Want to learn about stocks, cryptocurrencies, NFTs, and the metaverse? Join RichTV.io. How you doing today? I'm your host Rich, here on behalf of RichTV Live with our very special guest, the CEO of HPQ Silicon, Bernard Tourlillon. How you doing today, Bernard? Very good, very good. Very excited to have you on the show and we got some big things we're gonna talk about and it's always nice to have you back on the show and have updates. And I love the festive sweater. You're in the Christmas spirit. So first of all, I want to ask you with it's been a very productive year for HPQ. You transitioned to a tech company that was huge and started the QRR and onboarding the French R&D team. Can you go through these accomplishments for us and what it means for the company? Yes, well, the first one may not sound that big transforming for a resource company into a tech company but for HPQ it is. A lot of the investors that are interested in what we are doing, which is basically green tech, basically modernizing industrial processes making them up to what's required in the 22nd century and moving on in the future. Didn't like the connotation mining, okay? And in a certain way for what we were doing having mining operation attached to what we're doing I think was a negative in the sense that it would require a lot of capital, a lot of management time over dealing with a resource, dealing in resources as opposed to focusing on the technology. In our case, which is specifically we transform quartz into a silicone, it's the second most abundant element on the earth crust. So finding resources to use in our machine is not a problem right now. It's not a real issue. Now it's not gonna be an issue in the future. So for us converting or basically transitioning from a resource company to tech company made the story easier to sell, okay? And we also became a tier one, which is one level higher up into the quality on the TSX venture. And it opened doors. It was part of a strategy.  As a technology company, it's our technology, it's our patent, it is our know-how, it is what we're doing, where we are in the market that plays into it. So it opens the door to more potential investors coming in, all green funds, all those issues can come in. Okay, fortunately it happened in 2022 which was maybe not the year when everybody was really looking into it but the good thing it's behind us, it's done. Yes. And people don't realize it's actually as long and painful a process as going through an original listening because it is that, it's just that we, we just didn't stop trading. We continue to trade, we continue. For our shareholder, it looks seamless but behind the ceiling was not seamless. And you can see the transition now like Q3 was our first quarter as we were in mining companies and not a mining but as a technology company. So all the MDAs that the change, how we present our situation has to change. Our risk factors suddenly gain 10 times longer. So that was issue number one. Issue number two is after many, not issue but milestone number two that's really keen after many, many years of saying we would eventually, it's like being sort of like saying we're coming, we're coming, we've actually arrived. We've made the QRRs now working. We're now making silicone. We've now entered into the R&D development phase which is basically starting to make silicone improve it doing it. But there's a lot of milestone we reached this year. We proven that we, the technology scales up. Okay. That's a massive thing. The first machine was the size of a washer and dryer and this one is the size of a house. So there's a lot of impact when you start to do these things. Pyrogenesis has put a lot of team, a lot of personnel to work for us. They've actually hired a very top guide that knows a lot about silicone operations that's now running that part of the operation and we're seeing the advantage and what's that given to us. So I'm very happy that's finally done because it's a showpiece investment. Okay. I can now bring in, when I talk to investors from New York, when I talk from investors from Europe when I talk about them says, okay, you guys can come to Montreal, go take a look, sign in NDA, go take a look at the machine and they're gonna be impressed. And which brings me to my third subject because we've onboarded Novasium this year. And Novasium are people with basically three guys with a lot of expertise into silicone manufacturing, silicone purification, value added projects into it. And I brought them to do an audit for me of the QRR. And it came out very, very positive in the sense that they do realize it's a great machine, it's a great potential. And they're gonna help us, HPQ, Pyro, make it better and finding ways of generating even more potential value out of the system. So the onboarding of Novasium, it's only been six months. I, internally, I'm starting to see a lot of benefit to this, okay? And I truly believe that that will start to transpire more in the public realm during 2023. And I think that will be also a very surprising advantage to HPQ shareholders. Now, congratulations on all your milestones hit and all your success thus far. 2022 overall has been a really tough year for growth companies, large caps, small caps, penny stocks across the board when it comes to share price. What would you say to investors who are looking at HPQ today? If you look at what's gonna be the driving sector, okay, for the economy, it's greening of technology, it's reducing carbon footprint, it's rethinking. There's a lot of thematics, which HPQ checks all the marks. I'll give you an example.  The offshored into an area where the environmental controls didn't play that much. Basically, they were making it cheaper, but, you know, that wasn't good for the environment. Now we've gone into a concept where people are looking ESG principle, they're looking at those development and they now want to, and they're also realizing this was something that happened in 2022 is a strategic important where your raw material come from. Okay? Well, silicone is at the tip, the bottom of everything that's related to silicone electronics, everything that's related to solar, everything that's related to silicone, okay? And that expertise and that capacity to make that material has sort of disappeared, moved over to China. Same thing that's happening in silicone for batteries. All the manufacturing of the material is made of China. Well, there's a new law in the United States says if you're using not North American made material, doesn't count in your batteries, you lose out. So there's now a need for reshoring technology. It's really becoming a real issue everywhere, okay? And to do that, to reach that goal, well, we need a new processes to make the raw material, the new ingredient material, because people are not gonna want to bring back the old polluting plant. Well, HPQ, because we've been doing this in 2015, here we are, we're not talking that we're gonna say this is, this is where we have a pilot plant, we'll have a roadmap to go to commercial size, roadmap to be able to work with. People are interested in going to a polysilicon, the electronics, the solar, to all those materials. So we're perfectly positioned. So if you look at the overall of the market, it is massive. There's gonna be a million ton shortage of silicone in the coming years that will demand new plants. I believe that HPQ with our QRR we're well positioned to be one of the dominant player in that area, because our new plants, the new plants are gonna have to be based on new system and much more ESG compliant, and that's what we've been doing. Second of all, silicone for batteries. It's another big market, but it's at the beginning step. Once again, it's all Chinese made material that's leading the industry, but there's gonna be demand for more localized production. Because another issue that happens is, if you move the type of industrial production we do far away, then there's a carbon footprint of bringing it to the material. So there's a carbon footprint when you make it, carbon footprint when you transport. Well, people think that's not really that important. Well, the EU just said, there's a carbon tax, there's a tax now based on the carbon footprint of the material. So all