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  1. E

    Derivative of best approximation

    Say that we have a continuous, differentiable function f(x) and we have found the best approximation (in the sense of the infinity norm) of f from some set of functions forming a finite dimensional vector space (say, polynomials of degree less than n or trigonometric polynomials of degree less...
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    Function + tangent line = 0

    Thank you for your reply! Yes, f(x) is continuous. And indeed f(x) + g(x,y) is monotone. What I meant to ask was if there is a way to explicitly find that value for x at which f+g=0 other than the "brute force" way of inverting the expression? Or perhaps, more generally - does an equation of...
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    Function + tangent line = 0

    Say we have two functions with the following properties: f(x) is negative and monotonically approaches zero as x increases. g(x,y) is a linear function in x and is, for any given y, tangent to f(x) at some point x_0(y) that depends on the choice of y in a known way. Additionally, for any...
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    Max of sum of sines

    Hi! Consider the function \frac{d^n}{dx^n} \sum_{k=1}^m \sin{kx}, \quad 0 \leq x \leq \pi/2 . If n is odd this function attains its largest value, \sum_{k=1}^m k^n at x=0 . But what about if n is even? Where does the maximum occur and what value does it take? Any help is much...
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    Approximating function by trigonometric polynomial

    Thank you mfb for your reply! Yes, that was my original idea as well. If g is the approximation in the RHS of (1) , then I reasoned that the optimal result should be when (f-g) \perp f . However, (f-g, f) is a linear function in the coefficients a_n so there are no extrema (I am...
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    Approximating function by trigonometric polynomial

    Hi! Say that we wish to approximate a function f(x), \, x\in [0, 2\pi] by a trigonometric polynomial such that f(x) \approx \sum_{|n|\leq N} a_n e^{inx} \qquad (1) The best approximation theorem says that in a function space equipped with the inner product (f,g) = \frac{1}{2...
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    A tricky finite series!

    Hi! I've encountered the series below: \sum_{l=0}^{k-1} (r+l)^j (r+l-k)^i where r, k, i, j are positive integers and i \leq j . I am interested in expressing this series as a polynomial in k - or rather - finding the coefficients of that polynomial as i,j changes. I have reasons to...
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    Cauchy expansion of determinant of a bordered matrix

    Hi! It just dawned on me that any such matrices (I suppose there are only 4 places A could go ^^, ) are related by simple permutations. Since any permutation matrix has determinant + or - 1 then what you say must be true. Thank you for the enlightenment! =)
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    Cauchy expansion of determinant of a bordered matrix

    The Cauchy expansion says that \text{det} \begin{bmatrix} A & x \\[0.3em] y^T & a \end{bmatrix} = a \text{det}(A) - y^T \text{adj}(A) x , where A is an n-1 by n-1 matrix, y and x are vectors with n-1 elements, and a is a scalar. There is a proof in Matrix Analysis by Horn and...
  10. E

    Addition to a random matrix element

    Hi all! I have no application in mind for the following question but it find it curious to think about: Say that we have a square matrix where the sum of the elements in each row and each column is zero. Clearly such a matrix is singular. Suppose that no row or column of the matrix is the...
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    Properties of a special block matrix

    Thank you! I think I shall have to return to the drawing board for a closer investigation :)
  12. E

    Properties of a special block matrix

    That's a fair point. I was playing around with a different matrix - Hermitian and also symmetric about the anti-diagonal. Turns out that the eigenvectors are closely related to the eigenvectors of the matrix R above so I was curious about their structure. It seems reasonable that the upper and...
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    Properties of a special block matrix

    Hi folks! I've encountered the matrix below and I'm curious about its properties; R= \begin{pmatrix} 0 & N-S\\ N+S & 0 \end{pmatrix} where R, N and S are real matrices, R is 2n by 2n, N is n by n symmetric and S is n by n skew-symmetric. Clearly R is symmetric so the...
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    Prove the Lyapunov equation

