Finite Definition and 1000 Threads

I have lately been working with Numerical Analysis and I am using Finite Difference Methods for Ordinary and Partial Differential Equations by Randall J. LeVeque. It was recommended to me by a friend of mine (physicist) https://epubs.siam.org/doi/book/10.1137/1.9780898717839?mobileUi=0&...

Since there is no privileged inertial frame, I would have expected the first particles in the universe to have no particular bias in their momenta. Relative to an observer I would expect the distribution to be uniform and unbounded. The mean momentum of the initial particles relative to an...

I have a calculus 2 midterm coming up and given the exam review questions, this seems like this question can potentially be on it. I've tried to look it up, but I always find the famous painters example, which I don't find satisfying.

##1+x+x^2 = \dfrac{1-x^3}{1-x} = (1-x^3)\cdot \dfrac{1}{1-x} = (1-x^3)\sum_{k=0}^\infty x^k##. Isn't this a contradiction since the LHS has degree ##2## while the RHS has infinite degree?

Homework Statement This was in a problem set I found. Suppose that the proton in a hydrogen atom is not a pointlike object, but instead described by a uniform spherical charge distribution with charge e and radius R = 8.7 x 10^-16 m. Using Gauss's law and the definition for the electric...

In the formulation of Euler-Bernoulli Beam Theory, there are two degrees of freedom at a point, w and dw/dx. Typically, the finite element model of this theory uses cubic polynomial for interpolation of $w$ using a two noded element as given in Chapter 5 of this book [1]. This element is a...

Homework Statement Taken from a Ring theory class: Let ##A## be the ring of decimal number:$$ A= \{ n10^k|n \in Z, k \in Z \}$$ Show that ##x \in A## if and only if it's decimal expansion is finite. Here we assume that ##x \in R##. Homework Equations I have no relevant equations really...

If X is a metric space, what is the simplest sufficient condition for d(x,y)<∞, ∀ x, y ∈ X?

Hi! I have a code that solve the poisson equation for FEM in temperature problems. I tested the code for temperature problems and it works! Now i have to solve an Electrostatic problem. There is the mesh of my problem (img 01). At the left side of the mesh we have V=0 (potencial). There is a...

Consider a set $A$ and its subsets $B$ and $C$. It is known that $|A-(B\cap C)|=8$, $|B|=5$, $|C-B|=1$ and $|B\cap C|=3$ (here $-$ denotes set difference). How many subsets $X\subseteq A$ are there if $X\cap B\cap C\ne\emptyset$, $|X-(B\cup C)|\ge3$ and $|X\cap (B-C)|=2$?

If we start with a light mode in a perfect cavity, in a state such as ##1/2\hbar\omega_m + N\hbar\omega_m## , what happens if we introduce a small amount of roughness (something like 0.5% to 5% of ##\lambda_m##) ? Would it create a cluster of similar but non-degenerate discrete modes around the...

Homework Statement Consider the standard square well potential $$V(x) = \begin{cases} -V_0 & |x| \leq a \\ 0 & |x| > a \end{cases} $$ With ##V_0 > 0##, and the wavefunctions for an even state $$\psi(x) = \begin{cases} \frac{1}{\sqrt{a}}cos(kx) & |x| \leq a \\...

I do have a fair amount of visual/geometric understanding of groups, but when I start solving problems I always wind up relying on my algebraic intuition, i.e. experience with forms of symbolic expression that arise from theorems, definitions, and brute symbolic manipulation. I even came up with...

Homework Statement Identify the boundary ##\partial c_{00}## in ##\ell^p##, for each ##p\in[1,\infty]## Homework Equations The interior of ##S## is ##\operatorname{int}(S) = \{a\in S \mid \exists \delta > 0 \text{ such that } B_\delta (a) \subseteq S\}##. ##\partial S = \bar{S}\setminus...

Homework Statement Homework Equations Finite difference method The Attempt at a Solution I have tried two different approaches, but still i am wrong in the question. Can anyone guide me how to attempt this question? Thank you

Hi. I have written a code which solves a pde using finite differences. I won't post the code, because it's too long, and I want to discuss something specific on finite difference regarding theory. It can be proven that the forward finite difference first derivative can be obtained from the...

Ffom exercise 27 of Dummite and Foote: Let ##N## be a finite subgroup of ##G##. Show that ##gNg^{-1}\subseteq N## if and only if ##gNg^{-1} = N##. Why must the subgroup ##N## be finite? Isn't this result true for subgroups of any size?

