Uniqueness: The Physics Problem That Shouldn't Be Solved

The Uniqueness Theorem can PROVE that this problem only has one possible solution... so however we can find it (e.g. guessing), we know we've got the right one!
In this video, we'll be taking a look at how this uniqueness theorem is derived for the Poisson and Laplace equations in electrostatics.
To begin with, we'll recall what an electric field and potential field are in electrostatics. The electric field is a vector field generated by charged objects. It tells us about the forces experienced by other charged objects placed in the field. A potential field is a scalar field that also gives us similar information. In fact, the gradient operator applied to the potential field gives us (the negative of) the electric field!
Both the electric and potential fields follow Poisson's equation for electrostatics, if they are to obey the laws of physics. This equation relates the divergence of the gradient of the potential field, to the charge density in any region of space we want to study.
If we consider a particular region of space, such as a sphere, we can simplify our discussion if we assume there are no charges in this region. With this assumption, the charge density becomes zero and the Poisson Equation becomes the Laplace Equation. Also, we can assume we already know the potential values at the surface of our region of space since we measured them. (The potential field is the same as the voltage that we use when talking about electric circuits). The measured potential values make up our "boundary conditions".
At this point, we want to calculate what the potential field is everywhere in our region of space, and we know it must satisfy the boundary conditions (i.e. be the same as our measured values on the boundary).
We can now assume that there are two solutions that are different within our region of space, but the same on the boundary. And then we can subtract one from another to study a new quantity, which also happens to obey the Laplace equation. Additionally, this new quantity is zero on the boundary since our two assumed solutions matched our measured values on the boundary.
Then we can apply some general vector calculus identities to show that the value of our new quantity must be a constant. But since this value is also zero on the boundary, it therefore must be zero everywhere. In other words, our two assumed solutions must be the same as each other if they are to subtract to be equal to zero. We've just found that it is impossible for there to be two different solutions to our problem.
This theorem is the uniqueness theorem, and shows that there is only one possible solution to this problem. This is useful because it means we don't need to solve the problem exactly to figure out the right solution for our scenario. Even if we can GUESS the solution, and then plug it back in to our equations to check that it works, this is ok. The reason is that if we somehow find the solution, we know that it's the only possible one - so we've not accidentally found a solution that works, but doesn't apply to our scenario!
The discussion in this video limits itself to "Dirichlet" boundary conditions (potential known on the boundary). Read this page for more information on "Neumann" boundary conditions (electric field known on boundary) and mixed boundary conditions: en.wikipedia.o...
For a more detailed breakdown of the Poisson and Laplace Equations, check out this video: • Poisson's Equation for...
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Timestamps:
0:00 - The Problem That Only Has One (Guessable) Solution
0:45 - The Electric Field Generated by Charged Objects
2:04 - The Potential Field Generated by Charged Objects
3:05 - The Relationship Between the Potential and Electric Fields
3:45 - The Poisson Equation, Charge Density, and Boundary Conditions
6:05 - The Laplace Equation for Electrostatics (Charge Density = 0)
6:17 - Assuming There Are Two Possible Solutions (V1 and V2)
8:01 - Proving There is Only ONE Possible Solution (V1 = V2)
9:03 - The Uniqueness Theorem, and Why One Solution is Useful - Guesswork!
10:32 - Dirichlet, Neumann, and Mixed Boundary Conditions
10:56 - Summary

[@ParthGChannel]

Hi friends, thanks so much for your support! If you want to see a more detailed breakdown of the Poisson and Laplace Equations discussed here, please check out this video I made on my channel recently: kzbin.info/www/bejne/oWqUfHd_qd2cnJI Also, as always do let me know what other topics I should cover on this channel :)

[@georgerafa1775]

You should make a video about string theory, the basic ideas and some maths revolving around the theory!

