What is multiplicity: Definition and 78 Discussions
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root.
The notion of multiplicity is important to be able to count correctly without specifying exceptions (for example, double roots counted twice). Hence the expression, "counted with multiplicity".
If multiplicity is ignored, this may be emphasized by counting the number of distinct elements, as in "the number of distinct roots". However, whenever a set (as opposed to multiset) is formed, multiplicity is automatically ignored, without requiring use of the term "distinct".
Homework Statement
Consider a hypothetical crystal A consisting of 5 atoms, each of which has only two states, with energies 0 and ε. A microstate is described by the distribution of energy among the individual atoms; a macrostate is described by the total energy of the crystal. For...
This is probably a simple thing to do but it is driving me up the wall.
Say I had a box with M slots, and there are N particles inside this box, and each slot can hold, at most, 1 particle. (where N is less than or equal to M).
I am trying to calculate the multiplicity of this box system by...
Homework Statement
The energy eigenvalues of an s-dimensional harmonic oscillator is:
\epsilon_j = (j+\frac{s}{2})\hbar\omega
show that the jth energy level has multiplicity \frac{(j + s - 1)!}{j!(s - 1)!}
Homework Equations
partition function: Z = \Sigma e^{-(...
I was wondering how the general formula for the solutions of an nth order linear homogeneous ODE that had a characteristic equation which could be factored to (x-a)^n was derived(IE a set of solutions consisting e^(mx), x*e^(mx), ...x^(n-1)e^(mx)))?
For example the ODE,
y^(3)- 3y'' + 3y'...
Homework Statement
Find h in the matrix A such that the eigenspace for lambda=5 is two-dimensional.
A= [5,-2,6,-1] [0,3,h,0] [0,0,5,4] [0,0,0,1]
A-lambda*I(n) = [0,-2,6,-1] [0,-2,h,0] [0,0,0,4] [0,0,0,-4]
Homework Equations
The Attempt at a Solution
I'm not really sure how to...
Homework Statement
I have a matrix A [1 -1 -1 -1; -1 1 -1 -1; -1 -1 1 -1; -1 -1 -1 1], its characteristic polynomial p(t) = (t + 2)(t-2)3, and given value of lambda = 2. I need to find basis for eigenspace, and determine algebraic and geometric multiplicities of labmda.Homework Equations
The...
Homework Statement
Prove that if two matrices are similar then they have the same eigenvalues with the same algebraic and geometric multiplicity.
Homework Equations
Matrices A,B are similar if A = C\breve{}BC for some invertible C (and C inverse is denoted C\breve{} because I tried for a...
On the multiplicity of the eigenvalue
Dear friends,
Might you tell me any hint on the multiplicity of the max eigenvalue, i.e., one, of the following matrix.
1 0 0 0 0
p21 0 p23 0 0
0 p32 0 p34 0...
Homework Statement
Given matrix A:
a 1 1 ... 1
1 a 1 ... 1
1 1 a ... 1
.. . .. ... 1
1 1 1 ... a
Show there is an eigenvalue of A whose geometric multiplicity is n-1. Express its value in terms of a.
Homework Equations
general eigenvalue/vector equations
The Attempt at a Solution
My...
I am somewhat confused about this property of an eigenvalue when A is a symmetric matrix, I will state it exactly as it was presented to me.
"Properties of the eigenvalue when A is symmetric.
If an eigenvalue \lambda has multiplicity k, there will be k (repeated k times),
orthogonal...
Homework Statement
Let f has a zero of multiplicity k at 0.
Find the residue of f'/f at 0
The Attempt at a Solution
I'm kind of get stuck on this one. I got only this far: Since f has a zero of multiplicity k at 0, then f(z) = (z^k)g(z) :(
Thanks a lot for helping!
Homework Statement
Use the equation ln(q+N) to derived an equation similar to the equation, omega(N,q)=e^(N*ln(q/N))*e^(N)=(eq/N)^N only when q >> N, for a multiplicity of an einstein solid in the "low temperature" limit , q<<N
Homework Equations
ln(q+N)
ln...
Homework Statement
(a)
The formula for the multiplicity of an Einstein solid in the “high-temperature” limit,
q >> N, was derived in one of the lectures. Use the same methods to show that the multiplicity of an Einstein solid in the “low-temperature” limit, q << N, is
Ω(N,q)=(eN/q)^q...
Homework Statement
Consider a monatomic ideal gas that lives in a two-dimensional universe (“flatland”), occupying an area A instead of a volume V.
(a)
By following the logic of the derivation for the three-dimensional case, show that the multiplicity of this gas can be written...
Homework Statement
For C atom (2 electrons in 2p orbital) I get L=0,1 or 2 and S=0 or 1: so spin multiplicity/terms (in braket generacy of states) are 1S(1), 3S(3), 1P(3), 3P(9), 1D(5), 3D(15)...for a total of 36 states. I am sure I am overestimating because in reality those should be...
Consider an isolated system of two ideal, identical gases in thermal equilibrium. Gas A is occupying a volume A, separated by a wall from volume B, where gas B resides. They are in thermal contact. The volumes are the same, so are the pressures and temperatures, and it follows that the number of...
Homework Statement
Suppose A and B are similar matrices, and that (mu) is an eigenvalue of A. We know that (mu) is also an eigenvalue of B, with the same algebraic multiplicity(proved in class) Suppose that g is the geometric multiplicity of (mu), as an eigenvalue of B. Show that (mu) has...
Does the multiplicity of zero of characteristic polynomial restrict from above the possible dimension of the corresponding eigenspace?
For example if we have a 3x3 matrix A, and a characteristic polynomial
\textrm{det}(\lambda - A)=\lambda^2(\lambda - 1)
I can see that the eigenspace...
Can someone please explain multiplicity to me? I've been able to solve the problems involving it, but I'm not quite sure what it means in terms of the eigenvalue. Thanks.
1. What will be the multiplicity due to spin-spin splitting of the highlighted protons in the molecule: (C6H5)-CH2-CH2-CH2-OH?
Homework Equations
None
The Attempt at a Solution
I know that the ones surrounding the highlighted CH2 group are not equivalent. If they were, the...
Homework Statement
In stat physics, the multiplicity function is discrete but to find the max value, you can assume it is continuous around the max region hence use calculus. Why is it legimate to do that? That is approximate a discrete function as continous?
The Attempt at a Solution
Is...
I'm studying for a linear algebra final, and I'm looking over an old final our prof gave us and I've come across something I don't remember ever hearing anything about... Here's the problem:
Write down a matrix A for the following condition:
A is a 3x3 matrix with lambda=4 with algebraic...
In order to get a "feeling" of the sharpness of the multiplicity function, for a system of 2 solids (A & B) with N quantum oscillators in each, in the high-temeperature limit (that is the total number of energy units q is much larger than N), it is approximated as a Gaussian as [Shroeder, eq...
Hi all, I have a homework problem that I would like someone to check:
this relates to the eigenvectors: in the problem we are given characteristic polynomial, where I put x instead of lambda:
p(x) = x^2*(x+5)^3*(x -7)^5
Also given A is a square matrix, and then these questions (my answers)...
I'm reading a chapter on recurrence relations and they have a problem with the phrase: ...if r is a root with multiplicity 2. What does it mean for a root to have multiplicity? This is the first time I've heard of this and the book assumes I would know what they mean already.
Consider an ideal monatomic gas that lives in a two-dimensional universe ("flatland"), occupying an area A instead of a volume V. Find a formula for the multiplicity of this gas.
I arrived at this formula. Is it correct?:
1/N!(A^N/h3N)(pi^3N/2/3N/2)!(sqrt(2mU))^3N