What is Approximation: Definition and 761 Discussions

I'm reading a paper (Beamwidth and directivity of large scanning arrays, R. S. Elliott, Appendix A) in which the author starts from this expression: ##\frac{\sin\left [ \left ( 2N+1 \right ) u_0\right ]}{\sin(u_0)}\sum_{p=-P}^Pa_p\cos(p\pi)\left [\frac{\sin(u_0)}{\sin(u_p)} -1+1 \right ]##...

Hi, I am wondering if it is possible to demonstrate that: tends to in the limit of both x and y going to infinity. In this case, it is needed to introduce a measure of the error of the approximation, as the integral of the difference between the two functions? Can this be viewed as a norm...

Hi everyone, The four fundamental forces are gravity (I understand that G.R. does not look upon gravity as a force but I'm not worried about that here), the Lorentz force, the weak force, and the strong force. I'm familiar with the inverse square law for gravitation and the Lorentz force...

I'm trying to determine why $$ \lim_{N \rightarrow +\infty} ln( \frac {N!} {(N-n)! N^n}) = 0$$ N and n are both positive integers, and n is smaller than N. I want to use Stirling's, which becomes exact as N->inf: $$ ln(N!) \approx Nln(N)-N $$ And take it term by term: $$ \lim_{N...

Imagine you create a diffuse interface in space and determine which side of the interface you are on by a local scalar value that can be between 0 and 1. We could create a circle, centered in a rectangular ynum-by-xnum grid, with such a diffuse interface with the following MATLAB code: xnum =...

I am interested, in the context of my work, in the cross correlations between a spectroscopic probe (which gives a 3D distribution of galaxies with redshifts, which is also called spectroscopic Galaxy clustering, GCsp) and a photometric probe (which gives an angular distribution, that is to say...

Currently working on a project involving Monte Carlo integrals. I haven't had any prior studies of this method, so hence the following question. Consider the following expectation: $$E[f(X)]=\int_A f(x)g(x)dx.$$ Let ##X## be a random variable taking values in ##A\subseteq\mathbb{R}^n##. Let...

From This picture, I think the fish will be smaller but the problem is how small will it be? (Fish "L" is the image of fish "K") Let ##H## be the depth of fish "K", ##\theta## be the angle of eyes to y-axis and ##n## is the index of refraction of water.

Have I solved this linear approximation question correctly?

In a manner analogues to the linearization of functions of a single variable to approximate the value of a function of two variables in the neighbourhood of a given point (x0,y0,z0) where z=f(x,y) using a tangent plane. The tangent plane must pass through the point we wish to approximate z...

My understanding of barrier potential is as follows : Due to the holes and electrons combining in the diode a depletion region is formed. Essentially this region has no mobile charge carriers (as the holes and electrons have combined) and thus the ions (whose holes/ electrons have combined)...

I wanted to know what is understood as the dispersive approximation (or limit) in the context of the Jaynes-Cummings model for one mode of the field.

I had thought it would be failure of structural stability since in structural stability qualitative behavior of the trajectories is unaffected by small perturbations, and here, even tiny deviations using ##h## values resulted in huge effects. However, apparently that's not the case, and I'm not...

The speed of sound in a gas at temperature T is given to be ## v=\sqrt{\frac{\gamma RT}{M}}##, where ##\gamma## is the adiabatic exponent, R is the gas constant and M is the molar mass of the gas. In deriving this expression, we assumed that the compression and expansion processes were so fast...

There is an arbitrarily complicated function F(x,y,z). I want to find a simpler surface function G(x,y,z) which approximates F(x,y,z) within a region close to the point (x0,y0,z0). Can I write a second-order accurate equation for G if I know F(x0,y0,z0) and can compute the derivatives at the...

I get $$B_2(T)=2\pi N\int_{0}^{\infty} (1-e^{-\beta E_0((\frac{r_0}{r})^{12}-2(\frac{r_0}{r})^6)})r^2dr$$ as the coefficient. I was just unsure how to evaluate it numerically from here. Any suggestions would be appreciated. Thank you.

a) since np has to be greater than 5, n*p= 50*.5 =25 so yes, we can use this since it is much larger than 5. now, for mean, i believe the equation is saying that the mean is np, which is 25 but in this equation we do not have a q value, so this is where my issue begins... what should i use...

