2/3 Mathematical Physics of Hurwitz numbers
7 views
Formal Metadata
Title 
2/3 Mathematical Physics of Hurwitz numbers

Title of Series  
Number of Parts 
6

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. 
Identifiers 

Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2014

Language 
English

Content Metadata
Subject Area  
Abstract 
Hurwitz numbers enumerate ramified coverings of a sphere. Equivalently, they can be expressed in terms of combinatorics of the symmetric group; they enumerate factorizations of permutations as products of transpositions. It turns out that these numbers obey a huge number of relations represented in the form of partial differential equations for their generating function. This includes equations of the KP hierarchy, Virasorotype constraints, ChekhovEynardOrantintype recursion and others. Only a few of these relations can be derived from elementary combinatorics of permutations. All other relations follow from a deep relationship of Hurwitz numbers with moduli spaces of curves, GromovWitten invariants, matrix models, integrable systems and other domains of mathematics which are often referred to as `mathematical physics'. When discussing Hurwitz numbers in the talks, we consider them, thereby, as a sufficiently elementary but highly nontrivial model of all mentioned theories where all computations can be fulfilled completely, and all formulated relations can be checked explicitly in computer experiments.

00:00
Commutative property
Topology
Algebraic structure
Materialization (paranormal)
Thermal expansion
Kontraktion <Mathematik>
Similarity (geometry)
Partition function (statistical mechanics)
Basis (linear algebra)
Parameter (computer programming)
Axonometric projection
Number
Field extension
Recursion
Partition (number theory)
Partial derivative
Linear map
Recursion
Area
Spacetime
Forcing (mathematics)
Point (geometry)
Dualraum
Model theory
Moment (mathematics)
Group action
Functional (mathematics)
Bilinear form
Local Group
Vector potential
Formal power series
Spring (hydrology)
Wellformed formula
Oval
Combinatory logic
Relation <Mathematik>
Right angle
Coefficient
Units of measurement
Arithmetic progression
Transfinite Induktion
Directed graph
Inductive reasoning
Associative property
06:31
Point (geometry)
Commutative property
Topology
Building
Differential (mechanical device)
Multiplication sign
Algebraic structure
1 (number)
Sheaf (mathematics)
Mereology
Axonometric projection
Frequency
Perimeter
Partial derivative
Linear map
Recursion
Shift operator
Multiplication
Spacetime
Conservation of energy
Point (geometry)
Projective plane
Model theory
Expression
Basis (linear algebra)
Algebraic structure
Line (geometry)
Functional (mathematics)
Residual (numerical analysis)
Formal power series
Arithmetic mean
Computer animation
Combinatory logic
Hausdorff dimension
Linearization
Relation <Mathematik>
Coefficient
Family
Associative property
13:39
Topology
VirasoroAlgebra
Set (mathematics)
Multiplication sign
Connectivity (graph theory)
Element (mathematics)
Projective plane
Algebraic structure
Vector potential
Functional (mathematics)
Numerical analysis
Variance
Vector potential
Radical (chemistry)
Computer animation
Lecture/Conference
Relation <Mathematik>
Recursion
15:28
Zirkulation <Strömungsmechanik>
Direction (geometry)
Thermal expansion
Summation
Frequency
Manysorted logic
Moving average
Statistics
Divisor
Equation
Extension (kinesiology)
Units of measurement
Partial derivative
Shift operator
Constraint (mathematics)
Twin prime
VirasoroAlgebra
Model theory
Expression
Amsterdam Ordnance Datum
6 (number)
Thermal expansion
Functional (mathematics)
Connected space
Vector potential
Inclusion map
Computer animation
Equation
18:43
Musical ensemble
Product (category theory)
VirasoroAlgebra
Mereology
Functional (mathematics)
Grothendieck topology
Summation
Computer animation
Meeting/Interview
Moving average
Energy level
Differential equation
Equation
Divisor
Coefficient
20:51
Musical ensemble
Group action
Process (computing)
Product (category theory)
Sine
Mereology
Bilinear form
Maxima and minima
Summation
Meeting/Interview
Divisor
Equation
Units of measurement
Field (mathematics)
Units of measurement
22:32
Ocean current
Divisor
Decision theory
Drop (liquid)
Mereology
Variance
Bilinear form
Summation
4 (number)
Exploratory data analysis
Connected space
Lecture/Conference
Term (mathematics)
Oval
Divisor
Equation
Normal (geometry)
Subtraction
Multiplication
Product (category theory)
VirasoroAlgebra
Expression
Division (mathematics)
Weight
Bilinear form
Degree (graph theory)
Summation
Computer animation
Right angle
Pressure
26:02
Topology
Theory of relativity
VirasoroAlgebra
Division (mathematics)
Bilinear form
Variance
Power (physics)
Bilinear form
Connected space
Computer animation
Equation
Relation <Mathematik>
Social class
Equation
Coefficient
Volume
Relation <Mathematik>
Directed graph
Recursion
29:42
Point (geometry)
Direction (geometry)
Thermal expansion
Water vapor
Mereology
Shift operator
Grothendieck topology
Variable (mathematics)
Bilinear form
Explosion
Frequency
Roundness (object)
Lecture/Conference
Equation
Relation <Mathematik>
Social class
Recursion
Polynomial
Theory of relativity
Constraint (mathematics)
VirasoroAlgebra
Model theory
Line (geometry)
Functional (mathematics)
Numerical analysis
Vector potential
Spring (hydrology)
Oval
Logic
Function (mathematics)
Relation <Mathematik>
Theorem
Social class
Right angle
Volume
Series (mathematics)
33:40
Momentum
Thermal expansion
Parameter (computer programming)
Shift operator
Variable (mathematics)
Bilinear form
Manysorted logic
Shift operator
Polynomial
Conservation of energy
Variable (mathematics)
Functional (mathematics)
Bilinear form
Substitute good
Vector potential
Degree (graph theory)
Computer animation
Function (mathematics)
Thermal radiation
Theorem
Relation <Mathematik>
Social class
Right angle
Coefficient
Volume
