Pre-buildings as sheaves on the site of enclosures

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Pre-buildings as sheaves on the site of enclosures
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Given an ane apartment with action of the ane Weyl group, we can look at compact convex subsets whose boundaries are re ection hyperplanes. The category of these \enclosures" has a natural Grothendieck topology, whose sheaves constitute a natural environment in which to consider spaces which look like buildings or subsets thereof. This provides a technical tool, of a rather combinatorial nature, in our ongoing work with Katzarkov, Noll and Pandit on the construction of pre-buildings and versal harmonic maps.
Geometry Point (geometry) Group action Building Direction (geometry) Sheaf (mathematics) Parameter (computer programming) Student's t-test Mereology Grothendieck topology Group representation Many-sorted logic Term (mathematics) Different (Kate Ryan album) Körper <Algebra> Algebra Covering space Building Projective plane Curve Ordinary differential equation Dimensional analysis Infinity Algebraic structure Simpson, Carlos Grothendieck topology Computer animation Summierbarkeit Fiber bundle Object (grammar) Coefficient Higgs mechanism Family Spectrum (functional analysis)
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Building Group action Differential (mechanical device) Direction (geometry) Decision theory Water vapor Insertion loss Parameter (computer programming) Solid geometry Ordinary differential equation Grothendieck topology Subset Mathematics Many-sorted logic Different (Kate Ryan album) Ranking Physical law Algebra Rotation Process (computing) Reflection (mathematics) Closed set Moment (mathematics) Staff (military) Price index Degree (graph theory) Category of being 4 (number) Triangle Right angle Cycle (graph theory) Linear map Point (geometry) Geometry Divisor Parity (mathematics) Transformation (genetics) Characteristic polynomial Translation (relic) Element (mathematics) Product (business) Power (physics) Frequency Inclusion map Lecture/Conference Complex number Modulform Matching (graph theory) Validity (statistics) Weight Model theory Algebraic structure Line (geometry) Incidence algebra Axiom Cartesian coordinate system Subgroup Network topology Factory (trading post) Object (grammar) Hyperplane
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Building Theory of relativity Rational number Lecture/Conference Multiplication sign Insertion loss Open set Dimensional analysis Condition number
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Point (geometry) Covering space Projective plane 1 (number) Sheaf (mathematics) Mathematical analysis Basis <Mathematik> Insertion loss Mereology Grothendieck topology Theory Product (business) Category of being Goodness of fit Voting Lecture/Conference Logic Term (mathematics) Object (grammar) Körper <Algebra> Arithmetic progression Fiber (mathematics)
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much I'd like to thank the organizers for the opportunity to speak here all so what I'd like to talk about is that there is more of a little bit more down-to-earth about some kind of geometry on certain site and I hope you'll see many years later in the talk and there's some kind of algebraic structures which arise which are maybe a little bit of difference all and that the the ideas to 2 To view geometry as algebra in some way so that there is a lot of algebraic manipulations that we would like to which search sort of inspired by the small object argument in some way although that doesn't necessarily mean played a big role in itself in itself this is part of an ongoing joint projects with Woodman Kadyrov Alison Weld and on on and the concerns basically some stuff about per considered Simon .period view the point of view on on structure infinity Of cluster sororities along Calabi-Yau things since the wagon and also guided tour morning scheme at the spectrum words so but last summer I give a talk actually about the same subject all so it made you can see on the video probably something about the general framework of the project who similar to talk will be more about that the precise geometry so there the motivations now all described as a little bit later the moderation is that be if we have yeah yeah and didn't scare we have a family of 4 the ordered ordinary differential equations which will call them would see that directions on a bungled the bundle committee the fixed on remote servers X and we let Road TB the Monory representations then as he goes infinity from "quotation mark 1 of X India has the story absolutely there's Road St goes to infinity then the road the well it's going to be equal the 3 today but Miguel starlet and equals to is something many student as he goes very rose sort of approaches over which of them actions but I was on a building and there were using part and Burroughs steering and idea are project is that we'd like to understand you to know that I thank you we like to understand the building Starting from limiting data but for now and limiting data this is a skilled but sp roughly speaking Higgs field is sort of the leading term of Nabatiyeh it's the German which has coefficient and field has get gives a spectral it said that we have to go in particular the special curve of the so what's so special graffiti but just be extremely brief it's just that it's a rarified cover but that's continued the storm the so so bad for our purposes we can just see this as a multimillions section 1 x invited and what's called no 1 planned With this with the Commission's summation land idle 0 that's the trace
