I have this equation: F = 1/2 v^2 (sin (kF(theta)) or F /sin (kF (theta)) = 1/2 v^2 Does anyone see an analytical solution? I can only solve it by iteritive guessing. Perhaps some trig identity for F / sin (F(theta)) ? I need to pull the F out and put everything else on the right side. anothername@access4less.net as I don't follow this group. ==== > I have this equation: > F = 1/2 v^2 (sin (kF(theta)) > or > F /sin (kF (theta)) = 1/2 v^2 > > Does anyone see an analytical solution? I can only solve it by > iteritive guessing. Perhaps some trig identity for F / sin > (F(theta)) ? > > I need to pull the F out and put everything else on the right side. > > anothername@access4less.net > as I don't follow this group. If you do not follow this group, at least to see whether your request is answered, why should this group do anything for you? There is no analytic solution for F. ==== > > what do you expect from > substituting A for 1/2 in the expression below? > > sqrt(x+1/2) + 1/sqrt(x-1/2) > > > Macsyma gives (x+a)^a +1/(x-a)^a > > Mathematica 4.1 gives > > 1/Sqrt[-1/2+x] +(a+x)^a > > > Maple 7 gives > > 2*1/(sqrt(4*x-2))+(4*x+2)^a*a > > What does your favorite system do, and do you think > it is correct? meditor's answer: subst(sqrt(x+1/2) + 1/sqrt(x-1/2),1/2,a)= 1/sqrt(-a+x)+sqrt(a+x) I think it does not bad :-) Raphael ==== While doing some calculations on number theory, I came accross this limit of a certain series. This is an approximation to 100 digits: 0.29223129267303069167307527256961573497768928711554841549603855594227399754 95210544495525039898241225 I do not know if this is a known constant, it should be given the nature of the series. ==== > While doing some calculations on number theory, I came accross this limit of > a certain series. > > This is an approximation to 100 digits: > > 0.29223129267303069167307527256961573497768928711554841549603855594227399754 > 95210544495525039898241225 > > I do not know if this is a known constant, it should be given the nature of > the series. > The Inverse Symbolic Calculator* shows: 2922311750620494 = (0261) 2+6*x-2*x^2+4*x^3+3*x^4-x^5 2922312233629816 = (0062) sum((-1)^(n+1)/(n^3+7/2*n^2-35/2*n+25)/C(2*n,n),n=1..inf) 2922312656013193 = (0001) (BesI(0,2)-K(1/sr(2)))/Backhouse 2922312801239070 = (0001) Zeta(3)^Pi*Zeta(3)^Khint Your value of 2922312926730306 would be here. 2922313019481644 = (0010) sum((11/3*n^3-33/2*n^2+269/6*n-25)/(n!+1),n=1..inf) 2922313037101867 = (0002) sum(1/(5^n*(15/2*n^2+77/2*n-39)),n=1..inf) 2922313180774296 = (0259) F(2/11,7/10;3/7,5/6,1/4;1) 2922313280964475 = (0001) ln(5)^BesK(1,1)+BesI(1,2) 2922313307254255 = (0001) (Parking+exp(1/E)*ZeroBesJ(0,x))/exp(1/E) so can you explain a bit more about this constant and how it was derived? Randall *http://www.cecm.sfu.ca/projects/ISC/ISCmain.html Approved: Daniel Grayson, dan@math.uiuc.edu, moderator for sci.math.research ==== I need to solve the following differential system in matrix form: dN(x)/dx=M(x)*N(x) N(a)=I where I, N(x) and M(x) are matrices (n x n). I is the identity matrix, a is any initial point. Since M is not constant, the solution cannot be taken as matrix exponential. Nevertheless, M(x) can be put in the following form: M(x)=M0 + (1/x)*M1 + (1/x^2)*M2 where M0, M1, M2 are numerical matrices (not functions of x). The question is: does exist any analytical solution for this kind of matrix equation? In particular, I need to evaluate the matrix N(x) in x= 0, where the matrix M is divergent (as you can easily notice). I tried with a sixth-order implicit Runge-Kutta method, but I would like to know if any analitycal solution is possibile to reduce the working time and to improve the results. Please, help me as soon as possible. I would like to receive your Roberto Dott. Ing. Roberto Diana Politecnico