Relations Definition and 540 Threads

The definition of these relations as given in my textbook are : (1):- Reflexive :- A relation ##R : A \to A## is called reflexive if ##(a, a) \in R, \color{red}{\forall} a \in A## (2):- Symmetric :- A relation ##R : A \to A## is called symmetric if ##(a_1, a_2) \in R \implies (a_2, a_1) \in R...

Maxwell field commutation relations I'm looking at Aitchison and Hey's QFT book. I see in Chapter 7, (pp. 191-192), they write down the canonical momentum for the Maxwell field A^\mu(x): \pi^0=\partial_\mu A^\mu \\ \pi^i=-\dot{A}^i+\partial^i A^0 and then write down the commutation...

Homework Statement Firstly, I'm looking at this: I'm confused because my understanding is that the commutator should be treated like so: $$[a,a^{\dagger}] = aa^{\dagger} - a^{\dagger}a$$ but the working in the above image looks like it only goes as far as $$aa^{\dagger}$$ This surely...

Hi everyone. I wasn't sure where to post this thread, so I figured I'll post this under General Physics. Out of interest, I've been perusing online about connections that exist between statistical physics and theoretical computer science. For example, consider the following report by Pietro...

I find it difficult to believe that the canonical commutation relations for a complex scalar field are of the form ##[\phi(t,\vec{x}),\pi^{*}(t,\vec{y})]=i\delta^{(3)}(\vec{x}-\vec{y})## ##[\phi^{*}(t,\vec{x}),\pi(t,\vec{y})]=i\delta^{(3)}(\vec{x}-\vec{y})## This seems to imply that the two...

I'd like a comparison of how various CAD programs handle the task of creating relationships among objects that have been created independently and where the uses wants to change some parameters of one object and have the program adjust the parameters of the others automatically. I'm interested...

So the question I am trying to solve is this: Define a binary relation R on R as follows: R={(x,y)∈ R×R:cos⁡(x)=cos⁡(y)} Prove that R is an equivalence relation, and determine its equivalence classes. I've figured out the first two requirements for being a binary relation: 1. cos(x) =...

Is there good survey of known algorithms for solving recurrence relations ? By "solving" a recurrence relation such as a_n = \sum_{i=1}^{k} { c_k a_{n-k}} , I mean to express a_n as a function of n . In the case that the c_i are constants the algorithm based on the "characteristic...

Hello every one . A relation ( is a subset of the cartesian product between Xand Y) in math between two sets has spatial types 1-left unique ( injective) 2- right unique ( functional ) 3- left total 4- right total (surjective) May question is 1- a function ( map...

Hi All, The equation: ## v = \lambda f ## is presented as a dispersion relation (DR) for it is a formula that specifies the velocity of a wave of certain frequency. This equation seems to be the most famous DR in physics. My questions are the following: What is the second most famous DR? Which...

A new paper on arXiv today claims that relationsism allows one to evolve the universe through the big bang. Alas I am not familiar with relationsism, is it related to shape dynamics? can anyone explain? https://arxiv.org/pdf/1607.02460v1.pdf

Say you have set A with n elements and set B with m elements. If I recall, there are a total of 2nm relations between them. But my question is, does this count redundancies? What I mean is, if in the relation A~B = B~A. I don't want to count identical relations twice. Thanks!

Homework Statement Let X be a set and R ⊂ X × X. Assume R is an equivalence relation and a function. Prove that R = I_X, the identity function. Homework EquationsThe Attempt at a Solution Proof We know that R has to be reflexive, so for all elements b in X, bRb but b can't be related to any...

I am reading through a quantum optics book where they are deriving the equations for a quantized EM field and one of the paragraphs state: "In Section 6.1, the problem has been set in the Hamiltonian form by expressing the total energy (6.55) of the system comprising charges and electromagnetic...

Homework Statement Let X = {1, 2, 3, 4, 5, 6}. Determine the number of relations on X which are reflexive and anti-symmetric Homework EquationsThe Attempt at a Solution This problem looks a little bit hard. Approach: consider R={(x,x),... } If there is just one pair in the relation in the...

In another thread I quoted a paper Bill pointed me to. It included the statement "It is the measurement results that fluctuate, not the underlying object." Bill indicated that this was a misconception but would need a new thread to discuss it. So please discuss... Thanks Andrew

From my humble (physicist) mathematics training, I have a vague notion of what a Hilbert space actually is mathematically, i.e. an inner product space that is complete, with completeness in this sense heuristically meaning that all possible sequences of elements within this space have a...

Here is the question: By using the equality (for boson) ---------------------------------------- (1) Prove that Background: Currently I'm learning things about second quantization in the book "Advanced Quantum Mechanics"(Franz Schwabl). Given the creation and annihilation operators(), define...

