Hello,
I am taking some microwave engineering courses and was trying to explain the concept of reflection coefficients to my friend, but he asked me a question I am unable to answer...
So we know that given a transmission line with characteristic impedance Z_0 terminated with a load impedance...
Oh oops. Yeah it would be 1 / (p(n+1) - p(n)). But still, assuming the twin prime conjecture is true, there would be infinitely many cases where the slope is 1/2... Clearly this happens rarely, but does it compensate for the desired O(1/log x) order?
Well, that's what I want to do. My problem is how to rigorously prove that the slopes of the linear interpolation grow with order 1/log(x). The main issue is that pi(x) will be constant most of the time, except at the primes where it jumps by 1, so there will be infinitely many points where the...
I am trying to understand a condition for a nonincreasing sequence to converge when summed over its prime indices. The claim is that, given a_n a nonincreasing sequence of positive numbers,
then \sum_{p}a_p converges if and only if \sum_{n=2}^{\infty}\frac{a_n}{\log(n)} converges.
I have tried...
So I finally solved the problem, in a totally different (and way simpler) method -_-
I used an earlier result from another problem, saying that squarefull integers can be written uniquely in the form a^2b^3.
So to count the squarefull integers \leq x you can just do \sum_{a^2b^3 \leq x} 1 =...
It would be the frequency received. Oh I think I understand what you mean. It doesn't actually matter what the moving observer sees. He could just as well see gamma rays hitting the detector, but what matters is what the detector itself sees, which can be calculated knowing its velocity relative...
Hm I was thinking of using the photoelectric effect. The energy of photons hitting the detector would have to be larger than the work function of the material, and this would produce a current in the circuit. Since the energy of photons is proportional to their frequency, that would be a way to...
Sorry, I meant the threshold frequency of the detector. The circuit will turn on if the frequency of the light source photons is greater than the threshold frequency f_t.
Well, in this setup, the stationary observer is in the same frame as the detector (they're both stationary wrt the light source). The "paradox", as I see it for now, is that in a different frame, the apparent functionality of the device is not the same. Using the photoelectric effect mechanism...
My guess is that obviously the device will not explode in any reference frame. But then this would imply that the device's detection abilities are degraded as the observer moves faster, which also seems strange. A possible detector would be something based on the photoelectric effect, where the...
Hello, I just thought of something that looks like a paradox to me. Suppose you have a device which can detect a light source's frequency, and will explode if the frequency exceeds a threshold value f_t. Place this device stationary wrt the light source. Now, suppose there are two observers, one...
I'm doing the exercises from Introduction to Analytic Number Theory by A.J. Hildebrand (online pdf lecture notes) from Chapter 2: Arithmetic Functions II - Asymptotic Estimates, and some of them leave me stumped...
1. Homework Statement
Problem 2.14:
Obtain an asymptotic estimate with error...
Thank you for the advice. For the optimization, I found a fast way would be to use MATLAB. I found it has a function called "fmincon" which can solve nonlinear optimization problems with linear constraints (in this case, the requirement that probabilities add up to 1 and that they are between 0...
Hello,
I see, this seems like a good solution. So if I use the chi-squared function as my error measurement, I would compute \sum_{y=0}^{9}\frac{(E_Y(y) - O_Y(y))^2}{E_Y(y)} + \sum_{s=0}^{9}\frac{(E_{X+Y}(s) - O_{X+Y}(s))^2}{E_{X+Y}(s)} where E_{X+Y}(s) is replaced by the convolution sum in...
I am considering two discrete random variables, X and Y, with values in {0,1,...,9}.
From my data, I can obtain an estimate for the distribution of Y (the function P_{Y}(y) = P(Y=y) for y in {0,1,...,9}).
I can also obtain experimentally the distribution of the sum P_{X+Y}(s) (that is, P(X+Y =...