Explicit Definition and 147 Threads

I've worked that out in my Mathematica notebook "Lie-Algebra Matrices.nb" in directory "Extra Mma notebooks" in SemisimpleLieAlgebras.zip, but I must concede that some of my derivations seem rather inelegant, not using features of the algebras very well. The last comment in How should I show...

Exercise 2.1.5 in Berrick and Keating: An Introduction to Rings and Modules reads as follows: Let M be an abelian group with Mc = 0 for some positive integer c, and put c = ab for coprime integers a,b. Write 1 = ar + bs, and define endomorphisms \alpha and \beta of M by: \alpha (m) = arm...

Let e_i be a unit vector with one 1 in the i-th element. Is the following expression has a recursive presentation? $$y_N = \sum_{k=0}^N {\frac{{{X^k} e_i}}{\|{{{X^k} e_i}\|}_2}} $$ where X is a n \times n square matrix, and {\| \cdot \|}_2 is a vector norm defined as {\|z\|}_2 =...

I've found this paper: [physics/9712033] Closed Expressions for Lie Algebra Invariants and Finite Transformations and I've attempted to interpret it. In some representation, a general Lie-algebra element A = aiLi for generators L (repeated dummy indices summed over, like i here). det(I -...

Homework Statement x*e^(y/x) + y dx = xdy, y(1) = 0 Homework Equations The Attempt at a Solution To solve, I divide everything by x dx to put everything in terms of v. e^v + v = dy/dx dy/dx = x dv/dx + v e^v + v = x dv/dx + v e^v = x dv/dx e^v / dv = x/dx Flip both sides. e^-v dv =...

Hi. I have a little language problem. I'm studying in Germany, and my German is... nicht sehr gut, so I sometimes have problems understanding the exercises. The one I'm having issues right now has a part which says einen höchstens explizit zeitabhängigen Ope-rator I am Schrödingerbild. My...

Hi! Suppose we have two variables Y and Z that depend on a third one, X. We are given P(x), P(y|x) and P(z|x). The joint probability distribution P(x,y,z), according to the chain probability rule, is given by P(x,y,z) = P(x)P(y|x)P(z|x,y) But how can we compute P(z|x,y) with the given...

Accidentally I wrote in the wolfram f(x) = f(1/x) the the wolfram give me the solution for this equation (f(x) = Abs(log(x))). Hummmm, nice! Thus I thought: given the definition of derivative, ##f'(x) = \frac{f(x+dx)-f(x)}{dx}##, is possible to isolate f(x) in this equation? If yes, how? I...

Hello PF. Homework Statement A function f is defined by f(x) = 1 + 2x + x2 + 2x3 + x4 + 2x5 + x6... Find the radius of convergence of the series and the explicit formula for f(x).Homework Equations The Attempt at a Solution I know that the formula for the series is going to be similar to the...

Given a implicit ODE like F(x, y(x), y'(x), y''(x)) = 0, why your explicit form is y''(x) = f(x, y(x), y'(x))? Why a ODE is explicited always with y of higher grade?

$\Bbb{R}P^1$ bundle isomorphic to the Mobius bundle I'm trying to construct an explicit isomorphism from ##E = \{([x], v) : [x] ∈ \Bbb{R}P^1, v ∈ [x]\}## to ##T = [0, 1] × R/ ∼## where ##(0, t) ∼ (1, −t)##. I verified that ##\Bbb{R}P^1## is homeomorphic to ##\Bbb{S}^1## which is homeomorphic to...

Hi! I have encountered a little problem. I want to show that the explicit form of the Feynman propagator for massless scalar fields is given by: \begin{align} G_F(x) & = - \lim_{\epsilon \to +0} \int \dfrac{\mathrm{d}^{4}k}{(2 \pi)^{4}} \dfrac{1}{k^{2} + i \epsilon} \mathrm{e}^{- i k...

Hello, This actually isn't a homework question, more of a curiosity, but since its a book problem I decided to post it here. Homework Statement An object is released from rest at time t=0 and falls through the air, which exerts a resistive force such that the acceleration a of the the...

Solve the given differential equation by separation of variables y\ln{x}\frac{dx}{dy}=(\frac{y+1}{x})^2 I got it down to \ln{x}x^3-\frac{x^3}{3}=y^3+3y+y^2 At this point I had no idea how to solve having y^3 y^2 and y terms so I did what any good student would do and checked the back of the...

I need to find the explicit maximal solution of an IVP using exact Diff Eqs: The IVP is given as: {xexyy'-cos(x)+yexy=0 {y(0)=1 So I know at first I need to get the implicit solution by getting that: A(x,y) = xexy B(x,y) = -cos(x)+yexy I know I need to find the partial derivative of A(x,y)...

