Linked Questions

51
votes
3answers
16k views

Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using $\text{n}^{\text{th}}$ root of unity $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
7
votes
4answers
2k views

How do basis functions work?

Hopefully this isn't too broad of a question. I recently had it explained to me that the discrete Fourier transform was really a change in basis (thus the "dot product" like look of the sigma with ...
8
votes
4answers
152 views

Determinant of a $4 \times 4$ matrix $A$ and $(\det(A))^5$

Calculate $\det(A)$ and $\det(A)^5$: $$A= \begin{bmatrix}a&a&a&a\\a&b&b&b\\a&b&c&c\\a&b&c&d\end{bmatrix}$$ I found $\det A$ with Laplace expansion: $$a(...
4
votes
1answer
2k views

Autocorrelation of a Wiener Process proof

Given a Wiener process X, how do I prove this? $R_x(s,t) = E[X(s)X(t)] = min(s,t)$ There seems to be a trick with dividing to two cases of $s<t$ and $s>t$, but I can't figure out how this ...
8
votes
2answers
153 views

Proving that ${n+3\choose 3} =\frac{n+2}{2}\sum_{k=1}^{n+1}\csc^2\frac{k\pi}{n+2}$

Fancy physics predicts the equality $${n+3\choose 3} =\frac{n+2}{2}\;\sum_{k=1}^{n+1}\csc^2\frac{k\pi}{n+2}$$ which I can check (numerically and symbolically) for small $n$, but cannot prove for every ...
3
votes
5answers
426 views

An “isomorphism” between continuous and discrete mathematics

First of all, I should inform everyone that I am not a mathematician and my question might sound not at all rigorous or maybe even absurd to many of you. But I have been thinking about this problem ...
1
vote
1answer
820 views

Orthogonality of discrete sine functions

can somebody provide a proof of the attached result ? (excerpt from a thesis document). It seemed obvious but I got lost in the calculations. Many thanks Gerald orthogonality of discrete sine ...
0
votes
1answer
587 views

Diagonal approximation of the inverse of a sparse matrix

Given a large sparse matrix $\Sigma$, I want to find a (block) diagonal matrix $\Omega$ which approximates $\Sigma^{-1}$. In this specific problem, we know that $\Sigma$ is a block tridiagonal matrix....
2
votes
2answers
177 views

Find the spectrum of a compact operator

Consider the (compact) operator $T:C([0,1],\mathbb{R})\rightarrow C([0,1],\mathbb{R})$ s.t. \begin{equation} T(f)(x)=\int_0^1\min\{x,y\} f(y)dy \; . \end{equation} How could one find its spectrum? I ...
2
votes
2answers
58 views

Logistic function: where does it come from?

I read the book titled "Seventeen Equations that Changed the World" where it explains how the equation [A] $x_{t+1}=k \ x_t \cdot (1-x_t)$ where $x_t$ is the population of a certain species at ...
0
votes
0answers
122 views

A specific type of tridiagonal matrix?

$$\begin{bmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix}$$ In a talk on positive definite matrices, Prof. Strang mentions that he remembers the eigenvalues of ...
1
vote
1answer
87 views

Find the limit of $x_{n+1} = \frac{1}{2} x_n ^ 2 - 1, n \ge 1, x_1 = 1/3$ [closed]

I’ve already found that the limit $L = 1+\sqrt{3}$ or $L=1-\sqrt{3}$. Since this sequence does not totally an increasing sequence or decreasing sequence, it’s hard for me to except a root. Can anybody ...
0
votes
2answers
61 views

How do I intuitively process what's physically going on with the Planar Laplace/Poisson equations?

By definition the PDEs are used to model equilibrium phenomena, involving only spatial variables. Say for example I was given the two-dimensional Laplace equation,$\frac{\partial^2 u}{\partial x^2}+\...
0
votes
1answer
66 views

When finding the spectrum of $(Ku)(x)=\int^1 _0 k(x,y)u(y) dy$, where does $\lambda u(x)=\int^x _0 yu(y) dy + x \int^1 _x u(y) dy$ come from?

The integral operator $K:L^2 ([0,1]) \to L^2([0,1])$ is defined by $$(Ku)(x)=\int^1 _0 k(x,y)u(y) dy$$ Where $k(x,y)=min\{x,y\}$ for $0 \leq x, y \leq 1$ When finding the spectrum ok $K$ we let $Ku=\...
0
votes
1answer
32 views

How to properly define the “discretization of a functional”?

In the derivation of the path integral formulation of quantum mechanics, most Physics books end up finding the following (or similar) expression: $$K(q',t';q,t)=\lim_{N\to \infty}\int\left[\prod_{k=1}...

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