those big micro economic wins are in our back, are to our advantage, okay? So we're well positioned for this. We're well positioned to develop and we're well positioned to take advantage of this. Now, what we did in 2022, means that starting in January, 2023, we're even better positioned for that, okay? Yeah, go ahead. That actually leads me to my next question. What are your main goals for 2023 for HPQ? Well, main goal is the QRR demonstrating, getting it starting to produce higher purity material. That's, those are one of our Q goals, okay? One second of our Q goal is really refining our silicone for battery initiative, okay? We've been working a lot internally with Novacium, how are we gonna present this in packages and develop it? So that's gonna be another things. We also have a lot of exciting R&D initiative that are gonna come out of what Novacium is doing, which have application for where we're going. We're also gonna have the fume silica, which is another one of those very interesting, you know, value proposition we're gonna be coming to the market. That's gonna be, that's gonna be, you know, in 2020 is gonna be at the pilot plant phase. So we have these multiple areas. If I stopped it down to multiple things, the QRR demonstrating that we can make pure material cheaper, getting us all the numbers for going commercial scale. Battery initiative is working very, very aggressively to be able to become a North American suppliers of micro-sized silicone powders for battery space. And Novacium in addition to all of them is gonna be redeveloping the hydrogen part of the business where one of the ingredients used to make the hydrogen, because we're not going electrolysis, we're going hydrolysis, is gonna be silicone. So those are three big material. At the same time as we demonstrate technology is much more efficient so that people realize that if they wanna build, because I'm hearing, I'm in beginning discussion of a lot of people that wanna repatriate all the value change toward making electronic purity chips material, because, you know, you can't repatriate chip making if you don't repatriate the entire value chain. And the, basically the missing block to all of this, as surprising as it is is the one to make the first part of the value chain which is making the two-end silicone material for electronic to be used as feedstock. And that we have a very good proposition that we're working with Novacium to present. So I'm very optimistic about 2023, you know, green headwinds are not changing. There's gonna be funds, funds are gonna look to invest into green companies, companies with new technologies are gonna be looking for it. I'm very positive for it in a certain way. It was painful during 2022, but as we come into 2023, we get the QRR working, we'll have fume silica working and we'll have all those other projects. I'm very excited about 2023. If there was one thing that you would want shareholders to know about HPQ, what would that be? It was just one thing. Well, we were talking about earlier how you feel like right now you guys are better positioned than ever. Oh yeah, it's not even a question of better position. It's one thing I want people to understand that we've been saying for a long time that we would be here, okay? So I understand a lot of people have heard me say, you know, we will eventually, what we're there, we're making it now, okay? So we are here, we are at the beginning, we finally got ourselves into the, you know, the race track, everything, we're on the starting blocks, everything is really going in the right direction. And for the first time also, I got a feeling, because COVID actually created issue in regards to controlling your timeline. So that was also very big problem. Finally, get a feeling and get a good feeling that this is behind us. So I can get a better control over what's going on in regard to the development timeline. I also understand what we do. There are some projects that are massively long lead time, okay? And this other project have shorter lead time. So what I needed to do and what I worked very hard at doing is creating a balance between having long lead time projects are finally moving forward. As the same time, I have multiple shorter lead time that can issue more news so I can talk, I can create more excitement what we're doing. We were also doing a lot of marketing in Europe, a presentation in Europe, business development in Europe. Why? Well, contrary to Canada and the United States, silicone is a strategic material. So there's an even bigger listening to that. The market there are very big. Australia is also making a big spiel they wanna become also a big player in all those issues. So none of these people are competitors to HPQ, okay? It's like all the people that are doing silicone work for batteries are not competitors to HPQ. We're not doing animal material. We understand what's in the silicone for animal. We're not, we're there to offer them solution to make their product better as we make money doing that. What is the best way for anyone, any shareholders, future shareholders that have any questions about HPQ silicone to get in touch with the company? The best one is to send me an email through our general email at infoathpqsilicon.com. Either I will reply or a director of marketing communication will reply. To be totally honest, we try to stay far away of social media where we directly communicate just because you lose control over the communication. So we prefer to have these type of videos where we introduce where you ask us the question we answer. And then we answer directly. If we can't answer, we can't answer. We're always at a disadvantage when we're in the public toward a lot of people in the sense that whatever we say is recorded and the regulators look at what we say. So we have to be very cautious what we say. So I can only really talk about what's been done. I can give general indication where we're going and then we got to go through the disclosure through our website. And we'll have, I'm hoping Q1 will have a new website up and running. Fantastic. Here with the CEO of HPQ silicone, Bernard Turleo. Now I must remind everyone that Rich TV Live is strictly for information and education purposes.  In saying that, I believe this is a company that is grossly undervalued, under-appreciated and under-exposed. Take a look at the symbols. HPQ in Canada, HPQFF in America. Put it on your radar, put it on your watch list. Bernard, thank you for joining us. Have a Merry Christmas and a Happy New Year. And we wish you a wonderful and successful 2023 and beyond. Thank you. Always a pleasure. And thank you guys for watching. If you're not winning, you're probably not watching. Bring you the winners, bring you the news, CEO interviews, and we bring them to you first. Have a Merry Christmas and a Happy New Year from my family and from Rich TV Live to yours. And Bernard, thank you for joining us again. Have a happy holidays, and we look forward to seeing you soon. Thank you. Have a great day, everybody. We'll see you soon.


Merry Christmas, Canada! Today, our family joins Christians everywhere to celebrate the birth of Christ. This time of year is full of joy, light, and traditions, whether it's wrapping gifts, choosing the perfect tree, or keeping the roof clear for the reindeer. Some of us are getting together with the whole family. Others are sharing a table with friends. Maybe you're keeping the kids entertained in the snow or taking it easy and catching up with loved ones. Whatever we're doing, Christmas is a time to celebrate Christ's message of compassion and show people we love them. It's the season for giving and for giving back, from supporting folks in need in our communities to reaching out to neighbors who might be alone for the holidays. And we're especially thinking of our service members here in Canada and overseas who give so much to our country. Thank you for everything you do. You know, one of the greatest holiday traditions is taking care of each other, and it's something Canadians do all year round. Rain or shine or snow, we show up for each other, because that's who we are. So today, let's come together and let's move forward together in the new year. From our family to yours, Merry Christmas.


