# Search results

1. ### News Chalmers Johnson, how to sink America

1.) “There are three broad aspects to our debt crisis.” America is not in a debt crisis. Public debt as a percentage of GDP indicates how heavily a country is indebted in comparison to its ability to pay as measured by its GDP. Here is a Wikipedia article with the computed ratios for many...
2. ### Part time Jobs

Does anyone know a way for someone with good mathematical and computer science(including programming) skills to make some money doing part time work? I guess someone could look at my other posts to assess my level of mathematical competence. Something that would take around 10 hours per week...
3. ### Connections in differential geometry

I would start by examining an introductory book on Riemannian Geometry. One possibility is Riemannian Geometry written by Gallot, Hulin, and Lafontaine. I recommend doing some calculations and understanding some examples. The field of geometry is motivated by visual "intuitions", but not...
4. ### Parallelizable Manifold

R^(n+1)\{0} is a subset of R^(n+1). You can explicitly write down a global trivialization for R^(n+1) that restricts to a global frame for R^(n+1)\{0}. A global frame for R^(n+1) is... (1,0,...,0) (0,1,0,...,0) . . . (0,...,0,1)
5. ### Submersion btw R and S^1

Let f:S^1-->R be any smooth map and consider Im(f) subset of R. Let m=max Im(f). At any point x in the inverse image of m, the map f will not be a submersion. Take local coordinates c around the point x. Then, f compose c:(-e,e)-->R is not a submersion. Hence, f is not a submersion. The...
6. ### Submersion btw R and S^1

What you wrote is wrong and a bit difficult to read. 1.)A submersion from S^1 -> R does not exist. 2.)A submersion from R -> S^1 does exist. 3.)Your definition of submersion: "By a submersion from M to N, I mean a map f:M-->N whose tangent map is surjective." is missing a key element...
7. ### Theories of gravity

Do any theories of gravity exist other than general relativity that are capable of explaining the perihelion of mercury's orbit? In particular, I would like to know if a theory of gravity exists that does not impose the fact that nothing can exceed the speed of light in a vacuum. Newton's theory...
8. ### To distinguish if a critical point is a saddle point or not, . . .

This is the multivariate version of the second derivative test from calculus. If the second derivative is positive you are at a minimum. If the second der is Negative you are at a maximum. Let D be the discriminant matrix, and h a 2x1 column vector. At a minimum h^T*D*h>0 for all small h...
9. ### Vector field

The operations are linear on each fiber. So, if you solve w(Y)=0 and find one X such that w(X)=f, then X+Y is such that w(X+Y)=f. The question is not optimally formulated, and it is a little unclear why you are asking this question. Do you have an application in mind? Are you reading a proof...
10. ### Vector field

a differential 1-form on a manifold acting on a vector field on a manifold yields a function.
11. ### Vector field

Or none. Let w=0(the 0 1-form). Let f be a non-zero function.
12. ### Let M be a three dimensional Riemannian Manifold that is compact . . .

Let M be a three dimensional Riemannian Manifold that is compact and does not have boundary. I believe manifolds that are compact and without boundary are called closed. So, my manifold M is closed. I'm interested in knowing the answers to the following questions. Under what conditions is...
13. ### Tensor Newbie trying to find Kolecki's rhythm

When one index is raised and one index is lowered, the summation symbol is omitted. There is nothing more to it. The main difficulty is accepting that such a convention is actually useful. It is useful because Einstein cleverly chose to make some indices raised and some indices lowered.
14. ### Tensor product

As was mentioned previously, the physical significance depends on the application. Maybe this explanation will help. Let V be a three dimensional vector space with basis {e1,e2,e3}, and let W be a four dimensional vector space with basis {f1,f2,f3,f4}. Then V tensor W is a 12 dimensional...
15. ### Thurston Geometries

It appears to be essentially worked out in Scott's paper. I'll post a bit more about it when I've worked it out.
16. ### Thurston Geometries

