Knots and dynamics
Formal Metadata
Title 
Knots and dynamics

Title of Series  
Number of Parts 
33

Author 

License 
CC Attribution 3.0 Germany:
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. 
Identifiers 

Publisher 

Release Date 
2006

Language 
English

Content Metadata
Subject Area  
Abstract 
The trajectories of a vector field in 3space can be very entangled; the flow can swirl, spiral, create vortices etc. Periodic orbits define knots whose topology can sometimes be very complicated. In this talk, I will survey some advances in the qualitative and quantitative description of this kind of phenomenon. The first part will be devoted to vorticity, helicity, and asymptotic cycles for flows. The second part will deal with various notions of rotation and spin for surface diffeomorphisms. Finally, I will describe the important example of the geodesic flow on the modular surface, where the linking between geodesics turns out to be related to wellknown arithmetical functions.

Keywords  lowdimensional dynamical systems periodic orbits knots and links 
Related Material
00:00
Knot
Aerodynamics
Genetic programming
Position operator
01:03
Ocean current
Point (geometry)
Group action
Dynamical system
Presentation of a group
State of matter
Length
Logistic distribution
Multiplication sign
Water vapor
Mereology
Rule of inference
Frequency
Mathematics
Natural number
Knot
Circle
Differential equation
Körper <Algebra>
Nichtlineares Gleichungssystem
Physical system
Process (computing)
Mass flow rate
Coalition
Moment (mathematics)
Model theory
Projective plane
Ordinary differential equation
Basis <Mathematik>
Limit (category theory)
Flow separation
Quadratic form
Orbit
Chaos theory
Vector field
Object (grammar)
Game theory
Fundamental theorem of algebra
Resultant
08:09
Point (geometry)
Complex (psychology)
Dynamical system
Mereology
Frequency
Modulform
Differential equation
Fiber (mathematics)
Noise (electronics)
Process (computing)
Forcing (mathematics)
Model theory
Volume (thermodynamics)
Measurement
Numerical analysis
Orbit
Voting
Spring (hydrology)
Normal (geometry)
Theory of everything
Cycle (graph theory)
Object (grammar)
Figurate number
Thermal conductivity
Resultant
Spacetime
14:35
Axiom of choice
Dynamical system
Multiplication sign
Orientation (vector space)
Direction (geometry)
Range (statistics)
Kontraktion <Mathematik>
Insertion loss
Dimensional analysis
Explosion
Sign (mathematics)
Mechanism design
Coefficient of determination
Manysorted logic
Symbolic dynamics
Chromosomal crossover
Cuboid
Körper <Algebra>
Physical system
Rotation
Area
Process (computing)
Concentric
Mass flow rate
Sampling (statistics)
Measurement
Time domain
Functional (mathematics)
Sequence
Degree (graph theory)
Category of being
Vector space
Angle
Helix
Duality (mathematics)
Figurate number
Mathematician
Identical particles
Bounded variation
Spacetime
Point (geometry)
Suspension (chemistry)
Free group
Normal subgroup
Rule of inference
Theory
2 (number)
Product (business)
Frequency
Natural number
Modulform
Nichtlineares Gleichungssystem
Dot product
Numerical digit
Forcing (mathematics)
Projective plane
Basis <Mathematik>
Total S.A.
Incidence algebra
Limit (category theory)
Numerical analysis
Approximation
Film editing
Vector field
Network topology
Homomorphismus
Game theory
Pressure
33:02
Dynamical system
Presentation of a group
State of matter
Multiplication sign
Direction (geometry)
Dimensional analysis
Explosion
Coefficient of determination
Lattice (group)
Manysorted logic
Square number
Matrix (mathematics)
Cuboid
Körper <Algebra>
Series (mathematics)
Position operator
Physical system
Rotation
Area
Functional (mathematics)
Measurement
Connected space
Phase transition
Right angle
Identical particles
Spacetime
Point (geometry)
Suspension (chemistry)
Observational study
Graph coloring
Product (business)
Element (mathematics)
Scheduling (computing)
Goodness of fit
Hyperbolischer Raum
Causality
Complex number
Manifold
Knot
Modulform
Differential equation
Lie group
Acoustic shadow
Condition number
Module (mathematics)
Chemical equation
Forcing (mathematics)
Model theory
Basis <Mathematik>
Line (geometry)
Incidence algebra
Numerical analysis
Quadratic form
Computer animation
Homomorphismus
Object (grammar)
Musical ensemble
51:28
Logical constant
Dynamical system
Buckling
Multiplication sign
Direction (geometry)
Equaliser (mathematics)
Sheaf (mathematics)
Kontraktion <Mathematik>
Survival analysis
Water vapor
Parameter (computer programming)
Mereology
Proper map
Vibration
Dimensional analysis
Explosion
Mathematics
Manysorted logic
Matrix (mathematics)
Square number
Saddle point
Number theory
Körper <Algebra>
Position operator
Social class
Physical system
Fundamentalbereich
Potenz <Mathematik>
Theory of relativity
Process (computing)
Computability
Lattice (order)
Functional (mathematics)
Flow separation
Orbit
Connected space
Degree (graph theory)
Klassengruppe
Quantum
Right angle
Figurate number
Mathematician
Metric system
Arithmetic progression
Resultant
Spacetime
Point (geometry)
Functional (mathematics)
Link (knot theory)
Real number
Similarity (geometry)
Heat transfer
Distance
Rule of inference
Fraktalgeometrie
Theory
Product (business)
Power (physics)
Element (mathematics)
Pentagon
Frequency
Goodness of fit
Hyperbolischer Raum
Thermodynamisches System
Complex number
Term (mathematics)
Knot
Modulform
Nichtlineares Gleichungssystem
Analytic continuation
Diagonal matrix
Condition number
Module (mathematics)
Forcing (mathematics)
Prisoner's dilemma
Model theory
Physical law
Total S.A.
