Math.SE on Solarized Sublime Sekai

Deleted Question on Semi-Simple and Projective but not Injective Module

Thu, 14 Jan 2021 10:39:00 +0100

A backup of a deleted PSQ : https://math.stackexchange.com/q/3955443/290189

It has a detailed answer by Atticus Stonestrom. It’s pity that his post got deleted. As there’s no reason for undeletion, I’m posting it here so as to preserve the contents.

Question body

Is there a semi-simple and projective but not injective module? I will be glad if you help.

Response(s)

In the non-commutative case, the answer is yes. Consider $R$ the ring of upper triangular $2\times 2$ matrices over a field $F$, and denote by $e_{ij}$ the element of $R$ with the $ij$-th entry equal to $1$ and all other entries equal to $0$. We can decompose $R$ as a direct sum of left ideals $$Re_{11}\oplus (Re_{12}+Re_{22})=Re_{11}\oplus Re_{22},$$ so let $M=Re_{11}$. Then $M$ is clearly simple, and – as a direct summand of the free module $R$ – is also projective. However, $M$ is not injective; consider the map $f:Re_{11}\oplus Re_{12}\to M$ taking $e_{11}$ and $e_{12}$ to $e_{11}$. $Re_{11}\oplus Re_{12}$ is a left ideal of $R$, but there is no way to extend $f$ to a map $R\rightarrow M$, so this gives the desired example.

Ways to Draw Diagrams Displayed on Math.SE

Wed, 23 Dec 2020 11:46:47 +0100

I wanted to start a meta question, but I don’t see a point of that after viewing some related posts listed at the end of the next subsection.

Intended question

You may vote on your preferred way.

Ways	Advantages	Disadvantages
AMScd	supported on Math.SE for a long time	no diagonal arrows syntax less well-known Two-way arrows $\rightleftarrows$ look odd
`array`	supported on Math.SE for a long time easier syntax write basic diagonal arrows like $\nearrow$	fine tuning spacing is hard diagonal arrows only work for neighboring nodes
ASCIIFlow	WYSIWYG interface	lines are rendered as slashes in code
TikZ	well known syntax can draw pretty diagrams	not supported on SE, need to import as picture
others?

My Intended Trigo Answer

Wed, 01 May 2019 22:24:45 +0200

The Math.SE question $2\cos(2x) - 2\sin(x) = 0$ has attracted several answers from high-rep users.

I am expanding @rhombic’s comment into an answer.

$$ \begin{aligned} 2\cos(2x)-2\sin(x)&=0 \\ 2 - 4\sin^2(x)-2\sin(x)&=0 \\ 2\sin^2(x)+\sin(x) - 1&=0 \\ (2 \sin(x) - 1)(\sin(x) +1) &= 0 \\ \sin(x) = \frac12 \text{ or } \sin(x) &= -1 \\ x = \frac{\pi}{6}, \frac{5\pi}{6} \text{ (rejected) or } & \frac{3\pi}{2} \text{ (rejected)} \end{aligned} $$

JavaScript Copy Button

Fri, 26 Apr 2019 23:01:53 +0200

Goal

To create a copy button for my Math.SE comment template in order to save the trouble of copying and pasting.

My first attempt

I put the boilerplate inside a Markdown codeblock to prevent them from getting interpreted by Hugo’s Markdown parser. Under each codeblock, I placed the copy button.

Comment boilerplate goes here ...

Another comment boilerplate goes here ...

…

My page’s original layout

$(document).ready(function() {
  $('.copyBtn').click(function() {
    copy($(this).prev().children())
j  });
});

function copy(selector) {
  var screenTop = $(document).scrollTop();
  var $temp = $("<div>");
  $("body").append($temp);
  $temp.attr("contenteditable", true)
       .html($(selector).html()).select()
       .on("focus", function() { document.execCommand('selectAll',false,null) })
       .focus();
  document.execCommand("copy");
  $temp.remove();
  $('html, body').scrollTop(screenTop);
}

static/js/copyBtn.js at Git tag copyBtn0

My Dual Answer

Fri, 26 Apr 2019 18:01:13 +0200

Update: The question has been reopened.

I intended to answer 김종현’s problem on Math.SE. However, the programs in the question body aren’t typeset in MathJax. As a result, I downvoted and closed this question because found it unclear. From the proposed dual, it seems that I shouldn’t interpret the primal as a linear program. Anyways, without further clarifications from OP, I found no reason to look at this further. Here’s my intended answer:

Borel Cantelli Exercise 2019

Mon, 04 Feb 2019 16:24:40 +0100

I intend to post this for a Borel-Cantelli lemma exercise on Math.SE.

The target event is ${\exists i_0 \in \Bbb{N} : \forall i \ge i_0, X_i = 1}$, whose complement is

$$ {\forall i_0 \in \Bbb{N} : \exists i \ge i_0, X_i > = 0} = \limsup_i {X_i = 0}. $$

To apply Borel-Cantelli, one has to determine whether $\sum_i P(X_i = 0)<+\infty$.

Weak LLN Practice

Sun, 02 Dec 2018 08:11:56 +0100

My intended answer to a weak LLN problem on Math.SE.

Problem: Suppose $(X_n)$ is a sequence of r.v’s satisfying $P(X_n=\pm\ln (n))=\frac{1}{2}$ for each $n=1,2\dots$. I am trying to show that $(X_n)$ satisfies the weak LLN.

The idea is to show that $P(\overline{X_n}>\varepsilon)$ tends to 0, but I am unsure how to do so.

My solution: As in the accepted answer in OP’s previous question https://math.stackexchange.com/q/3021650/290189, I’ll assume the independence of $(X_n)$. By Chebylshev’s inequality,

Solution to a $p$-test Exercise

Tue, 20 Nov 2018 12:09:46 +0100

I intended to answer Maddle’s $p$-test question, but T. Bongers has beaten me by two minutes, so I posted my answer here to save my work.

The problem statement

This is the sum: $$\sum\limits_{n=3}^\infty\frac{1}{n\cdot\ln(n)\cdot\ln(\ln(n))^p}$$ How do I tell which values of $p$ allow this to converge? The ratio test isn’t working out for me at all.

Unpublished solution

The integral test will do.

$$ \begin{aligned} & \int_3^{+\infty} \frac{1}{x\cdot\ln(x)\cdot\ln(\ln(x))^p} \,dx \\ &= \int_3^{+\infty} \frac{1}{\ln(x)\cdot\ln(\ln(x))^p} \,d(\ln x) \\ &= \int_3^{+\infty} \frac{1}{\ln(\ln(x))^p} \,d(\ln(\ln(x))) \\ &= \begin{cases} [\ln(\ln(\ln(x)))]_3^{+\infty} & \text{if } p = 1 \\ \left[\dfrac{[\ln(\ln(x))}{p+1}]^{p+1} \right]_3^{+\infty} & \text{if } p \ne 1 \end{cases} \end{aligned} $$

When $p \ge 1$, the improper integral diverges. When $p < 1$, it converges.