Resultant
Series (mathematics)
36:43
Geometry
Direction (geometry)
Sheaf (mathematics)
Thermal expansion
Shift operator
Variable (mathematics)
Explosion
Latent heat
Term (mathematics)
Equation
Social class
Recursion
Shift operator
Polynomial
Surface
Division (mathematics)
String theory
Mortality rate
Functional (mathematics)
Function (mathematics)
Theorem
Social class
Coefficient
Volume
Series (mathematics)
39:00
Group action
Logarithm
Water vapor
Parameter (computer programming)
Mereology
10 (number)
Variance
Connected space
Fiber bundle
Physical system
Social class
Recursion
Process (computing)
Moment (mathematics)
Basis (linear algebra)
Coordinate system
Parameter (computer programming)
Thermal expansion
Bilinear form
Functional (mathematics)
Maxima and minima
Degree (graph theory)
Wellformed formula
Hausdorff dimension
Differential equation
Right angle
Bernoulli number
Resultant
Point (geometry)
Slide rule
Divisor
Connectivity (graph theory)
Modulform
Vector potential
Rule of inference
Number
Finite element method
Term (mathematics)
Wellformed formula
Equation
Normal (geometry)
Shift operator
Scaling (geometry)
Forcing (mathematics)
Model theory
Expression
Local Group
Numerical analysis
Vector potential
Mathematics
Formal power series
Number
Moment of inertia
Computer animation
Function (mathematics)
Autoregressive conditional heteroskedasticity
Social class
Fiber bundle
Coefficient
Flag
46:58
Spacetime
Connectivity (graph theory)
Model theory
Modulform
Sheaf (mathematics)
Bilinear form
Functional (mathematics)
Number
Frequency
Number
Wellformed formula
Manysorted logic
Combinatorics
Social class
Local ring
51:08
Point (geometry)
Topology
Transformation (genetics)
Logarithm
Modulform
Parameter (computer programming)
Infinity
Vector potential
Number
Integral transform
Connected space
Linear subspace
Fiber bundle
Isomorphism
Cluster sampling
Social class
Recursion
Spacetime
Process (computing)
Scaling (geometry)
Linear regression
Model theory
Content (media)
Operator (mathematics)
Parameter (computer programming)
Functional (mathematics)
Bilinear form
Number
LangevinGleichung
Computer animation
Vector space
Isomorphism
Function (mathematics)
Theorem
Social class
Bernoulli number
Coefficient
Directed graph
55:09
Topology
Multiplication
Spacetime
Presentation of a group
Operator (mathematics)
Algebraic structure
Differential operator
Vector potential
Numerical analysis
Maxima and minima
Integral transform
Formal power series
Frequency
Linear subspace
Computer animation
Isomorphism
Stress (mechanics)
Isomorphism
Negative number
Theorem
Recursion
56:59
Topology
Thermal expansion
Mereology
Vector potential
Grothendieck topology
Number
Integral transform
Goodness of fit
Linear subspace
Isomorphism
Equation
Units of measurement
Recursion
Spacetime
Scaling (geometry)
Theory of relativity
Elementary arithmetic
Model theory
Exponentiation
Operator (mathematics)
Exponential function
Grand Unified Theory
Line (geometry)
Bilinear form
Functional (mathematics)
Computer animation
Theorem
Differential equation
Game theory
Coefficient
1:01:56
Axiom of choice
Point (geometry)
Slide rule
Computer programming
Topology
Logarithm
Connectivity (graph theory)
Thermal expansion
Curve
Number
Inclusion map
Manysorted logic
Positional notation
Moving average
Equation
Normal (geometry)
Geometric quantization
Social class
Recursion
Numerical digit
Geometry
Forcing (mathematics)
Lemma (mathematics)
Critical point (thermodynamics)
Total S.A.
Term (mathematics)
Variable (mathematics)
Number
Computer animation
Doubling the cube
Function (mathematics)
Equation
Configuration space
Arithmetic progression
Series (mathematics)
1:05:02
Topology
Variety (linguistics)
Connectivity (graph theory)
Thermal expansion
Curve
Drop (liquid)
Mereology
Theory
Asymptote
Number
Mathematics
Term (mathematics)
Wellformed formula
Zustandsgröße
Normal (geometry)
Geometric quantization
Partial derivative
International Date Line
Units of measurement
Recursion
Covering space
Focus (optics)
Standard deviation
Logical constant
Process (computing)
Lemma (mathematics)
Moment (mathematics)
Coordinate system
Thermal expansion
Letterpress printing
Group action
Functional (mathematics)
Connected space
Computer animation
Function (mathematics)
Quadratic equation
Right angle
Coefficient
Identical particles
Family
Series (mathematics)
1:09:45
4 (number)
Topology
Enumerated type
Multiplication sign
Skewness
Curve
Group action
Functional (mathematics)
Subtraction
Number
Recursion
Series (mathematics)
1:11:37
Logical constant
Axiom of choice
Building
Group action
Musical ensemble
Differential (mechanical device)
Multiplication sign
Decision theory
Direction (geometry)
1 (number)
Water vapor
Parameter (computer programming)
Solid geometry
Mereology
Explosion
Mathematics
Roundness (object)
Manysorted logic
Square number
Extension (kinesiology)
Physical system
Spacetime
Process (computing)
Product (category theory)
Generating set of a group
Closed set
Moment (mathematics)
Basis (linear algebra)
Thermal expansion
Variable (mathematics)
Functional (mathematics)
Measurement
Bilinear form
Substitute good
Category of being
Proof theory
Arithmetic mean
Vector space
Hausdorff dimension
Duality (mathematics)
Equation
Quantum
Differentiable function
Bounded variation
Fundamental theorem of algebra
Weight function
Directed graph
Point (geometry)
Vapor barrier
Momentum
Presentation of a group
Transformation (genetics)
Symplectic manifold
Real number
Drop (liquid)
Event horizon
Open set
Power (physics)
Frequency
Goodness of fit
Lecture/Conference
Wellformed formula
Natural number
Term (mathematics)
Operator (mathematics)
Autoregressive conditional heteroskedasticity
Quantification
Lie group
Subtraction
Initial value problem
Hydraulic jump
Condition number
Addition
Shift operator
Multiplication
Military base
Forcing (mathematics)
Physical law
Diffuser (automotive)
Algebraic structure
Skewness
Total S.A.