that's the condition of the trace of is 0 because well he representation all we can assume that the determinant of the collection of trivial so so In terms of this nature of the spectral curve the way and eyes address the different values of both of them From fiber inside the code and among all acts so adding that set general point of exit we just have been different 1 4 1 at home hold the back and if anyone formed which in the jet were network and we make generosity assumptions so that for example will be assuming that this the curve is eerily similar if there are different than is more complicated OK so that yet the dialog of the this is my question this is filled this is well about Menachem we winning a blossoms here I made at the border with lots of different ,comma explaining that I will win as only way just referred to the video of that conference on the confidence of the world that we should go because of the the 1 in 4 months seems birthday considered for the conference but he so and so to get down to that now and that's all the motivation is that I so the account of the basic so basically real which so the whole Norfolk differentials are complex three-month service but we just take the real part going to get the real differential the fact that they have aggravated the corresponds to real parts of all Oregon forms just means that the harmonic and of course I'm saying this away from the ramifications ones the revocation points and items which will I will see the picture about minute and the sort of initial er proposition also which we been in our 1st a broadcaster grown on Monday is that the differential so but let me be the the building given my sorrow then we have a limiting map we From the universal cover of X into B. L. low LME citizens are building we have a lonely man Mike from Universal cover of action to be which is the core area for the action of by 1 the main point here is that debate that is equal to this vector of forms what's that mean the building's locally foreign minus 1 locally pieces on online 1 Dr unless 1 is just viewed the center vector points in or an who some equals 0 some constant so the Commission summation when Michael 0 means that the differential map into on minus 1 is given by and vectors whose some 0 so this is a massive allergies and this is the reason for this local wkt now this is actually about "quotation mark and we we discuss the proof in our paper but I guess it was you might say all we would have been well known In any case Freddy was to this missile theory is well known only discussed that me for no so Friday was to spend the Higgs field is just so then the spectrum of midfield but since the day when the 1 Lambert use something called 1 in Senegal 0 than Lampedusa's minus 1 1 so we just have a plus or minus a differential and that corresponds to this Vodicka square there we get a section of the cancer square of the canonical model so that's a quadratic differentials so you've probably heard the whole theory about the relationship between "quotation mark differential than maps to trees is England that's what's happening here In this case is treaty maybe I should say dimension of the Eagles so the map H so in this case the map that the building is a tree in the From
listed the factors this article we call this feeling the call this hoping the Ramsfield where the TV is the trail we Of the following defined by the differential form a real part of land along a real part of my 101 but remember that ramification for those 2 guys can switch around we just drop ramifications .period so as mortification 1 aquatic differential that is a point where there will be no choice said of the uniform choice of the square root of the "quotation mark differential so where under was going to be something like the the one-half the scene which is D of 2 3 hours of Boston now you try to draw the level set so that the leaves or sought real part conceded 3 hours the whole concept never tried draws together all this in the wrong direction well this classical bridges so the defoliation at addressing that defoliation look something like this and the space of waves is the tree which might run the opposite direction because points here corresponds to 2 leaves here the origin corresponds to these 3 line ones that come together this guy trying this guy In the money managers what production from experts in the next over this is the so what was being said you're basically is that the size of I'm not saying details about the size of the modern era transport functions for the connection if the connection has a large term which approaches acquired differential the size of transport from 1 point to another is sort of the it's the interval of this quarter differential from 1 point to the other this is a in the sense terminology this is like a major there's a major transfers to the Asian and the made the the distance and that major is exactly the the the growth rate of the the moderately I should say the Monroe meet the size of Monroe is he to the tee times that that major I know this well we wanted to do in our project is to L I understand this for and bigger than 2 and see how it relates to Kyodo more nite sky special outwards but and if so when we were here last summer summer 2014 we basically decided to to concentrate our efforts on the case and Eagles 3 for Miguel explained the reason for a moment so the goal of course is to understand the case of an arbitrary but but so far we avoid what you think is an Eagles 3 but what about this case is that in this case dimension the real dimension of active acts it's