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1. ### How to simplify this complex expression?

I added it in the 8th post .
2. ### How to simplify this complex expression?

##(e^{3i\pi}+e^{\frac{i\pi}{3}})=-1+e^{\frac{i\pi}{3}}= -1 +1/2 + \frac{(i \sqrt(3))}{2}=-1/2 + \frac{(i \sqrt(3))}{2} =e^{\frac{2i\pi}{3}}##
3. ### How to simplify this complex expression?

Yes the mistake is in the exponent##(e^{3i\pi}+e^{\frac{\pi}{3}})##. It's ##(e^{3i\pi}+e^{\frac{i\pi}{3}})##. I just plugged ##\sqrt[3\,]{e^{\frac {2i\pi}{3}}} +\sqrt[3\,]{-\sqrt 3 e^{i\pi/6}}= 3^{1/6} e^{-5i\pi/18}+e^{\frac {4i\pi}{18}}## into ##\sqrt[3\,]...
4. ### How to simplify this complex expression?

I think I've clocked it. ## \sqrt[3\,] {e^{3i\pi}+e^{\frac{\pi}{3}}}= \sqrt[3\,]{e^{\frac {2i\pi}{3}}}## ##\sqrt[3\,]{e^{\frac {2i\pi}{3}}} +\sqrt[3\,]{-\sqrt 3 e^{i\pi/6}}= 3^{1/6} e^{-5i\pi/18}+e^{\frac {4i\pi}{18}}## ##=e^{\frac {4i\pi}{18}}(3^{1/6}e^{\frac {-9i\pi}{18}}+1)## ##=e^{\frac...
5. ### How to simplify this complex expression?

If I write ##e^{3i\pi}=(e^{i\pi})^3=(-1)^3=-1## Then ##z=-\frac {3}{2} - i \frac{\sqrt 3}{2}=-\sqrt 3 e^{i\pi/6} ##
6. ### How to simplify this complex expression?

##(-\frac 32 - i \frac{\sqrt 3}{2})^{1/3}= \sqrt[3\,] {e^{3i\pi}-e^{\frac{\pi}{3}}}## and ##(-1)^{2\over 9}=\sqrt[3\,] {e^{3i\pi}+e^{\frac{\pi}{3}}}## When trying to solve ##(-3/2 - \frac{i}{2} \sqrt{3})=(a+bi)^3## I get ##a(a^2-b-2b^2)=\frac{3}{2}## and ##b(2a^2+a-b^2)=\frac{\sqrt{3}}{2}##. How...
7. ### How to simplify this complex expression?

How do I find the real part of ## (-3/2 - \frac{i}{2} \sqrt{3})^{(1/3)}## ? The cubic root makes it hard for me to see what to do here.
8. ### How to simplify this complex expression?

My attempt : multiplying numerator and denominator by ## (-1)^{2/9} + (-3/2 - \frac{i}{2} \sqrt{3})^{(1/3)} wehave \frac{(-1)^{4/9} +2(-1)^{2/9}(-3/2 - \frac{i}{2} \sqrt{3})^{(1/3)}+ (-3/2 - \frac{i}{2} \sqrt{3})^{(2/3)}}{(-1)^{4/9}+ (-3/2 - \frac{i}{2} \sqrt{3})^{(2/3)}} ##
9. ### How to simplify this complex expression?

I don't know how to start with the factorization. $$\frac{(-1)^{2/9} + (-3/2 - \frac{i}{2} \sqrt{3})^{(1/3)}}{(-1)^{2/9}- (-3/2 - \frac{i}{2} \sqrt{3})^{(1/3)}}$$ Any hints would be nice. Thank you.
10. ### I Why isn't this a Lie group?

So giving a different metric gives us the right topology to make H a Lie group. Thank you.
11. ### I Why isn't this a Lie group?

How can we write the topology change for H as a topological group ?
12. ### I Why isn't this a Lie group?

"The group can, however, be given a different topology, in which the distance between two points is defined as the length of the shortest path in the group joining to " What does this give us in formal math ?
13. ### I Why isn't this a Lie group?

How do we change the topology to make H a Lie group ?
14. ### I Why isn't this a Lie group?

I can't find a proper homeomorphism in ##U_2##.
15. ### I Why isn't this a Lie group?

1-a finite-dimensional real smooth manifold 2- the group operations of multiplication and inversion are smooth maps.
16. ### I Why isn't this a Lie group?

