Don't even act like you ain't know this was coming The YouTube team keep it clean what's going on is engraving here with another video and another episode of NFL questions from Subscribers and you knew as soon as you saw that Kyle Rudolph got released even before Kyle Rudolph got released when it was like buzz About him possibly being released a lot of Ravens fans was like But you knew as soon as it became official and we of course received several messages several DMs asking the question Should Kyle Rudolph become a Baltimore Raven should the Ravens pounce on this should they jump on this Opportunity and you know what let's just go ahead and get into it But before we get into it a question from subscribers is a series where you can ask me any NFL question Based off of any NFL team and we answered in video just like this now if you want the opportunity to be a part of NFL questions from Subscribers then you can send me an email to team. Keep it clean at gmail.com Don't send it any other way Don't send it to any other emails because if you do we can talk about it But it's not gonna be part of questions from subscribers But this special questions from subscribers is specifically on Kyle Rudolph and should the Ravens make the move So let's get into these questions first one came from my boy William J. He said Kyle Rudolph is Andrews 2.0 Yo, what's up in Graven? I just got I gotta get going right into it How do you feel about Kyle Rudolph being a Raven? He's got all of the tools to be the marvelous is number one weapon in this offense and surpass mark Andrews as target number one Or it could be just the joke that Andrews needs to set him into goat status. What are your thoughts? Have a good one boss and go Ravens. I Am I definitely wouldn't think that he could be the number one target in this offense He could be a significant target, but the number one I don't think he was a past mark Andrews And he would be a nice supplemental piece But bottom line for me the way I look at it if the Ravens got Kyle Andrews, yeah, that would be very Raven-esque of them But I would prefer them. I'll prefer them get somebody younger get somebody younger And I know he is a proven guy And I mean that could be nice because you could have Mark Andrews You have Nick Boyle, then you can have Kyle Rudolph. So you have a starter as your sort of third tight end. Oh, man That could be nice. That could be nice But um, I would just really want that that true field stretching tight end and And I mean he's been more of a playmaker than Nick Boyle, but he's also been involved more in offense than Nick Boyle Of course his role did change a bit last year because he did lose Kevin Stefansky to the Browns So honestly, I feel like he would hope that they would come calling first So he could reunite with his old head coach Kevin Stefansky and there's been some talk that the Browns could possibly move on from David Najoku So we'll see what happens there But I don't see him to answer your question about him becoming Lamar's number one weapon in this offense No, not that part. No. No, I can't get with that one. Next question came from my boy Joshua. Why he said hey, how's it going? I'm a fairly new subscriber, but a long time Ravens fan. Oh, welcome aboard team. Keep it clean He said I love your channel gives many fans a place to communicate and hear great ideas for the team Appreciate that. I was wondering what are your thoughts about Kyle Rudolph joining the Ravens after being released by the Vikings? Thanks for all you're doing best wishes to you and the fan Appreciate that Josh. So yeah, Kyle Rudolph if he came in here, like I said, nice nice supplemental piece And he could help a lot of the Ravens struggles at the tight end position And he could up the production at the tight end position Just looking at his numbers over the years Just last year he he started in 12 games and he had 28 catches for 334 yards So Vikings were like, hold up.  We can get that out of a rookie That's not any game-changing production. That's not any major production We can get that out of a rookie and if you look over the years of his career It's not like he's been producing a crazy amount of big numbers anything like that I mean his biggest season Was the 83 catches for 840 yards and seven touchdowns Mark Andrews. He bended that He bended that like that's that's old news for him and Mark Andrews still obviously a lot younger So, I mean if he could come in here coming to the Ravens with low Expectations would not expect him to be the guy and I mean again his role did change it with the Vikings So I don't think if he went to the Ravens, he'd be looking and he'd be like, oh, y'all got Mark Andrews He's getting ready to get paid soon. Oh y'all got Nick Boyle y'all paid him twice already. Oh I'm just I'm just here for the ride I'll get in where I could fit in I'll contribute where I can but I'm not I ain't gonna be Vikings Kyle Rudolph Hey, if you need me, I'm ready, but I'm not gonna be the guy as long as he went to the Ravens with that expectation Then it'll be cool and the last question on this episode of NFL questions from subscribers came from my boy the Sean Ready say a engraving. Hope all is well with you and the fam at this time of me typing this I saw that the dry kings just released Kyle Rudolph It seems like a Ravens type of move being though. He's a vet and not in his prime Sometimes we're just a wording of some of these questions may y'all be giving me man Because he could have said It seems like a Ravens type of move being though He's a tight end and the Ravens could go back to three tight end sets And he wouldn't count against the the comp pick formula He could have said that but he said being though.  So boom. There you go. Anyway He said I just want to know your opinion on him joining the Ravens personally. I wouldn't really mind that that's what he said Not me. Um, but yeah, I Like I said, I'd be I'd be cool with it I'd be cool with it And it is like if you were on Twitter yesterday when he got released you on Instagram yesterday when he got released Then yeah, you saw Ravens fans like oh here This is this is right up Ravens alley because it is it is and the thing with that too Why a lot of teams like going with veterans over young guys And you of course you got to have a mix of both But why a lot of teams like doing that is because it depends on if they're in win now mode or they just trying to get by the next season Ravens are obviously in win now mode. Um, and they're that Super Bowl window It's still open right now. It's been open for the past couple years How do we get a 14 14 and 2 team and we lose like that and the plushest thing about that but anyway, that Super Bowl window is still open still open and And That's why you will want to get guys that are ready right here right now to contribute Especially a certain positions other guys, of course, you still got a draft and have depth guys too but you want some guys that are ready and with with Nick Boyle coming back from injury you just You don't know how he's gonna be now They did sign Eric Tomlinson and I believe that was for that purpose for Nick Boyle as a just in case option like oh Just in case things ain't all the way there yet. We got we got Eric Tomlinson and he can be all back up So we'll count it off. Hey, if it happens cool, if it doesn't happen, trust me, I will be a okay There are other options. There are younger options. I mean, you can go through the draft You can get somebody with a little more speed. You still got Jacob Breeland.  You give him a shot But this yeah, this is one of those Ravens likes move. So will it happen?  So I would say but I would say the Browns have the one up on the Ravens on this move


In another development, the House of Representatives has urged the federal government to strongly condemn the coups in some part of West Africa, impose strong sanctions, and mobilize other nations and stakeholders to do the same. These followed the adoption of a motion of urgent public importance by Julius Ihovera, who had acquired a trend of military coups, describing the most recent in Burkina Faso as being part of a resurgence of a coup culture in West Africa. The lawmaker said if the trend is not immediately changed, it could erode democratic achievements and distort the emerging culture of constitutionalism. He added that it would promote opportunistic and undemocratic actors in the region and by extension the continent. Further notes that for the fourth time in only six months, Coz de Tar have precipitated the transfer of power in West Africa, most recently in Mali with the overthrow of President Ibrahim Qeeta, in charge with the takeover of power by the military following the death of President Idris Debi, in Guinea with the overthrow of President Alpha Konde, and in Burkina Faso a few days ago with the overthrow of President Raj Kabori. Also notes that this illegal takeover of power from democratically elected governments violates several national constitutions, international conventions and protocols established by multilateral organizations, donors, and development partners. The people are suffering. The manner of bad leadership that is pervading the African continent is something that we must look into while we are against military coup. It is now time for our leaders to leave up to expectation. It is now time for our leaders to ensure that the people, that they lead, that they are giving the effective, good governance. The worst form of democracy is far better than a benevolent dictatorship. For as such, we should always strive to jealously guard and protect our heart and democracy, especially in the West African suffrage.


