Relation Definition and 1000 Threads

Homework Statement suppose f~:~A \rightarrow B be a surjective map of sets. Prove that the relation a Rb \iff f(a)=f(b) is a equivalence relation whose equivalence classes are the fibers of f. Homework Equations The Attempt at a Solution I was able to easily prove that the...

Let's say we have a sealed container that has an adjustable volume, so the volume of the container can change from big to very small. When the volume is big, the container is filled with a certain amount of an ideal gas. After this the amount of gas is not changed but remains constant. Also we...

I'm learning ray optics and feeling so confused by the definition of "Hamiltonian of light". What I learned was that the "Hamiltonian of light" defined by H = n-|\vec{p}| = 0 indicates the momentum conservation, where n is refractive index and \vec{p} here is the canonical momentum. The...

Hey! (Wave) I have to find the primes $p$,for which $x^2 \equiv 13 \pmod p$ has a solution. That's what I have tried: $$\text{ Let } p>13:$$ We want that $\displaystyle{ \left ( \frac{13}{p} \right )}=1$ $$\left ( \frac{13}{p} \right )=(-1)^{\frac{13-1}{2} \cdot \frac{p-1}{2}}\left (...

'Slow-active suspensions realize small switching frequencies to control low-frequency body movements,such as roll pitch and lifting motions.Fully-active suspensions reach switching frequencies,like semi-active suspensions,greater than the natural eigen-frequencies of the vehicle.'- an excerpt...

Can anyone help explain this to me and solve this problem? I have gone over my textbook and I am having trouble understanding this. (Doh) Find the domain, range, and when A=B, the diagraph of the relation R. A={1,2,3,4,8}=B; a R b if and only if a|b. A.Domain {1,2,3,4,8} Range {1,4,6,9,15}...

Suppose we are given two functions: f:\mathbb R \times \mathbb C \rightarrow\mathbb C g:\mathbb R \times \mathbb C \rightarrow\mathbb C and the equation relating the Stieltjes Integrals \int_a^\infty f(x,z)d\sigma(x)=\int_a^\infty g(x,z)d\rho(x) where a is some real number, the...

Hey! :o I am looking at an example of the characteristic system of hyperbolic equations. One part of the example is the following: $\displaystyle{v=\text{ constant }, v=u_1+\sqrt{\frac{a}{b}}u_2}$, when $\displaystyle{\frac{dx}{dt}=\sqrt{ab}}$ $\displaystyle{x=\sqrt{ab}t+c \Rightarrow c=x-...

Hey there, This isn't a homework question, it's for deeper understanding. So I'm learning about unit normal/tangent vectors and the curvature of a curve. I have a few questions/points. 1) So my book states that we can express acceleration as a linear combination of the acceleration in the...

Do individual photons have some attributes which relate to EM wave frequency? In other words, is there any difference in photons composing a red and blue beam of light?

I'd like to know what exactly it's telling us. Does it mean that the more accurately we measure the energy of a system the less accurately we know for how long the system has been in that range of energies? Or does it mean that the more accurately energy is measured the less accurately we know...

So what if we have two thick lenses? Thanks in advance ,

Hello guys, Iam doing a project to find the tension of timing belt using a dial gauge indicator(consists of plunger and gear attached to it ie;rack and pinion mechanism) ,but i can't find a formula relating the displacement of the pointer and the tension (eg: 5° of dislplacement of pointer =...

Hi every one, Here is my question: In soil physics, knowing the relation between suction head, h, and the soil water content, S, one can derive the hydraulic conductivity, K, of that soil using a formula like: (ignore the superscripts "cap") where in my problem, τ=0.5, κ=1, β=2...

I'm in the first of 3 courses in quantum mechanics, and we just started chapter 4 of Griffiths. He goes into great detail in most of the solution of the radial equation, except for one part: translating the recursion relation into a form that matches the definition of the Laguerre polynomials...

(1) P_{l}(u) is normalised such that P_{l}(1) = 1. Find P_{0}(u) and P_{2}(u) We have the recursion relation: a_{n+2} = \frac{n(n+1) - l(l+1)}{(n+2)(n+1)}a_{n} I'm going to include a second similar question, which I'm hoping is solved in a similar way, so I can relate it to the above...

Homework Statement We say that two sets A and B have the "same powerfulness" if there is a bijection from A to B. Show that the relation "have the same powerfulness" is an equivalence relation between sets. Homework Equations An equivalence relation satisfy the following: xRx...

