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  • absorbing element | 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 3A30E20 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 3A30E20 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 absorbing element An element ζ ζ zeta of a groupoid G G G is called an absorbing element in French un élément absorbant for the operation if it satisfies ζ a a ζ ζ ζ a a ζ ζ zeta a a zeta zeta for all elements a a a of G G G Examples The zero 0 0 0 is the absorbing element for multiplication or multiplicatively absorbing in every ring R fragments

    Original URL path: http://www.planetmath.org/absorbingelement (2016-04-25)
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  • adding and removing parentheses in series | planetmath.org
    View active tab Coauthors PDF Source Edit adding and removing parentheses in series We consider series with real or complex terms If one groups the terms of a convergent series by adding parentheses but not changing the order of the terms the series remains convergent and its sum the same See theorem 3 of the parent entry A divergent series can become convergent if one adds an infinite amount of parentheses e g 1 1 1 1 1 1 fragments 1 1 1 1 1 1 normal 1 1 1 1 1 1 ldots diverges but 1 1 1 1 1 1 1 1 1 1 1 1 normal 1 1 1 1 1 1 ldots converges A convergent series can become divergent if one removes an infinite amount of parentheses cf the preceding example If a series contains parentheses they can be removed if the obtained series converges in this case also the original series converges and both series have the same sum If the series a 1 a r a r 1 a 2 r a 2 r 1 a 3 r subscript a 1 normal subscript a r subscript a r 1 normal subscript a 2 r subscript a 2 r 1 normal subscript a 3 r normal displaystyle a 1 ldots a r a r 1 ldots a 2r a 2r 1 ldots a 3r ldots 1 converges and lim n a n 0 subscript normal n subscript a n 0 displaystyle lim n to infty a n 0 2 then also the series a 1 a 2 a 3 subscript a 1 subscript a 2 subscript a 3 normal displaystyle a 1 a 2 a 3 ldots 3 converges and has the same sum as 1 Proof Let S S S be the sum of

    Original URL path: http://www.planetmath.org/addingandremovingparenthesesinseries (2016-04-25)
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  • addition formula | planetmath.org
    called the mother of all formulae x y x y 1 x y cos x y cos x cos y sin x sin y footnote The addition formula of cosine is sometimes called the mother of all formulae tan x y frac tan x tan y 1 tan x tan y 5 Addition formulae of the hyperbolic functions e g sinh x y sinh x cosh y cosh x sinh y x y x y x y sinh x y sinh x cosh y cosh x sinh y 6 Addition formula of the Bessel function J n x y ν J ν x J n ν y n 0 1 2 fragments subscript J n fragments normal x y normal superscript subscript ν subscript J ν fragments normal x normal subscript J n ν fragments normal y normal italic fragments normal n 0 normal plus or minus 1 normal plus or minus 2 normal normal normal J n x y sum nu infty infty J nu x J n nu y qquad n 0 pm 1 pm 2 ldots The five first of those are instances of algebraic addition formulae e g cosh x x cosh x and sinh x x sinh x are tied together by the algebraic connection cosh 2 x sinh 2 x 1 superscript 2 x superscript 2 x 1 cosh 2 x sinh 2 x 1 One may also speak of the subtraction formulae of functions one example would be e x y e x e y superscript e x y superscript e x superscript e y e x y frac e x e y Defines addition formulae subtraction formula subtraction formulae Keywords algebraic addition formula Synonym addition theorem Type of Math Object Definition Major Section Reference Groups audience Buddy List of pahio Mathematics Subject Classification 30D05 no label found 30A99 no label found 26A99 no label found Add a correction Attach a problem Ask a question Comments why addition formula is attached to persistence Permalink Submitted by mathforever on Fri 10 15 2004 20 01 May be I missed something but what is relation of this entry to the entry persistence of analytic relations Log in to post comments Re why addition formula is attached to persistence Permalink Submitted by pahio on Fri 10 15 2004 21 13 The attachment isn t so good Log in to post comments Re Re why addition formula is attached to persistence Permalink Submitted by mathforever on Sun 10 17 2004 16 10 Well the new attachment is also not that good to my opinion I wouldn t attach it to anything Log in to post comments Re Re why addition formula is attached to persistence Permalink Submitted by pahio on Sun 10 17 2004 20 03 OK if you are thought the thing thoroughly so I can remove the attachement I have not found any natural attachement BTW if you know some good additional exemples please tell me Jussi Log in to post comments Re Re why

    Original URL path: http://www.planetmath.org/additionformula (2016-04-25)
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  • addition formulas | planetmath.org
    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 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 3A26A99 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 3A26A99 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 addition formulas The addition formula of a real or complex function shows how the value of the function at a sum formed variable can be expressed with the values of this function and perhaps of another function at the addends Examples 1 Addition formula of an additive function f f f f x y f x f y f x y f x f y f x y f x f y 2 Addition formula of the natural power function i e the binomial theorem x y n ν 0 n n ν x ν y n ν n 0 1 2 fragments superscript fragments normal x y normal n superscript subscript ν 0 n binomial n ν superscript x ν superscript y n ν italic fragments normal n 0 normal 1 normal 2 normal normal normal x y n sum nu 0 n n choose nu x nu y n nu qquad n 0 1 2

