Control and stabilization of the incompressible Euler equation with free surface
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Formal Metadata
Title 
Control and stabilization of the incompressible Euler equation with free surface

Title of Series  
Part Number 
20

Number of Parts 
23

Author 

License 
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  
Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2016

Language 
English

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Subject Area  
Abstract 
The incompressible Euler equation with free surface dictates the dynamics of the interface separating the air from a perfect incompressible fluid. This talk is about the controllability and the stabilization of this equation. The goal is to understand the generation and the absorption of water waves in a wave tank. These two problems are studied by two different methods: microlocal analysis for the controllability, and study of global quantities for the stabilization (multiplier method, Pohozaev identity, hamiltonian formulation, Luke’s variational principle, conservation laws...).

00:00
Point (geometry)
Surface
Computer programming
Dynamical system
Presentation of a group
Euler angles
Multiplication sign
Mathematical model
Mereology
Time domain
Wave
Wind wave
Positional notation
Liquid
Aerodynamics
Equation
Free group
Subtraction
Standard deviation
Generating set of a group
Content (media)
Water vapor
Physicalism
Laurent series
Set (mathematics)
Asymptote
Bilinear form
Time domain
Liquid
Hausdorff dimension
Network topology
Equation
Figurate number
Resultant
Freezing
03:34
Pressure
Surface
Greatest element
Presentation of a group
INTEGRAL
Multiplication sign
Division (mathematics)
Mathematical model
Water vapor
Parameter (computer programming)
Time domain
Wave
Velocity
Gravitation
Diagram
Aerodynamics
Equation
Lie group
Free group
Subtraction
Units of measurement
Condition number
Constraint (mathematics)
Validity (statistics)
Chemical equation
Surface
Forcing (mathematics)
Gradient
Moment (mathematics)
Water vapor
Physicalism
Laurent series
Staff (military)
Mortality rate
Greatest element
Asymptote
Time domain
Functional (mathematics)
Vector potential
Maxima and minima
Liquid
Computer animation
Lattice (order)
Equation
Gravitation
Pressure
Marginal distribution
Surface energy
07:15
Pressure
Surface
Spacetime
Rotation
Multiplication sign
Division (mathematics)
Mathematical model
Infinity
Time domain
Wave
Mathematics
Gravitation
Moving average
Equation
Physical system
Condition number
Spacetime
Mass flow rate
State of matter
Mathematical analysis
Water vapor
Laurent series
Line (geometry)
Greatest element
Asymptote
Time domain
Bilinear form
Functional (mathematics)
Wave
Arithmetic mean
Liquid
Computer animation
Right angle
Pressure
Freezing
Surface energy
10:18
Classical physics
Complex (psychology)
Spacetime
Process (computing)
State of matter
Surface
State of matter
Mathematical analysis
Line (geometry)
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Frequency
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Manysorted logic
Mathematical singularity
Moving average
Hill differential equation
Film editing
Reduction of order
11:29
Point (geometry)
Complex (psychology)
Spacetime
Observational study
INTEGRAL
State of matter
Multiplication sign
Modulform
Kontraktion <Mathematik>
Complete metric space
Event horizon
Number
Wave
Frequency
Linear subspace
Arithmetic mean
Gravitation
Moving average
Equation
Differential equation
Normal (geometry)
Analytic continuation
Local ring
Condition number
Physical system
Observational study
Spacetime
Closed set
Neighbourhood (graph theory)
Water vapor
Mereology
Ultraviolet photoelectron spectroscopy
Twodimensional space
Price index
Limit (category theory)
Functional (mathematics)
Estimator
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Function (mathematics)
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Pressure
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Resultant
18:27
Logical constant
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Mathematics
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Chisquared distribution
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Hand fan
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Linearization
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Resultant
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Operator (mathematics)
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Operator (mathematics)
Limit (category theory)
Cartesian coordinate system
Standing wave
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Lagrange multiplier
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Coefficient
Pressure
Local ring
23:28
Divisor
Set (mathematics)
Multiplication sign
Modulform
1 (number)
Perturbation theory
Parameter (computer programming)
Inequality (mathematics)
Glatte Funktion
Wave
Sign (mathematics)
Conjugacy class
Harmonic analysis
Differential equation
Linear map
Newton's law of universal gravitation
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Metre
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25:05
Principal ideal
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Equaliser (mathematics)
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Wave
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Newton's law of universal gravitation
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Principal ideal
Operator (mathematics)
Parameter (computer programming)
Incidence algebra
Price index
Functional (mathematics)
Sequence
Sign (mathematics)
Computer animation
Function (mathematics)
Phase transition
Units of measurement
27:47
Existence
Observational study
Presentation of a group
State of matter
Multiplication sign
Time zone
Mathematical analysis
Parameter (computer programming)
Mereology
Inequality (mathematics)
Rule of inference
Food energy
Time domain
Wave
Duality (mathematics)
Manysorted logic
Uniqueness quantification
Equation
