# Search results

• Users: RUber
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1. ### I Area under a diffraction curve

Does this mean that ##I_{max} = 4\pi n I_0## all the time? If so, then that would be all the coincidence needed.
2. ### Contour Integral from Peskin & Schroeder Intro to QFT

Good point, Lev. The integral that you want to compute is the one around the branch cut. I think you would agree that the larger contour with the notch cut out encloses a region where the function is analytic, and therefore would have an integral equal to zero. Instead of using Cauchy's thm to...
3. ### How does ball A come to rest and Ball B remain stationary?

Give ball A some velocity and mass. Ball B and ball C have the same mass but zero velocity. Assume an elastic collision. Conserving kinetic energy gives something like: ##\left(v^2_A m_A + v^2_Bm_B+v^2_Cm_C\right)_{pre}=\left(v^2_A m_A + v^2_Bm_B+v^2_Cm_C\right)_{post} ## Since all the balls are...
4. ### Programs CE + Phys or CE + Math: Interested in Quantum Computing & AI

From what I've seen, AI is normally a blend of CE and advanced statistics (i.e. mathematics). I think that math would be the best 2nd degree, since it would be able to complement CE well and is a good springboard into the advanced physics needed to understand quantum computing.
5. ### Absolute value notation removal

The way you have this written is that -1 is greater than x AND x is greater than 1. Is that even possible? Remember when you negate an AND statement, like -1<x<1 which is read -1 is less than x AND x is less than 1, you will get an OR statement.
6. ### Solving PDE heat problem with FFCT

You should try to do this. At time = 0, you get ## u(x,0)=v(x,0) + h(x) ## Your cosine transform is just in terms of x, right? And for each n, you have an ODE to solve in terms of a function of t, which should be of the form: ## f'(t) = g(t) + c ## where your functions of x are treated as...
7. ### A Long wavelength limit

I am not sure that there is a standard, since it likely depends on your application. This is simply the Taylor approximation of the cosine function: ## \cos x = 1 - \frac12 x^2 + \frac{1}{24}x^4 - \frac{1}{720}x^6 ...## Therefore, you can cut off the higher order terms whenever you feel they are...
8. ### Convection Diffusion Equation

I believe this is usually handled by letting your C2 by a polynomial and solving for the boundary conditions.
9. ### Extending a Continuous Function a Closed Set

That makes sense.
10. ### I Positional Probability of Periodic Object Motion

You will probably want to be careful of your divide by zero errors when x = +/- a.
11. ### Solving PDE heat problem with FFCT

Oh, I see. I read that information as saying that ##f(0,t) = U_0, \quad f(L,t) = U_1##. In this case, you seem to have mixed boundary conditions then. ##\frac{\partial u}{\partial x} (0,t) = 0, u(L,t) = U_1 ## To use the separation of variables after your last step, you will find an appropriate...
12. ### Extending a Continuous Function a Closed Set

Got it. Since you didn't make that explicit in the proof, I was not following your logic. In that case, the only thing missing are the infinite endpoints. If ##\exists m \in \mathbb{E} \, s.t. \forall x \in E, x \geq m## and likewise if ##\exists M \in \mathbb{E} \, s.t. \forall x \in E, x...
13. ### Solving PDE heat problem with FFCT

To separate the variables, you assume that your function ##F## is a product of two functions, one dependent only on x, ##X(x)## and one dependent only on t ##T(t)##. If you let ## T(0) = 1##, then your initial conditions will fully describe ##X(x)##. In the case that ## \frac{\partial...
14. ### Extending a Continuous Function a Closed Set

In your write-up, you refer to ##f(a_i) ## and ##f(b_i)##. Since ##f## is only defined in ##E##, wouldn't this mean that all your points ##a_i, b_i \in E## as well? I gather that the function you defined is sort of a 'connect the dots' sort of linear interpolation between disjoint points. What...
15. ### Solving PDE heat problem with FFCT

@Aows, I have not worked much with the FFCT, but it seems like the method is much like that of other transforms. After you have rewritten the derivatives, you should separate the variables, setting F(x,t) = X(x)T(t) and use standard ODE methods to solve for T(t). What is unclear to me it that...
16. ### Solving the heat equation using FFCT (Finite Fourier Cosine Trans)

@Aows, The FFCT assumes that ##C(x,t) = \sum_{n = 0}^{N} a_n \cos \frac{n\pi x}{L}## Where ##a_0 (t)= \frac{1}{L} \int_0^L f(x,t) dx \\ a_n(t) = \frac{2}{L} \int_0^L \cos \frac{n \pi x}{L} f(x,t) dx.## Apply the transform to the PDE, as you have done: ##\frac{\partial^2 f}{\partial x^2} =...
17. ### Limit Points in a T_1 Space