Numerical analysis
Voting
Network topology
Combinatory logic
Theory of everything
Object (grammar)
Pressure
Coefficient
Freezing
00:00
It's a dead letter to introduce its CNJ a school of thought today knots and then Nymex she's got his Ph.D. said 1979 animated got the better was appointed Their position at and arrest he also of called the position 79 81 in Brazil in
00:24
Team bus and out of there he ran back to see an arrest in 90 days And I Spent a year at the University of New York and since 1987 symbol early on that
00:40
He also Calls to quotes the positions 1 their sent an hour corner month superior its Yance's is a member of French Academy of Sciences and sees it is a bit of a silver medal of & arrests and also Some of owners its hinges Watson an in degenerate aerodynamics
01:04
All the state dynamics and related topics actually some of this stuff became a today that makes you with the Book of Jesus methodical interest of That extremely boat he obtained fundamental results enjoyment of violations Globe actions actions my mind on homogeneous basis
01:27
Our modern vector fields Not applause edges of but of residence on the circle and many other fields Ed Sanchez has many students and in general he is very supportive of young limitations finally I would like
01:44
They're JCS you will see in a moment is fantastic speak its engineers How for how thank you much so evidence might die our won't discuss coalition between dynamical systems especially to and bottle and send for I think that there will be a year of Is that some it is what idea was think of a battlefield in These are the things on guys not Who length would begin finish it is mud Ambition to give a talk that Could be understood by Would you would you this implies bouts I spent a fair amount of time recording some very classical which envisions borrower the with is also implies that I Kind of like that the guy so about my presentation of the current state of research would be a very pessimistic photos however In the final part of my talk my plant would do want example that I want to destroy with some detail and I hope it will shed some new lights on the system comment from No. 4 hours go that ended begin my Jolie in 60th when they make their logistics and what was studying some guys there is simplified model for the portion the utmost them that his mom produced the following differential equations and ordinary differential equation it respects their white prompted the solution for this differential equation holiday is very modest computer off this in what he sold his August picture em what is most amazing with this picture is that it is very robust issued share politicians on his budget which issue out some extra what's more to the equation we used to see The same thing warned that tried to approach some Objects but is gold produced project fucked of thought but I will not discuss the bid is actual Island of this picture of the new through with emotion but when I can say is that when you see Picture and when you realize how is new judge want go standard yearend accuse you want to put it in mathematical NewWave remarkable objects world today with us job Lobbyist Lawrence picture has recalled body not picture of chaos theory bottles instant 58 is mail Monday where young must however warn of credit simple approach would be But should take such a differential equation for his face and began or what This will be ought to be a periodical bids After according to close And the Crows growth is not so bad the idea that Maybe we could approach topological is of this objects for the organizers of the periodic orbits that I don't think does have a look at some of these periodic orbits of the flow of water which is generous preview of from problems legal point of view Ms. harvests that want Also in more obligated for the topological border which was and enjoy away have to have this red which doubt be fresh or not Sold their idea woes Many would try to understand the nature of the knots which appear on this side The ruins of what is a collection of several of periodic orbits them when this is picture you say Well maybe we wind up proxy the Romans at Procter and think of it I think this is supposed to be Collection not so with is This picture In the back of your mind because you will see the game or on tour in this book so new mother was thought to have a Start Twice to endorse which Not purely which not don't appear inside the Procter and that fooled but the not In Norway's On their journey For instance the rule but the opera by the army off I will have positive signature but probably most of you don't know what is of but will buy for it the need to show you a collection of data not has some of the 1 of Williams fearing they genuine doubts among big talk handlers and 15 notched was
08:15
Math by or it for toe that crossing Then she writes of Dole's and normal and don't nor about issue little more crossings Bound for example 16 crossings which is much has been tabulated the on number of Not 16 crossings is 1 million 700 wooden wanted Thousand 900 6 not the among Damn White Jr But my computer objects only 21 of them appear The story the especially when just shorter 1 example This figure may not Tuesday on the right part of this is the 1st example of the not that you will never see the inside the right talked over How's or how does want pool such here where the twitch New A way young Was to use some kind of conduct or your approach they deal what is called a little hands and Giving formal definitions that ensure About was thought to war bones of paper is based that 204 assumed the spring IBM You for doing this side But A branch still face this brunch is rhymes longterm for that used to the middle of the spring this Put emotional object is embedded in space gallery is a natural dynamical systems I'm going to show you now As a model for this dynamical system is a semifinal pitcher for Middlebury intercepts logical segment exactly positive albeit of the bubbling up Usual Bala tries with digital enjoyed by some number between 0 and 1 direct doughnut is just ax moved to talk Ex modules So issue for periodic bids You want understand the apology Ellison would be by periodic orbits of the Dalai so for example onethird