[@jaiho8983]

Vector potential

[@chealseafc1433]

also on string theory

[@alwaysdisputin9930]

@Parth G Masterful explanation. TY. There is something I'd like you to explain please: Helen Czerski said: H₂O has a kink in it which means O - it forms bonds which means on the surface it's like an elastic sheet or skin called 'surface tension' / \ + H H + - O / \ + H H + - O / \ + H H + So I think when a wave starts, 1st of all the 1st molecule moves upwards - O / \ H H + + - O / \ H H + + - + - O / \ + H H + + - O / \ + H H + The 1st molecule pulls the 2nd molecule upwards - O / \ H H + + - O / \ H H + + - O / \ H H + + - + - O / \ + H H + + - O / \ + H H + So it's like if you have a cable lying flat & you pick up 1 end & lift it: the neighbouring parts of the cable will get pulled up because of the intermolecular bonds which I imagine are like springs. Ok so now let's say we have a rod with a lot of electrons congregating at the bottom | | | Rod | | | | | | | | | | - electrons | ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← | - - - - - - - | & like you say in the video, electric field lines form i.e. any positive particle's going to find the electrons attractive ¯¯¯¯¯¯¯¯¯¯¯¯¯ If we then move the electrons upwards | - electrons | ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← ← | - - - - - - - | the electric field lines will end up here | | | Rod | | | | | | | | | ¯¯¯¯¯¯¯¯¯¯¯¯¯ But! The field lines do not move upwards instantly. There's an intermediate stage that looks a lot like the water molecules & the cable: | - electrons | ← ← | - - - - - - - | ← | | ← | Rod | ← | | ← | | ← | | ← | | ← ← ← ← ← ← ← ← ←← ← ← ← ← ← ← ← ← ¯¯¯¯¯¯¯¯¯¯¯¯¯ So my question is why don't the field lines move up instantly? The water molecules don't move up instantly because they aren't a solid with rigid bonds like if we held a broom or pen out horizontally in front of us & lifted it. We could create a model of the water by connecting springs to masses: spring - mass - spring - mass - spring - mass etc Thus the H₂O molecules don't move up instantly because the springs stretch: a spring has to get stretched out & reach a point where it isn't prepared to stretch any further before it'll start lifting up the neighbouring mass. So is an electric field like a bunch of springs & masses?

[@SN26111]

Please please please make video on Drichlet and nuemen boundary conditions 🙏🙏🙏🙏

[@soorajsahani1603]

this topic 'Uniqueness theorem" was actually my guess when I saw "when a problem has one solution"

[@ParthGChannel]

Great guess :D

[@silvercraaken2164]

You found the one solution

[@girindrasinghrathore8418]

Warning ⚠️ : Physics is Addictive

[@rudraveer2112]

But this Addiction is good

[@localtitans4166]

😂😂

[@irri4662]

E= mc^□ 🤪

[@mjzudba5268]

@@irri4662 □=2

[@irri4662]

@@mjzudba5268 yep

[@jennymagnes7718]

This is the best video I have seen on the uniqueness theorem involving Laplace's equation. Most other videos mix up the boundary conditions for the potential and the potential function itself. Some of the comments disputing your logic forget that the Laplacian only holds for a charge-free region.

[@mzshmkn]

holy shit i cant believe why everything sounded familiar it was because this was exactly what i was studying a couple weeks ago in class. amazing!

[@petersmythe6462]

If there's only one configuration inside that produces that configuration on the boundary, then, at least in principle, the boundary contains all of the information of its interior volume... Which means that we've got perfect holographic representation of the field in a 3D space by looking at a 2D boundary of that space.

[@filippozar8424]

Nope. Boundary conditions alone are not enough to solve for an electrostatic system. You need to know charge distribution at every point in order to know which equation you are even trying to solve. What you said can hold true in a vacuum but then again you needed additional information that you are in fact working with a vacuum, an information which isn't contained in your boundary condition alone.

[@denniszhang9278]

@@filippozar8424 What? From perfectly described BC's there is only one charge distribution that could produce it (providing you don't have any enclosed conductive shells). I'm not sure what your argument is... the equation you would solve to obtain the inner charge distribution would be the inverse of Poisson's equation, which you could solve numerically with some sort of optimization algorithm.

[@philipphaim3409]

@@denniszhang9278 Correct me if I'm wrong, but I think there is a simple counter example to your point. If you take any spherically symmetric charge distribution and boundary, the resulting potential field will always be constant at that boundary (as the Laplace operator is also rotationally invariant). You can therefore create the same boundary conditions with different charge distributions. E.g. a point charge would create the same field at the boundary as a correctly scaled constant charge distribution.

