Gaussian Definition and 772 Threads

Homework Statement The random variables X and Y have a joint probability distribution of f_{XY}(x,y)= \frac{1}{2\pi \sigma _X \sigma _Y \sqrt{1-\rho ^2}} \exp \{ \left [ -\frac{1}{2(1-\rho ^2)} \right ] \left [ \left ( \frac{x-\mu _X}{\sigma _X} \right )^2 + \left ( \frac{y-\mu...

Homework Statement I know that in free space, the width of a Gaussian beam can be written as W=W_0\sqrt{1+(\frac{z}{z_0})^{2}}. However, I was wondering if it was possible to express this width as a function of refractive index instead (since I don't believe a Gaussian beam originating in say...

I was assigned a multistep task by my professor so I first made the molecule with ChemDraw, then I ran an optimisation using B3LYP/6-31G. I then ran a frequency scan and checked for any imaginary frequencies. There were no negative frequencies and my understanding is that this means that...

I posted this in two forums because the question might be too chemistry related for the people in the calculus forum. I'm a visual thinker so I struggle a bit to get my head around calculus concepts. So as an example, here's a potential energy surface: lets say this represents the structure of...

I'm learning about how Gaussian works and I'm reading about potential energy surfaces which Gaussian uses to calculate very properties of a molecule. Right now I'm reading about optimisations, I understand the concept that to optimise the structure of a molecule, it needs to find the global...

Homework Statement I must find the characteristic function of the Gaussian distribution f_X(x)=\frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{1}{2} \left [ \frac{(x- \langle x \rangle )^2}{\sigma ^2} \right ]}. If you cannot see well the latex, it's the function P(x) in...

Homework Statement I have to prove: ∫(-infinity:infinity) cos(pi*v/2L)*e^-((L-L_av)^2/sqrt(2pi)*sigma^2) dL proportional to cos(pi*v/2L_av)*e^-(t/tau)^2 tau is some constant, and sigma << L_av. The Attempt at a Solution i can change the integral to 0:infinity, since sigma <<...

Homework Statement For some Gaussian distribution, let's say e^{-x^2} times a constant, I want to find the expectation value of x^2. In other words, I want to evaluate: \int^{\infty}_{-\infty} e^{-x^2}x^2dx Homework Equations Integration by parts: \int udv = uv - \int vdu The...

Here is a link to a course which i am studying, http://quantummechanics.ucsd.edu/ph130a/130_notes/node89.html#derive:timegauss My problem comes from the k' term attached to Vsub(g) (group velocity). I used the substitution k' = k - k(0), factored out all exponentials with no k'...

Hey, Whenever I read Relativity I get stuck around Gaussian coordinates, i can't seem to find much out about them, do they have another name? does anyone know any good resources explaining them? or am I just in over my head? thanks guys,

Homework Statement let \varphi:\mathbb{Z}[i]\rightarrow \mathbb{Z}_{2} be the map for which \varphi(a+bi)=[a+b]_{2} a)verify that \varphi is a ring homomorphism and determine its kernel b) find a Gaussian integer z=a+bi s.t ker\varphi=(a+bi) c)show that ker\varphi is maximal ideal in...

I think everyone knows that Q(x)= P(X>x) where X is a Gaussian Random variable. Now I was reading about it and it says that Q(x) is bounded as follows Q(x)≤ (1/2)(e-x2/2) for x≥0 and Q(x)< [1/(√(2∏)x)](e-x2/2) for x≥0 and the lower bound is Q(x)> [1/(√(2∏)x)](1-1/x2) e-x2/2 for x≥0 Can...

Homework Statement Homework Equations The Attempt at a Solution Looking at equations 17 and 18, I don't see how that follows. If you substitute infinity for x you're going to get infinity divided by some real number which is infinity

We know that momentum is proportional to k so by adding more waves to localise our particle we are adding more waves with independent momentum values Upon measurement, the gaussian wavepacket must collapse into one eigenstate of momentum, but if we have a very localised packet there will be...

Hi, I'm studying about Chebyshev Quadrature and i found so little and confused information about this. I don't know if Gauss-Chebyshev Quadrature is the same of Chebyshev Quadrature. The only good information that i found was from Wolfram...

Homework Statement for some reason I am having real trouble trying to solve this system. Homework Equations The Attempt at a Solution a=(10,11,4) v1=(2,1,4) v2=(-1,-2,1) v3=(3,3,-1) the qstn does a belong to the span (v1,v2,v3)? so I do gauss elim but I keep running...

Homework Statement Find the convolution of g(x) = e^{-πx^{2}} with itself from -∞ to ∞ using the definition of convolution, not the Fourier Transform. The Attempt at a Solution See my attachment. My professor said that you have to use integration by parts, but I keep getting stuck...

