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    Distribution of balls in a box (with a twist)

    Suppose I have an box (set) containing two different colored balls, red and blue, say. Now, suppose the balls differ in size, where the size of the red balls has one particular distribution and those of the blue another. How can we describe the distribution of the balls in the box?
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    Combining Distributions (ex. Mixture distribution, copula)

    This is a vague question and I apologize in advance for not being able to explain it better. I'm combining r.v.'s from different populations (distributions). The resulting population can be thought to come from a mixture distribution. I think another way of describing the resulting...
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    Solve (-1)^x=1

    Ok. So the answer to finding the solution of (-1)^x=1 is clear. But say we didnt know it and wanted to solve it. One approach is to take the log of both sides x\cdot log(-1)=log(1)=0 But now the right hand side is defined where as the left is not! What am I missing?
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    Dielectric Boundary Condition Question

    Hi, I have a question regarding the boundary condition present for a dielectric immersed in a static field. I hope one of you physics guru's can shed some light on this. Suppose we have a dielectric in space subjected to some external static electric field. I have read (without explanation)...
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    Dielectric Sphere in Uniform Field

    Hi, One of the boundary conditions when solving for the potential, \Phi, outside a dielectric sphere placed within a uniform electric field is \lim_{r→0}\Phi(r,θ)<\infty Can anyone explain/prove why this so. Thanks,
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    Conditional Probability - Markov chain

    Hi, I was reading about Markov chains and came across the following statement: "The conditional distribution p(x_n|x_{n-1}) will be specified by a set of K-1 parameters for each of the K states of x_{n-1} giving a total of K(K-1) parameters." In the above we have assumed that the...
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    Acadamia vs. Industry - supervisors role

    Acadamia vs. Industry -- supervisors role Is there an incentive for supervisors to guide their Ph.D. students to follow an academic career rather than move to industry?
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    2nd order Linear DE

    Hi, When solving a 2nd order Linear DE with constant coefficients (ay''+by'+cy=0) we are told to look for solutions of the form y=e^{rt} and then the solution (if we have 2 distinct roots of the characteristic) is given by y(t)=c_1 e^{r_1 t}+c_2 e^{r_2 t} This is clearly a solution, but...
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    Dielectric Sphere in Time Varying Field

    Hi, I'm interested in how a dielectric sphere effects a (spatially) uniform time varying field. I'm sure I'm not the first to inquire about this very topic. Could anyone direct me to a resource? Thanks,
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    Dielectric Vs. Resistor

    Hi, Do material posses both dielectric as well as resistance properties? I imagine that when there is a difference in potential across a volume of some material, some current will flow (I=V/R <-- the material's resistance), but also the material may become polarized to a certain degree...
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    G(E[X],E[Y]) or E[g(X,Y)]

    Suppose X and Y are r.v. Suppose also that we get N samples of a r.v. Z which depends on X and Y. That is Z=g(X,Y). Which is a better estimate of the true value of Z? Z=E[g(X,Y)] or Z=g(E[X],E[Y])
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    Uniform Field & Poisson equation Mismatch?

    Hi, I'm getting some confusing results and cant figure out what is wrong Suppose we have a uniform field E=[0,0,E_z] in a dielectric media. By E=-\nabla\psi we can deduce that \psi(x,y,z)=-z E_z But, taking the Laplacian \nabla^2\psi=\frac{\partial^2 (-zE_z)}{\partial z^2}=0 does...
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    Mean of a function of a random variable

    Hi, I have a random variable X with some zero-mean distribution. I have a function Y of this r.v. given by something complicated Y=(a+X)^\frac{2}{3} Is there an explicit way of finding the distribution of Y or even its mean? Thanks
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    Attenuation of E-field

    Do electric fields attenuate in space? If so what causes the attenuation?
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    Working w/ Complex representation E-field

    Often a time varying E-field is represented in complex format. I have a simple E-field (uniform in space) given by \vec{E}(t)=E_o\cos(\omega t)\cdot\hat{k} or equivalently, the real part of \vec{E}(t)=E_o e^{\omega t}\cdot\hat{k}. We know the potential is the negative gradient of the...
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    Complex representation of electric field

    Confused.. please help! Often when an electric field varies sinusoidally with time, it is represented as a complex number. Say, \vec{E}(t)=A\cos(t) \cdot \hat{k} We know at any time, the magnitude of E is A\cos(t). Alternatively the same vector E is understood to be the real part of the...
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    Gauss's Law in Matter

    Hi, The Poisson equation (or Gauss Law) in a vacuum is given by \nabla^2\phi=-\frac{\rho}{\epsilon_0} where \rho \mbox{ and } \epsilon_0 are the charge density and vacuum permittivity or (electric constant of space). My question is what is the Gauss's Law in a dielectric material? Do...
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    Laplace Equation in Cartesian Coor.

