Gaussian Definition and 706 Threads

I thought that maybe it would be a good idea to do gaußian error propagation of the formular of the mean, this should give me the uncertainty of the average i calculate from the sample i have... And additionally consider the standard deviation Can someone maybe give me a detailed way to handle...

Hi! Please, could you help me on how to solve the following matrix ? I need to replace the value 3 on the third line by 0, the first column need to remain zero and 1 for the third column. I'm having a lot of difficulties with this. How would you proceed ? Thank you for your time and help...

I know a laser is generally treated as a coherent state (i.e. a plane wave). However, if a laser turns on and back off quickly, can the resulting light be treated as a gaussian multiplied by the plane wave that would have been emitted if the laser were continually active?

All resources I’ve found for grating resolving power assume uniform distribution on the grating and produce airy disks. Resolvance is determined by the Rayleigh criterion where the peak of one wavelength is at the minima of the adjacent one. This definition doesn’t seem applicable for Gaussian...

Let ##a, b##, and ##c## be real numbers such that ##a## and ##c## are positive and ##ac > b^2##. Evaluate the double integral $$\int_{-\infty}^\infty \int_{-\infty}^\infty e^{-ax^2 - 2bxy - cy^2}\, dx\, dy$$

Hi! Say i have two variables that have independent gaussian distributions of probability of being a certain value when i sample them, what is the likely hood that both will land on a 3 sigma value simultaneously? Is there an equation that easily determines that? Also for other combinations like...

Using this error propagation formula: I expressed the standard deviation (s) and the partial derivatives of s w.r.t. each data point as: This gives me an uncertainty of: , where m is the mean. Does this seem reasonable for the uncertainty of the standard deviation? I also found the thread...

Homework Statement:: difference between DFT and TDDFT in Gaussian 09 software? Relevant Equations:: hello , I hope this message finds you well Please I want to know what is the difference between DFT and time‐dependent density functional theory TD-DFT and why we use it in the measurement of...

I've been trying to find a way to calculate Gaussian curvature from a 4D metric tensor. I found a program that does this in Mathematica using the Brioschi formula. However, this only seems to work for a 2D metric or formula (I would need to use something with more dimensions). I've found...

Does anyone know a C# class that can return a value (0 - 100 percentage) of How close a perfect gaussian curve an 2D Matrix is? for example, these would all return a 100%:

I am confused why we don't take into account the lids of the cylinder since the Gaussian cylinder is of finite height L as shown in the image

I found this identity: ##x\int e^{-x^2} dx - \int \int e^{-x^2} dx dx = e^{-x^2}/2## by solving the integral of ##x*e^{-x^2}## and then finding its integration-by-parts equivalent. Is this identity useful at all?

A few questions about doing a Gaussian Fit : 1) Is gaussian fit and gaussian regression the same thing ? 2) I have a gaussian function that will return a list of gaussian numbers giving an initial list length. So if you input 5 you will get: 1,2,6,4,1. My question is if I have an image and I...

If I have a point charge q right outside of a gaussian surface, it makes sense that the flux is zero inside the surface because the electric field going in equals the electric field going out. However, how would the electric field be zero inside? Wouldn't it just take on the electric field of...

Hello! If I have N points (x,y) which I know they are described by a Gaussian i.e. y(x) is a Gaussian of unknown mean and standard deviation, and each y has an associated error of ##\sqrt{y}##, is there a general formula for the uncertainty on the mean of this function? Thank you!

Hi there! I have a few related questions on Gaussian curvature (K) of surfaces and simply connected regions: Suppose that K approaches infinity in the neighborhood of a point (x1,x2) . Is there any relationship between the diverging points of K and (non) simply connected regions? If K diverges...

Hello, there. Suppose a Gaussian beam is sent and is received at a great large distance, i.e., the propagation distance ##z \gg z_R## the Rayleigh distance. The Gaussian beam can be described by $$E_0 \frac {1}{w(z)} \exp \left ( \frac {-r^2}{w(z)^2}\right )\exp\left ( -i\left (kz+k\frac...

why does it say transforms? is there more than one Fourier transform?? we learned in class that the inverse Fourier transform of the Fourier transform of ##f## is ##f##, so there should be just one right? I'm uncertain of how to calulate this integral though.. Mr Wolfram showed me an indefinite...