I'm a bit lost, is ##[0,2\pi]-(\mathbb{Q}\cap [0,2\pi])## equal to H or ##\mathbb{P}## ?I've never worked with topological groups,how does a disconnected topological group or set become connected?
17. ### I Why isn't this a Lie group?

How do we change the topology of the subgroup from the original group? Could you show a simple proof of why it becomes connected ?
18. ### I Why isn't this a Lie group?

"The group given by ## H = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0 } \\ { 0 } & { e ^ { 2 \pi i a \theta } } \end{array} \right) | \theta \in \mathbb { R } \right\} \subset \mathbb { T } ^ { 2 } = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0...
19. ### I How to view conditional variance intuitively?

I am failing to understand the way to get this answer . Is there a simple computation or property that easily gives this answer ? I tried the formula for ##v_x(Q)## without success.
20. ### I How to view conditional variance intuitively?

We have a sample of X, a Normalized Gaussian random variable.We divide the data into positive and negative. Each will have a conditional variance of ## 1−\frac{2}{π}## . Can someone show how to get this result ? I found this problem here (page 3) ...
21. ### Entropy change when melting ice then refreezing the water

Yes, so we do have a gain in entropy.Thanks again.
22. ### Entropy change when melting ice then refreezing the water

##dS_2+dS_1=-\frac{dmL_f} {T_0}+\frac{dmL_f} {T_0+∆T}=-\frac{dmL_f} {T_0}+\frac{dmL_f} {T_0}(1-\frac{∆T} {T_0})= -dmL_f\frac{∆T}{{T_0}^2}=-\frac{dm(v_l-v_s)∆P} {T_0}=\frac{dm(v_s-v_l)2Mg} {ST_0}=\frac{(v_s-v_l)2Mgh} {v_s T_0} =∆S\; \; \;## since ##v_s=\frac{Sh}{dm}## I made another mistake while...
23. ### Entropy change when melting ice then refreezing the water

It's a mistake, it's supposed to be ##v_s##, the ##L##'s are ##L_f##'s also.

25. ### Entropy change when melting ice then refreezing the water

##dmL_f= Q \; \; ##,##∆T=\frac{T(v_l-v_s)∆P}{L} \; \;##,##\frac{dmL_f}{T_0}= dS_2 \; \;##,##\frac{dmL_f}{T_1}= dS_1 ##
26. ### Temperature change in a gas expansion

I guess so.Thank you for these clarifications.
27. ### Temperature change in a gas expansion

Question 2 : Express ##\delta T_S## as a function of ##\delta P_V##,T , P and ##\gamma## first then as a function of ##\delta T_v## and ##\gamma##
28. ### Temperature change in a gas expansion

Since δ##T_V## is a function of T and P there should be a way to change variables P and T in the formula for ##\delta T_S## and express it as a function of only δ##T_V## and ##\gamma##.
29. ### Temperature change in a gas expansion

Thank you but now I don't see a way to remove T and have ##\delta T_S## as a function of only ##\delta T_V## and ##\gamma ##.
30. ### Temperature change in a gas expansion

##\delta P_V =(\frac{\partial P}{\partial T} )_V \delta T_V =\frac{nR}{V}\delta T_V##. I don't know how to use ##TP^{\frac{1-\gamma }{\gamma}}=cst## to make ##\delta T_S## a function of ##\delta P_V##,T,P and ##\gamma##.
31. ### Temperature change in a gas expansion

1- ##\delta P_V =(\frac{\partial P}{\partial T} )_V \delta T_V## 2-##\delta V =(\frac{\partial V}{\partial T})_P \delta T+(\frac{\partial V}{\partial P})_T \delta P## so ##C_v \delta T=-P\delta V=-P((\frac{\partial V}{\partial T})_P \delta T+(\frac{\partial V}{\partial P})_T \delta P)## I can...
32. ### Low-Temp Liquid Helium System: Evaporation Heat & Temperature

If we change the pressure condition below a certain pressure some of the liquid will turn into gas no ?
33. ### Low-Temp Liquid Helium System: Evaporation Heat & Temperature

Perhaps it's an unstable equilibrium.I guess T is close to the evaporation temperature.
34. ### Low-Temp Liquid Helium System: Evaporation Heat & Temperature

We consider a system composed of liquid helium in equilibrium with its vapor at very low temperature T, each phase being considered extensive. We neglect the mass of the gas compared to that of the liquid, as well as the heat capacities of the gas and the walls compared to that of the liquid...
35. ### Entropy change in water

