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  • angle of view of a line segment | planetmath.org
    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 3A51N20 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 angle of view of a line segment Let P Q P Q PQ be a line segment and A A A a point not belonging to P Q P Q PQ Let the magnitude of the angle P A Q P A Q PAQ be α α alpha One says that the line segment P Q P Q PQ is seen from the point A A A in an angle of α α alpha one may also speak of the angle of view of P Q P Q PQ The locus of the points from which a given line segment P Q P Q PQ is seen in an angle of α α alpha with 0 α 180 0 α superscript 180 0 alpha 180 circ consists of two congruent circular arcs having the line segment as the common chord and containing the circumferential angles equal to α α alpha Especially the locus of the points from which the line segment is seen in an angle of 90 superscript 90 90 circ is the circle having the

    Original URL path: http://www.planetmath.org/angleofviewofalinesegment (2016-04-25)
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  • anticommutative | planetmath.org
    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 3A51F20 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 3A51F20 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 A binary operation normal star is said to be anticommutative if it satisfies the identity y x x y normal y x normal x y displaystyle y star x x star y 1 where the minus denotes the opposite element in the algebra in question This implies that x x x x normal x x normal x x x star x x star x i e x x normal x x x star x

    Original URL path: http://www.planetmath.org/anticommutative (2016-04-25)
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  • antiderivative of rational function | planetmath.org
    H x H x H x is a polynomial the first sum expression is determined by the real zeroes a i subscript a i a i of the denominator of R x R x R x the second sum is determined by the real quadratic prime factors x 2 2 p j x q j superscript x 2 2 subscript p j x subscript q j x 2 2p j x q j of the denominator which have no real zeroes The addends of the form A x a r A superscript x a r displaystyle frac A x a r in the first sum are integrated directly giving A x a d x A ln x a constant r 1 fragments A x a d x A normal x a normal constant italic fragments normal r 1 normal displaystyle int frac A x a dx A ln x a mbox constant qquad r 1 1 and A x a r d x A r 1 1 x a r 1 constant r 1 fragments A superscript x a r d x A r 1 normal 1 superscript x a r 1 constant italic fragments normal r 1 normal normal displaystyle int frac A x a r dx frac A r 1 cdot frac 1 x a r 1 mbox constant qquad r 1 2 The remaining partial fractions are of the form B x C x 2 2 p x q s B x C superscript superscript x 2 2 p x q s displaystyle frac Bx C x 2 2px q s where p 2 q superscript p 2 q p 2 q and s s s is a positive integer Now we may write x 2 2 p x q x p 2 q p 2 q p 2 1 x p q p 2 2 superscript x 2 2 p x q superscript x p 2 q superscript p 2 q superscript p 2 1 superscript x p q superscript p 2 2 x 2 2px q x p 2 q p 2 q p 2 left 1 left frac x p sqrt q p 2 right 2 right and make the substitution x p q p 2 t x p q superscript p 2 t displaystyle frac x p sqrt q p 2 t 3 i e x t q p 2 p x t q superscript p 2 p x t sqrt q p 2 p getting B x C x 2 2 p x q s d x E t F 1 t 2 s d t E t d t 1 t 2 s F d t 1 t 2 s B x C superscript superscript x 2 2 p x q s d x E t F superscript 1 superscript t 2 s d t E t d t superscript 1 superscript t 2 s F d t superscript 1 superscript t 2 s displaystyle int frac Bx C x 2

    Original URL path: http://www.planetmath.org/antiderivativeofrationalfunction (2016-04-25)
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  • antiharmonic number | planetmath.org
    to be an antiharmonic number is that the quotient σ 2 n σ 1 n 0 d n d 2 0 d n d i 1 k p i 2 m i 1 1 p i 2 1 i 1 k p i m i 1 1 p i 1 normal subscript σ 2 n subscript σ 1 n subscript 0 bra d n superscript d 2 normal subscript 0 bra d n d superscript subscript product i 1 k superscript subscript p i 2 subscript m i 1 1 superscript subscript p i 2 1 normal superscript subscript product i 1 k superscript subscript p i subscript m i 1 1 subscript p i 1 sigma 2 n sigma 1 n sum 0 d mid n d 2 sum 0 d mid n d prod i 1 k frac p i 2 m i 1 1 p i 2 1 prod i 1 k frac p i m i 1 1 p i 1 is an integer here the p i subscript p i p i s are the distinct prime divisors of n n n and m i subscript m i m i s their multiplicities The last form is simplified to i 1 k p i m i 1 1 p i 1 superscript subscript product i 1 k superscript subscript p i subscript m i 1 1 subscript p i 1 displaystyle prod i 1 k frac p i m i 1 1 p i 1 1 The OEIS sequence A020487 contains all nonzero perfect squares since in the case of such numbers the antiharmonic mean 1 of the divisors has the form i 1 k p i 2 m i 1 1 p i 1 i 1 k p i 2 m i p i 2 m i 1 p i 1 fragments superscript subscript product i 1 k superscript subscript p i 2 subscript m i 1 1 subscript p i 1 superscript subscript product i 1 k fragments normal superscript subscript p i 2 subscript m i superscript subscript p i 2 subscript m i 1 normal subscript p i 1 normal prod i 1 k frac p i 2m i 1 1 p i 1 prod i 1 k left p i 2m i p i 2m i 1 ldots p i 1 right cf irreducibility of binomials with unity coefficients Note It would in a manner be legitimated to define a positive integer to be an antiharmonic number or an antiharmonic integer if it is the antiharmonic mean of two distinct positive integers see integer contraharmonic mean and contraharmonic Diophantine equation Major Section Reference Type of Math Object Definition Parent divisor function Groups audience Buddy List of pahio Mathematics Subject Classification 11A05 no label found 11A25 no label found Add a correction Attach a problem Ask a question Comments a new article invisible Permalink Submitted by pahio on Sun 11 24 2013 18 41 Hi admins My new article antiharmonic number

