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  • criterion for maximal ideal | planetmath.org
    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 3A26A09 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 criterion for maximal ideal Theorem In a commutative ring R R R with non zero unity an ideal mathfrak m is maximal if and only if a R r R such that 1 a r formulae sequence for all a R r R such that 1 a r displaystyle forall a in R smallsetminus mathfrak m exists r in R quad mbox such that 1 ar in mathfrak m 1 Proof 1 superscript 1 1 circ Let first mathfrak m be a maximal ideal of R R R and a R a R a in R smallsetminus mathfrak m Because a R a R mathfrak m a R there exist some elements m m m in mathfrak m and r R r R r in R such that m a r 1 m a r 1 m ar 1 Consequently 1 a r m 1 a r m 1 ar m in mathfrak m 2 superscript 2 2 circ Assume secondly that the ideal mathfrak m satisfies the condition 1 Now there must be a maximal ideal superscript normal mathfrak m prime of R R R such that R superscript normal R mathfrak m subseteq mathfrak m prime subset R Let us make the antithesis that superscript normal mathfrak m prime smallsetminus mathfrak m is non empty Choose an element a R a superscript normal R

    Original URL path: http://www.planetmath.org/criterionformaximalideal (2016-04-25)
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  • criterion of surjectivity | planetmath.org
    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 3A13A15 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 3A13A15 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 criterion of surjectivity Theorem For surjectivity of a mapping f A B normal f normal A B f A to B it s necessary and sufficient that B f X f A X X A formulae sequence B f X f A X for all X A displaystyle B smallsetminus f X subseteq f A smallsetminus X quad forall X subseteq A 1 Proof 1 o superscript 1 normal o 1 underline o Suppose that f A B normal f normal A B f A to B is surjective Let X X X be an arbitrary subset of A A A and y y y any element of the set B f X B f X B smallsetminus f X By the surjectivity there is an x x x in A A A such that f x y f x y f x y and since y f X y f X y notin f X the element x x x is not in X X X i e x A X x A X x in A smallsetminus X and thus y f x f A X y f x f A X y f x in f A smallsetminus X One can conclude that B f X f A X B f X f A X B smallsetminus f X subseteq f A

    Original URL path: http://www.planetmath.org/criterionofsurjectivity (2016-04-25)
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  • crossed quadrilateral | planetmath.org
    in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit crossed quadrilateral A complete crossed quadrilateral is formed by four distinct lines A C A C AC A D A D AD C F C F CF and D E D E DE in the Euclidean plane each of which intersects the other three The intersection of C F C F CF and D E D E DE is labelled as B B B A complete crossed quadrilateral has six vertices of which A A A and B B B C C C and D D D E E E and F F F are opposite A A A E E E C C C F F F D D D B B B The complete crossed quadrilateral is often reduced to the crossed quadrilateral C E D F C E D F CEDF cyan in the diagram consisting of the four line segments C E C E CE C F C F CF D E D E DE and D F D F DF Its diagonals C D C D CD and E F E F EF are outside of the crossed quadrilateral In the picture below the same quadrilateral as above is still in cyan and its diagonals are drawn in blue E E E C C C F F F D D D The sum of the inner angles of C E D F C E D F CEDF is 720 o superscript 720 normal o 720 mathrm o Its area is obtained e g by means of the Bretschneider s formula cf area of a quadrilateral A special case of the crossed quadrilateral is the antiparallelogram in which the lengths of the

    Original URL path: http://www.planetmath.org/crossedquadrilateral (2016-04-25)
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  • cube of a number | planetmath.org
    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 cube of a number The cube of a number x x x is the third power x 3 superscript x 3 x 3 of x x x Similarly one may speak of the cube of an element x x x in any semigroup with the operation denoted multiplicatively cf general associativity The volume of a cube i e regular hexahedron with edge length a a a is a 3 superscript a 3 a 3 hence the name The cube function x x 3 maps to x superscript x 3 x mapsto x 3 from ℝ ℝ mathbb R to ℝ ℝ mathbb R is injective but not as a mapping from ℂ ℂ mathbb C to ℂ ℂ mathbb C one has x 3 y 3 superscript x 3 superscript y 3 x 3 y 3 always when x y 1 i 3 2 x y plus or minus 1

