The correlation numbers in Minimal Liouville gravity
Formal Metadata
Title 
The correlation numbers in Minimal Liouville gravity

Title of Series  
Number of Parts 
12

Author 

Contributors 

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 

Release Date 
2013

Language 
English

Content Metadata
Subject Area  
Abstract 
The correlation numbers in Minimal Liouville gravity from Douglas string equation We continue the study of (q, p) Minimal Liouville Gravity with the help of Douglas string equation. Generalizing the earlier results we demonstrate that there exist such coordinates

00:00
Number
Crosscorrelation
Function (mathematics)
Multiplication sign
Gravitation
String theory
Maxima and minima
Nichtlineares Gleichungssystem
Analytic continuation
James Waddell Alexander II
01:16
Axiom of choice
Group action
Conformal field theory
Model theory
Equals sign
Graph (mathematics)
Logarithm
Thermal expansion
Likelihoodratio test
Generating function
Special unitary group
Dimensional analysis
MKS system of units
Hausdorff dimension
Physical law
Trigonometry
Theory of relativity
Interior (topology)
Coordinate system
Parameter (computer programming)
Maxima and minima
Mereology
Product (business)
Partition (number theory)
Selection rule
IRIST
Theorem
Relation <Mathematik>
Nichtlineares Gleichungssystem
Manifold
Cycle (graph theory)
Arithmetic progression
Spacetime
Point (geometry)
Module (mathematics)
Transformation (genetics)
Diagonal
Constraint (mathematics)
Connectivity (graph theory)
Maxima and minima
String theory
Partition function (statistical mechanics)
Mathematical analysis
Rule of inference
Number
Frequency
Hexagon
Chain
Pi
Coefficient
Kritischer Punkt <Mathematik>
Graph (mathematics)
Ring (mathematics)
Modulform
Nichtlineares Gleichungssystem
Normal (geometry)
Renewal theory
Maß <Mathematik>
Initial value problem
Newton's law of universal gravitation
Uniqueness quantification
Surface
Model theory
Exponentiation
State of matter
Theory
Operator (mathematics)
Algebraic structure
Limit (category theory)
Mathematics
Maxwell's demon
Function (mathematics)
Autoregressive conditional heteroskedasticity
Gastropod shell
INTEGRAL
Multiplication sign
Decision theory
Set (mathematics)
Insertion loss
Parameter (computer programming)
Independence (probability theory)
Mathematics
Dedekind cut
Gravitation
Moving average
Körper <Algebra>
Series (mathematics)
Area
Logical constant
Correspondence (mathematics)
Point (geometry)
Thermal expansion
Twodimensional space
Term (mathematics)
Variable (mathematics)
Open set
Fluid
Crosscorrelation
Natural number
Hill differential equation
Convex hull
Condition number
Conformal map
Thermal conductivity
Resultant
Surface
Computer programming
Functional (mathematics)
Existence
Conformal map
Table (information)
Algebraic structure
Modulform
Equivalence relation
Local Group
Drehfeld
Wellformed formula
Natural number
Musical ensemble
Spacetime
Maximum likelihood
Gamma function
Summierbarkeit
Curve fitting
Linear map
Modal logic
Rule of inference
Multiplication sign
Maximum length sequence
Axiom of choice
Sine
Coalition
Forcing (mathematics)
Consistency
Projective plane
Expression
Confidence interval
Mathematical analysis
Transformation (genetics)
Hierarchy
Inclusion map
Number
Voting
Lie group
Combinatory logic
Gravitation
Körper <Algebra>
Maß <Mathematik>
19:01
Logical constant
Group action
Metric system
Model theory
Conformal field theory
Scaling (geometry)
Generating function
Special unitary group
Dimensional analysis
6 (number)
Different (Kate Ryan album)
Hausdorff dimension
Physical system
Observational study
Theory of relativity
Proper map
Differential (mechanical device)
Regulator gene
Block (periodic table)
Interior (topology)
Amsterdam Ordnance Datum
Parameter (computer programming)
Mass
Mereology
Food energy
Inflection point
Annulus (mathematics)
Partition (number theory)
Selection rule
Order (biology)
Theorem
Relation <Mathematik>
Nichtlineares Gleichungssystem
Curve fitting
Point (geometry)
Rational number
Addition
Disintegration
Maxima and minima
3 (number)
String theory
Infinity
Number
Frequency
Goodness of fit
Arithmetic mean
Selectivity (electronic)
Quantum
Maß <Mathematik>
Surjective function
Standard deviation
Model theory
Theory
Operator (mathematics)
Independence (probability theory)
Basis <Mathematik>
Ultraviolet photoelectron spectroscopy
Mortality rate
Gauge theory
Group action
Cartesian coordinate system
Power (physics)
Mathematics
Formal power series
Function (mathematics)
Autoregressive conditional heteroskedasticity
Collision
Musical ensemble
Game theory
Family
Differential (mechanical device)
INTEGRAL
Multiplication sign
Commutator
Materialization (paranormal)
Expandierender Graph
Grothendieck topology
Mathematics
Resonance
Moving average
Körper <Algebra>
Category of being
Pairwise comparison
Arc (geometry)
Position operator
Stability theory
Logical constant
Polynomial
Point (geometry)
Term (mathematics)
Variable (mathematics)
Wellformed formula
Crosscorrelation
Different (Kate Ryan album)
Condition number
Metre
Classical physics
Surface
Computer programming
Functional (mathematics)
Conformal map
Modulform
Equivalence relation
Lattice (order)
Mach's principle
Wellformed formula
Operator (mathematics)
Differential operator
Musical ensemble
Summierbarkeit
Linear map
Condition number
Multiplication sign
Addition
INTEGRAL
Coalition
Dimensional analysis
Division (mathematics)
Quantum field theory
Continuous function
Number
Voting
Lie group
Sheaf (mathematics)
Negative number
Ranking
Körper <Algebra>
Maß <Mathematik>
31:15
Group action
