Dynamical Systems on Point Processes and Geometric Routing in Stochastic Networks
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Dynamical Systems on Point Processes and Geometric Routing in Stochastic Networks
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Dynamical Systems on Point Processes and Geometric Routing in Stochastic Networks

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5

Number of Parts 
6

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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Release Date 
2016

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English

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Abstract 
Dynamical Systems on Point Processes and Geometric Routing in Stochastic Networks This talk is motivated by the study of geometric routing algorithms used for navigating stationary point processes. The mathematical abstraction for such a navigation is a class of nonmeasure preserving dynamical systems on counting measures called pointmaps. The talk will focus on two objects associated with a point map facting on a stationary point process Φ: * The fprobabilities of Φ, which can be interpreted as the stationary regimes of the routing algorithm f on Φ. These probabilities are defined from the compactification of the action of the semigroup of pointmap translations on the space of Palm probabilities. The fprobabilities of Φ are not always Palm distributions. * The ffoliation of Φ, a partition of the support of Φ which is the discrete analogue of the stable manifold of f, i.e., the leaves of the foliation are the points of \Phi with the same asymptotic fate for f. These leaves are not always stationary point processes. There always exists a pointmap allowing one to navigate the leaves in a measurepreserving way. Joint work with MirOmid HajiMirsadeghi, Sharif University, Department of Mathematics.

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so at time and on and on and on and on and on and on and on to the end of the year itself is Fantastique can produce little multiple clear OK so I felt they wanted to stressed that the this aficionados assignments amphitheater and the same Jim Simons who skated in from the left was exchanges in the woods and many other things that we discussed so and is a mathematician the the and very bright 1 of them and we did as skittered lots of interesting structure in the U.S. and worldwide OK so OK said about me and then the ecosystem on .period processes which is also home to media as in new book but in this would be on the media the difference between an assistant work with new meat units of Indian cuisine that day from domestic government action OK so the plan and sole 1st I start with motivations and because motivations so this is this is a project that will form the government as far as I am concerned that from very practical questions of halting the management works and we .period something that's too and then I go to school fuel fuel I don't think so .period positions and palpably because it didn't take it for granted in with you on the Net "quotation mark formula without then we learn look at the makers environment measure equation with defined .period maps which of this sort of walking into his arms the specifics of think and then we discuss the 2 main using things which of his book about for the teams and the point by Felicia and islands in the East the terminals and hope we surgeon on the sufficiency OK of motivation goes like this all this is so good that never again initially navigational point was a soul what is the point was it's on the measure but assume you it at a point seal the continuation of his accounting measure on you on and based on what you see you decide to move to another threat is as simple as that and so it is 1 of the simplest pops and navigation systems .period process which went colonialist Mabel drunken geographical think these things are used to in practice million and sold this is source of this nation and mutilated and egoism you want .period takes the view that tourism and you'll do was the dumbest thing you can think of of the the next hole on your halt to this nation is to go to the nearest among the nodes which are closed off from this nation than where you want to get your here you look only at the small which until the and half which on the point he said bluntly that bowl UP the closest the bowl of inside this is exhibit .period aside bold which at this point and utilities hundreds OK so ridiculous systems to J. Donald Tyson a person .period process for instance any conference is a fall there are infinite number of steps to this nation cancel this sort of simple and residents too it somewhere off on such as construction was itself
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illegal nation which called the election voting so you want to go to war in the west securing barracks and walk also a here in this picture of you and you want to go to listing Paris and so on and so what you do is the source of the big bold picture like this 1 pubic that book .period which the nearest to you which is saying on the half plane the he is on the west right and so would want to look at Seoul and then there was the idea of fun was the sort of previous Of course view of what we looked at with this point with would go to Y and his point with good this 1 and so for something you would send packets but this kid steps by looking at the local information which ought to be :colon something words networks to some destinations OK and that's the slide which really you have used wireless donation of it's just 4 of his formation sold pursuant to you're back transmitter knows the number of receivers who OK which are on individually sized corset with explained so we use a disease of mobile evoke network we welcome you want to send information from additional Nolte respect meters saying all to well to the east in that case the sole use of the Hong Kong follows with on which agenda Lawrence don't carry through just maturation someone threw there since Pittsburgh the law .period persist into categories those of which upholds the terrorists including the tactile speakers and those which are potentially secrets OK and so you pick care our Shannon's theory to the site of reception sold Of of it would be a sign of based so if I give you this set of transmitters it creates an entrance which we determine the chances of a auctioning the fact whether a given note with the potential receiver or receives the and signal profited by the on judgment another attack transferred .period tho this would be for instance among those which are potential receivers coach we receive on the basis of a sign off the inflation fury of this speculates that comforted by the fact of the and you select this created 1 which maximizes so progress to waltz newest against now because of 1 goes to OK so where the questions
