User:Lord Farin/Sandbox

This page exists for me to be able to test out features I am developing. Also, incomplete proofs may appear here.

Feel free to comment.

Over time, stuff may move to User:Lord_Farin/Sandbox/Archive.

Improvement of Sequence of Implications of Connectedness Properties
For brevity, let us introduce the following acronyms:


 * $\mathrm{AC}$: Arc-Connected
 * $\mathrm{UC}$: Ultraconnected
 * $\mathrm{PC}$: Path-Connected
 * $\mathrm{HC}$: Hyperconnected
 * $\mathrm C$:  Connected

Then the following sequence of implications holds:


 * $\begin{xy}

<-3em,3em>*+{\mathrm{AC}} = "AC", <3em,3em>*+{\mathrm{UC}} = "UC", <0em,0em>*+{\mathrm{PC}} = "PC", <6em,0em>*+{\mathrm{HC}} = "HC", <3em,-3em>*+{\mathrm C}  = "C",

"AC";"PC" **@2{-} ?><>(1.2)*@2{>}, "UC";"PC" **@2{-} ?><>(1.2)*@2{>}, "PC";"C" **@2{-} ?><>(1.2)*@2{>}, "HC";"C" **@2{-} ?><>(1.2)*@2{>}, \end{xy}$

What do you think? Now that I have created the moulds, it will be easy to adapt to the other 'Sequences of Implication'. --Lord_Farin (talk) 10:21, 31 August 2012 (UTC) It be noted that it will forever be impossible to endow diagrams (and indeed, any TeX rendered with MathJax) with internal links; sorry. --Lord_Farin (talk) 10:22, 31 August 2012 (UTC)

Pointwise Operations on Mappings
It is probably best to put up a specific page for every instance encountered; then for intermediate generalisations (that is, pages fully generalising a certain codomain (like is being done for $\R$ now), and pages fully generalising an operation (eg., addition could be generalised to groups, maybe commutative semigroups, after that it gets awkward to speak of 'addition')), and so on building up to the abstract generality of Definition:Operation Induced on Set of Mappings. Benefit is that most of the proofs can be short, referring to more general ones (giving a nice inherent bound on when a certain pointwise operation is fully generalised: when the proof of a further generalisation wouldn't be identically the same). This gives the potential of, I think, in the order of one hundred pages, if not more. Not sure if I can bring the patience to repeat the same exercise over and over again, but hey, there's no deadline, so what's the problem? Just checking now if I have sensed correctly that this is a desirable direction for PW (where again, I have in mind a reader who's only familiar with a handful of objects pertaining to his field(s) of interest; at least, I want PW to be accessible for people not experts in the terminology and concepts of abstract algebra (as I note I had to become one of the latter to pick up some of the defs on PW)). So, what do you think? --Lord_Farin 17:36, 6 April 2012 (EDT)

Apologies; I have a tendency for excessive parentheses (probably originating from my mind working faster than I can type). --Lord_Farin 18:18, 6 April 2012 (EDT)


 * Sounds good. I note the tedium of the repetitive nature of the task - I get round this by radical copypasta and waiting till my best time of day when there isn't something more interesting / immediate cropping up. --prime mover 19:01, 6 April 2012 (EDT)

Theorem on Sigma-Algebras

 * $\displaystyle \sigma \left({\bigcup_{i \mathop \in I} \sigma \left({\mathcal{G}_i}\right)}\right) = \sigma \left({\bigcup_{i \mathop \in I} \mathcal{G}_i}\right)$

That was the easy part; the proof is also easy. But the name isn't... Suggestions? --Lord_Farin 15:46, 16 May 2012 (EDT)

Functions and Measures
Given an appropriate $f: X \to \overline\R$ and measure $\mu: \Sigma\to\overline\R_{\ge0}$, one can form a measure $\nu$ by:


 * $\nu (E) := \displaystyle \int \chi_E f \, \mathrm d\mu$

for all $E \in \Sigma$. Now the question is what to name $\nu$; note this all is intimately connected to the Radon-Nikodým Theorem.

A quick search in my literature didn't yield immediate results. --Lord_Farin 17:45, 12 April 2012 (EDT)


 * I would suggest '$\mu$-test measure of $f$' or something like that, $f$ sort of plays the rôle of the test functions in distribution theory. What do you think? --Lord_Farin 17:35, 29 May 2012 (EDT)


 * Apparently exists as Definition:Measure with Density. --Lord_Farin 12:33, 6 August 2012 (UTC)

Magma of Sets
I just conceived of a name for an encapsulating notion that will allow me to cut the forest of theorems like Existence and Uniqueness of Sigma-Algebra Generated by Collection of Subsets. It is... magma of sets.

This is so named since it consists of a nonempty collection of subsets $\mathcal M \subseteq \mathcal P \left({S}\right)$ for some set $S$, such that $\mathcal M$ is a magma for a certain set of (not necessarily binary) operators, like countable union, binary intersection, and also in particular constant mappings like, e.g., $\mathcal P \left({S}\right) \to \mathcal P \left({S}\right), \Bbb S \mapsto \varnothing$.

The virtue of this is that any such type of magma of sets has the property that an arbitrary intersection of magmas stays a magma (provided the intersection is nonempty). Should I pursue this notion by creating pages for it or abandon the project immediately? --Lord_Farin 08:57, 10 May 2012 (EDT)


 * Can't get to Existence and Uniqueness of Sigma-Algebra Generated by Collection of Mappings, I'm done with it, and will put through this project. Names can always be changed if so desired. --Lord_Farin 17:35, 29 May 2012 (EDT)


 * User:Lord_Farin/Sandbox/Magma of Sets. --Lord_Farin 04:18, 31 May 2012 (EDT)


 * Now see Definition:Magma of Sets and Category:Magmas of Sets. --Lord_Farin 20:28, 4 August 2012 (UTC)