Finite Definition and 1000 Threads

Homework Statement From qualitative arguments, make a sketch of the form of a typical unbound standing wave eigenfunction for a finite square well potential. An unbound particle is one which has total energy E greater than the Potential V of the well Is the amplitude of the oscillation the...

Homework Statement Prove that a nonempty set T1 is finite if and only if there is a bijection from T1 onto a finite set T2. The Attempt at a Solution There are at least two different solutions to this problem that I found online...

Hi guys I wish to self-study the Finite Element Method. I am mostly interested in learning how to understand and implement the method rather than to investigate if a solution exists, i.e. I wish to follow the "engineering-approach" rather than the "mathematician-approach". Do you have any...

L={vwv : v,w elements of {a,b}* and |v| =2} I know that "v" can take either aa, ab, ba or bb as values. I also know that "w" can be any string containing "a" and "b". Overall, I know that the two first and two last characters must be identical. How would I show this in a DFA or even an NFA...

Homework Statement I have a designed a Finite Element code for a Poisson problem using Bilinear Element. - \DeltaU= 2\pi^{2}Sin(\pi X)Sin(\pi Y) U = 0 The exact solution is given by : UEX = Sin(\pi X)Sin(\pi Y). 2. The attempt at a solution On convergence analysis:Theoretically, if...

Can some one explain why the value for the electric field strength is ##E = 2\sigma/\epsilon_0## for a charged sheet but ##E = \sigma/\epsilon_0## if the sheet is infinitely large? I understand the procedure of using Gauss' law and creating a Gaussian surface of constant E; however, I...

These are finite solenoids/inductors and I only care about the field outside of the solenoids. Assuming identical coils and the distance on axis from the coil and the current through the coil is the same, will an air-core or ferrite-core inductor/solenoid create a greater magnetic flux...

How does one prove that for R commutative, a free finitely generated R-module has finite rank? If R is a field (i.e. in the case of vector space), then we can argue that given a finite generating set S={s1,...,sn}, if S is not linearly independent, then, WLOG, it is that (*)...

So if we have a finite collection of disjoint non-empty sets, one can show using ZF only(with no need of AC) there is a choice function. I understand the reason for this. My confusion is when one goes to non-finite collection of sets. For example if the index set is the Natural numbers, why do...

Homework Statement The MATLAB program below uses the finite element method to solve a Dirichlet problem for Poisson's equation. -a1u''(x) = 0, xl < x < xh u(xl) = 0, u(xh) = 1 Modify the problem to solve -u''(x) + u(x) = 1, xl < x < xh...

Hello, Can anyone help me with that? It's a problem taken from Wald book on General Relativity,in the section of Schwarzschild solution Thanks Show that any particle (not necessarily in geodesic motion) in region II (r < 2M ) of the extended Schwarzschild spacetime, Figure 6.9, must...

Homework Statement Let E be an extension of a finite field F, where F has q elements. Let \alpha \epsilon E be algebraic over F of degree n. Prove F \left( \alpha \right) has q^{n} elements. Homework Equations An element \alpha of an extension field E of a field F is algebraic over F if f...

I am writing a multi-region diffusion code. The two regions have different material properties, so the analytical solution shows a discontinuity at the interface between the regions. As can be seen here: The numerical code I am running is (Mathematica): While[converge > .00001...

Homework Statement A subspace N of a vector space V has finite codimension n if the quotient space V/N is finite-dimensional with dimension n. Show that a subspace N has finite codimension n iff N has a complementary subspace M of dimension n. Do not assume V to be finite-dimensional. 2...

Hi everyone, I have to prove that every element z of a finite field F is a sum of 2 squares. Really not sure how to go about proving this, though I've done some research and it is suggested to start with a function that maps F* to itself, defined by f(x) = x^{2} . I guess if I could show...

Homework Statement 1.) A particle of kinetic energy 50 eV in free space travels into a region with a potential well of depth 40 eV. What happens to its speed? a.) it stays the same b) it increases in the region of the well c) it decreases in the region of the well d) not enough information...

