Periodic Definition and 411 Threads

a) Proof: By definition, the potential energy ## V(x) ## is given by ## F(x)=-\frac{dV}{dx} ##. Note that ## \ddot{x}=-\frac{dV}{dx} ## where ## \ddot{x}=-x-\epsilon(\alpha x^2\operatorname{sgn}(x)+\beta x^{3}) ##. This gives ## \frac{dV}{dx}=x+\epsilon(\alpha x^2\operatorname{sgn}(x)+\beta...

Proof: Let ## f(x) ## be a function of the real variable ## x ## such that the integral ## \int_{-\pi}^{\pi}f(x)dx ## exists and if the Fourier coefficients ## (a_{n}, b_{n}) ## are defined by ## a_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos nx dx, n=0, 1, ..., ## and ##...

On the book, it says, "Let ## f ## be defined by ## f(4n)=f(4n+1)=0, f(4n+2)=2 ## and ## f(4n+3)=1 ##, for all integers ## n ##". (Other answers are possible). But I don't understand, how does this work in the problem? I know that it must has something to do with the period, which is ## 4 ## in...

Is there a function that outputs a 1 when the input is a multiple of a number of your choice and 0 if otherwise. The input is also restricted to natural numbers. The only thing I can come up with is something of the form: f(x) = [sin(ax)+1]/2 but this does not output a 0 when I want it.

I would like to print a nice PTOTE, written in Latex. So that I could make small modifications before printing it. I have checked on the Internet, but surprisingly couldn't find any really nice one. Ideally, it should contain information like the fusion point, possibly crystallographic...

or Are all naturally occurring crystals with periodic arrangement of lattices Bravais lattices? From two days, I have been trying to understand Bravais lattices and what it's importance is and after a lot of research, I came to know that they are a periodic arrangement of lattice points with...

First of all, all the physical quantities presented in this topic are unknown variables, and I require a functional relationship between these unknown variables. In a vast space that does not consider gravity , there are many ideal rigid balls moving freely. And in equilibrium. The ball is...

TL;DR Summary: Prove that a sum of trigonometric ratios is periodic but not not simple harmonic. We need to prove that ##x = sin{\omega t} + sin{2\omega t} + sin{4\omega t}## where ##x## is the displacement from the equilibrium position at time ##t##. I can see that each term is a SHM, but...

I wrote down the equation of motion for In(t) and I'm trying to match it with infinite spring mass system equation solution. In the spring mass system, we consider A to be the equilibrium length of the springs, and we can thus write Xn(t) = X(nA,t) and put it back into the equation of motion...

##\sum _{n=0}^{\infty }\:\frac{sin\left(2^nx\right)}{2^n}## I have to show the series is periodic, ##2\pi ## ( I think ), it is related to fourier - analyze fourrier course at academics, so I might be right, or just periodic, we learned only periodic of ## 2\pi ## ( of course also continuous...

I have plotted the function for ##T=15## and ##\tau=T/30## below with the following code in Python: import numpy as np import matplotlib.pyplot as plt def p(t,T,tau): n=np.floor(t/T) t=t-n*T if t<(2*np.pi*tau): p=np.sin(t/tau) else: p=0 return p...

As is well known, almost periodic functions can be represented as a Fourier series with incommensurable (non-multiple) frequencies https://en.wikipedia.org/wiki/Almost_periodic_function. It seems to me that I came up with an integral criterion for the degree of non-periodicity. The integral of a...

Could someone explain the geometry of this graph? Why does the radial distance vary non-uniformly? To-wit: Distance from origin to Nov 2020 is much larger than Nov 2020 to Nov 2021 Why are there two areas - one above and one below - the centre line...

I'm learning about Fourier theory from my lecture notes and I have a few questions that I wasn't able to concretely find answers to: 1. What's the definition of periodic extension? I think the definition is as follows ( Correct me if I'm wrong please ): for ## f: [ a,b) \to \mathbb{R} ## its...

From the statement above, since the ring is massless, there's no force acting vertically on the rings. Thus, the slope is null. ##\frac{\partial y(0,0)}{\partial x} = \frac{\partial y(L,0)}{\partial x} = 0## ##\frac{\partial y(0,0)}{\partial x} = A\frac{2 \pi}{L}cos(\frac{2 \pi 0}{L}) =...

