of fractions of R R R An R R R submodule 𝔞 𝔞 mathfrak a of T T T is called fractional ideal of R R R provided that there exists a regular element d d d of R R R such that 𝔞 d R 𝔞 d R mathfrak a d subseteq R If a fractional ideal is contained in R R R it is a usual ideal of R R R and we can call it an integral ideal of R R R Note that a fractional ideal of R R R is not necessarily a subring of T T T The set of all fractional ideals of R R R form under the multiplication an commutative semigroup with identity element R R ℤ e superscript R normal R ℤ e R prime R mathbb Z e where e e e is the unity of T T T An ideal 𝔞 𝔞 mathfrak a integral or fractional of R R R is called invertible if there exists another ideal 𝔞 1 superscript 𝔞 1 mathfrak a 1 of R R R such that 𝔞 𝔞 1 R 𝔞 superscript 𝔞 1 superscript R normal mathfrak aa 1 R prime It is not hard to show that any invertible ideal 𝔞 𝔞 mathfrak a is finitely generated and regular moreover that the inverse ideal 𝔞 1 superscript 𝔞 1 mathfrak a 1 is uniquely determined see the entry invertible ideal is finitely generated

    Original URL path: http://www.planetmath.org/fractionalidealofcommutativering (2016-04-25)
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  • Fresnel formulas | planetmath.org
    R 2 cos 2 φ i sin 2 φ e i φ d φ R 0 π 4 e R 2 cos 2 φ d φ subscript I 2 i R superscript subscript 0 π 4 superscript e superscript R 2 2 φ i 2 φ superscript e i φ d φ R superscript subscript 0 π 4 normal superscript e superscript R 2 2 φ i 2 φ superscript e i φ d φ R superscript subscript 0 π 4 superscript e superscript R 2 2 φ d φ I 2 left iR int 0 frac pi 4 e R 2 cos 2 varphi i sin 2 varphi e i varphi d varphi right leqq R int 0 frac pi 4 left e R 2 cos 2 varphi i sin 2 varphi right cdot left e i varphi right cdot d varphi R int 0 frac pi 4 e R 2 cos 2 varphi d varphi Comparing the graph of the function φ cos 2 φ maps to φ 2 φ varphi mapsto cos 2 varphi with the line through the points 0 1 0 1 0 1 and π 4 0 π 4 0 frac pi 4 0 allows us to estimate cos 2 φ 2 φ cos 2 varphi downwards cos 2 φ 1 4 φ π for 0 φ π 4 formulae sequence 2 φ 1 4 φ π for 0 φ π 4 cos 2 varphi geqq 1 frac 4 varphi pi quad mbox for quad 0 leqq varphi leqq frac pi 4 Hence we obtain I 2 R 0 π 4 d φ e R 2 cos 2 φ R 0 π 4 d φ e R 2 1 4 φ π R e R 2 0 π 4 e 4 R 2 π φ d φ subscript I 2 R superscript subscript 0 π 4 d φ superscript e superscript R 2 2 φ R superscript subscript 0 π 4 d φ superscript e superscript R 2 1 4 φ π R superscript e superscript R 2 superscript subscript 0 π 4 superscript e 4 superscript R 2 π φ d φ I 2 leqq R int 0 frac pi 4 frac d varphi e R 2 cos 2 varphi leqq R int 0 frac pi 4 frac d varphi e R 2 1 frac 4 varphi pi leqq frac R e R 2 int 0 frac pi 4 e frac 4R 2 pi varphi d varphi and moreover I 2 π 4 R e R 2 e R 2 1 π e R 2 4 R e R 2 π 4 R 0 as R formulae sequence subscript I 2 π 4 R superscript e superscript R 2 superscript e superscript R 2 1 π superscript e superscript R 2 4 R superscript e superscript R 2 π 4 R normal 0 normal as R I 2 leqq frac pi 4Re R 2 e R 2 1 frac pi e R 2

