## Search found 156 matches

Sun Dec 11, 2011 7:25 am UTC
Forum: Mathematics
Topic: motivation of abstract math
Replies: 14
Views: 2786

### Re: motivation of abstract math

Another thing is that these abstract structures (for instance, the notions of point-set topology) allow you to draw connections between different topics, and find fundamental similarities. So, we know what it means for a real-valued function to be continuous (the epsilon-delta definition). But the m...
Wed Dec 07, 2011 6:54 pm UTC
Forum: Logic Puzzles
Topic: Towers, Courtyard, and trees
Replies: 36
Views: 13847

### Re: Towers, Courtyard, and trees

This is really just blue-eyes in disguise, I think. Day 1: If Al can see more than 18 trees, he knows there must be 20 trees in total and both are released. Otherwise, both prisoners know Al can see at most 18 trees. If Bob can see fewer than 2 trees, he knows there must be 18 trees in total and bo...
Thu Oct 27, 2011 8:33 pm UTC
Forum: Mathematics
Topic: "Oh no! We forgot how to say... math... stuff!"
Replies: 294
Views: 92230

### Re: "Oh no! We forgot how to say... math... stuff!"

Also, even though 'open' and 'closed' are not mutually exclusive, the two concepts are dual to each other in a very concrete sense. The terminology helps reflect this duality. Also, since the terms are used so frequently, replacing them with significantly longer words (ie 'neighborhood complete') wo...
Thu Oct 20, 2011 7:10 pm UTC
Forum: Mathematics
Topic: "Oh no! We forgot how to say... math... stuff!"
Replies: 294
Views: 92230

### Re: "Oh no! We forgot how to say... math... stuff!"

I'd also kill the idea of independent/dependent variables as used in calculus in favour of ( \lambda x .<some expression in x>) 1 functions or some similar notation. I'm fine with 2(x+1) = 2x+2 having an implicit \forall x . 2 But then you start doing d/dx (2x+2) = 2 and that (2x+2) should by all m...
Mon Oct 10, 2011 10:00 pm UTC
Forum: Mathematics
Replies: 17
Views: 2799

Since there are only so many test questions that can be asked about this material, the more problems you do the more likely it is that the test problems are very similar to problems you've already done. For the problem you gave about the sequence of functions, I have two ideas. The first is, remembe...
Sat Sep 03, 2011 12:25 am UTC
Forum: Mathematics
Topic: SAT Practice Test Question
Replies: 3
Views: 1155

### Re: SAT Practice Test Question

Well, we can solve the second equation for p, then put that into the first equation:

Spoiler:
First we get p = (n - 4)/sqrt(2), this gives m = 3 + 2((n-4)/sqrt(2))^2 = 3 + (n-4)^2 = n^2 - 8n + 19.
Fri Jun 03, 2011 6:34 am UTC
Forum: Mathematics
Topic: A criminal hides in a room with 99 innocent people...
Replies: 13
Views: 4477

### Re: A criminal hides in a room with 99 innocent people...

More generally, if we have N people (one criminal), a lie detector with accuracy p, and upon testing them all we find that k of them test positive, the chance that the criminal is among that k is: This uses the same reasoning as SS: the probability that k people test positive and none of them are th...
Fri May 13, 2011 3:20 pm UTC
Forum: Mathematics
Topic: Formal mathematical definition of area
Replies: 10
Views: 2265

### Re: Formal mathematical definition of area

Intuitively, the idea of measure theory (lets do 2-dimensional are right now) is pretty straightforward. 1: We want a rectangle with width a and length b to have area ab. 2: If we know the area of region 1 and the area of region 2, and the regions don't overlap, then the area of region 1 and 2 is ju...
Thu Feb 10, 2011 5:48 pm UTC
Forum: Mathematics
Topic: Pebbles in buckets, a modified random walk
Replies: 26
Views: 2451

### Re: Pebbles in buckets, a modified random walk

My first thought is that you can model this as a type of random walk in n-1 dimensions. For instance, with 4 buckets, you have bucket 1 move you +(1,0,0), bucket 2 +(0,1,0), bucket 3 +(0,0,1) and bucket 4 +(-1,-1,-1). In general, for n buckets, the ith bucket (i < n) moves you +(0,0,...,1,...,0) wit...
Mon Feb 07, 2011 5:19 pm UTC
Topic: 0857: "Archimedes"
Replies: 137
Views: 35415

