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".
I ran across the following problem :
Statement:
Consider a gas of ## N ## fermions and suppose that each energy level ## \varepsilon_n## has a multiplicity of ## g_n = (n+1)^2 ##. What is the Fermi energy and the average energy of this gas when ## N \rightarrow \infty## ?
My attempt:
The...
Let ##P(x,y)## be a multivariable polynomial equation given by
$$P(x,y)=52+50x^{2}-20x(1+12y)+8y(31+61y)+(1+2y)(-120+124+488y)=0,$$
which is zero at ##q=\left(-1, -\frac{1}{2}\right)##. That is to say,
$$ P(q)=P\left(-1, -\frac{1}{2}\right)=0.$$
My doubts relie on the multiplicity of this point...
Hello! Can we have molecular transitions (not restricted to electric dipole) that change the multiplicity of the electronic level i.e. ##2S+1##. For electric dipole that is strictly forbidden. For magnetic dipole, we have a term in the operator of the form ##S\cdot B## and assuming the B is...
The peaks all look unfamiliar, no symmetry, or very little, and they all integrate to a small number of hydrogen while being surrounded by a large number.
Here are the images.1.)
2.)
3.)
4.)
5.)
Thanks for any explanations. I don't expect anyone to do it for me, I just have no other...
I was doing a question that wanted you to determine the structure given a molecular formula and an NMR spectrum. The following was the answer:
I'm unsure as to how the multiplicities were obtained. For ##X##, the neighbouring carbon ##Y## has 2 hydrogens, so this peak is a triplet (OK so far!)...
Question:
The multiplicity of an ideal gas is given by g(U) = A.U3N/2, where U is the energy of the gas, A is a constant and N is the number of particles in the gas.
Prove that the energy of the gas given a temperature T is U = (3/2).N.kb.T
Attempts:
My first thought was to...
Hello,
Few years back I was reading about calculating multiplicity using hypercube (n-cube). Multiplicity was normalized using this method. I wanted to read it again but I just cannot find it now. I tried every combination of keywords. I remember it was a Wikipedia link. Any help would be...
I'm trying to understand why the intersection multiplicity of two singular subvarieties is not equal to the complex dimension of the local ring but it is instead the Euler characteristic.
Is it possible to find an intuitive explanation?
I think that the following concepts need some...
Please have a look at the attached images.I am attempting a proof for the statement : The algebraic multiplicity of an eigen value λ is equal to dim null [T - λ I] dim V.
Please advise me on how to move ahead. Apparently, I am at the final inference required for a proof but unable to move...
For a map between vector bundles (which commute with a certain Lie groups like Sl2R or GL2R), what does it mean exactly for a fiber to be multiplicity free?
Eplanations would be good, but examples would be even better. Thanks in advance, Gauss bless you!
CM
Homework Statement
Consider drawing one card from a deck with no jokers or other special cards.
a) What is the number of microstates? (4/13/52/cant tell)
b) What is the number of macrostates?(4/13/52/cant tell)
c) What is the degeneracy of macrostate spade? (4/13/52/cant tell)
d) What is the...
Hi guys. Consider the problem of calculating the multiplicity (phase space volume) of N hard sphere gases each of whose center of mass is confined to a volume V. The spheres themselves have volume ##\omega## and do not interact with one another in equilibrium time scales. Then ##\Omega \propto...
Define the multiplicity of $f$ at $p$ and the interesction multiplicity of $f,g$ at $p$.
Let curves $A$ and $B$ be defined by $x^2-3x+y^2=0$ and $x^2-6x+10y^2=0$. Find the intersection multiplicities of all points of intersection of $A$ and $B$.
If we let $f=x^2-3x+y^2$ and $g=x^2-6x+10y^2$...
Hi. I am trying to understand the multiplicity function Ω!/L!(Ω-L)! where Ω= number of boxes and L= number of distinguishable boxes. I just want a simple intuitive explanation. I have seen a couple of these but none of them ever stick. The term that confuses me the most is the (Ω-L)!
Any help...