too which is equal to the real dimensions the building the tree case so that's why you have this quotient by the voyage in Madrid is the mention of building is 1 and the mention of access to so the map passed a contract one-dimensional fibers that every of the lead In the case and a bigger and bigger the Oracle to for that generically lease the image of the man is going to be some Yeltsin's some space of the building the case is much less clear how to get a hold of the points of the building but in this case at least the dimensions of the same so we can act as a to get a hold of some points of the building at least starting from a point of here you basically although all points of building at minute turned out that the Mavericks to fees rejected so you get all points the tree by just leaves of the voice we hear what's happening that's what I'm going to be talking about today was happening is that the image of the curve some subset of the building but it's not sort the whole building and that part of the question is how to recover the rest of the building I think so they're not with this is what we saw the notion of rebuilding the prevailing as there is the thing we get before the building so this is basically the image X told us in the field but the thing is we went we would like to have a similar factorization statement the offer is made is
supposed to be building it's worse was that it would like to have a similar situation there but let me just remarked What's that what's the characteristic of this situation with the treaty but we like to rubble gave to the point is that this map at X solidity that depends on some kind of complicated limiting process didn't a silver in Perot's working to choose an old Travolta you Dixon complicated woman on so that that that the building that you get there Opry are you haven't geometric control over what it looks like whereas in In the case of the Treaty of leaves a void nation we have a really is the argument somewhat easy geometric picture of but what this tree you what points of history represent and that picture only depends on the part of differential so we like to have a similar situation you would like to be able to construct a building where mathematics although to be the With Robert it really only dependence on geometric weight on the special on the different forms last year by the University has it then analyzed is a German of fire when soaked up Of course the fight for now once you look at an example was that we didn't have basing them on our 1st going have always say that still anonymous object onto a so so that's why you get this idea of a pre building which is going to be that essentially the image of plus maybe a little bit of extra staff and so yeah it's is is that the mainland moonlighted instructors and rebuilding then maybe the building and the governor appealing to the building there's sort of a small dig argument but for 1 thing it doesn't seem completely unique and maybe it's not completely essential so maybe the the real statement is that we have a factorization like this for years are mapped to the building age that may be the real the the real thing we would like to consider is a map from tilted to the prevailing the factory into the buildings but the privilege of sort of contained the structure of the building near the point of X told us at least so now the question is what is the prevailing so there's lots of definitions of buildings there's enough books and neighbors and everything discussing what water buildings and so on but we can see from this example that we really want understand subsets of buildings rather than just building and that the Axum Agfa buildings are usually pretty strongly related to the axiom that any 2 points are contained in a common apartment Wichern implies that were already looking at big are choosing I mean not to mention going these are choose the sort of going after infinity and in some of those aren't exactly directions which are not hit by but still so here's where we get to the question a sites awarded this aggression was a pretty buildings and so we we wanted to have a of that's the notion of of this type of object and furthermore it so it turns out that there is a whole process involved In the construction of this guide which all be discussing I and would like to be that process is something else Reg rather than something and with a lot of you know a lot of the points of view on buildings have 1 of metric stuff in there it's important to examine some of the major stuff obviously a crucial but and that has to try to stay in the world of algebra as much as possible so that's good so this was kind through this week to loudly the new so now Nelson would say so let's consider the case start building will consider the case and was 3 of our buildings are model on the standard apartment are to part you as I said we shouldn't be valid points X as the point X 1 X 2 x 3 some into own the text of the His extols the universal coverage that building the model on the center partner which is an hour to and the Valverde bats just as 3 when they are fine I this