Let me come to you Sanjeev, I mean, you know, you've probably got that style of funding right now, so it's probably very, very new with you having raised that big down from NASPERS. So, you know, as a social commerce company, and perhaps you are one of the first ones successfully doing it in India, how did it really come to happen for you? How did it work out? And I mean, you know, I know NASPERS from earlier and I know they, one of the main aims as a fund has always been to take companies global and to see that one idea can work in one market, it can work in other markets also. So, how do you propose to use this funding? And I mean, you know, what I've realized is that there are a lot of practices today which startups have for raising a CDJ or a pre-CDJ. But what about once one has raised a CDJ? How does this fund get deployed? You know, because obviously at right now the experimentation is now over, so you have to be a little more focused about how you use your funds. So, what are the best practices of deploying a CDJ investment to get bigger and better results for future? Those are a lot of questions, definitely. I'll try to answer them as I remember. So, the first one was probably how did we discover social commerce, why did we even start social commerce? Working on social commerce when there was no company in India doing it. I think one insight which has been very useful in hindsight is that ideally you should always pick a particular industry and then spend a lot of time there to kind of get hold of the core problem that that industry is facing. To give you example in our case, we didn't start with social commerce. We started with something which was hyper-local fashion marketplace. We used to now call it streaky for fashion, but what it just simply did was map stores around you and then you had option of trying before buying as a service. Now that didn't work out. We ran it for two, three months. We realized multiple problems. But the good thing about that was that we used to talk to those stockkeepers a lot and this was 2015 mid and they had started leveraging WhatsApp to stay connected to their customers. So, whenever they used to get new stuff and they would send it on WhatsApp saying, it's all new, if you're interested, buy it. So then we started exploring that social commerce is picking up now, where else is this happening? Definitely Facebook. There were a few shopping groups then with two lakh, three lakh members dedicated to buying and selling. So, both sellers and buyers added there, so it was a mini marketplace kind of thing. So that was when we realized that there is this thing called social commerce and this is picking up really well. The next step which is very important for startups to do is kind of before doing anything, building a product, even thinking of building a product, just talk to your customers, figure out what is their problem. So we had a lot of interactions with them. Then we realized that maybe payments is a problem for them. Maybe they need a micro website like Shopify. So then we built the first version, which was more like Shopify for India, which was only giving them micro websites. But then that scale decently well, not as good as we had expected. We went to Y Combinator with that. Then we realized, and again, talking to your customers is very important. So we kept talking to them, figuring out what are the new problems. And then we stumbled upon the core problem, which was for them to discover and fulfill the supply, because these smaller sellers on individual level have no leverage on suppliers. So how do you keep supplying fresh catalogs to them? So that is when the current Meshaw model that we are running now, that was born. So we did second pivot, which was in 2017 early, which is 5th 2017. And since then, we have started scaling really quickly. And that is after we raise Series A. So yeah, I think your second question, yeah, definitely. Yeah, Series A is decent amount of money. So I think in our case, we raised about 3.1 million lead investors. Before that, we already had a 40 people team. In tech, there were just five people, and I was a tech who founded. So yeah, it was a high performing team. And the remaining, whatever, 35, 40 people, all handling business, customer support, supplier support, everything. And so yeah, after that, we created dedicated teams, teams, and when I say team, it could be just a single guy, which could be just head of growth, which just started to build a team. Or in some cases, single-handedly pulling it. But yeah, so we then created dedicated teams for different verticals in the business, which could be, for example, growth, customer support, product. We brought the first product guy after Series A. Before that, we used to do it ourselves, yes, obviously. So yes, so I think creating dedicated verticals and figuring out and giving responsibilities to them, that is very important. I think the next thing, this happens, I think, when you're closing around as well probably, you have to kind of agree on certain metrics and projections of those metrics with your investors. So you have to kind of start tracking few metrics on a daily, weekly, monthly basis as they come. So identifying those metrics and making sure they are reflecting kind of long term growth of the business and not doing short term hacks. That is very important and I think making your investors agree to it and taking their buy-in is very important because at the end, it's your company, right? So you would probably, and you're spending so much time, very few people out there in the world would be kind of knowing, have such insights that you have about your company. So I think controlling the direction of the company is very important. And that too with taking your buying from your investors and definitely your team. So I think that becomes really important.


Felly, eingel, a wnaeth dataeth gydy'r wrth gweithio yn dechrau mewn gwirionedd y Comitee gan Niwethe Ffund Amendymint Scotland Bill. Fynt i ddim yn cael ei gweithio fel hyn i bwysig y sydd yn gymryd mwyaf o'r ffotoid. Yn mi yr ymarhwyng i ddim yn y dyma sydd yn rheid 점 yn eich gwestiynau ar gyfer mynedd ac yn ymgyrchiant agol. Beso'r dros honno, gan y CTC, rwyf meddwl iawn i'n gweld ar agwelch seяlineg mewn gweithio eich gymrydy o mynedd y mynedd. Felly, rwy'n dod yn ei gael i'w rhai allu a bryd, oherwydd ei sylfa ar hyn o'r hyn o'r clywed arall, fyddai ei fath o'i rhesymu. Felly, fe ddod y rhesymu, 9A.5.4A felly bod yn ei wneud am arweinydd o'r ddau'u gwahanol, ac mae'r ddweud o'r ni'n gweld i'w gwahanol a'r ddau'u gwahanol i'r ddau. Rhaid i'wheith 여러�od lle eu chweinidol wedi gwahod o hynny. I confirm that I do not have any declarable interests relevant to the remit of this committee and I also declare that I will act impartially in my capacity as a member of the writers to the Signate Dependence and Unity Fund Amendment Scotland Bill Committee and will base my decisions solely on the evidence and information provided to the committee. Can I now invite Alison Harris to make her declaration and declare her interests? I have no declarable interests and I declare that I will act impartially in my capacity as a member of the writers to the Signate Dependence and Unity Fund Amendment Scotland Bill Committee and will base decisions solely on the evidence and information provided to the committee. Can I now invite Tom Arthur to make his declaration? I can confirm that I have no declarable interests relevant to the remit of this committee however I will take this opportunity to amend the committee that I am the parliamentary liaison officer to the cabinet secretary for justice and I declare that I will act impartially in my capacity as a member of the writers to the Signate Dependence and Unity Fund Amendment Scotland Bill Committee and will base decisions solely on the evidence and information provided to the committee. Our second item today is to choose a convener. The Parliament has agreed that only members of the Scottish Conservative and Unionist party are eligible for nomination as convener of this committee. I am pleased to announce that Alison Harris is the Scottish Conservative and Unionist party nominee for this post. Are we agreed to choose Alison Harris as our convener? I congratulate Alison Harris on her appointment and I briefly suspend to hand over to the chair for the remainder of the proceedings. Cymru. I will now restart the meeting. Agenda item 3 is to decide whether to take agenda item 4 and the issues relating to our preliminary stage report in private. Agenda item 4 is to consider our approach to the scrutiny of this bill at the preliminary stage and to agree our work programme. The outcome of this decision will be publicly available when it is published on the bill web page on the Parliament's website. Are we agreed to take agenda item 4 in private? We are also to consider whether any future discussions in relation to issues relating to our preliminary stage report, including consideration of our draft report, should be taken in private. Are we agreed to consider any issues for our preliminary stage report and our draft preliminary stage report in private at future meetings? We now move on to agenda item 4 to consider our approach to scrutiny of the bill at the preliminary stage. As just agreed, we now move into private session. I will briefly suspend the meeting to allow any members of the public to leave the room.


Now the Electoral Hub under the leadership of Princes Harman-Obells has developed three research policy papers on citizens' right and impact of security on the 2023 general elections. Speaking at an event, the director of Electoral Hub, Princes Harman-Obells said the policy papers on the impact of security on the general elections will help to mitigate unusual situations during and after the election. At the preparation of elections, we need to showcase what public are doing. And I think since 2011, they've been doing their very best to prepare for elections, learning lessons from one election in a cycle to the other. But what has won't change is the environment of elections at this time. Why is that? I can just go on with the answers. So in this case, what we're going to see now are people trying to get at Ireland because of the period of the state to provide certain procedures that the state can provide. So my conclusion, therefore, is that, yes, we should be concerned about whether Ireland is prepared, but we should be more concerned about what the environment helps, what can do and I do to improve the environment within which elections are being conducted. Because that environment is beyond the control of Ireland. But in doing our own control, our citizens, to take action, to force government to comply in chapter three of our constitution, will provide a full space on the conditions for human security that all public authorities guarantee to the citizens. And I think our government about this might fit in doing that. Also speaking, senior research fellow, center for democracy and development, professor Adelaide Ginodou said that papers will provide political, economic and social environment and the state of Vynek in politics. To see how we can actually perform proper research, come up with recommendations that can guide all electoral stakeholders, not just Vynek. We've had to say that conduct for elections is not just Vynek. Being prepared for elections should be just Vynek for elections. Everybody has to be prepared for elections because everybody has a role to play. To ensure that elections are credible, elections have integrity and elections will accept it as the outcome of our will, as a will that meets that. Looking at insecurity and the election, it's very critical. And it's something that we should consider. And it's timely now because elections are just three years away. This calls for more participation and reflection in terms of looking at how we can we can make more conditions in terms of security and provide guides. So basically this means we're not the reason we're urging it. So see how we can perform some actually go recommendations that can guide security.


