Symmetric Definition and 539 Threads

I am currently doing a past paper question for my electromagnetism exam and I can't seem to figure out this problem, it is probably quite simple but I can't see a solutionHomework Statement Consider a spherically symmetric charge distribution: ρ(r) = ρ0(r/r0)-n for r>r0 ρ(r) = ρ0 for r≤r0...

Homework Statement I am working through MacLane/Birkhoff's Algebra, and in the section on Symmetric and Alternating groups, the last few exercises deal with generators and Defining relations for Sn (the symmetric group of degree n). These read: 11. Prove that Sn is generated by the cycles (1...

Homework Statement Let a = (a1a2..ak) and b = (c1c2..ck) be disjoint cycles in Sn. Prove that ab = ba. The Attempt at a Solution Sn consists of the permutations of the elements of T where T = {1,2,3,...,n} so assume we take an i from T. Then either i is in a, i is in b, or i is in...

λHomework Statement A symmetric confocal resonator with mirror spacing d =16 cm, mirror reflectances 0.995, and n = 1 is used in a laser operating at λ[o] = 1 μm. (a) Find the radii of curvature of the mirrors. (b) Find the waist of the (0,0) (Gaussian) mode. (c) Sketch the intensity...

So, I was studying coupled oscillations and came across a statement that I couldn't figure out. It was that a particular matrix was symmetrical by Newton's Third Law. I know what Newton's Third Law is, I know what symmetric matrix is. But, for example, a matrix like this: -2k/m...

Hello. I just wonder why the energy expression of Hartree Fock method is symmetric. I tried to find out the reason on the Internet but I could only find that: since the Hartree Fock energy expression is symmetric, it is variational. In Hartree Fock method, the repulsive energy between...

I am really stuck with how to prove that the only element in Sn (with n>=3) commuting with all the other elements of this group is the identity permutation id. I have no idea what I am supposed to do with it, i know why S3 has only one element that commutes but i don't know how to prove it...

Hello. I just wonder why the energy expression of Hartree Fock method is symmetric. I tried to find out the reason on the Internet but I could only find that: since the Hartree Fock energy expression is symmetric, it is variational. In Hartree Fock method, the repulsive energy between...

Let \begin{equation*} A=% \begin{bmatrix} 0 & 1 & \cdots & n-1 & n \\ 1 & 0 & \cdots & n-2 & n-1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ n-1 & n-2 & \cdots & 0 & 1 \\ n& n-2 & \cdots & 1 & 0% \end{bmatrix}% \end{equation*}. How can you prove that det(A)=[(-1)^n][n][2^(n-1)]? Thanks.

How to prove that the determinant of a symmetric matrix with the main diagonal elements zero and all other elements positive is not zero and different ?

Okay, I read that in the case of no air-resistance, projectile motion is symmetric; that the initial velocity will equal the final velocity, in magnitude; and that a projectile traveling upwards, achieving a zero velocity of the vertical component, will have to fall the same horizontal distance...

Homework Statement Consider a spherically symmetric charge distribution \rho = \rho (r) Homework Equations By dividing the charge distribution into spherical shells, find the potential \phi and the electric field strength \bf{E} in terms of \rho (r) The Attempt at a Solution The...

In order to prove my PDE system is well-posed, I need to show that if a matrix is diagonalizable and has only real eigenvalues, then it's symmetric. Homework Equations I've found theorems that relate orthogonally diagonalizable and symmetric matrices, but is that sufficient? The...

Hi all! I have a discrete 2D vector field with a particular characteristic: At every point, instead of having a single vector, I have two vectors which are in the opposite direction. For example, at point p(x,y)=p(0,0) I have two vectors: v1(1,1) and v2(-1,-1). And so on for all points. I...

Homework Statement I am on the final part of a question and I have to prove that the following is a irreflexive symmetric relation over A or if it is not then give a counter example. R is given as an irreflexive symmetric relation over A. Relation: {(X, Y) | X ⊆ A ∧ Y ⊆ A ∧ ∀x ∈ X.∀y ∈...

Homework Statement Find the vector, parametric and symmetric equations of a line that intersect both line 1 and line 2 at 90°. line 1: x = 4 + 2t y = 8 + 3t z = -1 - 4t line 2: x = 7 - 6t y = 2+ t z = -1 + 2t Homework Equations not sure. I am not asking for the answer...

