What is Approximation: Definition and 761 Discussions

Dear all, in what sense the tangent space is the best approximation of a manifold? The idea is clear to me when we think about a surface in Rn and its tangent plane at a point. But what does this mean when we are referring to very general manifolds? In what sense "approximation" and in what...

Homework Statement Firstly, I'd just like to point out that this is not actually a course related question. I have been trying to teach myself mathematics, and have been grappling with this for a couple of days. The book has no answer at the back for this particular question. Variables...

I took a LA course in the spring, and was interested by the least squares method for building models. I decided to practice this concept by attempting to build a model that would predict ticket sales for the Mega Millions lottery given the jackpot amount. I have 249 data pairs of jackpot and...

This problem arises in a paper on population genetics (Kimura 1962). 1. The problem statement Let f(p) = \int_0^p ((1 - x)/x)^k dx. For a small value of p, we have approximately f(p) = (p ^ (1-k)) / (1-k) How is this obtained? 2. My attempt at a solution I tried to expand the f(p) around p =...

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...

Find an approximate value of the number e-0.1 with an error less than 10-3 ı know that ex = Ʃ(from zero to ınfinity) xn / n!=1+x/1!+x 2/2!+... ı don't know how to use e-0.1 in this question.Do ı write -0.1 instead of x in ex series?

I am trying to understand the elctric dipole approximation when an atom interacts with an electromagnetic wave. I know that if the size of the atom is much much smaller than the wavelength of the radiation, then the dot product od the wavevector and the position vector becomes constant. I...

By chance I stumbled on this "almost" equality: \frac{1}{5}(1/2+2/3+3/4+4/5+5/6+6/7+7/8+8/9+9/10) ≈ √2 - 7.2×10^{-6} I'm just wondering, are these funny coincidences simply, well, coincidences :biggrin: or is there some kind of explanation? I've see a ton of other funny stuff like...

I created a method for both approximating a function and extending a it's domain from a Natural to a Real Domain. Does this have a name already or any interesting application? Basically. Add polynomial of degree 0, 1, 2, 3, etc. Making at the same time the approximation function equal to f(0)...

\frac{Ns-L}{L+Ns} What does that reduce to if L << Ns ? Obviously setting L to zero leads me nowhere since that argument above is actually inside a logarithm. I don't know how to perform the approximation. And the answer can't be zero by the way. Is there something I can do here? Usually the...

I was having this debate with a friend, and I wanted to know if I was correct. My friend was saying that SR is an approximation for GR (albeit a very good one) with the specific conditions of only inertial reference frames, and I was saying that SR is exactly accurate with GR, and so is directly...

y(4C) ≈ 7.3 + C + \frac{C}{3^{10C}} + \frac{C}{3^{20C}} + \frac{C}{3^{30C}} Would: y(nC) ≈ 7.3 + C\sum_{n = 0}^{\infty}{\frac{1}{3^{10Cn}}} Be an acceptable answer? If not, what am I doing wrong here?

Homework Statement I've attached the questionHomework Equations Pr(X<=x)= (x + 0.5 - n*p) / sqrt(n*p*(1-p))The Attempt at a Solution okay so n=1150, p=0.02 , Pr(X<23) =23 + 0.5 - 1150(0.02) / sqrt(1150*0.02*0.98) =0.105316 is that bit right so far. Because it is less than i thought x...

The relationship linking the emitted frequency Fe and the received frequency Fr is the Doppler Law: F_r = \sqrt \frac{1-\frac{v}{c}}{1-\frac{v}{c}} F_e The Taylor series for the function \sqrt\frac{1+x}{1-x} near x = 0 is 1+x+\frac{x^2}{2}+\frac{x^3}{3}+... On Earth, most objects travel...

The following component is connected to a reference voltage of Vref = 6V. The component is given the value (2C)16 to convert. Calculate Vout. http://img69.imageshack.us/img69/7396/figuringout.jpg Basically I treated this supposedly successive approximation converter like any other...

Hi all, I am reading now Zee's book "Quantum Field Theory in a Nutshell", there in Apendix 2 of Chapter I.2 the method of steepest descent is briefly described. The part where I have a question is almost self contained and half a page long, so I attached the screen shot of it (formula 19)...

Homework Statement Hi Whenever I read about the dipole approximation in QM, then the Hamiltonian is given as \hat {V}_{\text{dipole}} = -\mathbf{d}\cdot \mathbf{E} where E is the electric field and d the dipole operator. What I am wondering about is that d is an operator. Is it wrong to...