Hello, I'm Kathy Wieterholt. I'm the Fruit Project Manager here at the Carrington Research Extension Center. Welcome to our 2020 virtual tour. In this video, I'm just going to give you an overview of different plants that we have at the Research Center here. We'll jump around to all the different things that we have, and I'll explain a little bit about them. We're looking at black rot fungus here on a zestar apple tree. The spots on the leaves are the leaf form. They're called frog eye leaf spots, and let's see if we can take a closer look at them. They definitely have a little dot inside of them, and like I said they call them frogs eye. On the leaves, you can control this with a fungicide. This also shows up on branches as a canker, and will kill the branch eventually. That can only be controlled by pruning, but on the leaves and on the fruit, this can be controlled by fungicide. I just noticed these the other day. I've been gone for two weeks, and so it's really too late to control for these unless we get a lot more wet weather, although I am not exactly sure on the exact conditions that will allow the fungus to propagate. It has been very dry here in Carrington. We've only had about a quarter inch of rain in the last month, and only two inches total since the first of April. So just something to keep an eye on on your apple trees, and I want to also want to say this is the worst leaf I have found. Only about 10 to 15% of the leaves have a spot on them, and most have one to three spots. This is an exceptional leaf, and I'm not really worried about the tree. This is Aronia. We have 24 plants here at the research center. We've got some main varieties, a few seedlings. We have Viking, this kind of industry standard, and we have McKenzie, which was developed here in North Dakota. I'm just showing you the fruit here. It's actually looking like a pretty good crop. There's some yellow berries. I think it's kind of like the June drop in apples. Something's wrong with them, and they're just going to fall off. So it's no problem. We have quite a few fruits this year. I do want to show you some iron chlorosis. I think our soil pH is a little bit on the high side. We're near 8 here, maybe 7.8 at the research center, and you get a little bit of iron chlorosis. It's not too bad. It's really dry this year, so maybe I could apply some iron to help this out. But in general, we don't see a problem, and the plants don't seem to suffer from it. So yeah, we're looking for a pretty good crop. We are irrigating. We don't want to lose the crop. A few years ago, it was so dry that all the fruit fell off. And then the next year, we had so much fruit that it was ridiculous, and the plant couldn't ripen at all. So we don't really want to see that again. So we're irrigating a little bit, and hoping for a good crop in September. I'm here in the Hascaps. We have three kinds here. We have these Canadian ones in the background. They grow more wide than high, and they're very thick in the leaves. So it's kind of hard to get the berries out. These are the original ones from Saskatoon. They are borealis, tundra, indigo jam, and indigo treat. A little closer here, we have Japanese plants. These were planted in 2007. So they're pretty big. They're kind of rangy. We've thinned them out a little, so they're a little floppy this year. Here, let's look at the fruit on some of these older varieties. They are kind of small. But again, the cedar wax wings have done their job, and they have knocked off a lot of the bigger berries. But so that's the size of some of this fruit. And here's one of the newer fruits from the newest breeding about three or four years ago. So you can see the new fruit is a lot better. One particular problem for the Japanese Hascap is that the fruit tends to drop as it ripens. The plants bloom over a two-week period, and then they ripen over a two-week period, and then wind and the cedar wax wings tend to knock the fruit off. So we are looking for better plants. This is part of the specialty crop block grant that we have here. And we're looking for better plants that will keep their fruit in North Dakota. A little tighter, maybe not so big. The bigger plants actually will lose the bigger berries will actually fall more than the smaller berries. So these plants are from 2017, and it's their first year for a really nice crop. So we're looking at these, and we have more in the pipeline. We have quite a few more than working with the breeder in Oregon. Here's a shot of some of the fruit on this plant. I'm here with one of the new Hascap plants. You can see all of the berries that are on here. This is one of the most productive plants we will find. Just lots and lots of fruit up here on all of last year's growth. This plant is probably only going to be for homeowners and not commercial, because you can tell it's kind of soft. The branches are kind of thin, and they tend to droop with the high amount of fruit. We'll see what happens as the plants get older. But if they're going to droop towards the ground, they won't be conducive to harvesting for commercial purposes. But it is one of the most lovely and tasty berries out there. We're here looking at the Juneberries at the Research Orchard, and we have 100 plants. They're planted about five feet apart, and we have five varieties. They're all from Canada. They have the best elections for the most part. All just selections from the wild. Let's take a look at the berries here. Some of them are starting to swell a little bit. See if we can get this in the camera. Yeah, these are starting to swell, but right next to it, there's a bunch that aren't. Most of them look this way. Most are small. In this cluster, we can see some little leaf stippling on the leaf here, and that's from the lace bugs that are feeding on the backside.  This little brown area by my thumb is from another insect that feeds on top, probably a sawfly, but maybe there's other things too. It looks like there's going to be a pretty big crop here. We have had about seven pounds per plant, maybe four or five years ago, and there's going to be really a big crop this year if we can beat the fruit flies. It's going to be a little later than normal because of the long spring, but we I'm going to start spraying for the fruit flies this coming Monday, and hopefully we'll keep them in control this year. It would be very nice to get a good crop of June berries. We all love them. We're looking at the cherries here in the orchard. These are the Canadian hardy shrub cherries. They get about seven or eight feet tall, about the same width. They are very hardy, zone two hardy. The ones right near us are Juliet and the ones down the row are Romeo. I have to say I'm preferring the Romeo right now because they don't have all the suckers that Juliet has. If you look here, I've just mowed these off. These is just covered in suckers here. So that could be kind of a problem if you like to keep it tidy. And if you don't want branches to refill in your row, so you can keep it open for spraying for fruit flies. Romeo does not seem to have much fruit this year, although it blossomed quite well. Juliet, I would say Juliet has a moderate crop. Here's the fruit on it this year. It's going to be pretty nice fruit. We have not been able to get a cherry crop since about 2012 because of all the fruit flies here in at the research center. We have a brand new sprayer. We're going to try out this program this year and see if it can help us get a cherry crop. I love tart cherries and I hope you do too. They really make the best pie. We're here in the grape vineyard at the research orchard and I should call it the old grape vineyard because about three-fourths of the plants have been removed. Grapes take a lot of effort to take care of and we have so many other plants that are quite important to learn about here in North Dakota. NDSU at Fargo, Dr. Harleen, Hatterman, Valenti and her program is learning so much about grapes. They have huge orchard spaces. They are looking at what grapes do best in North Dakota. There is a whole breeding program that is trying to breed better fruits for North Dakota. So our program is really redundant. I want to show you the fruit here. It's looking really great. But once you do your pruning and you get this fruit, is your work done? No, it is not. These grapes require you to remove the lateral shoots to get the best fruit.  