    Original URL path: http://www.planetmath.org/fresnelformulas (2016-04-25)
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  • Fresnel integrals | planetmath.org
    3 3 1 z 7 7 3 z 11 11 5 z 15 15 7 fragments S fragments normal z normal superscript z 3 normal 3 1 superscript z 7 normal 7 3 superscript z 11 normal 11 5 superscript z 15 normal 15 7 normal S z frac z 3 3 cdot 1 frac z 7 7 cdot 3 frac z 11 11 cdot 5 frac z 15 15 cdot 7 ldots These converge for all complex values z z z and thus define entire transcendental functions The Fresnel integrals at infinity have the finite value lim x C x lim x S x 2 π 4 subscript normal x C x subscript normal x S x 2 π 4 lim x to infty C x lim x to infty S x frac sqrt 2 pi 4 0 2 Clothoid The parametric presentation x C t y S t formulae sequence x C t y S t displaystyle x C t quad quad y S t 2 represents a curve called clothoid Since the equations 2 both define odd functions the clothoid has symmetry about the origin The curve has the shape of a similar to sim see this diagram The arc length of the clothoid from the origin to the point C t S t C t S t C t S t is simply 0 t C u 2 S u 2 d u 0 t cos 2 u 2 sin 2 u 2 d u 0 t d u t superscript subscript 0 t superscript C normal superscript u 2 superscript S normal superscript u 2 d u superscript subscript 0 t superscript 2 superscript u 2 superscript 2 superscript u 2 d u superscript subscript 0 t d u t int 0 t sqrt C prime u 2 S prime u 2 du int 0 t sqrt cos 2 u 2 sin 2 u 2 du int 0 t du t Thus the length of the whole curve to the point 2 π 4 2 π 4 2 π 4 2 π 4 frac sqrt 2 pi 4 frac sqrt 2 pi 4 is infinite The curvature of the clothoid also is extremely simple ϰ 2 t ϰ 2 t varkappa 2t i e proportional to the arc lenth thus in the origin only the curvature is zero Conversely if the curvature of a plane curve varies proportionally to the arc length the curve is a clothoid This property of the curvature of clothoid is utilised in way and railway construction since the form of the clothoid is very efficient when a straight portion of way must be bent to a turn the zero curvature of the line can be continuously raised to the wished curvature Defines clothoid Keywords Fresnel integral Related SineIntegral Major Section Reference Type of Math Object Definition Parent complex function Groups audience Buddy List of pahio Mathematics Subject Classification 30B10 no label found 26A42 no label found 33B20 no label found

    Original URL path: http://www.planetmath.org/fresnelintegrals (2016-04-25)
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  • Frobenius method | planetmath.org
    x 0 confining us to the case of a second order differential equation When we use the quotient forms x x 0 c 1 x p x r x x x 0 2 c 2 x q x r x fragments fragments normal x subscript x 0 normal subscript c 1 fragments normal x normal assign p x r x normal superscript fragments normal x subscript x 0 normal 2 subscript c 2 fragments normal x normal normal q x r x normal x x 0 c 1 x frac p x r x quad x x 0 2 c 2 x frac q x r x where r x r x r x p x p x p x and q x q x q x are analytic in a neighbourhood of x 0 subscript x 0 x 0 and r x 0 r x 0 r x neq 0 our differential equation reads x x 0 2 r x y x x x 0 p x y x q x y x 0 superscript x subscript x 0 2 r x superscript y x x subscript x 0 p x superscript y normal x q x y x 0 displaystyle x x 0 2 r x y prime prime x x x 0 p x y prime x q x y x 0 1 Since a simple change x x 0 x maps to x subscript x 0 x x x 0 mapsto x of variable brings to the case that the singular point is the origin we may suppose such a starting situation Thus we can study the equation x 2 r x y x x p x y x q x y x 0 superscript x 2 r x superscript y x x p x superscript y normal x q x y x 0 displaystyle x 2 r x y prime prime x xp x y prime x q x y x 0 2 where the coefficients have the converging power series expansions r x n 0 r n x n p x n 0 p n x n q x n 0 q n x n formulae sequence r x superscript subscript n 0 subscript r n superscript x n formulae sequence p x superscript subscript n 0 subscript p n superscript x n q x superscript subscript n 0 subscript q n superscript x n displaystyle r x sum n 0 infty r n x n quad p x sum n 0 infty p n x n quad q x sum n 0 infty q n x n 3 and r 0 0 subscript r 0 0 r 0 neq 0 In the Frobenius method one examines whether the equation 2 allows a series solution of the form y x x s n 0 a n x n a 0 x s a 1 x s 1 a 2 x s 2 y x superscript x s superscript subscript n 0 subscript a n

    Original URL path: http://www.planetmath.org/frobeniusmethod (2016-04-25)
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  • Frobenius product | 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 3A34A05 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 3A34A05 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 Frobenius product If A a i j A subscript a i j

    Original URL path: http://www.planetmath.org/frobeniusproduct (2016-04-25)
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  • function $x^x$ | planetmath.org
    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 3A15A63 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 function x x superscript x x x x We list some properties of the real function x x x maps to x superscript x x x mapsto x x which may be called 2 tetration Cf also 1 The function is defined only for positive values of x x x and for negative integers x x x which we neglect in the following items cf fraction power and power function x x superscript x x x x can be composed of e t superscript e t e t and t x ln x t x x t x ln x lim x 0 x x 1 subscript normal x limit from 0 superscript x x 1 displaystyle lim x to 0 x x 1 as one sees by substituting t x ln x t x x t x ln x into the Taylor expansion of e t superscript e t e t x x superscript x x x x is differentiable Using the chain rule on e x ln x superscript e x x e x ln x one obtains d d x x x x x 1 ln x d d x superscript x x superscript x x 1 x frac d dx x x x x 1 ln x The function has absolute minimum value e 1 e 0 6922 superscript e 1 e 0 6922 displaystyle e frac 1 e approx 0 6922 and achieves all real values above this lim x e x x x lim x e 1 ln x x 0 subscript normal x superscript e x superscript x x subscript normal x superscript e 1 x x 0 displaystyle lim x to infty frac e x x x lim x to infty e 1 ln x x 0 cf growth of exponential function x x superscript x x x x has no elementary function as antiderivative 0 1 x x d x n 1 1 n 1 n n 0 7834305107 superscript subscript 0 1 superscript x x d x superscript subscript n 1 superscript 1 n