My my by Julie's drew But is The force only once felt 1 job back once of as however this is 1 vote To barked orders so that is why I sat on It was found for back toward that burned that is What did for 50 years for 50 did shift back to 150 off onefifth who finished 4th 5th produced barked onefifth of from a book is sold What B. Williams did was Put door who it on this picture and sound of told the not But created by this procedure the fact that this picture of dust will respond to the Romans with actually forgery originally put the war or screen of beer Isn't much noise results it has been proved by Warwick Tucker in 2002 For the Hill might put before that Berlin's lots of fiber about actually Is Recent photos they wait This Kind of ideas provided crew will Your after acidification of dynamical systems 1 could use others From measure all their dynamical system the complexity Of the not so that it don't think will For example on Top form this year I You could put to want full amazing examples of Robert who bruised talked to reverse a few injuries place which has saw where do they got they represent Possible And it is not It's just 20 who will give a talk and his Congress I think a true I recommend to attend however woke up Some dynamical systems of pretty complex with complicated dynamical dynamics and have a small number of the article Even some have no more like the although amazing over 15 of them would have no So many would change would involve point of view there the idea is that Why not only concentrate on periodic orbits with Ridgewood We should also look at Norton non This is an idea which is very old idea that I would like to concentrate under the name of the principle of shots on Saturday after Romania Costs that really it is what I knew was think of volume preserving measure was over dynamical systems as some kind of arson cycle
14:43
Ensure you a picture of what I had in mind he was on but by is over but few And I like rule you have a lot of albeit which is Don't but not closed So what did is to make it close is connected to end points by segments After blowing up to get across Provence basis depending on the time spent on the usual point not what you would like to see a boat when time is there And should be very points text of religion or points What you creating these ways in with approximation of the vector so I would like to write symbolic equation that the defer always thought as saying This is where it was fitting of our range of 1 of its yield Carry off UX for these equations used nothin Lloyd's multiply not not quite a number is a crazy that integrating Functioning with body was busily not seeing more crazy but they wear it is a very productive idea of a battlefield preserving a measure as being some kind of limits of origin values or not this is something like and then go on the point of view mixing dynamics and perform well that you want historical example wrote this approach was reproductive this uh was a product in Explanation of what is going on and this before we can mechanics and which is The Arson don't think linking Number But Before I told you I want to be by everybody and chased maybe not everybody knows what is thinking about Let me Don't ritual A pair of shoes seconds recalling what is thinking sort of have told promote coves oriented cross over space at 1 of the blue what they are these joint Yes it is And of course the projections on the screen on not this joy to concede that want point Think is although the rule of thumb on point of rule is over the big threat of concentrate on think or not crossing and I don't look at Hey what's happened you how basic killing possible local situation it could be that's alright So you are not defined Trust 1 of the 1st period And the us once the 2nd case You saw local renown goes to war Crossings On the eve edge of the rule and you get what's called the linking of the girls would of course is that this number not being on the way your project fuel from curls on the play and has value issue Weiss on thing between goes without Creating intersection points before somewhere in his did you have thinking about trust 1 because there's only 1 or more crossing of the pitch with a positive sign Told crossings on that they the rule with what it finds so that Thinking number is 0 4 days from elementary knows what Lincoln number is between to not his lots and since you that we should think of systems as they age limits of It is a very almost idea to told Computer thinking moved between their legal systems and this is what more and how to live on the was to compute thinking Number of overthrow with said So here and you dig of role let's say on Monday domain of offering you a to is resolving some rummaging measure you'll advantage and you think Mountvue points x 1 next And food finds she wanted to do which I would not review which were not clear of T1 X 1 case of Due to next through that used to be sure you're gone chilling can number It divide violates you want to to with the And I'll show what I got here and The broadest but this limit exist or what everywhere It is independent of the choice of Thanks for the next 2 or more and this produces the number Which is called helicity Photo when they would tour open question which has been Mentioned why what is this number Topological died Jaw What do you mean by that The following support half true Dynamical systems do froze fights you want fighting to which are preserving true measure you 1 Support knock on my sample no more than a page engages the Bulls orientation was other and sending you want The question is does about God a bear this it is off fewer 1 1 tool of the a fist It is I think my question should tell you why I emphasizing his question them up for the loss 1 is that I would be happy if somebody could answer because been prevented me from shooting forward Daniel Anderson of the southern from the 2nd reason is 40 more should use his dots Still have a look at any pictures of a How we alive or even for through numerical from simulation like this for example the floor to rolled by some obstacles were like this is kind of picture of you cannot prevent yourself on Quarries or I'll fight off of comments but life is so smooth and orderly we want to