[@Ender240sxS13]

@@philipphaim3409 so we actually had to consider basically this exact problem in my 2nd year university physics class. What you end up with is that in order to create the same field potential at the boundary the distribution of charges inside has to be identical to a point charge. Of course in practice things like sending resolution and the charge density within your boundary are hugely important. So if you could perfectly measure the field potential on a surface you could then identify the unique positions of every electron within the surface (ignoring quantum mechanics) of course in reality you cannot achieve infinite precision. So what this gives us is actually a really really powerful tool. It would be impossible for us to model every single charged particle in a given system, however we can describe a much simpler system that gives us boundary values that match up with what we measure at the resolution we are capable of measuring, the differences in the actual field vs the modeled field then are so small that they are beyond our capability of measuring. This allows us to model all kinds of complex systems. This uniqueness principal and the behavior of vector fields is hugely important not just for purely theoretical stuff. I study aerospace engineering and for aerodynamics we find that airflow behavior can actually be modeled by a scalar velocity field potential, the gradient of which gives the actual flow direction and velocity. The entire field of Computational Fluid Dynamics is built on the foundation of first describing the boundary conditions and then using numerical methods to solve the field equations and thus describe how the fluid flow through the region, allowing you to then calculate things like the pressure acting at any point on a surface and thus the lift and drag forces seen by that surface.

[@pinklinecabs3967]

Whenever i see your videos not only i get knowledge but to learn more about it.♥

[@radiuscubed7985]

Nice explanation! This literally just came up in my graduate E&M, so this really helped!

[@jensphiliphohmann1876]

04:27f > _...the gradient of the gradient..._ Not rather the divergence of the gradient?

[@rayeesabdulla1916]

You are so Good man. thanks a lot.. Thanks a hundred times for this beautiful video.. Which book do you follow to learn these concepts?

[@jessstuart7495]

The reason you must always define a boundary condition is because the classical electric potential blows up to ±∞ at r=0.

[@BrynSCat]

A single static charge is Q.G..to avoid its own collapse it has to exist as a kinetic energy eternal “ER-ER” bride. This avoids singularity & infinity Static charge is its own C.P. inverse. A 3-sphere.(Spinor)Geodesic 90Gly. The speed of light is Path of least action between these symmetries.(between 2 stereographic projection(cos^2) Ep = static charge. Solve the static charge problem in G.R..This is why we see so much duality ,we are causality of duality.

[@vishalmishra3046]

*Electric charges also create curvatures in space-time* Mass curves space-time and is also affected by curvature in space-time. The variation in gravitational potential is modelled as acceleration vector field at any space-time co-ordinate. This is similar to electric field being the negative derivative (gradient) of electric potential. Watch the motion of a charged sphere in an electric field (with no other mass or gravitational field around) to measure the space-time curvature caused by electric field. The resulting motion creates magnetic field, force and magnetic-curvature in space-time. A "more-General" Theory of Relativity (mGR) unifies electro-magnetic force with Gravity from "less-General" Theory of Relativity (lGR) explaining all forces that curve and stretch space-time (and explains how dark-matter curves space-time and dark-energy stretches space-time).

[@KaliFissure]

Great video as always although I think our convention of charge designation is backwards. The manifold is lopsided. Gravity says the next moment is more dense than previous. This is why cosmological constant was introduced (before Big Bang also required dark energy and dark matter). The next moment is more dense, this is an inflow, a convergence. The particle that signifies mass is proton so the charge of proton should also be convergent. Electrons have fields which are expansive and repellent filling all of space. Divergent. If there were only one atom of hydrogen in all space the proton would be a point and the electron field would extend to infinity. Negative charge flows towards positive. The future is more dense

[@rogerforsman5064]

Well not exactly similar! Gravity only attracts (as we know it atleast)

[@Sid_R]

Hey Parth, great video as always! However I have a small doubt, I’d appreciate it a lot if you would clarify it. You mentioned { ∇^2 V = -ρ/ε0 } there, it looked quite similar to gauss law { flux = net charge enclosed/ ε0 } to me. Tell me, does ∇^2 V represent electric flux density? I should mention, I only understand the topic to the level of high school physics. So please forgive me if I’m speaking nonsense, thanks again. :)

[@Hotmedal]

Yes. en.wikipedia.org/wiki/Gauss%27s_law#Differential_form

[@dogetendopain6171]

Left side of eqn is type of vector differentaial eqn.....nd gauss law is same but in integral Form

[@isaacsaxton-knight7708]

@@Hotmedal No, -gradV = E. And divE = ro/epsilon.

