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2 weeks ago

[ 49 - Self-interference in 1D box ... again ]

Random Motion but wavelike statistics --> 3 peaks

(With wave reflections at the boundaries)
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Now that's great! Does the pattern depend on the box length?

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3 weeks ago

[ 48 - Memory-induced Self Interference ]

Here we see the influence of memory on self-interference in a quasi-1D cavity.

The stronger we vibrate of the bath (While staying below the Faraday Instability Thresold)
-->The closer we are to the Faraday Thresold
-->The higher the self-interference effect
--> The more noticeable will be the statistical pattern on the position of the dot

NB : please notice the shape of the standing wave that is created from sec 20 and onwards
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3 weeks ago

[47 - Quasi 1D - particle Tracking ]

Another manifestation of self-interference in a confined geometry with wave reflections at the boundaries.

The cavity is a channel in which the dot can only move along a straight line (With a little transverse motion, hence the "quasi" prefix)

The motion of the dot seems random, but when you look at it statistically, you get a position distribution with peaks.
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Are these peaks reproducible? The strange thing is that the pattern is asymmetric.

[ 46 - Quasi-1D DIY Setup ]

We can do the same kind of experiment in a mono-dimensional channel.

Please notice the high-level sophistication of the experimental setup 😂🤣😅

3 weeks ago

[ 46 - Quasi-1D DIY Setup ]

We can do the same kind of experiment in a mono-dimensional channel.

Please notice the high-level sophistication of the experimental setup 😂🤣😅
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[ 45 - square cavity with wave reflections ]

3 weeks ago

[ 45 - square cavity with wave reflections ]

In this case, the walls around the square cavity emerge above the oil level. So, Waves are reflected by the boundary.

The motion of the dot seems random but it likes to dwell in some preferential places. This can be seen by the naked eye on the video, and this can be measured with particle tracking : the droplet trajectory and its position statistics show that a statistical pattern emerges from the random motion.

We can compare this behaviour with the previous slide [44] : there, without wave reflections at the boundary, no statistical pattern emerges.

So we may think that the statistical pattern build up is driven by the reflections of the waves at the cavity boundaries.
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4 weeks ago

[ 44 - Square cavity without reflections ]

One image is worth a thousand words, they say ...
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[ 43 - particle tracking ]

4 weeks ago

[ 43 - particle tracking ] The goal of the experiment is to track the dot's motion while it is trapped in the square cavity, so as to be able to compute statistics on its position and velocity .... ... See MoreSee Less

[42 - Self-interference - Memory induced chaos ]

4 weeks ago

[42 - Self-interference - Memory induced chaos ]

So a walker can walk on its own footsteps and thus interfere with its own past. I like to call this phenomenon self interference.

We have seen this in a confined geometry like a circular corral. I reproduced this in a square corral.

Please note that in this video the waves do not reflect on the border of the square cavity : there is a "beach" on each side, with a shallow depth of oil, on top of which the waves are vanishing.
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[ 40-41 Quantization In a Harmonic potential ] The central force is attracting the walker towards the center. So, if the walker has an initial velocity, it will reach a circular orbit, which radius depends on the initial velocity.

But if the memory of the bath is high enough, it will have a significant influence because the walker will walk on its own footsteps while going round the circle trajectory. And it will reach a place where the waves created a few time ago are still present. 

These waves, maintained from the past thanks to the memory of the bath, will interact with the position of the walker. 

The resultant memory-induced effect is that only certain radius can be observed if the memory of the bath is sufficient.

That is what can be seen on chart [40] : 
- On the X axis is the memory the bath
- On the Y axis, the radius of trajectory observed

--> When memory is increased, only certain radius can be observed.

But we can go further : by varying the stiffness of the spring, we can create different kind of trajectories for the walker : circular but also lemniscate (trefoil-like) ...

If the Memory of the system is high enough, only certain trajectory are allowed : radius and angular momentum (that is the speed at which the walker is going round) are both quantified. [42]

--> Double quantization in a central harmonic potentialImage attachment

4 weeks ago

[ 40-41 Quantization In a Harmonic potential ] The central force is attracting the walker towards the center. So, if the walker has an initial velocity, it will reach a circular orbit, which radius depends on the initial velocity.

But if the memory of the bath is high enough, it will have a significant influence because the walker will walk on its own footsteps while going round the circle trajectory. And it will reach a place where the waves created a few time ago are still present.

These waves, maintained from the past thanks to the memory of the bath, will interact with the position of the walker.

The resultant memory-induced effect is that only certain radius can be observed if the memory of the bath is sufficient.

