3/3 Mathematical Physics of Hurwitz numbers
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3/3 Mathematical Physics of Hurwitz numbers

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6

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CC Attribution 3.0 Unported:
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2014

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English

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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
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08:37
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25:51
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43:05
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1:00:19
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1:02:18
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1:07:55
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1:13:14
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1:15:21
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at the end and it's a new thing to playbills continued litigation still recalls state's Inc Bloomberg rare that the politically caution in my interpretation does India so I stopped near the Fermi as algebra the union and the scholars product and probably it's a surmise simple there is a basis and you'll song and there is a good thing if there is a remote beaches redefined the this way and that doesn't depend on the jurors was a and also a very considerable initiative dimensional space which is space of the real thing this is simply act expressed there is simplicity for which is defined by the formula that they wrote yesterday and also I fixed a surprising this space insert insulin into the middle party which is a just there were functions in age and also the poor parts which I do not buy L 0 but also on which uses a number that 1100 sup space vision government Italy's and there is a projections which originally by appealed to this substrates and also of the building it's not it's less important but less important but a whole foot drew some bridges where is and there is a dual with hero and project basis tool L 0 I obtained advises basis and In this place which is also fueled least 1 the company but the 2 of them so these long Dufour approaches oxide 0 my the vertical metal which is belongs to the conditions that El Islah grounds is the same as the biggest semantics condoleezza data holiday associated procedure to political incursion in as the examines the producers certain function In the were both 2 years and 1 extra but and I collect told MGM the Johnson which homogeneous institutional there isn't a politically caution is defined by the mustard questions for most interloper gracious In repeats year as a collision so if Europe to stones for a universal point of view so do all 4 of their coordinates the rebels you're over the function of commentary that as a as long as the coordinates for the space so this function ,comma treated as a function of the so until local coordinates Richard defines respect this political trails of the basis and in the only Utah use the particle .period the shift of origin so now I assumed that these are just numbers some fixed numbers during such a resident of this expresses starts with the terms of the greatest 1 and their leading joined was inevitable fate this time if I don't all this difference there is a then that's pure lies in May could change shift of wearables the Q I use the cureall for use telephone minus you go its socalled ownership and these equations defines the functions but also injured To put somewhere what this you want Delta probably it'll put you here is essentially as a differential of F vigilantism even in such a forum it's a little partially which itself the coordinates of the coordinates handled them by just like of this is the point all else this coefficient depends the Pentagon wants and also of these the richest purses which participated here is also so in different shades all parents to block multiplied by the Major Unit of the Britain Orioles a lot of as lawyer and functions this Villazon edge so this feels here multiplication in age and the loss of ingredients Delta to the rich wrote the formal yesterday no this loss 1 hour 8 x 1 time in this 1 plus some of the Alpha computer CSI also
08:37
star Clinton earlier title commands they was to hear so are right there he was told to use this expression you blasting Marlin went to see them as it is in New York functions say and then there's still rides a commission a form age then there is the usual children plus you'd square Is it but not at the movies I need to go what's belt of if I apply the definition of them the idea something Gallego some of Djelfa CSI Alpha Bloomberg the with a 450 is CSI also start so interested in the use of its functions and next these tests is Europol for increasing quarter on this here's a poll from cruising order from the multiplied you have jostled filed on him there is a convergence of it's not welldefined so it will arise this function the following wait let me go to Uribe's with a similar sound 4 of the Globetrotters substrates L 0 and then there's the remaining John side of this form so this is still here and there aren't enough for this political duties all what to call the roll their eyes at the airport and hero this is my division while Laura X on this "quotation mark there should be as overworked still put forth 1 the good it will cut its the problem was well the container from the books here and this because this is what holds 400 localization potential real I will allow surely related to the remaining part of the dorm just what they there also all 4 viewed the the because there are so this civilizations that a full inquiry I do it once for old Europe and then for any other ourselves I don't complain is what they did for a or for the 3rd definition but not before the the results of the presidential nominee isn't therefore the resulting function is a file formal functions 1 let me write as a function of Q when I arrived in the queue coordinates that its Independence Hall a choice of shifted origins dies at colleges of the coordinate system so I read if it isn't just the formal functional erectus spaces defined but if you are interested in particular correlated the functions until fusions all these sufficient expanded at some point to the specter some coordinate system for if you are interested in particular those coefficients mission days there the funds as well defined by their their below coefficients on this expansion in the .period Ecuador's visible minus theater which could raise and also aspect the new the chosen Gordon system so that this function is defined more or less invented but there this freedom in the Joseph Gordon lessens the domain of the function also discussed the controls in different ways origins um photos show that this change of the origins of defiance operation I