### Re: 0857: "Archimedes"

I've always been a fan of: Roses are red, violets are blue, I'll fuck you with a rake. It actually works with varying success with other phrases: When in Rome, I'll fuck you with a rake. There are two kinds of people in this world, I'll fuck you with a rake. Give a man a fish, I'll fuck you with a r...
Wed Feb 02, 2011 3:46 pm UTC
Forum: Logic Puzzles
Topic: Twins in a Maze
Replies: 81
Views: 21696

### Re: Twins in a Maze

Everyone's stated that there are algorithms for exploring and mapping the maze, but I'm not convinced. Maybe if they're planar, but consider the following two mazes: 1- A single path with a circle centered on it (think the London Underground logo). The ends of the path are connected to each other (i...
Thu Jan 27, 2011 12:12 pm UTC
Forum: Mathematics
Topic: Weird stuff near zero
Replies: 19
Views: 3125

### Re: Weird stuff near zero

Maybe we can get a meaningful answer by requiring the polynomial to be monic, ie the leading coefficient is 1. Because any nth degree polynomial that is not monic has the same roots if you divide all coefficients by the leading coefficient, so by making the leading coefficient very small you can get...
Thu Jan 06, 2011 8:08 pm UTC
Forum: Mathematics
Topic: Where to start studying math?
Replies: 13
Views: 2863

### Re: Where to start studying math?

I would recommend: 1- Learn some basic number theory (modular arithmetic, Fermat's little theorem, unique factorization) and Naive set theory. The former helps familiarize you with rigorous proofs, and the latter gets you accustomed to the language (and, in some sense, philosophy) of higher formal m...
Sun Dec 26, 2010 2:18 am UTC
Forum: Mathematics
Topic: A Probability Question
Replies: 10
Views: 1627

### Re: A Probability Question

A lot of people have posted correct solutions, I think I might offer some insight as to what is going on in problems like these. The first issue with calculating probabilities like these, is it is not like finding probabilities for, say, drawing a flush in a 5-card poker hand. In this case, there ar...
Sat Dec 18, 2010 6:04 pm UTC
Forum: Mathematics
Topic: Strange integral
Replies: 11
Views: 1900

### Re: Strange integral

The Lebesgue integral is more complicated to explain. It is a "better" integral in that any function that is Riemann-integrable is also Lebesgue-integrable, but there are some functions that are Lebesgue-integrable that are not Riemann-integrable. Your functions fall into the latter categ...
Tue Dec 14, 2010 10:54 pm UTC
Forum: Mathematics
Topic: Cantor Diagonalization Formula
Replies: 11
Views: 2847

### Re: Cantor Diagonalization Formula

There's also a cool bijection from N to the positive rationals where you take a natural number N and consider it's prime factorization. If p1^a1*p2^a2...*pn^an are all the primes raised to even powers, the numerator of the fraction is p1^a1/2*p2^a2/2*...*pn^an/2. If q1^b1*...*qm^bm are all the prime...
Thu Dec 02, 2010 1:11 am UTC
Forum: Logic Puzzles
Topic: Card guessing game
Replies: 18
Views: 12020

### Re: Card guessing game

A naive approach: Let P(R,B) be the probability of winning with an optimal strategy, given R red cards and B black cards remaining. Supposing there are R reds and B blacks, one strategy is to say "stop", which has probability R/(R + B) of winning. If you don't say stop,...
Fri Nov 19, 2010 9:25 am UTC
Forum: Mathematics
Topic: Have You Been Taught Things Which Aren't True?
Replies: 97
Views: 16312

### Re: Have You Been Taught Things Which Aren't True?

What irks me about the natural numbers starting from 1 is that, if we want to talk about the set of positive integers, we can just use the notation Z^{+} . On the other hand, if the natural numbers and positive integers are the same set, there is no simple widely used notation available for the nonn...
Thu Nov 04, 2010 4:18 am UTC
Forum: Mathematics
Topic: Geometric Proof of the Regular Pentagon Construction
Replies: 4
Views: 2102

### Geometric Proof of the Regular Pentagon Construction

Hey all. I've been wondering for some time if there was a construction of a regular pentagon that had a geometric proof that the construction really worked. The word "geometric" is fairly loose, but vaguely I mean using arguments like congruence/similarity of triangles, and angle arithmeti...
Thu Nov 04, 2010 3:21 am UTC
Forum: Logic Puzzles
Topic: To make a touchdown (American football)
Replies: 9
Views: 3293