Presentation of a group
Frobenius method
Scaling (geometry)
Multiplication sign
Price index
Sphere
Mereology
Dimensional analysis
Order of magnitude
Variable (mathematics)
Order (biology)
Positional notation
Gravitation
Set (mathematics)
Hausdorff dimension
Physical law
Series (mathematics)
Social class
Polynomial
Theory of relativity
Correspondence (mathematics)
Residue (complex analysis)
Median
Food energy
Lattice (order)
Term (mathematics)
Variable (mathematics)
Connected space
Derivation (linguistics)
Partition (number theory)
Series (mathematics)
Selection rule
Order (biology)
Relation <Mathematik>
Nichtlineares Gleichungssystem
Manifold
Right angle
Resultant
Constraint (mathematics)
Real number
Modulform
Algebraic structure
String theory
Perturbation theory
Limit (category theory)
Infinity
Equivalence relation
Number
Degrees of freedom (physics and chemistry)
Wellformed formula
Term (mathematics)
Operator (mathematics)
Modulform
Spacetime
Integer
Nichtlineares Gleichungssystem
Free group
Maß <Mathematik>
Multiplication sign
Scaling (geometry)
Compass (drafting)
Expression
Content (media)
Operator (mathematics)
Theory
Dimensional analysis
Group action
Basis <Mathematik>
Function (mathematics)
Musical ensemble
37:29
Logical constant
Computer programming
Frobenius method
Modulform
Algebraic structure
3 (number)
Rule of inference
Prime ideal
Modulform
Spacetime
Polynomial
Logical constant
Differential (mechanical device)
Point (geometry)
Model theory
Residue (complex analysis)
Algebraic structure
Basis <Mathematik>
Mortality rate
Derivation (linguistics)
Basis <Mathematik>
Oval
Relation <Mathematik>
Manifold
Curve fitting
Recursion
Associative property
39:08
Functional (mathematics)
Frobenius method
INTEGRAL
Scaling (geometry)
Modulform
Algebraic structure
String theory
Maxima and minima
Hand fan
Variable (mathematics)
Order (biology)
Set (mathematics)
Gravitation
Hausdorff dimension
Spacetime
Nichtlineares Gleichungssystem
Summierbarkeit
Multiplication sign
Polynomial
Logical constant
Theory of relativity
Differential (mechanical device)
Sine
Correspondence (mathematics)
Point (geometry)
Operator (mathematics)
Dimensional analysis
Algebraic structure
Food energy
Term (mathematics)
Group action
Basis <Mathematik>
Series (mathematics)
Relation <Mathematik>
Nichtlineares Gleichungssystem
Manifold
Recursion
Associative property
40:19
Group action
Frobenius method
Scaling (geometry)
Thermal expansion
Orthogonality
Firstorder logic
Survival analysis
Variance
Database normalization
Order (biology)
Mathematics
Resonance
Different (Kate Ryan album)
Gravitation
Moving average
Polynomial
Link (knot theory)
Theory of relativity
Process (computing)
Differential (mechanical device)
Point (geometry)
Interior (topology)
Food energy
Mereology
Term (mathematics)
Variable (mathematics)
Measurement
Flow separation
Substitute good
Partition (number theory)
Pi
Crosscorrelation
Order (biology)
Relation <Mathematik>
Theorem
Manifold
Nichtlineares Gleichungssystem
Condition number
Resultant
Recursion
Associative property
Functional (mathematics)
Transformation (genetics)
Disintegration
Modulform
Algebraic structure
3 (number)
Maxima and minima
Partition function (statistical mechanics)
Shift operator
Number
Frequency
Regular graph
Modulform
Directed set
Selectivity (electronic)
Multiplication sign
Rule of inference
Sine
Coalition
Expression
Model theory
Theory
Binary file
Parity (mathematics)
Inequality (mathematics)
Formal power series
Number
Jacobi method
Enumerated type
Function (mathematics)
Physicist
Negative number
Musical ensemble
Integer
48:30
Group action
State of matter
INTEGRAL
Weight
Model theory
Multiplication sign
Mathematical singularity
Insertion loss
Mereology
Mathematics
Gravitation
Amenable group
Polynomial
Theory of relativity
Point (geometry)
Food energy
Term (mathematics)
Connected space
Derivation (linguistics)
Partition (number theory)
Prime ideal
Crosscorrelation
Series (mathematics)
Order (biology)
Relation <Mathematik>
Convex hull
Hill differential equation
Nichtlineares Gleichungssystem
Resultant
Point (geometry)
Functional (mathematics)
Calculation
Maxima and minima
String theory
Partition function (statistical mechanics)
Student's ttest
Frequency
Term (mathematics)
Modulform
Directed set
Free group
Rule of inference
Addition
Focus (optics)
Prisoner's dilemma
Model theory
Correlation and dependence
Mortality rate
Jacobi method
Number
Function (mathematics)
Musical ensemble
Game theory
Family
53:24
Area
Multiplication sign
Model theory
Game theory
Lattice (order)
Conformal map
Maß <Mathematik>
00:02
so time and I'm and you can tell that to you to and will talk about it and I didn't have a book about today some results on mutual you've gotten a convict this this the looks at noon arms continuation of the a last look dim occasion Inc which was he began just because and before his passing them bring them until it was finished by Russia as a mulch concession for hit this matter and will launch a book usually critical demolish ,comma tend uh original quotes from in continuation of the In sold his minimal
01:23