Indeed,Is ##\Delta S_{th}=\mp\frac{Q}{T_0}=\mp\int_{T_i}^{T_0}\frac{C(T)}{T_0}dT ## correct ? If so then question b) is ## \Delta S_{univ}=\int_{T_i}^{T_0}C(T)(\frac{1}{T}-\frac{1}{T_0})dT ## when the water heats up and ## \Delta S_{univ}=\int_{T_i}^{T_0}C(T)(\frac{1}{T_0}-\frac{1}{T})dT ##...
36. ### Entropy change in water

for a)##\Delta S=\mp \int_{T_i}^{T_0}\frac{C(T)}{T}dT## and ##\Delta S_{th}=\int_{T_i}^{T_0}\frac{dQ}{T_0}dT## so ##S_{univ}=\Delta S_{th}+\Delta S##. What is ##dQ## equal to ? I don't know how to answer question b). Thank you for your help.
37. ### I Physical model of the roulette wheel

Thank you for this explanation,it's clear now.
38. ### I Physical model of the roulette wheel

Hello, I am having trouble understanding the two friction force terms from the ball rolling on page 4 of this physical model: http://www.dewtronics.com/tutorials/roulette/documents/Roulette_Physik.pdf What is the reason for the cos\theta term ? I think the frame of reference is made up of the...
39. ### I Schmidt decomposition and entropy of the W state

Hello, The state | W \rangle = \frac { 1 } { \sqrt { 3 } } ( | 001 \rangle + | 010 \rangle + | 100 \rangle ) is entangled. The Schmidt decomposition is : What would the Schmidt decomposition be for | W \rangle ? I am also intersted in writing the reduced density matrix but I need the basis...
40. ### Angular frequency of an ammonia molecule

Hello 1. Homework Statement The dipole moment of an ammonia molecule is ##d_0=5*10^{-30} C.m##.If we apply a static electric field of ##\mathcal { E }=1*10^{6 }V*m^{-1}## to an ammonia molecule initially in the state ## |ψG⟩## where the nitrogen molecule is considered to be on the left,we make...
41. ### Entropy change in an RC circuit

I think I have an explanation : The generator gives the system {resitance,capacitor} the electric energy : The capacitor charges itself by en absorbing half ( the work ##W=\int vdq##). The difference represents the lost energy through the Joule effect : This energy is received...
42. ### Entropy change in an RC circuit

I want to know how to express the entropy change , I think the entropy change of the universe is 0 but I'm not sure what the equation is.The RC circuit is getting heat from the reservoir and work from the generator,I'm guessing it's to make it a reversible process.
43. ### Entropy change in an RC circuit

Hello,I would like some help for a problem Homework Statement Initially:At t=0 [/B]the cylindrical capacitor of capacitance c=\frac{\epsilon s}{d} (d the distance between the 2 electrodes and s their surface; \epsilon = \epsilon(T) is the dielectric permittivity) is discharged and we close the...
44. ### Interval of the maximum solution of a nonlinear equation

Mentor note: Thread moved from the Technical Math section, so there is no template. @yamata1, , in the future, please post homework problems or exercises in the Homework & Coursework sections, not in the Technical Math sections. I have moved your post. Hello, I would like some help with an...
45. ### Angular momentum multiple choice question

Thank you for your explanations.The same test asked what the wavelength of the photon emmited for the transition between first and second excited states of the hydrogen atom was.The choices were : ##a##- < 652 nm ##b##- >652 nm ##c##- equal to 652 nm ##d##- >241 nm ##e##- equal to 241nm The...
46. ### Angular momentum multiple choice question

So ##e## is also correct because ##\mp2\hbar## implies j can only be ##2\hbar## but "if,and only if" would exclude ##b##.
47. ### Angular momentum multiple choice question

If j can take values 0,1or 2 and ##-j<m<j## ,how can knowing m be enough to determine a one-dimensional subspace of the state space ?
48. ### Angular momentum multiple choice question

since ##J_z|j,m>=m|j,m>## and ##J^2|j,m>=j(j+1)|j,m>## regardless of ##m## we still don't have ##j## ,it cannot be enough to determine a one-dimensional subspace of the state space ?
49. ### Angular momentum multiple choice question

Measuring ##J_z## does not give us the eigenvalues for the other angular momentum operators,so I think not.
50. ### Angular momentum multiple choice question

Could ##a##,##c## or ##e## ever be correct ?