Homework Statement A string of length a is stretched to a height of y when it is attached to the origin so making a triangle with length L=\sqrt{a^{2}+\frac{y^{2}}{a^{2}}} and therefore a length extension ΔL= \sqrt{a^{2}+\frac{y^{2}}{a^{2}}}-a which simplifies to...

Hi group, I'm a theoretical ecologist with fairly adequate training in applied math (ODE, linear algebra, applied probability, some PDEs). In my current work, I've encountered the use of adiabatic approximation to a joint probability distribution of two ever-fluctuating spatial variables. A...

Homework Statement Here's the setup. I have a gas of rod like molecules, where n*f(u)*dΩ is the number of rods having their direction in the solid angle dΩ pointing in the direction u. The problem says to consider rods pointing in different directions as separate species. Homework...

Hi, I'm having trouble with a programming problem for my numerical mathematics course. It's about the one-dimensional advection-diffusion equation a*u'(x)-epsilon*u"(x) = 1, on the interval -1 < x < 1. The boundary values are: u(-1) = u_r, u(1) = u(l) I have to approximate the solution...

Wiki says: Isn't this exactly what every A/D converter does? For a graph of Vin to digital output it basically approximates the nearest digital value to the continuous signal -> So I don't see the difference between them.

Homework Statement I've attached the questionHomework Equations x(n+1) = x(n) - f(x(n)) / f '(x(n))The Attempt at a Solution okay so x2= 1.3517323300 and I've already calculated x3 to be 1.3483949227 then how do i estimate the error in x2? do i subtract or something?

Homework Statement Prove that any function f(x) can be approximated to any accuracy by a linear combination of sign functions as: f(x) ≈f(x_{0})+ \sum{[f(x_{i+1})-f(x_{i})]} \frac{1+ sgn(x -x_i)}{2} Homework Equations The Attempt at a Solution Looks like taylors theorem...

i have found the area of sphere in two ways using the same approximation.but i get two different answers ;one the correct value 4∏R^2?how does this happen? i'm attaching the solution below?please refer to the attachments and give a solution?

Homework Statement Define f:R2→R3 by f(x,y,z)=(xy+z) ...(x2-yz) let p = (1,1,1)T and h=(δ,ε,θ) a)what are n and m? evaluate f(p) and f(p+h) b)Calculate the Jacobian Matrix Df(x,y,z) and evaluate Df(p) c) Calculate the error e(h) in the first order approximation to f(p+h) d) show...

Homework Statement I want to estimate f(x)=\ln (\frac{1}{1+x}) on the interval (1/10,1) with the error on the approximation being no more than 0.1. Homework Equations http://en.wikipedia.org/wiki/Taylor's_theorem#Example The Attempt at a Solution Following the example from Wikipedia, I...

Homework Statement The Type Ia supernova SN 1963p in the galaxy NGC 1084 had an apparent blue magnitude of B = 14.0 at peak brilliance. Then, with an extinction of 0.49 mag to that galaxy, the distance to the supernova is approximately d = 10(m - M - A + 5)/5 = 41.9 Mpc Homework...

Hi, I have to approximate an irrational number x by rationals r = p/q. Let ε>0 in ℝ, then, for almost all x exist α and r in (x-ε,x+ε) such that q ≈ c(x) ε^-α, c(x) in ℝ? I know, from Hurwitz theorem (and a conseguence) that α>2, if exists.

I got quite confused with the math in Thomas-Fermi's approximation. I thought it was supposed to approximate a length but the math from a textbook gives energy instead. I don't understand what is it trying to approximate. My professor told me that normal conductors screen electric field...

Homework Statement This problem will guide you through the steps to obtain a numerical approximation of the eigenvalues, and eigenvectors of A using an example. We will define two sequences of vectors{vk} and {uk} (a) Choose any vector u \in R2 as u0 (b) Once uk has been determined, the...

Diamond has a Debye temperature of Dt = 2000 K and a density of 3500 kg/m3. The distance between nearest neighbors is 0.15 nm. Determine the velocity of sound using the Debye approximation. I have no idea where to even start with this question. Most books don't even mention the Debye...

How do you come up with this approximation? [1+H(t-t_0)- \frac{1}{2}qH^2(t- t_0)^2]^{-1}\approx1+ H(t_0-t)+ \frac{1}{2}qH^2(t-t_0)^2+ H^2(t-t_0)^2 Is there a rule that leads to this approximation?

Homework Statement Approximate the function f(x)=sin(\pi x) on the interval [0,1] with the polynomial ax^{2}+bx+c with finding a, b and c. Homework Equations f(x)=a_{0}+\sum^{\infty}_{n=1}(a_{n}cos(nx)+b_{n}sin(nx)) a_0=\frac{1}{2\pi}\int^{\pi}_{-\pi}f(x)dx...

