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  • divisor function | planetmath.org
    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 3A11A51 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 In the parent article there has been proved the formula σ 1 n 0 d n d i 1 k p i m i 1 1 p i 1 subscript σ 1 n subscript 0 bra d n d superscript subscript product i 1 k superscript subscript p i subscript m i 1 1 subscript p i 1 sigma 1 n sum 0 d mid n d prod i 1 k frac p i m i 1 1 p i 1 giving the sum of all positive divisors of an integer n n n there the p i subscript p i p i s are the distinct positive prime factors of n n n and m i subscript m i m i s their multiplicities It follows that the sum of the z z z th powers of those divisors is given by σ z n 0 d n d z i 1 k p i m i 1 z 1 p i z 1 subscript σ z n subscript 0 bra d n superscript d z superscript subscript product i 1 k superscript subscript p i subscript m i 1 z 1 superscript subscript p i z 1 displaystyle sigma z n sum 0 d mid n d z prod i 1 k frac p i m i 1 z 1 p i z 1 1 This complex function of z z z is called divisor function The equation 1 may be written in the form σ z n i 1 k 1 p i z p i 2 z p i m i z subscript σ z n superscript subscript product i 1 k 1 superscript subscript p i z superscript subscript p i 2 z normal superscript subscript p i subscript m i z displaystyle sigma z n prod i 1 k 1 p i z p i 2z ldots p i m i z 2 usable also for z 0 z 0 z 0 For the special case of one prime power the function consists of the single geometric sum σ z p m 1 p z p 2 z p m z subscript σ z superscript p m 1 superscript p z superscript p 2 z normal superscript p m z sigma z p m 1 p z p 2z ldots p mz which particularly gives m 1 m

    Original URL path: http://www.planetmath.org/divisorfunction (2016-04-25)
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  • divisor theory | planetmath.org
    D is a free monoid on the set of its prime elements If the monoid mathfrak D has a unique prime factorisation mathfrak e is divisible only by itself Two elements of mathfrak D have always a greatest common factor If a product mathfrak ab is divisible by a prime element mathfrak p then at least one of mathfrak a and mathfrak b is divisible by mathfrak p 0 2 Divisor theory of an integral domain Let mathcal O be an integral domain and superscript mathcal O the set of its non zero elements this set forms a commutative monoid with identity 1 with respect to the multiplication of mathcal O We say that the integral domain mathcal O has a divisor theory if there is a commutative monoid mathfrak D with unique prime factorisation and a homomorphism α α maps to α α alpha mapsto alpha from the monoid superscript mathcal O into the monoid mathfrak D such that the following three properties are true 1 A divisibility α β fragments α normal β alpha mid beta in superscript mathcal O is valid iff the divisibility α β fragments fragments normal α normal normal fragments normal β normal alpha mid beta is valid in mathfrak D 2 If the elements α α alpha and β β beta of superscript mathcal O are divisible by an element mathfrak c of mathfrak D then also α β plus or minus α β alpha pm beta are divisible by mathfrak c α fragments c normal α mathfrak c mid alpha means that α fragments c normal fragments normal α normal mathfrak c mid alpha in addition 0 is divisible by every element of mathfrak D 3 If α α β β conditional set α normal α conditional set β normal β alpha in mathcal O vdots mathfrak a mid alpha beta in mathcal O vdots mathfrak b mid beta then mathfrak a b A divisor theory of mathcal O is denoted by normal superscript mathcal O to mathfrak D The elements of mathfrak D are called divisors and especially the divisors of the form α α alpha where α α superscript alpha in mathcal O principal divisors The prime elements of mathfrak D are prime divisors By 1 it is easily seen that two principal divisors α α alpha and β β beta are equal iff the elements α α alpha and β β beta are associates of each other Especially the units of mathcal O determine the unit divisor mathfrak e 0 3 Uniqueness theorems Theorem 1 An integral domain mathcal O has at most one divisor theory In other words for any pair of divisor theories normal superscript mathcal O to mathfrak D and normal superscript superscript normal mathcal O to mathfrak D prime there is an isomorphism φ normal φ normal superscript normal varphi mathfrak D to mathfrak D prime such that φ α α φ α superscript α normal varphi alpha alpha prime always when the principal divisors

    Original URL path: http://www.planetmath.org/divisortheory (2016-04-25)
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  • divisor theory and exponent valuations | planetmath.org
    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 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 divisor theory and exponent valuations A divisor theory normal superscript mathcal O to mathfrak D of an integral domain mathcal O determines via its prime divisors a certain set N N N of exponent valuations on the quotient field of mathcal O Assume to be known this set of exponents ν subscript ν nu mathfrak p corresponding the prime divisors mathfrak p There is a bijective correspondence between the elements of N N N and of the set of all prime divisors The set of the prime divisors determines completely the structure of the free monoid mathfrak D of all divisors in question The homomorphism normal superscript mathcal O to mathfrak D is then defined by the condition α i ν i α α  formulae sequence maps to α subscript product i superscript subscript subscript ν subscript i α α  normal displaystyle alpha mapsto prod i mathfrak p i nu mathfrak p i alpha alpha  1 since for any element α α alpha of superscript mathcal O there exists only a finite number of exponents ν i subscript ν subscript i nu mathfrak p i which do not vanish on α α alpha corresponding the different prime divisor factors of the principal divisor α α alpha One can take the concept of exponent as foundation for divisor theory Theorem Let mathcal O be an integral domain with quotient field K K K and N N N a given set of exponents of K K K The exponents in N N N determine as in 1 a divisor theory of mathcal O iff the following three conditions are in force For every α α alpha in mathcal O there is at most a finite number of exponents ν N ν N nu in N such that ν α 0 ν α 0 nu alpha neq 0 An element α K α K alpha in K belongs to mathcal

