User:Jshflynn

About
In time I'll learn to enjoy posting proofs.

For now I find it most relaxing adding sources and references :)

Onwards ProofWiki!

Quicker Typing Help
If you find yourself typing "Definition:" or some other key sequence repeatedly I have found the program AutoHotKeys to be very good:


 * AutoHotKeys website


 * There is also the link in "ProofWiki Specific" right below your edit pane. Saves learning random nonsense. --prime mover (talk) 07:15, 14 December 2012 (UTC)


 * The act of moving one's hand to the mouse, then the pointer to the link below, then click, then move hand back to keyboard is generally more expensive than simply typing the string yourself. Moreover, it presupposes that one does not use an external editor for PW (which I have started to do just a few days ago, the highlighting and improved search/replace functionality make it worthwhile). Thank you Jshflynn, I'll consider it later, when I get home. --Lord_Farin (talk) 12:04, 14 December 2012 (UTC)


 * Each to his/her own - but I have less patience with learning new stuff nowadays. Keeping up with tech is becoming a headache and more than. The links below the edit pane are just what I need. --prime mover (talk) 20:13, 14 December 2012 (UTC)

Thought
In graph theory I think that connectedness of vertices is an equivalence relation and the elements of the quotient set are the components of the graph. If that's true I don't know if it's worth proving. --Jshflynn (talk) 21:04, 21 December 2012 (UTC)


 * It seems to be correct (assuming that connectedness means "exists path" and the graph is undirected). Worth adding, I think. --Lord_Farin (talk) 21:58, 21 December 2012 (UTC)


 * Only in an undirected graph. Funny but I thought I'd already proved this. If not, then go for it because (though trivial) it's important. --prime mover (talk) 22:15, 21 December 2012 (UTC)


 * Aha - seems I have done. Graph Connectedness is Equivalence Relation. --prime mover (talk) 22:17, 21 December 2012 (UTC)


 * Excellent. Had I simply Ctrl F'd 'Equivalence' on the proof index page I would have found it and the corollary that alphabetically precedes it. If you would like a break from the topology section (which is coming along fantastically btw) here's another easy one: "The deletion of a cyclic edge in a complete graph is a complete graph" (the book I am reading at the moment though calls cycle's circuits and what you might think would be called a 'circuitous edge' is just a 'circuit edge', it is from 1981 however so there could have been a terminological shift since then). --Jshflynn (talk) 23:23, 21 December 2012 (UTC)

Deposit
I will have to deposit this here for the future:


 * $\displaystyle\left\vert{ \prod_{j \in J} \left\vert{ \prod_{i \in J} \left\vert{ \left({\mathcal{P}\left({S / \mathcal{R}_f}\right)_j}\right)_i }\right\vert }\right\vert}\right\vert$

Emerging Trends
I am keeping a note here of emerging phenomena on ProofWiki for me to think about. I am aware it is not organised.


 * Theorems that are special cases of other theorems.
 * Definitions that are special cases of other definitions.
 * Processes to generate new theorems (duality principles, the machinery of category theory (corollary spaces?)).

Other undeveloped thoughts:


 * Some theorems are not used in the proof of anything else. Perhaps have community consequence challenges? The players have to prove something else using the theorem at least once.
 * Lemma (or Exercise) namespace? The idea being that people store answers to questions in books so that more people can have a go of using LaTeX and wikis (the book in question of course not having the answers). Reward users by saying that their work has been "promoted" to the theorem namespace and let them know that they too can do mathematics. This way the theorem namespace isn't cluttered and it's getting others involved.

Antitransitive Implies Antireflexive
Let:


 * $xy,yz \vert \overline{xz}$

Suppose:


 * $xx$

Then:


 * $xx,xx \vert \overline{xx}$

A contradiction.

Hence:


 * $(xy,yz \vert \overline{xz}) \vert (\vert \overline{xx})$