Identify "explicit shape spec list" in the FORTRAN code Hi, I'm trying to do some source code analysis using the ROSE compiler infrastructure on some of the kernels in NAS Parallel benchmark. I get an error while parsing the Embarrassingly Parallel kernel. Digging further, I gathered that...

Hello all, I have been given a problem where I am asked to calculate "all" the components of the Riemann tensor in a gross non-diagonalized metric. I know there exists at most 20 independent components of Riemann, but I want to actually compose a list of these combinations. It is easy...

I have an implicit function [e.g. x+\frac{ln(y)}{y^2}=0)]. Is there any mathematical trick/technique that can convert this to an explicit function (i.e. y=f(x)) even within certain restrictions and conditions.

Homework Statement Write an explicit formula for the sequence determined by the following recursion formula. t_{1}= 0; t_{n} = t_{n-1} + \frac{2}{n(n+1)} The Attempt at a Solution t_{1} = 0 t_{2} = t_{1} + \frac{2}{2(2+1)} t_{2} = \frac{1}{3} t_{3} = t_{2} + \frac{2}{3(3+1)} t_{3} =...

Ok this question may be kinda stupid, but here goes. Do any surfaces exist for which a parametric form is possible, but cannot be described explicitly due to their highly irregular shape? (Or vice-versa)

Hey all, I am trying to see that in linearized gravity, choosing the transverse-traceless gauge is actually a valid choice to make. More specifically, I am trying to explicitly show what most textbooks just mention in passing, e.g. Maggiore: Where the \xi^\mu encode the gauge...

Homework Statement The sequence {en} is defined recursively as e0 = 3; ek = 4ek-1 + 7, for all k ≥ 1. Use iteration to make an educated guess at an explicit formula for the sequence. The attempt at a solution I spent all day on this one and I'm still lost. e1 = 4 x e0 + 7 = 4 x 3 + 7...

Homework Statement Consider the sequence n1, n2, n3, ... that satisfies the recurrence relation nk = nk-1 / k + 1 for all integers k ≥ 2 with the initial condition that n1 = 1. Find the explicit formula nk for the nth term of the sequence? 2. The attempt at a solution I calculated...

Let an,m be defined for the non-negative integers n and m such that n ≥ m. an,0 = 1 am,m = m! an+1,m+1 = (m+1) * an,m + an,m+1 Is there an explicit formula f such that f(n,m) = an,m? Here are the first numbers of the sequence: \begin{align} &m&0&1&2&3&4\\ n\\ 0&&1\\...

I want to do an analysis in ANSYS LS-DYNA. After applying element types, materials and loads to my model, I solved the problem. but after finishing the solution all of the deformations, plastic strains and ... are 0. (the applied force is very high and I expect to see rupture in a large area of...

Hi, I am working on an economics paper, and I find that the following first order condition for my variable of interest (\mu \in [0,1]): \frac{\lambda(1+\mu)}{(1-\mu)}=N-x\frac{(1+\mu)}{\sqrt{\mu}} I would ideally like to provide an explicit solution, but unfortunately, this would amount to...

Homework Statement Give an explicit formula for a function f : ℕ ⇒ ℕ that is a) onto, but not one-to-one. b) neither one-to-one nor onto. 1. The attempt at a solution a) The formula f_{2}(n)= ⌊\frac{n}{3}⌋. it's onto cause f_{2}(3n)= n for every n. but, it's not one-to-one, cause...

Homework Statement Consider a sphere centered at origin with radius R > z0. By calculating the total flux ϕ = ∫E . da through the sphere, explicitly show that ϕ = q/ϵ0 Homework Equations Gauss's Law The Attempt at a Solution I have a general idea of what to do, but I just want...

I have an equation (constraint) which I wish to solve explicitly in terms of gd (or more precisely a scaling factor of the vector gd) but I am unsure how to manipulate the equation σi(G) = |uiHgd|-1 Background: (please feel free to skips this. Much of this inforomation is I'm sure...

x = 10logx + 30 (log is log base 10) I cannot get to anything other than this implicit solution. By trial and error I can tell that x must be slightly more than 1/1000 but I would like to get an exact answer.

Hello, This is not a homework exercise, so I decided to post it here. Hopefully one of you could help. I would like to find an explicit expression for (I-A)^(-1), provided that A is a squared matrix (nxn) and A^k = 0. It is also given that I-A^k = (I-A)(I+A+A^2+...+A^(k-1)). I understand that by...