    Original URL path: http://www.planetmath.org/functionxx (2016-04-25)
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  • function of not bounded variation | planetmath.org
    of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit function of not bounded variation Example We show that the function f x normal f maps to x absent displaystyle f x mapsto x cos π x when x 0 0 when x 0 xcosπx whenx00 whenx0 displaystyle left begin array ll x cos frac pi x mbox when x neq 0 0 mbox when x 0 end array right which is continuous in the whole ℝ ℝ mathbb R is not of bounded variation on any interval containing the zero Let us take e g the interval 0 a 0 a 0 a Chose a positive integer m m m such that 1 m a 1 m a frac 1 m a and the partition of the interval with the points 1 m 1 m 1 1 m 2 1 n 1 m 1 m 1 1 m 2 normal 1 n frac 1 m frac 1 m 1 frac 1 m 2 ldots frac 1 n into the subintervals 0 1 n 1 n 1 n 1 1 m 1 1 m 1 m a 0 1 n 1 n 1 n 1 normal 1 m 1 1 m 1 m a 0 frac 1 n frac 1 n frac 1 n 1 ldots frac 1 m 1 frac 1 m frac 1 m a For each positive integer ν ν nu we have see this f 1 ν 1 ν cos ν π 1 ν ν f 1 ν 1 ν ν π superscript 1 ν ν f left frac 1 nu right frac 1 nu cos nu pi frac 1 nu nu Thus we see that the total variation of f f f in all partitions of

    Original URL path: http://www.planetmath.org/functionofnotboundedvariation (2016-04-25)
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  • fundamental theorem of ideal theory | planetmath.org
    varrho 1 ldots varrho m be a complete residue system modulo α α alpha cf congruence in algebraic number field Then α i α λ i ϱ n i i 1 r fragments subscript α i α subscript λ i subscript ϱ subscript n i italic fragments normal i 1 normal normal normal r normal alpha i alpha lambda i varrho n i qquad i 1 ldots r where the numbers λ i subscript λ i lambda i belong to mathcal O Since we have α 1 α r α α λ 1 ϱ n 1 α λ r ϱ n r α ϱ n 1 ϱ n r α subscript α 1 normal subscript α r α α subscript λ 1 subscript ϱ subscript n 1 normal α subscript λ r subscript ϱ subscript n r α subscript ϱ subscript n 1 normal subscript ϱ subscript n r α mathfrak a alpha 1 ldots alpha r alpha alpha lambda 1 varrho n 1 ldots alpha lambda r varrho n r alpha varrho n 1 ldots varrho n r alpha there can be different ideals mathfrak a only a finite number at most 1 m n 2 m m 2 m 1 m binomial n 2 normal binomial m m superscript 2 m 1 m n choose 2 ldots m choose m 2 m Lemma 3 Each ideal mathfrak a of mathcal O has only a finite number of ideal factors Proof If fragments c normal a mathfrak c mid a and α α alpha in mathfrak a then by Lemma 1 α α alpha in mathfrak c whence Lemma 2 implies that there is only a finite number of such factors mathfrak c Lemma 4 All nonzero ideals of mathcal O are cancellative i e if mathfrak ac ad then mathfrak c d Proof The theorem of Steinitz 1911 guarantees an ideal mathfrak g of mathcal O such that the product mathfrak ga is a principal ideal ω ω omega Then we may write ω ω ω ω omega mathfrak c mathfrak ga c mathfrak g ac mathfrak g ad mathfrak ga d omega mathfrak d If γ 1 γ s subscript γ 1 normal subscript γ s mathfrak c gamma 1 ldots gamma s and δ 1 δ t subscript δ 1 normal subscript δ t mathfrak d delta 1 ldots delta t we thus have the equation ω γ 1 ω γ s ω δ 1 ω δ t ω subscript γ 1 normal ω subscript γ s ω subscript δ 1 normal ω subscript δ t omega gamma 1 ldots omega gamma s omega delta 1 ldots omega delta t by which there must exist the elements λ i 1 λ i t subscript λ i 1 normal subscript λ i t lambda i1 ldots lambda it of mathcal O such that ω γ i λ i 1 ω δ 1 λ i t ω δ t ω subscript γ i subscript λ i 1 ω

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