Okay. Decided I'd leave the full title there just to freak people out, or the full introduction. So my name is Gavin Henrick, and I do weird stuff with people who want to use Moodle. And one of the things that they often ask me is about competency frameworks, and well, this presentation is actually going to be you deciding what I present, because competency frameworks are fun, weird, and sometimes quite overwhelming. So what is the competency? Well, lots of people define it really differently. Some people define it as a skill at which someone has to excel. Some people define it as a metric which people are used. And it really sort of, it needs to be discussed in context. So what I've got is I've got all of these examples which I'm going to talk through. I've got 25 minutes, so let's say you'll do 20 minutes. I'm going to choose three of these, but you're going to choose two of them. So I'm going to do the last one, which is the Masters of Pharmacy. So hands up. How many people want me to go through the first one? The key competencies for sales representatives? One. Okay. That's good. That was from 2003. It had four levels. Pretty scary. What about the world-class sales competency model? No. Put your hands up for the ICE competency model? No. Okay. What is ICE? Okay. We'll start off with that one then. There you go. Okay. So one of the key things in the last five, six years that competency models have started doing is identifying that people are different, to some extent at least. And to organizations often will have a set of competencies that all of their staff should have, and they call them core competencies. And then some of them will go, well actually after that then they have their speciality or functional ones. And the ICE is that. So it's the intercontinental exchange where they've lots of exchanges. And they decided that these are the core competency areas or domains that all of their staff should have. And all of this sort of stuff is available online and I'll put this presentation into the moodle site afterwards. So if you then just take one of them, and so how it becomes a framework is you've got these different levels. So in just under teamwork, you then have stuff like here, offers assistance to colleagues or fosters friendly and cooperative relationships. And that's sort of one level. I mean it's quite an interesting one that these are core and if we just jump back up. So they break it down into communication, teamwork, professionalism, and then problem solving. So most jobs are going to have those. But then that you have to recognize that different functions of the business are going to have different requirements. So for example, if you're in operations or if you're in sales and marketing, and these will have different skill sets or requirements for that particular role. So if we take one of them, product development. So then they break that area down into three sections, technical knowledge, productivity, quality, and design and innovation. And unlike their core set, which was quite light, they go another level again. And this is basically having a domain or top level area at competency. And then what do you call behaviors underneath that. So we go into design and innovation. And you see then they're getting more specific. Now this is like obviously here, identifies and effectively applies emerging technologies. So these are really, really key as to how somebody actually operates in their job. But can you get a grade in that? I mean, how do you start assessing those kinds of things? And that's one of the challenges with frameworks. And each framework is completely different in that respect. So one of the things I really like about this framework is the fact that it breaks it down into business areas. So that each business area, apart from those four initial ones, can be very specialist and recognize that their staff and the things their staff need to do are fundamentally different. So have you ever got any thoughts on that? Do you like the way that's done? Do you dislike the way that's done? But before I should go on, probably should have asked at the beginning, who here uses competency frameworks in their organization currently? 1234567. Okay. So it is starting to be much more popular at government level in financial sector, within a lot of the business and sales marketing, and especially within the medical area. So I'll just jump down to the next example, take you through it. So technology is one of them. Now competencies apart from being used within an organization for basically assessing whether somebody is good at their job or what they need to be good at their job and identifying training gaps. Because that's primarily what it is. Here is a dark board of 20 different things and competencies you should have. You're missing three of them. That's where your CPD or your continual professional development should be. And that's one way competencies are used. However, equally competencies can be used for your whole curriculum and competency-based learning. And that's this example. So if we look here, it's Virginia CTE Resource Center. And it's a very recent published framework. And here you can see, wow, they've got a lot of sections for a start. And it goes through quite a lot of different things. And because it's a full course we're talking about, and it really is comprehensive. And if we just take one of them using web technologies, again, they're going down that one level. And you can see here that they're talking about explain how evaluate, develop. These are more functional rather than some of the others. We're within an organization frameworks which are used for sort of post-learning for the assessment and ongoing evaluation. They're more holistic in how they view things. But then you can go down into the next level. So here, with the first one, explain how to locate resources and references. So what they do then, is they define how that competency can be attained, which is different to what a behavior is of what you expect to see from somebody. So this one here literally does specify how the teacher will decide whether that student has attained the competency. And that's an interesting one. So how do you decide that? Well, how they look at it is a level of competency for that. So can they teach others? Can they perform without supervision? So this is a movement away from just grades into can someone actually do this job that they're being trained to do or do this individualized task? And I really like this one. To a point, because I feel that it will be much better if that had been broken apart into sort of a higher level as well, and grouping these up a little bit. Because I think this is a little bit flat. Although you've got exploring programming concepts using algorithmic procedures, I think I'd like to have seen this breaking up at sort of a higher level. Because then you could go for core and then specialist, depending on what area of programming they're actually going to be looking at. But again, what do people think of this type of competency framework, which is then used for sort of tracing your learning as you go as you're going from zero to hero? Yeah, that's exactly. I mean, this is basically then their curriculum. This is what they actually work towards. So there's nothing that they do, which isn't meeting one of these needs. And as you saw here, this is exactly defining how it should be done. This is really taking away the teacher autonomy to a certain extent. And so therefore CBL or competency based learning can work very well towards something which can be delivered with less facilitation. It's basically show me your stuff. Have you done it? It is standardizing it. But again, their whole framework, I'm obviously only showing you two examples of parts of it. I do think it's incredibly detailed. And I really liked it from a technical point of view. Then we look at teachers. Now, I remember when I was doing work, a lot of work on Mahara, people used competencies and evidence for competencies in teacher training. So this was one of the areas which I looked at. And this is a general teaching council in Northern Ireland, which is a different place to Southern Ireland for those who don't know.  Is that in the UK? No, we left 100 years ago. Scotland, come on, get your act together. Anyway, moving swiftly along. So they've got only 27 competencies. So not that level, which is like the one with sales, which I think at 300, the first one which we didn't show. And they're broken into three general areas, which again, this is nice. This is a more modern framework. It's recent. And it's recognizing the whole CPD aspect or CE. So continuing education as the thing from continual professional development. And they also have, it's newly a rubric against it. So you saw there where for competency based learning, you had these different sort of levels of competency or level of skill set. Here they're looking at sort of phase exemplars or a description around how you expect them to be at different parts of their career. So if we look at, this is the key thing, and this is really where competencies have started to evolve, because it is a continuum. You suddenly don't start from nothing and then become competent at everything or competent being a teacher, it's constantly evolving. The whole challenge of being a teacher is evolving. And therefore, your competency evolves and your requirements behind that evolve. So it continues throughout the career. I think that's really important to none of the state. And this is where competency based evaluations and assessments yearly, both self and through your industry become really interesting. And you probably can't see that very well. But I tried to break it into something which was a bit smaller and a bit more manageable. But so it starts off and it goes through the competency statement, the aspect of the competency and then the different phases. So initial teacher education, like no one understand their commitment to learners, colleagues and other professions. And it moves then through induction, early professional development to continuing professional development, where they have to be prepared to proactively fulfill their commitments to learners, etc. So it's this continuum and it's clearly defined. And when you compare it to some of the early competency frameworks from 10, 15, even 20 years ago, they were much more, yes, no, they lacked this depth, this richness in understanding of what frameworks and competency and CPD really can be. And if we just take one of them, I mean, you can monitoring the pupils progress. Again, you've got these phases. I think it recognizes that what teachers need to be doing at each stage. So you could look at a teacher and nearly identify where they are on each competency. Are they still still at the induction phase? Because maybe they haven't had the chance to explore that aspect of their of their ability or of their skill set in that role. And that's one of the key things. Just because you have this competency framework, it doesn't mean that you're going to be able to develop every aspect of it every year. When once you qualify or once you're being continually and developing yourself, because not every job is the same. So for example, a pharmacist working in a pharmacy versus industrial research or a hospital, they're completely different in what they do on a daily basis. So, oops, come on. I'll briefly go through this one. This is another interesting one. This is used, the C-MALT is ALT, it's the Association of Learning Technologists in the UK. And this is our sort of professional standards framework for becoming a C-MALT sort of registered learning or certified learning professional. But it's peer based. So it's introducing that other aspect. It's not university based, but it is a peer based framework where you're actually creating the equivalent of an e-portfolio but against this whole competency system and professional values and so on. And again, it's about what type of evidence you produce against it. But again, it's a disease sort of phases are these levels. So if you were to look at when they actually then talk about it, it's like how to write the CPT, because this is where it's used on an ongoing basis, post graduation. And again, it's very clearly and reflection is a key part of this. So it's not as, this is where it's well beyond grades as such. Now, one of the ones which I was involved in, one of my clients came to me and said, Gavin, we've got this master's in pharmacy. And every year, these students go out into the workplace as part of their master's. And we have this competency framework that they have to self assess against up to six times a year. And their tutors have to assess them six times, up to six times a year. So I had a look at this document. And this is a very mature, it's a recent framework, but it's driven from the PSI, which is a regulatory one. And they've got a very, very rich bit of research behind it, but they break down a pharmacist into six set areas. And then when they go into that, each one of them has a set of competencies. And in each one of those competencies has a set of behaviors. So I think there's around 132 behaviors in total. But the behaviors are quite nicely done. So demonstrate a patient centered approach. But if you then look at the other few frameworks, we sort of looked at, okay, how do you then assess that? So they, they're sort of halfway between the two different systems of assessment on this. And this is for the work placement phase. So whether they are rarely meeting, whether they're sometimes mostly, or they're doing it consistently. So literally, six times a year, the learner goes in, and he chooses a drop down for every single one of these. There's not really many creative ways this can be visually enticing.  