Subtraction
Linear map
Standard deviation
Numerical digit
Observational study
Spacetime
Content (media)
Operator (mathematics)
Water vapor
Theory
Parameter (computer programming)
Group action
Evolute
Mechanism design
Computer animation
Universe (mathematics)
Duality (mathematics)
Equation
Theorem
Object (grammar)
Boundary value problem
Resultant
Group representation
30:13
Greatest element
Multiplication sign
Wave equation
Decision theory
Time zone
1 (number)
Bound state
Food energy
Summation
Duality (mathematics)
Equation of state
Dissipation
Process (computing)
Generating set of a group
Parameter (computer programming)
Time domain
Maxima and minima
Propagator
Arithmetic mean
Shooting method
Numeral (linguistics)
Theorem
Right angle
Thermal conductivity
Metre
Surface
Pressure
Neighbourhood (graph theory)
Inequality (mathematics)
Time domain
Wave
Wind wave
Lecture/Conference
Uniqueness quantification
Analytic continuation
Units of measurement
Linear map
Condition number
Time zone
Statistical hypothesis testing
Standard deviation
Neighbourhood (graph theory)
Mathematical analysis
Operator (mathematics)
Water vapor
Line (geometry)
Group action
Boundary value problem
Pressure
Group representation
35:50
Potential energy
Fluid
Summation
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Term (mathematics)
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Kinetic energy
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Food energy
Group action
Pressure
Time domain
36:31
Axiom of choice
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Fluid
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Lattice (order)
Numerical analysis
Equation
Right angle
Cycle (graph theory)
Game theory
Freezing
40:51
Existence
Standard deviation
Logical constant
Spacetime
Multiplication sign
Exponentiation
Moment (mathematics)
Mortality rate
Functional (mathematics)
Food energy
Computer animation
Lecture/Conference
Theorem
Pressure
Units of measurement
Associative property
Physical system
Newton's law of universal gravitation
43:15
INTEGRAL
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Estimator
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Quantum entanglement
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Physical system
45:33
Point (geometry)
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INTEGRAL
Set (mathematics)
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Calculus of variations
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Physical system
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Units of measurement
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49:21
Axiom of choice
Group action
Multiplication sign
Direction (geometry)
Decision theory
Insertion loss
Weight
Food energy
Variable (mathematics)
Mathematics
Sign (mathematics)
Gravitation
Square number
Damping
Differential equation
Moment (mathematics)
Mereology
Term (mathematics)
Sequence
Time domain
Functional (mathematics)
Lattice (order)
Oval
Tower
Hausdorff dimension
Approximation
Equation
Condition number
Identical particles
Physical system
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Point (geometry)
Divisor
Wage labour
Disintegration
Mathematical analysis
Limit (category theory)
Inequality (mathematics)
Wave
Frequency
Causality
Wellformed formula
Equation
Glattheit <Mathematik>
Normal (geometry)
Units of measurement
Standard deviation
Neighbourhood (graph theory)
Mathematical analysis
Dimensional analysis
Water vapor
Algebraic structure
Set (mathematics)
Local Group
Mathematics
Field extension
Summation
Lagrange multiplier
Computer animation
Boundary value problem
Coefficient
Pressure
00:04
the top and if and you and I and the thank you very much for the institutions and also to Mitchell's invitation that they appeared to be I apologize for because many of you have already seen about that talked about absurdity and contorted an apartment is about standardization of the court I am considering influences generation and absorption of what away and here we are going to consider the programming in a rectangular talent which means that we are considering abounded container with a friend inside and the container as Vatican awards which the containers about them which is flat and also you are right and go anywhere it's not right and the everywhere it was about to be show where this really freeing underlined with them to the tree for the sake of simplicity we are considering only to the waves can we are going to continue its gratuity way but in principle the result should also forestry dimension I am going to present results In collaboration with little that the and then had enquired about the content of city of to the attitude that it and also results about the fact that it intends standardization which apply for a 2 deals with the way we've always out surfaced the pontiff consolidated chance that utilization of what their whereabouts been today a lot of fun equation describing what awaited summer synthetic regime like the beans and the liquidation of the KGB equation of the southern and there are many many people that the walk this on this topic and to name a few I can work socalled Canedo offered international figure also pretty it was this there a difference with this With this walking that we are to in the form of that is we are considering the dynamics of winning compressible liquid In a free domain we a freeze of this is a key point we are considering the free domain which is timedependent and to fix notation we assume that at time t at a given time to use of the domain is that the following fall this is the set of part X and Y In 0 at times all those are wise because the minus X and smaller than either of 2 x and the fire that the domain of the freezer faces means that it down is an unknown to the public we are going to consider for domain which depend on time only goes to In abundant container in the freezer face these are the girls With the question of why it was itself to facts so the physical government year of the depths which is and the size of the domain which is at capital it consists
