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  1. K

    Compact manifold question

    I mean this with no patronisation intended, but you've answered your own question because in mathematics, a closed manifold is a special type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold.
  2. K

    What is the physical meaning of curvature?

    The curvature of a surface is (or at least can be) defined as moving a vector through parallel transport around a closed loop.
  3. K

    How cubed root of x is not differentiable at 0

    As LCKurtz writes, the derivative here is lim_{h->0}h^{-2/3} = lim_{h->0}/cuberoot{h^{-2}} = /lim_{h->0}(/frac{1}{\cuberoot{h^2}} -> /infty.
  4. K

    Integral of y=x^x

    Could approximate-around x=0 x^x looks like; x+x^2 ((log(x))/2-1/4)+1/54 x^3 (9 log^2(x)-6 log(x)+2)+1/768 x^4 (32 log^3(x)-24 log^2(x)+12 log(x)-3)+(x^5 (625 log^4(x)-500 log^3(x)+300 log^2(x)-120 log(x)+24))/75000+(x^6 (324 log^5(x)-270 log^4(x)+180 log^3(x)-90 log^2(x)+30...
  5. K

    Need help solving CDF (Cumulative distribution function)

    Ok so now you have the CDF in terms of the error function erf(/frac{1}{/sqrt{2}}) what is the associating PDF?
  6. K

    Help me to chose the right calculus textbooks.

    Tom Apostol - Calculus. the best textbook I have ever used.
  7. K

    How to evaluate this double integeration of a gaussian function?

    Perhaps try polar coordinates. x=r sin /theta x' = r cos /theta
  8. K

    Comparing normal distribution divided by normal distribution

    Stephen Tashi; Should your reply be directed to LeotheLion? I agree with your comment either way.
  9. K

    Supersymmetry n=8

    N=8 supergravity has exactly the same degree of freedom as N=4 Yang-Mills. The maximal number of supersymmetry generators possible is 32. Theories with more than 32 supersymmetry generators automatically have massless fields with spin greater than 2. It is not known how to make massless...
  10. K

    Matrix Equation

    Apologies, can now see the x_{i} are vectors. I'm at work and scanning articles when no-one is looking. I your reasoning has lead to to conclude that /boldsymbol{x_{1}^T}A_{1} = /boldsymbol{0} - How do you know this?
  11. K

    Matrix Equation

    Can any of the x_{i} or A_{i} be inverted? (i.e., do you know anything about their determinants?)
  12. K

    Powers of a superdiagonal matrix

    Whoops! Thought it was too easy. didn't read your laTex code correctly. Nor did I fully understand the terminology of "Super"diagonal. I'll have another think. My intial thought here is that you would be dealing with matrices in Jordan Normal form. Are you familiar? If not they are upper...
  13. K

    Collatz Conjecture - Bouncing Ideas

    Only got time to quickly look at your assumption paragraph beginning "If every +ve integer can be expressed as odd * 2^n..." But I can see straight away that you are not considering odd numbers at all in your tree branches as (any number odd or even) * (power of 2) = even. You do...
  14. K

    Question about GR

    My PhD c2006 concerned a lot about General Relativity. I don't claim to "know" it but rather I am intruiged by it. It sounds like you are asking yourself the question of whether I am intruiged enough to follow an advanced course. I am not familiar with what you have learnt so far but my...
  15. K

    Can time be quantized?

    Currently there is no theory to suggest that time can be quantized. However, in order that a theory of "Quantum Gravity" to exist, that is, the cohesion between Quantum Physics and Gravitiation, there is a suggestion that this may be achievable if one were to quantize time as it were. A nice...
  16. K

    Comparing normal distribution divided by normal distribution

    If, in your definition, X and Y are independent random vaiables you could consider their difference. Many many distributions of the type X-Y may show signs of normality. In this case if X has mean /bar{x}, variance /sigma^{2}_{x} and Y has mean /bar{y}, variance /sigma^{2}_{y} then you may find...
  17. K

    Powers of a superdiagonal matrix

    I think this is quite simple. You would simply raise each matrix element on the diagonal to the order of the power you are computing. If you are computing M^2 then the element in a_{11} position would be (n-1)^2, a_{12} = (n-2)^2 and so on down the diagonal. The then the (n-1)th power...
  18. K

    Math proof

    Surely one would assume a memoryless system to allow for the theory of queues and stochastic processes can be applied to your queueing system? I am sure it is mathematically allowed to assume (wlog) that your system is Markovian.
  19. K

    Math tricks for everyone

    Agentredlum. In your "proof" of the solutions to the quadratic equation you state: "Start with a general quadratic, do not set it equal to zero, set it equal to bx+c ax^2 = bx + c" The line above is not representative of what you intimated you would do. It should read "Start with...
  20. K

    Meaning of k in D. E. problem

    The parametric representation you give above is actually the parametrisation of the cycloid. The value of $k$ here (or $k^2$ if you will) represents the "height" of the humps in the equation of the cylcoid itself. If you are unfamiliar with the cycloid, check Wolfram...
  21. K

    Word for one-to-one correspondence between ideals and modules of an algebra

    For a left R-module M, you can identify certain submodules of M that are similar to that of prime ideals in a ring, R. With that definition there exists conditions on the module M which imply that there is a one-to-one correspondence between isomorphism classes of indecomposable M-injective...
  22. K

    Solving 3D matrix equation

    Yep - I think HallsofIvey is correct. The term we could use here is an underdetermined system.
  23. K

    Transposition of a matrix

    There are of course many ways to invert a matrix but thie is not the only use for the transpose. Systems of linear equations can be reformulated into matrix systems by looking at the equation xAx^{T} = b where x is a n x 1 column vector with entries {x_{1},...,x_{n}} and Z is a square matrix...
  24. K

    Why does the gradient vector point straight outward from a graph?

    Hi. The gradient vector measures the change and direction of a scalar field. The direction of the gradient is expressed in terms of unit vectors (in 3-dimensions, say) and points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of...
  25. K

    Group representations on tensor basis.

    Tensor representations are given by the direct (tensor) product of copies of the vector representation. They act on the set of tensors of a given rank, which is indeed a linear vector space. These tensors are called Lorentz 4-tensors. For example, consider (2, 0) tensors, that is, tensors...
  26. K

    Student t orthogonal polynomials

    As far as I can remember you should end up with the prthogonal polynomials taking the form /phi_{m}(t) = A_{m}/[1+/frac{t^{2}}{v}/]/frac{d^{m}}{dt^{m}}/[/frac{1}{1+/frac(t^{2}}{v}}^{/frac{v-1}{2}-m}/] Then, /int_{- /infty}^{+ /infty}...
  27. K

    Best model to fit data

    Ok - Local regression. I awsn't aware of it as a technique. I think you may be right though, looks like a good technique for my problem. An unknown function representing the data and my needs for a forecast. (Edit: I see that the technique is local since all we are doing are Taylor expansions...
  28. K

    Substitution of variables in improper integrals

    Without giving it a whole lot of thought I am going to say yes. I think that in some occasions it may be pertinent to change variable of integration so as to create an improper integral (for instance, when one considers smooth functions with compact support in distribution theory) as one may...
  29. K

    Best model to fit data

    I agree with your sentiment and accept that my philiosophy here is somewhat flawed and is certainly not perfect. My motivation for using the approach in my previous post is based on a "best case scenario" given some time series data that isn't really appropriate to answer the questions I am...