And, but when the student has done it, then the tutor can see what the student has done, and then they put in what they believe the student has achieved, and then they can discuss it. And this is in phases. And the very last one is a summative one where they both sit down and talk about it at the same time. So because, and this was done in Moodle, so we built a custom thing for them because Moodle doesn't really do competencies currently in this way. They could have used outcomes that would have been very difficult to have the students being able to assess at this level. But one of the reasons that they were using, they wanted to do in Moodle was they wanted to be able to leverage the whole Moodle logging and reporting systems within it. And so then this is actually how they were able to demonstrate to the students on a domain competency and behaviour level how they were able to develop their skill set over a year. So that you could literally see and compare it between the student and the tutor. So that they were able to see where they are in a sort of a radar based compared to the others. And then at the end of it they just dump the summative stuff out of it and they use that for the regulatory reporting and so on. So any thoughts or questions so far? Well this is the Moodle activity. So basically the way that they did this one was they have every tutor and student in a group by themselves with different roles and that role has different capabilities. So a student will go in and he will fill in their and their assessment and then a tutor will go in and see what students they have and be able to go in and then continue it. And if they have to swap tutors you just put a different tutor into the group and so on. So it becomes a it was just an activity and they wanted it within Moodle rather than building a standalone tool which probably would have been a lot cheaper and easier because they currently use Moodle for their whole masters. All of their courses, all of their tuition and assessment is done through Moodle so they wanted it to be done that way. So that was just an example of again if you have a framework and at the moment Moodle doesn't have something although going to come into that in a moment. Now any other questions before I move on? Everyone's still awake? No? Okay. Yeah. I mean frameworks are one of the most interesting aspects of education. Okay. A few chuckles. So I can't take it anymore. I'm out of here. Where's the pub? I'm so going to need to get a bit of it. Anyway so one of the interesting things that I've seen that even if you don't use competencies in your course I believe you can but what do you use them for? So one of the institutions that I've also done work with have started looking at it's like these business readiness or workplace readiness skills and these are just some examples. So every institution probably has their own set of 10 or 12 of these. What a student should be when they graduate apart from their qualification. These are just things that they have picked up along the way and you know make a cup of coffee, tea, get the paper. You know the important things and so these are just some of the examples and it's becoming much more important because businesses are looking for this. So even though the grades are what the modules give the students, one of the areas that's now being actively looked at and especially in Europe is using competencies to report on these kind of soft skills which are really valuable to give a report to the student to say this is how you have progressed and if you think of that and that sometimes rarely always kind of scale of how they demonstrate these that's the kind of thing you're looking at. Some use badges for these but I know quite a few who are looking forward to see what Moodle does with regard to competencies for that. So what about Moodle? Well Moodle has Moodle outcomes. Who here has been using Moodle outcomes in their in their courses? One, two, do I see three, four, five, okay going once, six six there going once, going twice, sold to the lady in the corner. So I love Moodle outcomes. I think they're absolutely brilliant. It is a start and you can still do a lot with them already by assigning them into different assessments and activities and so. I think that people just don't think about them because they are graded. They do block up the grade book a little bit but there's ways around that too because I mean if you see here you can, did they show up just right beside any of the actual items within it. So that's that's what they look like at the moment. So you also have a separate outcomes report which you which you can look at as well. And when you actually come to grade them so rather than just having the grade points you can actually then have these drop-downs for whatever scales you have selected for the outcomes. Of course outcomes can have different scales for each one. But at the moment there's a piece of work and a bit of research going on about competency-based learning for Moodle. So I thought I'd just drop this in here. So there's a specification on Moodle dots and there's a discussion on the specification and the requirements around it and there's also a survey. Martin is that survey still ongoing? Okay. Do you want to very quickly? Do you want to explain where it is at the moment Okay. Yeah. Yeah. I mean I think one of the key things is even if you just look at the examples I've shown you competencies can be used in so many different ways. And so having a robust framework management within Moodle will then give plugin developers an opportunity to how to leverage those competencies in different areas. So and that's really it. I just wanted to zip through this. I can always jump back. How much time do I have left? Quite a bit actually. Eight minutes. Okay. Well I'll jump back up and I'll give you one of the others. That's fine. Well if anyone has a question you want to ask now before I go through one of these. Anyone else got any questions? Anyone got any thoughts? Does it frighten you?  Can you just imagine having two? Let's go up to this one here. So this is one from 2003 which is a CPSA initiative at the Sales Institute. And so each one of these then has a set of competencies for it. And then you drop down another level and it's got another four, five, six. And then you drop down another level. So the pharmacy one at three levels, 132. This one has four levels. And the poor sales guy has to prove all of that. They've got enough on their plate selling but it's an interesting one. I mean obviously I would imagine that's 12 years old. I haven't seen an updated version but one of the reasons I showed this one here was this is a recent research based one where they're starting to try and make it look pretty but actually it's still got the same amount of areas. But it does look pretty doesn't it? But if you start counting all those little areas there's lots of them that there's lots of them there. But which shows that sales people are just really multi-faceted and incredibly talented in so many areas. So go on, questions please? No, fine. Well you see that's the thing. Sometimes that's not possible. So for example if you take about the sales guy a lot of those are practical ones and so therefore those competencies actually might be assessed in the workplace. It might be something where a sales that the mentor of a sales guy will have a form when he's sitting in on a sales call or a sales visit and he's filling in this form to say that the guy has yes he asked that question yes he did need analysis correctly yes he used reflective practice in in that and yes he returned got a sales quote he did all these different behaviors during it. So a lot of them in these frameworks are not something which you have evidence for apart from somebody else's word. In the ones where you self-assess again that's not really an evidence based. So it's a good question. It's a question is is there activities in Moodle which can represent them. So maybe an offline assignment or assignment which is has its information taken offline? I don't know. Yeah well I have seen occasions where they actually have done self-assessment and then based on that self-assessment it actually then shows up okay these are the six weakest areas that you have this is what you should try and focus on so it creates a dynamic learning plan or individualized learning plan for a student for the coming year and I think that's one of the really strong areas for it and that's actually I think where the CBL within Moodle could work well because then they can record evidence against those. Yeah yeah you know I mean again that's about once you have a framework management system that can manage these multiple levels and what they're called like some are domains some are categories some are competencies some are behaviors some are you know every single one of these depending on what culture depending on what specialization they're all called different things. Anyone else? Well basically the way that the Moodle framework is being proposed at the moment is that you would just be able to create a spreadsheet to upload the framework into Moodle and then decide how that is to be used just because I mean a framework as I said can be used for learning can be used for evaluation or assessment depending what plugins and what other stuff is around it and that's part of the survey so I strongly suggest read the specification take part in the discussion if you use competencies it's really important that the different levels are all contributed in there and take part in the next survey I think it's very valuable I mean it's a clear example of where Moodle is and Moodle HQ and Martin keep your ears closed actually is actively demanding from the community help direct us because they're not they they aren't using competence on a daily basis seven of you here are within your industry within your within your institutions and organizations so you're the ones who need to pass back that expertise you know I think actually I mean that's where I think it's really important that once you have that framework concept in there it's then people can leverage it much better so for example in that visualization tool I showed you we had to build our own competency framework management system into Moodle so when they create the activity in the Moodle course they upload the full framework as a CSV just within the one activity so they can literally have and again because in pharmacy you have core competencies and then you also have specialties now it's coming in as well so that you could have four activities in the Moodle course with different frameworks if you wanted to so we won't need to do that after Moodle has implemented it we'd be able to just point at that framework over there please which would have made development half the cost I think that's the key it's when it's there it's like a foundation stone for you to go I want to use it this way and then build it so and this is the page I mean I'm going to put all these links I want to tweak them out now in a moment anyway but this literally goes through as Martin oops I guess if I do that one nope here hold on yeah where is your your function oh it's function eight duplication there you go okay so there we go oh there we go so this is for example what Martin was talking about there that these are just it's a specification an example of how people could upload it so this is where it might be CBL where they're uploading evidence to support their claim that hey I got that competency and here's my proof or this is the assignment that I did which proves I did it so it's again but that's just one one way it could be a self-assessment tool that you build could be anything so read through it this is it's meaty I mean if you look here there's a lot in it but if competencies is your game invest the time okay anyway hopefully that didn't bore you too much thank you thank you Gavin


