Relation Definition and 1000 Threads

Homework Statement Given that the function f can be expanded in a power series of a and a^\dagger, show that: [a,f(a,a^{\dagger})]=\frac{\partial f }{\partial a^\dagger} and that [a,e^{-\alpha a^\dagger a}] = (e^{-\alpha}-1)e^{-\alpha a^{\dagger} a}aThe Attempt at a Solution I've tied using...

Homework Statement Pulsars are stars that have suffered gravitational collapse. They rotate rapidly and emit a narrow beam of radiation. The pulse lengths, at the earth, are ∼1ms and the periods are ∼1s. Within a few months of the discovery of pulsars distance estimates were obtained by...

Hello! (Smile) I want to find the exact solution of the recurrence relation: $T(n)=2T(\sqrt{n})+1$.$$m=\lg n \Rightarrow 2^m=n \\ \ \ \ \ \ \ \ \ 2^{\frac{m}{2}}=\sqrt{n}$$ So we have: $T(2^m)=2T(2^{\frac{m}{2}})+1$ We set $T(2^m)=S(m)$, so we get: $S(m)=2S \left( \frac{m}{2}\right)+1$...

can anyone convert the relation tan y=(Vsin(y)-gx)/Vcos(y) to an explicit function y=f(x) in terms of V, x and g? g is a constant V is the function V(x)= -aln(b/b-cx)-dx a,b,c,and d are also constants. Thanks!

Hello! (Smile) Let $R$ be an order of the set $A$. Then $R$ induces a strict order $S$ at the set $A$. $$$$ Let $S$ be a strict order of the set $A$. Then $S$ induces an order $R$ at the set $A$. The first sentence is proven like that in my notes: We define $S:=R-I_A$ and we can see that...

Homework Statement The radius for the inside of a curve is half the radius for the outside. With 2 cars of equal mass, car A travels on the inside and car B travels on the outside at equal speed. Which statement is correct? a. The force on A is half the force on B b. The force on B is half the...

Hello, I was wondering if H_{ii} (that is the ith diagonal element of a random matrix) has the same distribution with its corresponding eigenvalue, say \lambda_{i}. Thanks

Hi.. I want an explanation of the commutation relation. According to what I understand if two operators commute then they can be measured simultaneously. If they do not commute then the measurement of one depends on other as per the value of the commutator..I hope this is correct by far. In...

Hey! :o At the proof of the theorem that the expected time of Quicksort is $O(n \log n)$, there is the following sentence: We suppose that the partitions are equally likely, so the possibility that the sizes of the sequences $S_1$ and $S_3$ are $i-1$ and $n-i$, respectively, is $\frac{1}{n}$...

Mod note: OP warned about not using the homework template. I have read that 'a(t) determines the value of the constant spatial curvature'.. Where a(t) is the scale factor, and we must have constant spatial curvature - this can be deduced from the isotropic at every point assumption. I'm trying...

Homework Statement What is the commutation relation between the x and y components of angular momentum L = r X P Homework Equations None. The Attempt at a Solution I do r X p and get the angular momentum componants:L_{x} = (-i \hbar) (y \frac{d}{dz} - z \frac{d}{dy}) L_{y} = (-i \hbar) (z...

In the derivation of the generalized uncertainty principle (as pgs 1-2 of here), there is an anticommutator term that is dropped at the end, leaving just the commutator part...this is said to "strengthen" the relation, as both terms are positive. I don't understand this. So we basically have...

Hello! (Wave) I am looking at the proof of the proposition: $$\text{The relation } \epsilon_{\omega}=\{ \langle m,n \rangle \in \omega^2: m \in n\} \text{ is trichotomous on } \omega.$$ Proof: We define the sets: $T_m=\{ n \in \omega: m \in n \lor m=n \lor n \in m\},m \in \omega$ It suffices...

Hello! (Wave) Proposition Let $X$ be a set. The following are equivalent: $X$ is transitive Each element of $X$ is a subset of $X$ $\left( \forall x(x \in X \rightarrow x \subset X) \right)$ $X \subset \mathcal{P}X$ $\bigcup X \subset X$ Could you explain me why this: $X \subset...

Homework Statement [/B] Solve the recurrence relation (use iteration). an = an-1 + 1 + 2n-1 a0 = 0 Then prove the solution by mathematical induction. Homework EquationsThe Attempt at a Solution a1 = 2 a2 = 5 a3 = 10 a4 = 19 a5 = 36 The solution appears to be an = n + 2n - 1 How are we...