03:37
of domain inside the food domain the 3 motion is determined by it's a lie under city which is the rate offered on a major threat to answer a question by which motions to the timeline of when known through 2 centuries this out in Congress but I equation DTB president agreed in the present rate of speed this GUY is equal went to seal look at capital B is afraid pressure J His excitation of gravity but this is the usual but migration this is a incorporated duty constraint diagrams of these equipped inside the demand inside the freedom integration old and we have also bought condition on the water on the bottom and those of his office this light condition of the following 1st we assumes that the dollar and has agreed to 0 balls on the bottom and end the war what the usual certain what their condition then the problem is determined by 2 Monday conditional on the Pacific firstly we assume that all the freezer that is here we we no
05:03
longer have the antiquities we have instead deity of the diaries to square with of 1 breath the validity of heat up the EC squared times and so long as this is a kinetic equations that describe the margin of the Office of Fair and the 2nd equation is the balance of forces across the Pacific this is an equation for the difference between the pressure inside the fridge and the external pressure In we assume that the gentle pressure at the facetoface is proportional to the meeting the that is the moment the ecstasy "quotation mark minus couple times DX all the dikes divided by its growth of 1 blessed textbook the physical parameter G get on the surface tension proficient Pisa pressure and PSG is the external pressure In this talk we consider the case was the 1st tension which means that we assume that kept by equal to 1 and essential that remain it irritation the Government Museum this kind of this this is ultimate it dies of resource preservation the that I am free we assume that gonna be inadequate able and we can write to the as a gradient of some function fight which is the velocity potential diseases got a function again no I'm going to describe the problem of consumer the depression is unit to for explained with staff on the phone and there's a sense of mission
07:16
was to sold this is not a
07:24
problem is because as I will explain later we can extend his problem To prevent defined under 1 line yes we was have there is always the right and the right and the poor condition it is by the bridge has to work for 1 which is a good thing because that is where the real reason for him Europe's warned Wall rose what we're here all of you still wouldn't missed this winter .period pressure which was forced to the it's normal this means that there is no homes and making system I am making this assumption that this assumption is the plan is following you of a container with a fear that rest inside it OK and you want to buy some means to generate wave but starting from this really want to generate aware that it is you want to generate a freeze face you want to generate if motion inside and you want to be able to do that is sometimes get to generate a wave you of course many ways 2 items on the freedom container yeah we are going to consider what is possibly is the simplest way according to the mathematical analysis which is to blow but because of his office Of course if you blow above the fruits of a struggling to generate a when and at what question Is it possible to Brill but localized of confined portion of the price of engineering way more precisely the question foreigners mathematical questions of foreign given some time to that is given in advance given stayed at finality final in some space X development of our AB domain maker which is equipped in air find pressure value of the external pressure other forms of things since the function of the and ax which is about Indian village the Times of Omega said that a unique solution of the went away the question with a mission that the zeal he said that the solution coincided time capital to With the finest editors given in this is a complicated question but many of them observation that in this case when considered that the assumption that the it is a rotation that is meaningful because you start from something that is a rotation and have a conservative falls so the flow in irritation at any time after all this is a great
10:22
all the presence of U.S. doityourself to the case when there is no let alone walls and this is the meanest we do something of that case will only go to using Pfizer state of foreign accent lighting our lines all that was a lot of the minus entrance Morrison To do that we use the classical privatization instrument revision process with customer try the demand and then we extended by periodicity I suddenly languages with afraid which is better the swathe of course a lot of questions about the United because when you see tries recordings would use singularities without are some sort of question about secularity and the analysis of the crush on a matter like United Airlines I am not going to discuss this question this was but wizard recorded the complexity of the case without surface tension and Lausanne debated whether cut the profits from the case we have tension so I know I can stay
11:29
the 1st visit which is about consolidating and it is a look a complicated of twodimensional Connecticut Watson all that I just need to introduce 1 molestation limitation is editions the observation in the following an overnight stay in the talks since their divergent equal to the is a memory functions my mother the not name diesel on the wall as a about because of this was 1 condition so far is footage of my life is based face on the preserve of act and this is why it it is convenient means when he was a function side which is the evaluation of 5 on the life .period is that this is a function of the contracts where this function is defined as the main is free of the whereas this function is a function of time and X which is in the then the limitations that age new universal space for them you entity next and we leave that to the fact that functionalism magazine the statement of the
