http://twistedone151.wordpress.com Just another WordPress.com weblog Fri, 06 Nov 2009 09:21:56 +0000 http://wordpress.com/ hourly 1 http://twistedone151.wordpress.com/2008/04/11/physics-friday-15-hydrostatics/#comment-1818 Physics Friday 96 « Twisted One 151’s Weblog Fri, 06 Nov 2009 09:21:56 +0000 http://twistedone151.wordpress.com/?p=273#comment-1818 [...] Friday 96 By twistedone151 Suppose, as in this post, we have a fluid of density ρ, with some arbitrary three-dimensional object immersed in the [...] [...] Friday 96 By twistedone151 Suppose, as in this post, we have a fluid of density ρ, with some arbitrary three-dimensional object immersed in the [...]

]]> http://twistedone151.wordpress.com/2009/10/26/monday-math-94/#comment-1817 Monday Math 95 « Twisted One 151’s Weblog Mon, 02 Nov 2009 09:51:11 +0000 http://twistedone151.wordpress.com/2009/10/26/monday-math-94/#comment-1817 [...] that is nonzero only for , p prime and k a positive integer; in that case, . Thus . Now, using the relationship between Dirichlet convolution and Dirichlet series, we have that the Dirichlet series generating function for is thus the product of those for f-1(n) [...] [...] that is nonzero only for , p prime and k a positive integer; in that case, . Thus . Now, using the relationship between Dirichlet convolution and Dirichlet series, we have that the Dirichlet series generating function for is thus the product of those for f-1(n) [...]

]]> http://twistedone151.wordpress.com/2009/10/19/monday-math-93/#comment-1816 Monday Math 95 « Twisted One 151’s Weblog Mon, 02 Nov 2009 09:49:08 +0000 http://twistedone151.wordpress.com/2009/10/19/monday-math-93/#comment-1816 [...] is also zero in that case. Now, for n>1 with prime factorization , we have , with . I showed here that the Dirichlet inverse f-1(n) of a completely multiplicative function f(n) is the [...] [...] is also zero in that case. Now, for n>1 with prime factorization , we have , with . I showed here that the Dirichlet inverse f-1(n) of a completely multiplicative function f(n) is the [...]

]]> http://twistedone151.wordpress.com/2009/10/12/monday-math-92/#comment-1815 Monday Math 95 « Twisted One 151’s Weblog Mon, 02 Nov 2009 09:46:05 +0000 http://twistedone151.wordpress.com/?p=1730#comment-1815 [...] convolution, (Λ*1)(n)=ln(n). Note that since Λ(1)=0, the von Mangoldt function has no Dirichlet inverse. Now, let us consider a completely multiplicative function f(n), with Dirichlet inverse f-1(n), [...] [...] convolution, (Λ*1)(n)=ln(n). Note that since Λ(1)=0, the von Mangoldt function has no Dirichlet inverse. Now, let us consider a completely multiplicative function f(n), with Dirichlet inverse f-1(n), [...]

]]> http://twistedone151.wordpress.com/2009/10/05/monday-math-91/#comment-1814 Monday Math 95 « Twisted One 151’s Weblog Mon, 02 Nov 2009 09:40:14 +0000 http://twistedone151.wordpress.com/2009/10/05/monday-math-91/#comment-1814 [...] function is non-zero are , , with . There are ki terms for each pi, so , or in terms of the Dirichlet convolution, (Λ*1)(n)=ln(n). Note that since Λ(1)=0, the von Mangoldt function has no Dirichlet [...] [...] function is non-zero are , , with . There are ki terms for each pi, so , or in terms of the Dirichlet convolution, (Λ*1)(n)=ln(n). Note that since Λ(1)=0, the von Mangoldt function has no Dirichlet [...]

]]> http://twistedone151.wordpress.com/2008/01/07/monday-math-2-symmetry-and-integrals/#comment-1811 Physics Friday 95 « Twisted One 151’s Weblog Fri, 30 Oct 2009 09:22:02 +0000 http://twistedone151.wordpress.com/2008/01/07/monday-math-2-symmetry-and-integrals/#comment-1811 [...] substitution , we get . (the last step is due to symmetry, as the integrand is an even function; see here.) Now, by consulting a table of integrals (or via trigonometric substitution), we find that , and [...] [...] substitution , we get . (the last step is due to symmetry, as the integrand is an even function; see here.) Now, by consulting a table of integrals (or via trigonometric substitution), we find that , and [...]

]]> http://twistedone151.wordpress.com/2008/06/13/physics-friday-24-projectile-range/#comment-1810 Physics Friday 95 « Twisted One 151’s Weblog Fri, 30 Oct 2009 09:15:58 +0000 http://twistedone151.wordpress.com/?p=428#comment-1810 [...] not the distance along the ground to the impact point, and so differs from the h=0 case of Physics Friday 24. The only acceleration is due to gravity; so our basic kinematic equations tell us that and . [...] [...] not the distance along the ground to the impact point, and so differs from the h=0 case of Physics Friday 24. The only acceleration is due to gravity; so our basic kinematic equations tell us that and . [...]