I have a problem where I have to know the commutation relations for left handed fermions. I know ##\psi_L=\frac{1}{2}(1-\gamma^5)\psi## ##\psi^\dagger_L=\psi^\dagger_L\frac{1}{2}(1-\gamma^5)## and ## \left\{ \psi(x) , \psi^\dagger(y)\right\} = \delta(x-y)## So writing ## \left\{P_L\psi(x) ...

Homework Statement Given the mode expansion of the quantum field ##\phi## and the conjugate field one can derive $$\mathbf P = \int \frac{d^3 \mathbf p}{(2\pi)^3 2 \omega(\mathbf p)} \mathbf p a(\mathbf p)^{\dagger} a(\mathbf p)$$ By writing $$e^X = \text{lim}_{n \rightarrow \infty}...

Homework Statement Suppose that A = { 1, 2, 3} , B = { 4, 5, 6} , R = { (1, 4), (1, 5), (2, 5), (3, 6)} , and S = { (4, 5), (4, 6), (5, 4), (6, 6)}. Note that R is a relation from A to B and S is a relation from B to B . Find the following relations: (a) S ◦ R . (b) S ◦ S−1...

Homework Statement This problem is from Lahiri and Pal (2nd edition) Exercise 1.4: Suppose in a system there are operators which obey anticommutation relations ##[a_{r},a^{\dagger}_{s}]_{+}\equiv a_{r}a^{\dagger}_{s}+a^{\dagger}_{s}a_{r}=\delta_{rs}## and ##[a_{r},a_{s}]_{+}=0,## for...

Homework Statement For each of the relations defined on ℚ, either prove that it is an equivalence relation or show which properties it fails. x ~ y whenever xy ∈ Z Homework EquationsThe Attempt at a Solution Here's my problem: I am starting off the proof with the first condition of...

Homework Statement For the set ℤ, define ~ as a ~ b whenever a-b is divisible by 12. You may assume that ~ is an equivalence relation and may also assume that addition and multiplication of equivalence classes is well defined where e define [a]+[ b ] = [a+b] and [a]*[ b ] = [ab] for all [a],[ b...

Homework Statement We have a Gibbs Free Energy function G=G(P, T, N1, N2) I am not writing the whole function because I just want a push in the right direction. Find expressions for the entropy, volume, internal energy, enthalpy and chemical potential. Homework Equations Maxwell Relations...

I'm learning about Maxwell relations of Thermodynamics, but it's difficult for me to find more books about this in Vietnamese. So, I want to ask you about some english ebook about this. Thanks a lot!

I have a question about what I would call a relation inside a relation. Like: A={1,2,3) and B={a,b,c} R1={(a,1) ,(a,3), (b,2), (c,1,), (c,3) } R2={(a,a), (b,a), (b,c), (c,a) } R3=R1R2 Like this. I have 2 regular relations. Then I form another relation using these 2. How do I do that? Like...

I would like to prove that the angular momentum operators ##\vec{J} = \vec{x} \times \vec{p} = \vec{x} \times (-i\vec{\nabla})## can be used to obtain the commutation relations ##[J_{i},J_{j}]=i\epsilon_{ijk}J_{k}##. Something's gone wrong with my proof below. Can you point out the mistake...

Homework Statement My task is to find out what is the lowest # of elements a poset can have with the following characteristics. If such a set exists I should show it and if it doesn't I must prove it. 1) has infimum of all its subsets, but there is a subset with no supremum 2) has two maximal...

The generators ##(A_{ab})_{st}## of the ##so(n)## Lie algebra are given by: ##(A_{ab})_{st} = -i(\delta_{as}\delta_{bt}-\delta_{at}\delta_{bs}) = -i\delta_{s[a}\delta_{b]t}##, where ##a,b## label the number of the generator, and ##s,t## label the matrix element. Now, I need to prove the...

Hello! This post is strictly related to my previous one. Let's consider the same context and the same image. Regarding the oblique incidence of a wave upon an interface between two dielectric, all the texts and all the lectures write an equation like the following: e^{-j k_1 y \sin \theta_i} +...

Hello everyone, I am working on the Onsager reciprocal relations, more precisely on the demonstration of those relations. I try to understand the Onsager original paper (1931) but it's really not easy (although he says that the examples are "extremely simple"). I was wondering if any of you...

Homework Statement Hello, I'm having problems determining the relationships between delta a and delta c . I don't see how how delta C = 4/9 delta A [/B] http://imgur.com/a/B6eTx Thank you.