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structural of Roberts but what happens along Robertson castigation of altogether and we them in the simplest radicals setting which is that of a 1 . shiftsespecially .period presses and the soul which means that actually is received .period shift is just a a that an ecosystem on accounting measure and there would be no bells and whistles like exclaimed formation we just look at the configuration like like we did in the in the setting for instance on this 1 we just look at the point contravention to decide for 1 point on which makes went to go to case fair enough and there will be a few more conditions to be friendly .period shifts book so fully safe to introduce some of .period processes so the sources said accounting measures to measures Colorado measure also wants the obvious result accumulation of missiles on the of some locally finite digital value so 1 can be presented this new measure of the state of some of the at 7 points the end result final communication and sometimes the point some of the sometimes used in this report which is set of points with which that to be the case so that's what accounting measure aims some of them so you look at the space of counting mentioned something which they say you can put it is using magic wand that its intervention to political space and when you creep and with the fate of quarantine and so and sold let's coatings and sweet devising nephew of but on this 1 the space of countermeasures then you define a public space and what is the point was just 1 a modern viable ways values in the space and calligraphic and so on the Hong Kong Convention that's what upon processes and of Wessel .period was this week which appeared in the separate talks today is very simple examples what is especially .period was has sold in the setting of I will equip the party space with the floor the selection of of the of the addition will be on this point space which serve as the flying public is that if I look at my point was satisfied so it's a humble measure the face she still big of teteatete and look at the values it takes on be delayed taking fire will be gone on this set to be Constantine but and so is the this his view this would be the city gate so if you have as a steadily pointless says it will admit an intensity which would be the mean number of points so that it expects couldn't come back here OK so now comes at a very simple thing to understand what be which about review of especially .period was so William especially .period process duel so in good cases will evolution number of points and you can't make sense of what is a typical .period issue whether 5 them both .period he would take on the limitation on the parts and look up to the despondent imitation and that would be the point but the so what you do is lose following good think to define the quality which is the view of the pointlessness of .period processing from a typical .period bodegas definition which called about quality of an event a reason is isn't fall .period processor configurations .period you'll take any said to be with them digitally boiler a measure on the direct measure you'll "quotation mark will to order points in the city you check with other conservation scenes from this point and other property phases of what party on and and getting a again and and you divided a minimum minimum number of points in the 2nd and it can show that this doesn't depend on the city already because properties said the vigor of measure number generator and this is called about what way to define of for the point this view all the pointless as seen from a difficult OK and so the support of 1st of please you always contained in NC all his base of cutting measure reason Edelman diligent and inquiry because you should appoint process to all points at the end of the offices of the and the
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interpretation that they said there is a conflict can be seen that is mentioned with a 16 . to be caught conditional on our interpretation of the dissolution of the pointless as given that the origins including the point process In some cases made by is the whole distribution of the point and the the interpretation when the point was that we think is the amp illegal distribution of sequence fight complemented at year end for all appoints the head of fire in a lot became the and so now I can find .period his on HBO steadily .period positions so they said it the mass but incarnation of working with match match is it not each point of fight the pointlessness to another point of view without so this was that was going by the instead the much used in the recent work of Polaroid in Belarus which started in in the office and 5 and goes on the characters and you got positions bombs using location and OK fair enough so all of the soul let me put use them and they did that but he has said that I should add that the MoD initial motivation Allstate and Joycelin holloweyed and others and edges of thicket where essentially the the is Donald promised to bank calculus was here for motivation that I want to go home and assaulting algorithms will have led to the Soviets and the reason Digital childresistant tradition OK so let me introduce a few warned all examples and then we explain what the so the first one goes like that its goal is to .period shift to the party .period shift it was introduced in this paper valuable finale children and the minimum of tourism so it take so you take time off take a typical .period you here and you want to go to war again analysts school case awarded strip all of weeds to center of the .period OK which concluded diligent