Relations Definition and 579 Threads

Hello, all, the most important results that I know in this topic is the Gauss-Bonnet Theorem (and hence the classification of compact orientable surfaces) and also the Poincare-Hopf index theorem. But there are still some fundamental problems I don't understand. For example, is the...

How is this accomplished? How can one derive equations for stress in terms of strain from equations of strain in terms of stress or vice versa?

Hi, If we have an orthonormal basis, how can we show that the relation \sum|x><x| = Identity? I see this in Quantum Mechanics but I'm not sure how to prove it. Thank you.

Homework Statement Use the reciprocity relations and known transforms to compute the Fourier Transform of the given function. f(x)=\frac{1}{1+x^{2}} Homework Equations With the help of the table of Fourier transforms, write the given functions as F(f). The Attempt at a Solution...

Hey, I have had a lot of trouble understanding how one obtains a Maxwell relation. So let's say in general I know(from a specific problem) T ds = dE - F dL where F is a tension and L is a length, E is the energy T is the temperature and S is the entropy of a system. In a specific...

Hi, All: I am given two groups G,G', and their respective presentations: G=<g1,..,gn| R1,..,Rm> ; G'=<g1,..,gn| R1,..,Rm, R_(m+1),...,Rj > i.e., every relation in G is a relation in G', and they both have the same generating set. Does this relation (as a...

Homework Statement Let Z be the set of all integers. Then, S is a relation on the set Z x Z defined by: for (a1, a2), (b1, b2) belong to Z x Z, (a1, a2)S(b1, b2) <-> a1b2 = a2b1. Homework Equations The Attempt at a Solution The actual problem is about symmetry...

Homework Statement The relation R on the set of all groups defined by HRK if and only if H is a subgroup of K is an equivalence relation. Homework Equations Subgroup: has identity, closed under * binary relation, has inverse for each element. Equivalence relation: transitive, symmetric...

1. I am trying to practice solving recurrence relations but get stuck when it comes to generalizing the last part of them and would be grateful if someone could offer some help. I'm not very good with series which is why I may be having some problems with them. Here are a few examples if its...

\textup{A 2 x 7 rectangle has tiling with 1 x 1 and 1 x 2 tiles (singletons and doubletons).} \textup{How many such tilings of a 2 x 7 grid are there?} \textup{Let }a_{n}\textup{ be the number of tilings of a 2 x n grid using 1 x 1 and 1 x 2 tiles so that the} \textup{two rightmost squares...

Homework Statement Consider the following binary relations on the naturals (non-negative integers). Which ones are reflexive? Symmetric? Anti-symmetric? Transitive? Partial orders? a) A(x,y) true if and only if y is even b) B(x,y) true if and only if x < y c) C(x,y) true if and only...

I try to calculate refractive index of experimental spectral data using Kramers-Kronig relations but didn’t succeed. I need your expert advice and help to solve this problem. Data and expression for KK relation is give in worksheet, where alpha(omwga) is in cm-1. Solution through Matlab or...

Hello guys, I am new to this forum. I have a question: A relation can be subset of some other relation? For example? I have the relations X: A <---> B Y: B <---> C Z: A <---> C X...

I don't understand how a left (right) Green relation is a right (left) congruence. xLy <=> Sx = Sy (Green's Left Relation): where we join 1 to S if it doesn't have identity. Left Congruence: aPb ==> caPcb for some c in the semigroup S. Take this example table: * a|b|c a|a|b|c...

Hi, I'm struggling about with binary relations in sets. Can somebody check over and answer my questions about these sets: Given set A = {1,2,3} Provide one example each of a relation with the following properties where the cardinality of the relationship should be at least one in all...

If you have a Grassman number \eta that anticommutes with the creation and annihilation operators, then is the expression: <0|\eta|0> well defined? Because you can write this as: <1|a^{\dagger} \eta a|1>=-<1| \eta a^{\dagger} a|1> =-<1|\eta|1> But if \eta is a constant, then...

1. Let R be a relation on X that satisfies a) for all a in X, (a,a) is in R b) for a,b,c in X, if (a,b) and (b,c) in R, then (c,a) in R. Show that R is an equivalence relation. 2. In order for R to be an equivalence relation, the following must be true: 1) for all a in X, (a,a) is...