Homework Statement 2. The attempt at a solution I've tried using the relation Cp = T(dS/dT), isolating "T" for T = Cv2(dT/dS) and using the maxwell relations to reduce the derivatives, reaching, T = Cv2/D (dV/dS), but i don't think this is the right way to do solve this problem, i couldn't...

Reflectance, according to the Fresnel Relations, is given by ##R \equiv \frac{I_r}{I_i}##, and Transmittance is ##T = \frac{I_t \cos \theta_t}{I_i \cos \theta_i}##. Do these values depend on the wavelength of light? For example, if I have a beam of white light rather than a monochromatic...

Is there any relations of S and F? If any, I would like to know what the equation of this relationship will be. Thanks:wink:

Homework Statement I don't have a specific problem I'm trying to solve, I'm trying relate all the concepts for basic thermodynamics. I'm not entirely sure where I am misunderstanding 1. What is work 2. What is internal energy? 3. What is heat? 4. What is enthalpy? 5. What is entropy? Homework...

I'm trying to show that a function defined with the following recurence relations $$\frac{dZ_m(x)}{dx}=\frac{1}{2}(Z_{m-1}-Z_{m+1})$$ and $$\frac{2m}{x}Z_m=Z_{m+1}+Z_{m-1}$$ satisfies the Bessel differential equation $$\frac{d^2}{dx^2}Z_m+\frac{1}{x}\frac{d}{dx}Z_m+(1-\frac{m^2}{x^2})Z_m=0$$...

Where can I find and how can I derive the orthogonality relations for Hankel's functions defined as follows: H^{(1)}_{m}(z) \equiv J_{n}(z) +i Y_{n}(z) H^{(2)}_{m}(z) \equiv J_{n}(z) - i Y_{n}(z) Any help is greatly appreciated. Thanks

I am pulling my hair trying to find a straight answer to this, after looking it up in different books, websites: Say we have a many-to-many relationship in a RDB (Relational DB). Is there a standard way of creating a junction, bridge, etc. table? From what I know,. the bridge table will contain...

Due to the fact that the operators in the canonical commutation relations(CCR) cannot be both bounded, in order to prove the Stone-von Neuman theorem one must resort to the Weyl relations. Now the Weyl relations imply the CCR, but the opposite is not true, the CCR don't imply the Weyl relations...

Homework Statement Consider the Dirac Hamiltonian ##\hat H = c \alpha_i \hat p_i + \beta mc^2## . The operator ##\hat J## is defined as ##\hat J_i = \hat L_i + (\hbar/2) \Sigma_i##, where ##\hat L_i = (r \times p)_i## and ##\Sigma_i = \begin{pmatrix} \sigma_i & 0 \\0 & \sigma_i...

If someone could explain some of the steps needed to work out these 2 questions it would be much appreciated!

In historic face to face, Obama, Castro vow to turn the page http://news.yahoo.com/anticipation-grows-obama-castro-meet-saturday-panama-070657182--politics.html Seems somewhat similar to Nixon in China. About time!

Homework Statement : [/B] I am just learning recurrence relations, and they are proving challenging.Homework Equations -- [/B]Let's have xn = 3xn-1 + 6xn-2. I wanted to look at it with two scenarios. The first is x0 = 1 and x1 = 3. The second is x1=3 and x2 = 4 The Attempt at a Solution Is...

Homework Statement Consider left-handed fermions in two spacetime dimensions ##(t,x)##: ##\psi_L=\frac{1}{2}(1-\gamma_5)\psi_D## with ##J_0^\epsilon(t,x)=\psi_L^+(x+\epsilon)\psi_L(x-\epsilon)##. (a). Use canonical equal-time anti-commutation relations for fermions to compute...

Homework Statement 1. Range of the function ## \sqrt {x^2+x+1} ## is equal to? 2.ƒ:R---->R is defined as ƒ(x) = x2 -3x +4, then f -1 (2) is equal to?Homework Equations NA The Attempt at a Solution For the first one tried squaring on both the sides but that does not give linear x in terms of...

My understanding so far, critique appreciated: [1] workfunction closely relates to reduction potential Since workfunction is about boundaries and chemical reaction are mostly happening at the boundaries between bulk material, Workfunction should have a direct correlation with reduction...

Hello, I've to construct a proof of the following statement: Prove that if S is a set and S_1... S_k is a partition of S, then there is a unique equivalence relation on S that has the S_i as its equivalence classes. I'm really not sure how to go about this proof at all, so any help would be...

Prove that the following is an equivalence relation on the indicated set. Then describe the partition associated with the equivalence relation. 1. In Z, let m~n iff m-n is a multiple of 10.2. The attempt at a solution Reflexive: m-n = 0 0 ∈ Z, and 0 is a multiple of every number...