I am trying to prove the spectral decomposition theorem for normal compact operators. Now, my book says the space H is the closure of the direct sum of $F_{t}$ where we the $F_{t}$ are eigenspaces and we sum over all eigenvalues $t$.
My question concerns what happens when there are 2 linearly...
Hi, I was looking for a proof or explanation of this. From Schroeder's Thermal Physics, pg 56, explaining interacting systems in equilibrium.
The example in the text is two 3-harmonic oscillators with a total of 6 units of energy. So one macrostate is where each has 3 units of energy. The...
Homework Statement
Consider the multiplicity of a classical gas of N non-interacting molecules (not necessarily monatomic). Since they don't interact,their positions are not correlated, so the multiplicity of each will be simply proportional to the volume, with the result that the total...
I only have a doubt about which definition to use for the multiplicity of a two state system. Clearly the total multiplicity of a two state system is given by:
Ω=2^N,
but what about the definition:
Ω(N,n) = \binom{N}{n} = \frac{N!}{n!\cdot(N-n)!}.
Clearly:
2^N ≠...
Homework Statement
A function f has a simple zero (or zero of multiplicity 1) at x0 if f is differentiable in a neighborhood of x0 and f(x0) = 0 while f(x0) ≠ 0.
Prove that f has a simple zero at x0 iff f(x) = g(x)(x - x0), where g is continuous at x0 and differentiable in a deleted...
If two matrices similar to one another are diagonalizable, then certainly this is the case, since the algebraic multiplicity of any eigenvalue they share must be equal (since they are similar), and since they are diagonalizable, those algebraic multiplicities must equal the geometric...
In my last Linear Algebra class we saw Eigenvalues and Diagonalizations. It turns out that an n x n matrix is diagonalizable if its eigenbasis has n linearly independent vectors.
If the characteristic equation for the matrix is (λ - λ_1)^{e_1}(λ - λ_2)^{e_2}...(λ - λ_k)^{e_k} = 0 then 1)...
after finding out what geometric multiplicity was, I was surprised to notice that in every question I'd done it was always 1.
So I'm trying to prove an example with g.m. > 1 to see why it works.
I've found a matrix which definitely has an eigenvalue with g.m. = 2. I've checked everything with...
Could someone please explain to me (with an example if possible) what is the Geometric Multiplicity of Eigenvalues? I cannot understand it from what I have read on the web till now.
Thanks in advance.
A function f has a root of multiplicity $m>1$ at the point $ x_*$ if $f(x_*)=f'(x_*)=...=f^{(m-1)}(x_*)=0$. Assume that the iteration$ x_{k+1}=x_k-mf(x_k)/f'(x_k)$ converges to $x_*$. If$ f^{(m)}(x_*)≠0$, prove that this sequence converges quadratically.(We may use the Taylor's series, but I...
Homework Statement
I'm given this matrice 2 1 0
1 2 0
0 0 3
and I need to find it's eigenvectors
Homework Equations
The Attempt at a Solution
So I get the eigenvalues to be 1,3,3 with 3 being the one with...
Homework Statement
Prove without induction that Multiplicity of an eigen value , k = dim[ Null(T - k I)^( dim V) ]
Homework Equations
[(T - k I)^dim V ] v =0
[Thoughts]
i understand that normal eigen vectors with same eigen values may not be
linearly independent.
[(T - k...
Homework Statement
A particle can exist in three microstates, with energies E0 < E1 < E2. Consider N >> 1 such particles, fixed on a lattice. There are now n0 particles with energy E0, n1 particles with energy E1 and n2 = N - n0 - n1 particles with energy E2. We have that n_j >> 1 for j = 0...
Homework Statement
Let \vec{x} and \vec{v} be vectors in \mathbb{R}^3.
If A is a matrix such that A\vec{x} gives the projection of \vec{x} onto \vec{v}, then what are the eigenvalues of A and what are their algebraic multiplicities?
Homework Equations
Eigenvalue: A real number λ is an...
Homework Statement
In the book "Friendly introduction to analysis, 2nd Ed." by kosmala there is a definition of the root of a function and subsequent theorem and proof. Either the proof is not directly addressing certain important properties, or is flawed. The definition and theorem are as...
Hello Everybody.