That is what can be seen on chart [40] :
- On the X axis is the memory the bath
- On the Y axis, the radius of trajectory observed

--> When memory is increased, only certain radius can be observed.

But we can go further : by varying the stiffness of the spring, we can create different kind of trajectories for the walker : circular but also lemniscate ("trefoil-like") ...

If the Memory of the system is high enough, only certain trajectory are allowed : radius and angular momentum (that is the speed at which the walker is going round) are both quantified. [42]

--> Double quantization in a central harmonic potential
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[39 - the ferrofluid experiment ]
And this is another very extraordinary version of the walker experiment. I was luck enough to go in Paris and visit Yves Couder’s lab. He was with his PhD student Stéphane PERRARD, who was finishing his PhD by that time.

I call tis experiment a “Tour de force” because they manage to attach a spring to the walker.

Of course, not a real spring : they have used a ferrofluid and a magnetic field to place the walker in a harmonic potential : at any moment, the walker is attracted towards the center of the apparatus with a force proportional to the distance to the center.

You can see the apparatus with two big coils and a small moving magnet for tuning the strength of the magnetic field.

That is, the stiffness of the virtual spring attached to the walker can even be tuned.

4 weeks ago

[39 - the ferrofluid experiment ]
And this is another very extraordinary version of the walker experiment. I was luck enough to go in Paris and visit Yves Couder’s lab. He was with his PhD student Stéphane PERRARD, who was finishing his PhD by that time.

I call tis experiment a “Tour de force” because they manage to attach a spring to the walker.

Of course, not a real spring : they have used a ferrofluid and a magnetic field to place the walker in a harmonic potential : at any moment, the walker is attracted towards the center of the apparatus with a force proportional to the distance to the center.

You can see the apparatus with two big coils and a small moving magnet for tuning the strength of the magnetic field.

That is, the stiffness of the virtual spring attached to the walker can even be tuned.
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[38 - Circular corral DIY ]

1 month ago

[38 - Circular corral DIY ] I have reproduced this experiment with a circular cavity : but in my case, the boundaries of the circular cavity were exceeding the level of oil in the corral.

The difference in the boundary conditions have great importance because the waves do not behave the same. In a case they reflect at the boundary, on the other case they are simply vanishing on the region with shallow depth.

Nevertheless, I found a similar result with circular – like region where the walker was more likely to be found.

( Slides 42-45 will show a DIY reproduction of this experiment in a square cavity, showing the influence of boundary conditions )
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[37 - Circular corral ] Now :  one of the most intriguing and well know experiment, which was built together by the French and American team at Paris and MIT.

Put a walker in a circular cavity : it will follow a random trajectory inside the circular corral. But if you follow it long enough, you will notice that the walker is most likely to be found in specific region of the circular corral. This is what is depicted in the right picture : a statistical patterns builds up even though the droplet motion seems to be random

By the words of Professor Bush himself : “The observed statistical behavior is thus roughly analogous to that reported in the quantum corral experiments (Crommie et al. 1993) in which the density of electrons trapped on a copper substrate was found to have a wavelike pattern with the de Broglie wavelength,, and a form prescribed by the corral shape “

1 month ago

[37 - Circular corral ] Now : one of the most intriguing and well know experiment, which was built together by the French and American team at Paris and MIT.

Put a walker in a circular cavity : it will follow a random trajectory inside the circular corral. But if you follow it long enough, you will notice that the walker is most likely to be found in specific region of the circular corral. This is what is depicted in the right picture : a statistical patterns builds up even though the droplet motion seems to be random

By the words of Professor Bush himself : “The observed statistical behavior is thus roughly analogous to that reported in the quantum corral experiments (Crommie et al. 1993) in which the density of electrons trapped on a copper substrate was found to have a wavelike pattern with the de Broglie wavelength,, and a form prescribed by the corral shape “
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[36 - Orbital Quantization induced by path memory]

If a walker is submitted to Coriolis Forces, by rotating the vibrating bath, its trajectory is circular. 

But if the memory of the bath is strong enough, only certain radii are allowed.

This is a memory-induced quantization of trajectories

1 month ago

[36 - Orbital Quantization induced by path memory]

If a walker is submitted to Coriolis Forces, by rotating the vibrating bath, its trajectory is circular.

But if the memory of the bath is strong enough, only certain radii are allowed.

This is a memory-induced quantization of trajectories
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[35 - Tunnel Effect and semi-transparent mirrors ]This tunnel effect can be measured and played with.