This exposes the grapes to more sunlight and then they'll be tougher in the later summer when the sun is really hot and it's dry. They'll turn better color with the sunlight. So there's that thing. You won't do this just once though. We just did it today on July 3rd but you are also going to do this like one to two more times during the summer. In addition to that, we have all these suckers that are growing on the ground. These should be removed. They're just taking energy away from the fruit and it's best to cut them off or pull them off and then spray out all these weeds because the more heat you can get to the plants the better they will develop. One final thing that we can do for the grapes is comb them and this also takes a few times. These these are growing kind of across the plant but really you should try to make your canes grow vertically. So you should go through and pull everything that's crossing and growing sideways along the plant. Pull it out so that it grows up and down and hangs toward the ground. You'll have to do this a few times and it's a lot of work. Grapes, not as romantic as you think they are. I'm here looking at the Black Current Variety Trial at the Carrington Research Extension Center's fruit project and I just want to show you that we have taken out half of the trial. It just seems that people have not been able to get used to using these uh places that use fruit like jelly makers or wine makers still aren't so sure and maybe their customers don't really know about the fruit. I again say it is one of the most wonderful fruits on the planet. It's very helpful for you. It's tasty. It's just that we don't know about it here in the U.S. One of the world's most popular berries. Take a look at some of the berries that are developing here. Here's a nice cluster. Look at those. We've been irrigating a little this year because it's very dry. Some of the berries are getting a little bit of color on them and we'll be harvesting these at the end of July maybe early August. We're looking at hazelnut plants here. These hazelnut shrubs were planted in 2007. We got them as seeds uh rooted seeds from University of Saskatchewan. They're just a cross that was made nothing special um as far as selecting for nuts. They were just crossed to see what would happen. Some were going to be hardy, some weren't. Some were going to have a lot of fruit and big fruit. Some weren't so we were seeing the results here. These small plants were cut down this spring by me because they had eastern filbert blight. I've just noticed it in the last couple years and I didn't really realize what it was at first so cut those to the ground to remove the source of the inoculum. This nice plant here is uh it's about 10 feet tall. It has not had any blight and it has not had any winter kill. It's also a pretty good producer of hazelnuts so let's look a little closer at the nuts that are developing here. I think you can see that they are there in their frilly little coatings and there's quite a few of them on here this year. We actually have plants that are developed by Dan Johnson. He's been breeding them for over 20 years here in North Dakota. They don't really seem to have many nuts this year. It was really a crummy fall last year for everything so that may be the case of why they don't have any nuts. We had 20 inches of snow October 9th, 10th, 11th and then it melted got nice for a while and then went back to winter so a lot of a lot of woody plants were affected by this strange weather last year so all right this is hazelnut here at the research center. We're looking at one of our bigger pear trees and I've killed the central leader and I've kept it though so that I can use it to make better angles on those side branches there. One of these side branches is now going to be the leader of the tree and this will help keep it a little shorter and more manageable. We've got spacers in there to keep these branches downward a little and I just want to show you that I have painted the trunks white on these pear trees also the apple trees because they get sunscald in the winter. They warm up during the day in like say March and then at nighttime they cool off and the sap freezes and cracks and I'll show you that this was a big mistake on this tree that I didn't have it planted. Right in the center there it's got a crack that shows you the exposed trunk and it is healing so I hope this tree can recover and go on to a productive life. I think it can. We have two more specific videos on Hascap and Aronia and I hope you can watch those too. Good luck with your fruit growing this year and take care and we hope to see you next year. Thank you!


Okay, we're gonna take a look today at Sync Thing. Sync Thing is decentralized peer-to-peer file synchronization program that can sync files between devices on local networks or over the internet. It's completely free, open source, very actively developed and very secure. While it does synchronize files similar to own cloud or Dropbox, it does so without the need for a server. It is more or less an open source version of the BT Sync tool, but has a lot more features. Each device that you share with the folder will keep the same copy of the files that you sync. Now, Sync Thing supports Linux 32-bit, 64-bit, ARM, ARM64, Solaris, Windows, Mac, and Android. As of September 2016, we're at 0.14.6 version, secure, private, open development. A lot of great reasons to use this. Now, it's built on a framework referred to as the Block Exchange Protocol. And this is really, really well-written. This is not just well-written, it's well-documented. So this entire Block Exchange Protocol is the base that all of this runs on top of. Now, this whole project, I can't believe the level of documentation it has for it. I'm still gonna do a video on it, even though it's not hard to follow documentation to get things set up, but I just kinda wanna show it in action. So I got my demo machines fired up here and I already have it on here downloaded on a Linux box. I'm gonna quit Firefox and just show you how you open it up. So once you download Syncthink, you just have to run it. There's no config file you really have to do. There's plenty of options you can do, but we're just gonna start with the basics of just kicking it off and run. So you download, run, no wizard, Syncthink just starts. It's not running in a background. The first thing it does is launch a window directing your browser to HTTP, essentially local host or 127.001 with no username or password. It can only be accessed by you locally. It cannot be accessed against your external IP. So it's still secure, even though there's no login credential. Now from here we can go over to the settings and we can start customizing it. We can create a authenticated username if we want, authenticated password, use HTTPS, and you can pick your IP address to bind to so it can be externally accessible. Now if you wanted to do this on a headless server that does not have a web browser, can you launch Syncthink on a Linux box or a Windows box with a command line option telling it which IP to bind to and then go in and change all these settings to the web interface and customize it. Or you can edit the config.xml file directly, which is in Linux located under .config slash Syncthink.config.xml. And on Windows it's your username, local app data, Syncthink, and it's actually the same config.xml file. Now you can also set your rate limits, your outgoing limits, internal limits. So if you wanted to not have it saturate your bandwidth, you can do that. You can give your device a specific name. It's just gonna pull the name of the computer. This is a virtual box installation called Thomas Virtual Box. You can change the default listen address. Now this is some neat options here. So natural, virtual, global discovery, local discover, and relaying. What these things do is if you don't have a way to map your firewall and you wanted to connect devices and you just want it to be discoverable, you can use, and this is the default, their own discovery servers. Now your data does not pass through these discovery servers per se. The discovery servers just allow Syncthink's instances to sync with each other by registering with a global discovery server, sharing out your IP address to the global discovery server, and then allowing other people to connect to you. Now they can't just connect. They do need to use a key. So it's not like it's an unencrypted connection. And they do not need your username and password. They just need your unique key. And then even once they have the key, I'm gonna show you how you accept the share. So we have Syncthink running right here, and here is the default share folder, which is a unique folder ID that's randomly generated on here. Now whenever you're doing a share, you can call the folder label whatever you want. Default folder, whatever. But you have to leave the folder ID unique upon creation. It can't be changed. So once these are set, they're set. And we're gonna save and we can call this, this one's now referred to as default folder, whatever. Other options on these folders, you have a lot of different, how often you wanna scan it, you can take it whether or not this is a normal or master folder. A normal folder means whatever's in this folder is synced with all the other devices. And I say all. It's not just one or two devices you can sync with. It's many, many. Some of these people have a thousand devices that they're syncing with. Other option is master. And if you set it up as a master, anything you put in here just gets propagated to all the clients that connect to it and changes they make, don't go upstream back to this one. So we're gonna leave it here normal. File, pull order. You can say pull the newest, oldest, random or alphabetic list. So that's how, if you have a lot of them that are sync, which order do you want them to sync in? The file versioning is pretty cool. Now trash can versioning is pretty simple, which means when a file is deleted, throw it in the trash can. Now this is very important. If you're sharing this with someone else and they delete a file and they're completely synced with your folder, it deletes it on your computer too. But then you have an option to say throw that into the trash can. Now the downside of trash can versioning is you can set the number of days that it goes in there, but let's say we have a file called tom.txt and we delete it. Then we create another file called tom.txt and we delete it. It goes and overwrites it. That's where staggered file versioning, there's an option where it also synchronizes every incremental change and we'll show how that works. So we're gonna set this one to staggered file versioning. We're gonna go ahead and leave the days at 365. We'll leave the version path at default, but this is a neat option. You can take and have the staggered file versioning sent somewhere else. Now there's also simple file versioning, which just keeps five versions of the files. So every five edits of a file, it deletes the six edit. And full external versioning is you can set up a series of commands to execute every time a file is moved or deleted. And then that just gives you another level of customization. We're gonna use