@TedShifrin Know where I could read about this? Cartan's book on complex analysis, perhaps? I don't wanna ask my advisor yet in case its something stupid that he thinks I've already understood
Consider 4 trajectories parametrized by
$x(\theta)=r(\theta)\cos(\theta-n\frac{\pi}{2})$ and $y(\theta)=r(\theta)\sin(\theta-n\frac{\pi}{2})$, for $n=0,1,2,3$.
I have to show that each trajectory is defined by the following differenial equation:
$$\frac{d}{d\theta}r(\theta)=r(\theta).$$
I tri...
@Twink: The key part of the problem is in the words, not those equations. The words give the differential equation that governs the motion of the ants.
A good question for you folks here from a relative noob. A brand new poster put up a poorly written question and we chatted a bit. He really wanted to know how techniques for writing better expository type mathematics that is clearer to the reader using MathJax, as he puts all of his notes in that format. I prodded him to some success and eventually took his best edit and edited that to this end. How should I interperet this.
I liked the question he was "trying" to ask.
I am in no way bothered by the resulting judgement, I may have gone a bit overboard here even making this effort to get this general question of writing clear math into the fold. I was just surprised.
Given $x \in A \land (x \notin B \lor x \notin C) \land x \in C$, can I conclude that $x \in A \land x \notin B \land x \in C$ because of disjunctive syllogism?
@J.W.Perry It seems like a reasonable question if your interpretation is correct, and I'd vote to reopen it if it were edited like that. I probably would have rejected the edit if it had come up to me, since it's quite substantial (I do find it funny, though, that it was rejected both for being too major and too minor...) Why not share the link to the suggested edit rejection page, and ask OP to make the edit himself?
@T.Bongers Thanks, every word you said is what I was thinking. I guess the OP just has to do it himself, and @Omega I would say clearly yes, but I really want someone to tell me I am wrong because it tlloks so clear I am surprised you are even having a fit over it. Maybe it is the beer... bleh
I guess the thing I saw for a second was that it was not expressly saying in the middle $(x \in C \wedge x \not \in B \vee x \in B \wedge x \not \in C)$, but that is crap.
Or is it? hrm.
Fun!
Does $(x \not \in B \vee x \not \in C) \Rightarrow (x \in B \vee x \in C)$?
Seriously, clearly $x \in A \wedge x \in C$, but can we conclude that $x \not \in B$?
Ah rats, that is the disjunctive syllogism. Next question...
I tried searching online, but, eh... I don't really know what terms to look for. Sometimes I end up with exercises that are clearly many levels above mine..
Ok @Omega, I broke out some old notebooks. This is a good one. Let $P \Rightarrow Q$ be the statement. If $P is false$, and $Q$ is true, is the statement true of false?
I'm not playing man, I am reading some ancient crap I wrote. For example, "Implication is false only when antecedent is true and consequent is false" straight from a dusty box labeled 2008 hehe.
0.233333... has 2, 23, 23, 233, 2333, 23333, maybe more as prime prefixes. does the subset of [0,1) with an infinite number of prime prefixes have measure 1?
i mean is it a real paradox or do i have something wrong? i guess it might be related to the arithmetical hierarchy stuff they talk about in the wiki page?
The first of the sequences from these troll formulas are series coeffcients of a Lambert W function. I am wondering if the other two have Lambert W expressions too:
I am writing a draft of an answer to that question, also writing up an awesome original computation of the discriminant of cyclotomic fields I've patched together, and enjoying the weather.
@anon I am reading "Introductory Real Analysis" by Kolmogov and Fomin and sipping my lovely shiny coffee. Now reading about the Minkowski functional and stuff.
I use a product formula for resultant to relate the discriminant of $x^n-1$ (which can be easily be computed directly) to the discriminants of cyclotomic polynomials $\Phi_n$ plus some extra baggage (pairwise resultant factors that always yield prime powers). Then I perform Moebius inversion to compute the discriminants exactly.
Munkres defines a topological group as a $T_1$ space that makes composition and taking inverses continuous, but apparently it makes sense to drop $T_1$. $T_1\implies T_2$ is easy but I don't see how to do it with $T_0$.
Consider 4 trajectories parametrized by
$x(\theta)=r(\theta)\cos(\theta-n\frac{\pi}{2})$ and $y(\theta)=r(\theta)\sin(\theta-n\frac{\pi}{2})$, for $n=0,1,2,3$.
I have to show that each trajectory is defined by the following differenial equation:
$$\frac{d}{d\theta}r(\theta)=r(\theta).$$
I tri...