put you with a value of about some newly full in which cool useful for all coatings from dynamic from proindependence already got high after purported could invite in the army not topple degrees for get so that's why I believe is the question of what was question is all However Like Cary mathematician for political pressure simply so that's cool To a simple question that await or make it simple to move from their mentioned frame prevention so let me laugh Report their possible construction dynamical systems which is called suspension it is a way to move from dimension version free Suppose how much did more residents of the things that Edward F. might trigger a similar product over this into your back is based And you know The true face is by using the film office in this way you produce the foreign theory is that Who is wrong you have a field which is what they do your Ronald Toronto there which is such that old on this vector field sector with prawns this exactly like all bits of the film office So it is a way to produce from Editha won't affect called the suspension of amorphous now did not
23:48
On duty by the suspension of the because not all roles have the public is also what dating ways in the same direction how well the category of frozen did my suspensions is is already rather big categories uh that is true of studying and I think you understand that I want to tell you this question of Congo center is a case of flu which Obtained by suspension will build on what point with this Procedure is that you good to game instead of looking for flow Europe for these officers and differ on the basis of respecting the IRA is a Why wife go We can begin playing games with books not abroad So let me mention the sequence of heroes noble and then I will comment on them 1st Iran is your new Topalov torn sailed I did whites talk show Of the good of all things if dozens of this Which are Korea puzzling there which are the identity of the label of the bond In the hearings this doubt there is not your home From that little pavilion number I will come back to this Home office and later the 2nd theorem digital by now Midsize essentially notes Cannot is homomorphism is formally home office More precise in its is about the the should they don't Of have home amorphous which is normal subgroup of dish forget area this normal somewhat has normal almost on Saw the killer of cannot be is simple Brazil is a contact sheet between Colombia and had city along this irrefutable Norwood with myself The following should return to was over because we should think it's especially produces of interest in this Don't feel the has ahead this city as defined by on and the about the head this incident from the suspension of is for job This year just a simple observation lies observation which would you can use less if the next period of for pulled his or her myself and don't would go doubts the Callao a Home Office is a the logical which is that if you have Jew come off dozens of the Which are by a wholly owned Or it was over they end It is to show more officers of the have the same cannot be No. 4 if you put all this together You get doesn't booing go what the answer to our question in the special case of PROPOSED which John living by suspension each of this from office and A Many people honorable systems that live dogs Suspensions what kind of significant dual role Warburg Medical Systems think it's a good case that the origin of bridge loan or might well go ahead I cannot do you look for support for this I would not do so fire has limited the only thing I can do is to newissue Sheila They give some rough ideas white something like that could be tall and that would point is that Thanks to an effort to recover optimized definition of Get on his home and office Seoul Let me recall or tell you what is this Definition of a better start with film offers a this issue as before which is that there is no end that identity total Now I don't know that is more mobile force individual offices overblown is a contract The what's it It is connected and simply collected this means that you can go next You will more resemble the identity by some path efforts the bath starting from the identity and going to this path may all be each It is not with his homer but is even more since the war was concocted So now I realize this is your big for annual dues Asia And you look at them more shows on this point And off the ice sample paths of this drew points more and you look at the way I don't think there is moving and get some more audacious Osama look at a picture of you have True more sure of this Sorry on more of emotion that is not true Put it on warning and connecting them is what they think sometimes positively sometimes negative anomaly definition of Columbus homomorphism forfeited is about You just did a great ball for more For variation on the former list Based on the Points So if you don't know what were the only definition of cannot be be Was a little shaky little How the arranged over Points on the rotation angle my sample collected from offices of the identity Zulu That's You clearly homomorphism which is not clear is by the end of security that this is the only 1 well too Who's some of this box just say that Colombia's former officer a measure mold of work patient contained different Don't for them as well buoyed question Which I think you'll like John mother is another white questionable nature of the global War hallway on offices of the disk which are the identity of the Monterey Bay with which respect the question is is his move was simple No not issue will be priced hallway or by this the answer is no because cannot is homomorphism is a form wasn't there for the count of normal some war waiting to see who will is who use questions of war of mother would be to try to expand the definition of cut for washes away from differ with the Hollywood definitional what he makes it clear what you want to do what it wants measure the amount of work patiently 1 out The bar point is that when the function might not For your kind of lost For Mr. The question is something the 2nd guest Is it would come for normal on which has been created by bowl This is a very nice John removal Horn amorphous looms over this number but nobody knows it is almost double the figure or not this is gone some highly resumes of the disk that All Wis Take more about next south of us commanded to his door We probably discussed this kind of book let's try to your father gotten it Find more if it's really how Kellogg is final