David C. Bailey. "Not Normal: the uncertainties of scientific measurements." Royal Society Open 4(1) Science 160600 (2017). How bad are the tails? According to Bailey in an interview, "The chance of large differences does not fall off exponentially as you'd expect in a normal bell curve," and...

\begin{tikzpicture}[scale=0.6] %preamble \usepackage{pgfplots} \newcommand\gauss[2]{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))} % Gauss function, parameters mu and sigma \begin{axis}[every axis plot post/.append style={ mark=none,samples=50,smooth}, % All plots: 50 samples, smooth, no marks...

ok I was able to get the graph of P(z>1.28) \begin{tikzpicture} %preamble \usepackage{pgfplots} \newcommand\gauss[2]{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))} % Gauss function, parameters mu and sigma \begin{axis}[every axis plot post/.append style={ mark=none,samples=50,smooth}, % All...

ok I don't think MHB will process a newcommand but I don't know how to put this in the after \begin{tikzpicture} line the problem with posting pic here is eventually they get remove and OP is useless... this tikz code renders in overleaf but I also have many newcommands in preamble

Hello everyone. I have been recently working in an optimization model in the presence of uncertainty. I have read https://www.researchgate.net/publication/310742108_Efficient_Simulation_of_Stationary_Multivariate_Gaussian_Random_Fields_with_Given_Cross-Covariance in which, a methodology for...

I'd love have a little discussion about the Interior Schwarzschild Solution. Here's a diagram I slapped together to illustrate the key points. (I assume everyone reading this familiar with embedding diagrams, and using an axis to 'project' a value, in this case the spatial z-axis is replaced by...

We know that Dirac Delta is not a function. However, I just talk about the numerical version of it that we use every day. We can simply represent the Dirac delta function as a limiting case of Gaussian distribution when the width of the distribution ##\sigma->0##. $$ \delta(x - \mu) =...

Let us suppose we are given a Gaussian Distribution in the form of $$p(x,y) \propto exp(-\frac{1}{2}x^2 - \frac{1}{2}by^2 - cxy)$$ What are the equations that I need to use to obtain Mutual Information ?

I'd like to plot the normalized convolution of a Gaussian with a Lorentzian (see the definitions in terms of full width half maximum (fwhm) in the attached image). Here is my attempt, but the print statements with np.trapz() do not return 1 in both cases, but rather ##\approx##0.2. I'd also like...

a) First off, I computed the integral \begin{align*} Z(\lambda) &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} dx \exp\left( -\frac{x^2}{2!}-\frac{\lambda}{4!}x^4\right) \\ &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} dx \exp\left( -\frac{x^2}{2!}\right) \exp\left(...

For normalizing this wave function, I began by finding the complex conjugate of psi and then multiplied it with the original psi. Now what I am getting is A^2 integral exp(2cx^2-4ax) dx = 1 Now I am not getting how to solve this exponential term. I tried by completing the square method but it is...

I was reading the following webpage on Gaussian kernel but couldn't understand few details: https://www.imageeprocessing.com/2014/04/gaussian-filter-without-using-matlab.html . Would really appreciate it if you could guide me. Thanks in advance! Here you can find the high-res screenshot of the...

Hello, I've read that repeated convolution tends, under certain conditions, to Gaussian distribution. I found this description helpful, and Wikipedia's version of this says: The central limit theorem states that if x is in L1 and L2 with mean zero and variance ##σ^2##, then...

Hi, I have recently learned the technique of integration using differentiation under the integral sign, which Feynman mentioned in his “Surely You’re Joking, Mr. Feynman”. So, I decided to try it on the Gaussian Integral (I do know the standard method of computing it by squaring it and changing...

I have a Gaussian function of the form: def f(x,y): a=some number b=... c=... return 3*np.exp(-a*(-0.5 + x)**2+b*(x-0.5)*(y-0.5)-c*(-0.5 + y)**2)This is a Gaussian function symmetric around y=x, and I'd like to rotate it 45 degrees (counter)clockwise. Wikipedia gives an overdetermined...

Hello, I have to find an expression for the total movement of a bacteria ##s##, knowing that the bacteria is placed (centered) on a two side ruler at position ##x=0## (so a negative ##x## value means the bacteria has moved to the left of the ruler) and that the probability it moves to ##x## is...