Homework Statement Find: Homework Equations The Attempt at a Solution

Homework Statement Let ##R## be an ordered relation on a set ##A## that is reflexive and anti-symmetric. If there is a chain of elements in ##R## that begins and ends with the same element, say the element ##x \in A## is it true that all the elements of ##R## sandwiched in between the ones...

1. Derive (∂U/∂P)V=-T(∂V/∂T)S 2. I must use dU=TdS-PdV 3. Derivations are my weakest part of math. I checked many wikis about Total differentials, partial derivatives, Maxwell relations and derivations. I can use the Thermodynamic Square, I know how to find different Maxwell relations but I am...

Hello! (Smile) Let $S(m)=S(m-1)+m$. I want to show that $S(m)=\Theta(m^2)$. That's what I have tried: We suppose a positive integer $m>0$. We suppose that $S(m-1)=\Theta((m-1)^2)$, so it holds that $\exists c_1, c_2>0$ such that : $$c_1 (m-1)^2 \leq S(m-1) \leq c_2(m-1)^2$$ We will show that...

Homework Statement The Hermitian operators \hat{A},\hat{B},\hat{C} satisfy the commutation relation[\hat{A},\hat{B}]=c\hat{C}. Show that c is a purely imaginary number. The Attempt at a Solution I don't usually post questions without some attempt at an answer but I am at a loss here.

Homework Statement http://postimg.org/image/i4p19540z/ Homework Equations Resultant force on the control volume = Mass flow rate (Velocity outlets -Velocity inlets) The Attempt at a Solution I am just wondering if the 4cm is called depth, then what is the term for the "into the paper"...

hi guys my question is if you have few grams of alkali metals and vapored it , what is the mathematical equation that links between these variables density vapor , the mass and the temperture ?? can you help me ?

Hii,i'm new in electrical and much confused bcoz of complicated relation between V,I, nd P. If P = VI = cons. and increasing V,decrease the value of I, but since P is also equal to I×I ×R and r is cons.,so decrease in I will always cause P to decrease which is supposed to remain constant. And i...

This is more a conceptual question. So i am doing some self review of multi variate calculus and i am looking at functinal relations of the form F(x, y, z,...) = 0 In the book they talk about implicit differentiation. Now i fully understand how to do the mechanics of it, but i was trying to...

I know they are directly proportional but how? could anyone explain it graphically ? thanks in advance

Hello! (Smile) It stands that $R[A \cap B] \subset R[A] \cap R[B]$, since: $$y \in R[A \cap B] \rightarrow \exists x \in A \cap B: xRy \rightarrow \exists x(x \in A \wedge xRy) \wedge (x \in B: xRy) \rightarrow y \in R[A] \wedge x \in R[B] \rightarrow y \in R[A] \cap R[B]$$ But, it doesn't...

Hi again! (Smile) If $\rho$ is an equivalence relation, could you explain me why the following relations stand? (Thinking) $I_A \subset \rho$ $\rho^{-1}=\rho$ $\rho \circ \rho \subset \rho$

Hello! (Wave) Could you explain me why the following stands? (Thinking) If $R$ is a relation, then: $$R \subset dom R \times rng R \subset fld R \times fld R$$

I am learning some basic solid state physics idea, like density of state ...etc. For particle in a 1D box, E = n^2 (pi)^2 (h_bar)^2 / 2mL^2 But why it is written as E = (h_bar)^2 k^2 /2m does it means that energy eigenvalue E is related to momentum k ? I guess it is not because momentum is...

As we know \ a= \frac {F} {m} So does this equation prove that acceleration is directly proportional to the net force and inversely proportional to the mass of the object?

Hello! (Wave) I want to use the master theorem, in order to find the exact asymptotic solution of $S(m)=4S \left ( \frac{m}{2}\right )+m^3 \sqrt{m}$. $$a=4 \geq 1, b=2>1, f(m)=m^3 \sqrt{m}$$ $$m^{\log_b a}=m^{\log_2{2^2}}=m^2$$ $$f(m)=m^{3+\frac{1}{2}}=m^{\log_b a+ \frac{3}{2}}$$ Thus...

Technically I'm supposed to have a total of 8 optical modes but only 4 of them were seen in a solid (by spectroscopy). So I suspect there's some degeneracies and symmetries involved, but I don't know which ones. I have two sets of assigned degeneracies: frequency; degeneracy set 1; degeneracy...

what's the relation of the viscous hydrodynamic forces and swirl？

I know that the matrices {\Gamma^{A}} obey the trace orthogonality relation Tr(\Gamma^{A}\Gamma_{B})=2^{m}\delta^{A}_{B} In order to show that a matrix M can be expanded in the basis \Gamma^{A} in the following way M=\sum_{A}m_{A}\Gamma^{A} m_{A}=\frac{1}{2^{m}}Tr(M\Gamma_{A}) is it enough to...