translations product a true so this gives some kind of special F-1 transformations Gold scores despite the loss this is the apartment but what what is that what is geometry inside the apartment look like Georgia inside the apartment is controlled by 3 different directions is so there's 3 directions of reflection hyper .period separated by 60 degrees so these are the guidelines of the reflections on which the reflection that led the transposition the necessary our reflection so you're the guy that cycle texture no 1 acknowledged that in fact we have a little bit more structures so that so at any point we have 30 3 preferred a as actually 6 raised so the 6 but they're labeled the incident then divided into 2 groups they have appeared women so the thing about that they're the industry there's a rotation water 3 bidders on a rotation 6 so there's nothing which is again endurance by 60 agrees that the revisionists returns by 120 degrees which means that they the different that the it preserves the parity of the Of the vertices and so the the verdict that Florida the edges coming out from point of power parity and optional which is preserved the workers he on to groups there's the white Dodson why Godzilla but an element of S 3 as Teresa group which preserves a warden now invested preserves the parity of Of the edge there's no there's no element that goes from year to year there's only 1 thing goes from here you can rotate a reflector remember you saying about those things you know it L OK so much as soft side ,comma no no we have had in the way of half plans new emitted by these reflections however .period reflection wins in the are fine motor were any translates these are the kinds of jobs he said he wishes his populist policies words at the archery because there are that this is due to the motivation you're the the edge swings in our building are going to be sort of exponential ending look up the index moments of the size of Monrovia there's no reason to operate those should be inside Adobe and some subgroup maybe .period rank of going down on the way employed it's actually it's actually the period of vital girl 2 but that the sound real parts of periods of vital growth which are some subgroup of our troops the translation it was questioned all usually people called off I'm right in the eye OK our fun while you're presently this but isn't so we have reflections applicants fizzled the standard applications the half when they're doing it by translates to the 3 lines now an enclosure is just any comeback to intersection of close Southland so it's comebacks so it is drawn using a standard 1 might be a triangle world parole ground the decision here in the theater they we also have a point so what's called point P we have segments but that's about it but his Armenian in we sold so ANC is going to the category of enclosures :colon which I think is a category enclosures with with maps which were given by the elements of the affluent Bogra the inclusion of the all cultural allusions inclusion on inclusion the the matter the matches have to be linear maps but everyone your maps that are that don't if there were determined by an more 3 but not the 1 that was something yeah yeah ,comma into perhaps end In the department's analysts but in
the end zone for example but you know that you a triangle Mountain falling into a growing through side inside I that 1 nation up into the air at me in particular the point Pete maps and everybody a lot of relies all year the segment and we have the means and was has edges which amounts from segments into the enclosure the army news also amount in my there's also the part of an edge the only of schools and he he his visit with the reserved mentoring and that there is a need there you could imagine on this guy .period that's not actually magic which is useful for the weekly development and now so the point is that a a fairly simple not structure of site this is you might say inspired by what once I heard people saying the rigid geometry in rigid geometry maybe Tate had a site where you had a originally been sets when the point was that you would find it covers by regional concerns the committee and for denied that anyway this is maybe inspired by the idea which is that governs final cover 1st objections From fire short unions some of these conditions of because the quality of the analysis of this the generating covering the the generating covering generating failed the trouble is generated by covering which refines destroying union the way 0 also were allowed because we we did try to you they were not allowed to trial observers said the subdivided triangle into a all sequence of triangles like that and that would be a couple of years rejection you might you would have us ejection from the ERM former union of infinitely many trials onto a triangle but that's not all that's not a cover but the examples of in the museums In the end of the interior it was the who is the yield on the No no no no no maps from destroying unions stunning the news is that this is the only the variable the map from this would be a government that will make an overlap of that 1 Of course at that the point is not to have a lot points have them intersect just on the edge this guy not actually give me a burial for the covered because he can hold his fire in the hole in the wall notified to covering where there were arriving back because refined tour covering which is sort of standard in effect them suggest cutting by that cutting by mentioned 1 like that and so what 5 1 nite everyone finite all so now what are the shoes 1st of all supporting but why this technology said is not some but the the Federal Maritime yeah but so if we have an enclosure then there the appreciate represented by the enclosure is not a sheep but if we let the Jollibee The Associated she's then was he told 0 the maps f until you tell them it's piecewise these was outline maps we call these may be falling out so we do have a map from from From an exposure to different enclosure which are twofold long were only allowed to fold along these reflections don't involve things happen the work this is the sled physicists have finite these why that's why we wanted to stay with the final we only have found cover generating more trouble on the old original version of the plan to discuss knowledge is just about the peaceful the union of the 2 intervals the now be the case that you would use to do the tree to tree section 2 of the 4 we must have been you why OK said From the space after the space In effect I met and maybe that's a good
