Search found 145 matches

by MostlyHarmless
Wed Jul 21, 2010 11:01 pm UTC
Forum: Mathematics
Topic: How do we know i is well defined?
Replies: 16
Views: 2730

Re: How do we know i is well defined?

The k you chose is not the square root of -1. If you let k=i/\pi , then k^2=-1/\pi . There are actually two square roots of -1 , i and -i . To see this, suppose i and k were both square roots of -1. i^2=1=k^2 , so i=\pm k . Edit: To be fair, you could use any imaginary number you wanted as a "b...
by MostlyHarmless
Wed Jan 20, 2010 6:02 am UTC
Forum: Mathematics
Topic: Some Basic Analysis Questions
Replies: 13
Views: 2542

Re: Some Basic Analysis Questions

That looks good to me. The only issue I noticed is that you should call your new subsequence \{ a_{n_i} \} . (Or else define \{ b_{n_i} \} a little more explicitly, but it seems easier to use the same name as in the problem statement. This has the added benefit of looking exactly like the definition...
by MostlyHarmless
Tue Jan 19, 2010 11:49 pm UTC
Forum: Mathematics
Topic: Some Basic Analysis Questions
Replies: 13
Views: 2542

Re: Some Basic Analysis Questions

Weird. For some reason, I actually have the fifth and third editions, so now my problems and suchlike should be numbered correctly. You haven't covered the theorem jestingrabbit mentioned yet, but problem 26 actually gives you the special case you need (that if \{ a_n \} converges to a, then \{ \sqr...
by MostlyHarmless
Tue Jan 19, 2010 12:32 pm UTC
Forum: Mathematics
Topic: Some Basic Analysis Questions
Replies: 13
Views: 2542

Re: Some Basic Analysis Questions

I seem to have a different version than you do, so my section numbers may be a little off, but you should take a closer look at the theorems in section 1.3 (arithmetic operations on sequences). These theorems are exactly what skeptical scientist was referring to. The gist is that you can figure out ...
by MostlyHarmless
Thu Jan 14, 2010 4:28 am UTC
Forum: Mathematics
Topic: Mathematics and intellect
Replies: 85
Views: 9379

Re: Mathematics and intellect

I had a somewhat related discussion with one of my professors a couple years ago. He used to run some sort of biotech business, and he told me that he generally preferred to hire math majors rather than biology majors. While the biology majors may have had previous knowledge about the material they ...
by MostlyHarmless
Thu Jan 07, 2010 10:55 pm UTC
Forum: Mathematics
Topic: Shuffling Cards
Replies: 17
Views: 2596

Re: Shuffling Cards

Just riffle shuffle eleven or twelve times. Riffle shuffling is a lot faster than pile shuffling, and eleven shuffles should be close enough to random for a 60 card M:tG deck. This. The only real uses for pile shuffling are counting your deck and cheating. If you're just playing casually, the count...
by MostlyHarmless
Thu Jan 07, 2010 6:21 am UTC
Forum: Mathematics
Topic: Shuffling Cards
Replies: 17
Views: 2596

Re: Shuffling Cards

I don't know anything about the composition of a Pokemon deck, but I've shuffled a lot of magic decks before. In practice, I've found that seven riffle shuffles really isn't enough. This is mostly because, assuming you care at all about the condition of your cards, your riffle shuffles will be far f...
by MostlyHarmless
Wed Dec 16, 2009 5:40 am UTC
Forum: Mathematics
Topic: Closure of sets in a continuous function
Replies: 8
Views: 2383

Re: Closure of sets in a continuous function

Do you know any other definitions of continuous? In particular, have you learned that a function [imath]f: X\rightarrow Y[/imath] is continuous iff [imath]\forall V \in Y[/imath] such that V is open, [imath]f^{-1}(V)[/imath] is open.
by MostlyHarmless
Wed Dec 16, 2009 3:51 am UTC
Forum: Mathematics
Topic: Integration of e^( x^2)
Replies: 24
Views: 4130

Re: Integration of e^( x^2)

If you rename your constants, it's not bad at all. Let b = \frac{-\beta\epsilon}{\sigma\lambda-1} and a = -b\lambda\sigma . It should be: e^{a+br}(\frac{r^2}{b}-\frac{2r}{b^2}+\frac{2}{b^3}) It's at most twice that long when you plug in your limits, although it looks like it might factor pre...
by MostlyHarmless
Tue Dec 15, 2009 11:51 pm UTC
Forum: Mathematics
Topic: Readable Encryption
Replies: 15
Views: 1914