    Original URL path: http://www.planetmath.org/divisortheoryandexponentvaluations (2016-04-25)
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  • divisor theory in finite extension | planetmath.org
    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 3A12J20 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 3A12J20 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 3A13A05 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 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

    Original URL path: http://www.planetmath.org/divisortheoryinfiniteextension (2016-04-25)
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  • divisors in base field and finite extension field | planetmath.org
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    Original URL path: http://www.planetmath.org/divisorsinbasefieldandfiniteextensionfield (2016-04-25)
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  • duality of Gudermannian and its inverse function | planetmath.org
    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 3A13A18 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 3A13A05 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 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 duality of Gudermannian and its inverse function There are a lot of formulae concerning the Gudermannian function and its inverse function containing a hyperbolic function or a trigonometric function or both such that if we change functions of one kind to the corresponding functions of the other kind then the new formula also is true Some exemples gd x 0 x d t cosh t gd 1 x 0 x d t cos t formulae sequence gd x superscript subscript 0 x d t t superscript gd 1 x superscript subscript 0 x d t t displaystyle mbox gd x int 0 x frac dt cosh t

    Original URL path: http://www.planetmath.org/dualityofgudermanniananditsinversefunction (2016-04-25)
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  • dyad product | 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 3A26A48 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 dyad product A third kind of products between two Euclidean vectors a normal a vec a and b normal b vec b besides the scalar product a b normal normal a normal b vec a cdot vec b and the vector product a b normal a normal b vec a times vec b is the dyad product a b normal a normal b vec a vec b which is usually denoted without any multiplication symbol The dyad products and the finite formal sums Φ μ a μ b μ assign normal Φ subscript μ subscript normal a μ subscript normal b μ displaystyle Phi sum mu vec a mu vec b mu 1 of them are called dyads A dyad is not a vector but an operator It functions on any vector v normal v vec v producing from it new vectors or new dyads according to the definitions Φ v μ a μ b μ v v Φ μ v a μ b μ formulae sequence assign normal Φ normal v subscript μ subscript normal a μ subscript normal b μ normal v assign normal v normal Φ subscript μ normal v subscript normal a μ subscript normal b μ displaystyle Phi vec v sum mu vec a mu vec b mu vec v quad vec v Phi sum mu vec v vec a mu vec b mu 2 Here the asterisks mean either dots producing two vectors or crosses producing two dyads One can also allow the asterisks to mean empty in which case the vector v normal v vec v must be replaced by a scalar v v v the products Φ v normal Φ v Phi v and v Φ v normal Φ v Phi are dyads The dyad product obeys the distributive laws a b c a b a c b c a b a c a formulae sequence normal a normal b normal c normal a normal b normal a normal c normal b normal c normal a normal b normal a normal c normal a vec a vec b vec c vec a vec b vec a vec c quad vec b vec c vec a vec b vec a vec c vec a which can be verified by multiplying an arbitrary vector v normal v vec v and both sides of these equations and then comparing the results Likewise the

    Original URL path: http://www.planetmath.org/dyadproduct (2016-04-25)
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  • e is transcendental | planetmath.org
    xi 1 f xi 1 2c 2 e 2 xi 2 f xi 2 ldots nc n e n xi n f xi n 4 We shall show that the polynomial f x f x f x can be chosen such that the left side of 4 is a non zero integer whereas the right side has absolute value less than 1 We choose f x x p 1 p 1 x 1 x 2 x n p assign f x superscript x p 1 p 1 superscript x 1 x 2 normal x n p displaystyle f x frac x p 1 p 1 x 1 x 2 cdots x n p 5 where p p p is a positive prime number on which we later shall set certain conditions We must determine the corresponding values F 0 F 0 F 0 F 1 F 1 F 1 F n F n F n For determining F 0 F 0 F 0 we need according to lemma 2 to expand f x f x f x by the powers of x x x getting f x 1 p 1 1 n p n p x p 1 A 1 x p A 2 x p 1 f x 1 p 1 superscript 1 n p superscript n p superscript x p 1 subscript A 1 superscript x p subscript A 2 superscript x p 1 normal f x frac 1 p 1 1 np n p x p 1 A 1 x p A 2 x p 1 ldots where A 1 A 2 subscript A 1 subscript A 2 normal A 1 A 2 ldots are integers and to replace the powers x p 1 superscript x p 1 x p 1 x p superscript x p x p x p 1 superscript x p 1 x p 1 with the numbers p 1 p 1 p 1 p p p p 1 p 1 p 1 We then get the expression F 0 1 p 1 1 n p n p p 1 A 1 p A 2 p 1 1 n p n p p K 0 F 0 1 p 1 superscript 1 n p superscript n p p 1 subscript A 1 p subscript A 2 p 1 normal superscript 1 n p superscript n p p subscript K 0 F 0 frac 1 p 1 1 np n p p 1 A 1 p A 2 p 1 ldots 1 np n p pK 0 in which K 0 subscript K 0 K 0 is an integer We now set for the prime p p p the condition p n p n p n Then n n n is not divisible by p p p neither is the former addend 1 n p n p superscript 1 n p superscript n p 1 np n p On the other hand the latter addend p K 0 p subscript K 0 pK 0 is

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