For a single wavelength, the absorbance is given by: $$d=\log_{10}\left(\frac{I_{0}}{I_{t}}\right), \tag{1}$$ where ##I_0## is the light intensity incident on the material and ##I_t## is the transmitted intensity (so that ##I_t/I_0## represents the fraction transmitted). Many lasers don't...

It is said (hopefully no need to give references for such a common statement) that the electromagnetic field of a given charged particle is infinite in range (albeit converging to zero as the distance goes to infinity). However, given that charged particles apparently did not exist at the...

Homework Statement Working through Purcell (among others) as fun applied math/math modeling refresher. But, I have struggled all week in establishing from first principles that the potential/field/distribution for a configuration of two capacitive disks of radius 1 and separation s along the...

Homework Statement Prove that for any finite group ##G## there exists a sequence of nested subgroups of ##G##, ##\{e\}=N_0\leq N_1\leq \cdots \leq N_n=G## such that for each integer ##i## with ##1\leq i\leq n## we have ##N_{i-1}\trianglelefteq N_i## and the quotient group ##N_i/N_{i-1}## is...

Homework Statement Decide all abelian groups of order 675. Find an element of order 45 in each one of the groups, if it exists. Homework Equations /propositions/definitions[/B] Fundamental Theorem of Finite Abelian Groups Lagrange's Theorem and its corollaries (not sure if helpful for this...

I am working on modelling of a heat storage tank. More specifically, I need to find out the transient temperature variation through the tank height. My question is about the plume entrainment considering the case that hot water is injected from the bottom part of a heat storage tank. This...

Homework Statement Consider a line of charge stretching along the z-axis from -L to +L. Find the potential everywhere. What are the surfaces of constant potential. (The next question answers the previous question and says its a prolate ellipsoid. Homework Equations I will upload an image of...

Homework Statement For each ##n\in\mathbb{N}##, let the finite sequence ##\{b_{n,m}\}_{m=1}^n\subset(0,\infty)## be given. Assume, for each ##n\in\mathbb{N}##, that ##b_{n,1}+b_{n,2}+\cdots+b_{n,n}=1##. Show that ##\lim_{n\to\infty}( b_{n,1}\cdot a_1+b_{n,2}\cdot a_2+\cdots+b_{n,n}\cdot a_n) =...

In texts treating Hilbert spaces, it's usually given as an example that "any finite dimensional unitary space is complete", but I've found no proof so far and failed prove it myself.

Does anyone know of a great text that shows how to set up FEA in MATLAB in detail. I have wrote out the sort of pre pseudo code from John Andersons - Computational Fluid Dynamics now I just need to build it out in code. I am old school with the hand sketches, hopefully this does not get me a...

Homework Statement "If ##A_1,...,A_m\in O## and ##k\in ℕ##, show that the set of points in ##Ω## (the sample space) which belong to exactly ##k## of the ##A_i## belongs to ##O## (the previous exercise is the case when ##m=2## and ##k=1##)." Homework Equations Event space: O ##O\neq ∅##...

I'm trying to figure out how the finite volume version of Lax-Wendroff scheme is derived. Here is the PDE and Lax-Wendfroff scheme, assume initial conditions are given: $$u=\text{function of x,t}\\\hat{u}=\frac{1}{\Delta x}\int_{x_{i-1/2}}^{x_{i+1/2}}u\thinspace dx \text{ (the average flux...

I think the title sums up pretty well my doubts. I learned QFT from Peskin and Schroeder and other common sources, all implicitly defined QFT at zero temperature. Then I started learning about lattice QCD, how to define the action, how to find continuum limits, the importance of the dependence...

Hello, Does anyone know if a finite element skeleton source code exists. Finite element can be used for structural memebers, gas dynamics, etc. But the over all stepping software is the same and then just input the core equations. My application is for pseudo one dimensional compressible...

Problem: If ##H = \langle x \rangle## and ##|H| = n##, then ##x^n=1## and ##1,x,x^2,\dots, x^{n-1}## are all the distinct elements of ##H##. This is just a proposition in my book with a proof following it. What I don't get is the very beginning of the proof: "Let ##|x| = n##. The elements...

Hello, I am looking for an expression to calculate the electrostatic field outside a uniformly charged cylinder, radius R and finite length L. What interests me most is the radial field, E(r) in the plane z = L / 2. I found an expression of the potential V (r) in the case where L <<< R which...

