Approximation Definition and 705 Threads

Say that we have a continuous, differentiable function f(x) and we have found the best approximation (in the sense of the infinity norm) of f from some set of functions forming a finite dimensional vector space (say, polynomials of degree less than n or trigonometric polynomials of degree less...

Homework Statement Prove: Hint: Group the terms in the error as ##(a_{n+1}+a_{n+2})+(a_{n+3}+a_{n+4})+\cdots## to show that the error has the same sign as ##a_{n+1}##. Then group them as ##a_{n+1}+(a_{n+2}+a_{n+3})+(a_{n+4}+a_{n+5})+\cdots## to show that the error has magnitude less than...

Hello, I just recently found out that one could find the Friedman's equation in Newton's approximation (without GR) by assuming that the universe in homogeneous and isotropic simply by using F=ma and the conservation of energy. On can then find that the scale factor goes as t^2/3, as expected...

Homework Statement A monochromatic light source is used with a double slit to create an interference pattern on a screen that is 2.00 meters away. If the 2nd bright spot is observed 8.73 mm above the central maximum, can the small angle approximation be used? Show and/or explain your reasoning...

What is the physical interpretation of the Weisskopf-Wigner approximation, when it is applied in the neutral kaon system? I would say that the approximation means that a decay state has a small probability to suffer a transition (or be "transformed") into another decay state through weak...

Regarding interacting green's function, I found two different description: 1. usually in QFT: <\Omega|T\{ABC\}|\Omega>=\lim\limits_{T \to \infty(1-i\epsilon)}\frac{<0|T\{A_IB_I U(-T,T)\}|0>}{<0|T\{U(-T,T)\}|0>} 2. usually in quantum many body systems...

Hello! (Wave) Approximating $y'(t^n)$ at the relation $y'(t^n)=f(t^n,y(t^n))$ with the difference quotient $\left[\frac{y(t^{n+1})-y(t^n)}{h} \right]$ we get to the Euler method. Approximating the same derivative with the quotient $\left[\frac{y(t^{n})-y(t^{n-1})}{h} \right]$ we get to the...

Hello All, Light travels at 1,86,000 m/sec. i.e.2,99,338 km. The distance from Earth to Moon is 3,84,400 km. Is there any constellation or any physical object (to have an idea) which is not near to moon but falls within the range of 2,99,000 km? Just for curiosity. Thanks

Homework Statement The question is : Calculate the work done in joules when 1.0 mole of N2H4 decomposed against a pressure of 1.0 atm at 300 K for the equation: 3N2H4 (l)→4NH3(g)+2N2(g) Homework Equations None The Attempt at a Solution I did it like as: Assuming 100% dissociation of N2H4 4/3...

I am trying to determine the optimum length of the control vectors for a cubic Bézier approximation of a $90^\circ$ circular arc. However to my limited mathematical mind the equation cannot be solved algebraically. Wolfram Alpha even finds it too complicated. However I am told that it IS...

Homework Statement To rephrase the question, given a power series representation for a function, like ex , and its MacLaurin Series, when I expand the two there's no difference between the two, but my question is: Is this true for all functions? Or does the Radius of Convergence have to do with...

What is the physical meaning of this approximation? Qualitatively I'm finding it hard to know what differentiates it from other approximation methods...

http://www.mth.uct.ac.za/omei/gr/chap7/node3.html Shouldn't eq 45 have a minus sign, looking at eq 29. Although I'm confused because the positive sign makes sense when comparing with the Newton-Poisson equation. I can't see a sign error in eq 29. (I believe the metric signature here is...

The weak field approximation in the Newtonian limit shows that the coordinate acceleration along a geodesic is related to the gravitational force. The geodesic deviation equation relates the coordinate acceleration between adjacent geodesics to tidal forces. If I drop 2 balls together from the...

Hi all, I'd like to ask about " the vacuum saturation approximation " in a calculation as the decay width of ## B \to l \nu ## [hep-ph/0306037v2] equ. 1 till equ. 4, that why ## < 0 | \bar{u} \gamma_\mu b | B > ~ and ~ < 0 | \bar{u} b | B > = 0 ## ? and for ## [ \bar{l} \gamma^\mu...

Homework Statement ##\frac{\partial u}{\partial \phi}=-\frac{Me}{J^{2}}sin\phi + \frac{M^{3}}{J^{4}}(-\frac{1}{e}sin\phi+e^{2}sin2\phi+3e\phi cos \phi )##. Assume this to be ##0## at ##\phi=\pi+\epsilon##. Find ##\epsilon##? Homework Equations The only method I can think of is to expand out...