12:51
main was risk Benjamin and when the following give any time until it conducted by small and given a name MMT himself included in the old Zagazig and index and it's not enough and the number and the old smart enough that thought being nation that and any fine and that in the space age is 1 of and times said that the new although initially smaller than in the old and the new and the final point is smaller than and zeal you can fly an external pressure which is continuously in time with valuing HS stickball today was it would be times Omega that's not the question problem was that are at pains you given by the initial death as a unique solution which is continues in time with valuing edges plus 1 thousand pages which coincides with a time to be with the finance that is given look at this as a controllability resent coldbloodedly means that we are able to steal the state to this 1 the Financial Times look I means that we are walking the ball around zeal but to go by it's a local resident well walking in the neighborhood of zeal and 1 important point is that there is that this tool in any time soon the other 2 welcomes more times in the last time of course because you want to reach state in a smalltime users others and to reach a state no longer too he was also exactly where ends would depend on when the very small and the lyrics of the although the estimate on the pressure is yes everything goes up 120 goes to 0 this can be conquered by the port visit solemnity musicians for the holidays the only what do you want to stay get throughout the years ago constant depend on the list since it is no blunder as his universal uprooted something like 15 was here on the other side this was the largest polluters who we we personally involved it that a good question to try to find a pressure which is what but I'm going to say only a few of the vote approved the port users in crucial ways that we know a lot of things about it is a question we but the thing because many many studies which has been already describing produced also I adjusted display the key point is that this is the specific questions and we are using the crucial when the fight that the aggression is discussed this can be mastered it is seen as an event of Xilin arrived question if you neglect rugged to simplifies invitation if you neglect the long knocked down to consider arise question and if instead of walking with tight and ETA which shall be held by the functional you decide to walk with this complex body function you know which type minus the the 1 of the then you find that you satisfies its footing type aggression BTU press idea to this we are on view is equal to the extent bridge Close to control of this equation is extremely easy because you know explicit as a solution using the before comes for the starting point to pose to study the continuity of the 99 question is to use the scene of a organization of the 99 question and this can be done this is based on the stadium now proves this is based on this to dinner and coordinates which goes back to world by cycle of texted him and land and the fact that you can't completely but the equation and the fact that once you have completed by denies equation you consumerprice them and you can follow them all considered by moment but In the end we we can manipulate immigration quite easily and if you don't need to the result is that you can rewrite the went away system as a full integration did you bring this year few TXU preside the exodus will force acting until a few times there Israel for you is equal want to the extent that approach where the N. C French complete immigration similar to this 1 so as I said the dinner
18:29
I system and the region as constant provision can be controlled by Nina for United on many of them this is not enough for problems in the part cuisine to the place where nice not enough to know what that means for the linear right system and the region you cannot apply the implicit functions and walking only at the region you have to apply a scheme that is based on the analysis of the arrived equation in the neighborhood of the region so necessary you have to use the scheme and scheme has to be based on the analysis of immigration was viable retention and indeed we see the external pressure a limit of sedition to uphold cynical .period problems with viable petition we the cutting he some fans the main point is to study I mean I question with viable provision which are fixed at the dinner you consider a reference date you consider the operator of the which is the cheapest media experts I system where the and NC the previous function evaluated that Cuba we want to quantify disagrees To do so we are going to transform the operator PT a that the 1st step is to flatten the coefficient of the leading all that set introduced only as a change of viable With the change of Liability Act the placing of function page lets them all by age of the X and this gap of TX and it is convenient to mich apply nearby you put factor which has said that the economic prison by so doing you can find that the His conjugated says it's apparent that you which is the cheapest unhcr official undervalued expressed either this we are after all where are the the fathers the going further as assumed by some simple information that the mean value of the value is the it simple to conjugate dissipate after this 1 but it is not entirely trivial because the question is locals research and reliable and also because you you see if you have the cancelation of the sub principle time of 101 you go from the question of what is really out of 20 grand for and for this we are we should not but that's a principle that telephone 1 5 that confided that simply because which choose in an appropriate way that the so this is the 1st and the 2nd step in