Indicate if the following relation on the given set is reflexive, symmetric, transitive on a given set. R where (x,y)R(z,w) iff x+z≤y+w on the set ℝxℝ. It is reflexive because any real number can make x+z=y+w. It is not symmetric because if x+z≤y+w it's not possible for x+z≥y+w. It is...

Indicate which of the following relations on the given sets are reflexive on a given set, which are symmetric and which are transitive. {(x,y)\inZxZ: x+y=10} Tell me if I'm thinking about this correctly It is not reflexive because the only 5R5. It is symmetric because any xRy and yRx where...

Homework Statement Suppose g:(0, +∞) → ℝ is continuous, and consider F:ℝd\{0} → ℝd, where F(x) = xg(|x|). Prove F is conservative. Homework Equations F is conservative iff there exists a C1 function f:ℝd\{0} → ℝd, s.t. F = grad(f). (edit: Or is the codomain of f actually ℝ, so that it's a...

Hello My question is about the ground state of vibrations for a solid. I'm working with graphite and have found out that for k=0 (The Gamma symmetry point), the vibrational modes can be decomposed into irreducible represenations in the following way Vibration = 2 * E1u + 2 * E2g + 2 * A2u...

Why is it that our universe isn't perfectly symmetric? To demonstrate imagine that the 0 is the center of the universe: abc 0 cba Why doesn't the universe look like this? Why isn't there another me on the equally opposite side of the universe doing the same thing as I am right now?

Homework Statement Determine the orders of all the elements for the symmetric group on 3 symbols S3. Homework Equations _______________________________________ The Attempt at a Solution 3 symbols : e,a,b I don't know how to do the S3 table using just these 3 letters I can do...

If you consider the permutation representation of Sn in ℂ^n, i.e the transformation which takes a permutation into the operator which uses it to permute the coordinates of a vector, then of course the subspace such that every coordinate of the vector is the same is invariant under the...

Homework Statement Let G=S_6 acting in the natural way on the set X = \{1,2,3,4,5,6\}. (a)(i) By fixing 2 points in X, or otherwise, identify a copy of S_4 inside G. (ii) Using the fact that S_4 contains a subgroup of order 8, find a subgroup of order 16 in G. (b) Find a copy of S_3...

Homework Statement A) Let A be a symmetric, irreducible, tridiagonal matrix. Show that A cannot have a multiple eigenvalue. B) Let A be an upper Hessenberg matrix with all its subdiagonal elements non-zero. Assume A has a multiple eigenvalue. Show that there can only be one eigenvector...

Homework Statement What proportion of 2x2 symmetric matrices with entries belonging to [0, 1] have a positive determinant? Homework Equations A^{T} = A If A = [[a, b], [c, d]] Then det(A) = ad - bc. But A is symmetric, so c = b. So det(A) = ad - b^2 So, in order for A to have a...

Homework Statement What is the normalizer of the Sylow p-subgroup in the symmetric group Sym(p) generated by the element (1,2,...,p) where p is a prime number? Thanks Homework Equations na The Attempt at a Solution I know that the normalizer has order p(p-1). And I know that it has...

A and B are two symmetric matrices that satisfy: AB = - BA Which one of these statements are always true: a. (A-B)^2 is symmetric b. AB^2 is symmetric c. AB is invertable I tried to think of an example for such matrices, but couldn't even find 1...there must be a logical way to solve it...

Homework Statement Show that for a spherically symmetric potential \int _{all space} V(\vec{r})exp(i\vec{k}\cdot\vec{r})d\tau = \frac{4\pi}{r}\int_{0}^{\infty} V(r) sin(\kappa r)dr The Attempt at a Solution Given that the potential is spherically symmetric we have azimuthal symmetry...

Hello everyone, When we speak about the SU(2)L group (in electroweak interactions for example), about what group do we talk ? What is the difference with the SU(2) group ? And with the SU(2)R ? Why is the label so important ? I ask this because I see that a Lagrangien can be invariant...

First of all, I apologize if this is in the wrong place. I didn't really know where it should be placed and if it is in the wrong place I am sorry. This question was on my recent Linear Algebra I final exam and I had no idea how to do it when I was writing the exam and I'm still stumped by...

Homework Statement What is the easiest way of finding the symmetrical of a point relative to an arbitrary line? (I was asked on an exam to find the symmetrical of a point relative to the line y = x, but that's rather trivial - just switch the coordinates. How can I do it for any arbitrary...