Donc, aujourd'hui, il n'y aura pas de spécificité homogèneuse, donc il y aura un spécificité projectif, mais pas de spécificité homogèneuse, pas de cochon de Jimo de Lambda. Et ce sera un talk about stationary measures. First, generalities about stationary measures and Markov operators. And then, if we have time enough, I will discuss precisely the applications of the notion of stationary measures to result in the study of product of random matrices. The purpose of the introduction about stationary measures is two-fold. First, to introduce some general tools that we will need in the study of dynamics on homogenous spaces. And also to prove results about product of random matrices that we will also use in our study of homogenous dynamics. Ok, so let me recall you the general definition I gave you last time, so assume X is a locally compact space. And I will assume that it is second countable, which means, so this is very practical when you give the talk in English because you have this notion of being second countable. And there is no classical translation in French, so it's better. In French, it's something like metrizable with a sigma compact or something like that. Ok, so anyway, this is what it means. And also, I will define Markov-Feller. Markov chain, Markov-Feller chain, maybe, ok. C'est une réserve. Sorry? Maybe Feller Markov chain. Markov, ok, so Markov-Feller Markov chain, so everybody is happy. Or Feller Markov chain, so you choose the one you prefer. And this is a map X, map to PX, which goes from X to the set of probability measures. So I will denote usually by this P of X, I will denote the space of Borel probability measures on a locally compact space. And I will assume that this map, so I will ask for this map to satisfy the natural continuity property. So for example, such that, for every phi, so I looked at it carefully. Yes, I will add it, this is more or less, it's not very important. I assume that for any phi which is continuous and bounded on X, P phi of X, which is the integral over X of phi with respect to the probability measure dPX, this is continuous in X. So this is, so for me this is the random analog of a dynamical system, that is I have my space, it is maybe not compact. And when I start from a point, I will go to another point, but this point is not deterministic. Usually, for example, you can look, for example, you can take PX to be the Dirac mass at TX, where T is a continuous map. Ok, so, but now I will allow this point, this image to be a random image, which means that from X I will go to a point now, but which is random, that is distributed with respect to a probability measure. So, in fact, we could define the notion, for all I will say in general, I want to use continuity. So we could define the notion of a Borel Markov chain and everything would work, but maybe this is not very useful because all of the Markov chain we will encounter will be of this form. All I will say is that fact only depends of the Borel structure of X. So I could take X to be a standard Borel space, etc. It's not very, but this is just abstract formalism. So in this case, one would like to define, so one has, in this situation, so what I introduced is very classical in probability theory, one has a probability measure on the images of X. But one can look at the second image of X. So this gives a new probability measure. So what is this probability measure? This is average. So if I want to look at the distribution of the second image of X, the probability measure I get is something like the integral of Py, dPX of Y. So this is the distribution of the probability measure of the second image. And if I want to look, this is the distribution of the second image. But I can also look at the joint distribution of the two first images. So this is a probability measure on X times X, and this is just the joint distribution of the first two images. So this is the probability measure on this space whose projection on the first factor is Px and whose projection on the second factor is this probability measure. Ok? Because if I forget the first image, this is what I get. So in fact, this probability measure is integral of delta Y. When I take the first image to be Y, the second image is distributed with respect to Py. This is what I use there. So this probability measure is a fiber measure. This projection here is Py and the measure is Px and the measure on the fiber associated to Y is Py. So this is what I write formally as this measure. Ok? This is the joint distribution of the two first images. And now I can interact with this process. Ok? So I won't write the formulae, but you can look at the joint distribution of all the images. Ok? And this measure exists. This is the theorem. This is not very difficult. This is a standard fact from abstract measure theory in this situation. From the data of the distribution of the first image and then by a standard iteration process, I can define the joint distribution of all the images. So this will be a measure on the site W. So W is a set of trajectories. So I take copies of X and I want to put the distribution of the trajectories which starts from X. So there exists on this set, there exists a map which goes, so I keep this set with its product sigma, the product sigma algebra of the Borel sigma algebra of X. What I claim is that there exists a map from X to the probability measures on W which associate to one point, so that exists even in unique, which associate to one point a measure which I will denote by omega X. So the distribution of the trajectories which starts from X. And it has to satisfy an equation which is precisely, so this is a proposition. And it has to satisfy an equation which is as follows. When I look at omega X, this is the integral, the integral, the integral, I don't know, anyway it's not good English, so we don't care. So it is of what, yes it is, sorry, omega X satisfies an equation. It is, so the first coordinate, you start from X. So the coordinate at time zero is X. And then what you get is just the integral over X of the probability measure. When you look, the further coordinates, if you start from Y, they are distributed with respect, the sequence of the trajectories is distributed with respect to this probability measure. But I have to put another H over the X of Y. So this is a map that is Borel and which satisfies this property. And this is just, you write the natural distance distribution of the n-first coordinate and by standard arguments about measures and product spaces, you can prove that you have a limit and it satisfies this property. So this is called Markov measures. So in probability theory, they are denoted by this letter. This is by px. But sometimes I find this notation little confusing. So I like to denote it this way. My cursor doesn't share this point of view which makes sometimes communication difficult. So anyway, there exists such a map that is a distribution of the trajectories which starts from X. And in general, now if nu is a radon-measure, I will come to this point in one minute. If nu is a radon-measure on X, then I will denote by omega nu. We set omega nu as being the average of X of omega X dnu of X. So this is now, this is a measure, this is a measure, a radon-measure on W. Ok? So that is the issue. Now, this is not the distribution. You have the range, the starting point. You take a starting point that is itself distributed with the measure. Ok? And the proposition will answer the question. For any such nu, we have equivalence between saying that p, this measure, is invariant. So if you have an operator on functions, you have an adjoint operation on measure. Ok? So saying that the measure is invariant is exactly saying that omega nu is shift invariant. That is now, your measure, nu, nu is a measure on the space. Ok? So on the space, you have deterministic dynamics. You have a random dynamics. So for the measure to be invariant, saying that the measure is invariant is saying that it is invariant on average. Ok? So by the dynamics. Ok? So, when the Markov operator comes from the action of a group equipped with a measure, this is the property of being stationary. Ok? But now, this property of being invariant by the random dynamic, to the random dynamical system with a shift, you can associate deterministic dynamical system. You look at the shift map on the space of trajectories. That is when you fix all the trajectories. Now, the dynamics become deterministic and the dynamics is just forgetting the first letter. So, if you start with a stationary an invariant measure by the random dynamic, you can associate to it a natural measure on the space of trajectories and the property of invariant under the operator equivalent to the property that the measure on the space of trajectories is invariant by the shift map. Also, in general, this is not very useful because the shift space is a very large space. But this tells you why properties that are true for deterministic dynamics such as classification the fact that every finite measure is an average of ergodic finite measures, etc. Directly, we also hold random dynamics for Markov operators just because you have this nice bijection here. Ok? Be careful that I'm not saying that if you take any measure that is invariant under the shift map you will recover a stationary measure. This is just true for measures which have a very particular form, measures that are obtained by this process. Ok, so now we will speak precisely about ergodicity. So, I will say a definition. I will say that if I have a function I will say that nu is p, if nu is a measure that is invariant under the operator we say that nu is p ergodic if and only if for any function phi which is bounded and invariant then phi is constant almost everywhere, nu almost everywhere. Ok? So this is really analog to the usual notion of ergodicity for for transformations which preserve a measure and now again I claim that there is a link between the ergodicity of p with respect to nu with the ergodicity of the shift map t with respect to omega nu. So, if I want to draw a link I need to draw a link between functions that are p invariant which are sometimes called this is such a function sometimes it is called p-harmonic function Ok? So I need to draw a link between functions that are harmonic and functions and the shift space that are t invariant and this is not a direct link it needs the construction so the construction relies on martingale convergence or m so we need some more probability theory so to there is a general fact general fact that to such to a harmonic function you can associate a martingale so this is what I will construct now so a martingale so I will remember to remember nu what a martingale is so if you have say a probable x a mu which is a probability space and if bn is a sequence of sub-sigma algebras of a which is increasing I will say that the sequence phi n in the L1 space of the probability measure is a martingale with respect to the sequence of sigma algebras if and only if for any n when I take the conditional expectation of