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  • User account | planetmath.org
    Username Enter your planetmath org username Password Enter the password that accompanies your username Search form Search Home Articles Questions Forums Planetary Bugs HS Secondary University Tertiary Graduate Advanced Industry Practice Research Topics LaTeX help Math Comptetitions Math History Math

    Original URL path: http://www.planetmath.org/user (2016-04-25)
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  • field arising from special relativity | planetmath.org
    was created yet This can only be done by the original author Search form Search Home Articles Questions Forums Planetary Bugs HS Secondary University Tertiary Graduate Advanced Industry Practice Research Topics LaTeX help Math Comptetitions Math History Math Humor PlanetMath Comments PlanetMath System Updates and News PlanetMath help PlanetMath ORG Strategic Communications Development The Math Pub Testing messages ignore Other useful stuff Gallery Site Docs Corrections Info Owner pahio Added

    Original URL path: http://www.planetmath.org/node/88254/pmgroup (2016-04-25)
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  • field arising from special relativity | planetmath.org
    TeX increases you will probably want to edit this but it should be fine as is for beginners almost certainly you want these usepackage amssymb usepackage amsmath usepackage amsfonts need this for including graphics includegraphics usepackage graphicx for neatly defining theorems and propositions usepackage amsthm making logically defined graphics usepackage xypic used for TeXing text within eps files usepackage psfrag there are many more packages add them here as you need them define commands here begin document The velocities u and v of two bodies moving along a line obey by the special theory of relativity the addition rule begin align u oplus v frac u v 1 frac uv c 2 end align where c is the velocity of light As c is unreachable for any material body it plays for the velocities of the bodies the role of the infinity These velocities v thus satisfy always v Search form Search Home Articles Questions Forums Planetary Bugs HS Secondary University Tertiary Graduate Advanced Industry Practice Research Topics LaTeX help Math Comptetitions Math History Math Humor PlanetMath Comments PlanetMath System Updates and News PlanetMath help PlanetMath ORG Strategic Communications Development The Math Pub Testing messages ignore Other useful stuff Gallery

    Original URL path: http://www.planetmath.org/node/88254/source (2016-04-25)
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  • examples of fields | planetmath.org
    3 ℚ 3 2 i conditional set u v i w 3 2 x i 3 2 y 3 4 z i 3 4 u v w x y z ℚ ℚ i 3 2 mathbb Q sqrt 3 2 i u vi w sqrt 3 2 xi sqrt 3 2 y sqrt 3 4 zi sqrt 3 4 mid u v w x y z in mathbb Q mathbb Q i sqrt 3 2 every separable finite field extension is simple If p p p is a prime number then the p p p adic numbers form a field ℚ p subscript ℚ p mathbb Q p which is the completion of the field ℚ ℚ mathbb Q with respect to the p p p adic valuation If p p p is a prime number then the integers modulo p p p form a finite field with p p p elements typically denoted by p subscript p mathbb F p More generally for every prime power p n superscript p n p n there is one and only one finite field p n subscript superscript p n mathbb F p n with p n superscript p n p n elements If K K K is a field we can form the field of rational functions over K K K denoted by K X K X K X It consists of quotients of polynomials in X X X with coefficients in K K K If V V V is a variety over the field K K K then the function field of V V V denoted by K V K V K V consists of all quotients of polynomial functions defined on V V V If U U U is a domain connected open set in ℂ ℂ mathbb C

    Original URL path: http://www.planetmath.org/examplesoffields (2016-04-25)
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  • field arising from special relativity | planetmath.org
    plays for the velocities of the bodies the role of the infinity These velocities v 𝑣 thus satisfy always v c 𝑣 𝑐 By 1 we get c c c c v c formulae sequence direct sum 𝑐 𝑐 𝑐 direct sum 𝑐 𝑣 𝑐 for v c 𝑣 𝑐 so c 𝑐 behaves like the infinity One can define the mapping f ℝ c c S 𝑓 ℝ 𝑐 𝑐 𝑆 by setting f x c tanh x assign 𝑓 𝑥 𝑐 𝑥 2 which is easily seen to be a bijection Define also the binary operation direct product for the numbers u v 𝑢 𝑣 of the open interval c c 𝑐 𝑐 by u v c tanh artanh u c artanh v c direct product 𝑢 𝑣 𝑐 artanh 𝑢 𝑐 artanh 𝑣 𝑐 3 Then the system S 𝑆 direct sum direct product may be checked to be a ring and the bijective mapping 2 to be homomorphic f x y f x f y f x y f x f y formulae sequence 𝑓 𝑥 𝑦 direct sum 𝑓 𝑥 𝑓 𝑦 𝑓 𝑥 𝑦 direct product 𝑓 𝑥 𝑓 𝑦 Consequently the system S 𝑆 direct sum direct product as the homomorphic image of the field ℝ ℝ also itself is a field Baker 1 calls the numbers of the set S 𝑆 i e c c 𝑐 𝑐 the Einstein numbers References 1 G A Baker Jr Einstein numbers Amer Math Monthly 61 1954 39 41 2 H T Davis College algebra Prentice Hall N Y 1940 351 3 T Gregor J Haluška Two dimensional Einstein numbers and associativity arXiv 2013 Type of Math Object Topic Major Section Reference Parent examples of fields Add a correction Attach a problem Ask a question Search