Homework Statement How many elements of order 4 are in S6? (symmetric group with order 6) Homework Equations The Attempt at a Solution So, the different forms of elements with order 4 in S6 are (abcd)(ef), (abcd) from there I am sunk on how to calculate. I know there are...

Homework Statement If the graph of a differentiable function f is symmertic about the line x=a, what can you say about the symmetry of the graph f'? Homework Equations The Attempt at a Solution

Homework Statement Let A = {1,2,3,4}. Give an example of a relation on A that is symmetric and transitive, but not reflexive. Homework Equations Symmetric: if aRb then bRa Transitive: if aRb and bRc then aRc Reflexive: aRa for all a in A The Attempt at a Solution {(1,2),(2,1),(1,1)}...

Hi, I have currently been thinking about laser-plasma interaction and I have a simple model in mind. I am going to look for a cylindrically symmetric solution of a cylindrically symmetric laser beam (of radius R) hits a initially charge neutral plasma creating an electron beam in the plasma...

Hi everyone. My first post on this great forum, keep up all the good ideas. Apologies if this is in the wrong section and for any lack of appropriate jargon in my post. I am not a mathematician. I have a theory / lemma which I would like your feedback on:- Take a finite set S of integers which...

Hi guys, I have a bit of a strange problem. I had to prove that the space of symmetric matrices is a vector space. That's easy enough, I considered all nxn matrices vector spaces and showed that symmetric matrices are a subspace. (through proving sums and scalars) However, then I was asked...

Homework Statement I need to diagonalize the matrix A= 1 2 3 2 5 7 3 7 11 The Attempt at a Solution Subtracting λI and taking the determinant, the characteristic polynomial is λ3 - 17λ2 + 9λ - 1 (I have checked this over and over) The problem now is it has some ugly roots, none that I would...

Hello, Why do symmetric wave function has less energy than the anti symmetric wave function and how does it connect to the number of the nodes (why existence of a node point in the anti symmetric tells us that this is more energetic function?)

This is not a homework question, just a question that popped into my head over the weekend. My apologies if this is silly, but would you say that the symmetric group S4 is a subset of S5? My friends and I are having a debate about this. One argument by analogy is that we consider the set...

Hello all! I just had a question about combining elements of matrices. In the MATLAB documentation, there was a function called triu and tril that extracts the upper and lower components of a matrix, respectively. I was wondering if there was a way to copy the elements of the upper triangle...

Homework Statement "A gradient of a vector field is symmetric if and only if this vector field is a gradient of a function" Pure Strain Deformations of Surfaces Marek L. Szwabowicz J Elasticity (2008) 92:255–275 DOI 10.1007/s10659-008-9161-5 f=5x^3+3xy-15y^3 So the gradient of this function...

Discrete Mathematics -- Symmetric Closure Math help in Numerical Analysis, Systems of I can't seem to find the way to approach this problem. Because it has symbols I don't know how to type here, I have attached an image here instead. Please help me if you can. Any input would be greatly...

Homework Statement My question is how do you use the symmetric equation. For instance I have a question that states: A line L has parametric equations x=4+3t, y=3+4t, z=9-4t. Determine whether or not the points given lie on the line L. points (17, 14, -9). Homework Equations I know that...

Homework Statement aij is a symmetric matrix bij is a an anti symmetric matrix prove that aij * bij = 0 Homework Equations aij * bij The Attempt at a Solution any one got any ideas ?

Homework Statement Find parametric equations and symmetric equations for the line through the points (0,1/2,1) and (2,1,-3) Homework Equations The Attempt at a Solution I started out graphing the points and connecting them with a straight line. I called the first point P...

Hi, All: I am trying to see how to classify all symmetric bilinear forms B on R^3 as a V.Space over the reals. My idea is to use the standard basis for R^3 , then use the matrix representation M =x^T.M.y . Then, since M is, by assumption, symmetric, we can diagonalize M...

Just as the title says, suppose X is a symmetric manifold and \hat{S}(x) is the linear operator associated to \sigma_x\in G for some unitary irreducible representation, where \sigma_x is the group element that performs reflections around x (remember X=G/H for H\subset G). Now take the...

Homework Statement Relation which is reflexive only and not transitive or symmetric? Homework Equations No equations just definitions. The Attempt at a Solution I can find a relation for the other combinations of these 3 however, I cannot find one for this particular combination...