Hello everyone, I am currently trying to understand periodic boundary conditions for the mechanical investigation of mechanical properties of a RVE. I found a good video explaining the theory behind it: But something is unclear to me: At the above linked time step, the individual conical...

I have a problem with finding the energy of an electron in an FCC lattice using the weak potential method. We did that for a one-dimensional lattice during class, and I know that there was a double degeneration at the boundaries of the first Brillouin Zone. However, I'm not sure what...

So I've been programming the BDF methods and for some reason I have an issue with the Backward Euler technique. Given the differential equation y" + y = 0 (with y(0) = 2, y'(0) = 0), my backward Euler solution goes like this: Obviously this is not possible as the function should be a...

One dimensional Ising model is often treated as open chain system with free ends. Then when external field is added it is treated with cyclic boundary condition. Can someone explain me are those methods equivalent, or not?

I'm hesitated between the first and second choice

I am currently studying Electrical Engineering and I have this question: An energy band is formed by the overlapping of atomic orbitals of atoms coming close to each other.I suspect that if the energy of the atomic orbital of the valence electrons of a chemical element is less than the energy of...

Hello All There is much discussion on the existence of Dark Matter. Should we think of Dark Matter as having macro structure, ie comprising elementary particles, leading to atoms and a Dark Periodic Table? best regards ... Stef

Here is the Chen System I am given the initial condition (t=0) that a particle lies on the xyz-plane at a point (-10,0,35). I was notified that if I plugged in a=40, b=5, and c=30, the trajectory of the particle will be chaotic. On the other hand, if I retained the values of a and c, and...

Hello, The periodic table organizes all known chemical elements (total of 118) based on their atomic number and properties. My understanding is that the first 92 elements are commonly found in nature while the other 26 are either highly radioactive and/or artificially made. Radioactive elements...

Hi, I’m looking for ideas and guidance (maybe even formulas?) on making an on-planet situation match a hypothetical solar system. It’s for a fantasy role-playing, but the sentient species must necessarily care (and believably calculate) aspects about their solar system. (However, because it’s...

Consider the following function: $$f(x) = \begin{cases} 1 & \text{when} & -\pi<x<0\\ 0 & \text{when} & 0<x<\pi \end{cases}$$ Beyond ##-\pi## and ##\pi##, the function just repeats itself; it is periodic. I want to plot this function for values beyond ##-\pi## and ##\pi##. The graph should look...

I'm solving the heat equation on a ring of radius ##R##. The ring is parameterised by ##s##, the arc-length from the 3 o'clock position. Using separation of variables I have found the general solution to be: $$U(s,t) = S(s)T(t) = (A\cos(\lambda s)+B\sin(\lambda s))*\exp(-\lambda^2 kt)$$...

How do I get the wave dispersion for a 2D continuum unit cell subjected to a periodic boundary which is excited longitudinally? I'll be applying forces in ABAQUS with varying frequencies. I have come across Blochs theorem but I can't find any application of it in continuous systems. Every...

Find the only periodic solution for 𝑦′+𝑦=𝑏(𝑥) with 𝑏:ℝ→ℝ has a period of 2𝑇 and is 1 for 𝑥(0,𝑇) and −1 for 𝑥(−𝑇,0). The ODE is easy to solve: 𝑦(𝑥)=exp(−𝑥)⋅𝑐+1 and 𝑦(𝑥)=exp(−𝑥)⋅𝑐−1. But how can I find the 𝑐 such that the solution is periodic with a period of 2𝑇? The solution is...

i guess he is asking for the periodic time : $$Tension = \frac {λ*y}{a} $$ $$ \lambda= mg $$ $$y =3a$$ $$T = 3mg$$ $$F = T-mg\Longrightarrow F = 3mg-mg = 2mg$$ $$m{y}''=2mg$$ $$y'' = 2g \therefore\frac { dy'}{dt} = 2g \Longrightarrow y' = 2gt+c1$$ by applying the boundary conditions and...

Hi I have a project regarding micromechanics of composites. I'm starting my analysis on the Fiber Matrix RVE. Right now I'm trying to find the natural frequency of the unit cell. The Unit cell has some unique geometry which I will keep on changing to see how natural frequency changes. I have...

We're told that the electron shells give the relative reactivity/affinity to other atoms by their incompleteness, completeness or over-completeness, and arranged in the periodic table accordingly as columns. The shells are are concentrically arranged around the nucleus from smaller capacity to...

