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    Normalized correlation with a constant vector

    I am confused how to interpret the result of preforming a normalized correlation with a constant vector. Since you have to divide by the standard devation of both vectors (reference: ) , if one of them is constant (say a...
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    Vector difference metric that considers the variance of the components

    I am trying to match little square patches in an image. You can imagine that these patches have been "vectorized" in that the values are reordered consistently into a 1D array. At first glance, it seems reasonable to simply do a Euclidean distance style comparison of two of these arrays to get a...
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    Weighted least squares best fit plane

    I know that the plane through the center of mass whose normal is the eigenvector corresponding to the smallest eigenvalue of the scatter matrix of a set of points is the best fit plane. I now want to do a "weighted least squares" - would I simply multiply the...
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    Distance difference problem

    See the image in the attached document. I am looking for a function which will make f(d1-p)=f(d2-d1)=f(d3-d2) (see the very last part of the document) I thought it would be as simple as dividing by the angle between the lines, but that doesn't seem to work. Is it reasonable to do this...
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    Interpretation of power rule for integration applied to 1/x

    We all know \int \frac{1}{x} dx = ln(x) + c but if you try to apply the power rule for integration: \int x^n dx = \frac{x^{n+1}}{n+1} + c you get \int x^{-1} dx = \frac{x^0}{0} What can you learn from this/what does this mean? David
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    Wolfram answer for cubed root of -1

    At , if you type: 1) (-1)^(1/3) It gives a complex approximation. Isn't it exactly -1? David
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    Volunteers for an Experimental Teaching Tool

    I am starting a project to develop, with a very diverse team, an open source, modular, web-based teaching tool for math, science, and engineering courses. Here is a link to the full project description...
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    Fitting a quadric function to a set of points

    I have a set of points and I want to find a "best fit quardric surface" through the points. I did the following: 1) Assume the function is in the form: a x^2 + b y^2 + c xy + d x + e y + f = z 2) Make a nx6 matrix of the points put into (1), that is A=: x1^2 y1^2 x1y1 x1 y1 1 x2^2 y2^2...
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    Min cut on a graph

    If I have this graph: [Broken] Node 0 is the source and Node 3 is the sink. Is the min cut 11 (the minimum sum of capacities of edges cut to partition the graph into two parts)? There are two min cuts, correct? One that separates 0 from everything else, and...
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    Proving Inequalities

    I never have much luck when something says "prove something <= something else". I usually just fiddle around and occasionally get lucky and reduce it to a constant < an express I know can't be less than that constant. But most times I can't reduce it to something like that. Is there any kind of...
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    Combining independent probabilities in a meaningful way

    I have 1000 experiments on the same data, each of which is trying to decide the probability that the data agrees with a model. The problem is, even if each experiment REALLY agrees with the model (ie. p(d1) = .99, p(d2) = .99, etc), when I multiply these together to get P(d1 & d2 & d3.... &...
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    Infinite Integrals Numerically

    I want to integrate a function in c++ - so I dug out some numerical integration functions. However, they do not produce the correct results when the limits are infinite. Simply using 1e6 or something instead of infinity seems like a very "hack" solution... and I also don't know the function...
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    Interesting density function

    I am trying to make a function which is exponential for a while, and then turns gaussian: f(l,d) = \lambda e^{-\lambda d} , 0 < d < l and f(l,d) = (1-\int_0^l \lambda e^{-\lambda d} dd) \frac{1}{\sigma \sqrt{2 \pi}} e^{-(d-l)^2/(2\sigma^2)} , l < d < \infty (That is supposed to be...
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    Solving vv^T = M

    I have a known matrix M and an unknown vector outer product: ie v is 3x1 unknown and M is 3x3 known. Clearly there are 9 equations (each entry in vv^T must equal the corresponding entry in M) - but how do you solve this without manually writing those equations? Thanks, Dave
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    Determining the Likelihood function

    I was under the impression that the likelihood function was simply the probability density function but viewing the parameter theta as the variable instead of the observations x. Ie p(x|theta) = L(theta|x) However, the likelihood function is no longer a probability function See Example 1...
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    Solving AM = MB

    I would like to solve AM = MB where A,B,M are 3x3 matrices. What I came up with was to equate every entry in C to the corresponding entry in D (where C = AM and D = MB). You can then vectorize M (call it Mv) and figure out the 9 equations to fill a 9x9 matrix on both sides 9x9 matrix...
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    What is the purpose of the transpose?