staggered for our demonstration here. So save. All right, now we're gonna fire up the same thing on our Windows machine. So here's Windows, already have syncing downloaded here. Here is the location, users, username, app data, local sync thing. There's the config XML file, just like we have on the Linux machine under config. So much the same there and it runs and launches the web browser. Actually, to make this easier, I'm gonna go ahead and allow access to the other machine. So we do that by doing this. What's the IP address of this? 192.168.103. So we go into the settings and change this to 192.168.103. Save. Now, it's gonna warn me of the danger because there's no password on this and it says people can easily access Hackersbury, blah, blah, blah, I don't care. All right, so now we have it over on this device here. So this is our Windows machine and this is our Linux machine. We're going to show the device ID. Now, these devices are absolutely unique. You should keep them protected. That's one of the reasons I'm doing this in a virtual machine because knowing my device ID would be the step in towards accessing my existing machine. So we're gonna copy this and then we're going to add a device and what's the device ID? We're gonna give it that and it's the Windows 7Vbox. Now, if I knew the address, which we know, it's TCP colon IP colon port. Now that's how you connect directly. We're gonna go ahead and let it figure out what to do. We're also gonna go ahead and tell it to share this right away. So we're gonna hit save here and this will allow this device, which is the Linux box, to find and discover. And we're gonna go ahead and let it use the discovery servers and local discovery so it's gonna hit save. Now it's looking for it. Now this does not happen instantaneously. It's gonna take a second. And on here, we're gonna wait and we're gonna see a prompt come up that the devices wanna talk. Right now they're going through in a block exchange protocol is slowly pinging out and finding the other devices and syncing them up. And it's verifying with TLS security the keys that were generated and the way it verifies that we actually are trying to access this is because of that long ID when we into actions and show ID. And of course, if you're doing this with a phone, that's why it's got a QR code as well. So now the device is on a new device. This virtual box with this ID from here wants to connect. I like to add this device. I can say ignore it or add it or deal with it later. So it wants to know, okay, it's a dynamic address. This is fine. Do I wanna share the folder? I'm not gonna share my folder but they're gonna share it with them. So we're gonna hit save and we're gonna call this the Linux box. So we know that's the Linux one because we can have whatever device name we want. So save. Now it's gonna take a second because now the devices are connected with each other and we're gonna see this show up and turn here. This is on our Linux box and this is our Windows 7 box. This does not happen absolutely instantly. So I'm gonna go ahead and show you the delay. It does take sometimes 60 seconds before the two devices really start talking to each other. That is normal. Now it's found that says, hey, look, they want to share this folder. So we're gonna go ahead and add the folder. Now the folder path is pretty cool. I like the way this works. So you can say share it on C. What do we wanna say? We're gonna type users that auto completes for you. It is case sensitive. Whoops, users. Tom's. And we're gonna call it the Linux share. And if it does not have the folder, it creates the folder on there. Default folder, whatever. That was what we called it on the Linux machine. Default folder, whatever, from a Linux box. Then we're gonna go to advanced settings. Now the trash can setup is per device. So I set up a trash can for that share on the Linux machine, but we didn't set up yet on here. And so it's per device, per share, per folder share. So we're gonna go ahead and say we're gonna just do something simple here. Trash can file and versioning, keep 11 days worth and hit save. All right, now we have a share. And it says it's up to date. So this one's still unshared. And we're just gonna, for simplicity, we're gonna delete, because you don't need to have the default share. You don't have to have any share. You can just share with this. So we're gonna have, here's from the Linux box. It's under the Linux share. And here's my virtual box. Here's my folder for it. Now let's actually look what's in the folder. Let's put something in a folder. So we got desktop sync thing. Home, sync, right now it's empty. So we'll do this. And it's gonna rescan and find some files. I can force it to it now by default. It's set to every seconds, 37 files. And in a few seconds when this rescans, I'm gonna go ahead and force it to speed things up. It now has them indexed. It's gonna take a second and goes, hey, we're not up to date. This happens a little bit slow when you first connect them. Once they've been connected for a while, this happens instantly. It also happens faster if you specify the TCP ports. We've now transferred over to this one, 37 items, 11.8 megs. The protocol itself is using delta syncing within the block exchange protocol. And what that means is it syncs on the delta. So if you had a one gigabyte file and you change only a few bits into one gigabyte file, it can sync in 128 kilobit chunks the delta changes of that file. So the protocol is actually really fast. So now that these two machines are completely talking to each other and fast, let's go in and make some changes. On the Windows machine, let me drag it back over here. Here's the Linux share. And let's go ahead and just delete something like this. So we'll delete it out of here. Let me go back to the Linux machine. And now it's deleted out of here. So it's synced up and it's deleted. And now this, like I said, I've just turned on so we can see the hidden. Here's that ST versions folder. And there's that folder we deleted. So this is how it controls the versioning is for every share, there's an ST versions folder. And depending on the type of versioning you have set up, that's where the files go. So let's go ahead and create another file. So we have the readme text, make another copy of it and rename it just testing. Now let's make some edits on this. Some edit from the Linux box. Save the file. Go ahead and force a rescan now. By default it's set to 60 seconds. I'm not gonna set it to one second because it makes it kind of file intensive at that point. And we can last file receive updated testing.text. That means on our Windows box, which we'll bring back over here. There's our testing.text file. Some edit from the Linux box. Some edit from the Windows box. Save, oh, force a rescan real quick. Rescan files, make sure it's up to date. Yep, this one updated it. Some edit from the Windows box. So now you can go back and forth and as you save these. Now if we create a file conflict by doing that, it will automatically create a conflict. So we're gonna say start conflicting from Linux. Say, oh, don't wanna save it. So here's an edit ready to be saved and we're gonna simultaneously edit and save over here before they can do anything. So just make some edit here and we're gonna jump real quick and hit save, save. Then we're gonna force them both to sync and it's gonna give me a list of failed items because I tried to edit the same file at the same time. This one's synced first. This one's saying out of sync and trying to figure it out. All right, now for the version conflicts, I still saved them so fast that it did create it and here it is. We have a conflict one. So it tells you about the conflict of this. It doesn't do anything to the file itself. It just creates another version. So we can look here and see we have testing sync.conflict and it tells you the time that the conflict occurred. So we still have our original file and now we have one called conflict. Now also what was going on behind the scenes is in the version folder, we have another version from when we saved over the same file. So it tells you the date and timestamp of this version of that file. So if we edit this file again, so we're gonna go over here to the windows machine again and we're gonna open up that testing.text file and let's just clean it up.  Now it is mostly blank. File, exit, save. Then we'll go ahead and move this out of the way and we're gonna force a resync on these. Rescan, rescan. Go into our sync folder, look for this testing.text file. Now it is mostly blank. Go back to the actual root of the folder, look at this hidden stversions folder and here's the different variations. So if I wanna go back and look at what it was here or what it was here, it now kept all the version histories of them. So I synced it from, I edited it persistently on the windows machine and it creates all of these hidden ones. This is the staggered versioning. Definitely a great way to do it. That way everything happens to your files or something gets edited on one of the other computers. You need to go back. They're all nice and date stamped. So that's kind of the basics of how the syncing works. You just share that on there. This concludes sync thing for kind of a quick overview of how to get started and how to get it working. Now I'm gonna do another video. There's gonna be a follow up to this, taking a deeper dive in it, but this is just kind of an overview to get you started on this sync thing. It's a great tool. I really like it. It's definitely been a solid performer for me having all the files that I have in there. Thank you for watching this review of sync thing. If you like the content here, like and subscribe and I love to hear feedback of how you're using sync thing or any other tools that I talk about. Thank you.