phi n plus 1 with respect to bn this is phi n so you have you have experiments successive experiments and at each step you measure the result so the space x is the space of all the possible results and at each step you measure what you have so you determine some information so you restrict what is possible in the space but when on average when you measure the result of the experiment what you will get at step n plus 1 on average is what you had with probability 1,5 it gives you 1 and with probability 1,5 it gives you minus 1 you make the sum of all the other results since you have zero expectation if you look at the sum it satisfies this property but it is a martingale for example you can take at each step you can change the distribution as if you keep zero average and you still have a martingale ok? and there will come at once an example which is not of this form so if you have this property so there is a CRM so of course if you take a function I will denote by b the sub sigma algebra generated by all my intermediate sub algebras so of course if you have a function for example then if you take its conditional expectations with respect to the algebras bn it gives you a martingale and the CRM is kind of reciprocal to this statement so assume you have a phi n which is included in the LP space for p which is larger than 1 and that it is uniformly bounded and that it is a martingale then there exists some phi in LP such that for any n has a conditional expectation so this is a doob CRM of phi with respect to bn is equal to phi n ok? so there is a CRM is false for a p equal 1 and I'm not sure I can give you a counter example at once but I know that this is false for p equal 1 you need p to be larger than 1 for as it won't be a problem ok? but for p equal 1 it is not true I should write the sequence on the other side and phi n goes to phi mu almost everywhere phi nx of x converges to phi of x you have also convergence in the LP space except of course if p is infinity if p is infinity in general ok so this is a general fact from probability theory and now my claim is that from so I denote all the functions by phi this is not a very good idea I'm sorry my claim is that from probability space W équipé with Markov measure omega x for a given x and given harmonic functions I will define a martingale this is a very simple formula define phi n so this is a different phi this one is var phi and this is phi phi n of x of a sequence so this is a function on this space this will be my probability space ok I should define also the sigma algebra at W n this is the sigma algebra of functions this is very natural which depends on the n first coordinate so I set phi n of x of W just as being phi of W n ok so I have my point x then I throw the random walk so there is some W1 W2 I reach W n and I measure the value of the function so if the dynamics is deterministic saying that the function is harmonic is just saying that it is invariant of the dynamics so this is constant when the dynamics just comes when you have a deterministic dynamics that is you just have one map then since it is invariant what you measure as time n is what you had at time 0 ok of course you lose this when the dynamics is random but you keep the fact that this is a martingale and this is almost I claim that this is a martingale with respect to this sequence of sigma algebras and to this I need a formulae for computing conditional expectations so and then to apply this formula in this case for this function so indeed just and the formula for computing conditional expectations will come from the construction of the major omega x so this is not a product measure when you have a product measure it is easy to compute conditional expectations if you have a space x which is a product of 2 space y and z and you have a major nu a major nu, you make the product measure you have if you have a sigma algebras that is the product the tensor product of the given sigma algebras on y and z and you can see y as the sub algebras of x so if you compute the conditional expectation of a function of the 2 variables with respect to this sigma algebras y that is you want to average against functions that only depend of the coordinate y and what you get is that this is so you should evaluate it at some point y, this is now, this is only a function of y this is just the integral this is fubini sur m this is conditional expectation is a product case but you see what you need for this formula to hold is not really a product measure what you need is a fiber measure you have a measure, here you see this as a base here you see this as a fiber and you have a projection measure this is mu on the base and then you have a range on each fiber ok so of course, here the measure does not depend on the fiber but when the measure depends on the fiber which is the point of y, you still have a formula of this form what happens here is that so the first coordinate is x and if I fix a second coordinate as being y I very well know the measure on the fiber this is just the mark of measure of y this is what the formula says ok so if I do this at step n excuse me if I have a function phi which is a function L1 in L1 of this measure, here I have a formula for conditional expectation with respect to Wn which just comes from this property here the iterated version at time n of this property this is the fact that when you take conditional expectation with respect to Wn evaluated at some point so now I don't have to put the additional coordinates because this is just a function on the n first coordinate n plus 1 first ok so I have to remember the formula this mark of property the fact that this is just the integral over W over of phi evaluated at Wn Wn minus 1 et now I put some new point W prime, so this is fixed and I take this integral with respect to the mark of measure at Wn so why do so this formula this formula, these points are fixed and I want to build a new sequence this is a function on the set this function phi is a function on the set of sequences ok, and how do I build a new sequence I take my n minus 1 first coordinate and then I add a random sequence here and this random sequence is distributed with respect to the mark of measure at the last step ok, so this is exactly what they probably say when they say that this is a mark of process what happens after the end step only depends on where you are ok, the distribution you have the first step but the distribution of the future only depends on the point I have reached it does not depend on the past ok, this is why I have such a nice formula that is the distribution here this measure for which this is a conditional expectation always looks as something like that but the point here is that the measure that I put here it should depend on everything but since it is in Markov it only depends on the last step ok this is a Markov property step pour the end one is just a formula this formula here ok, so now I have a formula for conditional expectation and I can check that my sequence here is a martingale this is just this is just the fact that the function is harmonic so this is very abstract and formal probability theory but this will stop soon because it is just what I need why is this a martingale so I need to compute the conditional expectation of phi n plus 1 x respect to the sigma algebra Wn and to evaluate it at some point of this form so I take my formula this is just integral over W of phi n plus 1x but now what is this phi n plus 1x by definition this is just a small phi the function, the pin harmonic function evaluate it at coordinate Wn plus 1 and there is a prime so sorry this is not n plus 1 sorry now this is the coordinate 1 of the sequence W prime ok, it's just my formula and you have an integral so this is equal to the integral over W, large W of phi of W1 prime d omega omega n of W prime so W prime is distributed with respect to the Markov measure at Wn ok, and I only look at the first component ok, so by definition of the Markov measure the distribution of step 1 is just given by the measure p Wn so this is the integral of phi of say y dp this is just the definition of the Markov measure ok, that is what I'm saying is that I've reached step Wn and then there is a guy Wn plus 1 which I denoted by W1 prime at step Wn I integrate over all the possible values so this is just this is exactly p phi this is a very general fact you should take this function and what I'm saying is that if I look at the function phi evaluated at Wn plus 1 and if I take the average of all the possible values when Wn is fixed what I get is just p phi of Wn and this is exactly Wn which is phi nx of W so my sequence is a martingale that is when you have random invariance you build martingale when you have deterministic invariance you have constant functions along the trajectories when you have random invariance you have martingale along the trajectories this is really the key point is the Markov property now since I have a martingale and of course since my function I took, I was careful and I took a bounded function so what I have is a bounded martingale a uniformly bounded martingale so by doob's your M I know that it admits a limit that is when you take a harmonic function when you follow a trajectory there is a limit so by doob's your M there exists some function phi which is the function phi of x which belongs to an infinity of W equipped with a major omega x such that omega x almost surely phi of Wn converges to a phi x of W so I cheated you a little because from the beginning I use the fact that the property p phi equals phi holds everywhere but it does not hold everywhere my function is an invariant element of L infinity of x nu so this property holds nu almost everywhere with respect to nu which means that the conclusion I wrote here is not true for any x nu almost every x that is on a set of major 1 so for the moment nu is not finite so on the set whose complement has major 0 for nu when has this property so precisely the function x so the function which means nu for the moment I didn't assume nu to have total major 1 because I will apply this to random works on groups in 1 minute and when the group is not compact the natural invariant major is a hard major which is infinite I want to keep in mind that what I'm saying holds when you study random works on groups so I have the property so phi x in fact the dependence on x holds is present in the dependence on the first variable the first variable here is x ok and as you have phi of w which is the limit of phi of w n in particular phi is shift invariant ok so I define phi as being the Poisson transform of phi ok and the claim all this is very general there is nothing just formalism about Markov preparators ok I don't note maybe neveu maybe neveu the Poisson transform so this is a map that goes from infinity x with respect to the major nu and you take the p invariant functions in this space t is the shift map p is the Markov preparator so these are the functions these are the harmonic functions these are the bounded harmonic functions on the space ok and ok ok ok ok ok ok ok this is a bijective isometry and I almost gave the proof I just didn't give you the conversion the reciprocal it's just a reciprocal it's just, if you take a t invariant function phi you send it to the function phi of x which is the integral avec respect à la majeure de Markov. Vous avez la valeur d'un autre âge. Quand vous avez un gale martin, la valeur à la étape 0 est une expectation. C'est la computation de cette forme. C'est la naturelle. Si c'est la limite, pour récover la fonction originale, vous avez juste à prendre l'expectation. Ok. Ok. Donc, c'est... Vous avez aussi la valeur d'expectation. Quoi ? La valeur d'expectation. E5B1. C'est correct ? Oui, oui, oui. C'est exactement E5 avec respect à B1. Euh... 2B... 0. 