saw yesterday gave holders some purists properties of this function but before Dole go on let me user at this point is that my interpretation of also the logical order caution is to be the 1 images of more traditional riches was formulated yesterday I started all ideas for lost the Cold War that was written in the same place the yesterday at a local I saw there was classical singer standard to the usual treatment there notable political caution is defined in shuttling spectral untitled show you what the correspondence between the spectral trough certain this simplistic setting Sun Aachen Z the game the goal was to do so not visit car itself but they need only to lawless happens that is a small library holds the presidency of the critical points so our generic and forget about the existence of Of the cure for the global tour from dust or replace it by a dissident union of several small disks or even former Labour of .period so have and Joseph only careful this Our Carroll also have 2 functions best worries shifted Gordon such a that someone almost on 0 was at the center of the disks if you don't like it too last to shift the coordinates so both social life it takes 0 his origin but a local with different wiser local coordinate its long has long 0 do and the X has a critical point is simple critical point of more study before the pollsters local ramifications of fright covering call for a full and there is the inclusion rejected changes the branches of the function of X and on the list there that the ball was a prescription of political order cautions that defines the family of socalled differentials
17:15
also defined in this funny rates saw the fuel 1 is just Duval differential YDS where do you have Is this the 0 hour of order why Diaz has 0 out of order and the point a W 0 tolerance a 2 differential defines only on the their electoral curse so why 1 is the coordinator of the thrust the way to it's on the 2nd on the requirement that it's it's semantics and also it has the following me share would serve as a full 2nd belongs abandoned all and then after this fall's remaining part this whenever there a power series in 1 white 1 and why don't we also Semitic 1 and 1 . 2 and the requirement that uh to has such as sharply as an independent chase of coordinates if you will try to make certain change of analysis console on this coefficient wonder what remains the same it's all and 1 the conference you USA want producers my induction and the other differentials and the financial system but didn't finish I didn't faze the definition soviet duo the computer and differentials really the day the following the combination of these 2 of these all this the guys which has and plus 1 arguments and then the loser falling via basis functions and that instead of this arguments Blood 0 1 the coordinator of the trust's component and Seoul also the component of the 2nd order is replaced by a surge in value some spectacle which exchanges the branches of exits not the congressman variations in wonderful regional exchanges the branches of X and then all the remaining so this is a differential which contains some somewhat all types the 1 the initial divided or W 0 1 or 0 minus W 0 1 was about Is there any new multiplied by W 0 2 0 0 4 this year From the 20 places here is the year of the this was he was very the support for the 1 solar here it's a very important I didn't think there isn't you and this is what the people of things next step son now I would like but the editor of food makers could .period between this formula and and all salt or explain the little Beaver was hidden behind this complicated formulas I think that the bill would call you some serious about this and that most probably the specialists nowhere well but all know mentioned so let me no probably go here saw 1st of all it's age let me let me also use a Jew and all coordinates I will put X is equal its square full blue further there coordinators valuations doesn't with application by minor 1 no let me quote what's age this might sympathetic Space Age associated to this picture for the bench using this bristle all what I mean is expected in the USA while forms metamorphic their market 1 forms on the see so in amorphous news that they are allowed to have falsified border and the boys at the center of the Center this and also Of course we have laws of foreign sources there is no other polls and we don't discuss whether this local collection of Johnson grew to some global neuromorphic form because I consider the drums of Coors there no article forms and also a of the conditions that should know what descriptive pollution that means that if you have a form begins right is a bunch of also for defined owns it's all this is just a collection of wild forces the fund only is on his USX and uh I can define this and that my firm recorded as but the function a listener water function show me a 1 expansion or you warm especially compared all the UN polished of so the list I it's column functions there is a rated functions in so the prominence also in this case is just that coordinates reserve .period voices coordinates multiple attention and so on by raises is a there are idempotent in this area you in this case it's off normal and that utilizes the same for this this is a correspondent mountaineer explains how it works not at all see
25:54
how it works I concentrate myself on the the case and the ones of John cases does the same with differing by a more limited tensions and so that negotiators accused sulfur the gaze of legal consideration potential reaches a cosponsor there cure fridges just parabola X is global this clerical of Tuen and bicycle to justice In this case the the only civil jury threw the ball to the head of examples that make some more some sorry the real possibility that only so have we these and also the response in renewals here in Boston there is such a nice stranger all the user during current whether financial so I understand that this is not a member of the incentive award is didn't actually if you take a look carefully at the idea that there is a the former what is important for the course is nonexistent this is not the answer by differential but only USA all its parts after the evaluation and it didn't this function the give you just 1 example indigenous if you have just done at the W 0 believes this will belong the X the 1 by 1 the world 2 wars where oneliners wanderlust square None after these I understand the position you get something like why oneliners 1 . 