you look at this treatment of European defense .period industry that's the main point and and that's where you go and then you would be that that much soul between the 2 trajectories religiously so the 1 that they mentioned all the the symmetry of the 1 strictly Kenoza mixed midweek the woman mentioned is the AC direction .period shift so are you it is sent over here and the redeployment might have been and that's where you go and you did but he all the annoyance of surprising ones closes closes .period shift the points data in but like this 1 OK Seoul this is point at recess close .period this 1 and this is but the same 1 of reasons close 1 this 1 therefore I get an evaluation this 1 goes to this 1 and this 1 will go to the gate to the pollution and those points which are there are no Mitchell closes the book Dave but identity but creates it's a point shift its use scissors to decide on the local configuration seen from your configuration and decide where to go OK since Wenger it is the more fancy 1 which is used pretty much in the literature so this is the supercritical condoms reading sold this would be like the modeling the previous stock but in the percolation speculating Casey of a giant compliment OK and you apply on the staff only the idealism of which I call directional detection .period shift mainly in the main peak weight you goal tools that closes .period which is same east of you right and so long as the needle deputy coroner the digit jangle opponents is a very complex object with a lot of munchies and things which me and like this 1 and fucking address and you would get traps namely about the halt because it is my hope incandescent although there were structural instructor .period weather is basically nobody in the half and half discussion with the use of OK tradeoffs fields of few examples which give you the Riordan and now comes the definition of soul and suffered ecosystem the definition soul factor and and so on what is that's leading get .period Shiva said of it's a map on the support of the pointlessness to it's basically in those it goes in the whole nite it edge from any point to another point of point was perceive cooking and you you're interested in those things which are pulsation environment OK which affect most of the asserted that you think that they described earlier so and this is the formalism so the effect chief appointees the use effective which if you think it's a function best small "quotation mark appointment decision .period map which associates the 2 each .period in the support of of 5 OK s associates to but the configuration file was opened .period and simple bouquets and so this is the way did Major of the OK on it with this effort will be defined by balanced enough .period as it will you see where major was the always and OK and what do you do this to the fact of construction to Seawell .period deities mapped by the capital and what you do is the building .period tear into the vault by applying F composed by 2000 OK then this point as you will you apply and then you blink back appointed Chief Gates's he said construction and so are the result of that is that actually the image of a point is another point and if you want more of you know injured the .period process by any consolation than did image is more with this shift OK so this is this is a factor indeed any consistent viewpoint located Seoul neared the 1 more thing here which is the case although the did of St now think of the pointlessness so what you're seeing is that terminal in the needed on Belfast's when I was doing the picture of direction OK you'll be might prompt the picture this 1 OK Seoul and I think these 2 points that the same image came and I might not look at the point was says that they called F of 2 always older 1 folder time being all fight which are the set of points which entities 1 page that some points without payment can and marginally K is a set of points which have had escape reneges who cares and .period this point was this will be stationary namely to Teddy competence OK and this point was a factor most of which were reduced and in what follows OK so it is a good sign from the seventies by you wish to make it yeah will pulls that if there is an almost surely by 60 Fecteau .period shift then what whatever the .period persistent acts of it's associated .period map as preserved the pump would be this will create amazing result so mathematical if I push Of the bank would be used to buy today if by that keep the plant will so in the examples of that I gave only 1 of these .period shifts was with magic even old projector which it wasn't could close close neighbor can soul and send him a final to work assume I am about to be good .period and no 1 is my closest neighbor Phoebe cruises and had to use them from your neighbor and that then he faced the balance it a moved to you use the power OK so any by by 64 . 2 reservists the past cases look sole question now in order this interesting ideas and those which moved far away to move to Lost in You don't preserved you have no by sections of you don't presented upon the question of his with expression of what the answering of the frustration that they ask what happens on a hot OK so question the further I take F which is effective .period chiefly new navigation .period process which which is a function only .period process OK this is so with fractals all her of what is this set of quality measures which shoplifting valiant valued by 2 . 2 by the point not only to top of smaller as the case so far OK so don't make assailants a certificate presented his budget the vendor Parliament and the bundle and especially .period was the soft question but what happens in the interesting cases which were mind like this strip or direction
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indeed if you take the the the stick .period shifted skills at this picture this season coming you don't have projections we have 2 points and the darkening of in the same thing image by OK so