question it should say that the the the the sort of underlying set of points so and so all of the people live Florida don't have to this but in the social policies of my 101 A-minus on your desk this summer I think I mean an ally and think about it so interesting that there should be like a a piecewise linear version which is by actually well known or something like that of this whole discussion In this discussion were just limiting ourselves they were interested in buildings for this ballgame remaining ourselves to to maps whose which if it rotates installing given by but no 1 arrested 1 is a one-dimensional as 2 I mean for a general we group you'll have the vulgar but we're acting on the the Cotton and that's going to give you the transformation they want look at but that on some points of the enclosure is the appear appears that is the point evidence so we can think about the European become demented space and all so you're tomorrow 2 no this is warranted in emergencies were going together the pieces to form something like are building things them and for that um reintroduces the basic definition is not let me just explain that there is an example which relied as a rule out France some that example and so sorry so there are the objects would like to again will be then commuters she's on on and on the side of English now that examples of let me just a Silverstone 1st of all that if you inside the European if you not Europe's open said the memory appears just the usual space the points the corresponds to achieve what Scott you don't know if the set maps you till is a set of maps there Clinton until the such the FOP steps into you but this is not generally find it generated you have a triangle you might sort of just and there's some of that is something that is sort of tests ordered by some triangle the but they just have to get smaller and smaller as you sort of go tossed the boundary this was just another set assumptions no no no this in his usual topological space just trying to hold on at some of what we started talking about arbitration use and maybe not the only 1 I talked about for the top quality but for an enclosure itself that apology just the usual along with the people there are nuisances that wrong if his campaign too and then there is the issue of this election said the loans so she called unites with the chief of all the guys which go into that opens up which is it's a valid chief on the side of enclosures it's just not going to be very nice by and it's 1 which could actually happen inside our building in our building because because of the fact that we're talking about lines which could be arbitrary In procurement Roy small something like this could actually happen inside the building but what we like to avoid is the following things so that we like to avoid having something like the ones union the terrible Over open said you I want to do when we anyone might be equal the and where we might glued to go to copy those of triangle Over the same opens at this for the long haul policies as long as they this is not going to be a very good obdurate we have reviewed all the time the solution that's the slowing in exports at least help to I think so but so what we do to try to avoid the is we say that we have achieved that is why we generated if their existence rejection From a finite the striking of enclosures the
but basically this rejection the troubled topological said with the meaning judge has to be attractive navigation of ships a news she of the corresponding I uh we only had she had he said of the flexibly have investor reactions on beaches according to the American people to environment coherent she was With the lack of some people the she the remaining outside the residence the flower I get sentimental and as I said we are with this is not actually division which we usually want to remote because our building is not me from the generator there have been women you don't have a lot to itself was not included because I am asking voters to be comeback you could maybe discuss whether you wanted to go around but in particular are too young or too you have to know covered by but infinitely many a but we say inspired related the fire roared Freddie during loses mapping to the fiber His finally yeah so is and that I was able to prove actually but we would like to show that you get a house storefront topological space if we have 1 of these guns all of this is like the notion of growing coca growers that voters will use the I will do something which is a acquitted losses maybe we had to give a but anyway so so the definition is the construction is fine finally related issues it is an old she universal Gucci and this is an important 1 and I mean that traders used by Congress I think so maybe you could have infinitely many Indians like are with the idea that the goal is to get something which essentially recovers the case of our trees or buildings so what the dysentery that there had the idea would be taken to recover the theory of our oratory it is not an arbitrary collection of guys we that's the hope in any case the there is the construction of the definition is finally read related and now I can we can do things with the