Hello, I have a set of data on an x-y plot in Matlab and I'm trying to calculate the Gaussian distribution of best fit, I only want the right hand side of the Gaussian. I tried applying the least squares method but it gets messy. can you help me?

Characteristic energy "units" of primordial fluctuations if gaussian Correct me if wrong, but I think a purely gaussian distribution of the primordial fluctuations could be characterized by a certain unit of energy (which I'll express as mass). If so, then the observed fluctuations are...

\int_0^\infty e^{-x^2}dx \int_0^\infty e^{-y^2}dy = \int_0^\infty \int_0^\infty e^{-(x^2+y^2)} dxdy Under what conditions we could do the same for other functions? I don't get how Poisson (or Euler, or Gauss, whoever that did this for the first time) realized that this is true. It looks...

Which Gaussian surfaces have an electric flux of +q/εo through them? I think is b since in c, +q and -q cancel each other out, d is only -q is that the right approach? thanks!

Hello Colleagues, I am curious about a problem in Quantum Mechanics that incorporates the evolution of a Gaussian Wave Packet under the Gravitational Potential. What I am interested in is equation (3) in the following paper: "On the quantum analogue of Galileo's leaning tower...

Is there any circuit for generating Gaussian Noise with a mean and variance. we should have control to change mean as well as variance accordingly. thanks.

I would normally use Gaussian ELimination to solve a linear system. If we have more unknowns than equations we end up with an infinite number of solutions. Are there any real life applications of these infinite solutions? I can think of solving puzzles like Sudoku but are there others?

Is there a closed form expression for finding all the roots of the derivative of a k-component gaussian mixture model?

What is meant by phase of a pulse? (Gaussian or sech or any other pulse) How can we mathemaitcally express it and what is the physical meaning of it? I know about the phase of a wave but not sure about that of a pulse since it is a combination of waveforms of different frequencies. Thanks

Hi I have the following code: lorentz[A_, Ox_, Oy_, FWHM_, x_] := A (1/3.14) FWHM/((x - Ox)^2 + FWHM^2) + Oy; gauss[A_, Ox_, Oy_, x_, C_] := A Exp[-(x - Ox)^2 C] + Oy; Convolve[lorentz[1, 0, 0, 1, x], gauss[1, 0, 0, x, 1], x, y] It takes extremely long time for this to finish -- is it...

The following problem arises in the context of a paper on population genetics (Kimura 1962, p. 717). I have posted it here because its solution should demand only straightforward applications of tools from analysis and algebra. However, I cannot figure it out. Homework Statement Let z = 4...

I'm trying my very best to understand it, but really, I just couldn't get it. I read four books now, and some 6 pdf files and they don't give me a clear cut answer :( Alright, so this integral; ∫e-x2dx from -∞ to ∞, when converted to polar integral, limits become from 0 to 2∏ for the outer...

λHomework Statement A symmetric confocal resonator with mirror spacing d =16 cm, mirror reflectances 0.995, and n = 1 is used in a laser operating at λ[o] = 1 μm. (a) Find the radii of curvature of the mirrors. (b) Find the waist of the (0,0) (Gaussian) mode. (c) Sketch the intensity...

I put this under statistics because of your knowledge of the Gaussian. I have run into an elementary problem. I was considering what the average value, <x>, is for a Gaussian with an x offset, and got results which don't make sense to me. First, it is obvious that for P1(x)=e^(-(x^2)) the...

Homework Statement $$ \begin{pmatrix} -1&3&0\\ 2&0&-1\\ 0&-6&1 \end{pmatrix} $$ Finding the ImT basis of this The Attempt at a Solution I got it down to $$ \begin{pmatrix} 1&0&-1/2\\ 0&1&1/6\\ 0&0&1 \end{pmatrix} $$ I know that by the principle of having pivots as the only non-zero...

Homework Statement The larger context is that I'm looking at the scenario of fitting a polynomial to points with Gaussian errors using chi squared minimization. The point of this is to describe the likelihood of measuring a given parameter set from the fit. I'm taking N equally spaced x values...

Homework Statement Find the normalization constant N for the Gaussian wave packet \psi (x) = N e^{-(x-x_{0})^{2}/2 K^{2}} Homework Equations 1 = \int |\psi (x)|^{2} dx The Attempt at a Solution 1 = \int |\psi (x)|^{2} dx = N^{2} \int e^{-(x-x_{0})^{2}/K^{2}} dx Substitute y=(x-x_{0})...

