My Question Define subsets $G$, $G_0$, and $G_1$ of $\mathbb{R}$ by \begin{align*} G &= \{x:x=r+n\sqrt{2}\ \text{for some $r$ in $\mathbb{Q}$ and $n$ in $\mathbb{Z}$}\},\\ G_0 &= \{x:x=r+2n\sqrt{2}\ \text{for some $r$ in $\mathbb{Q}$ and $n$ in $\mathbb{Z}$}\},\ \text{and}\\ G_1 &= \...

I want to use the Lagrange inversion theorem from Whittaker and Watson, p. 133, to find the power series of $\sin^{-1}$ at the origin. It should be a valid procedure because $\sin^{-1}$ is analytic at the origin. The theorem is stated as follows: Let $f(z)$ and $\phi (z)$ be functions of $z$ ana...

Here is the binary operation $ *: R\times R \backslash (0,0) $ defined by $ (a,b)(c,d)=(ac-bd,ad+bc) $. My idea is that to show this is a group ($R\times R \backslash (0,0), * $), I need to show that $ * $ is well-defined and associative and then show it has an identity and inverse. I am struggli...

I have been doggedly searching for a direct proof of the following theorem: Theorem 1: Let $H$ be a complex nonsingular $n\times n$ Hankel matrix. Then $H$ can be factorized $H = V^\top DV$ where $V$ is a complex $n\times n$ Vandermonde matrix and $D$ is a complex $n\times n$ diagonal matrix. T...

I'm looking for a pairing function $f(x,y)$ which gives unique values for every combination of integers $x,y>0$ and and as special property 0 for $x=0$ $\forall y$. Or to be more general we can also replace $0$ with constants. We do know the max values of $x$ and $y$ can achieve We are allowed...

Code borrowed from here (This question has been asked there before.). A few Definitions that needed: Dirichlet kernel: $$\tag{77}D_N(x) = \sum_{n=-N}^Ne^{inx} = \frac{\sin\left( (N+\frac12)x \right)}{\sin(x/2)} $$ $$\tag{78}s_N(f; x) = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x - t) D_N(t)\, dt $$ ...

I have the following question here... Find an equation of a plane through the origin such that the intersection between the plane and the elliptical cylinder $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is a circle. Wouldn't this simply just be $ax+by=0$??? The plane has to pass the origin, so $d=0$...

We have to find the minimum and maximum distance of a point $(9/5 ,12/5)$ from the ellipse $4(3x+4y)^2 +9(4x-3y)^2 =900$ . I got to know the center of ellipse would be $(0,0)$ . after that I am not getting any start. I think there would be some trick to solve it .

In Artin's "Galois Theory" P38, he said the function $f(x) = \frac{(x^2 - x + 1)^3}{x^2(x-1)^2}$ satisfies the properties of $f(x)=f(1-x)=f(\frac{1}{x})$. Is the function given by some rational step or just by a flash of insight？ if $f(0)$ is a number, then $f(0) = f(\frac{1}{0})$. So that the do...

I'm trying to solve the ex 5.11 from "Probability Essentials", which asks to show that if X is Poisson(λ) then 𝐸{|𝑋−𝜆|}= $\frac{2\lambda^\lambda e^{-\lambda}}{(\lambda -1)!} $ . I've shown that 𝐸{|𝑋−𝜆|} = ${2\lambda^\lambda e^{-\lambda}} $ $\sum_{k=1}^\infty \left(\frac{k\lambda^k}{(k+\lamb...

I'm having some problems solving this integral: $$ I = \mathcal{P} \int_{-\infty}^{+\infty} \frac{1-e^{2ix}}{x^2} \ dx$$ where $\mathcal{P}$ is the Cauchy principal value. The exercise suggests to use the fact that: $$I_* = \frac{1}{2} \operatorname{Re} \left[I\right]=\mathcal{P} \int_{-\infty}^{...

Reference image containing the statement Can someone please explain the below text and what it means in simple words? From what I understood it means that if x=m/2^n and if m is odd then x can be represented in binary in two ways? But how is that possible. I tried with 1/2 but how can 0.5 be repr...