Quadratic form
Cartesian coordinate system
Vector potential
Residual (numerical analysis)
Spring (hydrology)
Combinatory logic
Universe (mathematics)
Dependent and independent variables
Game theory
Coefficient
00:03
so we have a right to think through much yesterday I have spoken to the courage and the very old and was beautiful political incursions subjects as I told you will have many striking similarities there is Nice space of solutions that will be nice group of services and income balls and so on and now similarities and also striking the people working in the area who presented the results in such a way that from outside would be but not be able to play so I will present my own version of the politically caution is actually equivalent to the standard 1 and the well I don't think this is my my version is better anyway fortunate to have a chance to become the case for instance yesterday demand but the caution fulfilled its numbers and the and was given after she spoke you're able to reconstruct some not cut some of his numbers from from the recursion so these kind of recursion appeared to be Maddox models and 1 of the 1st version of this recursion appeared to the papal Marilyn Monroe and an example of that papal to mothers models of another front and chair of its it happens you goal eruptions that disappeared and no it's often referred to in the and then we will be granted to In the caution saw b trying to draw during the political recursion just like these few minds so I don't think that I want to tell you that this nite I try to modify my call and it was completely after all it was completely broken so that I will be the only half of my computer experiments at achieve the last bomb doesn't walk at the moment but the we just did initial partition British works pretty well so what's into political protection so we we have made progress the data as compatible data related to problems connected to a single generating serious called the potential partition functions for the coefficients the amount of coefficients is the same as the amount of her all possible partitions so all this data collected from human race shares they have to several examples like or is serious written considers potential oil and also but the audience and forms for and they will many examples from and but the political discussion is a kind of bridge produces a coefficient of this 1 by 1 using some of their television viewing is as important this procedure but the contract and the caution after to goes not so far this year the function of the One Stop formal parameter religious schools which is responsible for the generals as for the party just who genius to undergo of dignity and so I do spoke Computing 1 by 1 in by of induction J and M 1 would use of force ,comma discussed the general accused of some of the 1st example filled with what to do and people belong and explain later the source of this spring this year you to the gym and the former huge yeah yeah there's sold subscription what do have to computer is not as a function of what is differential the addition of partial so is considered this differential the majors this combination of partial damages you just don't get materials used the financial officer went to jail for justice news as a new independent variable as a whole symbol In but emphasized that had treated just F16 I will do all that the different ways just excited CSI office that apply for details such functions you John life do which told of
06:36
the function of the small bridges and has a son Paul for a partial derivatives the Times so this amounts to an amount the formal linear combination of Excite handling of excised the model but also because this amount as the amount of the river spoke this course unity also the data which is you you so I consider you really residue new CSI you I call hear them buyers of the scheme because I don't want you to look at this later there expressed interest some of the policy and also by the projection of 4 this could be done for any function concussion that In implied here but not what what actually is used for the discussion of the following day to said that the the little space at the summit it is a clear structure as of show us how it works we have this discussion on the future also that data so that when very very isn't this what they did not waiting to mold is just the combination of the 2 stocks I offer the basis of the new space and this is the linear combination of his these coefficients depending on the so you can see that genetic point of the space to be so for me it will be a point of view but these coefficients raised depend on the way but anyway could considers the coefficients of the functions of the let him in when 2 :colon the Justice yesterday and I think let my interests like getting all these potential the only hope of for some of the key now just use it to change his anyone you originally IT and also need to understand that a magical shifts of oranges just need is particle .period of despair some fixed the act of the things that I have supported building should be inevitable part of the initial date of Russian and I know from where the conscience of the people here on black this apparent function the just 1 and lost 1 thank you the so how supported the Ed Sullivan made the call at this point of this coefficient depends a conceives of customer differentiate respect perimeter and then multiplied by CSI multiplicative structure in each of these here have .period means that multiplication Kansas brother age hearing but multiple abuse this will move brigades straight section and also some tensions and emotional and conservative and dimension of the problem arose immediately in the example of how it works so
13:29
huge expression Dole age so I used be installed the serious and the modification edge the
13:40
next step is to do all time so this is this is still an element of age and I want to relate the the new newly computer terminals watercolors but there is concern that there is a lesson cited is the element of the above the
14:03
inside the not so just ignore Xavi component of these all this
14:11
right hand side project just to the left of the 1st song this is huge so let me show you how it works it should have some examples of how it works so the 1st example of really the attention so incantation potential is the intersection on the fashion for the session number of cyclists socialist places of persons boys so this is implementation all this I I want to this is my implementation of these functions yeah know works this is a actual definition in some way to compute the side
15:29
knows for the actual competition you've lost restraints so this is a matter of constraints because of the strong constraints fall on the left side to him that the facial appearance which involves partial derivatives and also sometimes 2nd partial nudity and also I don't the factorial future the officials said almost all of the year except for for cases and equal to 1 1 and while the out of some connection yesterday this circumstance for the explanation of this generating function here are for the functions of the year to appear here 2nd apart from the other brother of bottled and also the another extension of period the unit's expansion also implies that they should here this fall upon the age anyway and this is the wellknown Minnesota Orchestra and or written consent potential no the following treated all these aggressions together in a single mom falling inquiry you a new model X news for what will ax and some of all these equations all question with this fact experts at a flower and plus 1 to play by less than that of the circulation the happened that is using this to right the question is Nice company so for instance was terrible safely summer the storm if some of this stuff future Europe the expression which is exactly function researchers this 1 so this summer will appeal if some after a shift of imports from Japan the same sort of summing up drawing this coefficient adopted what I do buy you here directions you yeah the