    Every book I've seen starts out with "to find the transpose, make B_ij = A_ji . However, they don't explain exactly why would would want to do this. Ie. they tell you the inverse is useful because if you have Ax = b, you can find x by writing b = A^{-1} x. The only thing I can think of to...
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    RQ decomposition from QR decomposition

    What I'm wondering is: Q and R in the QR decomposition of A are the same Q and R in the RQ decomposition of which matrix? I found some matlab code which will get RQ from QR, but I don't understand how you would do those operations FIRST, then find the QR decomposition. ReverseRows = [0...
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    Interpretation of A A^T

    In the SVD, we use the eigenvectors of AA^T and A^T A as the input and output bases for the matrix. Does anyone have any intuition about these matrices? ie. if I multiply a vector x by AA^T, what space (in terms of the column space, etc. of A) will it bring x to? Thanks, Dave
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    Geometric interpretation of SVD

    Ax = U \Sigma V^T x (A is an m by n matrix) I understand the first two steps, 1) V^T takes x and expresses it in a new basis in R^n (since x is already in R^n, this is simply a rotation) 2) \Sigma takes the result of (1) and stretches it The third step is where I'm a bit...
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    Generalized Eigenvectors

    So I understand that if an nxn matrix has n distinct eigenvalues that you can diagonalize the matrix into S\LambdaS^{-1}. This is important because then this form has lots of good properties (easy to raise to powers, etc) So when there are not n distinct eigenvalues, you then solve...
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    Signed angle between vectors

    The following discussion is in 2 dimensions: Take two vectors, A and B. Generally we find the angle (theta) between them by cos(theta) = dot(A,B) / (norm(A) * norm(B)) however, take vector A to be [0, 1] (straight up) If vector B is [1,1], the angle between them is pi/4 radians. If...
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    Where to put the transpose?

    if you want to find the derivative (gradient) of f(x)^2 when f is a vector, you would get 2*f(x)*del(f(x)) I never know where to put the transpose!! sometimes its clear because another term in the equation will be a scalar, so you know an inner product is needed, but if you dont have a...
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    Units of Fourier Transform?

    If i have a signal S(t) (the plot would be voltage vs time) and I take its fourier transform, what are the units of the vertical axis? The horizontal axis can either be frequency in hertz or in radians, but what about the other axis? I guess generally I plot the magnitude of the transform since...
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    Functions that integrate to a gamma function?

    maple syntax: int(theta^y * exp(-theta*(1-alpha) ) , theta) I have a distribution that I need to integrate, and I know the result should have a gamma function in it. The only thing I have found helpful is: My function is kind of in that...
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    Finding mimum of a function with 2 local mins

    I have a function that looks like this: [Broken] The function is non analytic, so I have to use something like the steepest descent method with numerical gradients to find the minimum. However, if my starting point is on the left of the tall peak, the min I...
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    Humidity of my lab WAY too low

    Everyone in my office (8.5 x 3.5 x 6 meters) was complaining of dry eyes/throat. We thought "hey we should check the humidity". We bought two separate humidistat's that both read 15% RH at 73 degrees F!! So we bought a humidifier. Just one of the small ones that holds 2.2 gallons and empties...
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    Symbolic Vector Manipulation Software?

    I would like to do something like a derivative with respect to a vector or really anything involving a vector (of known dimension, but unknown values) going in and a vector coming out: x= 4 by 1 vector A = 4 by 4 matrix diff( Transpose(x) * A * x, x) should return (1/2)*A*x Is there...