Homework Statement Proposition. Every subset of a finite set is finite. 2. Relevant definitions Definition. Two sets ##X## and ##Y## have the same cardinality iff there is a bijection between ##X## and ##Y##. A set ##X## is finite iff there is a bijection between ##X## and ##\{1, ... , n\}##...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some further help to fully understand the proof of part of Proposition 4.2.10 ... ... Proposition 4.2.10 reads as follows:In the above proof by Bland we read the...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.10 ... ... Proposition 4.2.10 reads as follows:My questions are as follows:Question 1 In the...

Let's suppose I have a finite potential well: $$ V(x)= \begin{cases} \infty,\quad x<0\\ 0,\quad 0<x<a\\ V_o,\quad x>a. \end{cases} $$ I solved the time-independent Schrodinger equation for each region and after applying the continuity conditions of ##\Psi## and its derivative I ended up with...

Let $\lambda$ be a positive number and $n$ a natural number. I want to show that $$\int_{-\infty}^{+\infty} x^{2n} e^{-2 \lambda x^2} dx<+\infty.$$ There is the following hint: $e^{\lambda x^2} \geq \frac{1}{n!}(\lambda x^2)^n$, thus $x^{2n} e^{-\lambda x^2}\leq \frac{n!}{\lambda^n}$.I have...

Hello! (Wave) We consider the initial and boundary value problem for the heat equation in a bounded interval $[0, \ell]$ with homogenous Neumann boundary conditions and $k=1$, and we suppose that the initial value $\phi$ is piecewise continuously differentiable and that...

I have 3 questions. After thinking about it I feel these questions may indicate that I have some misunderstanding in basic knowledge or some missed parts. 1. Why is the (time independent) wavefunction an exponential decay inside the potential barrier? I know the mathematical derivation, but I am...

There are two types of finite state machines (at least that I know about), mealy and Moore. What are the practical differences between them. I understand that mealy machines take the input into account for the output logic, but are the two machines used for different purposes? Or can the same...

I'm preparing for an exam and I expect this or a similar question to be on it, but I'm running into problems with using the Born approximation and optical theorem for scattering off of a finite well. 1. Homework Statement Calculate the cross sectional area σ for low energy scattering off of a...

Hi. I have been trying to solve this problem that has been keeping me up at night for a coupe weeks at least. If anyone can help me, I would be greatly appreciated. Hot air enters a cylindrical duct. The duct has some R-value and radiation and convection is being accounted for on the outside...

Homework Statement I am not having trouble with this question as such more trying to get to grips with the intuition of what the question is implying, also I believe there is a mistake in the question as the solutions give to not mange the given energy condition, they state that ##V_0>E## where...

I am looking at the integral $$\int_0^\infty dx \: e^{-iax} - e^{iax}$$ I know that this does not converge for many reasons, but most obviously because I can rewrite it as $$2i \int_0^\infty dx \: sin(ax) = -2i a [\cos(ax)]_0^\infty$$ which does not converge to anything. However the book...

I have to deal with this integral in my work, $$\int_{0}^{\infty} \frac{ k^4 e^{-2F^2k^2} }{ (k-k_0)^2 }dk$$ where ##F^2>0 , k_0>0## Is important to mention that it has a double pole in ##k_0## and as a consequence mathematically doesn’t converge. However I have seen before some...

Hi, I'm re-studying integrals and I got stuck with this problem. Actually the math beyond it is very clear but I still can figure it out. Take this function: ##f(x) = \begin{cases} 0, & x \lt 1 \\ 1, & x = 1 \\ 0, & x > 1 \end{cases} ## According to Spivak's Calculus I, a function is...

Hi PF, initially I would like you to focus on that link https://books.google.com.tr/books?id=Dkp6CwAAQBAJ&pg=PA389&lpg=PA389&dq=runge+kutta+method++is+tvd+proof&source=bl&ots=47ULQDVwcC&sig=e2zjdnXENJ7WxBbrf6hXkSouvLI&hl=tr&sa=X&ved=0ahUKEwjU5Z2XsbXZAhUMCMAKHWpnATQ4ChDoAQhKMAQ#v=onepage&q=runge...

Is infinity truly infinite if it has something else in it? Put differently, say there's an infinite volume of water that has some rocks in it, is the volume of water truly infinite? Though there's a place where there's no water?

I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ... I am currently focused on Chapter 2: Rings ... I need help with an aspect of the proof of Proposition 1.5 ... ... Proposition 1.5 and its proof read as...