Method Definition and 1000 Threads

Hello everyone. I'm currently trying to solve the damped harmonic oscillator with a pseudospectral method using a Rational Chebyshev basis $$ \frac{d^2x}{dt^2}+3\frac{dx}{dt}+x=0, \\ x(t)=\sum_{n=0}^N TL_n(t), \\ x(0)=3, \\ \frac{dx}{dt}=0. $$ I'm using for reference the book "Chebyshev and...

Problem: Suppose that the function $p : N \rightarrow [0, 1]$ satisfies $p >> n^{-1}ln(n)$ (i.e. $n^{-1}ln(n) = o(p)$). (a) Prove that as $n \rightarrow \infty$, the random graph $G(n, p)$ has minimum degree at least $\frac{np}{2}$ almost surely. Idea: Look at the degree of each individual...

Hello all, In high school physics, the magnitude sum of vector addition can be found by cosine rule: $$\vec {R^2} = \vec {F^2_1} + \vec {F^2_2} + 2 \cdot \vec F_1 \cdot \vec F_2 \cdot cos ~ \alpha$$ and its angle are calculated by sine rule: $$\frac {\vec R} {sin ~ \alpha} = \frac {\vec F_1}...

The recurrence relation was given as: ##p_k=2^{k+1}\cdot h_k## where ##h_0^2=2## ##h_{k+1}^2=(\frac{1}{2}h_k)^2+(1-\frac{1}{2}h_k \cdot \cot(2^{-k}\cdot \alpha))^2## and ##\alpha=\arctan(1)##. This is not exactly an original or noteworthy derivation, is it? I feel that it's been done...

Hello, I hope this is the right place to post about this. I live in a multi-unit condo, and I have a window that I am trying to lock and unlock from the outside. The window is from the kitchen and looks out onto a public-ish breezeway where I can access easily. I know this might seem a...

So for my scheme I obtained ##\frac{\mu}{h^2} U_{p}+(\frac{v_{1}}{2 h}-\frac{\mu}{h^2})U_{E}+(\frac{v_{2}}{2 h} - \frac{\mu}{h^2})U_{N} - (\frac{v_{1}}{2 h}+\frac{\mu}{h^2})U_{W} - (\frac{v_{2}}{2 h} + \frac{\mu}{h^2})U_{N} + \tau = f## however I am not sure this is correct. I am quite new to...

import math def poly(x): return (x**4 + 2*x**3 - 7*x**2 - 8*x + 12) def bisect(f,a,b,tol=1e-6): while (b-a)>tol: m=(a+b)/2 if (f(a)>=0>=f(m)) or (f(a)<=0<=f(m)): b=m else: a=m return (f(a),a) print(bisect(poly,-4,-2.5)) Here is...

Hi all I need to understand the following passage from Hall link page 78 : Some notation first: Basis for ##sl(2;C)##: ##H=\begin{pmatrix} 1&0\\0&−1\end{pmatrix} ;X=\begin{pmatrix} 0&1\\0&0\end{pmatrix} ;Y=\begin{pmatrix} 0&0\\1&0\end{pmatrix} ## which have the commutation relations...

Hi All I was wondering if there was a quick method of calculating the Second Moment Of Area about the Z axis shown below? I can quickly work out the Second Moment Of Area about the Y axis but the Z axis is proving very difficult and time consuming as the parallel axis therom needs to be...

Hi, In the question outlined in the images (apologies for the poor quality of the scans), the chosen solution has opted to use a symmetry argument and proceed from there. Question is from "Structures: theory and analysis" by Williams & Todd My question is: How could we approach the same...

In general, one could say the Finite Element Method is merely an interpolation method that could be used to solve field equations. Despite that, this question focuses exclusively on the FE Method and its use in Mechanical Engineering. ------------------- I have noticed that some schools now...

Good Morning, I have been doing computer practices in C ++, and for an integration practice, the trapezoid method converges faster than the Simpson method. The function to be integrated is a first class elliptical integral of the form: Where k is bounded between [0,1). I have been thinking...

Good afternoon,I am preparing a laboratory report on the study of the oscillations of a spring and the following questions have arisen:The script asks us to represent the mass against the squared period, in this case, the slope will correspond to the spring constant divided by 4Π^2 and the...

I attached below my attempt to solve the question.