21:18
the to conjugate this equation to the same results the debate down the value new salary we're sick of a those that the community itself With this idea since we are does the value of the x a is the although we want to eliminate system them by Nina from your computer if you seek a as differential product I can do that this is something we did with the provided for 2 set a standing wave public investment adviser the and we found on the bag of this fall a subset of the eventual buried them we're seeing bodies of the fall some longitude hue of the an exponential expansion of ID that TX sites the 1 With the function data in the face is the son of a physician depending on time an so badly in meeting Bond it is possible to find a committees because the body was mean by museum center with his operator you can conjugate this British immigration to this 1 Wisconsin's application that but reminded them of others Penney you want to again because the bank is it to the central parents of it it is absolutely central character insists exotic class as it will always always equal to 1 that 5 the digressions cuisine OK because if a coincidence instead the 789 question but the question to considers Benjamin immigration you find similar conjugation for example with the functional proteins that stand out from other classes you 1 so this is how we
23:31
proceed to scheme was approved plans for a missing studio cost just to explain to you that we reduce ourselves to constantly Confucian you we know the face of facilitation and time of the solution in this phase are the ones given by integrating this evolution equation L taking into account the fact that in the conjugation process we have an exponential factors that modifies the so in the end we
24:07
have used to the following the quality of harmonic analysis which is called again In .period beginning with the grand inequalities this is a promise that type inequality but which apply folks to the paddock function White prosperity parody functions in gonna stop said apparently that is viable X is seen as a parameter what can we do what with his involved interdependence and time the what it is that we are using is the following 1 gave only a value function the damages as smooth functions is real would use this phases of facilitation knew and of the which is the sign of a and integer times and To this we are plenty there's a bit of the times and to the 1 these are exactly the
25:07
fate the 1st edition of this to an end to this week of and
25:12
if you include use of face the time spent the 1 with and then
25:19
we can we can put the foreign resentful anytime soon I think it was the exact constancy of tea and that of things says that he be based more off in an infinite involves smaller than that then fall any effort to sequence the value added the integral in time as of the end of the original of this function controlled 0 2 divided this is like potential immigrated except that this phase in this function is not bad but it is the about this signal really to proceedings the only way to the Moses you can conquer explicit everything when I went to his lounges and some constant depending on if you consider only the highfrequency reconsideration of would grant of the constant you should consider a sequence where it not as incidence among them that have a very good columns constant when you consider to small not exponential factor also there is a lot for principal and L so this is being done tightening .period when you don't have me that this goes back to walk of being when it is not enough Cambodians them more than I will fall in the tent here we are forced to consider the case throughout the day as long as the old the fact that the synthesis of principal foreign because you different for the 1 but it is not better metastatic simply because as the leadership of the ANC has 1 of the main 1 is much more with this is not about to that is called but we can follow the given by all indications the Wenzel and adjusted to consider once you have this in piping equality where X is seen as a Parliament you apply for every X and integrating X and you obtain enough so that the it was and I am going to status
27:48
of .period so you consider them a Omega which is it today at a given time to to deal with it you assumes that this sort of you evolution in this evolution equation results will step as an initial that the Mets you that your great vision as the difference between sea and 1 is then you have this up 78 inequalities in the running time of the intake what although Miguel view of 2 x squared the is bigger than the energy which he the intake of the of the of the All X where appears this is an upset by beating quality by looking at the solution only about America you can have any information about the energy everywhere the inventiveness of this season of somebody did and this is indeed the people To conclude from other digital content today is based on the argument that is going to do that in the best method which is the duality argument going back to the walk object regionals the idea is very simple and very little his presentations you this inequality this you that you are requested did enough for so you can apply useful presentations and that they knew that something exists and seems that they exist is related to the fact that accompanies exists OK and since you have ended up for the universe capital which is localized the saying that exist we also the European so you have the existence of a controlled at his this is only a puzzling arise the question at some states from states in its rules and you off to Tupelo is similar to that in so spaces and you have to put some study testimony to prove that the scheme is conducting this is quite technical and I don't want to enter into the detectives is part of what they would like to do is to explain something instead about standardization which we also Indiana related to another way of thinking of 72 what I want to discuss