Re: Readable Encryption

What you're referring to is a specific case of steganography. The most basic way to do this would be through some sort of code word system. I'm not sure if that's what you mean when you say "encryption method" though.
by MostlyHarmless
Mon Dec 14, 2009 8:30 am UTC
Forum: Mathematics
Topic: How To Learn (Advanced) Math
Replies: 12
Views: 3770

Re: How To Learn (Advanced) Math

Oh no, leap right into differential geometry! If you've done diff eq's you should be all set for a curves and surfaces course. Classical differential geometry is awesome. I really liked do Carmo's book, but there are others. I think the style of differential geometry that comes with the full force ...
by MostlyHarmless
Mon Dec 14, 2009 6:21 am UTC
Forum: Mathematics
Topic: How To Learn (Advanced) Math
Replies: 12
Views: 3770

Re: How To Learn (Advanced) Math

All you should need for topology, linear algebra and number theory are good books. I don't know any particularly good linear algebra or number theory books, but Topology by James Munkres is an excellent place to start that subject. Real analysis is another area you might want to consider. (A good gr...
by MostlyHarmless
Fri Dec 11, 2009 1:50 am UTC
Forum: Mathematics
Topic: So, what can I do to keep myself entertained and interested?
Replies: 14
Views: 1942

Re: So, what can I do to keep myself entertained and interested?

You might want to look for a book about number theory. It really only requires algebra and some elbow grease (basically just the willingness to keep playing with problems until you figure them out). I wouldn't recommend anything that you're going to cover in the next year or two, since you'll still ...
by MostlyHarmless
Tue Oct 27, 2009 4:46 am UTC
Forum: Mathematics
Topic: What kind of function is this?
Replies: 9
Views: 1415

Re: What kind of function is this?

As far as I know, equations of that form don't have a special name (besides exponential, but that's pretty broad). I'm sure that if you refer to it as "the equation for an underdamped harmonic oscillator", people will understand. If you need to talk to people who haven't studied differenti...
by MostlyHarmless
Wed Oct 21, 2009 11:25 pm UTC
Forum: Mathematics
Topic: Euler's Elements of Algebra
Replies: 5
Views: 919

Re: Euler's Elements of Algebra

I haven't read it either, but a quick google search suggests that you'll be able to understand it just fine. Apparently Euler sometimes chose some strange subjects to focus on, but it's pretty similar to the algebra you've already done. Besides, you might as well take a look at any free book that st...
by MostlyHarmless
Sat Oct 17, 2009 9:17 pm UTC
Forum: Mathematics
Topic: Need help developing an algorithm!
Replies: 7
Views: 1015

Re: Need help developing an algorithm!

What about [imath]v=50\sqrt{s}[/imath] or [imath]v=\frac{50}{log(2)}log(s+1)[/imath] (where v is the value and s is the skill level)? These won't usually give you whole numbers, but you can just round down.
by MostlyHarmless
Sat Oct 17, 2009 9:01 pm UTC
Forum: Mathematics
Topic: Calculus Midterm Question
Replies: 3
Views: 832

Re: Calculus Midterm Question

Have you learned about tangent lines? A tangent line of a function f at a point a (also called a linear approximation) is approximately equal to f as long as you are close to a. Does that sound like part ii?
by MostlyHarmless
Sat Oct 17, 2009 8:15 pm UTC
Forum: Mathematics
Topic: Got a new calculator for Math 10, woot
Replies: 29
Views: 2765

Re: Got a new calculator for Math 10, woot

My high school was similar to RogerMurdock's. Everyone was supposed to have a graphing calculator, but if you couldn't afford one the school had spares that you could use whenever you needed to. They seemed very useful at the time, but I've found that calculators usually instill a lot of bad habits....
by MostlyHarmless
Sat Oct 17, 2009 9:33 am UTC
Forum: Mathematics
Topic: Definition of dy and dx?
Replies: 16
Views: 1864

Re: Definition of dy and dx?

If you're really curious, dy and dx are differential forms . (Specifically, one forms ). Don't worry if this goes way over your head. It's way over mine too, and I've been studying it. The basic idea is that calculus is a way to talk about infinitely small changes. One of the best ways to describe a...
by MostlyHarmless
Fri Oct 02, 2009 10:13 pm UTC
Forum: Mathematics
Topic: which asshat came up with infix?
Replies: 23
Views: 3309

Re: which asshat came up with infix?