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now I can include the dynamical systems and the answer to this question and we legal In 1 of Enders said poverty measure the set of Compton Michelson and 1 of and not to settle 40 missions on dumping measures was not committed the wouldn't would use its image and process OK uh Weizman depressed for conducting to the enemy defend of the first one he's that the 1 which looks out good by ended this than legal system at a time and it is the views that you .period process once you have it debated the matter to the West End Times figure of moved and time and times and you look at that point was scene from your point because that's the tip that does that so it brings back this point offenders 0 In the 2nd 1 is the law of the land OK so you push all of you push should take care of our initial low you push that low by by 2 . 6 and the end of this year this will be a distribution than on and not new is still that OK so the ecosystem What is the configuration and then step see from where I am and medical system to be the role of the latter the dissolution of OK so so here is a very interesting goes in the bottom of the definition so the point with state to be Benedick on the accounting measure 5 is that additional P that may depend on 5 a a subset folk a large enough and you have classical integration selected to God L the death of although edifying listed that as of 11 and the 2nd and the parts more important sole example would be for instance 1 .period cases which were the tops that they mentioned OK so you remember when I move on duty when I move Monday Citicorp want onto the under 100 yards which is better I would end up in points where the continuing mentioned in that case the major without said float and make the .period being all several political model was so this were traps and the 2nd 1 which is about more important is the notion of an operation Wednesday said that about operation if this image .period processes of older care when giggles but tend to and the measure OK so but hasn't eagerly down 0 . squeezing the many pages and already examples of that we say that they mentioned like straight or direction or do without so this measure the sort of non basic TVT all the maps of the capital F map OK so it concentrates if it contemplates you have less and less .period perhaps more massive should keep the multiplicity but in the end you will see no point for and that we use this property is said to be eyeing the set of configuration on and on such that basically fall on individual and which is nonnegative is at least 1 clear of older vintages of 1 . 4 and defensible position however is that for all affected .period and all stationary point was his days of operation hold the key thing of fight uh and even on the eve the productivity is this was Cecil OK so
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exemplars of Paris utility mentioned closes cruisers to direction or 1 an exhibit of appellation so steep and direction or whatever .period was on .period was enough no I can define it in legal systems and want to see a few things and then the first one is that the felony consistent is jokes very nice at 1st glance because said it's very ecosystem on accounting measure which is available to put the cost at its met it's as Aziz said you can't do this and you and you can they compete with their vehicle is therefore you can never nice to political space and fortunately this map is not continues today it is not continues but unless joined dimension 1 will create soul therefore you can't use and at the rationality of the political dynamics and it's why we concentrated on those soldiers proposition for you molds and 2 toward equal to 2 there is no point mark set to take his continues in the whole of an and therefore you have to concentrate on the ecosystem to which is that of overall and to study it's something then makes so you push P 0 by 2 . according to PF once so this is the view also .period process see form bonds which have pleaded injuries of although 1 or more With multiplicity OK and emotionally if you push PFM minus1 but declined to give different something which is of course pleased offense and that the same interpretation OK and so I know what we have heard we have haven't would then be consistent the real we go to the direction of job preserving them ecosystems this To carry and so far that we defined an era began in a set of keys he warned of the action of all of these 2 titans style and any any element of the Armageddon said many of the on subsequence but in this set of many of the measure passed away because F portability all .period map portability of it particularly if the sequencer cottages and it doesn't 2 examples of the day's the and and the direction of the halting vengeance recalled the appointment quality of care and didn't conceal OK so inwards soul but you that that is not a good the president and inconsistent that if you look at the how the role is pushed by the by the shift to die the soul atrocity the measures on not to compact on the on accounting measures on not a compact said poverty measure also consumers of I'm not Compaq said so space so you we have to be careful maybe said exists but if any limit exists you cannot therefore but subsequent so there exemplars of noconfidence subsequently so it's sort of clear you go to the border with more and more O'Hara points along OK so let me know know the results under this equity so that if the map is Benedicta than actually you you don't go way beyond the pump many finds its effect is 1 billion making them the quality of PC OS the wife of his exists in that case it settled continues was picked to please you all and the huddle naked David sold in that case so a court fight as Infiniti D I saw as they told you in that case you look at the images this images of K trees and delivered to recall that Infiniti and Nissan Infiniti is another tonight and that carries the goal of 1 billion this year and this is not only going to direct so you look up at the multiplicity of the point of the origin of this economic the point was divided by which would be 1 the fuel pump was would be simple addition OK and any addition of indepth case but it is not very surprising busy with which is the survey the thing obscenity interconnections with Victor pitiable sauce makers a question fair enough and nor is it just in case because this is the case with an operation so the system 2nd the alliance if you assume that fight evaporates and the objection of assumes that the authority exists but assumed that it satisfies makers equation then necessarily please at his singular was suspected and pool relies on exactly what we have seen the news that under these you'll ever I suppose you've already won whereas under the 0 eulogies of what it is you who can OK so what exists and it exists in the investment in many interesting numbers it's an object which is completely different from the public OK and so let me know all the environmentally questioned the solution of that so you for that to reproduce this is all sums a with respect to the measure defined OK and so if this is all sums due converge then of last week or peaceable to up half generally limit of this is a whole oranges and the sea and states that are in if you're there exists but subsequent Sweazy Endesa has