I found the linearization, L(x) = -0.0001x+0.2 and I found L(1/99) = 0.0199989899. Then I tried to put that value into my percentage error formula along with 1/99 and got: the absolute value of (1/99)-L(1/99) and then we divide that by our actual value which is 1/99, then I multiply everything...

I know that it is only an approximation to get an idea, but at times it works quite well (in class we solved the kinetics equation for a PWR reactor (point-reactor model) with MATLAB and then we plotted the solution along with the prompt jump approximation... It was very good). But I did not...

In order to use WKB approximation, the potential has to be "slowly varying". I learned the method from this video: But the Professor hasn't mentioned in detail what the measure of "slowly varying" is. What is the limit beyond which we cannot use the WKB method accurately?

hi, when we try to find the speed of a wave on a rope v = (F/u)^1/2, we use the fact that if the angles are small then sin x = x. I understand the approximation but not WHY we use the approximation. We say delta(Theta) is small (and then amplitude is small) then ... . So the proof is only...

Background of problem comes from Drude model of a metal (not necessary to answer my problem but for the curious): Consider a uniform, time-dependent electric field acting on a metal. It can be shown that the conductivity is $$\sigma = \frac{\sigma_0}{1-i\omega t}$$ where $$\sigma_0 =...

Hi everyone I hope you are well. Maybe as you know according to Behners-Fisher problem (unequal variance case of samples) there are some kind of approximations. I have recently covered the Satterthweiths Approximations and comprehended the logic of it. But I got stuck with the Cochran-Cox...

According to the Born-Oppenheimer approximation, what does the internuclear distance Req depend on? Atomic number Z? Rotational Energy of the nuclei? Electrons' kinetic energy? Coulomb interaction between the two nuclei? Coulomb interaction between the electrons? Vibrational energy of the...

Hi all, I'm having a problem understanding a step in an arxiv paper (https://arxiv.org/pdf/0808.3566.pdf) and would like a bit of help. In equation (29) the authors have $$R = \frac{\sigma}{\sqrt{\pi}} \int dk \ e^{-(k - k_0)^2 \sigma^2} \ \Big( \frac{ k - \kappa}{ k+ \kappa} \Big)^2$$ where...

Given the following vectors: how can i determine that Θ = Δp/p ? I can understand that p + Δp = p' but nothing arrives from this. Any help is welcome!

Hey! :o I want to determine an approximation of a cubic polynomial that has at the points $$x_0=-2, \ x_1=-1, \ x_2=0 , \ x_3=3, \ x_4=3.5$$ the values $$y_0=-33, \ y_1=-20, \ y_2=-20.1, \ y_3=-4.3 , \ y_4=32.5$$ using the least squares method. So we are looking for a cubic polynomial $p(x)$...

Problem: Assume $E$ is a measurable set, $1 \leq p < \infty$, and $f_n \rightarrow f$ in $L^p(E)$. Show that there is a subsequence $(f_{n_k})$ and a function $g \in L^p(E)$ for which $\left| f_{n_k} \right| \leq g$ a.e. on $E$ for all $k$. Proof: Maybe use?: $f_n \rightarrow f$ in $L^p(E)$...

Homework Statement The problem is taken from Morin's book on classical mechanics. I found out Lagrangian of motion. Now to solve, we need small angle and small x approximation. The small angle approximation is easy to treat. But how to solve small x approximation i.e how do I apply it...

i want to know about stirling approximation. why lnx! = xlnx - x

Homework Statement Approximate each integral using the trapezoidal rule using the given number for ##n##. ##\int_1^2 \frac{1}{x}dx## where ##n=4## Homework Equations Trapezoidal Approximation "Rule": Let ##[a,b]## be divided into ##n## subintervals, each of length ##Δx##, with endpoints at...

Because it holds that ##\displaystyle\int_{1}^{x}\frac{dt}{t} = \log x##, and ##\displaystyle\int_{1}^{x}\frac{dt}{t^a} = \frac{1}{a-1}\left(1-\frac{1}{x^{a-1}}\right)\hspace{20pt}##when ##a>1## it could be expected that ##\displaystyle\frac{1}{a-1}\left(1-\frac{1}{x^{a-1}}\right)...

The famous Stirling’s approximation is ##N! \approx \sqrt{2\pi N}(N/e)^N## which becomes more accurate for larger N. (Although it’s surprisingly accurate for small values!) I have found a nice derivation of the formula, but there is one detail which bothers me. The derivation can be found...

