Annals of Mathematics is one of the most prestigious journals in mathematics. My question simply is: Question: What should be the general characteristics of a paper to have a chance for being published in "Annals of Mathematics"? Should it be for example, too long, very technical or a break...

the problem states: "Show that $\cos(n\theta)$ is a polynomial in $\cos(\theta).$" Now, using De Moivre's and Binomial theorems i get that $$\cos(n\theta) = \sum_{k = 0, evens}^{n}\binom{n}{k}\cos^{n-k}(\theta)(\pm(1-\cos^{2}(\theta))^{k/2})$$ Note that the $\pm$ here does not mean both, but it ...

Does that sound about right? Assume that $X$ is uniform on the interval $[0,1]$. Plot $P\big(\big|X\big|\geq \varepsilon\big)$ and $\dfrac{E\big(X^2\big)}{\varepsilon^2}$ as functions of $\varepsilon$. How does this relate to Chebushev's inequality? Attempt: $ f(x) = \left\{ \begin{array}{l...

I want to use chebyshev interpolation. But i am a little confused for finding chebyshev nodes. I use the following figure to illustrate my problem. Consider i have a vector of numbers i depicted as a line In "A". In "B", the red points are the chebyshev nodes. How can i choose these points? (i ha...

For this question, I don't understand the highlighted part of the solution I thought it should be >5, but then 6?

For an arbitrary number of dimensions, I know that the mean minimizes the distance using the L2 norm and that the geometric median minimizes the distance function using the L1 norm (though I have yet to find a good proof of this). So what minimizes the $L_{\infty}$(or Chebyshev) norm? Context: D...

Suppose I have a sinusoid $f(t) = A \cos(\omega t + \theta)$ and I want to evaluate it at Chebyshev points of the second kind ($\cos(\frac{2 \pi i}{N}), 0 \le i \le N, i \in \mathbb{Z}$), and then take the DFT of it to form the coefficients of an interpolating Chebyshev polynomial. Is there any ...

Suppose the mean noon-time temperature for September days in San Diego is 24∘ and the standard deviation is 4.6. (Temperature in this problem is measured in degrees celsius) Using Chebyshev’s theorem, what is the minimal probability (in percents) that the noon-time temperature of a september day...

If you roll a dice twice, and subtract $ Result_1 $ from $ Result_2 $, in what interval with 97% probability will lie number of all zeros, if we will do this experiment 1200 times. I assume, I should use Chebyshev's inequality. For one experiment, the chance of getting zero is $ 6/36 $. How ...

The Chebyshev polynomials can be defined recursively as: $T_0(x)=1$; $T_1(x)=x$; $T_{n+1}(x)=2xT_n(x) + T_{n-1}(x)$ The coefficients of these polynomails for a function, $\space f(x)$, under certain conditions can be obtained by the following integral: $$a_n=\frac{2}{\pi}\int_{-1}^{1}\frac{f...

Prove that the gaussian Q function is bounded on the top by $\frac{1}{2x^2 }$, i.e. $Q(x)\le\frac{1}{2x^2}$ for $x\ge0$, using the chebyshev inequality and the nakagami-m distribution with m=0.5(that reduces it to half normal distribution). This is also known as the chebyshev bound I think. Can'...

Prove that the Gaussian $Q$ function is bunded on the top by $1/2x^2$, i.e. $Q(x)\le 1/2x^2$. for $x\ge 0$ using the chebyshev inequality and the Nakagami $m$ distribution with $m=0.5$ that reduces it to half normal distribution.

My question is: Find the best 1-degree approximating polynomial of $f(x)=2x^3+x^2+2x-1$ on $[-1,1]$ in the uniform norm(NOT in the least square sense please)? Orginially, as the title of the post suggests, I'm asking the general problem: given an $n$-th degree polynomial $p(x)$ on $[-1,1]$, find...

I'm trying to evaluate the integral of the Chebyshev polynomials of the first kind on the interval $-1 \leq x \leq 1 $ . My idea is to use the closed form $$T_n(x) = \frac{z_1^n + z_2^n}{2}, $$ where $z_1 = (x + \sqrt{x^2 - 1})$ and $z_2 = (x - \sqrt{x^2 - 1})$, giving the following integral: $...

Chebyshev approximation approximates a function by fitting a weighted sum of Chebyshev polynomials to it. It requires evaluating a function at $N$ sample points to form the weighting coefficients. Interestingly the weighting coefficients ($c_k$ in the link) approach zero as k gets larger, regar...