Homework Statement Hey guys, I have a small question on a proof in Struik's Differential Geometry book. It concerns the proof that all surfaces with K = 0 (gaussian curvature zero) are developable, i.e. all surfaces with K = 0 are ruled and the tangent planes along the rulings are parallel...

Homework Statement Apply Gaussian elimination to the following matrix 2 -3 0 3 4 -5 1 7 2 -1 -3 5 I understand how to get the answer. The Attempt at a Solution My question is why doesn't the following method work: Switch Z over into the first column, move row 1 to row 3, then exchange rows...

Homework Statement I'm having difficulty solving the following integral. \int_{-\infty}^{\infty} x^{4}e^{-2\alpha x^{2}} \text{d}x Homework Equations \int_{-\infty}^{\infty} e^{-\alpha x^{2}} \text{d}x = \sqrt{\frac{\pi}{\alpha}} \int_{-\infty}^{\infty} x^{2}e^{-\alpha x^{2}}...

Homework Statement Two bars, each 30cm long, and each having a resistance of 2-ohms, are connected to a 1,500 volt battery. The bars are attached to each other with 3 insulating springs, each having a spring constant of 9N/m. The two bars are initially at rest, 4cm apart. The switch is...

Homework Statement A cloud of electrons are drifting from a negative plate to a positive plate after being liberated by a laser pulse, (separated by a distance z = 10cm with an original potential difference of 15V) at an instant in time the centre of the cloud has traveled 25mm from the...

Homework Statement I will not post a specific problem, but rather, I would like to ask a general question. Say I am given a Gaussian wavepacket (function psi(x,t) ) and asked to find the expectation value for x-squared and momentum squared. Now, x-squared is rather straightforward...

Homework Statement Check the energy-time uncertainty principle for: \Psi(x,0)=Ae^{-ax^2}e^{ilx} using the observable x. Homework Equations \Delta{E}\Delta{t}\geq\hbar/2\\ \Delta{E}=\sigma_H=\sqrt{<H^2>-<H>^2}\\ \Delta{t}=\frac{\sigma_x}{\frac{d<x>}{dt}} The Attempt at a...

Hi everyone, I always have trouble on decomposing number into irreducible factors inside Gaussian integer field. I keep trying to express number as product as (a+bi)(c+di), and trying to solve a,b,c,d inside of integers (Z), then see if they are irreducibles, which of course end of very messy...

Homework Statement I am reading about electrodynamics, and in different formula, there is a SI form and a Gaussian form. Homework Equations What is the relationship in the different unit system for c or B or E and other relative variables? The Attempt at a Solution sometimes...

So we were taking measurements for an experiment in our radio astronomy lab. For the first part of the experiment, we recorded the intensity of a far away point source ( the signal from a TV satellite was used for the point source ) detected by commercial satellite dish and receiver. When we...

Homework Statement A grain of pollen shows Brownian motion in a solvent, such that the position x(t) on the x-axis varies with time. The displacement during one second, x(t + 1) - x(t), is measured many times and found to have a Gaussian distribution with an average of 0 and standard devation...

I need to factorise 70 into primes, how do I go about this? So far I have 2,5,7 as primes in Z. So I suppose I need to factorise these in Z[i]? 2 = (1+i)(1-i) How do I go around doing the other two, is it possible that they're primes in Z[i]? Edit: I have a corollary where if p is a prime...

Hey ^^, new here but I already have a question haha Does anyone here know how the coefficient (x-μ)^2 was derived in the following equation: σ^3=(1/√2∏)∫(1/σ)*(x-μ)^2*exp((x-μ)^2)/(2σ^2)) I know the general equation for density probability is (1/σ)*exp((x-μ)^2)/(2σ^2)) but in this case...

Homework Statement Consider two long coaxial metal cylindrical tubes, with radii a and b and length L. (You may assume a,b<<L. Also a<b.) Suppose the inner cylinder is given a charge +Q and the outer cylinder a charge -Q. Using Gauss' Law, compute the electric field for all r between a and...

Homework Statement Hi, the question is from a piece of coursework and before hand we were asked to find the Fourier transform G(K) of the function g(x)= e^(-∏(x^2)) (where g(x)= ∫ G(K)e^2∏ikx dx (integral from -∞ to ∞)). We were told to find G(K) by forming a differential equation in H(K)...

Hi folks, I'm trying to get from the established relation: $$ \int_{-\infty}^{\infty} dx.x^2.e^{-\frac{1}{2}ax^2} = a^{-2}\int_{-\infty}^{\infty} dx.e^{-\frac{1}{2}ax^2} $$ to the similarly derived: $$ \int_{-\infty}^{\infty} dx.x^4.e^{-\frac{1}{2}ax^2} = 3a^{-4} \int_{-\infty}^{\infty}...