18:39
present here altered by this goes
18:44
bikes the will tool to employers want double vectorial divided by critics of the warrantless you use 2 1 the level of the went to upon possible the partial site here is that Kelleher told the story you just product of 2 things like that so that
19:49
historic side bands Dawson function you the draw donors and the product of you the are square here here have his school this song containing several other possible narrative is a cruel blow people this bill died a tool function you so you you already contains just order partial beautiful if you apply is built on the obtained 2nd order narratives also on the pictorial part you appear to want to menu I think the coefficient CSI which is already knew that the disaster "quotation mark with your son the
20:52
music and the use of fracture most of the European Union Station now he has that algebra yielded age knowledge case this field there is
21:05
this regal floor from political and so the negative part of this book are part of the and we'll do that part I don't think that education action the girl whose love for Hong Kong and his today if you
21:45
reach it 3rd you want the value but with 1 of the possiblity I think this is wideopen storms that also of these jobs which is probably the most interesting so it's also the kind of some appearance if you from Opel over all possible in this in this society and this appears to be the product of 2 but lost to the trust you this country is partially due to fulfill and then there is the only 1 of its units which contains senior jobs in the form of a private security for this
22:46
is misspelled for CSI but if you take the problem but to I have
23:03
this from the campaign convinced much more dorms missile which is used here if some part of the of this product and also so the doors containing poised divorce of eggs so have some the attention of the summation of decision by the United States problem the deal to also increasing the boys .period division this CSI has been amended to followers of 620 multiplier of Boston's 4th definitive degrees the domes which they appear this
23:48
mission could spawn both on the partnership has 90 the difference is not this year but there is some good news from this term is a positive part of the concludes that if you apply these expressions or parts of the expression Council also is on path Israel so this is how it works so the computer that the side current but Hi drop positive blocked excellent way it for instance who look at the next storm but that's it in the is the place of party does exist but if you honestly feel dozens Roberts becomes much nicer 4 the pressure 1 you there are some of the world including forms foreign solidified divided by I am here for there just for the liberalization of the divide by 2 and the hero or alright could go to expand the company's square so I'll get the Commission's of this type these 2 possible ways to represent of emigration right the resulting for each part of the U
26:06
contribute 1 of New plus 2 my mind that this is what are doing quite this is In this particular case the and for the same aggression is the 1 that it should take you my plan why don't you "quotation mark you lost to security this a controlled there well up to the trial which is which is good to good this is in the form of the virus a Commission now if you'll just have these tags at the commodity spots in June and in the June foods the coefficient of the political power of original source homogeneous story of his division are the question to the Final Four .period the gusher caution and could be combined in the Commission for the city With a Boston with until the still
28:43
thinking so because donations holes for all June and except for some initials various which for jiggle to 1 and all but 1 and you hope to hear about all the things which are initial jostle Holocaust coffins will be given explicitly in a different way explained later so so what do I play claim that for all particle attended the general from France's of just followed by a formal manipulation recalled to present them with different for years this matter equation for this because of 1st questions just relation of the political percussion and these are just the same relations that
29:46
the the political attention from what my point here just before relations for a constraints OK so this is not this the
29:58
Commission that I hope you are able to apply because just come to the right hand side the the part things around part of this represents the next round of the percussion simple to the employment the best example and that they would like to
30:21
do presented is there the fall the function which is different from In the modifies written answers potential but my wife I think socalled couple classes so got colossus interesting glasses the models ways of your views also appear quite Hoffman some publication focusing June plus 1 consider the forget each you by just 4 of you think raising marking 1 of the market points in the yesterday was talk this could added to fight also is the universal cure this year and Mark points here a glasses say 1 site associated to this market for which the chair .period colossus of peace .period associated front line is my .period here also of explosives much the blast 1 off of this type "quotation mark exist for want of an hour later a little bit and so the words of his glasses which I here here we have seen this spring is just more of the same principle of the class size was about you were here was the models spread which of your New Jersey who has the board .period number I 1 bubble to separate directions :colon there some of the market .period so this glasses that could be laid at least 1 once but but the last 1 is just a completely new and there is no relation between these glasses glasses in the water it there for using in but produces you go logic about them we just call money the mark for glasses couple so is the performance of the buy or sell the car the I want to take people who bought sites to the cops got the key so I copper colossus
33:44
multiplied by a suitable and organizer with the radiation which is not known to be a serious balls to I use this as his formal parameters but would also if you don't continue the for this United's unions whose coefficients involved all portable intersection liners for a couple of glasses of key Cup Cup is the most prominent degrees of degrees to kill he acknowledged from his statement the functions could
34:51
be reduced to position potential buyer where simple substitution so consumer something but also in as wearable here just to make sure that the potential so that the variables to 0 2 1 announced shift about them which she shifted to by shift to the right by this Blumenthal this this is so all this complicated or serious could be come on this is resolved implicitly this is long form when you hear about we put forward explicitly discussed the with there's results special cases for instance there is a papal while this statement is sort of 40 people to hear right but this is this just slight momentum which occasional on the same arguments anyway so the good news is that this is just the same written conservation potential a shift of parliament parliament so don't think the National should work on the Saudis you so I could just couple some the gloss couple the
36:48
that's interpretation Joseph Freeman Georgia off surfaces so this is just the same as the class of the division will be sold the new sympathetic for up to foster reaches this I don't remember the computation of Paterson on school firmly that the terms of algebraic geometry adjusted the computation of the section and 50 involving just couple 1
37:36