    Original URL path: http://www.planetmath.org/antiharmonicnumber (2016-04-25)
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  • antiholomorphic | planetmath.org
    2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A11A25 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 3A11A25 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 A complex function f D ℂ normal f normal D ℂ f D to mathbb C where D D D is a domain of the complex plane having the derivative d f d z d f d normal z frac df d overline z in each point z z z of D D D is said to be antiholomorphic in D D D The following conditions are equivalent f z f z f z is antiholomorphic in D D D f z normal f z overline f z is holomorphic in D D D f z f normal z f overline z is holomorphic in D z z D fragments normal D assign fragments normal normal z normal z D normal overline D overline z vdots z in D f z f z f z may be expanded to a power series n 0 a n z u n superscript subscript n 0 subscript a n superscript normal z u n sum n 0 infty a n overline z u n at each u D u D u in D The real part u x y u x y u x y and the imaginary part v x y v x y v x y of the function f f f satisfy the equations u x v y u y v x formulae sequence u x v y u y v x frac partial u partial x frac partial v partial y qquad frac partial u partial y frac partial v partial x N B the place of minus cf the Cauchy Riemann equations Example The function z 1 z maps to z 1 normal z displaystyle z mapsto frac 1 overline z is antiholomorphic in ℂ 0

    Original URL path: http://www.planetmath.org/antiholomorphic (2016-04-25)
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  • antiperiodic function | planetmath.org
    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 3A30A99 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 A special case of the quasiperiodicity of functions is the antiperiodicity An antiperiodic function f 𝑓 satisfies for a certain constant p 𝑝 the equation f z p f z 𝑓 𝑧 𝑝 𝑓 𝑧 for all values of the variable z 𝑧 The constant p 𝑝 is the antiperiod of f 𝑓 Then f 𝑓 has also other antiperiods e g p 𝑝 and generally 2 n 1 p 2 𝑛 1 𝑝 with any n ℤ 𝑛 ℤ The antiperiodic function f 𝑓 is always as well periodic with period 2 p 2 𝑝 since f z 2 p f z p p f z p f z f z 𝑓 𝑧 2 𝑝 𝑓 𝑧 𝑝 𝑝 𝑓 𝑧 𝑝 𝑓 𝑧 𝑓 𝑧 Naturally then there are all periods 2 n p 2 𝑛 𝑝 with n ℤ 𝑛 ℤ Not all periodic functions are antiperiodic For example the sine and cosine functions are antiperiodic with p π 𝑝 𝜋 which is their absolutely least antiperiod sin z π sin z cos z π cos z formulae sequence 𝑧 𝜋 𝑧 𝑧 𝜋 𝑧 The

    Original URL path: http://www.planetmath.org/antiperiodicfunction (2016-04-25)
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  • antipodal isothermic points | planetmath.org
    GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A30A99 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 antipodal isothermic points Assume that the momentary temperature on any great circle of a sphere varies continuously Then there exist two diametral points i e antipodal points end points of a certain diametre having the same temperature Proof Denote by x x x the distance of any point P P P measured in a certain direction along the great circle from a fixed point and let T x T x T x be the temperature in P P P Then we have a continuous and periodic real function T T T defined for x 0 x 0 x geqq 0 satisfying T x p T x T x p T x T x p T x where p p p is the perimetre of the circle Then also the function f f f defined by f x T x p 2 T x assign f x T x p 2 T x f x T left x frac p 2 right T x i e the temperature difference in two antipodic diametral points of the great circle is continuous We have f p 2 T p T p 2 T 0 T p 2 f 0 f p 2 T p T p 2 T 0 T p 2 f 0 displaystyle f left frac p 2 right T p T left frac p 2 right T 0 T left frac p 2 right f 0 1 If f f f

    Original URL path: http://www.planetmath.org/antipodalisothermicpoints (2016-04-25)
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  • any divisor is gcd of two principal divisors | planetmath.org
    Primary tabs View active tab Coauthors PDF Source Edit any divisor is gcd of two principal divisors Using the exponent valuations one can easily prove the Theorem In any divisor theory each divisor is the greatest common divisor of two principal divisors Proof Let normal superscript mathcal O to mathfrak D be a divisor theory and mathfrak d an arbitrary divisor in mathfrak D We may suppose that mathfrak d is not a principal divisor if mathfrak D contains exclusively principal divisors then gcd mathfrak d gcd mathfrak d mathfrak d and the proof is ready Let i 1 r i k i superscript subscript product i 1 r superscript subscript i subscript k i mathfrak d prod i 1 r mathfrak p i k i where the i subscript i mathfrak p i s are pairwise distinct prime divisors and every k i 0 subscript k i 0 k i 0 Then third condition in the theorem concerning divisors and exponents allows to choose an element α α alpha of the ring mathcal O such that ν 1 α k 1 ν r α k r formulae sequence subscript ν subscript 1 α subscript k 1 normal subscript ν subscript r α subscript k r nu mathfrak p 1 alpha k 1 ldots nu mathfrak p r alpha k r Let the principal divisor corresponding to α α alpha be α i 1 r i k i j 1 s j l j α superscript subscript product i 1 r superscript subscript i subscript k i superscript subscript product j 1 s superscript subscript j subscript l j superscript normal alpha prod i 1 r mathfrak p i k i prod j 1 s mathfrak q j l j mathfrak dd prime where the prime divisors j subscript j

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