Homework Statement Hi guys, I'm having trouble understanding the finite potential well, in particular the boundary conditions The well under scrutiny has potential V(x)= 0 for |x|<a and V(x)=V_0 for >a Homework Equations \frac{d^2\psi}{dx^2}=-\sqrt{\frac{2mE}{\hbar^2}}\psi=-\alpha^2\psi...

Homework Statement A line of length 2a on the y-axis centred in the middle of the x-axis with its ends at a and -a on the y axis, has been charged with charge Q. Work out the electric field at point x on the x axis. Homework Equations E=kdQ/r^2 dQ=Qdy/2a The Attempt at a Solution...

I have in my algebraic topology notes, as a step in the proof of another theorem, that the product of a finite simplicial complex X with a single point (a 0-simplex) is isomorphic to the finite simplicial complex X, but I can't see why this is so. i.e. Xx{point} isomorphic to X Thanks

I think this subject may have been discussed here already; please point to the thread if so. I am unable to understand how Black Holes can form within the lifetime of the universe. If nothing can cross an event horizon in finite time, then it seems clear that the horizon can't form (at...

I say it is finite. That is, if we agree that the speed of light is constant irrespective of its surroundings. Let me explain what I mean. A car traveling 60 mph is only doing so relative to its surroundings on earth, such as trees and homes along the roadside. But what if the car had no...

Hello all: I'm new to the world of finite elements/finite differences. I'd like to understand the advantages of the finite element method. I read that the finite difference method cannot handle complex (e.g., curved domains, fractures) geometries. I have had no luck in understanding why...

Hello, I'm currently doing an undergrad project on this topic and I was wondering if any of you guys know what is the fastest algorithm (asymptotically) that has been discovered so far, for such purpose. Here is paper by Shoup (1993) which gave the fastest algorithm up to then...

Hello, I'm currently doing an undergrad project on this topic and I was wondering if any of you guys know what is the fastest algorithm (asymptotically) that has been discovered so far, for such purpose. Here is paper by Shoup (1993) which gave the fastest algorithm up to then...

I was wondering if there was a general way to find the sum of a finite power series: \sum_{n=1}^{N}{n^{m}} where m is a fixed integer. Now, there is some math folklore that a seven- (or ten-)year-old Gauss solved the m=1,\;N=100 case by realizing that by reversing the series and summing...

Homework Statement Hi everyone. This isn't homework.. I've been out of school for almost a decade; but rather this is for an opengl simulator I'm building. The integrals for the E and B fields of a finite line of moving-charge are well known and can be found in textbooks and online--even on...

f uniformly continuous --> finite slope towards infinity Homework Statement Given f:R \rightarrow R uniformly continuous. Show that \limsup_{x\rightarrow \infty} \displaystyle|f(x)|/x<\infty i.e. \exists C \in R: \, |f(x)|\leq C|x| as x \rightarrow \pm \infty. Homework Equations The...

Homework Statement There exists a set of all finite sets? Prove your answer. Homework Equations The Attempt at a Solution 1. Assume there is a set A of all finite sets 2. Take the power of set of A This is where i get stuck...what I'm thinking is that there exists a element...

Homework Statement If A=<1+i> in Z[i], show that Z[i]/A is a finite field and find its order Homework Equations The Attempt at a Solution Not sure where to start... Z[i]/A = {m+ni + A, m, n integers} ? is that right? And I don't know what else to do.

Hey, I want to solve a parabolic PDE with boundry conditions by using FINITE DIFFERENCE METHOD in fortran. (diffusion) See the attachment for the problem The problem is that there is a droplet on a leaf and it is diffusing in the leaf the boundry conditions are dc/dn= 0 at the upper...

Hey, I want to solve a parabolic PDE with boundry conditions by using FINITE DIFFERENCE METHOD in fortran. (diffusion) See the attachment for the problem The problem is that there is a droplet on a leaf and it is diffusing in the leaf the boundry conditions are dc/dn= 0 at the upper...

"finite cartesian product of connected space is connected" hi am not able understand the theorem that.. "finite cartesian product of connected space is connected".. what is a base point? how it is related to homeomorphism? can anyone explian?