so so for instance the generating Sears that Duma mentioned the last is just before the invasion specific stations for the ignore or couple colossus couple but gentle again it's the rating shares for pitiful going on is obtained from the middle of considers potential by shift of unions only reports here all year long and stood the are set to 0 realize where is simple and the protection to the intersection we wanted to obtain that this directive functions of we also require strings of the shift of the government does not change the rhetoric mentions the only thing that Shearer the partial notice of the fact that only the majority appears on this coefficient it was just too for the civil affairs here we have the but that by you it's a very simple creation the using the
39:01
same procedure can combine all this equations the right of the form 1 of the top of the advantage of the same service in Kansas and the same system basis it is the longest thing with changing his job as the tourists this ETA which called the shift the origins of still use the cupola still shift their origin so the polite if you're blood does the simulations with only nothing compared with the written of can you just so and that the 1 . example in what is
40:05
called and another modifications instead of long topical societal use the fairway land the plus side will continue generating function which contains all possible torching the group in the office in Monaco on possible communication from the cloth what is all but 1 of the land the glasses in the following way 1st of all is that I love the because I colleges component of the German charity for the whole of bundle of the closest drink glasses of water bundle and some of lumber over the German was passed John J. the as a result the bulk of mom which claim that the EU Is the the EU and competence of the joint shocked of the joint chart the I 0 so all in all competency waiting for you have only the director the new 135 points along the sides will use slide to dispose of the competence of the German chopped by this fact character from the kind of thinking I thought I discovered in the components of the German parked where this was his scaling factor which is building community for for further references for the Monfort formal so you to put it on hold him out of the country this is the political will to the new classes so you just and some and as a distinguished collection is a loafer or all of the problems because of the special and today generated function which contains all plausible communications all possible nonnationals and before you for every intersection of moment thing the this function by summaries reason the school that could result in generating function is called potential of economic Hopalong logical fueled fears of security equal to what appears to have considerable thought the case of an greater than 1 but for the people along the the use in knowledge and other than that the dimensions the mention of the of the leaders of each thinking only the Philippines Hideo falling carefully which Jason particularly if the aboard that of this Parliament of just 1 this Strong becomes just stop some of the most potential for single lumber colossus Richard Pearce used to formalize also from the dramatic part of the besides the issue of 4 particle accused China of of the Parliament the only son of a potpourri he was considered to have the force of the bomb attacks of my manipulation of of of the Commission but disappeared gusher on just supplementary parameters official not involved in the percussion part of the show it stated that there is a phenomenal function for his part and you I have in the a is probably not the best way to run businesses that the coefficient all of this issues here L Is the dignity of this model and also the left side of his her number year of action in here if you if you put them here football hero here but this is holding isn't as coordinator of the food this little thing a constant you might we have here if you can find 1 if you are some not all these terms is different this is my intention you with whom strong degree of dignity and of this expression discussed ways is but would numbers the it going on a number of calls here 2 computer specified by the way the constant the same as the cautioned the rules but just another and worry that I mentioned already known for its formula for the land the classes and this school there for more than that of this work expansion suspect and there are good but is described by bottle differential equations original was formerly not intersection but
47:00
Jones the colossus themselves and they could falls take the miracle you go to the MGM and this could be a little less than the full inquiry probably so it's just a full the glass which is a couple bridges the performance of science was 1 of the poorest plus 1 recalled b my the glass just below this forms the last 1 .period tight political to apologize so this is this closest supported on the longer the of the models because of local police draw it double .period it could be the the .period called his "quotation mark as a duo Isaac your friends for could be the core my mind still connected to but in way of these devices could be regarded as a its own models .period a mission needs to remark market .period numerous by industries and past 1 and a possible so it has many components and any accomplice in all the competences good regarded as a modern space to wear the market take this glass supported them this month .period bonded ways so it's just a matter of combinatorics from wanted these long fought for international from the section numbers of of glasses it's still there will get this form and the new form for
50:13
1 which sparked fears that is formed in the United States because it is an hour and a hearing is the function of these initial function which is of course the consumers it was not the sort of all these devices somehow the see In interest of justice the inventor of the device
51:09
the blast was not medical device here have more points good coming from the the and you have just points to have had lost points here .period here if you would like it was excellent and would going we could you here's a point in price from extra costs In front of the money cyclists at what some extra classes dependable numbers are your old article to heroes of the year is also good thing that the formation of written in these parameters so you're right for the initial but I'm the really it is for automatic article 2 0 ahead the concessions but about artists change agreed to resolve the question of fuel the solution of this linear regression schools coefficient is just exponential disappearing for articles you have to be contention that to fight this way so I hear here it's not the creation of jobs of not knowing what to expect but don't the function itself is usually the Jones from behind this way in this case and the creation of the pair of on the board during the year of the To me in the space of parameters know to change potential so it can no longer be occur in the space of our parameters in the store hold potential the going to
53:35
dysfunctional also and in the political percussion the same source it the slightly strange were using socalled "quotation mark lost 4 the majority of the formative years already and the local gene and also I use same drama say what this club clusters for years From my point of view a lot last transformers just leave it is more views of the space of around In the wearable In other words about these according to a more fuel efficient models you Papa and here they just take the same explanation of its support for but the scale the 2 of them both play and divided by 21 1 pictorial they're just linear spaces and being isomorphism of the spaces so if you want serious candidate presented to the left side of the entrance a dozen :colon that their coefficients the 2nd form by the way could be applied both boys affirmed that it is because of the
55:09