I would recommend starting with the definitions of T_1 space, infinite subset, and limit point. Use your definitions as fuel for your proof.
18. ### Limit Points in a T_1 Space

It sounds like you are assuming that there are an infinite number of points, and showing that there are an infinite number of subsets.
19. ### Solving partial differential equation with Laplace

@Aows, I second the recommendation that you type out your problem and work. This will make it easier for us to follow your work and provide assistance. In your initial conditions, do you have ## u_t (x, 0) = 0, u(x, 0) = 6\sin ( \pi x ) - 3\sin (4\pi x) ## It looks like maybe it is an ##n\pi x##...
20. ### Induction Inequality Question

That's exactly what I meant. It has to be true for all the terms up to and including the current one, then you can inductively prove it for the next one ##a_{k+1}##. If the floor function is confusing, sometimes you can just get rid of it. Remember that the floor function is always less than...
21. ### Induction Inequality Question

I recommend you look at it this way: ##a_k = a_{\left\lfloor \frac{k}{5} \right\rfloor} +a_{\left\lfloor \frac{3k}{5} \right\rfloor} + k## assume ##a_k \leq 20k ## Then show that ##a_{k+1} \leq a_k + 20 \leq 20(k+1).##
22. ### How to find standard deviation

You are not being asked for Z, and all your X values are given. What you are missing is the sigma, or standard deviation. In case your google search for "how to calculate standard deviation" didn't return any results, here is a start. Standard deviation is defined as the square root of the...
23. ### Inner Boundary Condition

@Aows, Let's take a look at where these things came from: We write things in dimensionless terms so we can talk about the math in general. Then, based on the physics, apply some assumption, like: Which tells you that the pressure is decreasing as you move away from the well. In your source...
24. ### MATLAB Weighting data points with fitted curve in Matlab

You can quote my username and PF as the source. I am currently an Asst. Prof of Mathematics.
25. ### MATLAB Weighting data points with fitted curve in Matlab

Fitting curves really is a black art. You can apply some standard methods, but when you want to apply some additional finesse, everyone will tell you it depends, so there are few authoritative standards. By definition, the un-weighted model will minimize your un-weighted error. By changing the...
26. ### MATLAB Weighting data points with fitted curve in Matlab

I have not used weights in Matlab before, but I think there is a need to normalize them. Using 1/SE seems to make the most sense, since SE is the standard error of the mean, which is what you are trying to plot. To normalize, I would recommend something like ## w = \frac{1}{1+|\overline{SE} -...
27. ### Jet fighter flying in a vertical loop

What if you were to let M be the mass of the aircraft? What would you get? Can you work out the acceleration components?
28. ### Inner Boundary Condition

The simplest answer is that your negative sign would have to be included in the definition of ##y_{ch}.## However, if you are trying to describe a different physical environment by switching the sign, (leaving the rest of the values unchanged), you end up with infinitely increasing pressure at...
29. ### Inner Boundary Condition

I am not sure what more to say. Depending on the problem you are working on, the right side could be any number of things. Each would have its own particular solution. If you consider that instead of -1 on the right, you had an arbitrary constant C, would that change things? Only a little. You...
30. ### Inner Boundary Condition

The reason is that it is convenient to do. Generally, the ##p_D## value is decreasing as your radial component increases. The first expression gives you an idea of all the factors that contribute to the specific rate. By lumping them all together, you can focus on only the part of the equation...
31. ### Converting mg/hr to ppb

That's what I get based on the conversions and approximations I used. It seems high, but I think that is why recently many consumer groups have advocated against ozone-producing air purifiers -- because they replace one type of pollutant with another. Ozone does naturally dissipate, so if you...
32. ### Converting mg/hr to ppb

ppb is parts per billion, right? I can walk through the process, but I cannot vouch for how accurate the approximations are. According to Wikipedia, air at 20 degrees Celsius is about 1.2 kg per m^3. And molar mass of dry air is 0.0289644 kg/mol. This means that ## \frac{1.2}{0.0289644} ##...
33. ### Calculate coefficients of expansion for vector y

For part a, I agree with Orodruin, there seems to be no such restriction on what t should be, other than non-zero. For part b, your development with the vectors you used was right. I suspect if you get the correct form for part a, you will get the correct solution for b. Try solving for the...
34. ### Basis of the intersection of two spaces