It depends on the size and speed of the walker,  and on the depth of the bath at the boundary. If you choose those parameters carefully, you can build an obstacle that behaves like a semi-transparent mirror to walkers.

1 month ago

[35 - "Tunnel Effect" and "semi-transparent mirrors" ]This tunnel effect can be measured and played with.

It depends on the size and speed of the walker, and on the depth of the bath at the boundary. If you choose those parameters carefully, you can build an obstacle that behaves like a semi-transparent mirror to walkers.
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[ 27 the DotWave Computer ]

Did I forget this amazing feature ??

I will now conclude the description of walkers with this surprising way to look at it : the whole system we described can be analyzed in term of a computer system.
At each impact, the dot writes information on the wave  : the local circular standing wave which is created during impact encodes information in the bath
The bath retains this information thanks to its wavelike memory because ehe standing wave is damping slowly.
When the dot bounces on the wave, its direction is modified by the local slope of the wave .The dot is reading information from the wave. The slope of the wave is the messenger

2 months ago

[ 27 the DotWave Computer ]

Did I forget this amazing feature ??

I will now conclude the description of walkers with this surprising way to look at it : the whole system we described can be analyzed in term of a computer system.
At each impact, the dot writes information on the wave : the local circular standing wave which is created during impact encodes information in the bath
The bath retains this information thanks to its wavelike memory because ehe standing wave is damping slowly.
When the dot bounces on the wave, its direction is modified by the local slope of the wave .The dot is reading information from the wave. The slope of the wave is the messenger
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[ 34 - tunnel effect ]

2 months ago

[ 34 - tunnel effect ]
Another analogy is the "tunnel effect."

The walker is trapped in a square box whose boundaries are made of submerged walls.

Because the oil is not deep enough on top of this wall, the waves associated with the walker are very quickly attenuated in this region, and so usually the walker is reflected by the wall and does not go through…most of the time… but sometimes it does go through … The walker has a certain probability to go through this barrier, which depends on its lenghth : this is similar to the quantum tunnel effect.
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[33 - Single particle creates interference after double slit ]

2 months ago

[33 - Single particle creates interference after double slit ]

As I told you, I did not manage to replicate the slit experiment.

But While trying, I managed to get this nice effect [ which is by the way the most successful video on youtube. 😉 ]

There is a walker coming from the upper left corner towards a 2-slit opening.

The waves of the walker pass through the first slit ,on the right

Then the walker passes through the second slit on the left.

Thanks to the bath memory, the waves which passed through the right slit are slit there after the walker has passed the slits.

And those remaining standing waves create an interference pattern with the waves created by the walker after it has passed the second slit.
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[30, 31, 32  - Slit Experiments ] People at the Bohr Institute at Copenhagen were intrigued by the sit experiment with walkers and managed to replicate it with a greater statistical signification : but they did not see any interference pattern.

Among the authors of the article, we can note the presence of Thomas Bohr : the grandson of Niels Bohr. 

Niels Bohr was one of the founding father of the Copenhagen interpretation of Quantum Physics, which opposes the pilot-Wave Theory developed by Frenchman Louis de Broglie.

For the record, [31] shows another replication of the slit diffraction experiment of 2006, that reported a failure …

The Belgium  Quandrop Team at Liège did replicate the experiment with a numerical simulation. [32]

Their simulation is based on the mathematical expressions presented before, with many approximations to take into account the reflections nears the all boundaries, which are very difficult to calculate completely, assuming many approximations.

They found an interference pattern, but also noticed “the fundamentally different nature of the Bohmian force upon a quantum particle as compared to the force that the surface wave exerts upon a droplet. “

In Bohmian mechanics, the particle momentum is the gradient of the phase of the wave function.
With walkers, the gradient of the wave field is used directly to calculate the acceleration of the walker.

“In view of this, it is not obvious to what extent the present classical analogy of quantum Wave-Particle duality can be maintained in more complex situations involving, e.g., more than one droplet. “

Further theoretical and experimental studies are clearly required to address this issue.”Image attachmentImage attachment

2 months ago

[30, 31, 32 - Slit Experiments ] People at the Bohr Institute at Copenhagen were intrigued by the sit experiment with walkers and managed to replicate it with a greater statistical signification : but they did not see any interference pattern.

Among the authors of the article, we can note the presence of Thomas Bohr : the grandson of Niels Bohr.

Niels Bohr was one of the founding father of the Copenhagen interpretation of Quantum Physics, which opposes the pilot-Wave Theory developed by Frenchman Louis de Broglie.