2 squared minus tunnel work also what I want plus weight to square the way was in the White told and then hit long square semantics is the respective 100 . but do not know the form of a branch of the this is their White also at stake at the dental Adela expression when applied to or why won 2 golds 2 0 we yeah yeah this is just as in the case of it incantation potential yeah don't it's a simple exercise on they're all here the German the progression on this will be written as follows that something like this new rated explicitly white 1 to the borrower to book barracks go to jail so I think that expansion this don't settle contained only boys to followers of the of Wait wait let's there where both white 1 Bill volume is that in a minute later in the coefficients we're all around function so after you buy tho Of these especially on top of the same with their breaks is that it doesn't blow semantic anymore no this is nadal these bikes or Kitty started and this is offwhite or and this is exciting OK the results star for for a long so this is a start not 1 just 1 some genetic progression 1 1 in its last summer yeah so this is an there is a school and this is not the medic anyone that's so in the canvassing consideration I think that we them the creation of a half the space of Mary Norfolk 1 farms and of course it these general forms and once at all user generated by Mike Taylor positions all the expansion tho there this negative part was that of your and the expansion to take of the expansion or the 2nd time around and their coefficients untreated functions soon in the 1st argument they will probably use more could you actually I would like to make some some demand more precise solid new book McCurry critical the picture so I said ages in the space of metamorphic 1 forms but this time we have to make it's uh the lawmakers led fortification of this definition than the current 1 actually let us consider a 1 forms after Cunanan's so that make some or all the forms if you want to solve its focus let me call to foreign
34:28
security and I will prolong this Olivia Cole told 1 false security interviews that differ by an even older girl sir 1 is equal why don't you if there is a difference is the goal and now using physical idle derided the definition of ages falls so this is 1 forms who was the bowler party is all about I cannot control the general apart but they can controls on it as it the bulletpocked abuses In principle part was appalled said the model the this musical citing allow always trade only duel and although longform III a considered so that it doesn't change of high art and you form you using the US dollar account right now the former that knowledge of this absolutely correct so now the lefthand side this is not actually senator couldn't resist it but if you optimistic that they belong to due to the space so I choose admitted basis in the space of regular 1 finds that over and all forms and that the local officials for the dual basis for instance this this don't differential as a kind of koizumi elements as it doesn't product of Of those spaces so the sites the coefficients of the and said that belongs to the bullies and the cohesion of the person said belongs to consider it there a hint and of course this go here as a sentimental did that in this way doesn't appendages of business but From is such a strange formal which is the rate here what does it so there of the Russian that this is nothing but the projection for L for remark so alpha goes still Over Was he won considerable tools what's the residue the energy mostly 0 we all thought it was you want 1 onehalf integral off from his unit is among these hunters address this is the 1 they want this source does stand for this In general this is a fine in the national forces following called retired David I'm just but it is onstage tries is just that there broke primitive is just primitive of this addressed the W 0 so this interracial from this forward history is adjusted the fare was just a fantasy images India's sole home get the hell do I understand this projection operator if it doesn't know all this formal so this projection if it takes into ghost over there some of these sales .period music sale their there basis in the space of a few in L. there is some confusion so so what are the coefficients here local officials here from you by dawn is here and it could become covered by the simplistic blaring called this offer the music of the duel with also not not I'm sorry she substituted all give executives because the initial the author of the report 0 0 0 2 yen because there's so what there is only 1 the book yeah this is so you should take primitives of the former this is exactly the way however and so on all the conclusions of as the differential forms defined by this aggression has the polling proposes that have bolstered their and points but more always they have some additional properties that the principal part of the ball is all it's not the fall what would you want to use the term and not yet not yet so only 0 1 another participated in there in this former is thus participated looked its hearing into it was it is a bit of a miniature of the pressure for the formal did you know you have this phase of the project and I think that now this is not some so after this adjusted and litigation students all With its moral is same and even more than makes them some government on the 1st property that image only
43:05
jails has ball and the critical points of all all of principles our all was the 2nd that's it we can contribute actually these forms optimistically owners with their cost of the lot change if you play this for a stop to this occurrence it this this positive aura cautioned that appear to be a all funds the final globally on but if you just consider the drums as at the points because change them freely of those occurrences of their induction in better prices of we'll exchange and Nelson this the alarms that because of this because of this Kusuma In this case you may replaced 1 of these zebari is invited by by the dual and if you take the dual that's invisible does change the assigned Sorokin you can actually move this this simple cited here or just pull them off into the woods that does the same that there is a scarcity Motiva the specs evaluation for each of their arguments at 1st the meeting here of the system we know this is the correspondence must still there is 1 more point if you consider this a caution is the 1 which I suggested yesterday rejected denoted by you and I think the leader the you you don't want .period what should you do we have face OK so that no here for a moment development for the bid the low myself relational traditional station but we will arrive here on the eve of the ball in the form of you know so bound and we compare these foreign countries form for the reasons it's about the numbers so that is 1 of follows I saw actually had to use this setting I'm almost form only just imagine and I consider them as a financal symmetric policy all the stress right for me it's a metamorphic longform respect each argument undertake projects of these so there was cemented and they're going to so far as to warrant militant in more explicitly coordinates some coefficients W he was jail for all fond of 1 of Britain's Oxford all right for all the way all in all it can to secure to arrived this summer the of 1 of the