    Original URL path: http://www.planetmath.org/cubeofanumber (2016-04-25)
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  • cube of an integer | planetmath.org
    2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A20 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 cube of an integer Theorem Any cube of integer is a difference of two squares which in the case of a positive cube are the squares of two successive triangular numbers For proving the assertion one needs only to check the identity a 3 a 1 a 2 2 a 1 a 2 2 superscript a 3 superscript a 1 a 2 2 superscript a 1 a 2 2 a 3 equiv left frac a 1 a 2 right 2 left frac a 1 a 2 right 2 For example we have 2 3 1 2 3 2 superscript 2 3 superscript 1 2 superscript 3 2 2 3 1 2 3 2 and 4 3 64 10 2 6 2 superscript 4 3 64 superscript 10 2 superscript 6 2 4 3 64 10 2 6 2 Summing the first n n n positive cubes the identity allows telescoping between consecutive brackets 1 3 2 3 3 3 4 3 n 3 superscript 1 3 superscript 2 3 superscript 3 3 superscript 4 3 normal superscript n 3 displaystyle 1 3 2 3 3 3 4 3 ldots n 3 1 2 0 2 3 2 1 2 6 2 3 2 10 2 6 2 n 1 n 2 2 n 1 n 2 2 absent superscript 1 2 superscript 0 2 superscript 3 2 superscript 1 2 superscript 6

    Original URL path: http://www.planetmath.org/cubeofaninteger (2016-04-25)
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  • cuboid with least surface | planetmath.org
    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 cuboid with least surface Let us determine among all cuboids i e rectangular parallelepipeds with a given volume k 3 superscript k 3 k 3 such one which has the least surface area Let the three edges of the cuboid beginning from a vertex be x x x y y y and z z z then we must start from the condition x y z k 3 x y z superscript k 3 xyz k 3 whence z k 3 x y z superscript k 3 x y z frac k 3 xy We get the expression f x y 2 y z z x x y 2 x y k 3 x k 3 y assign f x y 2 y z z x x y 2 x y superscript k 3 x superscript k 3 y displaystyle f x y 2 yz zx xy 2 left xy frac k 3 x frac k 3 y right 1 for the whole area of the surface of the cuboid Thus we have to make f x y f x y f x y a minimum when only the positive values of x x x and y y y can be taken into consideration The function f f f and its first order partial derivatives are continuous for all positive x x x and y y y According to the theorem of the parent entry a minimum can occur only when simultaneously f x x y y k 3 x 2 0 f y x y x k 3 y 2 0 f xxy yk3x2 0 f yxy xk3y2 0 displaystyle begin cases f prime x x y y frac k 3 x 2 0 f prime y x y x frac k 3 y 2 0 end cases These equations are true only for x y k x y k x y k

