On Saturday, February 2, 2019 at 1:40:29 AM UTC-7, Bruno Marchal wrote:
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> On 1 Feb 2019, at 21:29, agrays...@gmail.com <javascript:> wrote:
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>
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> On Friday, February 1, 2019 at 5:55:30 AM UTC-7, Bruno Marchal wrote:
>>
>>
>> On 31 Jan 2019, at 21:10, agrays...@gmail.com wrote:
>>
>>
>>
>> On Thursday, January 31, 2019 at 6:47:12 AM UTC-7, Bruno Marchal wrote:
>>>
>>>
>>> On 31 Jan 2019, at 01:28, agrays...@gmail.com wrote:
>>>
>>>
>>>
>>> On Wednesday, January 30, 2019 at 2:38:58 PM UTC-7, agrays...@gmail.com 
>>> wrote:
>>>>
>>>>
>>>>
>>>> On Wednesday, January 30, 2019 at 5:16:05 AM UTC-7, Bruno Marchal wrote:
>>>>>
>>>>>
>>>>> On 30 Jan 2019, at 02:59, agrays...@gmail.com wrote:
>>>>>
>>>>>
>>>>>
>>>>> On Tuesday, January 29, 2019 at 4:37:34 AM UTC-7, Bruno Marchal wrote:
>>>>>>
>>>>>>
>>>>>> On 28 Jan 2019, at 22:50, agrays...@gmail.com wrote:
>>>>>>
>>>>>>
>>>>>>
>>>>>> On Friday, January 25, 2019 at 7:33:05 AM UTC-7, Bruno Marchal wrote:
>>>>>>>
>>>>>>>
>>>>>>> On 24 Jan 2019, at 09:29, agrays...@gmail.com wrote:
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> On Sunday, January 20, 2019 at 11:54:43 AM UTC, agrays...@gmail.com 
>>>>>>> wrote:
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> On Sunday, January 20, 2019 at 9:56:17 AM UTC, Bruno Marchal wrote:
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> On 18 Jan 2019, at 18:50, agrays...@gmail.com wrote:
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> On Friday, January 18, 2019 at 12:09:58 PM UTC, Bruno Marchal 
>>>>>>>>> wrote:
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> On 17 Jan 2019, at 14:48, agrays...@gmail.com wrote:
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> On Thursday, January 17, 2019 at 12:36:07 PM UTC, Bruno Marchal 
>>>>>>>>>> wrote:
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> On 17 Jan 2019, at 09:33, agrays...@gmail.com wrote:
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> On Thursday, January 17, 2019 at 3:58:48 AM UTC, Brent wrote:
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> On 1/16/2019 7:25 PM, agrays...@gmail.com wrote:
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> On Monday, January 14, 2019 at 6:12:43 AM UTC, Brent wrote:
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> On 1/13/2019 9:51 PM, agrays...@gmail.com wrote:
>>>>>>>>>>>>>
>>>>>>>>>>>>> This means, to me, that the arbitrary phase angles have 
>>>>>>>>>>>>> absolutely no effect on the resultant interference pattern which 
>>>>>>>>>>>>> is 
>>>>>>>>>>>>> observed. But isn't this what the phase angles are supposed to 
>>>>>>>>>>>>> effect? AG
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> The screen pattern is determined by *relative phase angles 
>>>>>>>>>>>>> for the different paths that reach the same point on the screen*. 
>>>>>>>>>>>>>  
>>>>>>>>>>>>> The relative angles only depend on different path lengths, so the 
>>>>>>>>>>>>> overall 
>>>>>>>>>>>>> phase angle is irrelevant.