sides 1 trial for and this is just a matter of mutations but this is a senior editor of the social justice under the terms of the new corrected as a homogeneous functional Indiana after all but this is the inventor of the end of this presentation saw this as their usefulness as moral as the same as there were a whole me and Zuma collected partition data into genetic functions you can either choose an independent viable soon to come by and there is such a form or you die you go good competed as a submitted as a as a homogenous function and bowl combine this is the son a military that has the quality still there this is just 2 different ways to do alter the local the same collateral data and under these identifications or to let you make the Goshen it only if you draw school criticizing just look at this sitting uh real musician will and enough this procedure differential animals saw there was procedures 3 3 the potential ever but don't us just said in their yesterday at a local banana Air Force just a number of the associated to the UN cure but the aviation minister should the number about the power seizures so in that the setting it's not clear what other coordinates the untold story called altered from was you from a single the value of Culpepper produces a function and the told criminal descriptions chair which I couldn't it's but the fuel could different papers was alive as a on 1 or the other but it's not so here's a strong explicitly but that procedures soccer on so it is 1 of the prestigious Asturias a so have to feel very very the confidence in the introduced on board at the steering then jumped on the phenomenal number so it's a little bit function and other disability is just to make this identification the and the truth the coefficients the introduce some brands of space and audit function between this confusion says the government is so at the local officials of this functions and that the boss's produce due does the same answer just because of that on Wednesday all of the joy shift and as told you that the extension of the function of the newborn just is a criminal and the modification of the former leader of the function followed a different of this Utah will just he dollars of the romance of the Open the same the reason 1
51:43
exception attitude when change theater it's allowed to modify a bone in his direction so it it will not you know you will nautical strapped to their dependence on the whereabouts is your so it's happens quite often if you look at the paper said the examples of the competition examples they dependence on 2 0 is also is all often missed a slightly reconsider can only varies following day to heard a eyes here but there is an occurrence .period just to local local this coefficient thought what did not happen sure account more than that he was accused of middleincome fish potential relationship with this is deleted so I I hope support or Bushera and put the world on Oracle here so this this is the undersea addressed this is needed it no go to the want Bloomberg no notes that the quarterly serves future for this should to leader I have said it is being used for resolving also lost it's all in this sitting here they'll all are equal to answer probably should put some 5 to 6 years and integrated communications procedures 1st here OK so that it considers that in the case of the applicant's age potential Texas with me Powis Square toward the work and the way the costliest insuperable and also however there the the the 2 foreign ministers posted to the said very produces CSI cool latest school or when when slump pictorial don't forget the 2 were last 1 doubled from Total and 2 it's wonderful I'm lost long Diaz store so this is what they called so I think this is what I call by CSI and start the generate considerable on very clear and this is the same woman generates wealth which is this his held 0 but now so if from these by used this on the good will reproduced all this Nevada conforms and we're going to have them has a total coefficients stores the art auctions for option on 1 it's just consider this beleaguered yearround a total this the sum all this good directed 1 it's 6 miles kit that in itself With Vitamin K 1 K you 1 to Bloomberg this there this collaborators so it takes this semaphore arms which are obtained by this the car show and that the bill Rosa coefficients of this will expand this is the law of the land expand business and the coefficient by exactly the collaborators and in this procedure the the conference seeks is just a single parable it contains all information about above the cure but there is also often don't so this was a occurrence the cool excluded from work all white people blacks but now we define this using she said plus some all too clearly this could borrow 2 Kiddclass along the Jr minus plus federal something like this the Asian borrowers use most people are still and then this cooperation always independent now we have 1 and only already 2 jails this defensible sold to you and using a joint on the creation new car engines knowledge of June the presidency of the Union address again the same potential that you have that is the drive Gold told to computer model on the things that didn't deserve to win there conveyance d 0 further the premise of series somewhat toppled the sitting here it will be you yes you probably there is also a I mean that that is the idea that actually this is putting on or professional and its ability to put young congressional into 1 about the burdens on the here is more complicated by the procedures the Goshen procedures such that it does not deal is Infinity Resources it's all old but Illinois I want to some probably there are some new missing boys but more or less I explained the crisp onions and knowledge I live here let me come back to my sentence and real for this for a moment I thought of that concern the 1 all my books a problem if it was is going to make a small break for probably will read this I think you yeah will
1:00:30
durable so I repeat once again
1:01:01
dissenting so as does the same way that they wrote the landmark he also while the
1:01:26
savages film from BU cells are all this stuff the Lagrangian of the various he and the master question which is equivalent to the is customs procedures to compute this year jail which is
1:01:41
a differential of MGM the new show how I'm doing this so my sentence do you know that I consider this thing recital said Surroi simple tastes saw the algebra shipped on the cold Abdullah has also received and they also actually the formulas belonged to depends on the further to discuss my simple sort of attitude I have also competition for among some the
1:02:20
most simple case what's wrong I promise that they corrected Tennessee Ng please along the if it doesn't work when claims take a
1:02:57
hurdle for instance the results so this is how it
1:03:04