Where, on the internet, can one learn the method to solving equations modulus a number? I'd like to learn the method for finding such relations as this special case for the Erdos-Straus Conjecture, with n ≡ 2 (mod 3). Also, what is the technical name for finding the mod n relations in an...

First this is my first attempt at using latex to ask a question, so my appologies if the statements come out strange. I'll edit as needed. Homework Statement Let R and S be relations on a set A. Prove that if R \subseteq S, then R^{n} \subseteq S^{n} for all n \geq 1 Homework Equations...

Homework Statement A = {0, 1, 2, 3, 4 ,5} Let R be a binary relation on set A such that: R = {(0,1), (1,0), (1,3), (2,2), 2,1), 2,5), (4,4)} a. Make a Directed Graph for the relation R on A b. What must be added to R to make it reflexive/symmetric?

In Srednicki's book, he discusses quantizing a non-interacting spin-0 field \phi(x) by defining the KG Lagrangian, and then using it to derive the canonical conjugate momentum \pi(x) = \dot{\phi}(x). Then, he states that, by analogy with normal QM, the commutation relations between these fields...

Homework Statement I'm solving the quantum harmonic oscillator. And I'm solving Schrodinger equation. So I came up to one part where I have to use power series method of solving DE (that or Frobenius would probably work just fine). Now I have the recurrence relation...

Homework Statement [PLAIN]http://img530.imageshack.us/img530/6672/linn.jpg The Attempt at a Solution For parts (a) and (b) I've found the eigenvalues to be -\frac{1}{3} and -1 with corresponding eigenvectors \begin{bmatrix} -1 \\ 3 \end{bmatrix} and \begin{bmatrix} -1 \\ 1...

Homework Statement let A be any set of numbers and let R and S be relations on A. if S and R are symmetric then show S o R is symmetric. if S and R are antisymmetric then show S o R is antisymmetric. if S and R are transitive then show S o R is transitive. if S and R are...

Homework Statement For each of the relations on the set R x R - (0,0) (ie. no origin) : - prove it is an equivalence - give the # of equivalence cases - give a geometric interpretation of the equivalence cases assuming an element of R x R is a point on a plane a) {((a,b),(c,d)) |...

Hello all! I'm currently studying for a physics exam in regards to Newtonian mechanics. This chapter is about angular momentum, moments of inertia, etc. As I was studying, something dawned on me, and I wondered if someone here might be able to help me. The following relations are...

I know how to solve second order ones, but how would you solve third order ones? Because the characteristic polynomial would have a third degree so how can one find the roots? I have looked everywhere online to find out but I can't find anything. Please Please tell me!

Would be really grateful if someone helped me with 5a, and explained the ideas behind recursion relations. I don't know what a recursion relation is, and how to apply for forum ales given Homework Statement http://img839.imageshack.us/img839/2301/recurrencerelation.gif...

Homework Statement Provide an example that shows why the reflexive property is not redundant in determining whether a relation is an equivalence relation or not. For example, why can't you just say, "If xRy then yRx by symmetric property, and then using transitive property you get xRx."...

Homework Statement Let X = {a,b,c,d}. How many different equivalence relations are there on X? What subset of XxX corresponds to the relation whose equivalence classes are {a,c},{b,d} Homework Equations N/A The Attempt at a Solution So I wrote out all the possible "blocks"...

Homework Statement Assuming V is a function of P and T such that V = V(P,T) express the differential changes in volume due to differential changes in Temperature and pressure, what is the fractional/relative change? Homework Equations The Attempt at a Solution since V is a...

Maxwell Relations - derivations Homework Statement 1. Derive the Maxwell Relation based on the enthalpy. 2. Derive the Maxwell Relation based on the entropy. Homework Equations H=U+PV dU=dq+dw dw=-PdV dS=dq/T The Attempt at a Solution 1. I feel like I've gotten this one, but...

Homework Statement Verify the following commutation relations using \vec J = \vec Q \times \vec p and [Q_{\alpha},p_{\beta}]=i \delta_{\alpha \beta} I 1. [J_{\alpha}, J_{\beta}]=i \epsilon_{\alpha \beta \gamma} J_{\gamma} 2. [J_{\alpha}, p_{\beta}]=i \epsilon_{\alpha \beta \gamma}...

From the wikipedia page on http://en.wikipedia.org/wiki/Complementarity_(physics)" : As I recently noticed in the double-slit experiment there is a lot of time-uncertainty, not only position-momentum uncertainty. That is shown in the fact that the probability wave does not reach the screen...