I have a second order differential equation of the form (theta is a function of time): \theta ''=F\left(\theta ,\theta '\right) Turning them to two first order equations I get: \begin{cases} \theta '\:=\omega \\ \omega '=F\left(\theta ,\omega \right) \end{cases} And here's the algorithm...

Hey guys, So I have a question about the gauge invariance of the weak field approximation. So if I write the approximation as \Box h^{\mu\nu} -\partial_{\alpha}(\partial^{\mu}h^{\nu\alpha}+\partial^{\nu}h^{\mu\alpha})+\partial^{\mu}\partial^{\nu}h=0 then this is invariant under the gauge...

Hey! So we're deriving something in Daniel Schroeder's Introduction to Thermal Physics and it starts with this: \Omega \left( N,q\right) =\dfrac {\left( N-1+q\right) !} {q!\left( N-1\right) !} Both N and q are large numbers and q >> N. The derivation is in the book, but I am always...

Homework Statement I have a pretty simple question. I was going over an older exam when I encountered something that did not quite make sense to me. If \frac{(2s+5)(-s+0.5)}{(s+3)(s^2+0.1s+0.01)}, find a low order approximation for the system. I understand that the pole at s=-3 can be...

Hi, I am working on TDR (Time Domain Reflectometry). I send a 7GHz bandwidth fast rising edge (14ns) square wave into a coax. I get a return Signal. I have an ADC with 10Msamples/sec. I am using MPLAB IDE for coding the microcontroller. Now I would like to increase the Points on the...

Hi I am new member and I am new to the Signal processing so I hope I could get some help from the members to able to understand the concepts. I have a Signal. I have a 10Msaples/sec ADC. I view the Signal on an Oscilloscope which has 20Gsamples/sec sampling rate. The Point where I am...

Homework Statement I'm trying to understand an approximation Griffiths does (in his solutions' manual - exercise 9.18-b) and I'm not quite getting it. Let $$k = \omega \sqrt{\dfrac{\epsilon \mu}{2}} [\sqrt{ 1 + (\dfrac{\sigma}{\epsilon \omega}})^2-1]^{1/2}$$ He says that, because ##\sigma >>...

So, I ve been trying to add orientation to my model of the flight dynamics of a rocket but I ve been running into a lot of problems. I didn't bother actually doing the math for the moments of inertia and everything because I guess it really doesn't have that much of an effect on the general...

Homework Statement please check attachment Homework Equations please check attachment The Attempt at a Solution since the deviation from f_0 to f is linear then we can write f=f_0 + C where C is some constant this should be enough to prove the first question (i think so) for the second...

< Mentor Note -- The OP's question was in the thread title. They have been advised to please make more detailed and clear thread starters > thanks

if , then what will be . In fact I was solving the integral equation by the method of successive approximation.

Homework Statement With unity position feedbck, i.e. make K2=0, plot root locus as a function of pitch gain (K1). By imposing 2nd order system approximation, estimate settling time, rise time, peak time of the closed-loop system with 20% overshoot. Pic of system...

how does this aproximation reduse the integral

How is the Born-Oppenheimer approximation used in Molecular Dynamics simulations? And in which situations does it break down?

Homework Statement Find the limit of: ##\frac { \Gamma (n+\frac { 3 }{ 2 } ) }{ \sqrt { n } \Gamma (n+1) } ## as ##n\rightarrow \infty ##. Homework Equations ##\Gamma (p+1)=p^{ p }e^{ -p }\sqrt { 2\pi p } ## The Attempt at a Solution Mathematica and wolfram Alpha gave the limit as 1. My...

Dear friends, let us use the definition of Lebesgue integral on ##X,\mu(X)<\infty## as the limit ##\int_X fd\mu:=\lim_{n\to\infty}\int_Xf_nd\mu=\lim_{n\to\infty}\sum_{k=1}^\infty y_{n,k}\mu(A_{n,k})## where ##\{f_n\}## is a sequence of simple, i.e. taking countably many values ##y_{n,k}## for...

Homework Statement The stirling approximation, J! = √JJ+1/2e-J, is very handy when dealing with numbers larger than about 100. consider the following ratio: the number of ways N particles can be evenly divided between two halves of a room to the number of ways they can be divided with 60% on...

Chebyshev polynomials and Legendre polynomials are both orthogonal polynomials for determining the least square approximation of a function. Aren't they supposed to give the same result for a given function? I tried mathematica but the I didn't get the same answer :( Is this precision problem or...

