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...
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...
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...
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...
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...
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...
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...
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...
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!
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 +...
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...
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...