### Re: To make a touchdown (American football)

One approach (In which the football team cannot move backward): First, we examine the optimal strategy for getting a first down. Suppose we only have one down, and D yards to go for the first. If D \leq 2 , we can get the first with probability 1. If D>2, say we run a play that has Py probability of...
Tue Nov 02, 2010 3:49 am UTC
Forum: Mathematics
Topic: ways a line can cross itself (topology i believe?)
Replies: 4
Views: 2590

### Re: ways a line can cross itself (topology i believe?)

If I'm right, a formal way of stating this would be: Two line drawings are the same if you can deform one into the other without changing the number of intersection points or making an intersection point coincide with the endpoint of the line. Even more formally: f:I \rightarrow R^2 \sim g:I \righta...
Tue Oct 26, 2010 6:10 am UTC
Forum: Logic Puzzles
Topic: My write-up of the "Blue Eyes" solution (SPOILER A
Replies: 1368
Views: 420873

### Re: My write-up of the "Blue Eyes" solution (SPOILER A

It's actually an interesting philosophical point that if, say, a professor were to make an announcement in a classroom (or a guru on an island) that each person would immediately assume the unbounded recursion, ie that everyone knows that everyone knows that everyone knows... etc.
Mon Oct 25, 2010 7:22 am UTC
Forum: Mathematics
Topic: Quintic Equations
Replies: 12
Views: 4408

### Re: Quintic Equations

As someone mentioned before, the proof involves what is known as Galois theory, which relates group theory to field theory. The basic reason for "why 5?" is that the symmetric group on 5 elements is unsolvable, while the symmetric groups on 1,2,3,4 elements ARE solvable. Of course, if you'...
Sun Oct 03, 2010 5:49 pm UTC
Forum: Mathematics
Topic: I join the blagosphere
Replies: 28
Views: 2901

### Re: I join the blagosphere

If all of your entries are as lucid and enjoyable to read as the one you've posted, this blog is going to be awesome.
Thu Sep 30, 2010 6:44 am UTC
Forum: Mathematics
Topic: "Union-compatible" set properties
Replies: 5
Views: 1157

### Re: "Union-compatible" set properties

Not sure about the "special name", but here is an observation: Consider the property "not P", let's call it P'. Now if A has P', then every subset of A also has P'. For if B is a subset of A and B has property P then A = B U (A - B) must have property P. F...
Thu Sep 02, 2010 6:59 am UTC
Forum: Mathematics
Replies: 4
Views: 855

### Re: Question about Abstract Derivations

For the second question, if F has characteristic 0 and v_0^2 \neq 0 , then V \cong F[x] . Proof: First, note that any v satisfying Dv=0 must be c_0 v_0 for some constant c_0 , simply from our definition of D . In particular, since D(v_0^2) = 2 v_0 D(v_0) = 0 , we must have v_0^2 = c...
Mon Aug 30, 2010 9:00 am UTC
Forum: Mathematics
Topic: multiplicity of totient function
Replies: 17
Views: 2996

### Re: multiplicity of totient function

Well, I regret to inform you that your proof is flawed. Importantly, your assertion that the number of non-coprime elements to mn is not equal to the number of non-coprime elements to m times the number of non-coprime elements to n. Try some small cases for m and n and try to figure out where your p...
Sat Aug 28, 2010 6:46 pm UTC
Forum: Mathematics
Topic: Dimensions of related vector spaces
Replies: 9
Views: 1114

### Re: Dimensions of related vector spaces

As people before me have alluded to, there is a relation between the dimension of the domain of f, V, the dimension of the kernel of f and the dimension of the range of f, W (since f is surjective). In particular, you should make a basis of V by first making a basis for the kernel of f, then extendi...
Fri Aug 27, 2010 8:54 pm UTC
Forum: Mathematics
Replies: 4
Views: 855

Suppose we have a vector space V over F with basis v_0, v_1, v_2,...,v_n, ... and D:V -> V is defined by D(v_i) = v_{i+1}. Is there an associative commutative multiplication on V such that D is a derivation? (D(ab) = aD(b) + D(a)b). How many such multiplications are there? What if D:V -> V is define...
Tue Aug 24, 2010 9:27 pm UTC
Forum: Logic Puzzles
Topic: Three princesses
Replies: 446
Views: 205364