For the record, [31] shows another replication of the slit diffraction experiment of 2006, that reported a failure …

The Belgium Quandrop Team at Liège did replicate the experiment with a numerical simulation. [32]

Their simulation is based on the mathematical expressions presented before, with many approximations to take into account the reflections nears the all boundaries, which are very difficult to calculate completely, assuming many approximations.

They found an interference pattern, but also noticed “the fundamentally different nature of the Bohmian force upon a quantum particle as compared to the force that the surface wave exerts upon a droplet. “

In Bohmian mechanics, the particle momentum is the gradient of the phase of the wave function.
With walkers, the gradient of the wave field is used directly to calculate the acceleration of the walker.

“In view of this, it is not obvious to what extent the present classical analogy of quantum Wave-Particle duality can be maintained in more complex situations involving, e.g., more than one droplet. “

Further theoretical and experimental studies are clearly required to address this issue.”
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29- Single Particle Diffraction

2 months ago

[ 29- Single Particle Diffraction ] You can also place a slit in the path of a walker.

This is what Yves Couder and his team in Paris did way back in 2006.

They measured the diffraction angle of a walker after it has passed through an obstacle made of a submerged wall open in the middle by one or 2 slits.The result looks like the famous diffraction pattern observed in Young’s Experiment.

Please note that only 125 successful event were recorded, so the statistical meaning of this experiment is poor. Several hundred droplets were generated by hand with a toothpick, and only the good trajectory were kept : when a droplet enters the slit perpendicularly.

A similar result was obtained with two slit.

Here a video of my own replication attempt .... 😉
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By the way.....see you in London ?

2 months ago

By the way.....see you in London ? ... See MoreSee Less

 

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Wish I go. Will there be any transmission?

They have lots of video for EmQM 2015

[ 28 - Orbital Quantization ]

2 months ago

[ 28 - Orbital Quantization ] Now, the second part of this presentation : some analogies with the quantum world.

We have seen that walkers are a real life, human-scale realization of a pilot-wave system. That is, a system in which a particle creates an associated wave, and in which this very same wave guides the particle.

But we can now play with walkers, and see how they interact with other walkers, with obstacles, with external forces etc ...

A first interesting phenomenon is the interaction of two walkers : they can start gravitating around each other, like in this video.

It can be measured that the distance between two walker, when they go round each other like this, is a approximately a multiple of the Faraday Wavelength. (To a small additive offset )
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[26 - non-locality ] So , the trajectory of a walker does not only depend on its position and speed at a given time, but it also depends on its history. That is what we have called path memory.

This is a kind of temporal non locality : we need to consider the whole past trajectory if we want to calculate the future trajectory of the walker.

It can be shown that this is equivalent to a spatial non locality : without knowledge of the past trajectory of the walker, we can still calculate its future trajectory  

We need to know its position and speed of course, but also, on top of that, we need to have an exact and full knowledge of the full wave field at  a given time t.

This is a spatial non-locality.

2 months ago

[26 - non-locality ] So , the trajectory of a walker does not only depend on its position and speed at a given time, but it also depends on its history. That is what we have called path memory.

This is a kind of temporal non locality : we need to consider the whole past trajectory if we want to calculate the future trajectory of the walker.

It can be shown that this is equivalent to a spatial non locality : without knowledge of the past trajectory of the walker, we can still calculate its future trajectory

We need to know its position and speed of course, but also, on top of that, we need to have an exact and full knowledge of the full wave field at a given time t.

This is a spatial non-locality.
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[25 - Wave Equation ] : The wave equation describes how the wave fields evolves through time.

The wave field is the sum of the fields created at all previous bounces of the dot.

Each of these is a circular standing wave (Mathematician model this as a Bessel Function J0 ) that is slowly damped.

The damping parameter is the memory of the system : the less damping, the more memory.

From this, we can see that the evolution of the wave field depends on the whole history of the trajectory of the walker.

2 months ago

[25 - Wave Equation ] : The wave equation describes how the wave fields evolves through time.

The wave field is the sum of the fields created at all previous bounces of the dot.

Each of these is a circular standing wave (Mathematician model this as a Bessel Function J0 ) that is slowly damped.

The damping parameter is the memory of the system : the less damping, the more memory.

From this, we can see that the evolution of the wave field depends on the whole history of the trajectory of the walker.
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[24 - dot Equation ]And now for some mathemagics !

The equation of motion of a walker is split in two part : one for the dot and one for the wave.

The dot follows a classical Newtonian equation mass times acceleration equals the sum of force applied to the dot.

Among those forces, there is what we call a Wave force : the force applied by the wave to the droplet at the bounce. That force is proportional to the slope of the wave (What mathematicians call the gradient) , and thus, it depends on the local shape of the wave field at the position of the dot.