If you dont like infix, there is a simple way to fix that.

a+b=f(a,b)


To be fair, that doesn't solve the problem at all, since that's exactly what prefix notation does. Everyone just calls the function + instead of f.
by MostlyHarmless
Sat Sep 26, 2009 11:30 am UTC
Forum: Mathematics
Topic: What's the coolest thing you've ever learned in math class?
Replies: 58
Views: 6601

Re: What's the coolest thing you've ever learned in math class?

That's certainly the sticking point, since what I meant by "an explicit construction is possible" is that a mathematician could state it. I suppose I should just suck it up and suspend my disbelief when I start talking about infinitely many mathematicians.
by MostlyHarmless
Sat Sep 26, 2009 12:44 am UTC
Forum: Mathematics
Topic: What's the coolest thing you've ever learned in math class?
Replies: 58
Views: 6601

Re: What's the coolest thing you've ever learned in math class?

Depending on what you mean by "infinity," I think countable prisoners and hats is a lot more unsettling. This paradox bugs me because of the line "Using the axiom of choice, they select and memorize a representative sequence from each equivalence class." The Axiom of choice guar...
by MostlyHarmless
Tue Sep 22, 2009 1:28 am UTC
Forum: Mathematics
Topic: What's the coolest thing you've ever learned in math class?
Replies: 58
Views: 6601

Re: What is...

This. more generally, how bifurcations work.
by MostlyHarmless
Sat Sep 19, 2009 10:06 am UTC
Forum: Mathematics
Topic: Teaching myself higher mathematics
Replies: 30
Views: 3507

Re: Teaching myself higher mathematics

This looks like a good resource too. I have physical copies of several analysis books, so I've never used this one myself, but it looks very similar to other books I've used. (And it's free. That's the big part.) It's fairly dense, so it might be a little intimidating, but if you really want to lea...
by MostlyHarmless
Fri Sep 18, 2009 4:25 am UTC
Forum: Mathematics
Topic: Weird Ass-Function
Replies: 8
Views: 1442

Re: Weird Ass-Function

You should keep in mind that "the n th partial derivative" is only defined for n \in \mathbb{N} (and usually 0, since the 0th derivative of a function is generally defined as the function itself). This means that f(.5) doesn't really make sense. The formula wolfram gives you may wo...
by MostlyHarmless
Fri Sep 18, 2009 2:15 am UTC
Forum: Mathematics
Topic: Where did I go wrong on this question?
Replies: 26
Views: 2511

Re: Where did I go wrong on this question?

Gowers's weblog talked about a similar issue a little while ago. He was discussing problems where you're asked to show that a function is well-defined. Obviously, it's kind of a silly question. If f is a function, then f is well-defined by definition. Likewise, if you call f^{-1} "the inverse ...
by MostlyHarmless
Wed Sep 16, 2009 9:41 pm UTC
Forum: Mathematics
Topic: Teaching myself higher mathematics
Replies: 30
Views: 3507

Re: Teaching myself higher mathematics

If you want to learn something that you won't be covering in high school, I would definitely recommend topology or number theory. These subjects are very simple in the sense that they don't have very much prerequisite knowledge attached, so with some dedication you would be able to pick them up with...
by MostlyHarmless
Fri Aug 28, 2009 12:09 am UTC
Forum: Mathematics
Topic: Struggling with the concept of compact space
Replies: 40
Views: 4049

Re: Struggling with the concept of compact space

The important part isn't that you can find a finite cover. You have to be able to start with any cover and prune it down to a finite subcover. Basically, whenever someone gives you a collection of sets that covers (0,1), you have to be able to take away all but a finite number of them and still cove...
by MostlyHarmless
Tue Dec 16, 2008 10:06 pm UTC
Forum: Mathematics
Topic: D'oh
Replies: 15
Views: 1649

Re: D'oh

You can just take the discrete topology on any uncountable set, so this isn't a very interesting question as written. Good point. What I meant was totally disconnected. I was thinking more along the lines of ordinals. To be fair though, if someone just started analysis, the discrete topology in its...
by MostlyHarmless
Tue Dec 16, 2008 12:50 pm UTC
Forum: Mathematics
Topic: Negative Numbers Raised to Irrational Exponents
Replies: 20
Views: 9361

Re: Negative Numbers Raised to Irrational Exponents

It's actually discontinuous along the entire branch cut of the complex log function, unless you allow multi-valued functions (and excepting where b is an integer). Depending on your convention, this is typically when a is a negative real, which makes it a particularly bad definition when trying to ...
by MostlyHarmless
Tue Dec 16, 2008 12:45 pm UTC
Forum: Mathematics
Topic: D'oh
Replies: 15
Views: 1649

Re: D'oh

You're on the right track, but saying "list the neighborhoods" assumes that they are countable, which is what you're trying to prove. Have you tried using a contradiction? What would happen if they weren't countable? That's my standard operating procedure for these types of problems. Also,...
by MostlyHarmless
Fri Nov 07, 2008 9:52 pm UTC
Forum: Mathematics
Topic: Help with an infinite series.
Replies: 9
Views: 1465

Re: Help with an infinite series.