I think the answer is yes...but you have to be careful. If you are able to write a basis vector for one space in terms of basis vectors from another, then clearly, those vectors will be in the intersection. However, your space A is simply the xy-plane. It could have been written with other basis...
35. ### Integral quick q , integrate ((1-x)/(1+x))^1/2

I'm not sure if my definition of partial fractions is the same as yours, but ##\int \frac{1}{\sqrt{1 -x^2} }\, dx ## and ##\int \frac{x}{\sqrt{1 -x^2}} \, dx## both have known integrals.
36. ### Integral quick q , integrate ((1-x)/(1+x))^1/2

What if you multiplied by 1, i.e. ## 1= \frac{\sqrt{1-x}}{\sqrt{1-x}}## Then you might have something that looks like a known integral using trig functions.
37. ### Q about a Proof -- periods meromorphic function form discrete set

Right. Your set ##\Omega_f## has elements which are periods of ##f##. Say you have one element that you know for sure, ##\omega_0##. Then, if your set ##\Omega_f## is not a discrete set, then you can form a sequence of subsets. Assume there exists a sequence of open neighborhoods, radius =...
38. ### Q about a Proof -- periods meromorphic function form discrete set

They are periods because they are elements of Omega, the set of periods. So ##f(x + \omega_0) = f(x) ##. There is no assumption about the continuity of the function, only the periodicity. You are trying to show that ##\Omega_f## is discrete by proving (by contradiction) that it can only be...
39. ### Aerodynamics problem

Most of your steps look good. Remember that ##u^2 = f(x,y) ## implies ##u = \pm \sqrt{f(x,y)}.## This is important because you need to be consistent and keep ##\frac vu = -1##.
40. ### I Gradient of a time-dependent function

It looks to me like that came from the unprimed t in your original equation. ## t' = -\frac{R(t')}{c} +t## so ##\nabla t' = -\nabla \frac{R(t')}{c} +\nabla t## Is it reasonable to write ##\nabla t = \frac{-c}{R}\mathbf{R}##?
41. ### Q about a Proof -- periods meromorphic function form discrete set

There is no rule against a sequence being composed of one repeated entry...however, for some fixed ##\omega_1## and ##\omega_2## with ##\omega_1 \neq \omega_2##, there is a fixed distance ##|\omega_1-\omega_2|<0##, therefore there will, at some point be a high enough n, such that ##\frac1n <...
42. ### Circle inscribed in a triangle exercise

The part where the angles have equal measure is the assumption that this triangle ABC is isosceles. If T1T2 is parallel to AB, then you have all kinds of tools to fill in the measures of the angles. I suppose these assumptions are easy enough to justify based on the definition of the inscribed...
43. ### Remainder factor theorem: me reason this out

I don't have a complete solution at this point, but I think the fact that p is monic tells you to start with the leading coefficients. You should be able to constrain the possible q's based on that.
44. ### Q about a Proof -- periods meromorphic function form discrete set

Since you are using contradiction, you are assuming that the set is not discrete, right? If the set is discrete, then there is a radius around ##\omega_0## where no members of the neighborhood are in the set ##\Omega_f##. Because this is not true, you should be able to find members in...
45. ### Fourier Series

I must have lost track of the negative sign somewhere--your new solutions seem to be in the right neighborhood. Look at the coefficient on your ##a_n## integral. It is the same as the one you used on your ##a_0## integral. It should be twice as big.
46. ### Fourier analysis

It sounds like it might be asking about even/odd ... which coefficients will be non-zero in your Fourier expansion. Maybe even the period, like the argument of your sines and cosines, etc.
47. ### Fourier Series

You were right to use the half-period formula and notice it was an even function. You did not write it like one f(x) = |x|, not f(x) = x. For ##a_0##, it looks like you did 1/2pi, instead of 1/pi for the full period of pi. On the half-period formula, you want to double the result to get the full...
48. ### Fourier Series

Your work looks solid at first glance...maybe an algebraic error, I'll go back and look at the details soon. Remember, your final form should be something like: ## f(x) = a_0 + \sum_{n=1}^\infty a_n \cos 2n x ##
49. ### Logarithm question using compound interest formula

You seem to be mixing up the formula. Your base for the exponent should be 1 + the percentage growth for each time increment...in this case months. If you were given a yearly growth rate and asked to figure out monthly growth, then n would be 12, but in this case, n is 1, and you can let t...
50. ### Magnetic flux through a triangular loop?

I see what they did there. Since the wire in infinitely long, B is only dependent on r, the distance from the line. So, if you slice up the equilateral triangle vertically, along lines with the same distance to the wire, you get ##\int\int B \, dA = \int B h(r) dr, ## where ##h(r)## is the...