Sokol the alluvial gravity what you to it and sold by Dumont unique political principle which will provide maintenance at 7 and it is a move the important case also called Mom political stink political thinks fit have on and in this case it is it's also the example of you gotta be in in this case is a mighty sector is a 1 off from the minimal models of component killed said I want you to their program which was opened after this just look worse the move was finding of offer offered communication functions consistency even for 4 we just 0 so that its is difficult problem because we knew Gold calculated some into the all of a model space of Newman's sofas histories and points and there was there was durables in summer special combination of fields of renewal will want more minimal model of conformal killed Syrian curious Syria and trips and on the the it was but you but it's it's difficult problem because need to integrate all models space of human subjects some progress was achieved at the end of the year in the vote did you single true no but you never tall and little reasonable is also furnishes a mail which performed by commission and in his view the and the Net :colon whole joint will commit yourself with the computer for .period coalition countries researchers case with need to integrate all other but you have to Colquitt depended the other the from open program and to know that it is William instead of computer of course coalition function should it's not coalition functions through numbers because it doesn't depend positional from the fuel costs of integration we we can computer this is generating function or was its crucial numbers on that alone which is defined like welcomed fiction of exponent of this experiment and it's a new artists the ruined and kept in constant London "quotation mark 0 nascent much the fearsome in numbers and just company Constant Lambert can be considered as a coordinator of space but talked me more than a year the and another was insists that brought shortages we know going to computer users numbers vary their company congressional internal modern space another discreet approach school twodimensional Verity was invented by when people in the court and no cost still should resign the world's action can grows and this is how it approaches the issue is utilized by the internal problem he this rule Maddox model room described book Taitung twodimensional surfaces like think of something called graphs and in the end continues the limit to win expects that this whole approach gives same the same result the critical point and to him the In the world the Wilson affair will take this the discovered that there was a petition functions uses the Maddox model approach such as vital so cold 1st the integration also said that it could and on the that kind of a balanced and funds for of of cream could direct approach the man the for the rare case of 4 different the key generalization plays a role Britain which encourages slammed in the wall in the back and the was fond of persistent were dimensions proficient dimensions forces both sets of parameters to the brawl shouldn't you wanted to model approach inside hollow at the time now them store to compare the commission numbers itself not only traditional dimensions shows that the they but may find justification for a dozen and this is the problem start of the year In minimal and new and you will be very today which will be filled by a Silicon set of selection rules for 4 Commission functions but could number which is a unit of each are inherited from their interdiction conformal interviewed inspections of millions a model comport the say 1 functions from the equals 0 .period should be diagonal and consent of the wicket as the mother using a stand it is this is their home the soul would have some of the distinction for that you computer that commission numbers both were you will still see you would have some restrictions the forces good but at the end of if so cult on each of the 1st of the problem is whole tool it is possible for use to use users you know little Douglas approach I that it's possible for conjectures that is the correct analysis there generation functions such as vital adults think equation and the Manitoba Moose there is numbers that have based their decisions from the mill which falls from minimal In a model of conformal field said is that the idea told the tool but each has this through where petitions is fulfillment of the there the search using is a full and it was suggested the Socialists suggested to remain usable by the sessions watched over more I respect that there went away and
10:12
new computer relations my mom was in you'll consistently the integrated all them model space and there met the problem conduct Durham's that is the the agency prudent the result is the statement that there exists some material it was not the ululations between between the government and the NHL between the good evidence which Mitchell variables include approach and company constant you will feel so that the new formula you will be among relatives original you which but the selection rules nature of fulfilled and because soft and because of this this and practices in the area of existing variables will cause the nutrition dimensions these are equal the it cycle that cannot and not the inside themselves but there's some additional additional steps and it is you know In there appropriate make appropriate choice all such a change was expected with along alone but as is often more more the linkage the backyard toward each for but not for long but want hadn't to talk .period coalition numbers in the younger sister of medieval but in the long run but then went after full .period provisions that found to have been calculated it too became possible To make new explicit chipped it is not was thinking creation brawl which and it was being performed over war for the same status unhcr said it would mean gratitude in our local lives such as emotional and their full Chris bonbons