In the book "Understanding Analysis, second edition" by Stephen Abbot, the unboundedness of the set of natural number $\mathbb{N}$ is proven as the following proof:
Assume, for contradiction, that $\mathbb{N}$ is bounded above. By the Axiom of Completeness (AoC), $\mathbb{N}$ should then have a ...
Let $X \in R^n$ be a compact convex set, and $f:X \to X$ be a continuous function. Then, can we say that from all $x_0 \in X$, the fixed-point iterations $x_{k+1}=f(x_k)$ to converge to some fixed-point $\bar{x}(x_0) \in X$?
If not, what are the conditions that $f$ must satisfy such that the iter...
I'm studying with the book 'Numerical Linear Algebra' written by L. N. Trefethen, and I wrote code regarding the text below:
Here is a numerical example. Let A be a 200 x 200 matrix whose entries are independent samples from the real normal distribution of mean 2 and standard deviation 0.5/$\sqr...
I was wondering if the following integral has a closed-form solution?
$$I(x) = \int J_0(x)\sin(ax)\mathrm{d}x$$
where $a$ is a constant.
I know the answer for the case when $a=1$, see here.
I tried the similar method in that link but I was stuck.
Integrating by parts yields
$$
I(x) =x J_0(x)\sin(...
I have 6 (A…F) noisy 3D normal vectors <x_hat, y_hat, z_hat > and noisy point cloud points <x, y, z> that form a cube and are related by the following vector operations:
A cross B = B cross C = C cross D = D cross A
A cross E = E cross C = F cross A = C cross F
A=-C
B=-D
E=-F
(A cross B) dot E = ...
Let $A, B$, and $C$ be three points in $\mathbb{R}^2$ such that $A = (x_1,y_1), B=(x_2,y_2)$, and $C= (x_3,y_3)$ where $x_1<x_2<x_3$ and $y_1 > y_2 > y_3$. In other words, $B$ is to the upper left of $C$ and $A$ is to the upper left of $B$.
What is the curve $y$ of fastest descent that contains t...
If $a, b, c, d$ are positive real numbers such that $c^2+d^2=(a^2+b^2)^2$, prove that $$\frac{a^2}{c}+\frac{b^2}{d} \geq1 $$
and equality hold iff $ad=bc$.
I need some help with a question. I have to calculate
$$\lim_{x \to 3}\frac{x^2}{x - 3}\int_3^x \frac{\sin t}{t}dt.$$
If I'm not wrong, we can write
$$\sin(x) = \sum_{n = 0}^{\infty}\frac{x^{2n+1}}{(2n+1)!} \Longrightarrow \frac{\sin(x)}{x}=\sum_{n = 0}^{\infty}\frac{x^{2n}}{(2n+1)!},$$
then
$$...
Take the density of a generalized student-t, i.e.,
\begin{align*}
p( y_t | \sigma , \mu, \nu ) = \frac {\Gamma (\frac {\nu +1}{2})}{\Gamma (\frac {\nu }{2}){\sqrt {\pi \nu }}{\sigma }\,}\left(1+{\frac {1}{\nu }}\left({\frac {y_t- \mu }{\sigma }}\right)^{2}\right)^{-\frac {\nu +1}{2}}
\end{alig...
Original Problem:
Let $z$ be a complex number. The number $1$ is written on a board. You perform a series of moves, where in each move you may either
replace the number $w$ written on the board with $z$$w$
or
replace the number $w$ with a different complex number $w'$ so that
$$\max(\lvert\oper...