Homework Statement Estimate the magnitude of the fine structure splitting in H-α in THzHomework Equations Rydberg -- R_y \left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right) = \Delta E The Attempt at a Solution This isn't really a request for solution help, and more a justification. I know that if...

Hey! :o Having the following problem: $$(1): u_t=u_{xx}+f(x,t), 0<x<L, t>0$$ $$u(0,t)=u(L,t)=0, t>0$$ $$u(x,0)=0, 0<x<L$$ $$f(0,t)=f(L,t)=0, t>0$$ we do the following to find the general solution: We write the function $f(x,t)$ as a Fourier series: $$(2): f(x,t)=\sum_{n=1}^{\infty}{F_n(t)...

Hi all, Could someone please explain to me the process involved in converting an inhomogeneous recurrence to a homogeneous recurrence, I'm completely confused as to how it works.Thanks

Problem: Let $a_0$ and $b_0$ be any two positive integers. Define $a_n$, $b_n$ for $n\geq 1$ using the relations $a_n=a_{n-1}+2b_{n-1}$, $b_n=a_{n-1}+b_{n-1}$ and let $c_n=\dfrac{a_n}{b_n}$, for $n=0,1,2,\cdots $. Write a)Write $(\sqrt{2}-c_{n+1})$ in terms of $(\sqrt{2}-c_n)$. b)Show that...

My question is just to ask whether the operations like:- AUB is a relation or not? in our book it is written that the relations of two sets should be subset of the cartesian product of two sets but i think that relations are those which connects two sets and that can be AUB(A union B)...

We all know that quantum theory is based on the commutation relation and superposition principle. The trouble haunting me long time is that how to "get" the famous commutation relation? Could anybody give me an explanation?

What is the relation between sfc vs rpm in diesel engine and why does it happen like that ? what is relation between air flow vs rpm ??

Homework Statement I'm given a recursive sequence with the following initial terms: ##\begin{matrix} f_0(0)=1&&&f_1(0)=0\\ f_0(1)=2&&&f_1(1)=1 \end{matrix}## Now, I'm asked to justify that we have the following recursive relations: ##\begin{cases} f_0(n)=2f_0(n-1)+f_1(n-1)\\...

What is the Decay Width of a particle and how is it related to the particle's lifetime ?

Homework Statement I want to integrate \int_{0}^{a} xsin\frac{\pi x}{a}sin\frac{\pi x}{a}dxHomework Equations I have the orthogonality relation: \int_{0}^{a} sin\frac{n\pi x}{a}sin\frac{m\pi x}{a}dx = \begin{cases} \frac{a}{2} &\mbox{if } n = m; \\ 0 & \mbox{otherwise.} \end{cases} and...

This thread does justice to a question put forth online several times and, as far as I can tell is only answered in part. I believe this question warrants a distinct and succinct answer. What I'm finding online is summarized below, and as one can see... there is something missing. I've been...

Dear Friends, I carried out an experiment of sudden release of oxygen (open nozzle) from an oxygen cylinder used for medical college and hospitals. I found that pressure drops quite rapidly and cylinder surface cools from outside such that water droplets accumulate on its surface. This...

Homework Statement The Attempt at a Solution I know what relations are individually but what do I do to represent the composition of both? Is it some matrix operation? Would I multiply them, but instead of adding I use the boolean sum?

Does atoms in a crystal have energy states like electronic states? it is according to Einstein relation for harmonic oscillator.E=nhw

Hi guys, I was thinking about the relativistic effects a little bit, and I have a question regarding relative simultaneity. Time dilation and length contraction grow as a function of the speed of the observer, and become noticeable and large on speeds close to the speed of light. By this...

We let C be the set of Cauchy sequences in \mathbb{Q} and define a relation \sim on C by (x_i) \sim (y_i) if and only if \lim_{n\to \infty}|x_n - y_n| = 0. Show that \sim is an equivalence relation on C. We were given a hint to use subsequences, but I don't think they are really necessary...

Digging in the wiki, I found this relation between 'arc-functions' and 'arc-functions-hyperbolics" \\ arcsinh(x)= i \arcsin(-ix) \\ arccosh(x)= i \arccos(+ix) \\ arctanh(x)= i \arctan(-ix) https://it.wikipedia.org/wiki/Funzioni_iperboliche#Funzioni_iperboliche_di_argomento_complesso...