works uh at the moment that home but it's the result of my computer that they did this late last nite so if you have done for some particles also Over the initial data and then is the result is that you you during his or her own political analysts so foreign little unless is very associate but then I have expanded these In the basis of CSI all and then I had a lot of Clay also weighed in the so was even here virtually call all of tools along political also index but conceded In this form for this is the calling tool around putting on also and then they placed every called Richard without CSI offered by itself American this summer assume that this is a differential function and the American recovers function itself so this is the result of some sample computation and this is for instance the Doolittle upon equation and the company has condition Shmuel that might the cost of procedures contradictory so there the thing is that makes partial narratives in where the way the Internet there the quality of responsible to suggests the Quran tho the signature of the the WGA under suspect all arguments as to recruit things they may I ask you to go back to the park programs that do not worry could you quickly place with this year during all of all it will probably make it work but Jr sir 0 In the structure all appropriate training once again so I started this computer and some internal definitions the difference of anyway so SH Suria Z go officials of the foreign leader in what in this case sorrow those 0 2 or more ages there is room degree while is 1 In order to make the media In the be inaudible and the on terms school Mitchell admitted here just show them in their their underwear then the also viewed that as a former this which produces developmental steps please zone requirement had today that it from a semantics would be offered me at the bills the all far hours the same according to this line and that and accept that it could be chosen maturity against that I have this CSI definition of CSI which is viewed as Butte there are some folks total confusion due to this and that is basis and so on so in response to what is now the number is assumed in these standards on the basis of the wellgrounded it's defined by this be but in the words of a former middle stretch then there there is a projection operators and projection operator looks very simple trade the expansion songs
1:08:04
some of thanks to the power of the exclude applied and
1:08:15
plus 1 and it won't last 1 Donald pictorial there's some confusion in and last year the path then when they make a positional just as they this tale this after projection with goes exactly to sum of 8 and CSI CSI and and then I keep the formal in this former delivered by the powerful next to keep it in this formal form we will also for me the employers by no details this is how shows this is really and this is
1:09:10
actually the aggressive as it is the implementation of the customs procedures don't you think is the projection of the oral divided by eat this is initial jails of of the procedure and this is the In that then this In I can use in this or the homogeneity conditions there equation the Doolittle immigration their appearance but this 1 is was more efficient and the fastest so for Evans positive I knew this 1 but this but this gives you the differential in the function of themselves for the drawbridges initially should before what because 1 into and equal to 0 so I have compatibility conditions the Doolittle aggression so there's at the expense of parameter produces the coefficient of Jim understood possum the and the component of the narrative in the direction of the we to done the and then it crashes all the 2 boys a cholera cautions I already mentioned this for me that contains potential that they actually could be regarded as a kind of the formulation of a restaurant type equations so if you that the question square and the widest guys need them for the 1st time since time to your work that greed contribution so if you have no rights the Commission is there's as open as the master questioned this form and just although less concerned this is the only consider this a quantity and I agree that this according to host age and I consider 1 model the Newfoundland so I may consider the coefficient illequipped as Europe was a coefficient of particle so all so the the concerns the quantum displays and the coefficients said the coefficient all CSI also wants the coefficient of Ksar for the confusion ask selfrenewal In this 1 right this is just
1:13:16
coefficient Excel from Fuentes finish 1 here then for instance whether the coefficient of CSI hero I will on day the square of fuel and that the results of the building the brother of partial
1:13:40
narratives resumption coefficients the coefficients is told the identified you think CSI also excited people divided by Toledo and projects that this is a big the project or just break with a dual event of the dual ways so this is it this giant produces a combinatorial just order partial would the same this this during produces that some combination of 2nd order derivatives all of a half his With this 1 is coefficients the furor 2nd derivative just over this lyrical effusions and so on and so on as what
1:14:30
the combination of this regression of these forms of music the type progression said involves do you which solid of order 1 told some posted linear coefficients and sometimes service costs of gold futures and so on so far in the sense that this is another couldn't wait to represents topological recorder from some you may think that the politically cautious just and efficient way tool right down the assorted In many cases so this is the occurrence because it a form of the political caution what so thank you so much wastage of the now
1:15:29
this was just said the remarks concerning the caution and knowledge past was the main subject of 2 they talk that study the dependence of this potential to politically caution on the Lagrangian grace sorry there the space potentials all the political which is just the same as Lagrangian this menu excuse me for years as a 6 for this process homogeneous there is accused group Sumatra's saddled with precedents simply to yeah cellular the incidence of such a condition of locally when you make the salt also so you make mimic if still transformations and sometimes integrates and so on this simplistic the group also political formations that have reported victims also mentions the elder role for him in the infamous done that fuels and lyrical assignment on different fuel school begun by quadratic medallion function and so this not a result is the same as the algebra the algebra Of course redundancy friends of the as usual berated the lyrical lilt of operation so this yields are excellent USM expresses so the music and film potentials the question how to request that this action the answer is especially simple have especially Nice the is that this curable the concert because what's called .period a conservation of