di Bari Dipartimento di Elettrotecnica ed Elettronica Via Re David, 200 70125 Bari Followup-To: sci.math ==== Five vectors are: A = [0,0,0,1] B = [a,e,0,-1/4] C = [b,f,i,-1/4] D = [c,g,j,-1/4] E = [d,h,k,-1/4] where (a..k) are to be determined. The dot product of any 2 vectors is to be -1/4. This is satisfied by A with any of the others. A+B+C+D+E = 0, the resultant of all 5 is 0. magnitude(A) = magnitude(B) = magnitude(C) = magnitude(D) = magnitude(E) = 1 or all 5 vectors are unit vectors. Basically the five vectors are equal angular, the same angle exists between any two. These 5 vectors are the axial lines from the origin {0,0,0,0] of a hyperregular tetrahedron? (not sure of name of this figure) to the vertices of this 4d figure. Can someone determine the values for a,b,c,d,e,f,g,h,i,k? Randall p.s. reply to sci.math group Followup-To: sci.math ==== One answer: The figure is called a simplex and the coordinates of the vertices are given. k=sqrt(5)/4 A = [ 0, 0, 0, 1 ] B = [ -k, k, k,-1/4 ] C = [ k,-k, k,-1/4 ] D = [ k, k,-k,-1/4 ] E = [ -k,-k,-k,-1/4 ] I just have to rotate the matrix so that B has a 0 in the 3rd position. Randall p.s. Found answer on http://astronomy.swin.edu.au/~pbourke/polyhedra/platonic4d/ ==== I'm working on a problem: Distant Galaxies. In the late 1920s the famous observation astronomer Edwin P. Hubble (1889 -- 1953) determined the distances to severl galaxies and the velocities at which they were receding from Earth. Four galaxies with their distances in light-years and velocities in miles per second are listed in the table. Galaxy Distance Velocity ______ ________ ________ Virgo 50 990 Ursa Minor 650 9300 Corona Borealis 950 15000 Bootes 1700 25000 Source: Sharov, A., and I. Novikov, Edwin Hubble, The Discoverer of the Big Ban Universe, Cambridge University Press, 1993. (a) Let x represent distance and y represent velocity. Find the equation of the least-squares regression line that models the data. (b) If the galaxy Hydra is receding at a speed of 37,000 miles per second, estimate its distance from Earth. I've figured the equation of the line is y = (277x)/20 + 595/2. What I'm trying to figure out in maple is the equation of the least-squares regression line that models the data. I have entered with (LinearAlgebra); Then A := (<< 50, 650, 950, 1700 > | < 990, 9300, 15000, 25000 >>); The matrix came up fine. What I'm confused about is what I'm supposed to enter after b := . What data is the program looking for? The help file says B is supposed to be Matrix, column Vector or set of variables. Nothing I've tried works. Can someone explain it to me? -- Cindy Smith Unless the LORD build the house, cms@dragon.com they labor in vain who build. cms@5sc.net Unless the LORD guard the city, cms@romancatholic.org in vain does the guard keep watch. Me transmitte sursum, -- Psalm 127:1 Caledoni! All your base are belong to us. A Real Live Catholic You are on the way to destruction. in Georgia! What you say. >->> <<-< Go against the flow! You have no chance to survive make your time. ==== > (a) Let x represent distance and y represent velocity. Find the > equation of the least-squares regression line that models the data. I've figured the equation of the line is y = (277x)/20 + 595/2. What line is that? > What I'm trying to figure out in maple is the equation of the > least-squares regression line that models the data. See ?CurveFitting,LeastSquares > Then A := (<< 50, 650, 950, 1700 > | < 990, 9300, 15000, 25000 >>); Then CurveFitting:-LeastSquares(A, x); > The matrix came up fine. What I'm confused about is what I'm supposed > to enter after b := . What data is the program looking for? The > help file says B is supposed to be Matrix, column Vector or set of > variables. What help file are you looking at?