[@satyamkumarmal7783]

Charge density is not equal to the charge 🙄

[@satyamkumarmal7783]

But they can be inter related .

[@manikdas1429]

Wow great video mate. Make one for Magnetic vector potential too

[@ParthGChannel]

Thank you very much! I've discussed magnetic vector potential a little bit in this video already :) kzbin.info/www/bejne/j37NdWugmtuHi9k

[@kushagrac-2517]

Brother just out of context any tip u wanna give or make a video on it for young ones like me to become a great physicist ??

[@Chewbacca0702]

Hi Parth, you are an inspiration. I am a career changer who is going into education, and I am taking notes on your lecture style. If you can simplify and communicate Poisson's equation for an engineer, it gives me hope said engineer can be as clearly spoken on topics like newtonian mechanics. Thank you for your contributions to science on KZbin.

[@beaconofwierd1883]

Does this mean that if you put a huge metal sphere around the earth and measured the potential everywhere on the sphere you could (theoretically) know exactly how all electric charges on earth was distributed? :O Doesn’t this also mean that we only need 2 dimensions to express charge distribution in 3D :O

[@aienbalosaienbalos4186]

Well, this result is true for the Laplace equation, meaning when the charge density is 0. The charge density inside earth is not 0. So the result as stated in this video does not apply when the region of space has charges inside it. That being said, I have no idea if it applies more generally to that case or not.

[@beetlesstrengthandpower1890]

No :(, you then have to solve Poissons equation, not Laplace. Since the charge desnity is not zero inside.

[@tvit]

Ok, this was quite nice explanation! Earlier I found some of your videos quite not explainy enough, or then I just wasn't ready for those topics :p

[@wayneyadams]

I could never get use to calling it nabla, that always seemed like some kind foreign word. "Did you nabla this morning?" "I was late for work, so I'll nabla on my coffee break."

[@ayush-ind]

Superb explanation!! ✨

[@Aritra-M4Glacier]

Ok you consider the Dirichlet boundary condition that's why you get V1=V2... On the other hand if you use Neumann's condition you will get V1-V2=constant.

[@jlpsinde]

Your videos are so helpful and high quality!

[@lock_ray]

It's surprising how common this "guessing" method is as a foundation for solving differential equations. Though usually you'd have guess a certain family of solutions instead of the specific one.

[@ParthGChannel]

Yeah it pops up quite often!

[@SN26111]

Please please please make video on Drichlet and nuemen boundary conditions 🙏🙏🙏🙏

[@TheSandkastenverbot]

So which physics problem should I not attempt to solve? 🤷‍♂️

[@Aritra-M4Glacier]

Problems on method of electrical images

[@anthonyguerrera191]

Can you do a video on the Dirac delta equation? When finding the divergence of an electric field due to a point charge at the point charge the value goes to infinity, and at any other point, the divergence is 0 without using dirac delta.

[@gabrielferreira-pu6ep]

This was really good I wish I could like a video twice

[@user-sg8my2nm3r]

thanks, this problem appeared in princeton university physics competition !!!!!

[@chealseafc1433]

Great video. parth can you explaine to me what string theory is? i mean what property it detemines about the particle that we deal with the mass or.. what ever it is or it creates the particles themselves

[@helenabc219]

Hi , thanks for your useful videos , why Cullen's law is inversely related to the square of the distance? I mean why distance should be square?!

[@DeclanMBrennan]

The potentials on the 2D boundary unqiuely determining the potentials of the 3D volume enclosed seems somewhat reminiscent of a duality and the holographic principle.

[@philipphaim3409]

This ONLY holds if you first fix your charge distribution! In this video, he demonstrated it for rho=0, if you look at the potential created by a different charge distribution, the difference of the fields doesn't have to be zero, even if the values at the boundary are identical.

[@DeclanMBrennan]

@@philipphaim3409 OK, but as far as I understand it :-) , if there are no enclosed charges, then the boundary potentials imply the distribution in the enclosed volume and visa-versa which is sort of a 2D3D duality very slightly reminiscent of the holographic principal (although admittedly I'm making too big a deal of it).