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villages which converges to result Abkhazia were kidnaped and if the stars continues that deserved to die then this is village subsequent sauce makers look OK so let me say a few words so all we have done this construction on the street .period shift in direction .period Schaffel porcelain off too using this comical for property of personal when processes and in that case I sold .period deed of portability in question the that .period probability exists there is evaporation is uniquely defined and it's also because equation but it's something completely different from about measure so you have to understand that this thinking goes along the lines of 0 so you have to pull this infinitely and this past to see configurations the meeting of the emissions given by the weekend and you cannot see this week claiming that any local appointed when process so it is really a property such as yet infants OK so I can only find the formulation which is the 2nd and most intriguing object soul I come back to the initial distribution gives you all for your you could appoint was sold noise stopped looking at what happens attention to him and we look at something different which is indicative of a stable many Ford of the statement OK so let me define a little bit of what they need the no as a member of commercial space said and the matching which is so much of our own leaks Seoul I will introduce legal installation level and we had barely essentially they resisted sets sold my notion of recruits will be the support so he could use a to B. G equivalent to why these days is the need ,comma and says that he supplied the map g and plans to weeks all too widely and that at the same point His integration and sold the G 4 edition of inks call as kg of it is simply the cautioned of explained this Mexican Spanish the chickens class will be put forward and actually issues Inc a little bit about its authority is the enemy of the state where many foreign devoid of thinks the 74 of exquisite Victor G. OK so this is a set of points taking saying Look at this set of points of MySpace says that if I apply the dynamic at the same number of times to both .period signed up at the same event that the same fate will take the indictment exactly in that case is a case in point for the stable money for usually for and makes you would like that delineate of the different stances all but you will be diskettes spaces and we think this assumption so actually there is another article installation and partition soul you may say that leaks and wiring the same G connected component if they exist to individuals under you don't insist that they they meet at the same time so that you want to say that the same compliment you get used to it themselves mn and sets up the general view is that you know what OK and so is defiant another politician which is called a set of connected components and the partition of affordability fragment of the partition of copper sort of classical crossover take so began the final you want more used to that of Gothic recorded Capital G. we settled this is 6 inches between each extension and connected component of the and the physician is this a partition of the committee components based on what OK so and Europe's with affordable 6 and sees Unix the collected competent booking I got to fully share the city with a point not that we learn the use of both of them sold the first one is a partition of the support of so you fight and you read you will decomposes to of which is it's clear the set into its forests with respect to the capitalist map and is going to come but you can also work on ends you all known configuration and speed and which is used space In 2 In it's 2 guys pollution all its declared company OK the soldiers saw the Phoenicians optimistic that administratively than defense Fleisher won the support of acquittal .period wasn't so queso enough to mostly consumed for a share of the support of 5 and for village is the subject so the red object so this is a point OIG so you see on the map that goes to war on easy .period she's so you go on this trip you always find the next point because it's how it's defined bouquet of and if you look at the following indices a set of points which have exactly so too .period citing the same forward is by applying the same number of times that may depend of shelf .period that make you adapt to a common point in the future right and soul yeah you see their state lines of gold from points of the 42 to point of difference in front of 0 so of sub process and there is a complete partition of the whole point was this indicator is only 1 component into such foreign and so this is the year that this is a wonderful addition stands
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out and if you go to to perhaps the Williams tentatively entertainment OK so perhaps I will give