I understand how to find an implicit description if given the span of, say, two vectors. How do I go about finding an explicit description of a plane as the span of two vectors? For example, where would I start if the plane equation was: 3x+2y-z = 0 Thanks!

Homework Statement (y+2)dx +y(x+4)dy=0 Homework Equations The Attempt at a Solution ∫(1/x+4)dx + ∫(y/y+2)dy=0 ln(x+4)-2ln(y+2)+y=lnc Here is where I get confused how do I make this into an explicit solution, that +y really bothers me. I was thinking ln [...

Homework Statement Let X= \{a,b,c\} and Y= \{d,e\}. Write down and explicit bijection N_{|X×Y|} → X×Y The Attempt at a Solution Well I came up with the easiest method, just giving one value to each member of N_{|X×Y|} so I was just wondering whether there is another way of doing it not by...

My friend and I have been getting all confused about the following problem with a Lagrangian. It comes from David Tong's online notes on QFT, but given it is about the Lagrangian, I figure it does well in this section. Ok, Tong is talking about Noether's theorem, and using the example of...

Hi all, I was having a bit difficulty understanding the term scleronomic constraint. From what I have read, it is a type of holonomic system(which means there is time dependence). However, the difference between the two types(scleronomic and rheonomic), is that although scleronomic...

Hi folks. So I've found in multiple places the formula R=R_{0}[1+\alpha(T-T_{0})] with qualifications that for a given material it will only work for certain temperature ranges. However, I've never seen it turned into a differential equation and solved explicitly. It seems like a perfectly...

Homework Statement Consider the following implicit scheme: y_{n+1}=y_{n}+\frac{\Delta t}{2}\left [f(y_{n+1})+f(y_{n})] By linearization one can obtain an explicit scheme which is an approximation to this - with approximation error O(\Delta t^{3}) Homework Equations The solution is...

Hi there, I have the plane in an explicit form: 12x + 2y – 20z = -56 How do I make it in parametric form? Thanks in advance

Hi all, Can anyone advise me on the following...I'm trying to get a more intuitive feel to classical strings and have always found following exercises in textbooks and online lecture notes to be useful. However, I'd like to have a sanity check on some of my solutions and ask for some help with...

Does anyone know when is the Potential explicitly time-dependent? Can you give me an example?

Homework Statement So i think i found the general solutions to both these separable equations, but I am not sure if I am suppose to simplify any further to get it in explicit form, and how i can even do that. Homework Equations The Attempt at a Solution 1. \frac{dy}{dx} -...

I read from a book that the "total energy is not preserved when the potential depends explicitly on time", i.e. U(x,t). Can anyone show or prove it? Many thanks.

The equation is: x2 dy/dx= y - xy IC (initial conditions): y(-1) = -1 (This is used to solve for C) Must first separate the variables x and y and then integrate them and solve for y, but I got stuck... x2dy = (y - xy)dx x2dy = y(1-x)dx dy= y(1-x)dx/x2 <-- not sure what to do with the y now...

I don't really know why, but I'm having trouble actually building deformation retractions, although I understand the concepts behind homotopies, etc.For example, when constructing a deformation retraction for \mathbb{R}^n-\{0\} to S^{n-1}, I found that you could define the mapping F(x,t) =...

Homework Statement I'm trying to comprehend \hat{P}_t^{-1} \left( \vec{\sigma} \cdot \vec{A} \right) \hat{P}_t = \ \cos{\Psi\left(t\right)}\left( \vec{\sigma} \cdot \vec{A} \right) - \sin{\Psi\left(t\right)} \sigma \cdot \left[ \hat{a}\left(t\right) \times \vec{A} \right] +...

Hi, I am trying to write a code for an equation which looks like this: \frac{\partial{y}}{\partial{t}}=f(x,y,t) -y(x^{n+1}-x^{n}) where n is the time step number. I have no idea how I can go on about solving (approximating the solution) to this problem. Any hints would be appreciated...

I would love to know how we can contruct SO(n) matrices. I know we get them from Clifford algebras somehow. I am especially interested in SO(8). Could anybody help? thank you

Hello, I'am new here and happy to find this great forum! Here's my first question: there's an explicit function as follows: y=2.sin(x)-1 The transformation to polar form (r=3cos(\theta)) - x=r.cos(\theta) - y=r.sin(\theta) So I get: r.sin(\theta)=2.sin(r.cos(\theta))-1 Now you see...

When solving differential equations numerically with finite difference methods, textbooks get to the point of solving: A psi_(n+1) = B psi_n (with A, B some matrices, typically complex conjugate of each other) and advise on using LU decomposition to do so. My question is, why not...