Search found 66 matches

by CFT
Thu Jan 08, 2009 9:25 pm UTC
Forum: Science
Topic: Light Reflection
Replies: 15
Views: 2237

Re: Light Reflection

The question of why metals but not other solids reflect this way is my favorite student question ever ("Why are metals shiny when salts are not?" with nothing leading into the question). I had to go to a solid-state colleague to get the answer, which is both simple and profound. In a meta...
by CFT
Wed Jan 07, 2009 1:51 pm UTC
Forum: Mathematics
Topic: Prove that two curves in a square intersect
Replies: 76
Views: 6772

Re: Prove that two curves in a square intersect

Assume the curves don't intersect. Define a loop to be the the horizontal line starting on the left edge, until it ends on the right edge, and then close it using the bottom of the square. Redefine the vertical line so that it starts the last time it intersects the bottom of the square. Now the vert...
by CFT
Thu Jan 01, 2009 1:28 pm UTC
Forum: Science
Topic: Inteference is not forever?
Replies: 17
Views: 2630

Re: Inteference is not forever

1. QM has nothing to do with your question. Interference exists in all waves, even water waves. Your question applies equally to water waves and so QM is overkill. 2. You made a series of contradicting assumptions: cut a hole a wider than the slits near the center but at a point where the interferen...
by CFT
Sat Dec 27, 2008 6:27 pm UTC
Forum: Science
Topic: Time Dilation questions go here!
Replies: 77
Views: 10482

Re: Time Dilation questions go here!

if i accelerate at X meters/second^2 for Y seconds as observed by a stationary observer how much time do i experience? Assuming flat spacetime(special relativity), and your spacetime coordinates are given by x(s), y(s), z(s), t(s), s 0 <= s <= s 1 , then the time you experience is: \tau = \int_{s_0...
by CFT
Sat Dec 27, 2008 10:26 am UTC
Forum: Mathematics
Topic: The applicability of Godel's incompleteness theorems.
Replies: 5
Views: 1722

Re: The applicability of Godel's incompleteness theorems.

The key words are "effectively generated". Here is an example: Take the true statements of number theory as your axioms. Then you have a consistent and complete theory. But you can't determine whether a statement belongs in your theory or not. Godel's incompleteness theorem assumes that yo...
by CFT
Sat Dec 20, 2008 8:50 am UTC
Forum: Mathematics
Topic: Using the graph of a polynomial to prove it has no zeroes
Replies: 6
Views: 2129

Re: Using the graph of a polynomial to prove it has no zeroes

Sturm's theorem gives you how many zeros a polynomial has in a given interval. Multiple zeros are counted once.
by CFT
Tue Dec 16, 2008 11:02 pm UTC
Forum: Science
Topic: Is there a limit to temperature?
Replies: 36
Views: 3606

Re: Is there a limit to temperature?

Temperature is often defined as the derivative of the entropy with respect to the energy: T = \frac{dE}{dS} . The limit on temperature depends on your system. For example: A two state system, where every element can either be in the ground state or an excited state. When all elements are in the grou...
by CFT
Sat Dec 13, 2008 7:01 pm UTC
Forum: Mathematics
Topic: Galois group of some roots.
Replies: 1
Views: 642

Re: Galois group of some roots.

1. If the coefficients of the polynomial are rational than ab has to be rational. I'm not sure what you mean by your notation, but I don't see a way it will be rational, so I think you have an error. 2. The Galois group is a subgroup of S 3 . So it must be one of: S 3 , A 3 , a cyclic group with 2 e...
by CFT
Thu Dec 11, 2008 12:15 pm UTC
Forum: Mathematics
Topic: group theory question
Replies: 14
Views: 2127

Re: group theory question

Let a be a generator of the subgroup. Let b be an element not in the subgroup. What are the possible values of bab -1 ? Calculate ba and ba n . Show that b 2 is in the subgroup, and therefore b 2 = a n . If n is even define c = ba -n/2 . If n is odd define c = ba -(n+1)/2 . Calculate c 2 . U...
by CFT
Fri Nov 28, 2008 3:44 pm UTC
Forum: Mathematics
Topic: Modules
Replies: 13
Views: 2302

Re: Modules

Show that Zmn/C' is the dual of C, and that C and its dual have the same number of elements. You can do that with the structure theorem for Z modules.
by CFT
Mon Nov 24, 2008 4:19 pm UTC
Forum: Mathematics
Topic: *please* don't hate me for this...
Replies: 16
Views: 2954

Re: *please* don't hate me for this...