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Moses classification before going to so this is the this is the most important on that part which is a classification the fueled halting reason ballpoint maps it was the categories each year take a property components so you never know and on the exemplars that they took the reports on .period was is not too steep and directional is only 1 copy of the concludes that using about 1 ideas but they won't put up to gape at the general determination for each competency Of the GOP that I just mentioned this is the Cottagers sought on the street .period shift cares the move was gaffe Of all the affiliation of these scuffle defoliation of the .period map is in 1 of the 3 classes which offers 1st class it is the finite frantically so ceasefire not in each of his 4 is fine and it's likely 1 did is the 1st case and so are we give example in 2nd and a more precise than the 2nd classes I have seized invented in each of his 4 years finance soul which means that actually there will be a new all for instance the last day of classes and II case which is the most surprising 1 season she need and all its forces but spell out what these Fuerman season gave a few examples if if class so India 5th case but so is it said finite and each of his 4 is finite and actually the the existing digital and which would be the number of foreigners and you can pose at sea a unique cycle of links and so example of would particularly of the clothes may drop it introduces this it is the 2nd fantasies of the cycling season the revelation in that case and this is what said that the last line the servicing of nearest nearest .period changed all the rest .period chief also and those in his class when poor soul Nos .period chief ends up into the nearest neighbor newest bears which are the sort of looting cycles and you get sort of of finite arms that and then there was the 3rd to the information which is defined psychos it seems seems set on course was seized on leafy descending chains that actually composer of business operations for distributing cycles of the what remains of the pointlessness when you when you eat it if he OK so in a sense this is a classical consensus the case where you would end up with something which is the people you dig some of the some peace and therefore you end up with something which essentially bottom altered because of the 2 parliaments there's .period making a single law that OK 2nd case much more interesting already it's not the most interesting but I have sold their 2nd possibility is that season fitted and each of its and for his fine In that case sees the city as a click each Floyd a junior forward which is a guaranteed so the guilty of the deceptive for Azizi and not ended OK I that infinity of sees the EU is the unique but believe in the past is a unique being from past 0 1 Bassett goes on to observe of the hold of the treaty but in that case it's a beautifully and there is only 1 of its 2 to end the day at the past and this is if you anything you've seen and that it does not ever going to see because if introduces an attempt so and when it would example of that is difficult to see but actually on base on street .period shift you can hold that's the point shift as a single component the falsely .period his treat and you can't take a tree travel social history will mention as a single entity which is the result of the common ancestor of the water the tree which at infinity you don't see that all of the branches of case and therefore you can cover of this trees by any older yours I mean I have to take care of where you'll have a succession of flying all traversal of the treaty and did you get the fall but past that privacy is the whole point which is a byelection which is part of some of trees OK so it's a by section there was a point that case every for it is limited to a single point and the subject of infertility is just seats which is or when process for OK the example and that's the most interesting case because that's really In that case things until you only see yet Infiniti the Class II carries season Finley all insofar as I did then seized basically and Evans gives isn't so they have a point OK so now come in quite surprising things to accused of the 6 the point she's ever .period see even figure of Class II OK so I end so this is this it means the following things so as a even the fright the soul if appellation as soon as you point of origin you pause at necessarily there isn't fit and if there is a single compensate that there is an injury number of 410 and each for an is innocent for gay and you can also show that in that case the number of all pretty images of points is the notion that Seoul is exactly the notion of interpolation he is equivalent to 2 things you remember it was easier VAT the fight the of the pointless as isn't it all right that's few of images of financial donor OK so it means you have a single tree the tree as only 1 end at this Infiniti and all branches of fine OK interpolation is equivalent weight which states all of this stays on the 1 end and this is true in that case as soon as you have that you know that infinitely many foreigners with the older of those of a mentioned and each for his Infiniti governor OK so OK so let me conclude that we wins last season power which is so this resulted in the worst state its policy and he and and this will be a statement in the case and for that they have to go back
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to this is something that they might
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start middle by words and has quite amazing when you see them this picture soul this says the following so you can't know this past so it goes on this point as I point and this this points are the points of the 4th will carry on there was a fight which is a stable many foreign of Rapunzel OK and actually what happens is that because was preserved by motion by actually by section on the forehead and particularly by the motion erasable motion which is created by the to go from 1 point to the next something older so you would you would look at your mother's social issues say efforts to what your father you would look at your boss of 1st then you look at the 2 1st cousins than you go 2nd cousins and these kids at this case and all the on the floor in the UK and so the amazing fact what is that PC always preserved by this motion the ticket call effort of another vessel to the Noughties motion on the phone book which means that 5th 1 October at of is that with the can exemplars were affordable known is indeed there would be you can apply genetics alignment of and and the cost would be and for those will it with series reporting those who've amenable hopes and complex of villas sets to make coverage is indeed very fat saturated but with this show says is that all you can actually