Hello! (Wave) I want to prove that $T(n)=4 T \left ( \frac{n}{2}\right )+n^2 \lg n=O(n^2 \lg^2 n)$,where $T(n)$ is constant for $n \leq 8$, using the following method: "We choose a specific function $f(n)$ and we try to show that for an appropriate $c>0$ and an appropriate $n_0 \in...

Hello! (Wave) I want to find an asymptotic upper bound for the recurrence relation: $T(n)=9T \left (\frac{n}{3} \right ) + n^2 \log n $, $T(n)=c, \text{ when } n \leq 9$, using the following method: We choose a specific function $f(n)$ and we try to show that for an appropriate $c>0$ and an...

Hi! I understand that if you have a stator with a a three phase electric signal going into it, both the frqyency and angular velocity of the magnetic field and the electric entry will be the same. Now when you feed it with 2 groups of three phases electric signals you get four poles, or at least...

What method should i use to know if a recurrence relation is increasing or decreasing? i was given the following relation: A1 = 1 An=(An-1)^5 - 3 I know for sure it actually decreases since every term for n>=2 is a negative number raised to and odd number, but i don't know how to demonstrate...

Hi! (Wave) Let $R$ be a relation. Show the following sentences: $dom(R^{-1})=rng(R)$ $rng(R^{-1})=dom(R)$ $fld(R^{-1})=fld(R)$ $(R^{-1})^{-1}=R$ That's what I have tried: Let $x \in dom(R^{-1})$. Then $\exists y$ such that $<x,y> \in R^{-1} \Rightarrow <y,x> \in R \Rightarrow x \in...

1. The problem I am trying to prove the following relation in cartesian coordinates. We were given a hint to use integration by parts, as well as the fact that we know $d \vec r = dx\,dy\,dz$ (volume integral). $$\int f(\vec r)\ \nabla \cdot \vec A(\vec r) \, d \vec r = -\int \vec A(\vec...

Hey! (Cool) According to my notes, each relation is a subset of an ordered pair. How can it be that each relation is a subset of an ordered pair, knowing that a relation is a set of ordered pairs? (Thinking)

Hi Physics Forums, The solubility of a gas according to Henry's Law depends on partial pressure. Would an increase in pressure in a system increase the solubility of a specific gas, even if the partial pressure of that particular gas doesn't change? The system described above increases in...

Homework Statement Calculate the commutator ##[\hat{L}_i, (\mathbf{rp})^2]## Homework Equations ##\hat{\vec{L}} = \sum\limits_{a=1}^N \vec{r}_a \times \hat{\vec{p}}## ##[r_i,p_k] = i\hbar\delta_{ik}## The Attempt at a Solution Okay so here is what I have so far: $$ \begin{eqnarray}...

Hello! (Wave) I want to prove by induction, that the solution of the recurrence relation $T(n)=2T \left ( \frac{n}{2} \right )+n^2, n>1 \text{ and } T(1)=1$ is $n(2n-1)$. We have to suppose that $n=2^k, k \geq 0$, right? Do I have to prove the solution by induction with respect to $n$ or to...

Homework Statement [/B] A particle of mass m is moving in the +x-direction with speed u and has momentum p and energy E in the frame S. (a) If S' is moving at speed v, find the momentum p' and energy E' in the S' frame. (b) Note that E' \neq E and p' \neq p, but show that...

I'm in calc 1 and want to make sure I'm understanding the reason that we find derivatives. From what I understand, a derivative is simply an equation for the rate of change at any given point on the original function. Is that correct? And the tangent line at point (x,y) is obtained by using...

I am given a formula in explicit form and as a recurrence relation. It is asked to derive the recurrence relation from the explicit form. How is this done?

Hey again! (Wave) I have to determine if $g(n)=O(g^5(n))$ is true or not. I thought that I could use the definition of Big-0h, but I don't know how to begin, formulating it. From the definition, we have that $\exists c>0$ and $n_0 \in \mathbb{N}_0$ such that $\forall n \geq n_0$: $$0 \leq...

An observer in an inertial frame finds that at a point P the Electric field vanishes but the Magnetic field does not. This implies that in any other inertial frame the electric field E and the magnetic field B satisfy: [these values in vectors] 1. |E|2 = |B|2 2. E . B = 0 3. E x B = 0 4. E = 0...

Hello all, this is my first post! Hopefully I can gain some valuable insight. Homework Statement Find the relation among T, P and mu for the system with the given equation U = b S4/NV2 I let b equal the several constants stated in the problem. Homework Equations T=dU/dS P=-dU/dV mu=dU/dN The...