How to make faceless videos. Now I don't mean if you don't have a face. If you don't have a face you won't be able to do any of this so you might as well just click off now.  I'm going to show you 10 ways to make videos without having to show your face. A lot of people would love to make videos but they've got maybe a fear of the camera for one reason or another they just don't want to show their face and they may think well since I don't want to be on camera there's no way I can make videos and that's just not the case because in this video I'm going to give you 10 ways you can do it and you don't have to be on camera. Hey I'm JR Fisher and I've been selling online since 2009. I've sold both physical and digital products and on this channel I show you the tricks and tips that I've learned along the way so that I can help you out. I've even authored a couple books and I have several courses on how to make money online. I want to take this opportunity to remind you you need to subscribe hit that little button below there and subscribe to my channel make sure you turn on notifications too so that I can let you know every time I've got a new video. Now I've got a special surprise for you at the end of this video I'm going to give you access to a $97 course absolutely free so you're going to want to stick to the very end. Now have you been struggling to make videos online but you don't want to get on camera I want to hear maybe some reasons why you don't want to be on camera stick them in the comments below I'd love to hear about it. Okay let's jump right in now the simplest and easiest way to make videos without being on camera is to just record your screen you can record your screen and get your message across and there's a lot of softwares to do it with but I really prefer Camtasia Camtasia is awesome I use Camtasia for adding music and adding sound effects and putting clips together and pictures and it's just an awesome piece of software I highly recommend it now there are some free editing softwares you can get maybe with Microsoft you can get Movie Maker or if you have an Apple operating system you can get iMovie and both of those are pretty good but the problem you run into is they're limited as far as the number of tracks now what do I mean by tracks a track is let's say music that's a track your voice recording is another track a picture is another track and then if you put images on screens that's another track and all of these start to stack up and with some of the free softwares they only allow you a handful of them and you really can't do that much editing now if that's all you have to start out with go ahead and start with it but I do recommend Camtasia on down the line now the second way you can make videos online without putting your face on the screen is through product reviews now how could you use this in your business well let's say you have a plumbing business and you want to show them the type of products you're using you want to maybe do reviews on pipes or you want to do reviews on different types of faucets and the type of things you install that's an excellent way to do a video and all you're going to have to do is really show the product and maybe your hands holding the product and pointing to things a lot of people love these product reviews matter of fact they search for them on a daily basis another couple ideas for that product review thing is you know let's say you have a hair salon and you want to show different people's hair styles you could film in the salon and show the different haircuts that you did if you have a restaurant you could film some of the dishes you have that can be very appetizing people love to watch food videos I know I watch them all the time now number three is slide shows slide shows are awesome now on a weekly basis I do a coaching call and I use slide shows I'm not on the screen I go from slide to slide and I can actually make the words come in when I want by using animations you can do the same exact thing by making your videos now personally I use PowerPoint and PowerPoint's great but there's a lot of free ones out there you can use you can use Google slides Google slides is really awesome because once you made your presentation it'll be there online so you can access it from anywhere so that's a really good alternative to doing slide presentations the next thing you can do is use static images now a lot of people who have podcast or radio shows or whatever that are broadcasting online don't want to actually be on the camera maybe they're doing it from their house but you can put an image with your logo on it what the subject's about and just have that on the screen and then talk over that for your video the next thing you can do is use stock images you will want to check the usage rights because some images you can use and some you can't when you go to Google you can actually search for images that you can use without rights so you don't have to give any credit to anybody or anything like that okay I'm here at Google right now and I'm going to search some images so I can show you how to find images that don't have a problem with copyright so you want to be really cautious about this so let's pick something I will put in blue car let's see what I get there's several steps to this now once you've searched in google you're going to have all these tabs across the top here this is images videos shopping news more settings tools so we want to go to the one right here that says images and I got that one and then we come up with all these images and you go yay I got all these images I can use any of these you cannot and if you do use the wrong ones you can run into all kinds of problems who suing you want money from you so you want to prevent that across the top here where you have all these things there's last tab here says tools click on that that one there is going to drop down this other one here which is actually pretty useful too you can you can sort by size small medium large you can sort by colors but the one you're worried about is right here says usage rights now by default the google is going to give you all the images so it's not filtered at all by any license the next tab here is labeled for reuse with modification and that means you're going to have to change it the next one is labeled for reuse so that's that's more open you can do whatever you want with it it's an image that you can take and put on your site or whatever the next one is labeled for non-commercial reuse with modification so you couldn't use this for business and you would also have to modify it and then the last choice here is labeled for non-commercial reuse so that would just be your personal use so I would always go with this labeled for reuse I'm going to click that what that does is change all the images here so you just sort through find the image you like what do we find here there's so many cool blue cars I want to go with this like kind of comic book one and what other thing you can do I want to add in here too if you don't want to have a background type in here PNG and it doesn't clear out all of them but it does clear out a lot of them so that like if I had a car like this let me click on it and see all the little checker board right there you see that so that if I put this on my site here there's no background to it so that's pretty darn helpful this one's the same way even though it's got the shadow and everything there's no background and then what you would want to do is you would want to right click on it and you would just click save image as and then call it whatever you want and then you've got your images now you can use these images and splice them together to tell your story in video form as a matter of fact there's a channel that is built around stock images like that on youtube called all time tens it's cool channel and there's millions of subscribers that go there every single day and watch their videos you can also use stock videos now there's several sites you can get them from but I like to use video blocks at video blocks there's thousands and thousands of clips of videos that you could put together to tell your story and you could just voice over it or you could use words on the screen along with music the next type of video you could do is called point of view video now point of view is I guess the best way to explain it is somebody who uses like maybe a gopro where they're actually doing something and you can see what they're doing maybe they're wearing the camera on their head or their