33:07
Well about nothing more as a formal more for you usage of combined and now the lawns Budget like concept which seems to be more and more You will see full endemical system is which is a concept Polish with Liam The CDC definition is As you have in your mouth the remarks color from Myanmar to are 1 said is that it is a quasar if Cairo gum monument to child younger 1 might just how much is bonded it should be zeal is a form of bonded is resigned once it is that this is quite more for modules It satisfies Cargo gamut is and find type of gum Boost Concepts for a wonderful feeling cold bonded go home model which has been busy could lot By Michel I would really love to have oust Discussed and I cannot so that fortunately I'm much because they were you talk in Congress might be put on no I think it's next to live and uh I'm sure he will discuss connections which initially For the production of this concept but as an appetizer stole that you just mentioned tool here and that you can find as ever Crew would idea of what The main of this kind of object simple example is this if you a For you the for World or is minimal Many don't know that This concept is not useful We line officer from IBM who are thought when they are home where did order since it is a new thing as home home office so but is not the answer Or I'm wrong sort of raw however Richard Dicker food millions 1 billion For more generally a oneoff item this concept is knew we had a age and this Faisal not of formal Jin resigned was gone off hyperbole who is if a dimension so you have to Kind of opposite situation is not a million things happen having no not with office in the upper case in case a war that I should say Ozal Ohio right lattice is amazing what is your music and what some will have no choice and that did I want to discuss it For the hope that drug tested They were on hold Field All of these which are the identities of the wandering which offer a preserve it don't now that this shot on the Hyperbole who did however faces Hall modules of if dementia I forward I would just short picture the idea is just next do to the same thing as we did with Kaleb homomorphism but instead of taking 4 points that as they want points for example He was a chill using 6 . 4 0 0 On the motherboard this thinks 6 points And the want as as their warship path she evolves from zeal to 1 This export more removed is that this quarter and was soon brought 6 Show that off course Might not come back to the ordinal points however What you can do is exactly what we we did in the case of Our mode has visited in Uganda how hard it is to have the moral Still similar to the box the rights to which the . com mark of the apology was initially just segment The That's you would use of brave It's a collection of 6 shows stopping 26 points in among themselves sold With this type of construction is you have a way of constructing a right objects right and is your enough to find square like more residents on the way will be and we get a new disk with office cause To produce no and is no you them instead of on must face fails yuan or the histories of pentup but this is the wrong idea the will construct Quiz I'm on the Book of Korea preserving the amorphous souls opens got homomorphism than measure of the amount of work they should each column him from office but none of those measure the amount of writing Danger of course in the his and the we ever brave was dream a way of toast the only thing you get our continent is rotation number of number Wedgwood get what kind of a When I'm office and I was worried was on the way and the key the signature signatures of traditional classical died of raising knots and comes at the presiding officer which is Very useful in this context as no father best tried to find more ways and more Whether I could speak for hours on this topic and this not the point in this painting so Oh For some reason it seems that we buy more brought farming These days and I think anything time to describe what his new objects so I decided to choose only tourism constructions and that Was don their so The frost never want to mention the beautiful wife and both were which where the study The goal of area preserving detail offices this year And what they did is not constructive ionized more Karzai going from this past to authority out of which is presiding officer and 2nd of how wonderful report to that if my chair the film office of the studio has forgotten that has called box report contained did this For this morning's Of There are a lot less than onehalf of the forms them They got what they wanted to number of path His thing for me to come up with a number of the restriction of the disk So it's a nice way of constructing the quarries I'm office Pollack of the Mideast war has homomorphism defined or more These but I don't know I would love to go into details despicable that for a long time This It is Lies Function function of the global office in this field which is in mourning their lot Polls See hectic model before launching I would have they would form a moral duty I would but I would have to speak of your home mortgages that we did too long however what I have to do something about The construction of what they would do not fit the Prudential case they have sit on construction for more sympathetic manifolds going for his form this year to that pointed phase of mentioned for example it is not known whether or not You can find such a way that for
42:16