    Original URL path: http://www.planetmath.org/fieldarisingfromspecialrelativity?destination=node/88254 (2016-04-25)
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  • $n$-section of line segment with compass and straightedge | planetmath.org
    B see compass and straightedge construction of parallel line These points divide the line segment A B A B AB in n n n equal segments Proof For clarity we prove the theorem only in the case n 3 n 3 n 3 p p p A A A B B B B 1 subscript B 1 B 1 B 2 subscript B 2 B 2 A 1 subscript A 1 A 1 A 2 subscript A 2 A 2 A 3 subscript A 3 A 3 The line A B A B AB intersects the parallel lines A 1 B 1 subscript A 1 subscript B 1 A 1 B 1 A 2 B 2 subscript A 2 subscript B 2 A 2 B 2 and A 3 B subscript A 3 B A 3 B and thus the corresponding angles A 1 B 1 A subscript A 1 subscript B 1 A A 1 B 1 A A 2 B 2 A subscript A 2 subscript B 2 A A 2 B 2 A and A 3 B A subscript A 3 B A A 3 BA are equal Similarly the angles A A 1 B 1 A subscript A 1 subscript B 1 AA 1 B 1 A A 2 B 2 A subscript A 2 subscript B 2 AA 2 B 2 and A A 3 B A subscript A 3 B AA 3 B are equal Because of the equal angles the triangle A A 2 B 2 A subscript A 2 subscript B 2 AA 2 B 2 is similar to the triangle A A 3 B A subscript A 3 B AA 3 B with the ratio of similarity 2 3 normal 2 3 2 3 Therefore A B 2 2 3 A

    Original URL path: http://www.planetmath.org/nsectionoflinesegmentwithcompassandstraightedge (2016-04-25)
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  • a condition of algebraic extension | planetmath.org
    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 3A51M05 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 3A51 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 3A51 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 a condition of algebraic extension Theorem A field extension L K L K L K is algebraic if and only if any subring of the extension field L L L containing the base field K K K is a field Proof Assume first that L K L K L K is algebraic Let R R R be a subring of L L

    Original URL path: http://www.planetmath.org/aconditionofalgebraicextension (2016-04-25)
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  • a pathological function of Riemann | planetmath.org
    value of t t t a jump saltus equal to 1 1 1 being in these points continuous from the right but not from the left For every real value t t t one has 0 t t 1 0 t t 1 displaystyle 0 leqq t lfloor t rfloor 1 1 Let us consider the series n 1 n x n x n 2 superscript subscript n 1 n x n x superscript n 2 displaystyle sum n 1 infty frac nx lfloor nx rfloor n 2 2 due to Riemann Since by 1 all values of x ℝ x ℝ x in mathbb R and n ℤ n subscript ℤ n in mathbb Z satisfy 0 n x n x n 2 1 n 2 0 n x n x superscript n 2 1 superscript n 2 displaystyle 0 leqq frac nx lfloor nx rfloor n 2 frac 1 n 2 3 the series is by Weierstrass M test uniformly convergent on the whole ℝ ℝ mathbb R see also the p test We denote by S x S x S x the sum function of 2 The n th superscript n th n mathrm th term of the series 2 defines a periodic function x n x n x n 2 maps to x n x n x superscript n 2 displaystyle x mapsto frac nx lfloor nx rfloor n 2 4 with the period 1 n 1 n frac 1 n and having especially for 0 x 1 n 0 x 1 n 0 leqq x frac 1 n the value x n x n frac x n The only points of discontinuity of this function are x m n m 0 1 2 fragments x m n italic fragments normal m 0 normal plus or

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