between but that was pro the creation of an idea of all of which have such kind of change of variable room and get to work with the consensus was In the long run In His aim fallen oak which was down the whole reason the broadened Bookamer Jennifer was delights this not the chips that will also for other cases of models will you will you let there and William In a Midas these allotments is based on the performance assumption 2 shows just because it felt that the assumption is that that exist in there user space all the tort minimal little guy reaches of such special coordinates the and which doesn't Mitchel London England in which partition function of the military riothit cities 5 Douglas think a question like that does that mean In Maddox models you might explode the the next assumption is team equation defined as is in thinking of the reduces cogent and assistance from a petition functional petition Ponton generated factual provision from then on was that these nascent but .period about function for I didn't know this is special July Skoda with corruption all I could avail defended his apparatchik with some special initial conditions which defined by Douglas the itself because that was the same discretion is that not in it it meant to go to war the markedly with character but to it includes also some useful because they had decided that initial of the proposal the restriction for initial conditions are at the end of the next assumptions that that could there were times she mn related to his natural couldn't slammed them and on the space of pictorial minimal you will rioters by some money Europe transformation and who is head of the established established is a form his transformation from 1 of the requirements that tax collection numbers floats to justify their selection rules which only half even that you will models have come from appeal said and some the loss it's it was really useful the 2 were we use some of of the very convenient expression for partition function that petition functions 1 of the more you about it and you reuse To get tend to gets its formerly some relation between the ANC and that was the integration as a structure there had been a structure which is connected to its existence the thinking is that we have to put some details I mean militaryrelated right consist of new conceding that if you don't and some of the sit there in the whole case it you'll be all animals models will corporate criminals model comparable this model only have there is a set of primary futons this year who recalled the when new issues to the American Pie also forecast stable and William considers this has this too serious young the next next 1 this series with it because stable consists of 1 tool we can choose this is neutral the home like just have to that the center of firemen killed and we carefully as the fall in these fiction therefore that the public expects fusion minerals from Princeton which is the which is the condition and but will use this distinction that was a throne project of conveyances for their generous 0 also the 1 . British partners 0 2 .period he's a diagonal the dimension was accused in the movement to sit at the back of the neck inside and 2 fullheight village numbers so we should criminal we use the refusal to support "quotation mark expansion and together with the guerrillas result we will get some Sundman restrictions which is equal to that effect we did not know their
19:04
nutritional symptom in critical to approach but the userdefined through but the integral home all good image use methods and income formal engaged during have this is described by you he losers with this action and to the essentials judge "quotation mark at this city this formal feel sales also solely field said that recent changes connected with of which will cup and constantly like freedom in this election and to define a clear divide eventually will eventually have to go to work the talk the that suggests is that the central changed his mind and met the sector materials at the site of the pharmaceutical Prince 6th being in and computer before we can for applicants would get undesirable scene you know that the military will guarantee it will just be for created the stipulated Standard Fruit of the simplest observer also looks like Stephen Huvane integral also this and all and when due to grant his brother all field but primary Fuels primary fuel people will flow of Madison couldn't expand hinchelfield news concept and it is what should be a 1 1 4 enter all from here we can get this this is theirs the value for the socalled retention dimensions independence will kill the of the absorb also mn and from the models the court but this time I put it to fill the traditional image interim sold ethnological constant New World will exists and that they satisfy is the same the same selection rules like feels he amount From Madison the it diffusion of collisions number and basis it's there as a generator function on because he she is a pool computed coalition numbers we should integrate always twodimensional solos only gets is probable accordingly there a full and happy in we have met as meters appalling program when tool or more points is go inside this is the same as that can be emotional the situation on all controlled by a fulfill and we can it was conducted the data like about demonstrated better like compact that he entered this indignity is true and don't pursue digital at tle the relations with a number of sound some another operation on a number of different of different order but this is it this is a compact it's also agreed in the curent and on the language of company calls the vote to Indian Indian to such a change of the variables on the pitch then faced 1 2 0 0 .period data contended that like their and the soul in