(as the classes of increasing bounded series of rationals) I assume by classes here, you mean equivalence classes? And the equivalence relation is something like (a n )~(b n ) iff the sequence a n -b n converges to 0? I think Konstantin refers to Dedekind cuts . This is a method of constructing the...
by CFT
Mon Nov 24, 2008 8:40 am UTC
Forum: Science
Topic: E&M Puzzler for all you physics geeks
Replies: 8
Views: 1479

Re: E&M Puzzler for all you physics geeks

Maxwell's equations, as they are taught for undergrads, are only correct in certain frames of reference in flat space-time. I don't think you can generalize statements like "Accelerating charges radiate" to curved space-time. My guess is: 1. A charged object on a table will not radiate, ev...
by CFT
Tue Nov 04, 2008 8:09 am UTC
Forum: Mathematics
Topic: Euclidean Geometry's Relation to Spherical Geometry
Replies: 6
Views: 1283

Re: Euclidean Geometry's Relation to Spherical Geometry

The reasons are mostly historical. Spherical geometry was researched because it was more interesting and simpler at the time. Today more general approaches are researched. For example, spherical geometry and euclidean geometry are special cases of Riemanian geometry of curved spaces.
by CFT
Wed Oct 08, 2008 10:22 pm UTC
Forum: Mathematics
Topic: Where do cross product comes from?
Replies: 5
Views: 1730

Re: Where do cross product comes from?

There are several ways to generalize cross products. One such way is based on the exterior product . If you have n variables x 1 , ..., x n the exterior algebra is made of expressions: \sum_i c_iM_i Where M i are of the form x_{i_1} \wedge x_{i_2} \wedge \cdots \wedge x_{i_k} , where \wedge (wedge) ...
by CFT
Wed Oct 08, 2008 9:36 pm UTC
Forum: Mathematics
Topic: Implicit differentiation is a lie!
Replies: 39
Views: 4650

Re: Implicit differentiation is a lie!

The implicit function theorem says that under certain conditions, a local solution to the problem must exist. In order for a local solution not to exist you need {\partial f \over \partial y} = 0 It is possible for a global solution not to exist if y "falls of the manifold". In the case of...
by CFT
Wed Oct 08, 2008 2:54 pm UTC
Forum: Science
Topic: Wanted: People who know QM
Replies: 67
Views: 6554

Re: Wanted: People who know QM

I'm not sure what you mean by the comment about spin 1/2 particles. Why do you use three dimensional vectors in your example, and then switch to two dimensional vectors? There is an ambiguity in the use of the word spin. An object's intrinsic spin refers to the way the object behaves under rotation...
by CFT
Tue Oct 07, 2008 11:42 pm UTC
Forum: Science
Topic: Wanted: People who know QM
Replies: 67
Views: 6554

Re: Wanted: People who know QM

I've still yet to come across an explanation of spin on the internet that doesn't degenerate into hand-waving over magnetism or Pauli's exclusion principle. Try learning the math involved. Its the only way to understand these things. The problem is that most people run away in terror when they see ...
by CFT
Tue Oct 07, 2008 10:34 pm UTC
Forum: Mathematics
Topic: Galois (?)
Replies: 8
Views: 1329

Re: Galois (?)

To add to that. Consider p = x 3 + 1 in F 2 [x] Claim: Gal(p) = Gal(s(p)), for all s in S 3 . Proof: char(F 2 )=2, so 1 is always a solution. I think you are wrong here. The splitting field of x 3 + 1 is a quadratic extension, while the splitting field of x 3 + x and x 3 + x 2 is F 2 . Let p(x) = x...
by CFT
Tue Oct 07, 2008 1:12 pm UTC
Forum: Mathematics
Topic: Simultaneous equations
Replies: 11
Views: 1544

Re: Simultaneous equations

uh...this doesn't require much brute force (or linear algebra) 1. eliminate p I'd eliminate s. The sheep have the smallest price, and you are therefore likely to have a larger number of them than the other animals. After you do that, take the equation modulo the higher coefficient. It gives you one...
by CFT
Mon Oct 06, 2008 9:39 pm UTC
Forum: Science
Topic: Wanted: People who know QM
Replies: 67
Views: 6554

Re: Wanted: People who know QM

I've still yet to come across an explanation of spin on the internet that doesn't degenerate into hand-waving over magnetism or Pauli's exclusion principle. Try learning the math involved. Its the only way to understand these things. The problem is that most people run away in terror when they see ...
by CFT
Sun Oct 05, 2008 9:17 pm UTC
Forum: Mathematics
Topic: Naughty Functions
Replies: 59
Views: 6281