Note to moderators... this is actually a serious question... The reference to Adams' book was for amusement... but the rest is serious... after all... The Answer to The Ultimate Question Of Life, the Universe and Everything...and... i'm really interested in the answer... o:)
17. ### Thurston Geometries

Hi, Every Thurston Geometry (X,Isom(X)) with the exception of the geometry modeled on S^2xR can be achieved as a 3D Lie group with a left-invariant metric. That is, the space X=G, where G is a 3D Lie group with a left-invariant metric. After picking a left-invariant frame field consisting of...
18. ### A smooth manifold

You have not reformulated the problem correctly. I suggest looking at the definition of a differentiable manifold and figuring out exactly what it is that you need to prove.
19. ### A smooth manifold

Let T be the subspace topology of M inherited from the plane and let T' be the subspace topology of the x-axis inherited from the plane. Clearly, (x-axis, T') and (R,standard topology on the real line) are the same. Projecting M onto the x-axis gives a bijective correspondence of M and the...
20. ### A smooth manifold

Since Halls of Ivy designated this as a homework problem, I'm not sure what the appropriate protocol is. I am new to these forums. Let me rephrase the problem so that it is slightly clearer. Let M={(x,y)|y=|x|}. M inherits the subspace topology from the plane. Denote the subspace topology...
21. ### Lie Groups

I solved the problem. The functions can be found explicitly... it is fairly straightforward, but the calculation is a bit tedious. The functions involve hyperbolic sines and hyperbolic cosines.
22. ### Practical effects if it is proved that P=NP

I remember something like this too... but I didn't have a source... Nice reply with the other post... good to know that there is a guiding principle that says the algorithms don't usually scale well and the probability declines with the size of the scale... funny... best use of advanced...
23. ### Lie Groups

Oh... it occurred to me that it may be possible to do this using infinitesimal generators... which basically means exponentiating a matrix... I'll post again after working it out...
24. ### How do I estimate complex eigenvalues?

If your off diagonal entries are small, maybe you could use Gershgorin's Circle Theorem... :shy: http://en.wikipedia.org/wiki/Gershgorin_circle_theorem ps... this may not be what you're looking for...
25. ### Finding determinant of Vandermonde matrix

This Wikipedia article might help. http://en.wikipedia.org/wiki/Vandermonde_matrix The formula is given, but a proof is not given. To prove the formula given in the article, replace the ith row by 1 x x^2 ... x^(n-1) Then, take the determinant. The determinant is a function of x...
26. ### Lie Groups

It appears someone fixed the LaTeX issue... Thanks... By the way, the functions alpha, beta, gamma, and delta comprise the automorphism phi. phi is a map from R to the automorphisms of R^2. For some values of lambda, theta, and sigma; I know these functions explicitly. If anyone believes that...
27. ### Lie Groups

Let G be a 3-dimensional simply-connected Lie group. Then, G is either 1.)The unit quaternions(diffeomorphic as a manifold to S$^{3}$) with quaternionic multiplication as the group operation. 2.)The universal cover of PSL$\left( 2,\Bbb{R}\right)$ 3.)The...
28. ### Will studying fourier analysis prepare one for string theory and QP?

I would recommend studying theoretical mathematics. Differential Geometry and Functional Analysis would be quite useful. Fourier Analysis is also quite useful. Many, many things are quite useful... E=mc^{2} :rofl:
29. ### Interacting with people after lots of math

I find that after exercising for 5 hours, I find it difficult to walk and do other physical activity. Maybe your brain is tired after working so much... ps... I haven't actually exercised for 5 hours at one time...
30. ### Practical effects if it is proved that P=NP

The question in the OP asked what would be the practical implication of P=NP NOT the practical implication of P not equal to NP, but the mistake is understandable given that many researchers believe that P does not equal NP. I imagine your post meant that if P does not equal NP, then there is...