negative that'll Victoria Falls negative what number is well defined that's just the way formal authority who didn't know what was applied for for positive and negative USA multiplication years the
55:33
political aggression use multiplication this place and this isomorphism of course does preserve the multiplication so it's not Baltic mitigated isomorphism when I speak about multiplicative structure you know space age
55:48
I don't use the presentation of this place on the residence of the nicer parental multiplication ways here if you don't just multiplication ways you it becomes also simple hereabouts here her from stress confront the partners because of the concern the storms to be a differential operator 1 consider the slump last as far follows you can it's not considered to have we have the special foreign political but the just didn't have the patience of the financial for the new Woodbury New the data we introduced for the floor specified on the extra the financial affairs of ordered the financial period None of
57:01
precipitously the fall the advises by users with simple if you use the right side of lost for started here reviewed by the exponent of thought small are on the same as part of my family so update Of this serious issues that could it's possible there where have the numbers you're the space which did all right and although I can say this is just the Houston the old enough conservative from the scale of the coefficient the case .period Inc basic veterans X coordinates in order to update my I want to buy multiplied by a suitable model year was inside the Federation of the Blind has operated the lives of so obvious but modernized some of the other branches in Rector from the reference site to just looks realize reference lop off form but in the lessons that you should you do Strange talked of put the fusion and you do I will speak about what they mean by ability and with it if you wish story playing the usual good photographs riches part the
59:15
borrower's you hear in the this could spawn tho before minus 1 or 2 line the poll this includes 4 of the last 1 With reference to which responds except I'm here so that these are units the In the Lord have balls so please these years spun off Of the closest homages to pursue this does the same as before this as a space of their functions it's but the song it was just coordinates deployment of spares about now it's not coordinated just made fun of the character which is slated up space in the space of promoters years and also the precautions taken were required .period the trust of the view that the Utah and it is equal to according to my attention something like CSI why which is to explore it's also some information of than of the functions extended its incantations initiate forks just same With the relations with relations they work some these implants percussion a game users usually if you take a few days ago this Dr. Johnson jeered agency then you obtain this because a procedure to recover the
1:01:59
Hodges component of this potential and in
1:02:06
particular if you will for for the
1:02:11
choice of the various using the numbers you have specific location of these for a scorching intervals along the molded the single and the classes chorus numbers and this is what do you do mentions his talk at the
1:02:29
socalled before conjecture the most money conjecture claims that the it's sort of a caution for quarter's numbers for the same proportion of girls if you remember it was related so there the most the related to
1:02:51
certain spectral cure for the structural from this incursion is twofold longer Lambert cure for use a slightly my station is so different from those used to total of viewer was floated shoes of variables so I used that this is a landmark draw I use Gordon assistance that the critical point of the progression to the next slide is the origin so if you expand the it's what the synthetic squirt was cooler full climbers when you toss the plus 1 force 4 1 said something like this and inclusion the exchange's privileges of protection the the program takes double Digital coverage so there is the a Jewish new chose to points out that many homes in this notice how some bite my this is slightly misleading attention because has nothing to do with the complex configurations however is used to provide the papers devoted to this guy percussion let me introduce an old coordinates his new by these equations for this notation of Asmara physical to minus as for this is Arturo and this is
1:05:14
how the functions while other is coordinate becomes just a squirrel book if you look carefully at a cost of 4 Nova ignorant and then have that it depends only on the all components of all of this function life so you can freely modifies the competition them not participating consider corruption the there was a sense focus there is a huge variety of persimmon adjust the change the spectacle for His produces does the same percussion few modified it the composition of the gusher vision Villamizar said so I did the slumber ,comma just as a job a for the origins if you write this way is a terrible
1:06:13
isn't like stands as a global coordinator a fine coordinate on this land but the question you're still a drop of the covers origin so you just hours sitting in computing coefficients unit only 1 part of the the computer also do would for this or blocked the I the heading mysterious the relative their share of the bayou In my from auditions by just lost here to functions for the 1st quarter this by the coefficients the 2nd 1 is viewed by the duty which affords or partial bottles chords idle for 1 the couched in terms of the special character of the problem the next but just to make the case for the focus 1 of the stranger the kind of identity and I use the reference being if you think that this is a family that at this remote corner of the career of this 1 I observed slew of strange but part of the guys from Mathematical Institute Institute in soil this is exactly that this language is foremost from this paper this paper is I think the chorus numbers this is the political discussions you have to have a disco officials which appear here it slightly different from the other related to the subject of our talks deals is as simple as expansion of government functions of state and Victoria there is no standard formula something like it was a part right and then there are connection to and simplistic expansion which you here forget about this is your moment but this is In I mean if not not exactly political to the theory goes furious appears to have the asymptotic credits in terms of the synthetic
1:09:46
functions and city informally people a borrower and the lender society observed for which is close to what was used in the book Duma wanted all of the above on the stock you have your own you as
1:10:21
I do the following observed the caution for 4th quarter
1:10:26
numbers I that this year so that I pulled this coefficient enumerators and also in the rest of the In has Coulter Missouri what they labeled the Library of thickens and committed differences and they gave me this reference to his religion because of the the to hold on to his death from the but I don't know of course so I think that the the issue is my example and now ready to if you the more general forwards accounts the Commerce moralists all the cases you have to buy their own we still have time probably will start in the definition of the problem you know
1:11:56