Re: Naughty Functions

Functions that have the property that they can be expressed by convergent Taylor series are called analytic functions. Equivalently, these are all the functions that can be approximated using polynomials(using uniform convergence). What? On a compact interval, all continuous functions can be approx...
by CFT
Sun Oct 05, 2008 6:05 pm UTC
Forum: Mathematics
Topic: Naughty Functions
Replies: 59
Views: 6281

Re: Naughty Functions

Functions that are continuous and differentiable are so uncommon, in fact, that they can always be written as an infinite (or sometimes finite) polynomial. Careful when you say that... the function is 1/(1+x 2 ) is about as nice as you could ever want a function to be on the real numbers (infinitel...
by CFT
Sun Oct 05, 2008 2:00 pm UTC
Forum: Science
Topic: Wanted: People who know QM
Replies: 67
Views: 6554

Re: Wanted: People who know QM

Bell's theorems all concern spin. I feel like the Bell-like theorems saying "hidden variable theories don't work" really say "we don't understand the nature of spin." Bell's theorem is about probability. EPR paradox is about certain properties of vector spaces. Spin and polarity...
by CFT
Sat Sep 27, 2008 2:35 pm UTC
Forum: Mathematics
Topic: Summing an infinite series (sin(nx))/n
Replies: 24
Views: 13142

Re: Summing an infinite series (sin(nx))/n

... Hello topic. Anyway, @CFT: Do those converge individually? My knowledge of complex numbers is still a little lacking so I don't know how to apply geometric series to them, but just trying in a haphazard way using: \frac {a(1-r^n)}{1-r} (Where r is the common ratio and a is the first ter...
by CFT
Sat Sep 27, 2008 8:59 am UTC
Forum: Mathematics
Topic: Summing an infinite series (sin(nx))/n
Replies: 24
Views: 13142

Re: Summing an infinite series (sin(nx))/n

So if I can show that sum(sin(nx)) is bounded for any x, then we have pointwise convergence, since 1/n is monotonically decreases to zero. Use sin(nx) = (e inx - e -inx ) /2i. Then sum(sin(nx)) becomes the sum of two geometric series and you can calculate it. You will have to calculate the special ...
by CFT
Sat Sep 27, 2008 8:30 am UTC
Forum: Mathematics
Topic: The 1/7 trick in other bases
Replies: 2
Views: 1419

Re: The 1/7 trick in other bases

Assuming n and k are co-prime: The length of the repeating part of 1/k in base n is the order of n modulo k. = It is the smallest a s.t. n a = 1 mod k. For l/k the length of the repeating part is the smallest b s.t. n b l = l mod k. In order for these digits to be the same, we need that all of the r...
by CFT
Wed Sep 24, 2008 10:25 am UTC
Forum: Science
Topic: Projective Geometry and Physics
Replies: 2
Views: 1145

Re: Projective Geometry and Physics

To be honest, I'm not really sure what projective geometry is and how it's different from topology. Take a manifold without a metric or local affine geometry and all you have is a topology. Geometry isn't introduced until we define a metric or affine connection. According to wiki, projective geomet...
by CFT
Tue Sep 23, 2008 1:28 pm UTC
Forum: Mathematics
Topic: Question about Infinity
Replies: 72
Views: 5421

Re: Question about Infinity

samian wrote:are there any larger infinities than the number of reals? can anyone think of any bigger sets? or is it known that there are none?

For a set S, the power set P(S) consisting of all subsets of S is of greater cardinality than S.
by CFT
Tue Sep 23, 2008 11:25 am UTC
Forum: Science
Topic: Large Objects and Spin
Replies: 18
Views: 2467

Re: Large Objects and Spin

Half integral spin particles have the property the after a 360 degrees rotation, their wave function changes sign: from f(x,t) to -f(x,t). However, observation values do not depend on a constant phase factor. That is the functions f(x,t) and af(x,t) for |a| = 1 will give the same observation. So 360...
by CFT
Thu Sep 18, 2008 10:45 pm UTC
Forum: Science
Topic: Superconductor confusion
Replies: 8
Views: 3006

Re: Superconductor confusion

Superconductors expel all electric and magnetic fields from inside them, therefore the potential difference V is always 0. So Ohm law: V = IR is trivially satisfied. There are several ways to induce electrical current in superconductors: 1. Connect it to a battery. In this case the internal resistan...
by CFT
Tue Sep 16, 2008 4:24 pm UTC
Forum: Science
Topic: String Theory?
Replies: 35
Views: 3869

Re: String Theory?