    Original URL path: http://www.planetmath.org/cuboidwithleastsurface (2016-04-25)
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  • curvature determines the curve | planetmath.org
    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 The curvature of plane curve determines uniquely the form and size of the curve i e one has the Theorem If s k s maps to 𝑠 𝑘 𝑠 is a continuous real function then there exists always plane curves satisfying the equation κ k s 𝜅 𝑘 𝑠 1 between their curvature κ 𝜅 and the arc length s 𝑠 All these curves are congruent Proof Suppose that a curve C 𝐶 satisfies the condition 1 Let the value s 0 𝑠 0 correspond to the point P 0 subscript 𝑃 0 of this curve We choose O 𝑂 as the origin of the plane The tangent and the normal of C 𝐶 in O 𝑂 are chosen as the x 𝑥 axis and the y 𝑦 axis with positive directions the directions of the tangent and normal vectors of C 𝐶 respectively According to 1 and the definition of curvature the equation d θ d s k s 𝑑 𝜃 𝑑 𝑠 𝑘 𝑠 for the direction angle θ 𝜃 of the tangent of C 𝐶 is valid in this coordinate system the initial condition is θ 0 when s 0 formulae sequence 𝜃 0 when 𝑠 0 Thus we get θ 0 s k t 𝑑 t ϑ s 𝜃 superscript subscript 0 𝑠 𝑘 𝑡 differential d 𝑡 assign italic ϑ 𝑠 2 which implies d x d s cos ϑ s d y d s sin ϑ s formulae sequence 𝑑 𝑥 𝑑 𝑠 italic ϑ 𝑠 𝑑 𝑦 𝑑 𝑠 italic ϑ 𝑠 3 Since x y 0 𝑥 𝑦 0 when s 0 𝑠 0 we obtain x 0 s cos ϑ t 𝑑 t y 0 s sin ϑ t 𝑑 t formulae sequence 𝑥 superscript subscript 0 𝑠 italic ϑ 𝑡 differential d 𝑡 𝑦 superscript subscript 0 𝑠 italic ϑ 𝑡 differential d 𝑡 4 Thus the function s k s maps to 𝑠 𝑘 𝑠 determines uniquely these functions x 𝑥 and y 𝑦 of the parameter s 𝑠 and 4 represents a curve with definite form and size The above reasoning shows that every curve which satisfies 1 is congruent with the curve 4 We have still to show that the curve 4 satisfies the condition 1 By differentiating the equations 4 we get the equations 3 which imply d x d s 2 d y d s 2 1 superscript 𝑑 𝑥 𝑑 𝑠 2 superscript 𝑑 𝑦 𝑑 𝑠 2 1 or d s 2 d x 2 d y 2 𝑑 superscript 𝑠 2 𝑑 superscript 𝑥 2 𝑑 superscript 𝑦 2 which means that the parameter s 𝑠 represents the arc length of the curve 4 counted from the origin Differentiating 3 we get because ϑ s k s superscript italic ϑ 𝑠

    Original URL path: http://www.planetmath.org/curvaturedeterminesthecurve (2016-04-25)
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  • curvature of Nielsen's spiral | planetmath.org
    right frac a cos t t 3 y d d t a t sin u u d u a sin t t superscript y normal d d t a superscript subscript t u u d u a t t displaystyle y prime frac d dt left a int infty t frac sin u u du right frac a sin t t 4 and hence the second derivatives x a t sin t cos t t 2 y a t cos t sin t t 2 formulae sequence superscript x normal a t t t superscript t 2 superscript y normal a t t t superscript t 2 x prime prime a cdot frac t sin t cos t t 2 quad y prime prime a cdot frac t cos t sin t t 2 Substituting the derivatives in 2 yields κ a 2 cos t t cos t sin t sin t t sin t cos t t t 2 a 2 cos 2 t a 2 sin 2 t t 2 3 2 normal κ normal superscript a 2 t t t t t t t t normal t superscript t 2 superscript superscript a 2 superscript 2 t superscript a 2 superscript 2 t superscript t 2 3 2 kappa a 2 cdot frac cos t t cos t sin t sin t t sin t cos t t cdot t 2 left frac a 2 cos 2 t a 2 sin 2 t t 2 right frac 3 2 which is easily simplified to κ t a κ t a displaystyle kappa frac t a 5 The arc length of Nielsen s spiral can also be obtained in a simple closed form using 3 and 4 we get s 1 t x 2 y 2 d t 1 t a 2 cos 2 t t 2 a 2 sin 2 t t 2 d t 1 t a t d t s superscript subscript 1 t superscript x normal 2 superscript y normal 2 d t superscript subscript 1 t superscript a 2 superscript 2 t superscript t 2 superscript a 2 superscript 2 t superscript t 2 d t superscript subscript 1 t a t d t s int 1 t sqrt x prime 2 y prime 2 dt int 1 t sqrt frac a 2 cos 2 t t 2 frac a 2 sin 2 t t 2 dt int 1 t frac a t dt i e s a ln t s a t displaystyle s a ln t 6 Note The expressions for x superscript x normal x prime and y superscript y normal y prime allow us determine as well d y d x y x sin t cos t tan t d y d x superscript y normal superscript x normal t t t frac dy dx frac y prime x prime frac sin t cos t tan t which says that the sense of the parameter t t t is

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