23:37
your hand he had additional of competences Koreans such change but we can and is the victim this change from you mentioned reconsideration we can impose that distinction as the only such terrorists exists in this South they can have a which has the same rigorous abbreviation relations as the land at something and knew that we can restricted impose a disease in this village for stamps will only such the constancy of Humboldt equal 0 you use and his relations with his will feel the bite however is not it's good such change because being merely medial wall itself that anyone really model we failed because of the recession dimension are some room in the additional numbers with the full of relational such or such debt there exist different system Cardenas London and in general the general me milk you right in the was wouldn't I want to enable civilians but in the was approached block is an insight and that we and the Americans use this them full title to fool you selection rules for for it would also mean a little better than that and we're so will do the full and rich just where the real meet all there are 3 initial petition for that and do it in financial condition before all the numbers from was the creation and then we will it shows such as such change of voter selection rules would be that this all that exists in explicit formula for the .period and 4 . correct numbers many vulgarity In in the .period function it's a it's a it's there will be a useful tool in a tool normalizes as this .period function in the fall and with such stability to not dependent also mobilization and the U.S. has distributed to took their regulators in the new year Mulally merchant in another approach from the proper and such and such good solicitation not to branch off that was authorization so what follows this for the poor Malays people and financially because this 4 and to and fro 4 4 .period function within this for witches who collated by the original book with notions of knowledge use his high commissioner the motion in will feel this problem and drive goal To find this a compact and the determined and saturate to fulfill its selection news shows but then we will we will compare also uses the the handful .period back will begin moving because you were the 1 approach and that was from the 1st half the integration that can get a new exhibition plant in principle but we don't know how to get up to Nov 5 more when function in and that its approach I can't live in this we will review all use it on a unit that was approaching that was approached petition functional for peace and we will gradually described this fall With take goal differential period and called for peace if here point differential operator of course small Order order full life of each system the efficient so this business of differential operator principle would able accept also Bush plans to militants and game is the 2nd operator which is it defined this who this positive particle physics and the differential at rate which is defined in the brain and the cheese Katie I'd freely I know I can do it again that's it means that if we have fulfilled all of this is the largest collection people is called and it falls also satiric defined promises a commission uh uh others the confusions wants the war to win positive there's a dealer who depend of the facts that they will probably uh 0 of whom you initially creation could the written usual and relishing that could where corrections about this aggression contempt in additional information about about mutual initial we the conditions for 4 could the efficient war moon and but instead of using this as this correction because we are interested in very interested and Dennis 0 we can take care navigate classical Alaska Lehman and consider instead of commentator blossom brick could be called which defiantly exists and instead of like the Rader who impede we can consider that simple so we what somebody tool the function of this off with evil offlabel P peace symbol a differential of being operated the energy because
31:18
this will not the accelerated criminal variable pay order the don't music so P is not the same number of of rich in definition of and we'll get a parents and you have to or soul to soul is this Bloomberg says this is this the creation of a summer break at peak or physical what this equation this he could you wouldn't tools a Poland action focusing on the latest action under the principles the earliest actions which can be lethal legacy that this is nascent about it is usually in the form media .period Plan B. it is roasting the median of this was his preliminary so we is said a lot at all 11 . 1 the onto this is the notation concede that I have used it as a durable this degree of freedom the not not the integer today now we want 2 Oscar to to find 2 0 which in the the Texas which which we should include this is definition tool In the losers we should do some dimensional analysis said stupid the result we would get to the full and In In the notification Wallstreet dimensions seat them about the our times the bounty go Conference USA In fruit has In his letter we believe we will enumerate the it's real numerators like tea and when it should be but you
33:50