Homework Statement Solve the Mass-Spring-Damper Differential equation mx''+bx'+kx=exp(-t)cos(t) (Where x'' is d2x/dt2 etc, don't know how to do the dots above :confused:) I understand how to solve this problem, but the thing that confuses me is that the right is in terms of "t"...

Homework Statement The set A has 5 elements. 1. How many relations exist on A? 2. How many of those relations are symmetric and reflexive? The Attempt at a Solution Some of the parts of this question are harder than others. 1. By simple counting, there are 2^(5^2) or 2^25 total relations...

Homework Statement I am not an expertize at algebra, and recently I was considering derivisions of equations based on postulates already known to physics. I just wanted a little guidence if any of the equations are inconsistent, which I am sure probably quite a few are, knowing my own...

My professor did this in lecture, and I can't figure out his logic. Can someone fill in the gaps? He went from: dS = \left( \frac{\partial S}{\partial P} \right)_T dP + \left( \frac{\partial S}{\partial T} \right)_P dT (which I totally understand; it just follows from the fact that...

I have two questions: i) Does a distinct equivalence relation on a set produce only one possible partition of that set? ii) Can multiple (distinct) equivalence relations on a set produce the same partition of that set? In other words, given a set S and two distinct equivalence relations ~...

The relation to be proved looks pretty simple. However, there is no evident point from whih I can start. How do I start relating quantities that are under root? The cyclic order of the sides on the left side is reminiscent of the cosine rule, but that's just it! Plus the right hand side...

Homework Statement We propose to study simple processes of liquid alcohol. At T_1, the molar volume of alcohol is V_1 and its molar heat capacity at constant pressure is C_p,m. We assume that its isobaric coefficent of thermal expansion a, and the isothermal compressibilty coefficient B are...

Hello My first post here and I don't know if this is the right place but I'm trying to get a grip on form factors when it comes to deep inelastic scattering. My tool is Gordon Kane's Modern Elementary Particle Physics. Exercises 5 and 6 chapter 18 seems like good exercises but I need som help...

Homework Statement Given is the set X. The set of functions from X to [0,1] we call Fun(X,[0,1]). On this set we consider the relation R. An ordered pair (f,g) belongs to R when f^{-1}(0)\setminus g^{-1}(0) is a countable set. a) Prove that R is transitive. b) Is R an equivalence relation...

Can we always calculate the commutation relations of two observables? If so, what’s the commutator of P (momentum) and H (Hamiltonian) in infinite square well, considering that the momentum is not a conserved quantity?

Really struggling with this question guys, any help/advice will be greatly appreciated. Thanks in advance! Homework Statement Sketch dispersion relations for, (i) a proton (ii) a free electron (iii) an electron, in a crystalline solid. Comment on the similarities between (ii) and...

I mentioned in a thread on here almost a year ago now that I was considering making a change into math, after entertaining several other options and figuring out that I really enjoy math, and especially probabilty, I have decided to to move toward getting either another bachelor or a masters in...

Homework Statement Show that {x_i, x_j} = 2*y_ij* I for i = 1; 2; 3 and j = 1; 2; 3. where y_ij: N x N -> {0,1}, such that y_ij = {1, if i = j ; 0, if i not = j Homework Equations The Attempt at a Solution I'm confused about exactly what I'm supposed to do here. Do i do all the...

I have a complicated recursion replation, which I'm sure is unsolvable. (By "unsolvable" I mean that there is no closed form solution expressing \xi_1, \xi_2, \xi_3, etc. in terms of \xi_0.) It goes \frac{(k+4)!}{k!}\xi_{k+4} +K_1 (k+2)(k+1)\xi_{k+2}+ [ K_2 k(k-1) +K_3] \xi_{k} +K_4...

I'm really puzzled about this one. Say you have a discrete, nonnegative random variable N where the probability pn = P{N=n} satisfies the recurrence relation p_{n+2} + r p_{n+1} + s p_n = 0 for n = 0, 1, 2, ...; p0 and p1 are given. How do you find the expectation E[N] without solving...

I have a quick question about the relationship between the complex Fourier coefficient,\alpha_n and the real Fourier coefficients, a_n and b_n. Given a real-valued function, I could just find the real coefficients and plug them into the relation below, right?Fourier Coefficients for periodic...