>>>>>>>>>>>>>
>>>>>>>>>>>>> Brent
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> *Sure, except there areTWO forms of phase interference in Wave 
>>>>>>>>>>>> Mechanics; the one you refer to above, and another discussed in 
>>>>>>>>>>>> the 
>>>>>>>>>>>> Stackexchange links I previously posted. In the latter case, the 
>>>>>>>>>>>> wf is 
>>>>>>>>>>>> expressed as a superposition, say of two states, where we consider 
>>>>>>>>>>>> two 
>>>>>>>>>>>> cases; a multiplicative complex phase shift is included prior to 
>>>>>>>>>>>> the sum, 
>>>>>>>>>>>> and different complex phase shifts multiplying each component, all 
>>>>>>>>>>>> of the 
>>>>>>>>>>>> form e^i (theta). Easy to show that interference exists in the 
>>>>>>>>>>>> latter case, 
>>>>>>>>>>>> but not the former. Now suppose we take the inner product of the 
>>>>>>>>>>>> wf with 
>>>>>>>>>>>> the ith eigenstate of the superposition, in order to calculate the 
>>>>>>>>>>>> probability of measuring the eigenvalue of the ith eigenstate, 
>>>>>>>>>>>> applying one 
>>>>>>>>>>>> of the postulates of QM, keeping in mind that each eigenstate is 
>>>>>>>>>>>> multiplied 
>>>>>>>>>>>> by a DIFFERENT complex phase shift.  If we further assume the 
>>>>>>>>>>>> eigenstates 
>>>>>>>>>>>> are mutually orthogonal, the probability of measuring each 
>>>>>>>>>>>> eigenvalue does 
>>>>>>>>>>>> NOT depend on the different phase shifts. What happened to the 
>>>>>>>>>>>> interference 
>>>>>>>>>>>> demonstrated by the Stackexchange links? TIA, AG *
>>>>>>>>>>>>
>>>>>>>>>>>> Your measurement projected it out. It's like measuring which 
>>>>>>>>>>>> slit the photon goes through...it eliminates the interference.
>>>>>>>>>>>>
>>>>>>>>>>>> Brent
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> *That's what I suspected; that going to an orthogonal basis, I 
>>>>>>>>>>> departed from the examples in Stackexchange where an arbitrary 
>>>>>>>>>>> superposition is used in the analysis of interference. 
>>>>>>>>>>> Nevertheless, isn't 
>>>>>>>>>>> it possible to transform from an arbitrary superposition to one 
>>>>>>>>>>> using an 
>>>>>>>>>>> orthogonal basis? And aren't all bases equivalent from a linear 
>>>>>>>>>>> algebra 
>>>>>>>>>>> pov? If all bases are equivalent, why would transforming to an 
>>>>>>>>>>> orthogonal 
>>>>>>>>>>> basis lose interference, whereas a general superposition does not? 
>>>>>>>>>>> TIA, AG*
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> I don’t understand this. All the bases we have used all the time 
>>>>>>>>>>> are supposed to be orthonormal bases. We suppose that the scalar 
>>>>>>>>>>> product 
>>>>>>>>>>> (e_i e_j) = delta_i_j, when presenting the Born rule, and the 
>>>>>>>>>>> quantum 
>>>>>>>>>>> formalism.
>>>>>>>>>>>
>>>>>>>>>>> Bruno
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> *Generally, bases in a vector space are NOT orthonormal. *
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Right. But we can always build an orthonormal base with a decent 
>>>>>>>>>> scalar product, like in Hilbert space, 
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> *For example, in the vector space of vectors in the plane, any 
>>>>>>>>>> pair of non-parallel vectors form a basis. Same for any general 
>>>>>>>>>> superposition of states in QM. HOWEVER, eigenfunctions with distinct 
>>>>>>>>>> eigenvalues ARE orthogonal.*
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Absolutely. And when choosing a non degenerate 
>>>>>>>>>> observable/measuring-device, we work in the base of its 
>>>>>>>>>> eigenvectors. A 
>>>>>>>>>> superposition is better seen as a sum of some eigenvectors of some 
>>>>>>>>>> observable. That is the crazy thing in QM. The same particle can be 
>>>>>>>>>> superposed in the state of being here and there. Two different 
>>>>>>>>>> positions of 
>>>>>>>>>> one particle can be superposed.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> *This is a common misinterpretation. Just because a wf can be 
>>>>>>>>> expressed in different ways (as a vector in the plane can be 
>>>>>>>>> expressed in 
>>>>>>>>> uncountably many different bases), doesn't mean a particle can exist 
>>>>>>>>> in 
>>>>>>>>> different positions in space at the same time. AG*
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> It has a non null amplitude of probability of being here and there 
>>>>>>>>> at the same time, like having a non null amplitude of probability of 
>>>>>>>>> going 
>>>>>>>>> through each slit in the two slits experience.
>>>>>>>>>
>>>>>>>>> If not, you can’t explain the inference patterns, especially in 
>>>>>>>>> the photon self-interference.
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Using a non orthonormal base makes only things more complex. 
>>>>>>>>>>
>>>>>>>>> *I posted a link to this proof a few months ago. IIRC, it was on 
>>>>>>>>>> its specifically named thread. AG*
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> But all this makes my point. A vector by itself cannot be 
>>>>>>>>>> superposed, but can be seen as the superposition of two other 
>>>>>>>>>> vectors, and 
>>>>>>>>>> if those are orthonormal, that gives by the Born rule the 
>>>>>>>>>> probability to 
>>>>>>>>>> obtain the "Eigen result” corresponding to the measuring apparatus 
>>>>>>>>>> with 
>>>>>>>>>> Eigen vectors given by that orthonormal base.