In the Drude model of the free electron gas to explain the conduction of a metal, the relaxation time approximation that the electron has a collision in an infinitesimal time interval ##dt##is ##dt/\tau##. It can be shown that the mean time between collisions is ##tau##. If we choose an...

Homework Statement w!/(w-n)! = number of ways of distributing n* distinguishable particles in w distinguishable states w = number of distinguishable states n = number of indistinguishable particles How many ways are there to put 2 particles in 100 boxes, with no particles sharing a box...

Homework Statement Hello guys, I fail to understand a mathematical approximation I see in a solved exercise. The guy reached a partition function of ##Z=\sum_{l=0}^\infty (2l+1) \exp \left [ -l(l+1) \frac{\omicron}{T} \right ]## and he wants to analyze the case ##T>> \omicron##. He states...

First, I already know that when we have to do linear approximation of ##f(x, y)## if ##\Delta z = f_{x}(a, b)\Delta x + f_{y}(a, b)\Delta y + \epsilon_{1}\Delta x + \epsilon_{2}\Delta y ##. and ##\epsilon_{1}## and ##\epsilon_{2}## approaches to nought wneh ##(\Delta x, \Delta y)## approaches...

I have seriously stocked in the subject below. According to Ashcrift & Mermin (chapter 13): If the electrons about r have equilibrium distribution appropriate to local temperature T(r), g_n (r,k,t)=g_n^o (r,k)=\frac {1}{ exp^{(\epsilon_n (k) -\mu (r))/kT} +1} (formula 13.2) then...

Use the binomial approximation to derive the following: A) γ=1+.5(β^2) B)1/γ=1-.5(β^2) C)1-(1/γ)=.5(^2) I know the approximation is 1+(.5β^2)+(3/8)β^4+... A) is self explanatory but not sure how to derive B) and C)

Hello, I am having a hard time getting my errors to come out to what the book says the answers should be. My approximations are correct, so I think I'm just misunderstanding how to find K. Q.a) Find the approximations T8 and M8 for ∫(0 to 1) cos(x2)dx I found these to be T8=0.902333 and...

Hey guys, I have just a few more questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. I'm only asking about 1ab, ignore 2abc please: So for the first one, I used cos(30) as the estimated value to approximate L(28). Then I...

If: ##\hat{H} \psi (x) = E \psi (x)## where E is the eigenvalue of the *disturbed* eigenfunction ##\psi (x)## and ##E_n## are the eigenvalues of the *undisturbed* Hamiltonian ##\hat{H_0}## and the *disturbed* Hamiltonian is of the form: ##\hat{H} = \hat{H_0} +{\epsilon} \hat{V}...

hi everyone , i don't understand these steps for Taylor Expansion , it has used for state space equations the equations are the approximations for sin and cos the equation for Taylor series is ( i don't understand at all ) please help me if you can

I'm stucked in a passage of Particle Physics (Martin B., Shaw G.) in page 41 regarding neutrino oscillations. Having defined E_i and E_j as the energies of the eigenstates \nu_i and \nu_j, we have: E_i - E_j = \sqrt{m^2_i - p^2} - \sqrt{m^2_j - p^2} \approx \frac{m^2_i - m^2_j}{2p} It...

I'm trying to understand why the WKB approximation doesn't seem to work in the following case. Suppose you have a particle of mass ##m## in a potential ##V(x)=q m\cos(2mx/\hbar)##, where ##q\ll 1##. Consider then the stationary solution with energy ##E=m/2##. The Schroedinger equation is then...

In a paper that I'm reading, the authors write:- N_e \approx \frac{3}{4} (e^{-y}+y)-1.04 ------------ (4.31) Now, an analytic approximation can be obtained by using the expansion with respect to the inverse number of "e-foldings" (N_e is the number of "e-foldings"). For instance, eq...

Homework Statement A dipole is placed next to a sphere (see image), at a large distance what is E proportional to? 3. The Attempt at a Solution or lack thereof I'm having trouble figuring out what's happening in any variations of these. How does the dipole affect the sphere's charge...

Homework Statement Find an N so that ##∑^{\infty}_{n=1}\frac{log(n)}{n^2}## is between ##∑^{N}_{n=1}\frac{log (n)}{n^2}## and ##∑^{N}_{n=1}\frac{log(n)}{n^2}+0.005.## Homework Equations Definite integration The Attempt at a Solution I began by taking a definite integral...

I know that at early stage (around 1999), GW implementation uses Matusbara frequency to help calculating self energy, and then apply analytic continuation to change it to real frequency for subsequent calculation. I don't know whether this scheme has been superceded by other implementations for...