helmet or their bike and they take you through the journey of what's going on point of view videos work awesome and you don't have to be on screen to do it okay so here I am at video blocks and we'll take a look at this one here now this is somebody who's riding their bicycle but you know you could do this making a cake doing recipes you know fixing and repairing stuff whatever it is these point of view things are pretty cool you still don't have to be on camera the next type of video you can do is doodle videos that's right doodle now you've seen these before where somebody's actually hand drawing out a story and they're telling the story as they draw it out this is done with a simple software you can google it just google doodle videos and you'll find places you can get software really inexpensive where you can do these drawing type videos they entertain people they love to watch them and it's something you can do once again you don't have to be on the screen the next type of video you can do without being on screen are how-to videos now on how-to video is actually showing you doing something this could be anything from cooking where it shows you cutting up the ingredients and cooking everything to making crafts maybe you want to set up a table in your living room and you want to put a light over it and you want to make crafts and to show your hands doing the crafts now what you can do for this is actually just run the camera take a video of you doing all these different tasks and then later on voice over that video and show people the instruction of how to do that I've seen people do it with cameras and they're showing all the different settings on the cameras and they're videotaping themselves setting all those little settings on the cameras those are very helpful to people and they search them every single day the next type of video you can do big fancy word here kinetic typography videos now what is that actually it's actually you just talking and the words coming up on screen as you talk something like this and as the words come up it's entertaining enough to keep the people occupied and watching your video now there's probably a ton more ways to make videos out there without getting on screen but I just want to prove to you you don't have to be on screen it's not necessary some people never go on screen and they have huge channel these are just some of the ways that you can make videos without getting in front of the camera I hope you enjoyed that if so I want you to subscribe below click that button below don't forget to click the little bell too because that will give you notifications every time I do a new video now as promised I'm going to give you a free e-commerce course with a value of 97 dollars look in the description down there right below click that link and you can get access to this awesome course a bunch of videos that's going to teach you how to grow and run your online business I really appreciate you watching and if you've tried some of these things or you've got ideas for ways to make videos I want you to put that in the comments below I'd love to see that that would help out a lot of people if you've done some of these ideas what kind of results did you get what happened when you did this I appreciate you listening and I'll see you in the next video hey thanks for watching my video don't forget to subscribe to my channel and click that little bell right there so you can be notified every time I do a new video also click on one of those videos there keep watching on my channel


I was going to say it's working in free probability and as Percy Diakonis said last week I think everybody who works in random matrices should know free probability and the simple reason is that often if you meet one random matrix you will meet another one and once you have two you need to know some free probability to know what to do. So hopefully Dima will explain you how to do that. All right, well thank you very much again, I think in the interest of time we should just get going and then once the boards are up they could be placed. So at the end of this morning's talk you honored Amitru, she told you that if you take A, a self-adjoint Gaussian random matrix, so it was like that, with Gij normal, right? And they're as independent as they can be. Then you can actually make a computation. You can look at the expected value of the trace, the normalized trace of some moment of N and this she computed and that was the Catalan number, but also it was computed as the number of certain paths. So you started with one M and then you looked at certain walks in this graph, right? And then the idea was that this walk could only backtrack once, right? So every edge could be traversed exactly twice. So there is another way of thinking about such a thing. This is simply the number of ways to parenthesize M variables. So what do I mean by that? So I am supposed to take two M parentheses, half opening, half closing, and then I can just, I'm allowed to write any legal expression in terms of these parentheses, okay? And if you remember how these walks were described saying that this is the same thing as doing a kind of the depth first traversal of a tree, then you see that this is exactly how a computer algebra system or something like this parses an expression like that, right? The various parentheses tell you how deep inside must you go inside the recursion. And now this is just the same thing as the number of what are called planar non-crossing pairings. So what these are is that you look at M points, one, two, up to M, and you're looking at all ways to pair them up in such a way that these pairings when you draw them are non-crossing. So an example of such a pairing, let me just try to do something that corresponds to that bracket, that bracketing there is something like this. You see, all I've done is I've just recorded which bracket cancels which bracket. And it's not very hard to see that if you actually have a diagram like this, which tells you who cancels whom, and a diagram like that, which tells you which bracket closes which, then they are the same, right? And precisely, the fact that this is a legal bracketing states that this pairing is non-crossing. If you try to write down a bracketing expression corresponding to a pairing like this, you quickly see that you end up with too many closed brackets for the numbers that you've opened, right? Because you open one bracket, open another one, but then you try to close this one, right? Which is prohibited. Okay. Now, as Ali said in her introduction, where free probability starts is where you have more than one matrix. So imagine now that we have a whole bunch of them, say a1 of n, ad of m of n, so they're d matrices, and they're entries as before. So what I mean is that each individual matrix is just an iid copy of this matrix here, okay? So the entries of the first matrix are exactly like described here. The entries of the second matrix are independent of the first one and so on, okay? Well, you can do the same thing. You can try to look at the expected value of some kind of a power of a matrix, but now you have d matrices to choose from, and you can write a much more complex power than that, like that. Namely, you can look at something like you can take one of them, say the a first matrix, and then you can put it to some power, let's say m1, and then you can put, say, take some other matrix and you put it to the power m2 and so on. So you can do some computation like that, all right? And again, you would take the product of these matrices, you take their normalized trace, and you look at its expected value. And what you find is that again, this converges, I'm sorry, this should have converges as n goes to infinity. By the way, please interrupt me because I sometimes, by accident, sometimes purposefully will make mistakes. So if you're not paying attention, then, you know, anyways, what will, it will converge to is a very similar kind of thing. You will have, you'll be counting some kind of diagrams drawn on, so let's say that m is the sum of these mk's, it's the total degree, so we have m points over here. And then what you're looking at are exactly these kinds of non-crossing diagrams, so maybe this is paired with that, and maybe this is paired with that, and maybe this is paired with this, and maybe something like that. But now, all the points receive colors. So point number one receives color i1, point number two receives color i2. This is color i3 in all the way to, oh, you know what, let me erase