[@michaeljburt]

Parth, is there a "uniqueness theorem" equivalent when solving for the canonical ensemble in Liouville's theorem (under equilibrium conditions) in stat. mech.? See section 2.2 of Pathria Stat Mech if interested. I always wondered if there was a mathematical way to show that the canonical ensemble was the ONLY solution to this PDE. Specifically, that, [p, H] = 0 Yields ONLY, P(E) = Aexp^(-E/kT)

[@timeussubanath3900]

Hii!! Parth G... Can you answer to this question.. Are Physicist a scientist?

[@snehasismaiti342]

Great physics content

[@dogetendopain6171]

Can u please make a video on electrostatic energy....nd how to locate electrostatic energy between pair of charges

[@jairathod5322]

The 'guess' makes it iterative...but with sufficient knowledge of the behavior of the system we can make less iteration...

[@keerthi1346]

omg i was looking everywhere to try to understand this!

[@jnhrtmn]

With every charge issue, there is also a mass issue. That bothers me, because the declaration of a constant can become socially solidified after a few hundred years. What if the Strong force was invented to keep charge constant?

[@PrivateSi]

The Electrostatic Force is an interesting one. I am not sure how well the field can be measured seeing as an electron and positron only briefly form Positronium before annihilating (or whatever..... dark matter candidate, cough, cough - if QCD turns out to be as fundamentally flawed as it seems!).. Positron + electron forms Hydrogen in a short amount of time. I see an electron as a ball of Strong Mass Spin loops surrounded by an in-out field vibration standing wave of charge. -- I don't buy the magnetic field-like structure commonly shown, I see spherical vibes turning into longitudinal vibes in a flux tube connected to each central, tiny mass spin energy ball.. In the case of hydrogen, 2 flux tubes from 1 electron connected to each up quark in the proton that sits or orbits in proton standing wave energy level shells.. Again, in-out compression-stretch vibes in the quantised EM field wave medium with +ve and -ve shells blending into each other.. Proton energy shells come from vibrating up-quarks causing spherical standing waves.

[@javaidnasir1460]

Hello I am javaid are you explain me how to electric field rotate

[@graybot8064]

I understood none of this. I don't get the aspect of the conclusion. So V1 and V2 look the same from the outside, so they can be treated the same, from the outside? Is that what you're saying?

[@mal2ksc]

No. If V1 and V2 look the same from the outside, they _necessarily must be the same._ Not just treated as the same due to a lack of sufficiently precise information, but identical.

[@rammycanales3784]

Maybe I didnt get it right, but is this also some kind of proof that any electric field can be only generated by one single configurartion of charges? And if it is like that, then everywhere else, whenever a vector field is used, this also applies. But i dint really get how do you determine “boundries” in anything else but on electric fields.

[@andrin1248]

No not directly. In a givem volume you can have the same electric field generated by different charge distributions outside of that volume. What it says is that if you have a volume where you know the field on the boundary, then you know the field inside.

[@MohdIrfanZ7]

Brilliant Explanation.Love It.

[@rustycobalt5072]

Q: If you must draw a sphere around charges, but that sphere is infinitely thin (and assuming charges are point particles) Then what would you get if you always drew it as the smallest sphere containing them, or simply always pass in the maximum possible density?

[@andrin1248]

You need to calculate a limit. Since you divide the charge through the volume, which goes to 0, you get 0/0. But this has to be done in a limit, so this "0/0" can become any real number (or a distribution).

[@SAAARC]

Acha laga. Jiyo mere bhai

[@vijaysinghchauhan7079]

At 3:28 the vectors drawn after taking the gradient of V is misleading and not exactly accurate. For example, look at 4 at the top right corner it should have pointed to 8 and not 1. I love Parth's videos anyway 💕.

[@TheLevano22]

E = -∇V, the minus sign is essential.

[@t0k4m4k7]

Is it me or this video has an explation working principle of the faraday cage?

[@iakhtakh4248]

How the title related to the content?

[@theartisticactuary]

Also can be used in the world of finance. Why should I jump through hoops deriving a formula for the price of a call option on a non-dividend paying stock when I can guess the formula and show that it (i) satisfies the Black-Scholes partial differential equation, and (ii) meets the boundary conditions?