    Solving the Laplace equation in Cartesian Coordinates leads to the 2nd order ODEs: \frac{X''}{X}=k_1, \qquad \frac{Y''}{Y}=k_2 \qquad \frac{Z''}{Z}=k_3 In each case the sign of k_i will determine if the solution (to the particular ODE) is harmonic or not. Hence, if two people solve the...
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    Simple limit

    Hi, If k_3 \cdot (A_1 \cos k_1 x + A_2 \sin k_1 x)\cdot(B_1 \cos k_2 y + B_2 \sin k_2 y)\cdot(C_1 e^{k_3 z}-C_2 e^{-k_3 z})=E in the limit as \sqrt{x^2+y^2+z^2}\rightarrow\infty Can we infer anything about any of the constants?
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    Soln of Leplace in Cartesian Coord

    Hi I'm getting confused solving the Laplace eqn in Cartesian coordinates. The equation can be solved by solving each of \frac{X''(x)}{X(x)}=-k_x^2, \qquad\qquad \frac{Y''(y)}{Y(y)}=-k_y^2, \qquad\qquad \frac{Z''(z)}{Z(z)}=k_z^2 and then substituting into the equation...
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    E.Field - E.Potential conversion

    Hi, I'm given a sensitivity threshold for a system in terms of the electric field. Is there any way of finding what the threshold is in terms of electric potential? I know the relation E=-\nabla\Phi , but I dont see how I can apply it here. ex. Given E_{t} \mbox{ in } [\frac{V}{m}] what...
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    Perturbation of a uniform electrostatic field by a dielectric cube

    Hi, Is there any way to analytically calculate the perturbation of a uniform electrostatic field by a dielectric cube. I know a solution exists for dielectric spheres but I haven't been able to come across the solution, when dealing with a cube. Ohh.. and I'm assuming the simplest case...
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    MATLAB MatLab structures

    Hi The following structure has 2 elements. people = struct(... 'name',{'bob', 'john'},... 'numKids',{0, 2}, ... 'kidsage',{[],[12,9]}); each element people(1), people(2) has three frields (name, numKids, kidsage) Instead of declaring the structure in one line as above...
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    MATLAB MatLab Matrix Help

    Hi, I have a quick question regarding MatLab. I have 3 matracies A, B and C. For every entry equal to 1 in B, I want to let the corresponding entry in A be an integral of x from 0 to C. ex. for all (i,j) if B(i,j)==1 then let A(i,j)=quad(@(x) x, 0, C(i,j)) A quick way to do...
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    Solving a cubic sort of!

    Solving a cubic.... sort of! Hi, Can the equation \sqrt{Ax-x^3}+\sqrt{Bx-x^3}=C be solved explicitly? All of MathLab, Maple and WolframAlpha seem to give an explicit solution but they dont show how they come to it. I'm afraid they may be missing other possible solutions. Thanks,
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    I once came across a Wikipedia page describing a system where

    I once came across a Wikipedia page describing a system where indifferent of the initial starting position, at some final time t=T the system would always reach the same equilibrium position. Does anyone know what the name of such a system is? I recall there was an animation of 4 balls each...
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    Representing a function in a different space

    I have an implicit function f(x,y,z) which represents a surface in the XYZ Cartesian reference frame. I would like to change this current XYZ reference frame by a matrix M. ie. M: XYZ \rightarrow X'Y'Z' If I have a vector v in XYZ, then v'=Mv is my representation of v in the X'Y'Z'...
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    Complex notation of periodic functions

    I have found in physics literature a periodic function of time is many times written in complex form. For example,f(x,t)=g(x)e^{i\omega t} As a non-physicist this has proven a bit confusing. Is it generally understood that the function we are really interested in the real part of the...
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    Kirchhoff across dielectric

    Hi Everyone, I'm told that the following formula represents Kirchhoff's current law g_1 E_{1 n}+\varepsilon_1 \frac{\partial E_{1 n}}{\partial t}=g_2 E_{2 n}+\varepsilon_2 \frac{\partial E_{2 n}}{\partial t} where the first term on each side is Ohm's law and the conductive current and...
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    Integral homework problems

    Hi everyone, Can anyone show me how the property \frac{1}{2\pi} \int ^{\infty} _{-\infty} e^{i\omega x}d\omega= \delta(x) holds. Thanks,
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