I verified with others the equation below is an Euler method as well with ##a## can be any value such that it give the same ##\frac{dE}{dv}=-1.4\frac{p}{v}## but with ##a## other than one, it have no meaning in physics. For anyone that already understand Euler method can omit the part i have...

I know that you can get the answer through using Fs as 18 and solving for K, then subbing it into the equation for elastic energy. I was just wondering why another method wouldn't work. I tried doing it using the concept that Work is an equal to the Change in Elastic Energy, therefore Ee=xF...

Exercise statement Find the general solution for the wave equation ftt=v2fzzftt=v2fzz in the straight open magnetic field tube. Assume that the bottom boundary condition is fixed: there is no perturbation of the magnetic field at or below the photosphere. Solve means deriving the d’Alembert...

I'm having a hard time grasping the concept of reducing the two recursive relations at the end of the frobenius method. For example, 2xy''+y'+y=0 after going through all the math i get y1(x) = C1[1-x+1/6*x^2-1/90*x^3+...] y2(x) = C2x^1/2[1-1/3*x+1/30*x^2-1/630*x^3+...] I know those are right...

[SIZE=14px][FONT=Noto Sans]So I was trying to understand Horner's method. I understand that you can take a polynomial and factor out the x's and re write it as multiple linear functions recursively plugged into each other and that this makes evaluating a polynomial easier because you just...

What is yours?

Does anyone know a simple implementation in FORTRAN 90 of collisions of particles?

Good Morning, I am using the Jacobi diagonalization method for symmetric matrices and I have realized that as the number of iterations progresses, the speed with which the larger element (in absolute value) outside the diagonal of the diagonal becomes smaller Matrices are increasing (graphical...

In a typical quantum course we learn how to approximate the ground state of a particular Hamiltonian by making an educated guess at an ansatz with a tunable parameter then calculating the expectation energy for the ansatz. The result will depend on the tunable parameter if done correctly. Then...

Homework Statement: The Task is to calculate the Transmission coefficient with the WKB Approximation of following potential: V(x) = V_0(1-(x/a)²) |x|<a ; V(x) = 0 otherwise Homework Equations: ln|T|² = -2 ∫ p(x) dx I have inserted the potential in the equation for p(x) and recieved p(x) =...

Hello all I am trying to analyse a truss using the method of joint process i.e I am trying to determine whether each member is in tension or compression. I have the following truss that I want to analyse using the method of sections. Just focusing on joint A, I have created a FDB and...

Since the Newton's method is as follows: $$x_{n+1}=x_{n}-\frac{f(x_{n})}{f'(x_{n})}$$ $$x_{1}=x_{0}-\frac{cos(0)-1}{-sin(0)-2}$$ Is this correct? What should I proceed on from here?

Summary: different methods give different results. They are not consistent. Summary: different methods give different results. They are not consistent. I use two different methods to detect whether a matrix is singular. The result of calculating the determinant of a 9-order square matrix is...

I am looking at a paper (ref below) that uses a method to precipitate calcium carbonate on a steel pipe. However, the paper gives no method because the reference it uses is linked to a thesis that has not been released. The setup is below... Now, I kind of understand that it uses a method of...

I have the following PDE I wish to solve: \frac{\partial u}{\partial t}=D\frac{\partial^{2}u}{\partial x^{2}} With the following boundary conditions: \frac{\partial u}{\partial x}(t,1)+u(t,1)=f(t),\quad u(t,0)=0 Now, I wish to do this via the Crank-Nicholson method and I would naively...

George makes a paper boat with his brother Bill and goes out in the rain to play with it. It falls in the stream along the curb, racing towards the sewer. Let t be measured in seconds, p be the velocity of the paper boat in meters/second and g be George’s velocity, measured in meters/seconds...

Use Euler's method to approximate the value of y at $x=1$ on the solution curve to the differintial equation $$\dfrac{dy}{dx}=x^3-y^3$$ that passes through $(0,0)$, Use $\Delta x = \dfrac{1}{5}$ or 5 steps $\quad x_{1}=x_{0}+h=0+\frac{1}{5}=\frac{1}{5}$ $y\left(x_{1}\right)=y\left( \frac{1}{5}...

I have the following problem. P-n-p problem is often referred to when talking about camera localization(https://en.wikipedia.org/wiki/Perspective-n-Point). It’s a mathematical problem also a computer science problem. Lambda Twist is one of the state-of-the-art...

How are Savitzky–Golay second derivative different from Savitzky–Golay smoothing?

