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  • asymptote of Lam\'e's cubic | planetmath.org
    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 asymptote of Lamé s cubic We will show that the Lamé s cubic x 3 y 3 a 3 superscript x 3 superscript y 3 superscript a 3 displaystyle x 3 y 3 a 3 1 where a a a is a positive constant has the line y x g x y subscript normal x g x y underbrace x g x as its asymptote Because the equation 1 of the curve is symmetric with respect to x x x and y y y the curve is symmetric about the line y x y x y x From the solved form y a 3 x 3 3 f x y subscript normal 3 superscript a 3 superscript x 3 f x displaystyle y underbrace sqrt 3 a 3 x 3 f x 2 of 1 we see that every real value of x x x gives one point of the curve psaxes Dx 9 Dy 9 0 0 3 5 3 5 3 5 3 5 x x x y y y a a a a a a psplot linecolor blue 3 511 x 3 exp sub 1 3 div exp psplot linecolor blue 13 50 x 3 exp 1 sub 1 3 div exp sub Lamé s cubic y a 3 x 3 3 y 3 superscript a 3 superscript x 3 y sqrt 3 a 3 x 3 blue The difference Δ f x g x normal Δ f x g x Delta f x g x represents the distance of a point x y x y x y of the curve and the point of the asserted asymptote y x y x y x with the same abscissa x x x We multiply the numerator and denominator with the expression a 3 x 3 3 2 x a 3 x 3 3 x 2 superscript 3 superscript a 3 superscript x 3 2 x 3 superscript a 3 superscript x 3 superscript x 2 sqrt 3 a 3 x 3 2 x sqrt 3 a 3 x 3 x 2 for being able to utilise the polynomial formula u v u 2 u v v 2 u 3 v 3 u v superscript u 2 u v superscript v 2 superscript u 3 superscript v 3 u v u 2 uv v 2 u 3 v 3 getting Δ normal Δ displaystyle Delta f x g x absent f x g x displaystyle f x g x a 3 x 3 3 x 1 absent 3 superscript a 3 superscript x 3 x 1 displaystyle frac sqrt 3 a 3 x 3 x 1 a 3 x 3 3 3 x 3 a 3 x 3 3 2 x a 3 x 3 3 x 2 absent superscript 3 superscript a 3 superscript x 3 3 superscript x

    Original URL path: http://www.planetmath.org/asymptoteoflamescubic (2016-04-25)
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  • average value of function | planetmath.org
    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 3A53A04 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 3A53A04 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 average value of function The set of the values of a real function f f f defined on an interval a b a b a b is usually uncountable and therefore for being able to speak of an average value of f f f in the sense of the average value A V a 1 a 2 a n n j 1 n a j j 1 n 1 fragments A normal V normal subscript a 1 subscript a 2 normal subscript a n n superscript subscript j 1 n subscript a j superscript subscript j 1 n 1 displaystyle A V frac a 1 a 2 ldots a n n frac sum j 1 n a j sum j 1 n 1 1 of a finite list a 1 a 2 a n subscript a 1 subscript a 2 normal subscript a n a 1 a 2 ldots a n of numbers one has to replace the sums with integrals Thus one could define A V f a b f x d x a b 1 d x formulae sequence A V assign f superscript subscript a b f x d x superscript subscript a b 1 d x A V f frac int a b f x dx int a b 1 dx i e A V f 1 b a a b f x d x formulae sequence A V assign f 1 b a superscript subscript a b f x d x displaystyle A V f frac 1 b a int a b f x dx

    Original URL path: http://www.planetmath.org/averagevalueoffunction (2016-04-25)
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  • basal units | planetmath.org
    3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26D15 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 3A11 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 3A11 00 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql

    Original URL path: http://www.planetmath.org/basalunits (2016-04-25)
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  • base and height of triangle | planetmath.org
    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 3A15A03 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 base and height of triangle base h 1 subscript h 1 h 1 base h 2 subscript h 2 h 2 Considering the area of a triangle one usually names a side of the triangle to be its base For expressing the calculation way of the area of the triangle one then uses the height a k a altitude which means the perpendicular distance of the vertex opposite to the base side from the line determined by the base In the above two triangles the heights h 1 subscript h 1 h 1 and h 2 subscript h 2 h 2 correspond the horizontal bases One calls foot of the height the projection of the vertex onto the line of the base The rule for the calculation reads area base times height divided by 2 In the below figure there is the illustration of the rule The parallelogram A B C D A B C D ABCD has been divided by the diagonal B D B D BD into two triangles which are congruent by the ASA criterion see the alternate interior angles Thus the both triangles have the areas half of the area of the

