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  • boundedly homogeneous function | planetmath.org
    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 boundedly homogeneous function A function f ℝ n ℝ normal f normal superscript ℝ n ℝ f mathbb R n to mathbb R where n n n is a positive integer is called boundedly homogeneous with respect to a set Λ normal Λ Lambda of positive reals and a real number r r r if the equation f λ x λ r f x f λ normal x superscript λ r f normal x f lambda vec x lambda r f vec x is true for all and x ℝ n normal x superscript ℝ n vec x in mathbb R n and λ Λ λ normal Λ lambda in Lambda Then Λ normal Λ Lambda is the set of homogeneity and r r r the degree of homogeneity of f f f Example The function x x r sin ln x maps to x superscript x r x x mapsto x r sin ln x is boundedly homogeneous with respect to the set Λ e 2 π ν ν ℤ fragments Λ fragments normal superscript e 2 π ν normal ν Z normal Lambda e 2 pi nu vdots nu in mathbb Z and with degree of homogeneity r r r Theorem Let f ℝ ℝ normal f normal subscript ℝ ℝ f mathbb R to mathbb R be a boundedly homogeneous function with the degree of homogeneity r r r and the set of homogeneity Λ 1 1 normal Λ Lambda supset 1 Then f f f is of the form f x x r f 1 ln x f x superscript x r subscript f 1 x displaystyle f x x r f 1 ln x 1 where f 1 ℝ ℝ normal subscript f 1 normal ℝ ℝ f 1 mathbb R to mathbb R is a periodic real function depending on f f f Proof Defining g x f x x r assign g x f x superscript x r g x frac f x x r we obtain g λ x f λ x λ x r λ r f x λ r x r λ 0 g x λ Λ formulae sequence g λ x f λ x superscript λ x r superscript λ r f x superscript λ r superscript x r superscript λ 0 g x for all λ normal Λ g lambda x frac f lambda x lambda x r frac lambda r f x lambda r x r lambda 0 g x quad forall lambda in Lambda Thus g g g is a boundedly homogeneous function with the set of homogeneity Λ normal Λ Lambda and the degree of homogeneity 0 Moreover define f 1 x g e x assign subscript f 1 x g superscript

    Original URL path: http://www.planetmath.org/boundedlyhomogeneousfunction (2016-04-25)
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  • boundedness of terms of power series | planetmath.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 3A15 00 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 3A15 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 boundedness of terms of power series Theorem If the set a 0 a 1 c a 2 c 2 subscript a 0 subscript a 1 c subscript a 2 superscript c 2 normal a 0 a 1 c a 2 c 2 ldots of the terms of a power series n 0 a n z n superscript subscript n 0 subscript a n

    Original URL path: http://www.planetmath.org/boundednessoftermsofpowerseries (2016-04-25)
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  • brachistochrone curve | planetmath.org
    the equations of the brachistochrone curve i e of the curve γ γ gamma along which a point of mass m m m moves under gravitation only from the origin to a given point x f y f subscript x f subscript y f x f y f in the shortest time It is supposed that x f 0 subscript x f 0 x f 0 and y f 0 subscript y f 0 y f 0 On the curve γ γ gamma having an equation y y x y y x y y x with y 0 y 0 y leq 0 the equality 1 2 m v 2 m g y 1 2 m superscript v 2 m g y frac 1 2 mv 2 mgy is in force by the work energy principle Thus the velocity of the mass point on the curve has the expression d s d t v 2 g y d s d t v 2 g y frac ds dt v sqrt 2gy and accordingly d t d s v d s 2 g y d t d s v d s 2 g y dt frac ds v frac ds sqrt 2gy Hence the total time elapsed on γ γ gamma is given by the path integral T 0 0 x f y f d s 2 g y T superscript subscript 0 0 subscript x f subscript y f d s 2 g y T int 0 0 x f y f frac ds sqrt 2gy taken along γ γ gamma restricting to differentiable functions y y x y y x displaystyle y y x 1 this reads T 1 2 g 0 x f 1 y 2 y f y y d x T 1 2 g superscript subscript 0 subscript x f subscript normal 1 superscript y normal 2 y f y superscript y normal d x displaystyle T frac 1 sqrt 2g int 0 x f underbrace frac sqrt 1 y prime 2 sqrt y f y y prime dx 2 The function 1 should be determined such that the integral in 2 would attain its least value Since the integrand f y y f y superscript y normal f y y prime of 2 does not depend explicitly on the variable x x x the Euler Lagrange condition of this variational problem reduces by the Beltrami identity to f y y y f y y y 1 y 2 y y 2 y 1 y 2 1 y 1 y 2 constant f y superscript y normal superscript y normal subscript superscript f normal superscript y normal y superscript y normal 1 superscript y normal 2 y superscript y normal 2 normal y 1 superscript y normal 2 1 normal y 1 superscript y normal 2 constant f y y prime y prime f prime y prime y y prime frac sqrt 1 y prime 2 sqrt y frac y prime 2 sqrt y cdot 1