I remember these two things when I first heard about it, which hopefully can be corrected and or disproven or explained. 1) String Theory might explain why gravity is weird. Apparently gravity has some weird properties through our observations, and I thought it had something to do with gravity seem...
by CFT
Wed Sep 10, 2008 9:24 pm UTC
Forum: Mathematics
Topic: Separable PDEs
Replies: 3
Views: 1636

Re: Separable PDEs

If your equation is linear then a sum of two solutions is also a solution: f 1 (x)g 1 (t) + f 2 (x)g 2 (t). However, in many cases(e.g. the wave equation) the general solution is given by: \sum_ia_if_i(x)g_i(t) or \int a(\omega)f_\omega(x)g_\omega(t)\,d\omega
by CFT
Wed Sep 03, 2008 4:24 pm UTC
Forum: Mathematics
Topic: n-dim matricies
Replies: 15
Views: 2195

Re: n-dim matricies

This rule generalizes to products of more than 2 vector spaces. Additionally the tensor product (V \otimes U) \otimes W is canonically isomorphic to V \otimes U \otimes W . The isomorphism takes the basis element (v_i \otimes u_j) \otimes w_k to v_i \otimes u_j \otimes w_k . Since t...
by CFT
Wed Sep 03, 2008 2:28 pm UTC
Forum: Mathematics
Topic: n-dim matricies
Replies: 15
Views: 2195

Re: n-dim matricies

This rule generalizes to products of more than 2 vector spaces. Additionally the tensor product (V \otimes U) \otimes W is canonically isomorphic to V \otimes U \otimes W . The isomorphism takes the basis element (v_i \otimes u_j) \otimes w_k to v_i \otimes u_j \otimes w_k . Since t...
by CFT
Wed Sep 03, 2008 2:20 pm UTC
Forum: Movies and TV Shows
Topic: Firefly & Serenity (Use Spoilers!)
Replies: 918
Views: 198238

Re: Firefly

Did anyone read Herman Hesse Siddhartha? The courtesan there has her own bodyguards.

This and the Hindi theme always made me think that Inara is based on her.
by CFT
Mon Sep 01, 2008 9:34 pm UTC
Forum: Mathematics
Topic: n-dim matricies
Replies: 15
Views: 2195

Re: n-dim matricies

Here is a quick introduction to tensors from the mathematical perspective. Let V, U be vector spaces and \{v_i\}_{i=1}^n , \{u_j\}_{j=1}^m bases for the vector spaces. then the tensor product V \otimes W is the nm dimensional vector space with basis \{v_i \otimes u_j\} i = 1, ... n; j = 1, ... m. Th...
by CFT
Sat Aug 23, 2008 8:43 am UTC
Forum: Mathematics
Topic: Infinite Sum of ((-1)^n)/(n^2+1)
Replies: 8
Views: 3854

Re: Infinite Sum of ((-1)^n)/(n^2+1)

This is called Poisson summation . It has some interesting uses in number theory. Try the same method for e^{i\alpha x} . ( \alpha real and the function continued periodically with period 2\pi ) It should give something like {\pi \cos \pi \alpha \over \sin \pi \alpha} = \frac{1}{\alpha} + \sum_{n = ...
by CFT
Thu Aug 21, 2008 10:22 am UTC
Forum: Serious Business
Topic: Marxism/Socialism/Communism
Replies: 263
Views: 23918

Re: Marxism/Socialism/Communism

[...]Russia, China, etc don't count because they run almost contradictory to the way the theory should have been applied. What communists don't seem to understand is that most people don't want them in charge. The evidence is how poor they perform in democratic elections. This is why communists mus...
by CFT
Wed Aug 20, 2008 7:45 pm UTC
Forum: Serious Business
Topic: Religion
Replies: 2000
Views: 189439

Re: Religion

First, let me state that I join this discussion late, and there is no way that I'm going to read 35 pages. So I have very little clue what I'm talking about. Since the wages of sin are death, you take it out on something innocent-- in the days of the Bible, that would be a sheep or something. If you...
by CFT
Wed Aug 20, 2008 5:01 pm UTC
Forum: Serious Business
Topic: Marxism/Socialism/Communism
Replies: 263
Views: 23918

Re: Marxism/Socialism/Communism

wrote: existential_elevator wrote:Government monopoly is better than private monopoly Capitalism supports competition, not monopolies. What you wrote is simply a straw man argument. That is, quite literally, putting words in my mouth. Nicely ironic, though. I apologize for making such a breach of e...

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