collaborating with onions many the falling that's going to commit on there this year span of the following Maloney also have died as a beall you also Cucuta Delphic you theater there also be beaten so what the response to this little analysts so Monday applies a slight shift toward such space supporters following the fall of the the Abdulhamid on and that the field physicals for newcomers going on there but the social changes so it'll be the challenge good Britain as a differential appear at a hearing this case this is just multiplication all 4 of the virus the the same quadratic when I mills and lay idle youths here a few coordinates which are shifted coordinates the respected to as do dawdled shifts respect the interior most as origin to the points table created tasteful to eat up more beautiful to here source the 1st of the differential operator you Peter J. or and human health 2nd of different the tossed DQ Alpha distributor so this director a Paredes efforts have told conscious of fuel and the function of your visit just there the editors exponent of along the page fearful you and curious shifted so if you this correspondence of jobs solves function effort effort for themselves this be the described as follows so let this crap told slightest 1 by 1 so the Trust Law 1 thank you all for to Is this a parent or is that just modifies therefore 0 White good tactics jostle dysfunction and I said that I don't want to book the beard uh my sending this is all 0 so I don't want to consider the fills the 1st line of the correspondence all this on the other hand for newcomers dynamic changes most their way there is there as the articles of this substrates this golf view the devil please so it's probably real called it's countered the consideration situation that been all salt dome the Villagomez freezing and this is the old affect socalled collaboration which makes the difference between the 7th and the ones consists potential so just modifies the this was the toast of division of DVDs 0 but for a moment I don't use for the next line well so this is the alpha Cucuta this was to and these preserves advances of space so as does the change of the medicine Wheel already discussed there is the potential is there it doesn't change when you it is assumed by a relative advise the defined and the only way but we don't what happens if you to make a change of basis is just as produce this is also the just linear chain show all of this coordinate system for this is just the change in the romance also the problem is all the potential environment the survey was the last 1 which is more probably more interesting it all this is the potential buyers it is those infinitesimal changes so the potential which has the following 4 and it is the 1st time 1 this age all QB blasts during the cureall for you don't to so here we have such a strange the summation of the potential omission all Sekimoto derivatives knowledge in explaining what does it mean it's true so let me from lists local artists so let me take a certain equal to 0 sorry that make just Wilkinson general 0 part of their potential John visit here says that we have just this summer if you know all you hear from you risks Estonia no 1 was sure this news that that was a Lagrangian there 1 graph all just 0 bottles of understanding of the potential is just turns falling victim Young is just move follows the for the moment you so it takes the differential offer of 0 is a solid Lagrangian will solve our activities not the uranium and then only applies sympathetic those formations electrons endurance but it remains Lagrangian saw Is it regained quoted presented as differential of certain functions of accuse President the way Howard Texans 0 auditoriums for call intervention use extension of this formula for the use of this summer the next coral is as follows that name chose the specific brazenly CSI so as defined up to treasure of the change of coordinated so that for example the particular common new case is the called the please stop divided by the thousands the to the floor of the faithful flocked to double total like went over to the standard Maine and also respectively CSI the new kid it wishful the Times cumin 1 kid was 1 of the poorest of the poor people saw the in this is now Our this the In court sorry memory the basic rentals might doubling the this is and now all 1st live reconsider the case when we have just the coordinates of space in 3 of the space 4 I understand Nevada this case is that and socalled to potential all of the political field here associated age the news of falling political skills you is a coalition of homomorphism Coleman fumes but in general some obesity the lawsuit June of semantics homomorphism I think used see Shearer stands for 0 dimensional homology College of about so that the political field here uh Excel still to go ,comma logical feels he refuses Kolko ,comma Lord of the model spaces but in this case for the book therefore home From results which is given by explicit formula jail 12 81 they end physical toll the U.S. scholar product of anyone of their products times this settlement occasions the continue the this is just such a number now we produced the potential who stole coefficients are these good fences at the intersection of our usual intercession number others with a modest pace the curently you currently have Duke conduct that falls with that is main consideration because of this topology glory of topological field it said that there is little consolation potential so that it doesn't give you an exact into provisional his formal the police blocked for all was there to charitable the it's incantation potential replaced by the following strange combination here are new year the James In new so now this element is element was New Age some replied black you to this potential the games of then with the element of then after southern this killing and paintings this revealed this gives you a number this is anyway that this is a potential for more logical of topological fuel serious so combination the area around the southern information off the same data from the social injustice of cyclists which participated the middle conservation potential Ceausescus expressed in terms of the written consent potential this state known is just another reformulation of the same Oracle's France it's just worries at the Czech slot much harder than the original it's the it's just occurred original statements of that in the case of chemical belongs the rethink onstage and so this is what happens 1 the holders Europe let me change should be so in general case the John Jessica really define his abuse of change of the vehicle did as falsely keep their people and the American just shift just days that there was there is no there there was contemplated in this direction so it could be that doesn't ejected from from the original but not until this 1 could be realized as a given volume flow give the full formal