Hi all, In S. Weinberg's book "Cosmology", there is a derivation of the slightly modified temperature of the cosmic microwave background as seen from the Earth moving w.r.t. a frame at rest in the CMB. On Page 131 (1st printing), an approximation (Formula 2.4.7) is given in terms of Legendre...

Hello, I am trying desperately to find the solution indicated in this question : If I compute the equations on the 3 axis, I can't get the flow to be directed along ##\vec{e_y}##. I have only : ##\dfrac{\partial v_{z}}{\partial t} = -\dfrac{1}{\rho_0}\dfrac{\partial \delta P}{\partial...

Homework Statement Consider the standard square well potential $$V(x) = \begin{cases} -V_0 & |x| \leq a \\ 0 & |x| > a \end{cases} $$ With ##V_0 > 0##, and the wavefunctions for an even state $$\psi(x) = \begin{cases} \frac{1}{\sqrt{a}}cos(kx) & |x| \leq a \\...

Hello I have tried to resolve the problem below Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.040000 cm thick to a hemispherical dome with a diameter of 45.000 meters. My procedure was: the volume of the sphere is V=4/3 pi r^3...

How is the "geometric optics approximation" exactly defined? Given all the source of visible radiation's parameters, all the apparatus, instruments, screen, etc, specifications, how can I know if, e. g. there will be diffraction, interference or other wave properties or if I'll be able to...

I am very new too Matlab and how it all works but I am having trouble understanding at what axis the numerical integration is occurring from on the graph that I plotted. So I am currently doing an experiment in gamma ray spectroscopy and due to issue with the software we found it hard to...

An artificial lake is made up of 5m width and 100m length in dimension. The depth of the lake varies every 20m length as recorded in the following table. Use Simpson's rule approximation to estimate the volume of the water in the lake. Distance (m) 0 20 40 60 80 100 120 Depth (m) 2.0...

What is the preferred method of measuring how accurate the normal approximation to the binomial distribution is? I know that the rule of thumb is that the expected number of successes and failures should both be >5 for the approximation to be adequate. But what is a useful definition of...

Hi I'm reading through a Quantum Mechanics textbook called Quantum Mechanics by Book by Alastair I. M. Rae and in the opening chapter it talks about the Heisenberg uncertainty principle and talks about how a measurement of position of a particle causes an uncertainty from the momentum due to the...

I came across the following working in my notes and would like some help understanding how the step was done. Many thanks in advance! The following is the working, and we assume that ##\beta## is small $$\frac{1}{1+ \beta \hbar \omega /2 + (7/12)(\beta \hbar \omega)^2 +...} \approx 1 - (\beta...

Rehashing this topic because I believe a clear misconception is stated in many threads. Classical mechanics is an incorrect ( by the definition of correct ) theory which is only an approximation that uses incorrect assumptions ie. Constant time but yet makes accurate predictions in its regime...

Homework Statement Homework Equations [/B]The Attempt at a Solution Can somebody explain to me how did we find the function in red? Thanks

So I thought I understood something well, and then I went to explain it to someone and it turns out I'm missing something, and I'd appreciate any insight you might have. If I think about Bloch's theorem, it states that ψk(r)=eik⋅ruk(r) where uk has the periodicity of the lattice. If u is...

Hello! (Wave) I want to prove that each continuous function $f$ in a closed and bounded interval $[a,b]$ can be approximated uniformly with polynomials, as good as we want, i.e. for a given positive $\epsilon$, there is a polynomial $p$ such that $$\max_{a \leq x \leq b} |f(x)-p(x)|<...

Homework Statement A plane z=0 is charged with density, changing periodically according to the law: σ = σ° sin(αx) sin (βy) where, σ°, α and β are constants. We have to find the potential of this system of charges. Homework EquationsThe Attempt at a Solution [/B] I...

Hi there. I have a question about the damped pendulum. I am working on an exercise where I have already numerically approximated the solution for a simple pendulum without dampening. Now, the excercise says that I can simply change the code of this simple situation to describe a pendulum with...

I'm preparing for an exam and I expect this or a similar question to be on it, but I'm running into problems with using the Born approximation and optical theorem for scattering off of a finite well. 1. Homework Statement Calculate the cross sectional area σ for low energy scattering off of a...