Consider the following periodic function: ## f(t) = \sin(ωt) + \cos(2ωt) + \sin(4ωt) ## What is the time period of the above periodic function? The following is given in my book: Period is the least interval of time after which the function repeats. Here, ##\sin(ωt)## has a period ##T_o =...

In molecular dynamics people use periodic boundaries to confine particles being simulated. I read here that they are used to simulate large "infinite" particle systems. How can I know that the periodic boundary is simulating actual molecular outcomes for a finite particle system that had a large...

Can I sum up the potential due to all positive line charges and all negative line charges separately, with the boundary condition being at the edge of my unit cell, the potential should be the same and inside the metal there is a contant potential?

I was working on a proposal for a spacecraft , and suddenly realized that the ideal orbit may be a high inclination type of near-Earth coorbital called a "retrograde satellite" or RS orbit. Do you know of: * A person who can compute 100 years of coorbital stability using three body (sun...

Hi guys When I ran the Chemical Equilibrium Code from NASA (grc.nasa.gov) to predict the products of a chemical reaction and their concentrations, it says it does not handle certain elements in the database from the periodic table. Is there a computer code that predicts the products of a...

According to the semiclassical approximation, in response to a constant electric field I would get a periodic motion of the electron, like this: The sinusoidal type function is the velocity, while the function that goes to infinity is the effective mass. Thus I was wondering, since ##v## also...

I know it is something simple that I am missing, but for the life of me I am stuck. So, I used the identity ##sin(a)sin(b)+cos(a)cos(b)=cos(a-b)## which gives me $$\int^{\infty}_{-\infty}dx\:f(x)\delta(x-y)=\int^{\infty}_{-\infty}dx\:f(x)\frac{1}{2L}\sum^{\infty}_{n=-\infty}\lbrace...

We have a periodic function ##f: \mathbb{R} \rightarrow \mathbb{R}## with period ##T, f(x+T)=f(x)## The statement is the following: $$\frac{1}{T}\int_0^T f(x)dx =0 \implies \frac{1}{T}\int_0^T\frac{d}{dx} f(x)dx =0$$ Can you give me a hint on how to prove/disprove it? The examples I tried all...

Summary: Transform the periodic table of chemical elements (periodic table) into a universal way of storing and transmitting information using spectral analysis. I propose a concept in which the basis for working with information (conservation, transmission in networks) is to use spectral...

I do not know how to declare time t and unit steps. The image of problems to get a better understanding as well.

Hi all, What is the general set notation for specifying a vertical asymptote and domain for a periodic function? For example, if I have a periodic function which has a period of pi/2, and within that period, a vertical asymptote occurs at pi/4. The domain is R, excluding that vertical...

The Periodic Table is 150 years old sometime this year (I could not find its exact birthday). Good job Mendeleev! Here is a Science magazine news info graphic on how it has changed over time (before and after Mendeleev). The graphic came out a while ago, but was not working then. Now it does.

Homework Statement Prove the discreet signal is periodic: Homework Equations for periodic funtions: x[n] = x[n + N] The Attempt at a Solution I made an equality (im going to leave the sigma out for simplicity): 2^(-abs(n-2m)) = 2^(-abs(n+N-2m)) I don't know what I need to do from...

Homework Statement Ql: Which sound wave will have its crests farther apart from each other - a wave with frequency 100 Hz or a wave with frequency 500 Hz? Homework Equations Frequency= 1/ periodic time The Attempt at a Solution I did it like that: I just found the periodic time for each...

Dear Everyone, I do not know how to begin with the following problem:Suppose that $f$ is $2\pi$-periodic and let $a$ be a fixed real number. Define $F(x)=\int_{a}^{x} f(t)dt$, for all $x$ . Show that $F$ is $2\pi$-periodic if and only if $\int_{0}^{2\pi}f(t)dt=0$. Thanks, Cbarker1

1) Are there any periodic alternating series functions other than sine and cosine (and series derived from them, like the series for cos(a) * cos(b))? 2) What is the following series called when x is (0,1) and (1,2]? Quasiperiodic? Semi? \sum_{n=0}^\infty \, (-1)^n \...

Chart: https://periodicity.io/ Kadin Zhang's code on github: https://periodicity.io/ edit: https://github.com/kadinzhang/Periodicity The app is very useful for beginning chemistry students, IMO, but is even better for learning the vue.js toolset for web development. Code example itself is...

periodic potential how can i show that period is d