Any money for the 2nd exam what I wanted to mention he is a federal boat trip Deserve which is essentially the same thing except except gradually which was down by Selftest combat itself a higher bid for combat office of jeers GE deposit did Uh if a yet Doris Hydrogen sofas and the fears is and that is why we have causing more reserve on the World war funding how we don't know more wasn't Not so with the same What as before if the film half has a support continued this them is homomorphism coincides with Colombia home office work instead of going for although I do you want more and more and more Dozens and Questions I think it's time to concentrate on schedule Auburn questions directions that might be interesting for the future and mention a shield the subs would be The hope that maybe it's possible to conduct from mission shortly Ancient trade not forget confirmation And we came back to dimension which will buy because I was forced to do so You're among us that Colombia's homomorphism what kind of a weaker version of this incident and this On the suspension is cannot be be additional forces so What would be their knives used is to be able construct value After Which Did the cable suspensions would lead you wish the value and I just explain what would be even nicer is that is balance prove would be topologically died I told you I am on by the way in and there were is this construction of mental forgo wattage shows that Maybe Roger seeing the tip of the iceberg but usually not restrict that also brought mention like to walk free We had more Then there was to go higher dimensional we after June the 3 of us eclectic 100 boys and dynamics and dimension so really there need for See it off construction highdimensional why don't you I I would prefer the stop clear the presentation of the present state of affairs because that would be technical and this is the kind of things that I want to make a break Told them that I want presented 1 example Would point is that it is Fortunately for those of you walk that I'm lost Call to all that the Treasury's I wanted it want Example of a wellknown dynamical system that I want to look at it with his new wife I want to look at it with eyes of a dog launched this then dynamical system used for more demanded off On Lesotho module as is the before I destroyed the dynamics I will begin by describing Before use They use them It's Porgy but and finally By now so that this phase is to model was so issue like holding your happier when it if the caution each of the originally what to buy dormitories is the dominant 1 YBasub more of these goods from which injured were coefficient so on the basis of provisional and not if you want to get the better of his position Both aware this Is the of doctors is off to While spending spree what I a Mac is of is just storm this could sum of October which is isomorphic It's is Want so you have a picture of left and right showing 1 example of a matter of course if you again cause for me by matrix as 2 are given all letters of marijuana and this will you get Warren matters None of this based on matters of RAI want is a homogeneous Based on the reaction of isn't too far And the But that is Of the latter's Z Jew is of course has sent to Z by definition so really are marginal as it is nothing more than this Phase of lattices of Korea won the bet for Blue Note the basis for what would stop or what has been known for Manila was probably 150 this based on double space is wholly Norfolk for the compliments over 24 a lot the freezer Explain other than from a mini ways of seeing that meant the newest thanks to lead No. 2 with a 4 point of view which is easiest way explain its the shadow letters long Additionally people from members during castles 2 20 Gedo and engine afraid they are defined by either series So did to 6 5 4 of the 6 My here is 16 times the store initially or for nonzero elements On the matters of When they get to the minors for See the complex number will indeed 40 It is a It was 140 with omission only minus 6 hours or elements of the That it is and that they fear a 19th century chewing on number his Don't number bacterise More Suddenly the father is the following the not sure not for each lot is among U.S. also took gold bricks number was due to injuries don't don't doubt that This is not which is by objection each between the base of doctors is on the left lefthand side and that the pace of a complex numbers which are way a from the sole orders brokers where due to shoot would door 27 GE foryou square So this is bloodless is wholly more Compliment is His wife go for Gin would 27 inches square the summer look at a picture of This represents a simple Bloomer walks As it Gedo threequarters of the optical remained boxes His due for it is equal to 0 Can now go to concede is right Due to June is equal to 20 70 square But we cannot hold city is awful For this picture that for the national picture the Copley brew waxes is actually a prop your seat or if you prefer a quarter of all tool this is true mention them if you want to look at this base on lattices of what you have to import according 21 condition which means in practice that you have to slice this picture by UN Studio in tool which will affect the don't go which is of cult just office which will intersect don't go On the not this Not try for store since his picture was very important on understand I wish or you The film initially
51:30
The picture of what you see here the for battle to save the Dylan nothing objectionable Christian this is what they in now News perform not Genito tumors it would 27 jittery square Ruiz you wanted to act Pentagon greedy using the other acts tho not only have a space we kind of understood its topology not dynamical systems That makes is easy to show how would like to see you Altschuler he just just forms form by diagonal matrix Ito 18 0 0 0 into the mind of novel sorry for this former defiance of dynamical system The space on Madison which is SN toe or what wasn't only which is the condom fought not injuries she this the want legal system I want to know what I want to look at other door you were using gasses formidable just so I want to look at a very early makes the way they think together The way they constitute not set said So what's all the of periodic orbits from this particular chemicals as well I must say that his political base I have a very very long tradition yet led me back Photo you'll have the Lord for you how to construct and ticket to my matrix essay Lesotho As a drummer Richards Woods was the coefficient for my definition this tomato matrix results The into were but don't know are still Madrid's baby is hyperbolic Part of what it means The process As moderate At least for it What