30:13
minister of Britain on display
30:15
this is this is the only consistent quality the Agency for assistance this consists quality yes for wave equation for sweating like which involved only question that's evident but is the inequity of that time you look at the solution it's a pointed that the unit continuation reserves of religious tradition vanishes above 0 at time makers that the solution is the way they were the confident it has to do is wait generation which is of course very important to standards is also in a way that another thing which is very important which is way that's option the reason is very simple and the reason is that many of what awaits you may want to study the behavior of weighing propagating and bundled them like the way you can see and Japan and that followed use reasons numerical experiments and analysis of the what wave equation wines to to walk into bonded them men Of course so you have to effect you have to do something to stimulate emboldened demands that walking in bonded them the American not to approach is that that can be used as a you can prompt a the domain by introducing an artificial boulders and you see for artificial bully conditions was to walk for buying with knife all you can also try to dead outgoing waving and not something zone surrounding the computation of abundant of you on the computer and they are surrounding the competition a domain where you down doing with this is for the American and that is the case I'm going to present something which has to do
32:04
is an experiment tested want away of Kansas City I which show you moving this was his small was done at the cost of fundamental visited the condition went from Munich was able show awaits I'm kind of lonely too to show this morning when this morning you are going to see way status the swimming they just giant swimming pool shooting at us the Committee on the idea of something like 25 meters the depth of the Boston spies muted and you have to side of the waste time on 1 side you generate went on the offensive line you generate waves and on the right hand side the government to generate waves like this well on the open sea and if you are going to dump way because otherwise it will not going from the left to the right and then the bounce back in the Batson decision "quotation mark the new generation yesterday so this is a picture of this year this is a picture of the screening process is Of the town where the studded with the used to generate and on the right side of tent cities and absorbing device which is extremely simple this is a device which you introduce an artificial beach like the beginning of this year and the "quotation mark waiver according to his estate and agreeing to blow out of "quotation mark likened and blowout dissipates in Shh I'm going to show you a picture a schematic pictures of it is OK so that it is a picture of the from above it is here if it doesn't touch the bottom of the way through the air and sea waves generated by the way the others from the left to the right these are my only No way the following and you did unity in the nation based on their energy when they reach the beach users is that nothing is going back to this is because a special test to date of what a away of course this is not something you are going to do to stabilize mathematically on is the weather wave equation because we don't understand Of course if you want to standardize the great anathema to carry on something much simpler to do which is to dissipate energy in the surrounding by using against the idea of going above the Pacific he has a picture of the way generating the waste and as about a unique way of traveling to the right and then we want to find a way to blow the 50 faces that we disagreed saying that "quotation mark we want folks what win the neighborhood of Mexico and by means of an excellent conducting pressure the plan is
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the following at the dinner by age ineligible for that time to which is the sum of 2 terms of potential energy and the Qinetiq and with international is a of people privacy In the letter under the domain of this growth of 1 president at square and the kinetic energy is 1 I of the to of ability and the going to find a few feedback along for the pressure P X since then yet would be included in to combat it supported in a Maine is
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and it could be but that this year is
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the NRC is decreasing as is interKorean type of the energy will be controlled by a constant thing zeal and what they forgotten to 2 right is that we want the feedback loop that since we want that to be axed is defined as a function of the dumpsite the plates of passageway it is a function of the dump site by contrast was a political problem so why do we want to make netted a decrease of course because we want to dampen endusers condition what is the reason of this condition simply because if this is true and in ages decreasing age of the smaller than the main of age but this was in 1 of the divided by 2 years into a lot of each of the which she is because of the smaller than candidacy divided by capital the agency which means that you have a dumpling whenever capital is bigger than the Of course the age of 15 will be smaller than some fraction died age of 0 which means that in fact you have exponential decay simply because you can eat their agencies age at time and blanketed the would be smaller than capital city guided by teachers it times the initial that the whenever you this browbeaten and whenever the exist on long enough tenants and value of exponential decay but isn't it which is standardization so now the question is to find a feedback loop that those to obtain this book the the 1st question is to find it did not know at a loss to dissipate imagines this can be done take full of using Yanaton and formulation yelling find minute and then there the fact felt that survived back cycles that the equation on it and they can be written under this fall did you of the died but the the analogy with respect to sign the job society but minus the date accusing energy with respect to eat up minus the extent Bridge then the the directive of the can be written in this form of cross you use a cancelation coming from the fact that the question is an Internet and gaming always spot but the line at the meeting of the G . 