Hey! 😊 If we want to calculate the nodes $x_1, x_2$ and the weight functions $w_1, w_2$ for the Gaussian quadrature of the integral $$\int_{-1}^1f(x)\, dx\approx \sum_{j=1}^2w_jf(x_j)$$ is there a criteria that we have to consider at chosing the weight functions? I mean if we use e.g...

hi guys i am trying to solve the Gaussian integral using the power series , and i am suck at some point : the idea was to use the following series : $$\lim_{x→∞}\sum_{n=0}^∞ \frac{(-1)^{n}}{2n+1}\;x^{2n+1} = \frac{\pi}{2}$$ to evaluate the Gaussian integral as its series some how slimier ...

Could you help me about the derivation of inverse gaussian distribution? During my search I encountered that it was derived by schrödinger as a result of differential equation solution but I can not find his derivation on internet...

Here is a tough integral that I'm not quite sure how to do. It's the Gaussian average: $$ I = \int_{-\infty}^{\infty}dx\, \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}\sqrt{1+a^2 \sinh^2(b x)} $$ for ##0 < a < 1## and ##b > 0##. Obviously the integral can be done for ##a = 0## (or ##b=0##) and for...

The equation above (from Wikipedia), assumed that the Gaussain beam has polarization in x-direction, as I know that the polarization means that the oscillation direction of the electric field and so the intenisty... so how we get circular intensity in every direction which means in x and y...

I'm trying to solve the inequality: $$ \int \limits_0^1 e^{-x^2} \leq \int \limits_1^2 e^{x^2} dx $$I know that $\int \limits_0^1 e^{-x^2} \leq 1$, but don't see how to take it from there. Any ideas?

The Schrödinger equation I need to prove is this one And the Gaussian wavepacket is found here Thanks for your advice. JorgeM <Moderator's note: upload images to PhysicsForums. Do not use external image servers.>

I tried plugging Psi into the right of the Schrodinger equation but can't get anything close to the solution or anything that is usable. How should I solve this?

I am looking for the expectation of a fraction of Gauss hypergeometric functions. $$E_X\left[\frac{{}_2F_1\left(\begin{matrix}x+a+1\\x+a+1\end{matrix},a+1,c\right)}{{}_2F_1\left(\begin{matrix}x+a\\x+a\end{matrix},a,c\right)}\right]=?$$ Are there any identities that could be used to simplify or...

Hello I am reading some introductory laser cavity stuff and I am a bit confused about the existence of gaussian beams in the Fabry-Perot interferometer. If you solve the stability condition for a cavity (i.e. asking for the q parameter to reproduce itself after one round trip) you get that in...

## \int_{-1}^{1} \int_{-1}^{1} e^{-(x^2 + y^2)} cos(2π (x^2 + y^2)\,dx\,dy ## ## I = \int_{-1}^{1} \int_{-1}^{1}f(x,y) \,dx\,dy \approx \sum_{i=0}^{n}\sum_{j=0}^{n} w_i w_j f(x_i, y_j) ## ## = w_0 w_0 f(x_0, y_0) + w_0 w_1 f(x_0, y_1) + w_1 w_0 f(x_1, y_0) + w_1 w_1 f(x_1, y_1) ## ## w_0 =...

This problem arose in modeling camera focusing movement, such as a control system might do. It assumes a simple (thin) lens, rays close to the optical axis, and monochromatic light. While most camera lenses are not simple, this is a first approximation. Camera lenses project an image of a...

Hi, I ask for a clarification about the following: consider for instance a 10 x 12 homogeneous linear system and perform Gauss elimination for the first 8 unknowns. Suppose you end up with 5 equations in the remaining 12-8 = 4 unknowns (because in the process of the first 8 unknowns elimination...

Consider a gaussian wave packet whose wave function at a particular instant of time is Its time dependence is implicit in the "constants" A, a, <x> and <p>, which may all be functions of time. But regardless of what functions of time they may be, these constants will take on some values at...

Hi, I've the following doubt: consider an homogeneous linear system ##Ax=0## with ##A## a singular square matrix. The resulting matrix attained through Gaussian elimination will be in upper triangular or raw echelon form ? Thanks.