is your prefer As you Madrid's is diagonalizable In this case you can to by by some change of placed on May 20 big there's image which makes a dilemma for you have it These are the P debut in those Brussels minus 1 another metrics so issued final Bell From individual or not Esposito sure Of course The that is not is fixed by exponentially By that is with another point So against young riders were filmed joining this sought They correspond to a periodic show is based on matters but looked kill a simple it is not a single point is moving along the harbor book hyperbolic But then Is soul Not lowest for each matrix a message within which is how the body by creating a cruel the rule in the space of an artist is that Mentioned just offshore Cases where this period war as a cheerleader mathematics the appeal course continuity classes of part what elements in his aunt was really correspond to growth Joe deserts On the module sofas they are coowned by Ideal classes what brought fields or brought to real big correspond to the trust traditional indefinite whether what did they were false Z these wars a point of view of golf's 1 can reform within in terms of constant Fox News but said I think What would you have met you 1 way or another later this year it's more like vote now it's time for me to go to a question I want to address For each matrix that medical used quantities across Co Is across go in the space of Not to see this river about this is the complement of afford not sold questions What kind of not to go for a week or them for 4 more modern what I'll do not that you get by these procedural freeze a 2nd question probably with the pressure for each Madrid's you have not space is a complement of turf or not How can computer number of this with the transfer the function of the Matrix throughout southern look at some examples of this new Madrid's 1 1 1 2 who wore yellow as usual the press for not agree the corresponding Not want with not his free your but is not situated in the same way as the 1st example this is a triumph for not the right Situation is not I want His each matrix to get past the the computer 2 door Dorward this pictures I will make a comment on my talk on making lawful distance as you pick your unknown matrix toe you want understand that we bought Hall also questioned 1 Then yours of course Before I tell you what kind of what we get that can you hold a link with the try for use as a C or the number of betraying the Associated permit a lukewarm The Honda Marcus financially well New might said Who knows what are the last of she is so I Schroeder They call what is on the suspension Michael at sea water warned of paper or did they can deduct function as this is written in production that they're thinking that function is want although most don't did functions In mathematics so I thought maybe somebody new ideas could have gotten definitional dedicated definition The new recorded just he was wrong so they're making it tough function is defined by this issue its product eat Delft Is the product of exponential Pied Piper divided by 12 times this Product but you get the screen how was a complex number You go behalf space you Marjorie toppled posited the must forge board is not function Is that its 24 Power is marginal function which means that if you evaluate the 24th it On being told was Divide my seat belt was did you get don't 24th or want pride by the model fuck door to door was deemed 24 hour this is true for them Twice in that war matrix This image has warned that All this . French novel As you simple definitional not functioning did the product of non So this nor abolishing function which is a horrible thing on function on our behalf place and their behalf playing is simply connected So it does make sense of the word object of them Because you have not vanishing forum of function simply the 1st log in front of his more joy question and you get this 24 upon rubber ball brass and legal remote to postwar This is a new wanted worldwide Deep bent on ondemand and his 4 behind the market's French It is not sure functional defined Peterson was It was a very not sure from the point of your model function for this Aug definition that 1 finds indeed of unpublished manuscript we might be up to just of it wonderful privileges would command is Natsios paper he gives set definition offset functions contests and and then only in the middle of the year of the end is some of that Conditions are really amazing different How do with conduct or of dealing with numbers Here are some of the was as a function some of was top Vogel dimensions dimensions Dimension for a wander Mark equal of course has got to it would do this kind of passenger so proposal this talk is to add a definition It is not difficult to see that his heart of office functions is kind of Figure more and this is to focus for now you know 1 harder muscles budget is so let me did you not approved but don't get He did not catch from For the full force of the here and now that they're thinking number of gay and 24 bodies in what I told you so The key role because job Jeff pulled the bill below kind relations between the functional the founder of the candidates for the award are holding the issue to the Z cross How was it is to us you go Chu It's due to trade their computers tool She won 27 different square and know where that is essentially the 24th borrow Darfur So this for either between need not functioning with ended this kid your mouth But I hope you will were due to join forces June 4 square is it with Brazil Is the equation you perform well also home Ramon sequentially The underdog the complex number is basically the same thing as argument So if you have the show of it in the proper amount of time not and if you want to know joints thinking number with 24 not if you have to do it They did to show my 20 That is square metal and are initially onto the bar additionally argument is essentially the same thing of liberation of the log of this complex but the rock is don't know the same of the role of financially by depot Q. the ranking number of KTA using Related to die from shaking their heads toward office that was fast that promise it is not difficult but let's have a look to make it clearer if his