]]> http://twistedone151.wordpress.com/2009/10/19/monday-math-93/#comment-1807 Monday Math 94 « Twisted One 151’s Weblog Mon, 26 Oct 2009 08:20:02 +0000 http://twistedone151.wordpress.com/2009/10/19/monday-math-93/#comment-1807 [...] here I showed that the Dirichlet series generating function for |μ| is . I also demonstrated here that |μ| and the Liouville function lambda(n) are Dirichlet inverses, so we can see that [...] [...] here I showed that the Dirichlet series generating function for |μ| is . I also demonstrated here that |μ| and the Liouville function lambda(n) are Dirichlet inverses, so we can see that [...]

]]> http://twistedone151.wordpress.com/2009/09/07/monday-math-88/#comment-1806 Monday Math 94 « Twisted One 151’s Weblog Mon, 26 Oct 2009 08:19:04 +0000 http://twistedone151.wordpress.com/?p=1675#comment-1806 [...] which have Dirichlet series generating functions and , respectively. You may also recall that here I showed that the Dirichlet series generating function for |μ| is . I also demonstrated here [...] [...] which have Dirichlet series generating functions and , respectively. You may also recall that here I showed that the Dirichlet series generating function for |μ| is . I also demonstrated here [...]

]]> http://twistedone151.wordpress.com/2009/10/12/monday-math-92/#comment-1802 Monday Math 93 « Twisted One 151’s Weblog Mon, 19 Oct 2009 09:26:12 +0000 http://twistedone151.wordpress.com/?p=1730#comment-1802 [...] Math 93 By twistedone151 Last Monday, I discussed the Dirichlet inverse. I showed that the Dirichlet inverse can be found by the [...] [...] Math 93 By twistedone151 Last Monday, I discussed the Dirichlet inverse. I showed that the Dirichlet inverse can be found by the [...]

]]> http://twistedone151.wordpress.com/2009/08/14/physics-friday-85/#comment-1799 Physics Friday 93 « Twisted One 151’s Weblog Fri, 16 Oct 2009 08:15:24 +0000 http://twistedone151.wordpress.com/?p=1620#comment-1799 [...] neglecting the ground state. First, let us examine the spacing between the energy levels. For our quantum particle in a box, we have energy levels , where nx, ny, and nz are positive integers. Using , we rewrite as . The [...] [...] neglecting the ground state. First, let us examine the spacing between the energy levels. For our quantum particle in a box, we have energy levels , where nx, ny, and nz are positive integers. Using , we rewrite as . The [...]

]]> http://twistedone151.wordpress.com/2009/05/22/physics-friday-73/#comment-1798 Physics Friday 93 « Twisted One 151’s Weblog Fri, 16 Oct 2009 08:14:23 +0000 http://twistedone151.wordpress.com/?p=1515#comment-1798 [...] the energy. For T>Tc, we have as before, . For T<Tc, we have . From this, we find the heat capacity at constant volume by . Below the condensation temperature, we take the derivative of the above, to [...] [...] the energy. For T>Tc, we have as before, . For T<Tc, we have . From this, we find the heat capacity at constant volume by . Below the condensation temperature, we take the derivative of the above, to [...]

]]> http://twistedone151.wordpress.com/2009/10/09/physics-friday-92/#comment-1797 Physics Friday 93 « Twisted One 151’s Weblog Fri, 16 Oct 2009 08:10:51 +0000 http://twistedone151.wordpress.com/?p=1724#comment-1797 [...] Friday 93 By twistedone151 Continuing from last Friday, we now find the behavior of an ideal Bose gas at a temperature below the condensation temperature [...] [...] Friday 93 By twistedone151 Continuing from last Friday, we now find the behavior of an ideal Bose gas at a temperature below the condensation temperature [...]

]]> http://twistedone151.wordpress.com/2009/10/05/monday-math-91/#comment-1789 Monday Math 92 « Twisted One 151’s Weblog Mon, 12 Oct 2009 09:16:25 +0000 http://twistedone151.wordpress.com/2009/10/05/monday-math-91/#comment-1789 [...] Math 92 By twistedone151 Last Monday, I discussed the Dirichlet convolution. We noted that it is commutative, associative, and has an [...] [...] Math 92 By twistedone151 Last Monday, I discussed the Dirichlet convolution. We noted that it is commutative, associative, and has an [...]

]]> http://twistedone151.wordpress.com/2009/10/02/physics-friday-91/#comment-1786 Physics Friday 92 « Twisted One 151’s Weblog Fri, 09 Oct 2009 08:29:05 +0000 http://twistedone151.wordpress.com/?p=1712#comment-1786 [...] Friday 92 By twistedone151 Last week, we began modeling the ideal Bose gas, including showing that it possesses the same classical limit [...] [...] Friday 92 By twistedone151 Last week, we began modeling the ideal Bose gas, including showing that it possesses the same classical limit [...]

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