>>>>>>>>>>
>>>>>>>>>> I’m still not sure about what you would be missing.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> *You would be missing the interference! Do the math. Calculate the 
>>>>>>>>> probability density of a wf expressed as a superposition of 
>>>>>>>>> orthonormal 
>>>>>>>>> eigenstates, where each component state has a different phase angle. 
>>>>>>>>> All 
>>>>>>>>> cross terms cancel out due to orthogonality,*
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> ?  Sin(alpha) up + cos(alpha) down has sin^2(alpha) probability to 
>>>>>>>>> be fin up, and cos^2(alpha) probability to be found down, but has 
>>>>>>>>> probability one being found in the Sin(alpha) up + cos(alpha) down 
>>>>>>>>> state, 
>>>>>>>>> which would not be the case with a mixture of sin^2(alpha) proportion 
>>>>>>>>> of up 
>>>>>>>>> with cos^2(alpha) down particles.
>>>>>>>>> Si, I don’t see what we would loss the interference terms.
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> *and the probability density does not depend on the phase 
>>>>>>>>> differences.  What you get seems to be the classical probability 
>>>>>>>>> density. 
>>>>>>>>> AG *
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> I miss something here. I don’t understand your argument. It seems 
>>>>>>>>> to contradict basic QM (the Born rule). 
>>>>>>>>>
>>>>>>>>
>>>>>>>> *Suppose we want to calculate the probability density of a 
>>>>>>>> superposition consisting of orthonormal eigenfunctions, *
>>>>>>>>
>>>>>>>
>>>>>>> Distinct eigenvalue correspond to orthonormal vector, so I tend to 
>>>>>>> always superpose only orthonormal functions, related to those 
>>>>>>> eigenvalue. 
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> *each multiplied by some amplitude and some arbitrary phase shift. *
>>>>>>>>
>>>>>>>
>>>>>>> like  (a up + b down), but of course we need a^2 + b^2 = 1. You need 
>>>>>>> to be sure that you have normalised the superposition to be able to 
>>>>>>> apply 
>>>>>>> the Born rule.
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> *If we take the norm squared using Born's Rule, don't all the cross 
>>>>>>>> terms zero out due to orthonormality? *
>>>>>>>>
>>>>>>>
>>>>>>> ?
>>>>>>>
>>>>>>> The Born rule tell you that you will find up with probability a^2, 
>>>>>>> and down with probability b^2
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> *Aren't we just left with the SUM OF NORM SQUARES of each component 
>>>>>>>> of the superposition? YES or NO?*
>>>>>>>>
>>>>>>>
>>>>>>> If you measure in the base (a up + b down, a up -b down). In that 
>>>>>>> case you get the probability 1 for the state above.
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> * If YES, the resultant probability density doesn't depend on any of 
>>>>>>>> the phase angles. AG*
>>>>>>>>
>>>>>>>
>>>>>>> *YES or NO? AG *
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> Yes, if you measure if the state is a up + b down or a up - b down.
>>>>>>> No, if you measure the if the state is just up or down
>>>>>>>
>>>>>>> Bruno
>>>>>>>
>>>>>>
>>>>>> *I assume orthNORMAL eigenfunctions. I assume the probability 
>>>>>> densities sum to unity. Then, using Born's rule, I have shown that 
>>>>>> multiplying each component by e^i(theta) where theta is arbitrarily 
>>>>>> different for each component, disappears when the probability density is 
>>>>>> calculated, due to orthonormality. *
>>>>>>
>>>>>>
>>>>>>
>>>>>> That seems to violate elementary quantum mechanics. If e^I(theta) is 
>>>>>> different for each components, Born rule have to give different 
>>>>>> probabilities for each components---indeed given by the square of 
>>>>>> e^I(theta).
>>>>>>
>>>>>
>>>>> *The norm squared of e^i(thetai) is unity, except for the cross terms 
>>>>> which is zero due to orthonormality. AG *
>>>>>
>>>>>>
>>>>>> *What you've done, if I understand correctly, is measure the 
>>>>>> probability density using different bases, and getting different values. 
>>>>>> *
>>>>>>
>>>>>>
>>>>>> The value of the relative probabilities do not depend on the choice 
>>>>>> of the base used to describe the wave. Only of the base corresponding to 
>>>>>> what you decide to measure. 