This is one of the important distinctions between finite series and infinite series. When you add two numbers, it doesn't matter what order you do it in, you'll always get the same answer. 1 + 2 = 2 + 1 = 3. When you take an infinite series, though, changing the order changes the sum. You can take t...
by MostlyHarmless
Tue Nov 04, 2008 5:43 am UTC
Forum: Mathematics
Topic: Maximum Principle for Parabolic PDEs
Replies: 0
Views: 651

Maximum Principle for Parabolic PDEs

I have a homework problem due tomorrow for my PDE class, and I've been beating my head against it for a while now. Assume that \Omega is bounded, \partial\Omega is of class C^2 and that u,v\in C^2(D) \cap C^1 (D) . (Note that D = \Omega \times (0,T] and \overline{D} = \overline{\...
by MostlyHarmless
Wed Oct 22, 2008 5:58 am UTC
Forum: Mathematics
Topic: Help for trig.
Replies: 36
Views: 2506

Re: Help for trig.

My favorite mnemonic is still Some Old Hippy Caught Another Hippy Tripping On Acid. Probably not allowed in a lot of high schools though.
by MostlyHarmless
Wed Oct 22, 2008 5:34 am UTC
Forum: Mathematics
Topic: A Simple question
Replies: 3
Views: 679

Re: A Simple question

Does your calculator have a button labeled "log"? If so, then yes. log(x)=y means that y is a number such that 10^x = y. In your case, you don't want to solve 10^a = 1234, but your calculator doesn't have a log-base-6-button, so you have to resort to a little mathematical trickery. One of ...
by MostlyHarmless
Tue Oct 21, 2008 11:55 pm UTC
Forum: Logic Puzzles
Topic: Random Questions
Replies: 10
Views: 2841

Re: Random Questions

Spoiler:
She took two. That way, half of everything she took is just one, so they all give her back all the fruit she gives them and she keeps both of them.
by MostlyHarmless
Fri Oct 03, 2008 9:04 pm UTC
Forum: Mathematics
Topic: Constraints on the derivative and the nature of a function
Replies: 12
Views: 1384

Re: Constraints on the derivative and the nature of a function

Well, the original post just said continuous and differentiable everywhere, so I was technically within the rules =). But you're right, that "for instance" is wrong. I'm too lazy to check right now, but I believe you need f to be continuously differentiable on a connected domain. I can't a...
by MostlyHarmless
Fri Oct 03, 2008 8:04 pm UTC
Forum: Mathematics
Topic: Constraints on the derivative and the nature of a function
Replies: 12
Views: 1384

Re: Constraints on the derivative and the nature of a function

In case it's not obvious, you can do exactly the same thing if f' < e for some e<0. Sadly, it's not enough to say that the derivative is bounded away from zero (unless you specify that f is continuously differentiable). For instance, f: R\{0}-->R s.t. f(x)=|x| is no good. And I was going to say Lips...
by MostlyHarmless
Fri Sep 26, 2008 1:20 am UTC
Forum: Mathematics
Topic: I might have an opportunity to help teach math next semester
Replies: 6
Views: 1255

Re: I might have an opportunity to help teach math next semester

As long as you know them both fairly well, I would recommend teaching calculus. I've found that it's far harder to teach more introductory classes than it is to teach advanced classes. When someone doesn't understand integration by parts, I can just think back to when I was learning the subject and,...
by MostlyHarmless
Thu Sep 11, 2008 12:41 am UTC
Forum: Mathematics
Topic: Mathematics and Beauty
Replies: 42
Views: 5517

Re: Mathematics and Beauty

For the most part, I agree that surprising yet simple proofs are beautiful: the kind where you read it and suddenly the theorem fits perfectly and you can't remember why it seemed complicated before. I also find some theorems beautiful in themselves. For instance, the maximum principle (more of a co...

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