looking in the saddle points where this Discriminant 0 you look at the of form where distributors what number Is now putting mention It is still face is not and not just off his board is supported as header If a log Therefore not unconstructive in front of you there Zico is was based on points wrote this on the points where the discriminant is a positive real number Nite you'll probably have not this and is moving in this I'm a big Gomez audio crossed The life of So let's call now look on the combination of this picture of How or not She didn't guns and not carry would you do the robbing is based and which this sector From time to time the vital gun intersected from these To To the never worn or from the agenda 1 1 1 of quantum when there are June Sean minus 27 square goes from Now Imagine Imagine advance of the other way around so this not this section agenda not and the still face you can't 1 . wanted it when they get point according to the size of intersection There is no No we aren't Discussing is Joe the total number of right intersection between the knot The the vie for He'll for example used to when more than 1 That blue nor by depressed Response finds 1 What only basically so question told the question is what it would cost warm what on his watch and the answer is that back The beginning bore the longlost are exactly the same as those Robins Co will not That there is a big sometime I will just explain to you were kilo toward films and with my hands The port has improved since wanted from North Wanted on of the same You have to find somewhere all face is in Warned Vigilance and do with this with false We know that no not this is the place which haven't had that will launch a tender offer that this in a onedimensional object to be good you get what they did in a new direction for the space on matters which are designed the socalled them look at the based on matters is whose fundamental domain is wrong with the Harlem were home where the parliament is more of the home that you were told to move from 60 to 120 degrees this in industries of matters Before you miss your future this to be seized a socalled and this interval to gather 28 objects which is Risen during the show little if it does And I think onedimensional think Bush if you leave out there more direct direction off their model offer water with me a suburb This is the report not putting the right position was something and the Koreans onedimensional objects and their push the unstable direction to create this object which looks like leading role in Now Understand Hey We want to show that want I'm really for the drug is big enough and you want to push Them To put them in the right position each month 10 Foley was good and I sort of you won't oppose everything to the who I will not how time but although you will grow who was number theory it is nothing more than a dollar production process for Just put it in the right position or for those you would prefer a model self face and just contract in the model still face on I Joe desert chronic that was points anyway Sound triggered a hooch auto jails ends and while what war too Friends of the as poor and then he'll formation And your is dubbed the Goal still attempt after the accommodation is exactly the position of explained to you For the rise of Guillaume example that gray want more complicated of Jaws his major cities in such a way that would cause point not kind of nice this is a for performed on field therefore mission joined with example led to that really is not done here I thought Dole toward Rashid the case of the law and the his sole example the more complicated matrix till fold The album show you example together to show you that Not only is it for now but thoughtful approved for links You should take together several water article Ugandan similar values not them to for food and spend and who picture A toe the news the final ritual is a way for you or the beginning keep his joined buckle your mind business Well What's next Where was next is 1 happy with this uh Iran because We have connections referring toward different lots sold it would be nice to go from 1 bottle the other parts that I did some progress recently is mentioned the standing of vibration is Tourism for that Some of the wish I had planned however we did some A long way from midday Warejee told good number of food that now makes no apology and some that gave that admission is by A quick summary sure to comment the summary is that the best is to say with 20 would be the beginning of what I think is a good idea to think of a dynamical systems and make some kind of or not His voice on new inside dynamics I think for the nor look I'm sorry to visitors for the fun of comments about the value that you want the gold I met my eyes and nonMuslim dish but his name is It is also less is a mechanical engineer is Mathematical guard background is less according to hear chose the University however wrong doing mathematics and put nice picture Iowa subjected to good use Of Web site for you with V things initiative orgy of fractals where 1 of his images have won a competition they want to be restarted the collaboration together and the feelings that you have be in it or are really you really do is under the joint I wanted to thank him for that not only I had the feeling that the last mathematics and rolling out into path Nor did she would reduce which also can be disconnected and I think it is our duty to promote environment will call to understand Better Wholesale but not only between mathematician dishes I knew that It is the duty of Enron mathematician what they it's field Tool will work hard to get contact with no mathematicians Itself This is a condition for survival Send I wanted to find joy of 4 You don't look to support this last a socially I wanted to hide myself between talk with patients for whom this Congress 160 succeed 106 euros of gold when he said the following mathematical theory is not to be considered complete plans to lean so clear that you can't explain it to the man in the street also said something which is very Didier Julia fairways The shortest summary of white on mathematics for What is clear easily comprehended attracts very complicated repair costs Thank you A toe