>>>>>>
>>>>>>
>>>>>>
>>>>>> *This cannot be correct since the probability density is an objective 
>>>>>> value, and doesn't depend on which basis is chosen. AG*
>>>>>>
>>>>>>
>>>>>> Just do the math. Or read textbook. 
>>>>>>
>>>>>
>>>>> *Why don't YOU do the math ! It's really simple. Just take the norm 
>>>>> squared of a superposition of component eigenfunctions, each multiplied 
>>>>> by 
>>>>> a probability amplitude, and see what you get !  No need to multiply each 
>>>>> component by e^i(thetai).  Each amplitude has a phase angle implied. This 
>>>>> is Born's rule and the result doesn't depend on phase angles, contracting 
>>>>> what Bruce wrote IIUC. If you would just do the simple calculation you 
>>>>> will 
>>>>> see what I am referring to! AG*
>>>>>
>>>>>
>>>>>
>>>>> Bruce is right. Let us do the computation in the simple case where 
>>>>> e^i(theta) = -1. (Theta = Pi)
>>>>>
>>>>> Take the superposition (up - down), conveniently renormalised. If I 
>>>>> multiply the whole wave (up - down) by (-1), that changes really nothing. 
>>>>> But if I multiply only the second term, I get the orthogonal state up + 
>>>>> down, which changes everything. (up +down) is orthogonal to (up - down).
>>>>>
>>>>> Bruno
>>>>>
>>>>
>>>>  *Fuck it. You refuse to do the simple math to show me exactly where I 
>>>> have made an error,  IF I have made an error.  You talk a lot about Born's 
>>>> rule but I seriously doubt you know how to use  it for simple 
>>>> superposition. AG *
>>>>
>>>
>>> *If you take the inner product squared (Born's rule) using an 
>>> orthonormal set of eigenfunctions, you get a sum of the form (a_j)^2 + 
>>> (b_j) ^2  where A_j is the complex probability amplitude for the jth 
>>> component, A_j = a_j + i * b_j. The cross terms drop out due to 
>>> orthonormality, and the phase angles are implicitly determined by the 
>>> relative values of a_j and b_j for each j. *
>>>
>>>
>>> If you have prepared the state, so that you know that the state of your 
>>> object is given by 
>>>
>>>  phi = A_1 up + A_2 down, say, then, if you decide to measure the 
>>> up/down state, and use the device doing that, you do not need to make the 
>>> inner product between phi and phi, but between the base state up and/or 
>>> down to get the probability given by the square of phi * up (to get the 
>>> probability of up) and the square of phi*down, to get the probability of 
>>> down. They will both depend on the value of A_1 and A_2. They are 
>>> respectively (A_1)^2 and (A_2)^2. Of course, we suppose that we have 
>>> renormalised the state so that (A_1)^2 + (A_2)^2 = 1 (which makes them into 
>>> probability of getting up and down).
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>> *The question then becomes how do we calculate the probability density 
>>> with the phase angles undetermined.  Are we assuming they are known given 
>>> the way the system is prepared? AG*
>>>
>>>
>>>
>>> Yes. The Born rule, written simply, is only that if phi = A_1 up + A_2 
>>> down, (so the state has been prepared in advance) then if you measure if 
>>> the object is in up or down, you will find up with a probability given 
>>> respectively by (A_1)^2 and (A_2)^2.
>>> All probabilities are relative to the state of the object and the choice 
>>> of what you decide to measure. It is always simpler to write the state in 
>>> the base corresponding to the measurement, so that the “simple” Born rule 
>>> above can be applied immediately.
>>>
>>> Bruno
>>>
>>
>> *For reference I repeat my last comment and add a significant point:*
>>
>>  If you take the inner product squared (Born's rule) using an 
>> orthonormal set of eigenfunctions, you get a sum of the form (a_j)^2 + 
>> (b_j) ^2  where A_j is the complex probability amplitude for the jth 
>> component, A_j = a_j + i * b_j. The cross terms drop out due to 
>> orthonormality, and the phase angles are implicitly determined by the 
>> relative values of a_j and b_j for each j. The question then becomes how do 
>> we calculate the probability density with the phase angles undetermined.  
>> Are we assuming they are known given the way the system is prepared? AG
>>
>> The question for me is how the phase angles are related to interference. 
>>
>>
>> But that is explained by may calculation above. You calculation does not 
>> make sense to me. You compute an inner product of the wave to itself? I 
>> don’t see the relation with your problem. 