    Original URL path: http://www.planetmath.org/baseandheightoftriangle (2016-04-25)
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  • basis of ideal in algebraic number field | planetmath.org
    α 1 subscript m 2 subscript α 2 normal subscript m n subscript α n m 1 alpha 1 m 2 alpha 2 ldots m n alpha n where the m i subscript m i m i s run all rational integers form precisely all numbers of mathfrak a One has also α 1 α 2 α n subscript α 1 subscript α 2 normal subscript α n mathfrak a alpha 1 alpha 2 ldots alpha n i e the basis of the ideal can be taken for the system of generators of the ideal Since α 1 α 2 α n subscript α 1 subscript α 2 normal subscript α n alpha 1 alpha 2 ldots alpha n is a basis of the field extension K ℚ K ℚ K mathbb Q any element of mathfrak a is uniquely expressible in the form m 1 α 1 m 2 α 2 m n α n subscript m 1 subscript α 1 subscript m 2 subscript α 2 normal subscript m n subscript α n m 1 alpha 1 m 2 alpha 2 ldots m n alpha n It may be proven that all bases of an ideal mathfrak a have the same discriminant Δ α 1 α 2 α n normal Δ subscript α 1 subscript α 2 normal subscript α n Delta alpha 1 alpha 2 ldots alpha n which is an integer it is called the discriminant of the ideal The discriminant of the ideal has the minimality property that if β 1 β 2 β n subscript β 1 subscript β 2 normal subscript β n beta 1 beta 2 ldots beta n are some elements of mathfrak a then Δ β 1 β 2 β n Δ α 1 α 2 α n or Δ β 1 β 2 β n 0 formulae sequence normal Δ subscript β 1 subscript β 2 normal subscript β n normal Δ subscript α 1 subscript α 2 normal subscript α n or normal Δ subscript β 1 subscript β 2 normal subscript β n 0 Delta beta 1 beta 2 ldots beta n geqq Delta alpha 1 alpha 2 ldots alpha n quad mbox or quad Delta beta 1 beta 2 ldots beta n 0 But if Δ β 1 β 2 β n Δ α 1 α 2 α n normal Δ subscript β 1 subscript β 2 normal subscript β n normal Δ subscript α 1 subscript α 2 normal subscript α n Delta beta 1 beta 2 ldots beta n Delta alpha 1 alpha 2 ldots alpha n then also the β i subscript β i beta i s form a basis of the ideal mathfrak a Example The integers of the quadratic field ℚ 2 ℚ 2 mathbb Q sqrt 2 are l m 2 l m 2 l m sqrt 2 with l m ℤ l m ℤ l m in mathbb Z Determine a basis α 1 α 2 subscript α 1

    Original URL path: http://www.planetmath.org/basisofidealinalgebraicnumberfield (2016-04-25)
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  • Bernhard Riemann | planetmath.org
    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 3A11R04 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 3A11R04 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 3A06B10 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 3A06B10 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

    Original URL path: http://www.planetmath.org/bernhardriemann (2016-04-25)
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  • Bernoulli equation | planetmath.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 3A01A55 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 3A01A55 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 Bernoulli equation The Bernoulli equation has the form d y d x f x y g x y k d y d x f x y g x superscript y k displaystyle frac dy dx f x y g x y k 1 where f f f and g g g are continuous real functions and k k k is a constant 0 absent 0 neq 0 1 absent 1 neq 1 Such a nonlinear equation is got e

    Original URL path: http://www.planetmath.org/bernoulliequation (2016-04-25)
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  • Bernoulli polynomials and numbers | planetmath.org
    x 1 b n t dt x n 1 The constant term of b n x subscript b n x b n x is the n th superscript n th n mathrm th Bernoulli number B n subscript B n B n The Bernoulli polynomial is often denoted also B n x subscript B n x B n x The uniqueness of the solution b n x subscript b n x b n x in 1 is justificated by the Lemma For any polynomial f x f x f x there exists a unique polynomial g x g x g x with the same degree satisfying x x 1 g t d t f x superscript subscript x x 1 g t d t f x displaystyle int x x 1 g t dt f x 2 Proof For every n 0 1 2 n 0 1 2 normal n 0 1 2 ldots the polynomial g n x x x 1 t n d t x 1 n 1 x n 1 n 1 fragments subscript g n fragments normal x normal normal superscript subscript x x 1 superscript t n d t superscript x 1 n 1 superscript x n 1 n 1 g n x int x x 1 t n dt frac x 1 n 1 x n 1 n 1 is monic and its degree is n n n If the coefficient of x n superscript x n x n in f x f x f x is a 0 subscript a 0 a 0 then the difference f x a 0 g n x f x subscript a 0 subscript g n x f x a 0 g n x is a polynomial of degree n 1 n 1 n 1 Correspondingly we obtain f x a 0 g n x a 1 g n 1 x f x subscript a 0 subscript g n x subscript a 1 subscript g n 1 x f x a 0 g n x a 1 g n 1 x having the degree n 2 n 2 n 2 and so on Finally we see that f x a 0 g n x a 1 g n 1 x a n g 0 x f x subscript a 0 subscript g n x subscript a 1 subscript g n 1 x normal subscript a n subscript g 0 x f x a 0 g n x a 1 g n 1 x ldots a n g 0 x must be the zero polynomial Therefore f x f x displaystyle f x a 0 g n x a 1 g n 1 x a n g 0 x absent subscript a 0 subscript g n x subscript a 1 subscript g n 1 x normal subscript a n subscript g 0 x displaystyle a 0 g n x a 1 g n 1 x ldots a n g 0 x i 0 n a i g n i

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