that's the theory that the ninthseeded my fear was the story the potential although political occassional good buried them explicitly the following it's "quotation mark blamed frozen original potential of the political serious where is the the all the flawless of quantized gone as tho for dialog the Gladys medallions Beijing buyer samples all the details the old computer the altered the DEA 1 the key unity so so that it applies exponent of this settlement 2nd order differential and this prejudice there is a potential itself so what we are 1 the Orange part and I don't remember from tree with 1 structural problems with the management of the instrument so we started is 1 potential for each Reno's answer then the people I the flow of the Hambletonian and the obtains a final answer and that's fine is that loss able to implement this a showing the competitions at sites it very efficient formal for the for the potential so in general have gives the following situations that they have to alternative way to compute this potential Isaac applied this explicitly this formula or just applies stepbystep notables included cautioned that too race of competition of doing the actually Of of the parents of the 2 1st the Davis a definition that from this definition you can during the master questions Jordan aggression and so on only parties this is the result and I'll admit so this is makes some similarity is that what happened soon indeed the series of food fuel could the whole variety of solutions that it's folks realize there's huge an isolated group of doing all the group is where is so data and this is very convenient you can imagine what kind of the potential to "quotation mark Hicks obtained In this situation on the other hand there is nothing that isn't there is no local structure like the Bulls from cosponsors but anyway it's action doesn't so just makes evidence the goddess of this fact so unfortunately them intermediates as a variant of the book the statement that for anthropological the precautions for any potential tuples according caution with coefficients are determined by the officials be also viewed the by the section oversold cyclists it's go observes that there is actually is a stranded Is it is a then shall there was evidence shows that potential the potential for it Bush generalized the cordial told logical views to field the the way ,comma logical feels news of falling further just any connections region section of the validation group In court is aged for all those know sumo funeral so that they want us to go all the way the field theory requires some additional for the competent so few of I don't want to pursue there that these bodies will properties no called this situation In good now idea neither decision that there was no no protection from from the universal care nothing but anyway if you just look at the formal there condensation of the plutonium than that observed that for instance if you take it norm "quotation mark D squared off all that it agreed to be done it could respond the the movement does the modification of these fields here by the following think you put should other the contribution of the devices which response to the across double .period which hit songs 4 something like that for a new ones the 1 new tool kit or so 40 1 the kid on the new 1 1 you sorry currently the Paul Casey ones the current fall to 204 this spring devices it's the model this assertion of silicon have devised consist of similar costs which goes points and cyclist frustrated because the land bundles of food In the same way so foreign also security you want to 1 do you or do you kit or just that please this 1 these principles projections from the universal cure insertion of couple yeah sorry I'm lost 1 the priority of services for workers .period and so on so we area overrepresented participates it that the definition of this section of simplex a group called the modeled as a settle modification of this had the political field if you can call the company explicitness of this we looked at this for this for those who knows the result actually belongs to them the house call to the village according the this financial forward gunmen job presented computer explicitly in terms of summation autographs some the actions of the sixpoint could be their last summation um along graphs and the summation will and this formal test interpretation on 1 hand this summation autographs described this crisis is visited what kind of also on what kind of the I want to fight it was spilled Kurras's medical .period complicated mentioned cooperative and there and what about the the sport contribute so we have completed the configuration which cemented Boisgrimot appalled some of the Alpha Butte officials some combination of cyclists and of the of the group .period and integrate all together and this produces explicitly way how this exponential selects and the space of Cornwall illegal fuels you on the other hand the same formula could be applied at the level of formal with operation despite our serious the physicists and all the direction action of the 2nd or the financial operator of the schools and confusion squandered computed .period the summation or thereafter there is something convolution or graphs formal so this is formula is a moralists committed did good but didn't explicitly so reserves future the original which immediately called formula but not we have this potentials and some of the potential fall into this room most of this event Respondent journalists ,comma local filthy but there is some fire which corresponds to the economists mortgageable will fuel theory which allowed which satisfy all this excellent of gold will fill the area it is funny ,comma donors are socalled longhidden hidden potentials and those all hidden potentials it's also a very muted tho described in this language of of this correspondence for the granting of potential so formal the potentials are described as in the unit Dulcia it said I still believe Lagrangian plans which satisfies certain additional signatures recalled the given substrates sluggish so by definition the good news is that preserved the title way Xenia yours this was a strange definition but that was the hole hit the Puritans but it was also given Gold understands this crisp understatement it has for this and this is incidence of what was performed so my case I used Lawrence functions take it medicine each Gonzalo all plus formed from makes this the same but this and that respondents as usual few have conceived to the club did use a society forever the extra support came to me and gave me some confusion this so this is just has a more fuelefficient than spaces this complicated apparently becomes just the shift of the multiplication by that Arkansas so if you want to speak about the ball articles from the multiplicative structure and from there this presentation is also the convenient this case follows of course is not multiplication