In second order case we should rewrite the equation in terms of 2 first order DE's. So I wrote, $$dx/dt = wx$$ $$dwx/dt = -GMx/r^3$$ and $$dy/dt = wy$$, $$dwy/dt = -GMy/r^3$$ Now I guess there's two ways to do it in 4th order RK method. I would either do it component by component or just in...

So I was reading Jackson's discussion on Image charge method of a grounding sphere. He first assumed an image charge q inside Sphere with radius a, so the potential for real change and image charge is . The by set potential equal to 0 at x=a, he solved q' and y' Then he can get potential...

I understand the idea of the method of images, and its clever use of uniqueness to determine V(x,y,z) for non-trivial systems. My question now is simply about guidance for obtaining the effective "image" of this system, as it is clearly more complicated than the 2-plane analogue (in which there...

Summary: Worth teaching in secondary school? - or too bewildering? The mathologer video made me aware of Lill's method for solving polynomials with real roots. Although I'm not involved in secondary school teaching, I can't help wondering if it is a suitable topic for that level. Perhaps...

So far I've got E=1/2m(dx/dt)^2+mgh this can be rewritten as (dx/dt)^2=2(E-mgxsin(Θ))/m Would there not be a positive and negative solution? dx/dt=±sqrt(2(E-mgxsin(Θ))/m) or do we discard the negative one? Why would we do so, if that's the case. So far I've only solved the postive solution and...

Recently i read, that GEO orbit actually isn't enough for a space elevator, since its weight would pull it down, either it needs constant thrust, or build it much taller than 33.000 km, so upper GEO parts pull it up. That further lowers its plausibility level. Any other methods? Build a tall...

On the right paragraph it says "The trial divisor 1200 goes into the dividend 13952, 8 times" Clearly 1200 goes into 13952, 11 times. I don't understand why 8 is (arbitrarily?) chosen. Please help. Thanks.

Since Hubble's Law has been around a long time, so, after almost 100 years, to challenge its validity looks like dumb and stupid. So let me be that dumb guy, or maybe the bad boy that spoils the whole thing. Here let me start with the Redshift in general. There are three Redshifts proposed till...

I'm trying to choose a protocol for estimating GLutamate Cysteine ligase assay. I've two of them. Reaction: L-glutamate + L-cysteine + ATP gamma-glutamyl cysteine + ADP + Pi #Protocol 1: Dasgupta 2007 Though this method, the author has estimated GCL activity by measuring a blue coloured...

I am attempting to solve the following PDE for Ψ representing a stream function on a 2D annulus grid: (1/s)⋅(∂/∂s)[(s/ρ)(∂ψ/∂s)] + (1/s2)⋅(∂/∂Φ)[(1/ρ)(∂ψ/∂Φ)] - 2Ω + ρ(c0 + c1ψ) = 0 I have made a vertex centered discretization: (1/sj)⋅(1/Δs2)⋅[(sj+1/2/ρj+1/2,l){ψj+1,l - ψj,l} -...

Crystal acts as a diffraction grating.Although I don't know what that means.

I am having difficulties computing the magnetic moment for an even-odd (proton-neutron) nucleus. The formula is: $$\mu_J=g_J\times j\times \mu_N$$ I checked this helpful post...

I have posted this here because I am not sure it is relevant in any of the main boards. I have a question about what science can and cannot observe. Maybe this is too philosophical but I am more interested in a specific empirical matter. As biological creatures, our entire experience of the...

In case of overhanging beam with point load at the end. For example: (here RA-reaction is negative) The equation will be as follows (by double integration method): , as we can see the equation will not have Point load (10kN) term in it. 1) How the influence of the point load is accounted in...

Unlike in electromagnetics, the nonlinearity of mechanical structures is not only due to the material property. It could be due to large deformation and contact as well. Even though I will be implementing the existing and popular methods, I am still scared and I feel it is not a job that can be...

So, Heavside's method confused me, I mean, but we can't divide by zero, can we? So, my mind sees it as cheating, and to multiply both sides by zero to cancel zeros, and for the example above(from Thomas Calulus), when x = 1, before you multiply by x-1(=0), you get: 2/0=A/0+B/-1+c/-2, and I think...

This attractor is unusual because it uses both the tanh() and abs() functions. A picture can be found here (penultimate image). Here is some dependency-free Python (abridged from the GitHub code, but not flattened!) to generate the data to arbitrary order: #!/usr/bin/env python3 from sys...