So I'm reading a paper which assumes the following statement but I would like to be able to prove it. Let S denote the symmetric group on the natural numbers. If \emptyset\subset A \subset \mathbb{N} then S_{\{A\}}=\{f\in S:af\in a,\;\forall{a}\in A\}$ is a maximal subgroup of S...

After hours searching the web I was not able to find the unidimensional case "a finite barrier inside a finite well". I would appreciate if someone could give me a reference about it.

I am doing BE in mechanical engineering. I'm looking for a comprehensive book on finite element methods preferably with a lot of solved numericals . Suggestions, please?

Homework Statement Let F be a finite field of characteristic p. As such, it is a finite dimensional vector space over Z_p. (a) Prove that the Frobenius morphism T : F -> F, T(a) = a^p is a linear map over Z_p. (b) Prove that the geometric multiplicity of 1 as an eigenvalue of T is 1. (c) Let F...

What is a finite-difference discretization for the partial differential term: \frac{\partial^4\phi}{\partial x^2\partial y^2} Thanks in advance.

Ok, I am a mechanical guy, mechanical engineering degree. With an argument with my Engineering physics wife. 1. Imagine 2 particles (lightest particle with any finite mass) 2. separated by the the entire distance of the universe 3. the particles are attracted (due to gravity) to each other...

Homework Statement Are there countably many rational numbers in the interval (0,1) or are there finitely many? Homework Equations The Attempt at a Solution I am confused. There are countably many rational numbers in the interval (0,1). Does this mean I can list them all in such a...

Homework Statement prove that every finite group has exponent that divides the order of the group Homework Equations The Attempt at a Solution Given G is a finite group and x \in G. Suppose x has order m, then <x> = {e,x,x^2...x^(m-1)} and so \left|<x>\right| = m so by...

i was told that high emergy gamma rays had a finite range in water. how can this be if they are both massless and chargeless. is it because they are absorbed/produce electrons/positrons thanks

Homework Statement Solve for the allowed energy values E of a finite square quantum well of depth U0 = 25eV, width a = 0.5nm that contains an electron of mass m (I'm presuming that m = 9.11*10^-31kg, the question doesn't indicate a specific value to use). I'm defining the interior potential to...

Hi, I am trying the solve the Poisson equation in a domain with curved boundaries using the Finite Difference Method (second order accurate). I need to apply the Neumann condition on the curved boundary. I have used bilinear interpolation to do this but this causes the resultant scheme to be...

Homework Statement The problem is to find a well's depth Vo that the electron which is trapped inside has two stable states. Well starts at x=0 and ends at x=L. Homework Equations The Attempt at a Solution I tried to solve Schrödinger equation for each area (x<0 0<x<L x>L) but...

Hi All, I want help to solve a problem: A point charge is placed at a distance 'L' from the surface of an infinite dielectric slab (permittivity: k1 and thickness 'd'). The entire region of space (outside the slab has a permittivity k2). First, I need to solve for the energy of the system...

Homework Statement Find an example of a martingale (X_n, (\mathcal{F}_n)_{n=0}^\infty, P) such that P(X_n = a \mbox{ } i.o.) = 1 for a = -1, 0, 1 and sup_n X_n < \infty . (i.o. = infinitely often) Homework Equations We must have sup_n X_n < \infty . The Attempt at a Solution...

Hello, I am preparing for a screening exam and I'm trying to figure out some old problems that I have been given. Given: Suppose \mu is a finite Borel measure on R, and define f(x)=\int\frac{d\mu(y)}{\sqrt{\left|x-y\right|}} Prove f(x) is finite almost everywhere If I integrate I...

This is a problem I encountered in Martin Isaacs' 'Finite Group Theory'. It's located at the end of Chapter II which deals with subnormality, and the particular paragraph is concerned with a couple of not so well-known results which I quote for reference: (In what follows F is the Fitting...

Homework Statement Hi guys Is there a difference between finite element methods and finite volume methods? Or do they both rely on each other? Niles.

i am a 3rd year civil engineering student, currently taking a course in which i need to use finite element software for assignments. the course is 100% theory, and i need to learn how to use the software on my own, i am looking for FREE software with FE capabilities, and preferably one that...