2B0. Maintenant, c'est la fonction d'un autre âge. Et c'est avec respect à la majeure de Markov. Avec respect à la majeure de Markov. Ok. Donc, c'est l'expectation, oui. Ici, toutes ces expectations conditionnelles sont élevées avec respect à la majeure de Markov. Quoi que vous construez, ce n'est pas un gale martin avec respect à cette majeure qui n'est éventuellement pas finie. Quoi que vous construez, ce n'est pas un gale martin avec respect à la majeure de Markov. Et puis vous laissez varier un set de majeures 1, un total de majeures, pour vous. C'est ce que vous faites. Ok. Donc, peut-être que c'est tout à fait pour vous, comme il m'a dit la première fois que j'ai rencontré ça. Mais je veux juste emphasiser le fait qu'il y a une fermeture. Quand vous avez un Markov opérateur, il y a des processus limites sur les trajectories. Et il y a cette poisson transformée qui vous permet de changer le point de vue. Si vous parlez des dynamismes randomes sur X, vous parlez, vous parlez sur les dynamismes déterministes et sur le espace de trajectories. Et ce sont des points de vue duales. Donc, la corollerie de ça. Donc, avant de parler de corolleries, je vais me justifier la terminologie de poisson transformée. C'est juste de la fax qu'il y a un exemple. Donc, un exemple, on prend X pour être le disque unique, à l'intérieur de C. Et je ne le vois pas comme le disque unique, mais c'est le point carré de la plaine. Le disque point carré, peut-être. Et si vous définissez P d'une fonction évaluée sur X, comme donc vous fixez un radius R, et comme 1 par 1 par 1 par une surface cette disque dans formule r l'intasto sur le disque centre.x selon formule r d'une faille par respect à et la mesure surface. D'une şekilde surface je dois dire la surface n'est pas la distance ukrainale ce n'est pas la distance hiperbolique C'est le majeur, qui vient du point carémétrique, et c'est le majeur qui est associé au point carémétrique. C'est le Riemannian majeur associé au point carémétrique. Vous pouvez dire que c'est le PSL-2R invariant métrique pour la action naturelle de PSL-2R sur le disque. Pour le P11, peut-être, si vous êtes familiar. Hyperbolic, radius, hyperbolic. Et maintenant, le poisson transforme, identifie le poisson transforme dans la façon classique. C'est-à-dire, quand vous avez une trajectorie pour ce processus de Markov, quand vous avez une trajectorie, le staff d'un point x, il va avoir un point limité dans la boundary.  Et ce que vous avez à la limite, c'est juste le poisson transforme, le poisson transforme, la fonction harmonique. Et la raison n'est pas, c'est juste de la facture, je mets ici, je ne veux pas étudier le Markov chaine avec un temps continu. Pour une simple raison, je n'ai absolument pas de compétence. Mais ici, c'est juste une version discrétisée de la motion de Brownian. Bien sûr, quand vous voulez comprendre les fonctions harmoniques, vous avez l'application, vous n'avez pas un opérateur discret, vous avez un paramétre semi-groupe. Mais tout ce que j'ai fait pour un opérateur, je ne pouvais pas le faire pour un paramétre semi-groupe de opérateurs. Et dans ce cas, vous devez prendre un paramétre semi-groupe, qui est l'exponential de l'application. Ce que vous avez, c'est exactement, c'est exactement le poisson transforme, c'est le sens classique. Mais ici, ce n'est pas de la différence. Ce que vous avez, c'est le poisson transforme utile. Le poisson transforme classique dans le disque unique, c'est le cas particulier de tout ça. C.R.M, je l'ai mentionné, ça peut aussi être dit comme le fait que quand vous avez une fonction harmonique, et quand vous avez une trajectorie de la motion brunienne, quand vous followz la fonction harmonique along la trajectorie de la motion brunienne, c'est une motion hyperbolic brunienne. C'est une motion hyperbolic brunienne. Puis, en dehors du cas classique, il y a une motion hyperbolic brunienne, il takes an infinite time to reach le boundary. Where as the classical one hit the boundary at some point. And when you follow the value of the harmonic function along the trajectory of the brunienne motion, it has a limit. It is linked to the boundary value of your harmonic function in classical potential theory. In this theorem on your left, you define the formulaic Poisson theory or Poisson theorem without defining Poisson boundary. Yes, because Poisson boundary, what Poisson boundary is here, is just the space of ergodic components. The space of, but in general, for example as a measure is infinite, this is not very clear what the space of ergodic components is. So I prefer to speak because if you take the space, the sigma algebra of thin variant subset, it is not clear whether the restriction of the measure omega nu to this sigma algebra is sigma finite. You have this kind of problems but you can always speak of this space. This space makes sense. In general, if you give a nice motion of a space of ergodic components, then you know that the infinity, when the measure is finite, when the measure is finite, for example, but this is not the case you want to consider in group theory, of course, in the case of random works of groups. If the measure nu is finite, then you can see this space as just the L infinity of the space of ergodic components. In this case, this space of ergodic components is the Poisson boundary and what you get is the classical notion of Poisson isomorphism, etc. Ok. Ok, so precisely, and the corollary of this is that P is ergodic with respect to nu if and only if T is ergodic to respect to nu. Ok. Ok. Donc, maybe this was a little abstract. I'm sorry. But now I will be more concrete. So, so now we will focus on a particular case and this is a case of group actions. So, again, if I have now, I still have a space X which is a locally compact and second count table. But now I have also a group which is of the same species that is it is also locally compact and second count table and I have an action of G on X, a continuous action. So that if I equip nu, if nu is a probability measure on G, I can define a necessitated Markov operator P mu, so P mu of a function phi evaluated at some point X is just integral over X of phi over X over G. You are always right if mu is a probability measure or you are always right this little difficult for me. D mu of G. Ok, so I take instead of acting by one transformation I have my point, I take a random transformation and I jump to GX but G, there are several possibilities and I take another edge over all the possibilities. So, which means that P mu can be seen as a map where we send watts point X to the measure which is a convolution product of mu with respect to delta X to the Dirac mass at X. Ok. So, this is a case of this is an example of Markov operators that I will always study. And in this case so there is a very classical example and this is so the existence of this example is the reason why I have always been using I wanted to maintain the case where mu was infinite and that is a case where X is G and you take the right action of G on itself so you could also take the left action of course this is exactly the same but for some reason yet we see in one minute I want to take the right action and then a harmonic function is just a function which satisfies a harmonic function is a function which satisfies that for say so you keep G with hard measure which is a so you tell maybe we can assume that G is an immodular in this case to simplify which is not very essential but assume G is an immodular that satisfies that hard almost every G in G you have phi of G that is equal to the integral over G of phi of G H in the mu of H so to be very very formal since I assume all my actions to be left actions I should put an inverse here but see if I want the right action to be a left action just follow me but I will put I won't put an inverse of course this is the same if you are not happy you can replace mu with a mu tch it is in measure under the inverse map ok anyway this is just kidding so you have harmonic functions and the construction I just made identify the harmonic functions so you have the space W here but what is W so there is another way of seeing W so in general when you have this situation there is another description of the measure omega X in this case because from the measure on G you can form the space so I will denote so in all the sequence all the talks B will denote the set of sequences in G so this is against not a standard notation and but when we started working on this topic this is Benoît for us this was the state this was the Bernoulli space so we denote it by B this is for Bernoulli sequences ok sorry Benoît space ah ok this is a Benoît space it's a very good idea ok and you will keep it the point is that you could say this is more or less the same as W for example in case of random works a group on itself but I don't denote it by the same letter because I don't put the same measure now this is not this is not the set of trajectories this is a set of trials that is before moving in the space I first select all the elements of G that I will let act ok and then I can move in the space which means that you have a measure you have the space X cross B you have a map if you fix some X in X sorry given X in X you have a map from B to W which sends B to the trajectory so the first point is the trajectory is X you start from X ok so this measure I will denote it by beta usually and then you go to the first point B1 X then I choose the second point X point B because this is a map from B to W ok but X is fixed ok so for given X there is such a map because I have to fix my starting point when you don't fix I want to fix a point and to describe omega X for the moment there is no measure on X there is no measure new for the moment I have to fix starting point so there is a map of this form etc etc ok this is a trajectory and it is not very surprising that this is a it is a lemma or a sublemma that omega X so define this map I don't know by PX lemma that omega X is just the image measure by PX of the major beta ok the Markov measure can be recovered from the measure and the space of trials ok so of course if in the space X there are stabilizers this map is maybe not a major space isomorphism but in all the cases we will study this is a major space isomorphism in general you can there are cases where you forget something when you smash ok but let me be careful in all the cases we will study this will be a major space isomorphism ok and you also have so this is the third point of view when there is a group action so now if you have a measure new which is a measure on X then you confirm you say that it is new stationary if the convolution power of new with respect to new is equal to new ok if it is defined because when new is infinite it is even not clear that it is defined but when this is defined and equal to new you say that the measure is stationary and the following equivalence that saying so again this is formalism that saying new is stationary amounts to say and this is almost the same result as a move that when you take the measure new and you make the product with the measure beta beta is this measure in the space of trajectories it is invariant under the map that sends X B to B1 X T B where B is a shift map so this is almost my voice is lost so this is almost the same theorem as before when these maps here are measure space asomorphism when there are no stabilisers this is the same theorem yeah B1 because I want the coordinate 0 to be X so these are the sequences with indices at least one this is just a matter of convention ok you want the starting point to be the coordinate 0 ok so this is just formalism this is more or less the same theorem and saying that this one is ergodique this is the same ok so in this case if I come back to poisson to harmonic function of the full group so what I am saying is that I have this dynamics this here so let us go back to the example where X is D and you have action on the right ok so you you have a dynamics on B cross D or G cross B maybe and what I am saying is that again the set of harmonic function so you have infinity of D and you take the P harmonic functions P new harmonic functions and what I am saying is that this is the same of the set of T invariant function so maybe you know this map by T tilde of T twiddle invariant functions here ok so this is the same theorem you have poisson as a morphism and here but here the point is that when one speak of poisson boundary of groups the point is that there is an additional structure here which comes from the fact that the group acts on itself on the left and this action commutes with the Markov operator on the right and this is a key feature of groups that actions on the right and on the left commute to each