    The sufficiency can be obtained by considering B=\int_{0}^{∞} e^{A^τ t} Q e^{A t} dt Inserting into the Lyapunov equation gives AB + BA^{T} = A \int_{0}^{∞} e^{A^τ t} Q e^{A t} dt + \int_{0}^{∞} e^{A^τ t} Q e^{A t} dt A^{T} = \int_{0}^{∞} \frac{d}{dt} (e^{A^τ t} Q e^{A t}) dt = [e^{A^τ...
  15. E

    Zero as an element of an eigenvector

    Thank you for your reply! Is there a way to determine from the matrix whether a zero will appear without calculating the eigenvector explicitly?
  16. E

    Zero as an element of an eigenvector

    Quick question on eigenvectors; Are there any general properties of a matrix that guarantee that a zero will or will not appear as an element in an eigenvector? Thank you!
  17. E

    Help with Kinetic Energy concept of two masses that move the same distance

    Don't forget that the KE depends on the speed of the blocks, not just the masses and certainly not (at least not explicitly) the distance they move. Hint: consider instead the amount of work done by the force on the masses.
  18. E

    Roots of linear sum of Fibonacci polynomials

    Thanks for your reply! Interesting observations. Yes, you're right! The product of the solutions will be 2 since G can be written as the characteristic polynomial of a matrix with determinant 2. Since the solutions come in complex conjugated pairs this suggests some pretty strict bounds. The...
  19. E

    Roots of linear sum of Fibonacci polynomials

    For what complex numbers, x, is Gn = fn-1(x) - 2fn(x) + fn+1(x) = 0 where the terms are consecutive Fibonacci polynomials? Here's what I know: 1) Each individual polynomial, fm, has roots x=2icos(kπ/m), k=1,...,m-1. 2) The problem can be rewritten recursively as Gn+2 = xGn+1 +...
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    Implicit Hyperbolic Function

    Hi Mute! Thanks for your reply. The problem is actually a result of a polynomial of degree n, which has been rewritten in it's present form. The coefficients of all n+1 terms are non-zero integers dependent on a, except for the leading term. I could probably do a more thorough analysis on the...
  21. E

    Implicit Hyperbolic Function

    This is not a bad idea. All in all I can boil things down to tanh(ny) = cosh(y) which has the expected roots (found numerically). Solving for n is straight forward but inverting seems impossible, at least in terms of standard functions. If n is a positive integer, what can be said about y...
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    Implicit Hyperbolic Function

    Hi tiny-tim, Thanks for your reply. I did try this and it cleans things up a bit. In particular it becomes clear that a=-2 is a convenient choice since we get 0 = (4+2a)sinh(n*y)sinh(y) + 2a[sinh(n*y) - cosh(n*y)cosh(y)] after expanding cosh((n-1)y). However, it is not clear to me how to...
  23. E

    Implicit Hyperbolic Function

    Hi all, In studying the eigenvalues of certain tri-diagonal matrices I have encountered a problem of the following form: {(1+a/x)*2x*sinh[n*arcsinh(x/2)] - 2a*cosh[(n-1)*arcsinh(x/2)]} = 0 where a and n are constants. I'm looking to find n complex roots to this problem, but isolating x...
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    A slice causes a tennisball to spin and deviate from its normal parabolic path.

    This is actually a fluid dynamics problem. The balls spins - thus, as it moves through the surrounding air the flow of air will move faster around one side of the ball and slower on the other. By Bernoulli's principle a faster flow means a lower pressure. Thus we end up with a reduced air...
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    Quantum Numbers

    n=2, L=m(l)=0 should be correct for the reasons you have outlined above. If you are being asked for the corresponding Set(!) of quantum numbers I believe you want to include both the spin-up and spin-down states in your answer.
  26. E

    Solve it using energy conservation?

    Since you know the initial velocity, why not try to solve it using energy conservation?
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    Three coupled pendula

    Thanks a bunch. That did the trick =) Of pure curiosity, what would happen if we in fact had a degenerating force?
  28. E

    Three coupled pendula

    Homework Statement Derive the equations of motion for three identical pendula A, B and C, of mass m and length L coupled together (A to B and B to C) with two identical springs of low spring constant k. Can't quite appreciate the forces acting on these pendula as they all should be...
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