First post here. This question has two parts. (1) Connecting the dots between the Impulse-Momentum Theorem and the Law of Conservation of Momentum and (2) Book recommendations for a more theoretical treatment of classical mechanics? (1) Difficulty reconciling the Impulse-Momentum Theorem...

Hi, I have a few questions regarding the experimental outcome of the stern-gerlach experiment. Let's suppose the following setup: We have a magnetic field whose field-lines point towards the positive z axis and the intensity of that field becomes stronger towards the positive z axis, so there...

Homework Statement a.) Is symmetric and transitive, but not reflective: b.) consists of exactly 8 ordered pairs and is symmetric and transitive: The Attempt at a Solution If the question asks me to define some relation, do I need to define some math property like power of some number or...

Under the effect of an electric and magnetic field the momentum in the Hamiltonian becomes the canonical momentum, p-qA where p is the linear momentum and A is the vector potential so H=(1/2m)(p-qA)^2 + qV where V is the scalar potential. I am trying to find [H,(p-qA)]. My main question arises...

Homework Statement Find ##[L_i,L_j]##. Homework Equations [x_i,p_j] = \delta_{ij}i\hbar The Attempt at a Solution [L_i,L_j] = \epsilon_{ijk}\epsilon_{jlm} [x_jp_k,x_lp_m] = \left( \delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl}\right)[x_jp_k,x_lp_m] = [x_jp_k,x_jp_k] - [x_jp_k,p_kx_j] =...

How to solve this question. Please explain step by step.

I asked this question at AskMrPhysics and received no response so...either it's a really stupid idea/question that doesn't deserve to be answered or no one can help me with a response. I'm hoping that someone here will either let me down gently and tell me to stop asking dumb questions or help...

I have a question about the derivation of the formula for relation between Specific Conductance and Equivalent Conductance i.e. Eq. Conductance = k.V where, k= Specific Conductance ,V=Volume in ml Given link explains the derivation...

Hey again! :p Let $f:[0,+ \infty) \to \mathbb{R}$ strictly increasing and continuous at $[0,+\infty), f(0)=a$(I am not sure,if it is $a$,it could also be $0$ (Blush)) and let $\lim_{x \to +\infty} f(x)=+\infty$.The range of $f$ is $[0,+\infty)$ and the inverse function $f^{-1}:[0,+\infty) \to...

[a, a^{+(n)}] = na^{+(n-1)} 1) What's the name of this relation if it has any? 2) I tried to prove this by induction, I started by saying that for n=1, this holds since [a, a^{+}] = 1 (as we all know and as we can all prove) then I assumed it true for (n-1), but I didn't go too far...

Homework Statement Using the Debye dispersion approximation, calculate the heat capacity of a harmonic, monatomic, 1D lattice. Next, find the temperature dependence in the low temperature limit. (Assume that the longitudinal mode has spring constant CL = C, and the two transverse modes both...

We know how to find S_{x} and S_{y} if we used S_{+} and S_{-}, and after finding S_{x} and S_{y}, we can prove that [S_{x}, S_{y}]= i\hbarS_{z} (Equation 1) and [S_{y}, S_{z}]= i\hbarS_{x} (Equation 2) and [S_{z}, S_{x}]= i\hbarS_{y} (Equation 3) but can we, starting from Equations 1...

According to the theory, E= -dv/dx or E.dx = -dv So if both are positive, the potential drop should increase. But as we know, if a positive charge is placed, as the distance from it keeps on increasing, field strength starts decreasing and potential drop should increase But this is...

Hi, It is a well known fact that in an inverse linear problem low condition numbers have low noise amplification and therefore decrease the error. So I wanted to test this: I draw random (skinny) matrices A, calculate y=A*c where c is a known coefficient vector, add some noise and...

Hi, I have a transitive relation and wana build a complete set of pairs that reflect all (direct/indirect) relations among the pairs. Ex.: suppose I have this relation R = { (1,2), (2,3), (3,5), (5,7), (3,4) } I wana to produce this relation R oper R = { (1,2), (1,3), (1,4), (1,5)...

Homework Statement Evaluate the following series ∑u(n) for n=1 → \infty in which u(n) is not known explicitly but is given in terms of a recurrence relation. You should stop the summation when u(n) < 10^(-8) u(n+1) = (u(n-1))^2 + (u(n))2 with u(1) = 0.5, u(2) = 0.6 Note 1:The lecturer...