direction we can sort of construct things with construction so on many an accessory definition is that a takeover aberrations so maybe I am an accessory definition is that I think we need this zydeco vibration it is a map of constructions is is an injection of shaves such that enclosure mapping into Efron the trace of the enclosure of the intersection with F is finally done but this is assumption of so again we like to not have uh you know and inclusion of an open centers of going down and then for example Bush out Sonko vibrations preserved constructions and this is the basic operation which we like to see you so mean it's to time to stop him the only just get back to the case of extolled the rebuilding on so getting back to R situation dramatic situation we can write extolled as a man as the union of small open sets and sort of divine we can find enclosure for EDI freeze you the men modified open set so that they look In terms of the Gordon system given by differentials so they look like enclosure and then all you the you intersecting you given the good relations among
which among the young and from there we do that then we glued these together moved together to get F
0 which an initial construction so the the 1st up to define opera building is going to be too to good other little pieces corresponding to 2 small open sets of X told the hearing loss was really made to cover experts are locally finally that's maybe I was thinking about the case of a because of voyage entries always is actually more complicated in the cases at an eagle-3 because that is an eagle-3 since the dimension of axes evil the mention of the building at work and in the end that gluing together infinitely many pieces if we try to do this for for the formation of a treaty we will not my already get into trouble because this gluing relations could sort of we together and on many things if you take a rational slope defoliation on on tourists we try to do something like this we're going to end up going together infinitely many so this evaluation you need some condition that says that the leaves Nova back arbitrarily close coastlines themselves it ended in this two-dimensional case uh just looking at things geometrically you sort glue together the pieces call might now know that the only problem which is Lofton no interim time to discuss what are now the basic bombers that have 0 is not mingle with that's where the algebraic that's where those you
might see some kind of higher algebra comes into play so we just explain why F 0 would not be made weaker 4 they there is an area of curvature and conditions this is seen if you look at points of our summer construction then there's an notion of global spherical construction which is a graph an inside the graphic of hexagon on the graphic corresponds to something flat the the it's kind of the local directions the gravel have vertices each edge coming out and then edges for each to sell we have hexagon that's something flat we have said opted that's going to be negatively if we have something of wing for some undersea going to draw the I we might have something like this when we do that we things together and generally this will happen all over the place we start doing things together according to ruling by will open sets of acts extolled my Cypriot anything can happen in particular saw a picture like this will happen so to parallelograms who together along to edges at this point but there is at this point that the grass looks like this the graph has only 4 that's a that's a positive weaker .period just dementedly conceived above the water .period and so the basic algebraic promised in fuel and both of you go to the courts the AIG open until we get that about two-thirds of those surveyed so I said now that the the that the problem defiantly buildings of rebuilding would like to be negative but ,comma the essential the thing which is halfway between the notion of a construction and the notion of a building is to ask for some kind of negative curvature and so the basic brawl just explained the problem here without finding the solution that that that the basic problem here is that we have a well fourfold vertex like that well is clearly wanted a wonderful fold together to the pieces of soaring have to be gluing together and stuff the difficulty is that we don't know which way good together right you have like this choice of children you can win link together like battered who together like this so some additional data is needed to determine the age of the vertices which direction to choose so there's the procedure of Bloomington of sort of collapsing together but positively vertices to like to you and collapsing together these 2 there's a procedure collapsing together positive weaker vertices but you need additional information you need some information of marking of the the whole picture telling you which direction you should leverage .period and that the information which is going to come from the original Ramon services on and so that's what we