    Original URL path: http://www.planetmath.org/additionformulas (2016-04-25)
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  • algebraic numbers are countable | planetmath.org
    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 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 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 3A26A99 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 3A26A99 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 algebraic numbers are countable Theorem The set of a all algebraic numbers b the real algebraic numbers is countable Proof Let s consider the algebraic equations P x 0 P x 0 displaystyle P x 0 1 where P x a 0 x n a 1 x n 1 a n 1 x a n assign P x subscript a 0 superscript x n subscript a 1 superscript x n 1 normal subscript a n 1 x subscript a n P

    Original URL path: http://www.planetmath.org/algebraicnumbersarecountable (2016-04-25)
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  • algebraic sines and cosines | planetmath.org
    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 3A03E10 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 3A03E10 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 algebraic sines and cosines For any rational number r r r the sine and the cosine of the number r π r π r pi are algebraic numbers Proof According to the parent entry sin n φ n φ sin n varphi and cos n φ n φ cos n varphi can be expressed as polynomials with integer coefficients of sin φ φ sin varphi or cos φ φ cos varphi respectively when n n n is an integer Thus we can write sin n φ P sin φ cos n φ Q cos φ formulae sequence n φ P φ n φ Q φ sin n varphi P sin varphi quad cos n varphi Q cos varphi where P x Q x ℤ x P x Q x ℤ x P x Q x in mathbb Z x If r m n r m n displaystyle r frac m n where m n m n m n are integers and n 0 n 0 n neq 0 we have P sin r π sin n r π sin m π 0 Q cos r π cos n r π cos m π 1 formulae sequence P r π n r π m π 0 Q r π n r π m π plus or minus 1 P sin r pi sin nr pi sin m pi 0 quad Q

    Original URL path: http://www.planetmath.org/algebraicsinesandcosines (2016-04-25)
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  • algebraic sum and product | planetmath.org
    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 3A11C08 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 3A11C08 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 algebraic sum and product Let α β α β alpha beta be two elements of an extension field of a given field K K K Both these elements are algebraic over K K K if and only if both α β α β alpha beta and α β α β alpha beta are algebraic over K K K Proof Assume first that α α alpha and β β beta are algebraic Because K α β K K α β K α K α K fragments fragments normal K fragments normal α normal β normal normal K normal fragments normal K fragments normal α normal β normal normal K fragments normal α normal normal fragments normal K fragments normal α normal normal K normal K alpha beta K K alpha beta K alpha K alpha K and both factors here are finite then K α β K normal K α β K K alpha beta K is finite So we have a finite field extension K α β K K α β K K alpha beta K which thus is also algebraic and therefore the elements α β α β alpha beta and α β α β alpha beta of K α β K α β K alpha beta are algebraic over K K K Secondly suppose that α β α β alpha beta and α β α β alpha beta are algebraic over K K K The elements α α alpha and β β beta are the roots of the quadratic equation x 2 α β x α β 0 superscript x 2 α β x α β 0 x 2

    Original URL path: http://www.planetmath.org/algebraicsumandproduct (2016-04-25)
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  • algebraically solvable | planetmath.org
    7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A13B05 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 3A13B05 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 algebraically solvable An equation x n a 1 x n 1 a n 0 superscript x n subscript a 1 superscript x n 1 normal subscript a n 0 displaystyle x n a 1 x n 1 ldots a n 0 1 with coefficients a j subscript a j a j in a field K K K is algebraically solvable if some of its roots may be expressed with the elements of K K K by using rational operations addition subtraction multiplication division and root extractions I e a root of 1 is in a field K ξ 1 ξ 2 ξ m K subscript ξ 1 subscript ξ 2 normal subscript ξ m K xi 1 xi 2 ldots xi m which is obtained of K K K by adjoining to it in succession certain suitable radicals ξ 1 ξ 2 ξ m subscript ξ 1 subscript ξ 2 normal subscript ξ m xi 1 xi 2 ldots xi m Each radical may contain under the root sign one or more of the previous radicals ξ 1 r 1 p 1 ξ 2 r 2 ξ 1 p 2 ξ 3 r 3 ξ 1 ξ 2 p 3 ξ m r m ξ 1 ξ 2 ξ m 1 p m cases subscript ξ 1 subscript p 1 subscript r 1 otherwise subscript ξ 2 subscript p 2 subscript r 2 subscript ξ 1 otherwise subscript ξ 3 subscript p 3 subscript r 3 subscript ξ 1 subscript ξ 2 otherwise normal normal otherwise subscript ξ m subscript p m subscript r m subscript ξ 1 subscript ξ 2 normal subscript ξ m 1 otherwise displaystyle begin cases xi 1 sqrt p 1 r 1 xi 2 sqrt p 2 r 2 xi 1

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