once you have the numbers achieved III clarity the objects the final confirmation which looks at the very least the past which is used for it and if you make a note of that particular item that for you end up with is the solution of course when the ceiling when he did that would lead to refine the expectation if it is a feast for this shift to the default of Eastern is not a good you will find a conditional expectation respecting significantly so you can apply at the costume order from wonderful which is quite surprised OK and so what is the last resort that they wanted to mention this dimension would default over the years because his motion and that forward and the result is here it's a new world order for the same dimensions soul Of let's put it that way so no L pink tulips on page 2 work continuation fire and let or n the unique conditions the assume that this to a continuation of the same forces that if you apply to their fault of another and times to the 1st to get a 2nd OK and that this several notice those things which are depicted my picture OK and so actually would you look at it is that on well you know so it means that you look at your cousin of older and on the fire this is this would be this would be the center of Europe and then you look at the images all eulogy in and did a major bypass often because of all the interviews on the next 1 OK and you let them go to unity and discipline which is delivered just because you and which I 2 say is that there is is that this limit exists and has limited knowledge of its positive and final civilian and 1 of his movies this fall but therefore it did so with the subject of a complex objects but in a sense that the same dimensions OK so if you look at get government he said this will be on many people are in between the majors off to take 2 points Friday to offend you look at the images and you will you will find limit which is which is known to located Seoul hunting with 10 minutes the sorts of
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course but I think this obligation
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to work carry burden to ensure interested in this book the party at the plant the Philippine question the results were based it is apparent in the U.S. Howard in the futures of OK so we will use this case here in the as U.S. assault they said sold for instance 1 conjecture so this was for to sees itself at this point maps are very chilled any consistent you don't have any of them that there is no single mom about that you're working on the media OK so you look at a controversial points to do something OK it's like your sequence of bits of on the medical system you multiplied by 2 1 news get something and not move more something and you can get it so this this academic awards and there is no longer but if you take the example of so we pulled down we pulled respect their tender and you'll need that this wireless incarnation of directional change when you tune earlier Aloha in the right way as a positive and when you tune them going the wrong way as a wilderness OK and so on we are struggling on understanding why and I think with this tour we might in the future it when we have will have extended their tools 2 .period pursues mocks the minisummit song unless we never handled once of this question I think that the answer tool for for instance the case where the evidence it is you will hear the case where the best .period did but not for the tee off but others said Shannon the maximum published novels and history would have the structure where all the writers you know points and on the left you see actually something very specific that we've managed to this is the I said the I said so there will be it's something like it was someone was since the tragic tho is it implements the I'd set whose name means that you have an easterly which is infinite OK but on the heights stays away to move OK Donnell points to move right soul this other situations likely to be collecting them in the so this would be when fading Israeli wasn't physical decay than there is no there is no way whatever wait sample the fading in your heart to move to 1 1 so this would be the picture could this conjecture of "quotation mark map portability for these wireless problems who cares we know where the point that the what would we have the wireless structures it's quite difficult to pull that and we have not done it but we have a very good and link up which is the the point man for all stadium directional voting result wireless encouraged Christian the OK so the if you only of offered to answer your question of the directions halting this board is empty the manager who could you move here and they didn't take aboard center here and I know that this is oriented for carrying an apartment next point here From here the Bornmann expungement brought here but this tho it isn't it OK so this tells you is that if you always denied out of number of times infinitely many timely receipt that sort of specially distribution and there will be a half and there will be some kind of which aren't just point and you will be in the pulled off 1 of the and what to do I know that no points there who claim their fault when you rebuild your border you will make progress and progress will be bigger send out of the park 3 days you'll hear what type of configuration of the medium you will be at infinity and dose used to compute tweed that's what we did the speed and don't get voted against Atletico speed is the equivalent of that's the view that sort of history and in this case useful course always not to staple Hardison quality that picture like that where you can trust the Polo you 2 pools at the object exist and they give you a handle on it and for all those where you have no progress where there's been this indeed the quality we expect to see objects of desire so therefore the duties very very talkative unit the 2nd think the gasification is just because as beautiful as it gives you the general structure that there won't be any other upward shift than the industry category right and discover the general so whatever .period process whatever there are only loosely categories and once you know that when you stop playing with objects in bunches are ubiquitous you find fantastic things