I give this because it's going to affiliated looks like so I just think that if you just 1 ways forget the youth of this followed the repeated many many times have you principle of the whole of this almost 2 years before the according to according the the general principle of the new ideas come along because people believe stopped the belief this book because comes a new generation of additional full discretion when it works for them the people of the U.S. this year by the way it does also before the before I go further than the they're play a little bit of this especially closely and fall that is the strength of the nation good in all of this the utility of the call for you but this is Bryant formerly the falling for it is the hope of your but you in office that he will the mark this financial mutual combines drawings of different nature so this is 1 of this is a point of space so I Nevada to do you the more you're going to compute the dance of this expansion about different wages as they belong to different spices I don't like to pull them together about the furious you've got balls include the good consideration is what good does correspond to the presentation of W 0 for where is it so so if you will compared with the bulk of initiative and that this is his 4th of the Colonel what do you want to log into the continue consideration was the the right thing to make it was metaphor so might the formal as it appears in the Bible all the related to you should go .period someone alive Is it it's painful to watch some extra jobs so you can combine these the transformative who did all this time former leaders "quotation mark defined the former is hero don't and that this job include will be included but I still prefer not to do this for me Holly you this year told steals your there not such jumps well I think is about these someone 1 of 8 x I did race but the letter the existing of case it is the concierge and led the drive to compute built lofty what is built of so far but I missed a drop of this is a belt of the alleged infiltration can build of 2 some modification of the 0 . what do you Air bridges will last long the Victorian with the power of the Times India the wonderful or it was the duty of you if you your CSI stop for a while is a global cell phone and the 0 2 feature so if you have a therefore it will be by definition this is 6 I just come so of the come then you can rent the England this 1 . 4 4 this kind of thing because it's should been like this you works for the potential what I do you want to call it the serious side to keep the 4 20 I of Europe this so now we played a little bit these examples I think that your idea the the initial so now I you down the condition of the political precautions foreign start to 1st of all started some it is a fine dimensional community will also review and also acute business product which is rare for a move multiplication which is due by the functions so is 1 of the very little is known "quotation mark at 4 but I thought at the time for dimension of this spring In all my previous examples age was just the onedimensional and the loyal considerable difficulties the typical example support signed the premier cells of which has been submitted for offered them so when of which I it wants to use them some multiplication like the fight square musical for the this is for the rest automatically but it's not the 1st of all the use of your place you want to the new awareness is proportional to the I don't remember well 1 the I don't remember the conventions prefer some may think variables which purport stop like you and also and also that it reduce the arches of regions this article built up please it is the 3rd part of the impetus for the advice of whatever this from expanding the base and that officials of the French all of this this is 1 of the typical examples in the case which responds to the precaution of friends and around them how it appears to practical is it most papers response to that at least get some or all of the Church of the 4 them all right but that happens in the petitions users who have so far as to the source Of the unity of the physical .period some Lodi so detention of capital of the so it's the convention than all the builtup Isaac equal to 1 is not the physically meaningful a typical example is called knowledge and call you go moldering cost some congressmen defaulted this is the way forward appears in Quantum ,comma 4 minutes ills the fix for once and for all considers that the formation of the initial date of political caution and therefore Minnesota but also known the deformed fetus at once and for all the next step is to consider the algebra the place used my examples age this is this spice of bouncy as you no 1 variable discussions I I hope for the sprays of water the serious work in and you can multiply multiple multiply using multiplication as of this spring the becomes it began the report also directed for this spread due by the following the formal so if you dig 8 x to the borrower can do what the physical demands 1 quarter of the year this case is a the planning of a and B at times the coefficient some extra coefficients regions president and the shriveled his father taught to plus 1 some because of this property it's skews future players can sign the negative side but there a more inward and white you have to learn to work with them the following is a due thank you you get a possible the boss was labeled as eurozone Booker plus 1 plus 1 different buyers I made his substitution my substitutions that affect people for the same and that it is primitive of these the primitive difference defined up to recall some of the constant doubled contributed residue the power to define the the basis of the schismatic working them the next fall and the spring the thereby away by the various this if you this was a vector for without Runkle produced many commendable resistors basis 1 of the possible of awareness of the following so user the uh bestowing Texas the but your case really which is is something new times lyrics to the power game To the minus column for the Fort Portal you would expect new can this stuff disaster this want them to keep 1 so if you smell it that so the burden because the sheer neurons for 1 dimension of the spread and came from 0 2 so if you take the playing with him reactors the same investors to get 1 up to 5 0 and all other cases where you yes there is not presentation of the data using what love for my cousin played G so I what please stay the part of the picture for real so Space View it is the 1st is just a part of this the moment I have not they plan to use for the caution because all of these can so it can be used for any for a year the fees that is the next year or the year sources note it's important the reason festival this is just the opening and the coordinates of space consisting of all around .period opening of 1 or more I know and by definition always organizers of subscribe you good so for these coordinates of where members but by the reference is just 1 of the most examples but of course there are much Romania more different .period mountains of spaces the a booming precise phony the basis of race CSI all the bases you know you and also some of the start of you ways the fall of the subjective many of these soft basis during the 1 another for the summer sympathetic some where the heroes of a deal with security the together former double always and then build space age the question the explicit form of this business is not so important but for the mother of computations for so to make some the competition and also the future of the bases no where it change 1 of the bases of the producers the changes of the other with for instance like users according this is the end the actual could do about it except for for all this there you the betterment total I means that of space protracted by quadratic forms solid from this and the following week folk space complimented players and all sorts of use the coordinated protests of 0 Algeria's according to spend it was you before for which could consist of money but not all of it so here we use which have chosen the way this CSI all 4 she overheard you always have which makes also there is a basis here which is also due police monitors but the bread and butter furious battles heroes of do not at this point the decision the difference between these 2 measures mutual what additional article last struck vertical so