Homework Statement In a manufacturing process for electrical components, the probability of a finished component being defective is 0.012, independently of all others. Finished components are packed in boxes of 100. A box is acceptable if it contains not more than 1 defective component...

Moderator's note: This thread is a perfect example of what not to do in the homework help forums. It is unacceptable for the opening poster not to work through the problem and to demand answers. It is inappropriate for the helper to give out those answers, or tell the poster exactly what to do...

Hello! I was wondering if the following statement is true for large n: \sum_{i=1}^{n} \ \left( 1 \ - \ \frac{i}{(n+1)} \right) \ \approx \ \lim_{n \rightarrow \inf}\ \sum_{i=1}^{n} \ \left( 1 \ - \ \frac{i}{(n+1)} \right) \left( \frac{1}{(n+1)} \right) Firstly, the RHS is an integral...

I've been reading the book "why chemical reactions happen", and according to my understanding, it seems as though orbital hybridization is just an "approximation" and not real, as in there is no such orbital, while MO are (real). Is my understanding correct? Or are MOs also just approximations...

Homework Statement Problem 1.8 here (Link to Google books) Clarification: C[0,1] are the continuous functions on the interval [0,1] and let S denote the set of points in the problem, as it is stated (can't tell if it's a S or a P in the book). Homework Equations Have I understood the...

Hello, I am reading a paper, and the author claimed that in asymptotic sense as M goes to infinite: \sum_{i=1}^M\sum_{l=0}^L|h_i(l)|^2=M where: \sum_{l=0}^L\mathbb{E}\left\{|h_i(l)|^2\right\}=1. How is that asymptotic follows? Thanks in advance

Find the linear approximation to the equation f(x,y) = 3 sqrt((x y)/4) at the point (2,8,6), and use it to approximate f(2.28,8.22). I know you take the derivative of fx(x,y) and fy(x,y), I think I'm taking the derivative wrong. Then after that you put x and y in the equation and solve for...

Homework Statement So in my biology class, my professor wants us to use the Nernst equation without using calculators. I personally think this is stupid. However, I have no choice, so today, I tried coming up with approximations of the log function. Homework Equations We start with loga(b) =...

I've asked this question before. But still I got some unanswered ones. I am really tired, but I cannot sleep if I got something laying there, tingling me. http://pokit.org/get/cfb750b79f49cfc12dc51a74a37f576e.jpg This is digital ramp. In attachments I added a full circuit, from...

Homework Statement Hi! I have a couple of problems on Taylor Series Approximation. For the following equations, write out the second-order Taylor‐series approximation. Let x*=1 and, for x=2, calculate the true value of the function and the approximate value given by the Taylor series...

y(t) = y0 + \int^{ t}_{t_0} f(s, y(s)) ds. Picard’s method starts with the definition of what it means to be a solution: if you guess that a function φ(t) is a solution, then you can check your guess by substituting it into the right-hand side of equation (2) and comparing it to the...

Homework Statement A particle is in a region with the potential V(x) = κ(x2-l2)2 What is the approximate ground state energy approximation for small oscillations about the location of the potential's stable equilibrium? Homework Equations ground state harmonic oscillator ~ AeC*x2...

This isn't a coursework problem. I'm on winter break. Homework Statement A common approximation used in physics is: (1+x)n ≈ 1+nx for small x This implies that lim(x→0) (1+x)n = lim(x→0) 1+nx which is a true statement. However, lim(x→0) (1+x)n = lim(x→0) [(1+x)1/x]xn = lim(x→0) exn This...

Homework Statement The interatomic force in the chlorine molecule Cl2 may be represented by the Lennard-Jones potential: with e = 1.79.10-19 J and r0 = 0.2 nm. (i) Find the interatomic distance at which V(r) is minimized. What is the interatomic force at this separation? (ii) Calculate the...

Homework Statement Suppose the point (pi/3, pi/4) is on the curve sinx/x + siny/y = C, where C is a constant. Use the tangent line approximation to find the y-coordinate of the point on the curve with x-coordinate pi/3 + pi/180. Homework Equations TLA: f(a) + f'(a)(x-a) Where a is...

Homework Statement I need to find the approximation to: X = m_N\>\bigg[\frac{m^2+\mu^2}{m_N^2 - (m^2+\mu^2)}\>\mathrm{ln} \bigg(\frac{m_N^2}{m^2+\mu^2} \bigg) - \frac{m^2-\mu^2}{m_N^2 - (m^2-\mu^2)}\>\mathrm{ln} \bigg(\frac{m_N^2}{m^2-\mu^2} \bigg) \bigg] for m^2 \gg \mu^2 ...