I have a rather simple question, which still kept me thinking for two hours without any result.
If we want to determine the multiplicity in the microcanonical ensemble we just divide the volume of the shell containing the accessible microstates over the volume of one...
Hi. I've got a theoretical doubt: why is the geometric multiplicity more or equal to 1?
Couldn't happen that the eigenspace is the null vector?
Thanks!
Say I have a 4x4 matrix and I know 3 eigenvalues and the 3 corresponding eigenvectors.
Is there a fast way to calculate which one has multiplicity 2 without calculating the characteristic polynomial(too time consuming for a 4x4 matrix) or without determining the dimensions of (A-λ I) for each...
http://dl.dropbox.com/u/33103477/question.png
I have determined the eigenvalues which are -2, 2 and 1 respectively.
I'm pretty sure that the one with multiplicity of 2 is the, the eigenvalue = 2 as it occur's twice in the diagnol. But I don't think that's a concrete enough reason.
Any...
Homework Statement
The matrix A = \begin {bmatrix} 0 & -1 & 0 \\ 0 & -1 & 0 \\ 0 & 1 & 0 \end{bmatrix} has two real eigenvalues, one of multiplicity 2 and one of multiplicity 1. Find the eigenvalues and a basis of each eigenspace.Homework Equations
N/AThe Attempt at a Solution
I've done...
hi friends please help in finding out the ans for this . A 3x3 matrix was given , am asked to find algebraic multiplicity of it ! how to find algebraic multiplicity of 3x3 matrix ??
Okay, I am not sure if this is the right subforum, but here goes:
An einstein solid is a solid composed of N quantum harmonic oscillators, which can store evenly spaced energy units q.
Now suppose we have an einstein solid of N oscillators with q energy units, where q>>N. My book wants to...
Homework Statement
We have a surface that can adsorb identical atoms. There are N possible adsorption positions on this surface and only 1 atom can adsorb on each of those. An adsorbed atom is bound to the surface with negative energy -\epsilon (so \epsilon > 0). The adsorption positions are...
Can the natural logarithm in the statistical mechanics formulation of entropy , S = k ln W be based on both multiplicity and an integral ? The extensive property and multiplicity explanation : That for any given macrostate , the total entropy of two interacting
systems is the sum of their...
Homework Statement
A 20 × 20 matrix C has characteristic polynomial (λ^2 − 4)^10. It is given that ker(C−2I), ker (C − 2I)^2, ker (C −2I)^3 and ker (C −2I)^4 have dimensions 3,6,8,10 respectively. It is given that ker (C + 2I), ker (C +2I)^2, ker (C +2I)^3 and ker (C +2I)^4 have di-
mensions...
I didn't really know where the proper place was for this, but this is an intro thermodynamics class and I'm really confused over this math question (it's not strictly physics-related).
Homework Statement
Consider rolling N six-sided dice. Define a microstate as the number showing on...
Homework Statement
\lambda=0 is an eigenvalue of
A=
|1 1 1 1 1|
|1 1 1 1 1|
|1 1 1 1 1|
|1 1 1 1 1|
|1 1 1 1 1|
Homework Equations
Find the geometric multiplicity of \lambda=0 as an eigenvalue of A
The Attempt at a Solution
I row reduced it then got the last four rows of all 0s...
Homework Statement
For a single large two-state paramagnet, the multiplicity function is very sharply peaked about N_{\uparrow} = N/2.
(a) Use Stirling's approximation to estimate the height of the peak in the multiplicity function. (I am fairly confident in my answer here)
(b) Use the...
Homework Statement
Let A = a 2x2 matrix row 1 [3/2 ½ ] row 2 [-½ ½]
a) Find the eigenvalue(s) of A
b) Find the algebraic multiplicity of each eigenvalue.
c)Find the geometric multiplicity of each eigenvalue.
Homework Equations
The Attempt at a Solution
I found the...
Homework Statement
I am having trouble finding an expression for W (multiplicity) for a chain that can move in all possible directions (3 dimensions)
Homework Equations
Multiplicity is the number of possible states over total states.
The Attempt at a Solution\
I understand for a...