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  • Eisenstein criterion in terms of divisor theory | 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 3A11J82 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 3A11J81 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 3A11J81 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 Eisenstein criterion in terms of divisor theory The below theorem generalises Eisenstein criterion of irreducibility from UFD s to domains with divisor theory Theorem Let f x a 0 a 1 x a n x n assign f x subscript a 0 subscript a 1 x normal subscript a n superscript x n f x a 0 a 1 x ldots a n x n be a primitive polynomial over an integral domain mathcal O with divisor theory normal superscript mathcal O to mathfrak D If there is a prime divisor mathfrak p in D such that a 0 a 1 a n 1 fragments p normal subscript a 0 normal subscript a 1 normal normal normal subscript a n 1 normal mathfrak p mid a 0 a 1 ldots a n 1 a n not divides subscript a n mathfrak p nmid a n 2 a 0 not divides superscript 2 subscript a 0 mathfrak p 2 nmid a 0 then the polynomial is irreducible Proof Suppose that

    Original URL path: http://www.planetmath.org/eisensteincriterionintermsofdivisortheory (2016-04-25)
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  • elementary function | planetmath.org
    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 3A13A05 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 elementary function An elementary function is a real function of one variable that can be constructed by a finite number of elementary operations addition subtraction multiplication and division and compositions from constant functions the identity function x x maps to x x x mapsto x algebraic functions exponential functions logarithm functions trigonometric functions and cyclometric functions Examples Consequently the polynomial functions the absolute value x x 2 x superscript x 2 x sqrt x 2 the triangular wave function arcsin sin x x arcsin sin x the power function x π e π ln x superscript x π superscript e π x x pi e pi ln x and the function x x e x ln x superscript x x superscript e x x x x e x ln x are elementary functions N B the real power functions entail that x 0 x 0 x 0 ζ x n 1 1 n x assign ζ x superscript subscript n 1 1 superscript n x displaystyle zeta x sum n 1 infty frac 1 n x and Li x 2 x

    Original URL path: http://www.planetmath.org/elementaryfunction (2016-04-25)
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  • elimination of unknown | planetmath.org
    m a i y x i 0 b x y j 0 n b j y x j 0 a x y i0mai y xi 0 b x y j0nbj y xj 0 displaystyle begin cases a x y sum i 0 m a i y x i 0 b x y sum j 0 n b j y x j 0 end cases 1 in two unknowns x x x and y y y where e g m n m n m geq n It is possible to eliminate one of the unknowns from 1 i e derive an equivalent pair of polynomial equations f y 0 g x y 0 fy 0 gxy 0 displaystyle begin cases f y 0 g x y 0 end cases First we form the polynomial c x y b n y a x y a m y x m n b x y fragments c fragments normal x normal y normal normal subscript b n fragments normal y normal a fragments normal x normal y normal subscript a m fragments normal y normal superscript x m n b fragments normal x normal y normal normal displaystyle c x y b n y a x y a m y x m n b x y 2 the degree of which is less than m m m When x 0 y 0 subscript x 0 subscript y 0 x 0 y 0 is a solution of 1 then it satisfies c x y 0 b x y 0 cxy 0 bxy 0 displaystyle begin cases c x y 0 b x y 0 end cases 3 On the other hand when x 1 y 1 subscript x 1 subscript y 1 x 1 y 1 is a solution of 3 then 2 implies that

    Original URL path: http://www.planetmath.org/eliminationofunknown (2016-04-25)
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  • empty product | planetmath.org
    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 3A13P10 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 3A12D99 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 3A12D99 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 empty product The empty product of numbers is the borderline case of product where the number of factors is zero i e the set of the factors is empty The most usual examples are the following The zeroth power of a non zero number a 0 superscript a 0 a 0 The factorial of 0 0 The prime factor presentation of unity which has no prime factors The value of the empty sum of numbers is equal to the additive identity number 0 Similarly the empty product of numbers is equal to the multiplicative identity number 1 Note When considering the complex numbers as pairs of real numbers one often identifies the pairs x 0 x 0 x 0 and the reals x x x In this sense one can think that the Cartesian product

    Original URL path: http://www.planetmath.org/emptyproduct (2016-04-25)
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  • empty sum | planetmath.org
    the addends is an empty set One may think that the zeroth multiple 0 a 0 a 0a of a ring element a a a is the empty sum it can spring up by adding in the ring two multiples whose integer coefficients are opposite numbers n a n a n n a 0 a n a n a n n a 0 a n a na n n a 0a This empty sum equals the additive identity 0 of the ring since the multiple n a n a n a is defined to be a a a n copies subscript normal a a normal a n copies underbrace a a ldots a n mathrm copies In using the sigma notation i m n f i superscript subscript i m n f i displaystyle sum i m n f i 1 one sometimes sees a case i m m 1 f i superscript subscript i m m 1 f i displaystyle sum i m m 1 f i 2 It must be an empty sum because in i m m f i superscript subscript i m m f i displaystyle sum i m m f i 3 the number of addends is clearly one and therefore in 2 the number is zero Thus the value of 2 may be defined to be 0 Note The sum 1 is not defined when n n n is less than m 1 m 1 m 1 but if one would want that the usual rule i m n f i i n 1 k f i i m k f i superscript subscript i m n f i superscript subscript i n 1 k f i superscript subscript i m k f i displaystyle sum i m n f i sum i n

    Original URL path: http://www.planetmath.org/emptysum (2016-04-25)
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  • entries on finitely generated ideals | planetmath.org
    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 3A05A19 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 3A05A19 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 3A00A05 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 3A00A05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader

    Original URL path: http://www.planetmath.org/entriesonfinitelygeneratedideals (2016-04-25)
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  • envelope | planetmath.org
    GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A13C05 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 3A13C05 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 3A16D25 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 3A16D25 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 envelope Two plane curves are said to touch each other or have a tangency at a point if they have a common tangent line at that point The envelope of a family of plane curves

    Original URL path: http://www.planetmath.org/envelope (2016-04-25)
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  • equal arc length and area | planetmath.org
    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 equal arc length and area We want to determine the nonnegative differentiable real functions x y maps to x y x mapsto y whose graph has the property that the arc length between any two points of it is the same as the area bounded by the curve the x x x axis and the ordinate lines of those points The requirement leads to the equation a x 1 d y d x 2 d x a x y d x superscript subscript a x 1 superscript d y d x 2 d x superscript subscript a x y d x displaystyle int a x sqrt 1 left frac dy dx right 2 dx int a x y dx 1 By the fundamental theorem of calculus we infer from 1 the differential equation 1 d y d x 2 y 1 superscript d y d x 2 y displaystyle sqrt 1 left frac dy dx right 2 y 2 whence d y d x y 2 1 d y d x superscript y 2 1 frac dy dx sqrt y 2 1 In the case y 1 not equivalent to y 1 y not equiv 1 the separation of variables yields d x d y y 2 1 d x d y superscript y 2 1 int dx int frac dy sqrt y 2 1

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