[@alwaysdisputin9930]

Cool

[@jcughan]

Exactly what I was thinking. I’m full of sh*t I have no clue what you just said.

[@PMA65537]

What about Hugh Everett's correction for counterparty risk?

[@theartisticactuary]

@@PMA65537 Not something I'm familiar with. But the point of my comment is to back up the point that if you can magic up from nowhere a formula that fits the differential equation and boundary conditions then nobody should care how the formula was derived.

[@TheHumanHades]

Hi Parth. Please reduce the volume of background music in the beginning as it is very distracting. Otherwise awesome video as always 🙂

[@ParthGChannel]

Thanks for the feedback - I'll definitely reduce the volume for the next video :)

[@TheHumanHades]

@@ParthGChannel 👍🏻👍🏻

[@Kabitu1]

That's all very interesting stuff... but what the hell does any of it have to do with the video title? In what sense is there a problem here that "shouldn't" be solved?

[@shafiphy3204]

what is reciprocity thorem

[@algorithms_mit]

Bhai konsa aap use krte ho is simple animation k liye????

[@earendilthebright5402]

"making a guess is very much possible" Schrodinger whole heartedly agrees lol

[@JuniperHatesTwitterlikeHandles]

Idk man you seemed to solve it and explain it and we arent dead so I think we're fine.

[@user-sn7yq8ch4j]

I love these videos.

[@helio3928]

brb gonna solve the uniqueness theorem

[@ernestooscarpatrono9070]

Thank you!

[@fluffyteddybearsbottoms3308]

Sorry for the geometer‘s nitpickyness: The sphere is a boundaryless object. So, if you access the sphere, there is nothing more „inside“ to access. You are talking about the surface of the ball versus it‘s inside. Is that maybe the reason, why physicists and mathematicians sometimes do not get along well? ;-)

[@vfwh]

At 6:45 you say "Poisson's equation" but you display Laplace's equation on the board. Given what you just explained before, it's clearly Poisson's equation, not Laplace's that should be written there.

[@andrin1248]

Well it's still a poisson equation, just with ρ=0.

[@adamtaylor2142]

It is a Poisson equation, in the same sense that a square is also a rectangle ;)

[@theothertroll]

Someone get this guy a hair cut 😱

[@sq4nTQO]

Cool

[@Imnothere59]

Hey parth, what's your name ?

[@nehaseth2793]

His name is Parth OK?

[@Imnothere59]

@@nehaseth2793 no, I want full name, is he indian citizen ?

[@prashantlale4976]

Well this was certainly non intuitive but the video was great Actually regular watches

[@piyushgalav6483]

Can you please tell one example o making such guess?

[@andrin1248]

Mirror charges. You can look it up if you're interested.

[@pathfinder4all]

If they look same on the outside, they must be same on the inside too...😱

[@SCHLMF]

Physics in a nutshell: just guess the correct answer and you'll be fine.

[@phrozenwun]

The zero field solution is only mandated to be a constant in the static solution. The real world is dynamic, the rest is left as an exercise of the reader.

[@gurleensingh1895]

Understood nothing.😔😔

[@davidepierrat9072]

"gradient of the gradient" made me very mad

[@danielgilder8672]

challenge accepted

[@1019jatin]

Good video but clickbait title.

[@GaryFerrao]

ahh the long winded explanation and application of NP problems.

[@rogerforsman5064]

Elaborate

[@patricktilton5377]

But what if Schrodinger's Cat is inside that sphere, alive or dead or maybe both at the same time? Hm? Or, did I just break Physics?!

[@andrin1248]

Well Schrödinger's cat is definitely not only a electric potential. If you want to go down to quantum mechanics you also need to do quantum mechanics (quantize EM field etc.), and this video was about classical mechanics. And in QM there is certainly no uniqueness theorem ;)

[@timeussubanath3900]

Hii!! Parth G... Can you answer to this question.. Are Physicist a scientist?

[@timeussubanath3900]

Hii!! Parth G... Can you answer to this question.. Are Physicist a scientist?

[@timeussubanath3900]

Hii!! Parth G... Can you answer to this question.. Are Physicist a scientist?

[@timeussubanath3900]

Hii!! Parth G... Can you answer to this question.. Are Physicist a scientist?

[@timeussubanath3900]

Hii!! Parth G... Can you answer to this question.. Are Physicist a scientist?