    Original URL path: http://www.planetmath.org/brachistochronecurve (2016-04-25)
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  • Briggsian logarithms | planetmath.org
    logarithm of a positive number a a a is the logarithm of a a a in the base 10 i e log 10 a subscript 10 a log 10 a nowadays denoted by lg a lg a lg a probably from the Latin logarithmus generalis The term is due to Henry Briggs 1561 1630 Before the electronic calculators and computers the tabulated values of logarithms were used for performing laborious numerical calculations multiplications divisions powers roots E g in the high schools of Finland the use of logarithm tables was teached still in the begin of the 1970s There was several wide tables of Briggsian logarithms e g the well known five place tables of Hoüel and Voellmy Since the logarithms of rational numbers are mostly irrational the logarithms in the tables are in general approximate values Because lg 10 a lg a lg 10 lg a 1 lg a 10 lg a lg 10 lg a 1 formulae sequence lg 10 a lg a lg 10 lg a 1 lg a 10 lg a lg 10 lg a 1 lg 10a lg a lg 10 lg a 1 quad lg frac a 10 lg a lg 10 lg a 1 moving the decimal point one step to the right resp to the left increases resp decreases the Briggsian logarithm by the integer value 1 the decimals of the logarithm do not alter Thus the tables give only the decimals of the logarithms of positive integers For example the table gives for the logarithm of 8322 only the five decimals 92023 Since lg 1 0 lg 1 0 lg 1 0 lg 10 1 lg 10 1 lg 10 1 and the logarithm function is increasing we can infer that lg 8322 0 92023 3 lg 8322 0 92023 3 lg 8322 approx 0 92023 3 lg 832 2 0 92023 2 lg 832 2 0 92023 2 lg 832 2 approx 0 92023 2 lg 83 22 0 92023 1 lg 83 22 0 92023 1 lg 83 22 approx 0 92023 1 lg 8 322 0 92023 lg 8 322 0 92023 lg 8 322 approx 0 92023 lg 0 8322 0 92023 1 lg 0 8322 0 92023 1 lg 0 8322 approx 0 92023 1 lg 0 08322 0 92023 2 lg 0 08322 0 92023 2 lg 0 08322 approx 0 92023 2 lg 0 008322 0 92023 3 lg 0 008322 0 92023 3 lg 0 008322 approx 0 92023 3 When one expresses logarithms of numbers as sum and difference in the way as above the decimal part is called the mantissa and the integer part the characteristic of the logarithm A positive caracteristic is joined to the mantissa e g 3 92023 but a negative characteristic is held apart e g 0 92023 3 0 92023 3 0 92023 3 It s clear that the mantissa of the logarithm of a number does not depend on the position of the decimal point

    Original URL path: http://www.planetmath.org/briggsianlogarithms (2016-04-25)
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  • calculating the solid angle of disc | planetmath.org
    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 3A65A05 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 3A65A05 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 3A01 08 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 3A01 08 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

    Original URL path: http://www.planetmath.org/calculatingthesolidangleofdisc (2016-04-25)
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  • calculation of contour integral | planetmath.org
    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 3A51M25 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 3A51M25 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 calculation of contour integral We will determine the important complex integral I C z z 0 n d z assign I subscript contour integral C superscript z subscript z 0 n d z I oint C z z 0 n dz where C C C is the circumference of the circle z z 0 ϱ z subscript z 0 ϱ z z 0 varrho taken anticlockwise and n n n an arbitrary integer Let s take the direction angle of the radius of C C C as the parametre t t t i e t arg z z 0 assign t z subscript z 0 t arg z z 0 Then on C C C we have z z 0 ϱ e i t 0 t 2 π formulae sequence z subscript z 0 ϱ superscript e i t 0 t 2 π z z 0 varrho e it quad 0 leqq t leqq 2 pi and d z i ϱ e i t d t z z 0 n ϱ n e i n t formulae sequence d z i ϱ superscript e i t d t superscript z subscript z 0 n superscript ϱ n superscript e i n t dz i varrho e it dt quad z z 0 n varrho n e int whence I 0 2 π ϱ n e i n t i ϱ e i t d t i ϱ n 1 0 2 π e i n 1 t d t I superscript subscript 0 2 π superscript ϱ n superscript e i n t i ϱ superscript e i t d t i superscript ϱ n 1 superscript subscript 0 2 π superscript e i n 1 t