2 months ago

[24 - dot Equation ]And now for some mathemagics !

The equation of motion of a walker is split in two part : one for the dot and one for the wave.

The dot follows a classical Newtonian equation mass times acceleration equals the sum of force applied to the dot.

Among those forces, there is what we call a Wave force : the force applied by the wave to the droplet at the bounce. That force is proportional to the slope of the wave (What mathematicians call the gradient) , and thus, it depends on the local shape of the wave field at the position of the dot.
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[23 - Path Memory ]

2 months ago

[23 - Path Memory ] Add a droplet on top of this bath with memory, and you will get : path memory.

Each time the droplet touches the bath, it creates a circular standing wave that will continue to oscillate a certain time.

This will repeat 30 times per second, so that the field created by the droplet all over the bath is the superposition of the fields created at each bounce.

In the wake of its path, the walker creates a trail, made of a string of localized circular standing waves, that continue to oscillate as long as the memory of the bath allows (See 22 - bath memory)

All these waves interfere and that interference pattern can be observed in the wake of the walker.
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[22 - Bath Memory]

2 months ago

[22 - Bath Memory]

In 2011, 6 years after the initial Discovery, a new important step was made in the study of walkers. The Paris Team showed that the bath on which the droplet were moving had a memory.

Oh a strange kind of memory : a Wave memory !

Look at the surface of the bath, when it is vibrated below the instability threshold : its surface is still. Lets make a small perturbation with a tooth pick : this perturbation will create a local circular standing wave, that will take a certain time to dissipate.

That time is the amount of Memory of the system : the bath memory.

We can tune it by adjusting the vibration amplitude closer and closer to the instability threshold without exceeding it.
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2 months ago

[21 - Bound state ] Here we can see two different views of the same phenomenon : the bound state of two Walkers, through a wave mediated interaction.
- With a visible vertical dynamics : "boucing walkers"
- With a filtered vertical dynamics : "levitating walkers"
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2 months ago

[20 - Stroboscopic Effect] But the vertical dynamic of the droplet, or the study of its different ways of bouncing, is not the main interest of Walkers.
The main interest is : the association of the dot and the wave.

Lets look at this very same video [19], but we keep only frames corresponding to the times where droplets are on the top of their trajectory. Doing so, we can erase the vertical dynamic through stroboscopic effect, as with any periodic motion.

The dot vibrates vertically 30 times per second. If we take a movie at 30 images per second, we will get the impression the dot is not moving vertically : it will float .

And its associated local Faraday wave field will follow. But we will not see it vibrate.

In all videos that will come up next, we will now mostly use this stroboscopic effect.
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2 months ago

[19] The dot creates the wave. The wave guides the dot

Let's see very concretely how a walker is made : all you need is ...a tooth pick 😄

Dip it in the vibrating bath and remove it quickly : a droplet will form and the magic will begin.

The Wave-Particle duality of this phenomenon appears :
- Without the dot, there is no wave. The dot is creating the wave.
- Without the wave, there is no motion. The wave is guiding the dot. It is the local slop of the wave at the point of contact that is guiding the dot.

That is why walkers are called a realization of a pilot wave system.

What you can also notice here is the bounce of the droplet : it is not a single drop as you may think at first: the droplet touches the bath twice quicky, and that can only be seen with a fast camera. This one of different kind of bouciong for droplets....
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[18] The walking droplet behaviour occurs only for certain combination of the forcing (or driving) acceleration and of the droplet size

Now,  take a look a these two phase diagrams. One is from 2006 and the other from 2013 : we can see how science has made progress in between those two dates ! 
- On the horizontal axis, we have the acceleration of the vibrating bath : the forcing, or driving, acceleration.
- On the vertical y axis, the size of the droplet (Or the vibration number, which is a more elaborate adimensionnal number containing information about the droplet geometry and physical characteristics)

There is only a small portion of this parameter space which allows for the formation of a walking droplet.

2 months ago

[18] The walking droplet behaviour occurs only for certain combination of the forcing (or driving) acceleration and of the droplet size

Now, take a look a these two phase diagrams. One is from 2006 and the other from 2013 : we can see how science has made progress in between those two dates !
- On the horizontal axis, we have the acceleration of the vibrating bath : the forcing, or driving, acceleration.
- On the vertical y axis, the size of the droplet (Or the vibration number, which is a more elaborate adimensionnal number containing information about the droplet geometry and physical characteristics)

There is only a small portion of this parameter space which allows for the formation of a walking droplet.
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