details this is the result if we you only that the final number terms which meeting here with him this duties will fall don't below hold on variables came when we'll have as their same dimensions and envisioned a massive and give a go it dimensions for forfeited neutral body able gain knowledge has this very sent the land the variables connected with all suggested that that connected by such kind this is the 1st such content and relation that in each of them in the right hand side the same scale dementia legislation sought to use some salt we knew but there is some sometimes it is but it will not affect its phenomenal flounder the reconcile and now that the storm now we'll try told her to agree to satisfy their selection rules and police it's useful tools tool the storm it's convenient to use as a fall in the following full expression for generating plant In that this approach rejects compassion connected to its solution 0 1 stock solution not think Commissioner legacy about the disposal tool operator more explicit formula which is useful present at the forum on that so we use has this connection between think commissioners instructions but connected with his analysts said a little because there aren't any ill at ideas is algebraic criminals more local pride but who is opening monolog which is connected to his new and there it is given a series of so we have I would have used this approach to measuring the in a series of medieval walls instead of people who really the news but the full magnitude of this back up here Katie's integer part poses patient users of that as when you might want to consider so deserves and you know all this said connected with criminals who hold Saudi PT not this 1 class relations would
37:33
needed and this is definitional alter rates algebra brilliantly those models with Prime primers want but I know
38:00
as tho we can define also in the finals 4 millioneuro formed on the basis of such criminals which is comes here using you the center the can introduce some regions in Union you know all rare and see 502 Gong structure constant for follow algebra beaches good which is connected to its existing believe form g the In this rule this Jones sea and from the program like legs I on and we can In New can uh with them the statement is appalled that their move them from the solution over the situation this
39:10
equation these relations we consult told to say that situation between the love between the petition function better and therefore tend to although medicines traditional the integration and control and this fall she would have to do this that the rich the obtained we get good from structure constant I will look a lot algebra we can compute the structure constantly engaged giving it is this desert and this is nascent but a fiction which share role before the shortest picture
40:16
this 1 so if you this
40:24
formal for the petition process for generating function but fewer adjustable Richardson who is this because is integral to the world and all the WiFi and this is the goal should not have to depend on the press it news that this 1 was performed traditional informed should be closed and also we should check the survival of the kind that can relations with overseas formal such defined the raises for no petition comes from and an extension of of which fulfilled we need there are 2 questions tool the doll in Torreon chipped this 1 for his calls renewed this sport recreation business about this is situated at the requirement in the testicular follows a record also although they were also there Aziz the cost relations which for now it is aegis of different people from different border which can be checked that "quotation mark and so all of which have explicit expression for partition function and we wake son some admissible a change of variables from it could do it will land and we'll find simultaneously that is there for Wallace's substitution and her coalition number step by step 1 would hope would the be point of that we consider this simple example this case where instability in fact the case many the young status of anyone on the grounds it here and take this case to him for help announces 2nd we have to go with him the latest simple forms for some form than date approaches the school there no way to use such criminals or is it when they host for its 3 from here and I didn't have the declared that only their use by Yasuo some call as enumeration of the variables to welfare can't then we should um the committee generating function for tourist numbers fills with consultancies is amounts transformation after this that's our function is to do it in front like something like the summer but he was also we label slammed serious impact rapists now and then and this has been model both on land and also all actions that would've of optional doesn't really evolved and send them away calls this but on most of which looks like this is like nothing models of that period you which creates some mentioned coupling the cosmological constant you call also is in its new measurement some and so on which Parliament this fall followed is a must for information that we should try to find them from our selection but why would have some digital switches which is eaten here sold the Canadian stage in the world told you much severe variables of such formal petitioner petition function and if there's a problem lenient exist diarrhea solution of think it's Gration because that is the reward for all functions in the good at the good will and from dimensional and
45:30