    Original URL path: http://www.planetmath.org/calculationofcontourintegral (2016-04-25)
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  • calculation of Riemann--Stieltjes integral | planetmath.org
    g x k quad mbox for x c quad g x k alpha quad mbox for x c Then a b f d g f c α superscript subscript a b f d g normal f c α int a b f dg f c cdot alpha Let f f f be continuous on a b a b a b a c b a c b a c b and the function g g g be otherwise continuous but have in x c x c x c a step of magnitude α α alpha Then g g g is sum of a continuous function g superscript g g and a step function h x 0 for x c h x α for x c formulae sequence h x 0 formulae sequence for x c formulae sequence h x α for x c h x 0 quad mbox for x c quad h x alpha quad mbox for x c and one has a b f d g a b f d g h a b f d g a b f d h a b f d g f c α superscript subscript a b f d g superscript subscript a b f d superscript g h superscript subscript a b f d superscript g superscript subscript a b f d h superscript subscript a b f d superscript g normal f c α int a b f dg int a b f d g h int a b f dg int a b f dh int a b f dg f c cdot alpha Suppose that g g g can be expressed in the form g g h g superscript g h g g h where g superscript g g is continuous and h h h a step function having an at most denumerable amount of steps α i subscript α i alpha i in respectively the same points c i subscript c i c i on the interval a b a b a b as the function g g g If f f f is Riemann Stieltjes integrable on a b a b a b then a b f d g a b f d g i f c i α i superscript subscript a b f d g superscript subscript a b f d superscript g subscript i normal f subscript c i subscript α i displaystyle int a b f dg int a b f dg sum i f c i cdot alpha i 1 Suppose that g g h g superscript g h g g h as above has a finite amount of steps α i subscript α i alpha i in the points c i subscript c i c i of the interval a b a b a b but f f f does not have same sided discontinuities as g g g in any of those points Then f f f is Riemann Stieltjes integrable on the interval and the equation 1 is

    Original URL path: http://www.planetmath.org/calculationofriemannstieltjesintegral (2016-04-25)
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  • cancellation ideal | planetmath.org
    commutative ring containing regular elements and 𝔖 𝔖 mathfrak S be the multiplicative semigroup of the non zero fractional ideals of R R R A fractional ideal 𝔞 𝔞 mathfrak a of R R R is called a cancellation ideal or simply cancellative if it is a cancellative element of 𝔖 𝔖 mathfrak S i e if 𝔞 𝔟 𝔞 𝔠 𝔟 𝔠 𝔟 𝔠 𝔖 formulae sequence 𝔞 𝔟 𝔞 𝔠 normal 𝔟 𝔠 for all 𝔟 𝔠 𝔖 mathfrak ab ac Rightarrow mathfrak b c quad forall mathfrak b c in mathfrak S Each invertible ideal is cancellative A finite product 𝔞 1 𝔞 2 𝔞 m subscript 𝔞 1 subscript 𝔞 2 normal subscript 𝔞 m mathfrak a 1 mathfrak a 2 mathfrak a m of fractional ideals is cancellative iff every 𝔞 i subscript 𝔞 i mathfrak a i is such The fractional ideal 𝔞 r a r 1 a 𝔞 assign 𝔞 r conditional set a superscript r 1 a 𝔞 mathfrak a r ar 1 a in mathfrak a where 𝔞 𝔞 mathfrak a is an integral ideal of R R R and r r r a regular element of R R R is cancellative if and only if 𝔞 𝔞 mathfrak a is cancellative in the multiplicative semigroup of the non zero integral ideals of R R R If r R r R r in R then the principal ideal r r r of R R R is cancellative if and only if r r r is a regular element of the total ring of fractions of R R R If 𝔞 1 𝔞 2 𝔞 m subscript 𝔞 1 subscript 𝔞 2 normal subscript 𝔞 m mathfrak a 1 mathfrak a 2 mathfrak a m is a cancellation ideal and n n

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