far although the sponsor the belongs to the so this is president of the would you be so you here None of analysts city the subways so claimed here on clay this opera 0 is assuming a single and all 4 rounds for all the barriers which he said the 1 thing that you know you can use different measures but not for a moment the business here and the was due to the spot but respect the differences of race that actually In just the before you T some of also some Utah some practical somebody told him .period of friend the system itself this is some specific cases of these 1 and it's 1 of the following 4 so the tube From time to time 3 the European Union on possible traces but this year the from you don't this have the year for infants for the cost potential was different for me then and 4 for maintenance for veil but the salute was like but further the harsh occasion cases was different the kasha now I From later the main percussion after 2 years this is a mail commissioner so what here while the potential of told lies with the addition of the bank's structures don't you also multiplication .period different the coefficients this specter parameters of the multiple Cup multiplied by this CSI of using multiplication image the question could In composite form like like must I don't think that I promised the Delhi was initially make of this all of this the caution and the initial date of cautioned acquire the port here are some special joy of this year you but the thing that I remember was leaving it to fix you on what you do for the child Church of it 1 support you ate he is a continuous Gold thank you we have to this year 1 this is because of the falling for some the people it's all so Excel for excited the boss functions therefore of multiply you have to to be still more often don't function of fuel increases that a assault would be above the law Of the functions but is getting greater and greater Fourier carefully defined 2 founded and later joined this contribute the Commission due to the human I will speak more about that for the of the initial data but now from this year so long it was just applied directly to the because all of these Church of also contains the definition of you 1 1 and you 0 the venue but applies this discussion you find your June for all Jews there for this you Aegean it is universe the 2nd statement is the falling so the the quality of this year's Eugene In uses itself this is the claim that the display on knew that there are only 5 men job here on final this summer's final action on the fundamental nature John and each individual coefficients Of this expansion is putting on to on for each gene of a heavily funded in real terms so there you for the for the conditional the point of his narratives it could it is a difference these properties depend heavily on defense some space is Lagrangian immoral this condition is the initial conditions also used where much of the proof of this condition of this property if you just but slightly initial convention conditions that this property for for a stranger to the British stopped all books were sure but the to the profit from difficult but this is so far as to use caution could be applied for any for initial conditions because Yanukovich saw it is the 1st of the 2nd landowners doesn't use it hundreds of property of but the Serbs properties in the property this is crucial since all this implies that you're going to recover have gym so as to truly huge uses differential for instance you can use them better form I what is important is also believed that 42 because differential function that can occur the function of of optical some of the examples of a convention before 4 in the form of accomplishment and potential this constant force holders 0 the purpose that we cannot assume that the soul of therefore session this summer break to recover the constant jumping on the cold here capital that this potential as many as it was proper to someone obviously look a the way socalled by socalled dual equation which said if you will defenestrated it all literature observers from Geneva to condition imposed problem because we're deficient ones so here are this is Jr minus close by G so this call we already defined formed and little to do this year so it's all hold for the villages of well is defined this is going to hold together humanistic wanted the door jeez you in the same way so defiance In defiance of the company might be here to John mind soldiers who assumes this month in this isn't but the initial here reported here for would you mind to box and positive 0 0 0 2 0 0 1 2 0 0 2 and also want you're all of this year the no properties for the potential the Royal discussed :colon on the variation of the Lagrangian plans so long as there is a quantification sympathetic consideration of whether to on you because you tomorrow I'll will discuss his dependence on the choice of what to buy the knowledge in mentioned several on the property that's why we fall on the way but to what extent the Church of the day is important for the 1st probative of the different Utah leaves per shift all durables social aggression that all of the combination you might and to minus the offer was made so a different churches of EPA useful just shift of the timber will absorb the fact you the the death of the weighting functions the shift of the terrible says the Community tool there is a currency store the differences of the 2 at the center of the property this this is what happens if the troops a different ways drew different tiers of on the basis yeah war tho case findings if you look at his formula no it was meaning that the actual children of the business is not involved so this falls so so this constituted is all the there is a formal functions well defined and the the Bernardino of the terms of the baby the what you're the feeling in the event but developed a local officials to new diffuser coordinate so there's that particles just as the presence of the political coordinate system on fears of new defense bases but if you just consider the a formal functions so well defined on the and even is not too that's not the depend on the on the nature of anyways but more more all takes expansion of this function of origin expert so .period what problem I some of the functions that was a defined but In the what this is a space and we don't know how to expand this functions origin but not politics but this is the point of view when he had different shades of Utah but shift of foreign so this condition conditions on the community that if you change the terms their good responding footage of the functions of all defined the shift of the government and the direction of these notes is welldefined operations the function I think it's putting on his direction some of the children who knew that this potential has the burden meaning the it's essentially as determined by a Lagrangian point of essential part of the definition of the potential plan and all although think it's that just the coefficient which are used in them for the political interests of communities at the end of this the meaning of this actually is unique determined by then there the property of its creation the specter and also in the view of this I feel wall 2 hours thinking much b question no I think but to postpone the creation of the next book is the next stop will be mostly devoted to Huish huge amount of new different applications of foreign firms give and very few places not I think people want to believe that political leaders written so you're speaking on the eve of the Office of talking to kids Griffin's cautioned that these are the ones you want for varies from here's here because of what used to be that of the you won't it its close all the political clout of the you think that people be 1 or to times in 2 the fact that the rest of people over years and also professionals from different religions or what he said he had have the effect of general book few