9 6 per so you want that this is negative to dissipate and that it was simply truce for example because the function Chi minted blinded you do that then of course be but at the end of it with the the age would be negative and this explains that the choice feedback OK this battle it is widespread In the the numerical analysis of what the wave whenever you want to dump in energy to dump doing weight in the surrounding area and the question is to understand why which is localized and goes to prove that nobody gives I am going to show you an American experiment that was amended on it is extremely easy because it is an experiment flopped in their eyes question they just the committee finalized the question because you can solve the question using for its wonderful we are going to see that it works very well here we are generating weighed by blowing each about a freeze of based on the left and we generate about the quake training that is happening to the right and right way using the dunking it is down and it was when we are able to produce cutting wasteful packaging from the left to the right and evidencing only not so much reflection I'm going for it will clearly whether this American League the next statement
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saying that it will close consolidate this system and explaining of the previous standardization statement is the following U.S. use that as the ball did that pressure is equal to X by and the miners stunned by independent function just to ensure that the I think that is it where can I get some wellchosen then you can prove 2 things festive this is to positive can them better seeing depending on on the physical Barnett which G greedy capitalist of attention provision age is adept and on the size of the container so that each the Dow and its 1st debated in infuriating almost and dental and Yves association exist on the timing zeal C z I schism the age of the old clause an age zeal divided by 2 I thought energy larger than the other candidates with the of the ICC for energy so you left details imagining but this rate of cross a question of is it possible to have a solution which exist deployment of there was and is it possible to ETA decide the amount of an exponential decay and discretion sold at least partially by the 2nd statement there a consensus that said that he the mission that are small enough in this space when an accident then the solution is the enemies of size of deciding on the timing of out of sight system of a divided by its so you can put the existence of this edition of final time intervals because I'm going to say just a few words about approved Thursday somewhat about the a crucial let that's the
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world about the fact that I think the guy to be which show that at this moment How I
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choose guys fully after from the function capped by the legitimate function ends in a defined circuit of function can Telefonica the function gap bolted the minus and even I'm equal to 1 on the last then a defined function and which is multiplier which is equal to acts couple of X the guy of X is equal to 1 minus the of this that function guy which is not as usual as usual you were defined something like the disease this is the kind of function Williams
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OK I know a lot about the crucial program as above we are walking in a container leasing try we paradise so we asked readers to what's a good support for the weather or integration we that and With the producer manager you can think of said that with with system of this equation did you because you have huge exuberance I didn't occupancy guy off as the exuding quotas if you want to call that solution we say that sign an existing entanglement of size 1 about China you need to pull as usual uniform estimates in its Idaho Falls this equation you divide this them by the side of the at the worksite aren't because I am assuming that he depends linearly so you see that as a damping down does not add for the studios approach a problem because in fact the ad at 10 with I have future singled out and they have to be thank but but you know you ought to take it's not stifle ought to put that in his sights set my spine is 1 of outside it's not difficult but it is not insightful find entering into a sedated because political and
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which is in a much more interesting from the from my point of view he stressed that is the studies and I said that being to put something that is decreasing as excommunication because it is based on the ultimate and foundation of the question but his mind pressing supposedly energy converted to seal it is loaded the gun because the question is questioning nonlocal and at 1st wasted his program my motivation was To is poor firstly move on but opposed by the French factories and Michael Palin medicines for them but that's because I wanted to discuss with engineers people walking in future because I wanted on purpose to the move Michael Kennedy and I wanted to use instead is instead of walking away from the accident site walking only with integral function of eating up some function and by so doing I found the book that I will not be able to find using friendship that "quotation mark this is the important thing to remember is that the book is based only