### Re: Three princesses

Actually, the above problem as stated is impossible (whoops). How about this: 5 sisters, exactly two random, at least one truth teller, at least one liar. I'm not sure if this one is solvable either, but it might have a better chance... Reason why the original problem is impossible: Suppose the thre...
Tue Aug 24, 2010 9:15 pm UTC
Forum: Logic Puzzles
Topic: Three princesses
Replies: 446
Views: 205364

### Re: Three princesses

As this problem seems completely and utterly solved, how about this variant: There are 5 sisters, and each of them is either truthful, false, or random (answers "yes" and "no" with equal probability, if you like, replace "randomly" with "maliciously"). If you ...
Mon Aug 23, 2010 7:34 pm UTC
Forum: Logic Puzzles
Topic: Light Switches!
Replies: 5
Views: 2302

### Re: Light Switches!

Given a desired output, the first 2 switches uniquely determine the rest (since the 2nd light is now only modified by the 3rd switch, and once that's determined, the 3rd is only modified by the 4th, and so on). Since the number of switch positions equals the number of light positions, i...
Tue Aug 17, 2010 10:58 pm UTC
Forum: Mathematics
Topic: Is every electric field possible?
Replies: 2
Views: 972

### Re: Is every electric field possible?

Here's the basic idea (to your general problem): If E is a curl-less vector field, then p = e0*div E is the charge distribution you're looking for, where e0 is the electric constant. Why? Well if E' is the electric field generated by the charge distribution p, gauss's law says div E' = p/e0, hence d...
Fri Aug 13, 2010 10:04 pm UTC
Forum: Mathematics
Topic: Revised Simple Proof of Beal's Conjecture
Replies: 29
Views: 4209

### Re: Revised Simple Proof of Beal's Conjecture

The only way that (A+B)(A^2-AB+B^2) stands a chance of being equal to C^z is if and only if A and B share a common prime factor which will give (A+B)(A^2-AB+B^2) a chance to become a positive integer raised to a positive integer power. I don't see this. Certainly A + B can be a positive integer rai...
Thu Jul 22, 2010 8:29 am UTC
Forum: Logic Puzzles
Topic: Twisted Wires
Replies: 8
Views: 3640

### Re: Twisted Wires

Spoiler:
Oddly enough, the biggest hint for me was that the puzzle was, in fact, solvable. I spent the first minute trying to prove it was impossible, then looked at the quoted text, then came up with an answer pretty quickly.
Thu Jun 10, 2010 6:28 am UTC
Forum: Mathematics
Topic: 5th degree polynomial equations
Replies: 2
Views: 1253

### Re: 5th degree polynomial equations

The proof that a 5th degree polynomial cannot be solved by radicals relies on the fact that the permutation group S5 is not solvable, meaning we cannot have a chain of subgroups {e} = G0 < G1 < ... < Gn = S5 such that each group is normal in the next and the quotient groups are all cyclic. This is b...
Mon Jun 07, 2010 4:19 am UTC
Forum: Logic Puzzles
Topic: 100 Princesses Problem... and dating
Replies: 9
Views: 4039

### Re: 100 Princesses Problem... and dating

Well, perhaps the love of your life comment was misleading. My take on it is this: For the 100 princesses problem, your best strategy is to reject the first 37 (calculated by N/e, where N=100), then accept the first princess that has the best suitability up to that point. Assume you meet pa...
Tue May 25, 2010 6:13 am UTC
Forum: Mathematics
Topic: Cardinality of uncomputable numbers
Replies: 48
Views: 6098

### Re: Cardinality of uncomputable numbers

Since computable numbers are a subfield of the real numbers, you COULD have a basis for the reals over computables which would consist of 1 and uncomputable numbers. This seems like it pretty much satisfies what squareroot is looking for. However, as has been pointed out, the use of said basis to ca...
Fri May 07, 2010 7:53 am UTC
Forum: Logic Puzzles
Topic: Palindromic Odometer
Replies: 19
Views: 3630

### Re: Palindromic Odometer

Spoiler:
I wouldn't call that 169. Tricky wording, maybe. But the answer is definitely gettable, and makes sense.
Fri May 07, 2010 7:09 am UTC
Forum: Mathematics
Topic: Simple combination problem
Replies: 3
Views: 622

### Re: Simple combination problem

The easiest way I can see to do this, to make absolute sure you don't double-count anything, is to break it up into cases. Using your more recent problem: Case 1: Answer 3 of the first 5, 2 of the last 5 Case 2: Answer 4 of the first 5, 1 of the last 5 Case 3: Answer 5 of the first 5, 0 of the last ...