>>
>
> *Obviously, you don't know how to apply the rule you speak so highly of, 
> Born's rule. To calculate the probability density of wf function psi, you 
> must calculate <psi, psi>.  Do you dispute this? *
>
>
> Yes, you need to put some projection operator (corresponding to some 
> eigenvalue you intend to measure) in between. 
> <psi,psi> is the amplitude of probability to go from the psi state to the 
> psi state, and should be equal to one (psi being normalised).
>

*Wrong! Not equal to one. I see you like to talk the talk, but refuse to 
walk the walk. Just read the 4th paragraph of the Wiki link. I correctly 
calculated the probability density for orthonormal eigenstates! AG  *

>
>
>
> *How the phase angles relate to interference is another issue, which I 
> think Phil explained. AG*
>
>>
>> The calculation above shows that the cross terms drop out due to 
>> orthonormality. 
>>
>>
>> Do it again, explicitly. Take the simple state phi = A_1 up + A_2 down. 
>> Up and down are orthonormal,
>>
>
> *Up and Dn are NOT orhonormal.  AG*
>  
>
>> but phi is not orthonormal with either up or down. If “up” means go to 
>> the left hole, and “down” is go the right hole, the amplitude A_1 and A_2, 
>> if not null, will interfere, even if only one photon is sent.The wave go 
>> through both silts, and interfere constructively along some direction and 
>> destructively along other direction, making it impossible for that photon 
>> to lend on those last place, like anyway, by the laws of addition of 
>> sinus/wave.
>>
>> But IIUC these are the terms which account for interference. 
>>
>>
>> I am not sure what you say here. The interferences comes only from the 
>> fact that we have a superposition of two orthogonal state, and that 
>> superposition is a new state, which is not orthogonal to either up or down.
>>
>> Thus, applying Born's rule to a superposition of states where the 
>> components are orthonormal, leaves open the question of interference.
>>
>>
>> That does no make sense. The Born rule just say that if you measure 
>> (up/down) on phi =  A_1 up + A_2 down, you get up with probability (A_1)^2 
>> and down with probability (A_2)^2. But if you do any measurement, the state 
>> beg-have like a wave, and the amplitudes add up, constructively or 
>> destructively.
>>
>> If you don’t understand that, it means you begin to understand quantum 
>> mechanics, as nobody understand this, except perhaps the Mechanist 
>> Philosophers …(which predicts something at least as weird and 
>> counter-intuitive). 
>>
>> Bruce wrote that the phase angles are responsible for interference. I 
>> doubt that result. Am I mistaken? AG
>>
>>
>> Yes, I’m afford you are. The relative phase (in a superposition) angles 
>> are responsible for the interference. A global phase angle changes nothing.
>>
>
> *If I am wrong, it's just because I assumed all interference comes from 
> the interactions due to the cross terms -- which cancel out for orthonormal 
> component states. Also, I never introduced a global phase angle in my 
> calculation. If you would do my calculation, or at least understand it, 
> you'd understand Born's rule.  I don't need to read Albert's book to 
> understand Born's rule. AG*
>
>
>
> Once you say that up and down are not orthonormal, I am not sure you have 
> studied the QM formalism correctly. Any two distinguishable eigenstates of 
> any observable are orthogonal (and normalised). 
>

*Right. I was mistaken. AG *

>
> I have no clue what you don’t understand in my use of the Born rule. You 
> definitely need to study Albert’s book, I think.
>

*Wiki shows I correctly calculated the probability density. Also I agree 
with Phil, and noted the error I made (not in any calculation, but in 
interpretation). Didn't you read it? AG  *

In your other post you mention wikipedia. No problem there? Actually you 
> can see that they do put the projection operator at the right place. You 
> can help yourself with a dictionary, but books and papers are better.
>


*You could help yourself by reading plain English. SEE PARAGRAPH 4 OF WIKI 
LINK. THEY CALCULATED THE PROBABILITY DENSITY AND DIDN'T PUT IN THE 
PROJECTION OPERATOR!  AG*

>
> Bruno
>
>
>
>
>
>
>> I really wish you to read the first 60 pages of David Albert’s book. Its 
>> exposition of the functioning of the interferometer is crystal clear. I am 
>> still not sure if you have a problem with the formalism or with the 
>> weirdness related to it. Read that piece of explanation by Albert, and if 
>> you still have problem, we can discuss it, but it would be too long (here 
>> and now) to do that here. 
>>
>> Bruno
>>
>>
>>
>>
>>
>>
>>
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