so for instance this interpretation although space is used to the proposal didn't volunteer preserved the notation of the variables disease stands for is the same but the paper solve goes sympathetic form has a different form Shearer saw in the jails of the function the simplistic form also something coexistence the integral should take careful neither the GOP and then take that there is institute consideration due to the so that that's the different form lot but anyway books like this no In response then the anywhere else in respect of these apparatus described to reasons so misconstrued or all Space View multiplied by the interviewers sorry the business of specimens from is that since got I mention such spaces Susan Circle of I'll probably the reason but the reason is that the not all yeah yeah yeah yeah have records so that it's more even tho I take Els and multiply by using the which contains this 1 and saw that Mitchell somewhere the new this is amazing so it's determined by serious Xinyi union serious video dealing in the 11 In the Matter system in the Baltimore fuels age the 2 of them are human well and also more tomorrow was that the fact that allies like France and with the the identity of mind is the latest car overseas dueled respect accessible to 1 this is near dull property and all this former group 1 these a announced and the whole of space because such brands and the boys you know that you know where the formed this so if it takes the this lakeside Newell Europe these this is the very exciting can you clearly just as you have problem once the CSI 0 up to so fucked the signs 4 this 1 so the call what grounds of substances determined by the this is a main and all defiant formal review potentials Our for associated his cold on the event in the history of Celtel consideration of these uh organizers substrates satisfying these additional property of them there the whole group of girls commercials reduces tho these small 1 for this as substrates as this a near arts for a subgroup the space of a group of simplistic personal exonerations Ridge is good that is as characterized by the conditions as they should commute with universal they preserved the substrates softened during the whole such but this is only a small part of the hall displays of Lagrangian and honest but a small part of those potentials that could be obtained by a the Baldrige politically cautions now all potentials they saw in general if you study general potentials of the political order cautioned the letter available Apostolos foreign doesn't is not very useful but it especially when you're dealing with a group of futile potentials that threatens inside could be your so this is smaller notice thinks so that new Member of the time is over remember if we want to see 1 of them other qualities of this although the 1 of the coordinators of the USA existence of Of the Velasco wants cultural series socalled property that Evgeny is a pretty long known that Tunisia is Morris explicit bonding for the video of this prolonged because Columbia it's quite homogeneous vehicle not be greater than the dimension of the model of space and Jim and some other properties feature moralists the which simplifies the fuel just look at on this side of the story we're live everything centurymuch so holds that you have the precipitated the compared to a different on different points of view that the politically cautioned that after the summit international get some patients have thinking thank you and that is it it was like all this is the move there is most bottled water to this small USA for now the or this in the virus or the search for special why you so there are of those that's right ideas as rises of good ideas and that the 2 units virus aggressions of the system Eliza although with concessions that the the politically cautions that stability is around local vet all potential to politically cautioned this is a very nice mark the leaders of the conditions of our true but the program was aware that 1 of the benefits of it the flight but that is a broader but there's still only broader acted within their rules then when I saw that make will comment on the offer still owns a special corrosive and you take a spectral can and she exists W 0 1 W 0 told they're not so important what's important all part of these and the accounting change freely is even part and those this gives you a huge flexibility miniatures of the spectral Crawford and it's not clear what for jurors to can in the global 1 so I can feel we have some freedom the justice of the spectral traffic and for some choices the converted to the global on sometimes not since the were on the injuries and the uniform he was of this that was just 1 fossil of the new USRussia arrested among more than a single that don't here is not my invention but my interpretation what their autumn different products they the only inventions of the language the results themselves are not new I I think you can get loans through it the way you when you look at yourself over to the villages well maybe only of villagers here and there is the jury of basis sigh of relief and then there is this the thinking is that the news that extra leaves that is more than a year get them was granted on there is also a few comparison from of fraud and this is accepted the ladies In this is inhibitors and work with the lowest form of equipment for last for years it seems to me all these variations are monitoring the area no not at all so 1st of all my statement that this summation of what grass doesn't it I have nothing to go on this this is remove hidden potentials as could be done for it the political potential but if you win a specified for the particle 1 should you should take of this political basis and to describe basis you used "quotation mark of possible for him to In the event of former U.S. even take busses we want to do something and the of these reportedly replaces you by the powerful like losses and then the this pictorial double for the local officials of the Laplace transform the USS fallen given us form at the end of the students are probably a furor right then there you so in Europe so there is a lot lost souls form so for instance if you Texas you stop CSI the new this regional body CSI new clearly start annotation firms Anderson said this case .period sorry we just so you're probably right for this we will continue to use as little in the way of the political affiliation of the area because when I speak exist side by developed I tried to avoid to consideration of of particle based but if show diligence this practical 1 problem with this interpretation is the singlestory belongs proprietor fury it was also the year people's or forceful some of us into this is not to be delivered by party at that it does exist as a collection of commentaries come to my computer programs I promised to on Poland title make a PDF file from this computer programs to help available so few and