representation

1. Difficulty finding power series representation of a function

As the title says, I'm having a lot of trouble finding a power series representation of a function. Here it is: And here's my attempt at a solution: I had a lot of trouble trying to reindex the summations such that I'd have a single summation at the end. This is what I ultimately got...
2. Matrix representation by function

I have one matrix (103x6) with 5 inputs (real numbers) and one output (output can take integer values from 0 to 4 ). I need to represent matrix in form of function. Any ideas?
3. Matrix representation of multimodal function

I have matrix (5 inputs and 1 output)-I need to represent it via multimodal function but I don't have idea how to do it. Thank you very much for help,
4. Representation for A4

Can you give me some hint about how to represent alternating group 4? I know we can define { a b c d } to four 3-cycle and then find the relation between the four 3-cycles. But it is really troublesome. Another thought is: since S4 is isomorphic to the 4*4 permutation matrix. We can use...
5. Do the cosets of a subgroup always form a representation of the group?

G = {g1, g2, g3, g4, g5, g6, g7, E} Lets say the cosets of a subgroup H1 of G are: {g1, g2} {g3, g4} {g5, g6} {g7, E} so G = { {g1, g2}, {g3, g4}, {g5, g6}, {g7, E} } = { a, b, c, d} so the representation { a, b, c, d} has order 4 and the subgroup H1 has 4 cosets My question is...
6. Diagonalization of the matrix representation of D(p(x))=(p'(X)).

"P is the space of polynomials degree 3 or less. Find the eigenvalues for the representation matrix of the linear transformation corresponding the derivative, D: P -> P. Is it diagonalizable? " I solved for the representation matrix: 0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 With the base...
7. Proof involving diagonal matrix representation with respect to some basis

Let $V$ be $n$ dimensional vector space, $T:V\rightarrow V$ a linear map, and let $B=\{b_1,b_2,...,b_n\}$ be a basis for that vector space. Suppose $[T]_B$ (the matrix representation of $T$ with respect to $B$) is diagonal. I'm trying to prove that $T(b_i)=a_{ii}b_i$, where $a_{ii}$ is the...
8. Find matrix elements and expansion for operator in position representation

In 1-D let TL be an operator defined on the position eigenstates |x> such that TL|x>=|x+L>. Find the matrix elements TL(x,x')=<x'|TL|x> and construct an explicit expansion for this operator in the position representation. Show that “in the position representation” <x|T_L|\psi >= \psi (x-L)...
9. Why is this series representation wrong?

Hello, I am asked to represent \int \frac{e^x}{x}dx as a power series and I proceeded as follows: We know: e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} So, \frac{e^x}{x}=\sum_{n=0}^{\infty} \frac{x^n}{x\cdot n!} = \sum_{n=0}^{\infty} \frac{x^{n-1}}{n!} Then, \int \frac{e^x}{x}dx = \int \left...
10. Exercise on representation of free groups

Hi to everyone, I had to deal with an exercise on representation of free groups. I tried to solve it, but I'm not sure. Could you help, please? Below you can find the exercise and its solution. Let be $G$ a group which representation has $n$ generators and $s$ relations. Show that if $s<n$...
11. Help on parametric representation

Hi all! Can anyone help me with the following question on parametric representation: A circle of radius r is placed at the top of a ramp of negative gradient and angle θ = π/6 rad. At t = 0 the circle is at rest with its centre at coordinates (0, r) and then it starts to rotate clockwise...
12. Representation of p-adic integers using primitive roots

Let p be an odd prime and let r be any positive integer that is a primitive root module p^2. Let X_p = \varprojlim \Bbb{Z} / (p-1)p^n \Bbb{Z} be the inverse limit of the rings \Bbb{Z} / (p-1)p^n \Bbb{Z}. This is useful because for any unit u \in \Bbb{Z}/p^n\Bbb{Z}, there exists K \in \Bbb{Z} /...
13. IF Statement Representation

Apologies, if I have put this in the wrong section but I don't know enough about Maths to know what kind of question it is! I need to represent an IF statement formula in an alternative way as the software I am using doesn't allow IF statements. It does however allow all of these functions...
14. Representation of a given polynomial in factorial notation

Hi, Represent f(x)=$2x^4 -12x^3 +24x^2 - 30x +9$ and its successive differences in factorial notation Solution:Let f(x) =$2x^4 - 12x^3 +24x^2 -30x +9$ =$2x^{(2)} +bx^{(3)} +cx^{(2)} +dx +9$ =2x(x-1)(x-2)(x-3) +bx(x-1)(x-2) +cx(x-1) +dx +9 where b,c and d are...
15. Permutation representation argument validity

Hello, I would like to check if the work I have done for this problem is valid and accurate. Any input would be appreciated. Thank you. Problem statement: Let G be a group of order 150. Let H be a subgroup of G of order 25. Consider the action of G on G/H by left multiplication...
16. Representation of a group acting on finite sets?

Hi, I'm just doing a course in the Representation of Finite Groups, and in the notes, we are given the following definition of a representation which is not actually given a name: Let G be a finite group, M a finite set, and let G act on M. For F some field we can construct a representation of...
17. ODEs, Adjoint forms and matrix representation

Hi MHF members, I was looking up in the books I have but I could not go anywhere. Consider the higher-order ode L[y]:=y^{(n)}+\sum_{k=0}^{n-1}p_{n-k}y^{(k)}=0, which has the matrix form \pmb{Y}^{\prime}+\pmb{A}\pmb{Y}=\pmb{0}, where \pmb{Y}=\left( \begin{array}{c} y \\ y^{\prime} \\ \vdots \\...
18. Power Series Representation

I understand fully part a and b. But for part c they just plug in 1 , why is that? Why didnt they plug in 2 or 3 because those are between -5 and 5? Then how do they calculate the sum by plugging in 1? Thanks
19. Power Series Representation

I need help finding the power series representation of ln(root(4-x2)). I have gathered that I pull out the root 4, so I'll have ln(2)+something, but I'm not sure where to go from there. I know that ln(1-x) is -xk/k, but I don't know how to manipulate it as far as the root part.
20. Need help for a proof, related to decimal representation of a rational number

The problem appears in Tom Apostol's Calculus, Volume 1, pages 31,32. This is introductory material, related to foundations of real number system. Quote: No proof is given and as often happens to me, contrary to the claim, I could not do the thing easily. So I need a proof that sup(S)=x. For...