Wikipedia:Reference desk/Archives/Mathematics/2011 October 11

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October 11

Jacobson density theorem: topological interpretation

I'm struggling to remember the following so that I can refine the Jacobson density theorem article: let D be a division ring with discrete topology, and U be a D vector space with the product topology. Then the D linear transformations of U to U can use at least two topologies: the subspace topology inherited from U^U, or the compact-open topology. I'm rusty enough not to remember which topology is the appropriate one for Jacobson_density_theorem#Topological_characterization. Thanks for the help. Rschwieb (talk) 19:47, 7 October 2011 (UTC) — Preceding unsigned comment added by Rschwieb (talk • contribs)

I think I remember the situation, and my question has changed. I think it's that U gets the discrete topology and End(U) gets the subspace topology of U^U, and then the embedding is dense in that topology. My question has changed then: Herstein says it's dense in the compact-open topology... does these two topologies coincide in this case or is Herstein smoking something? Rschwieb (talk) 02:36, 11 October 2011 (UTC)

I'll assume you mean the product topology when you said U^U. Since U is discrete, the compact subsets are exactly the finite subsets, so it's easy to see for every element of a basis neighborhood H of the compact open topology there's a product basis neighborhood containing that point and is contained in H. So yes the compact open is the same as product (but only because U discrete; in general compact open will be finer). Money is tight (talk) 08:57, 11 October 2011 (UTC)

Thanks very much :) Rschwieb (talk) 13:17, 11 October 2011 (UTC)

Use of Mathematical symbols for "or" in solutions to equations and inequalities, etc.

When describing the solution or root to an expression, inequality, equation, or any other similar function that has more than one solution (or range of solutions), should one use the "or" symbol that looks similar to a capital V (⋁, $\vee$), or the one that looks similar to a capital U (⋃, $\cup$). For example, the solutions to $3x^{2}+2x+1=6x$ are ${\frac {1}{3}}$ or 1. In this case, would one use $\vee$ or $\cup$? Hmmwhatsthisdo (talk) 02:56, 11 October 2011 (UTC)

Mathematicians hardly ever use $\vee$, but that would be the appropriate symbol. Really, just list them out, either separated with commas or with the word "or". --COVIZAPIBETEFOKY (talk) 03:45, 11 October 2011 (UTC)

So, $\vee$ is technically right, but $\cup$ is more prevalent? Hmmwhatsthisdo (talk) 04:05, 11 October 2011 (UTC)

No, $\vee$ is technically right, but "or" is more prevalent. In fact, I've never seen $\cup$ used for "or", despite union naturally identifying with or when moving from power sets to boolean algebras.--Antendren (talk) 04:08, 11 October 2011 (UTC)

If you want to make it look more mathematical (though I think I'd just use 'or' here) you can write them as a set with squiggly brackets like {1/3, 1}. If you want to be technical 1 and 1/3 aren't logical formulae so you'd have to write x=1 ⋁ x=1/3 though nobody will mind if you don't do that. Dmcq (talk) 09:16, 11 October 2011 (UTC)

I just popped open my Calculus and Linear Algebra books to see what they use and both use $x=1,{\frac {1}{3}}$. -- kainaw™ 12:54, 11 October 2011 (UTC)

Writing it informally would be best. But you can do this symbolically if you like:

$x=1/3\vee x=1$

$x\in \{1/3\}\cup \{1\}$

$x\in \{1/3,1\}$

So there are ways of using either one (or neither), but they are not interchangeable.

CRGreathouse (t | c) 17:25, 11 October 2011 (UTC)

Math refresher

SO, I'm being prevented from getting maximum enjoyment out of reading stuff because a lot of what I've been reading assumes mathematical knowledge that I just don't have. I'm not reading anything really hardcore or specifically about math, mostly computer science and physics stuff (for laymen). What I'm interested in is a good, not-too-huge intro, or intros, to math topics. Wikipedia is great, but I need to do some problems in order to get my head around some of these topics. I haven't taken any math in a long time, and the farthest I went was an intro calculus course. I know a little bit about sets and groups from studying linguistics, but that's about it. Any recommended books, websites, etc? Thanks! Matttoothman (talk) 04:30, 11 October 2011 (UTC)

What do you know about groups? I want to make sure I understand what you know before giving recommendations. CRGreathouse (t | c) 17:26, 11 October 2011 (UTC)

After reading Group theory, I have to say that I don't understand the difference between groups and sets very well. I know what the respective definitions are, and yet.... Matttoothman (talk) 20:05, 11 October 2011 (UTC)

It appears to me that you are reading topics that include discrete mathematics. That is very common in computer science (in fact, discrete has been a requirement for computer science in every university I've worked at). There is a lot of set and graph theory involved. Perhaps it would be best to get a discrete math book from your local library or university book store and see if you can just dive into it and make sense of it all. I started discrete before I started calculus and the only problem I had was a heightened annoyance with my eventual calculus professor who proclaimed that anything outside of calculus is not math. -- kainaw™ 17:40, 11 October 2011 (UTC)

A quick browse down through discrete mathematics reveals lots of topics both interesting and relevant. Thanks for the tip! Matttoothman (talk) 20:05, 11 October 2011 (UTC)