right rights just because of the cargo the classification it's I don't think it's practical with its more than practical concerns excuses such a result work which which had seen in years the have fantastic number of voters currently we are an extension of the Constitution to union money or the from was among reverse some local soul solving the details of the violence of the the but there also has castigated seasons of artistic injuries and that you how practical they are themselves say the 1st part was partly small it's missile most of the potential consequences them With that you understand the authority you understand there standard speed Of the angle is OK and the acidity speed even nobody would bet that way this case the ascended the peace bid would be easier if you don't you know correctly nobody would that it was the result of situation we tried don't yet know that we are sure we have proof that it so that it clogs that that means that even severely fit you anything fades you got to any point which proved the big progress due to your destination fades hello the depend depend depending on how you tune Aloha there is a face transition you if you if you are always too aggressive subdued even as it is easier this is this is but to understand why specialty goods because cities are hot and Alyssa Milano the media you we voted on the media and and the universe as revolver medium you'll the medium changes because you have selected certain parts of the media and you end up in areas which are very special and that's a very special things that was ongoing process further story which we never seen possible that's the pursuant to the working and iv your blocks it's the only possibilities if you be skillfully fades and the and Anaheim's Odonnell pointed show and according to these results and the its regulations .period must know what is getting to see if this is the time around Station No I don't think so because the ministry said the easily dynamics the base of packet of pockets on them and so it's not in the mandate of the it's not just but it's not the 1st message preparations so it is in dynamics and his models to the dynamics of packets initially that's well magicians speed the and wearable brackets constantly and the thing is that they will end up being the result of a visible at all because you don't see them at any descends on initiate a their old markets we for the end up at the point Texas which is sort of a bit said so it's like nascent so severely that have been off pitcher with the going on in this this week in itself these book about this month the blessings of this module holdings are interesting in many applications including a homey because you you we won't have disobedience expecially with data you take decisions based on Saroken local information and they're not passing minute away might be opposed by the objects of the site the only thing is that we thought it would be for the II less cast shows up as soon as you have in the mood numbers and so if there is no called horrid and is economists and point was that you will be II class if you go somewhere OK so the idea class which is a the main the menu objective this line of thought is an amazing thing and it will be the norm for other reasons good cannot be local all down equipment so my hoping that they don't progress if you want to progress you have to be numbers at the Sherman directive as soon as you are the least speaking numbers the up case but it once you on a possible emphasis and that is enough so intimate as your amendment movement of which should cover the spot for most of the nation's maybe going the original the creation of new something of a similar pressures processes the release of a national phenomenon process so that is the essence of a good point very very essential to actually being of course so the subject no the subject is resisting the party is not at all of wanting positive hooker but it has lots of common things to eat for instance you pick 1 .period new peak 1 point here in the middle and you look at the descendants assault at a cost what happens on the right of the Indian sister calls for an ad that is very far out there and and so EDI case which you know is that you will have a finite number of descendants OK but the really is a tunnel so it goes to a dominant on his attention who but so it's opposite indeed if you look at the minute the mean number of points but also to be good note this mean evolve .period of the BankBoston until the 1 and actually marginally the number of dozens of older and he's also want OK so it's in terms of just about expectation there is no independence Day special coalition is extremely complex but it behaves like a pretty good watching posters with parameter 1 OK and he and his wife anything is possible and that's a general statement depending on the statistics and what you got so this case and we die it because I tasted we but will surely it's not the wanting process because they specialize if you have a special accommodations nevertheless he does on the characteristics in terms of means that as of wanting posters we so expectation exactly 1 it's widely depending on the final status May is also vital that India gazing died social OK on Net debate case you have exactly 1 this and in the fall you never died the yield a sensitive about and where you have to be cheap but the sources the hour this is an extension of the COE of critical wanting but we suspicious fragile because his fall any point but so you you may face in that way so you look at launching is all special bonding process is being formed you have only 1 facet right everybody has only 1 fast and and therefore you look at as sort of a gun tree descendants of something over and over and you know that you will go if you not a to land on which the region is indeed very only passed right but infinity and so there is a classification of launching processes of that died among the Sisley classic that's what it says if you want the case if you want to look at the glass and see it as a bonding was but it's a good energy and it is what it is what pushed us to look at the beauty model glass and for instance media downloaded them Watson please use model got but this is a vaginal gel silly because governors or dependency so what agents the act