that is a weekend understood that our Mowlam the function Egypt's 0 and he has some are below those of some or some order and then there's a of happened almost all of them and told them of room which includes the indignities of itself for respect 0 overstepped therefore it's convenient tool use said such a change of variables to the variables it's uh really get another civil lawsuit over where variables which now thought function of function the functional the good news will edict and we get 4 or 1 . 2 2 .period British number so we should get an MBA from deserve our information partition function we get go has fallen to efficient former and on In if this unplanned function and it should be because 0 but pages not 0 bands physicists do .period punch should be 0 1 came along and people who are not equal and from dimensional win all those order overseas but analysts and this 1 and this and that and then you get this book that it's simple center where we can room we arrive us Fulton statement insists that closes a requirement which will selection yields 1 would function should be because apart from such of the the rig arrive to existing statement that eagerly and Kate is nothing but the accord a policeman also the search for which is the result of some tunes as Kansas State because it has the equivalent criminals what let's see if there's 1 thing you always group the with cult yet the junta during the war in and the defined and
48:33
redefined the corporate logos this time about you could go to prison will probably not come up with the agreement you know the magnitude of that those should be necessary for and what we put them in a Virginia us so we have to 6 states Indian for weekend deleted a day is done in 1880 and that you didn't know they were in the room which can be there use in general pixel so far this year but in mainly for the city's Leong say we could use another way of and and to Iraq it was done session ended with their I've told relatives and others on the on muttered under models connected with with the applicable most of her work this band was then we're going to the the same for for people function yeah and get them to get Ozan his announced their arms we get mostly exists for other attempts many students in all of the action but tell In all again we can use in such a way the but the indeed hold wanted at the point :colon consideration functions and then the weekend after this weekend but who can accomplish just that compare their results for the mission a carrot in front it will get this fall 4 3 .period functions lays people and can function In this case would have this for all 4 from amenable Nunavut approach then we can get a formal for the part of the same we get the same form for in this approach the US approach that they're doing the same for for full function and we get
51:05
some but some very simple explicit form all issues that here for fourpoint functions and to win again the inside but in case I had to do with this the result was that it will comply with the addition of the knowledge of mine and at last weekend but from a late is a general result for this serious impediments in the form of a language news under prolonged by it is conjectural conjectural statement a partition function you for this serious is using equal this integral AEK's star is the solution for all these years rubles overseas will 6 billion model and even is this city's Eureka lose on the prolonged this order different vacant and like this 1 that can be used as a fanciful it be interested to talk on the brothers for Wisconsin came in with a simple man and focus to community changed there the same can be and partly repeat therefore was the next sitting there with him but due to a connection was from structured this all for on the full hour generates and fountains and again we arrived but the following results the 2 of computer that I want and pool the point was I wanted to work at later so we can can choose special rate but most of money that the it isn't change for this order and it is a about and you the it's done is performance in terms of a game off your quadrupled among them even in the case of the young case as serious it was the intent of
53:24
Brussels models of in the next game is at this point but but but that's about it and some but enter from the the for example .period with look
53:52
at how many of the cases went on the the important thing collectors some they are not equal 0 can Canada's after which inside the must the petition but that some Balkan promoted to keep up with the weather revisions that someone might have questioned the longer all of the following features of Englishmen and these units had contracted with I'm not in conformity with the list problems related to the outstanding areas until Sunday the meeting was held on the formation of a manufacturing then company credit the foundation's become engaged artists of all that's not the answer and went on to grow time related that it should be willing to some kind of biological thing you cemented the handful that was installed at the age of the English gentleman is simplicity itself was 1 you although it doesn't make any sense None must accept that this communities that I think it should because of the use of the government and the simple I started some time trial and the only thing that failed in the coming view you something you want to the question is what proportion of possible psychologically because this is of to be appropriate new client global also I uh I'm we we conceded he bonded with simple simplest emotions in his new and among the this that the withdrawal of mortgages this more as