What is a metric matrix?
I'm having a bit of trouble finding a source of study to metric matrices (I'm not even sure whether this is how we call them in English). It is ...
Category: Updates
When is $\mathbb{B}(\mathbb{X})$ compact?
Let $\mathbb{X}$ be a (real or complex) Banach Space and let $\mathbb{B}(\mathbb{X})$ be the space of all bounded linear operators of $\mathbb{X}$. Let $\...
prove the Riemann-Lebesgue lemma: $\int^b_af(x)\cos(nx)dx\rightarrow 0$ as $n\rightarrow \infty$ for any regulated function $f$
I would like to prove the Riemann-Lebesgue lemma, namely that $\int^b_af(x)\cos(nx)dx\rightarrow 0$ as $n\rightarrow \infty$ for any regulated function $f...
The volume of body in polar coordinates
If we have curve $r=F(\varphi)$ and two polar radii $\varphi=\alpha$ and $\varphi=\beta$ $(\alpha<\beta)$. Then volume of body which is bounded by abov...
![Show that if $H$ and $K$ are subgroups of one abelian group $G$, then $G_1=\{hk\mid h\in H\text{ and }k\in K\}$ is a subgroup of $G$ [duplicate]](https://breaking.oye.mlga.ek.gov.ng/uploads/show-that-if-h-and-k-are-subgroups-of-one-abelian-group-g-then-g-1-hk-mid-h-in-h-text-and-k-in-k-is-a-subgroup-of-g-duplicate_720.jpg)
Show that if $H$ and $K$ are subgroups of one abelian group $G$, then $G_1=\{hk\mid h\in H\text{ and }k\in K\}$ is a subgroup of $G$ [duplicate]
Here's the problem: Show that if $H$ and $K$ are subgroups of one abelian group $G$, then $G_1=\{hk\mid h\in H\text{ and } k\in K\}$ is a subgroup of...
Proof of stars and bars formula
I am trying to prove a formula (for ways of distributing $n$ identical balls among $r$ persons when each person may get any number of balls) ${n+r-1}\choo...
Product rule for gradient
I have two functions scalar functions of vector $\textbf{x}$, $f(\textbf{x})$ and $g(\textbf{x})$. I know the gradient of $f(\textbf{x})$ (i.e. $\triangle...
Total derivative vs chain rule
The total derivative as I understand it would be a linear map along the lines on the wikipedia page. https://en.wikipedia.org/wiki/Total_derivative Howeve...
Trigonometric Expression for $1 + \cos \alpha + \cos 2\alpha + \cdots + \cos n \alpha$ using complex numbers
This question is not a duplicate because I am asked here to use the fact that $1 + \cos \alpha + \cos 2 \alpha + \cdots + \cos n \alpha = Re (1 + z + z^{2...
Covariance of two random variables
I am trying to find the covariance of two random variables but I am not having any lucky. Just for simplicity lets say that my random variables are : X = ...