on about quantity we seek exact identity using but quantity than using 6 principles the fastest evaluation that principle for what the wave of languages so managed to use them to buy of matter which is also the principle of a quick politician affinity but I identities was ejected in a manner that we use it as far as possible consolidation loans and we and formulation as it is I use of principle and I am going to explain only this the Tobias I don't want to the with the way the thus the wet isn't a method in July insisted misstated a student discipline it is used to compute this quantity you want to computing behind diamonds in the let us consider for example the 1 you integration with Jacob condition when the wind which from date of alike you have slightest identity that can be obtained by multiplying the question by X the it's you you must abide by existing and new integrated by In time and space indictments and what do you obtained Europeans simply on in time objects you evaluated that I seek 1 1 is equal to 2 times this quantity at 99 as it's going get this is in decline in times of union by her before the wedding questions today to obtain an expression forcing the unintended Xanadu which which depends only on the bond 98 Healthways something localize it is localized on exit when 1 added and that can be neglect white neglected this intrusion is very simple this is the integrally tied in together because you working and this doesn't depend on buying this is a difference at 2 different values so when to very loud this quantity would be very large wives this 1 we have a fixed sites so can rob system so you can compare theatergoing time of energy with something which is localized what this is a keeper so the key
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identity for went away with but where it's surprising that you can obtain also an exact identity where all ordered them this simple expression using them to play that instead
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of walking here with the weight which is X I walk with the
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weights which is a general sinking into function vanishing at both and and the ensuing were in the early Wednesday you have some degree efficiency than the whole of the pending in a very simple way and then died and subsets with this way if you neglect of attention to a simple statement you have this collection is in decline in time of the Energy blasts some dumped you I would commodities acquired to something depending on the pressure about 90 100 times an observation tower and so it would comment on the stand quickly what is QQ is a positive step so you can neglect it is budgeted you neglect because it is was a good sign this is an observation why because wine minors NO X and X minus and combo sequences you will 1 0 and Maine as that of the migrant as he was there in the direction by the choice of the weight so this is an observation it depends only on the solution need exit .period and what is interesting is that they are evidently up with which means that the identity of the same puzzling arrived equation the only dampening from the 1990 is this 1 the Pentagon about a few domain of the dating of all times extend my wife at this moment in them can be very easily compare do is imagine because a concluding 5 square this latest so that only a cubic nonlinear there is this 1 and it can be political base in energy easily provided that all effects using an infuriating nonsmokers and 1 of which is the belief the neighborhood assumptions on this was commissioned an instant newspaper the quality of life of the use of chadors or in some way to make youth yes the end of the meeting was planned I know that information we did l specifically when they mention that maybe not this 1 but this 1 is very specific twodimensional mentioned to you have to use my column that analysis In this not Delaware the city into recently was the 1 you can you can flatten the coefficient is also in the invention to to flatten the collection is something that you can do it is worse than normal for example falls disappointingly sets of assumptions for them by 2 but apart from and I wanted to do his think strikeforce teammates forward they want to raise specific twodimensional 1 is defined exist to the different In became a disease you have to use Michael smart directions to communications and and to you have to to to do something and sending them that but there are a lot of Michael insisted you can difficulties to pull low frequency because even England that a difficult to polling lowfrequency that there are some tricks that you can use in their emotional and dimension to your to do something the changes the formula the related to the change of labor journalist and a general liability in exactly the same thing for funding I question considers wedding I question with equal efficiency and if you want to move to flatten the provision of to make of that for this would generate a double although 1 which would be a collected debating which is with an imaginary coefficient which is quite weak but it is and divided it is can be conducted by saying that at the beginning of the move by using pocket that factor in the prefecture shares in such a way that agenda by about the change of venue presented to me which is the case if we you make only the change of so if you make on the agenda viable units under President and also you know the kids in the structure of the group it was not just with the amendments more than that is the sum of its cause the world of U.S. the kind of guy comes from the fact that they amusing emitted by a method to put the standardization the animal infected also welcomed the choice of the guys loss of the season in what they have not explained later out to use users inequality to post that the decision is known at some point you need much more identity and for 1 of its identity I am forced to choose going this way the the few B