If $R$ is a left Artin ring, then any non-zero left module $M$ must contain a simple submodule. Since $M$ is not zero, we can take a non-zero element $m \in M$ and thus we obtain the finitely generated module $Rm$. Since $R$ is a left Artin ring, the finitely generated module $Rm$ must have finit...

Today, in a casual discussion, the following problem came up: suppose $a_n$ is a sequence converging to $1$ from above (i.e., $a_n > 1$ for all $n$). Then, consider the series $$\sum_{n=1}^\infty \frac{1}{n^{a_n}}$$ Does this series converge or diverge? It seems like it diverges, since it's ever ...

A function $f : \Bbb Z \times \Bbb Z \rightarrow \Bbb Z$ is defined as $f(u,v) = 3u + 6v.$ Is the function surjective? Prove it. I had the following proof. Proof Pick $x = 2$, then $3u + 6v = 2 \Rightarrow 3(u + 2v) = 2$ Let $y = u + 2v$ $\exists y \in \Bbb Z \times \Bbb Z$. Thus $3y = 2 \Rightar...

Consider the Mellin integral $$K(s)=\int_{1/2}^1 \zeta\bigg(-\frac{1}{\log x}\bigg)~x^{-s}~dx $$ Where $\zeta(\cdot)$ is the Riemann zeta function defined for real $1/e<x<1.$ $K(s)$ is holomorphic on $\Bbb C$ due to the convergence on $-\infty<\Re(s)<\infty$. We have the classical product repres...

Take a partition of $\Bbb R^2_{\gt 0}$ by the union of functions indexed by real $t\ge 0$ $$\mathcal F:=\bigg \lbrace \mathcal M[\chi_t(x)]\cup \mathcal M\bigg[\frac{1}{1-\chi_t(x)}\bigg] \bigg \rbrace$$ where $$\mathcal M[\chi_t(x)]:=\int_{(0,1)} \chi_t(x)x^{s-1}~dx= 2\sqrt{\frac{t}{s}}K_1(2\sq...

Let $W$ be the subspace of $\Bbb C$ linear combination of the following functions:$$f_1(z)=\sin z,\qquad f_2(z)=\cos z,\qquad f_3(z)=\sin2z,\qquad f_4(z)=\cos2z.$$ Let $T$ be the linear opeartor on $W$ given by complex differentition.Which of the following statements are true? $1$.Dimension of $...

I don´t know how to derive the reduction formula for this integral: $$\int \frac{1}{x^m\sqrt{1-x^2}} \, dx$$ I know you have to use integration by parts, but I have tried everything I can come up with and I don´t get anywhere. These are some of the ones I've tried: $u=\frac{1}{x^m},\mspace{0.4cm}...

I saw a post on Fourier transform of a continuous Kaiser window. But I need a compact formula for DTFT of a Kaiser window. When I plotted the DTFT, I noticed that the nulls for DTFT occurs at slightly different frequencies in $[-\pi ,\pi]$ compared to continuous Fourier transform. $w[n]=\frac{I_0...

This is more of a fun puzzle than a problem born out of necessity, but I would like to know if there is a way to represent the more fundamental operations of addition and subtraction without using the operators + and -, or some "coverup" function that does just that behind the scenes, so to speak...

There is a longer integral for which integration by parts $\displaystyle\int udv=uv-\int vdu$ was attempted as it came across in research: $$\frac i{2\pi}\int_0^{2\pi}\underbrace{\ln\left(1+\frac{e^{-i t}+1}a\ln\left(1-\frac1be^{\frac{e^{it}+1}a}\right)\right)}_ud\bigg[\underbrace{\ln\left(e^\fra...

the problem We call a binary sequence of length $n$ a string with $n$ digits of $0$ or $1$. For such a sequence, $A$, of finite length,$f(A)$ represents a transformation where every 1 in $A$ becomes $0,1$ and every $0$ of $A$ becomes $1,0$. e.g $f((1,0,1))=(0,1,1,0,0,1)$ Find the number of pairs...

Definition. Let $f: \mathbb{C} \to \mathbb{C}$ be an entire function. The order of growth of $f$, denoted by $O_G(f)$, is defined as \begin{equation} O_G(f) := \inf \left\{r > 0: \exists A, B > 0 \,(\text{depending on } r) \text{ such that } |f(z)| \leq Ae^{B|z|^{r}} \text{ for all } z \in \m...