other and in particular when you speak when you have a harmonic function then you can regularize it on the left so you can always assume that it is left continuous ok what happens is that in this space the left, the vectors here you have an action you have an action this is not b1x this is b1 because the random walk act on the right ok so now you have the action of g on the left on x and what happens is that in this infinity space there is an action of g it is not a strongly continuous action that is in general because it is in action of infinity it is in general not strongly continuous but the vectors which are strongly continuous for a weakly dense substance of infinity due to the fact that when you have a harmonic function you can regularize it on the left and this is a weak approximation for the weak start topology of infinity so forget about this technical details but what happens here is that you have another space which is a space of functions which are continuous for the g action here which are vectors which are continuous for the g action here ok and what happens is that here you have a cistar algebra structure it is an infinity space so what I am describing is a subsistar algebra product of continuous functions product of two continuous functions is continuous so this cistar algebra this is the algebra of continuous function on some compact space which is in general very large ok and this is what is called the Poisson boundary you take the set of continuous vectors here and this is the sigma algebra so as you have an action this space is equipped with a continuous action of g ok so this is the g space and this is what is called the Poisson boundary ok but in general you don't have this kind of additional structure ok so the Poisson boundary comes from the fact that you have left action on g which commutes with a mark of operator this is very formal we will come back to product of random matrices so for example if g is just a free group i.e. a two generator nu is a measure a simple random work of the free group the Poisson boundary here this is the boundary of the free group the Grumov boundary in general when you have a hyperbolic group under mild conditions if you have a nice random work on a hyperbolic group the space you recover here is exactly the Grumov boundary ok this is this is just the link of what we are doing when we handle stationary measures with a classical this is a link there are closed links with the theory of Poisson boundary ok so so let me now go back to the case of group actions and now i will again i want to study stationary measures on homogeneous spaces so now i will always assume a measure i will study i just so i try to give a general description of stationary measures even in the infinite case because i wanted to draw a link with Poisson boundary of groups etc but now i want to study dynamics so i am interested in probability measures so now i want to study probability measures in all the sequels now the stationary measures will have finite measures will be finite measures so for the study of stationary measures there is a another way this is one of the richness of the structure of stationary measures of finite stationary measures is that when you have a stationary measure there are two ways so one way is as a probability measure that is invariant on average the other way is to see it as a family of probability measures that satisfies an equivariance property so for the moment it has a strange and i will precisely say what i mean so this is a there is a transformation on the set of stationary measures and this is totally analogous to Poisson transformation and this is why i mention click i bit it will be simpler so this is a proposition by first and back this Poisson transformation for stationary measures so what does it say it say that when you have d at an x you have two probability measures a new stationary so again new new probability measures then there exist a map from b to the set of probability measures on x which is Borel which is a Borel map such that every continuous function with compact support x every continuous bounded function makes no difference for beta almost every b in b when you take of phi of b1 b and x over x over what over x d new x this converges to the integral of our x of phi with respect to new b which means that so you have so this is very strange because you see I want my group to act on the left ok so I want to first act by b1 then by b2 by b3 etc so the product is not written in the right order so this is not linked to the action of the random work but somehow to the converse of the action and you will see one minute where this comes from so this tells me that I have my measure on my space x there is some distribution the measure new and then I push it by b1 bn and it has a tendency to concentrate on something else of course if the measure is really invariant under the group then this something else is just the measure but it tells me that when the measure is not invariant something happens at the limit and it is not very clear what it means even for me but I can prove you why it is true this is something wonderful which is really due to first and then this idea of starting from a stationary measure transforming it to something else ok and the proof is just the Poisson isomorphism it's just from the fact that in this situation when I have a stationary measure I can look at the function of the group which sells some element to the group d to to what to the integral over x of phi no no this is a small b sorry this is a measure which depends on the parameter ok what happens is that this function is harmonic on the left on the right exactly where stationarity tells you so since it is harmonic on the right you have this property this is just martingale theorem ok this is Poisson boundary all the construction I made ok this is why I wanted to be very careful and to let the group act on the right in the case of group theory because here when you act on the left and x you get something on the right ok and this is the action Poisson isomorphism theorem but applied just as a neutral element ok in fact of course what happens that I could put any g ok but the limit measure would depend I would get a translate but this holds but if this works at g this works at the neutral element for the function composite by g the point here is that ok so that stationarity holds not almost everywhere on the group with respect to our measure but it holds everywhere ok so you have the properties that this is a martingale from this function you can build a martingale in the usual way just this holds everywhere for every starting point in the group in particular at the neutral element ok so this is harmonic ok so this is another in fact this is another way this is Poisson theorem for measures but it comes from Poisson theorem on the group ok so and now what do my measures so what are these measures newbie but you know that what happens when you shift b so when you shift b you forget the first letter here b by b 1 minus 1 so this is just almost surely for b almost every b you have this equivalence property if you want to compute if you shift on the shift space then you just add you multiply your measure you let one element of the group act on your measure ok and what is new and this is just Poisson theorem the harmonic function from the limit martingale what you do is that you take expectation so new is just the integral over b of newbie ok and in fact this is equivalent again this is some Poisson theorem that tells you that if you want to give a stationary measure you can give un mini of measures and b which satisfies this property and takes expectations and this measure if you have this property you directly have that this measure is stationary so there are 2 point of views what you lose when you take stationary measures what you lose is invariance of course so invariance would imply that this would be constant but anyway you still have something at the limit when you studied ergodicity of measures when you take harmonic functions they are not invariant but there is something at the limit and for stationary measures for stationary probability measures you have the same kind of property that is you lose invariance but you get a limit object and you can either study the stationary measure in itself or study the family the equivalent family of measures and it turns out that all the study of homogeneous dynamics I will proceed to this is what I will study I won't study the stationary measure in itself I will study the limit measures here so now I will focus on the case let me check if I didn't forget anything ok so now we will go back to Lee groups and I will describe what happens and you see this was the origin of first hand bear interest in this theory the description of product of random matrices so are you ok with all this ?


In this video, I wanna do another question involving some simple interest type problems, but this one's a little bit different than the ones we saw before here. So we have an individual who's gonna be investing $10,000 into two separate accounts. The first account has an APR of 5%, and the second account has an APR of 6%. Now tell me where those accounts are, I would love to find that. In 2020, those are amazing interest rates. But anyways, we have these two accounts. Now let's suppose that for whatever reason, our individual cannot invest all 10,000 into the 6%. So some is placed in the 5% account, some is placed in the 6% account. If our person made $560 of interest in that first year, how much money did the individual invest in the two accounts? So notice we actually have like two unknowns in this situation. We have some X value, which we'll say is the number of dollars say in the 5% account. And then we also have another variable Y, which is the number of dollars in the 6% account. We have two unknowns in this situation. This isn't something we've seen a lot in this series already, but we can handle this nonetheless. Cause some things we know are the following. The total amount in the two accounts is $10,000. So we know that X plus Y equals 10,000. More specifically, this tells us that Y equals 10,000 minus X. So we really don't need the variable Y because we know the other number is just gonna be 10,000 minus X. So because of the simple interest formula, we know that interest is equal to principal times rate times time. And so here X is that principle we placed in the 5% account. So we're gonna have 5% times one year times however much we invested, which is X. Then we have the 6% account, which also invested for one year. And we times that by Y, but Y is just 10,000 minus X. And the interest is gonna equal 560. So this equation right here then tells us how much money, well, this gives us an equation of interest. And if we solve for X, which is a linear equation, we can then find out how much was invested in the 5% account. There are some decimals playing around here. So notice that this thing will look like, well, if you don't want decimals, you can always times both sides of the equation by 100. You just have to make sure you do it to both sides, which basically has the effect that you're just gonna move the decimal over by two places. Like so, if you don't want decimals, that's a very nice way to fix this thing up. So we end up with 5X is equal, 5X plus six times 10,000 minus X this is gonna equal 560 and then two more zeros. So 56,000, that's a way of working that way. And so then distribute the six right here, we're gonna end up with 5X plus 60,000 minus 6X is equal to 56,000. So I would combine the X's together.  Subtract 60,000 from both sides, you're gonna get negative X equals negative 4,000 and divide by negative one is the X equals 4,000. This means that our individual invested 4,000 into the 5% account, which means that then they invested $6,000 into the other account. Remember, 10,000 take away 4,000, 6,000. And so we then can conclude here that this person, they invested, this person invested $4,000 at a 5% rate, whoops, apparently it bumped the page down there, at a rate of 5% and invested $6,000 at a rate of 6%. I guess I should say they invested, it already happened, right? So our individual here invested $4,000 at a rate of 5% and $6,000 at a rate of 6%.