plan on paper at least that's what we're actually have it's not it's not a theorem sort of a program of power you should be able to do that as usual follow through this procedure so we should think of this as a sort of great procedure of forming a coalition of this construction according to some additional information the surface this year will fall that the the map comes from midfield men were using the geometry of the map the way the launches this is the story and basically I as enjoining if we had it if if the front program came from a piece of the amount service in the back program came from a different piece agreement service for some WK be reading we know that we're not supposed to fold the Ramon surface itself that tells us worse was switch like that France was twofold on this 1 the rumors that once you once you start folding then you get you get the edges which are priori don't necessarily come from come from pieces of Montserrat that's actually where the guy 2 more nite sky Buchholz spectra summoned to come into force the only Back in the old users this is a a list of mishandling and his clothes Laszlo yet this is what I'm trying to say actually give you liked it means that the original was risk-sometimes general procedure like a small ordered argument was that allow us to to map X still detonated weaker space but turns out to be that the small argument is sort of the disorder control as you move along in some ways because of you need to control which direction you collapse at each point there is some sort of that some sort of control of the argument there needs to be done as you as you proceed I mean the ad said that the the very 1st ideas you might as a wise Modric argument we sort of take the the universal David Wheater building and a weaker premium being negatively term construction which is prevailing weather map from excellent mapping to but without giving differential the point is that universal guy doesn't actually exist of reward an fact end it in exactly the cases where there are a spectral network where there's BPS states in the spectral at work but you can see that the demand actually due to possible choices but but now the point is that it won in frigid weather Janata generosity conditions were hoping to prove that that doesn't happen this so it's kind small or big argument but we with piloting along the way the need to maintain the data that tells you at each step what you need to do 2 authorities in the field and he has the legal problems of the people and the use of uh that's a good question what's the category of points of the temple the only 1 which would be the point peace but the fact that not only are unnatural at the Met and maybe like germs that enclosures at the point the following list but maybe there's there's a the .period P S their viewpoint which the germ of an edge in a point which is the germ triangle so maybe there's report possible but that guy and our held that the U.S. is finding is that there is this is about as believable as it is defined by finally come this society aggressive about their violent they found this
is the site of a ride on I think this site has filed 1 the you can intersecting enclosures when she began hold that products rather than by ironic don't have good property part of the construction field here this is social over the people of the progress of his vision was in which the government for use of my questioned during the probe was the same as those people as a the following issues the case of this so I call on them to fill this was no I I I don't know the many whose logo on I don't know yet it's a very sad theory simple-minded you might see examples of their use of the the flexibility those proposals the cops on the basis of fire burning in the world you this is it is located the Soviet was for the this is is probably the only guy was closed on Friday this is the 1st of all I think so yes this is 1st of the visit Gaza 1 of these is the fiber rides of basic got enclosures are fiery related upstream the use of new but there but I mean I don't see any reason why that would not you get the offices police say there is no fraternizing with the priorities of our society and the political analysis about I subsidies for cotton proposal ,comma good going because of Article yeah but I think the blight upon is the smaller and smaller ones I mean he small arbitrarily small In quotas you the odds arbitrary lodging houses being yobbish Royce Mongolia's invitee infinitely many because a few fix some uncountable collection and you can get all possible joints that although our history and cover girl who finds the words as they the good news it's a good thing that would used this very this is the "quotation mark the proposal so the In the those who grows worse is this also a nullified is not finally generated no without the product in the image of loss of the project to Rogers is not finitely generated you can see that is not the it's not that they couldn't go through with the name nuclear fuels and on the side of the year In the lead in here this is a planned and vote yes or that that's probably not true because that is the the money that will he only at 280 Arroyo again two-dimensional objects and you find take the product of 2 different two-dimensional objectives are to be afforded object but it's not that's not going to disarming it made the fire that them in the product of the she's obviously exists as a sheet it just doesn't have it's not very good dozens of various public property you have to look more like the section with a fiber products so that by writing to the all of ironic actually works pretty well its use it was like intersection with a product but not a fiber part of this call he was the 1st thing his young idea among other things 1 should be careful of the point P is not all the same as the empty product then she bright star know that the the constant chief would values equal to 1 . is not a good obligates not finally related for example home so he doesn't have to be object assumed what it there is a term logic in the in the in the troubles but but it's not our construction analyzer it has found from the police way solicited father a look at the creditor Safire yes this is a good summer as