all the m's because it's equivalent, I can just repeat the i's, let me do that. So m is just k in this example, i4 and so on. And then the rule is that the colors must be matched, so you must match colors, okay? So now you're counting not all diagrams that you can draw in a non-crossing way, but only ones which are compatible with a specific coloring of the partition. So for example, if all the colors here are different, what is the number of partitions like that? Zero, right? And indeed, if all the numbers are different, that means that if you look at any entry of this k-fold product, it's going to be a product of k independent Gaussians, and of course that's going to have zero expectations, so for sure the trades will have zero expectations. Anyways, if you think a little bit about what happens in this case, is that we have defined for each p, what's called a non-commutative polynomial, we have defined some expression which will write tau of p, and this will be by definition the limit as n goes to infinity, the expected value of the normalized trace of this p evaluated in my matrices a1 through ad, okay? Right? So here the specific polynomial I took was p, let's say the variables are t1 through td is ti1 through tid, okay? So the point is that random matrix models, assuming convergence, give you these kinds of linear functionals on spaces of non-commutative polynomials. So we have a linear functional tau from all non-commutative polynomials in the variables to c. So formally this is very reminiscent of probability theory because you can in a very algebraic way interpret a usual expectation functional as a way of assigning a number to a polynomial or to a function, a more general test function. So this leads to the definition of what's called a non-commutative probability space. So we are dealing with some kind of a unit algebra, one example of course is this one, the algebra of non-commutative polynomials that I've advertised like over there. But what is very often done is to make the assumption that this algebra is actually represented on a Hilbert space. So why is that a good thing to do? Well, if your algebra is an algebra of operators on a Hilbert space, H is a Hilbert space, B of H is the algebra of all bounded operators in that Hilbert space, what is nice is that this allows you to apply complicated functional calculi to your elements. So for example, you know that if you take an operator in B of H and F is analytic on the complement of the spectrum of T, then it makes sense to talk about F of T as a contour integral by a Cauchy integral formula, right, where this is the spectrum of T and you have chosen some contour gamma outside of that spectrum, OK? So let me think here on a neighborhood of the spectrum of T, right? And then if moreover if T is self-adjoint or more generally what's called a normal operator, then you have the full spectral theorem. You can write T as an integral of spectral projections where E from the spectrum of the operator, which is a subset of R, to projections in when your Hilbert space is a projection-valued measure. It's a measure that assigns to Borel set a projection and so this is the formula for your T. And then you can apply any Borel function on the spectrum by simply this formula over here, OK? So the moment that your algebra is represented in a Hilbert space, you're able to talk about complicated functions of the space going beyond polynomials and that's very useful. Just to give you one example, we can talk about resolvance. So Z minus T inverse is defined for any Z not in the spectrum of my operator. So for example, if Z is sufficiently large and my operator is bounded, the spectrum is some compact subset of complex planes. So that's immediately well-defined because the function 1 over Z minus lambda is analytic for lambda sufficiently big, OK? All right. Now, this linear functional that we have, tau, we will often assume that it's actually implemented by vector. So we will assume that this algebra A, M, which I call in the slide, is sitting in B of H and I have this tau from M to C. I will make the assumption that tau is given by evaluation on a vector. So here, C is some normal unit vector in my Hilbert space. What this ensures is that this plays very nicely with functional calculus. So for example, if I have this decomposition of T, then if I'm interested in tau of F of T, then I can just apply this expression to a vector over here and I see immediately that this is the integral of F of lambda C E D E lambda of C, which is the integral of F of lambda D mu of lambda, where mu is simply tau composed with D E, right? So if I compute, compose this projection valued measure with this vector linear functional tau that I've written down, then I will get a scalar valued measure, OK? And this is very useful because knowing that the functional is given by vector evaluation tells you that everything here actually converges. So we don't have any problem with writing this integral or anything like this because of course this is a bit of a subtle matter as to what this integral actually means, right? And then the last thing that we will often assume is that our linear functional restricted tau algebra is a trace, so tau restricted to M satisfies tau of x, y equals to tau of y, x. One motivation for it is that if you actually look at things that come from random matrix theory, of course that does work because of course anything that you write down will have the cyclic permutation property. If you take a monomial and you permute it cyclically, the answer will be the same because the trace in matrices does this, right? But there are other reasons just simply from the point of view of the general theory. In this case you can attach a very nice LP space to M, and if you assume also the tau is faithful with the definition written above, then actually you have that M sits inside the LP spaces and so on. So you can even talk about the space of measurable operators. These are called sort of measurable relative to M. These are unbounded operators for whom all the spectral projections live in M. And it turns out that in this case this is a very nice algebra which is in the Abelian case like the algebra of measurable functions. Which I should mention by the way, in the Abelian case this is just the usual probability spaces. That's the case where I take M to be essentially bounded functions on some measure space which I can view as bounded operators on L2 of that measure space by sending a function to the operator of point-wise multiplication by that function. And in that case my expectation functional is just again point evaluation. So in this kind of abstract situation we simply are looking at a generalization of the Abelian situation where we're permitting algebras like that to the situation where we simply permit non-Abelian algebras. So in my notes I've written a kind of a lecture zero which I'm not going to give which tries to give you a little bit of the flavor of what are called phenomenon algebras which are the correct framework to work in in this case. These are exactly the correct subalgebras of bounded operators in the Hilbert space. And actually what surprises me all the time is that if you want to define what a classical measure space is it's actually faster to do it using phenomenon algebras just to say that this is an Abelian subalgebra of bounded operators in a Hilbert space satisfying some properties than to go through the usual axioms of measure theory. It's quite fun to actually see it done that way. Anyway so the the carry out from this is that there are these non-commutative algebras like the algebra of polynomials. We typically want them to be represented in a Hilbert space and what we're interested in is we're interested in computing things like that. The trace of some polynomial or more general function like a resolvent of operators in that algebra. Okay so one thing I want to define is what it means to have the to what a non-commutative law means. In one way to say what is a non-commutative law is to say what does it mean to be the same in law. So what does it mean classically that two random variables x and y have the same law? Well one way to say it is that if you apply arbitrary test functions to them then you get the same answer when you take the expected value. So if you take the expected value of f of x that's the same thing as the expected value of f of y okay for all f. Now what functions can you take? Well the largest set of functions you can take is simply all functions that have expectations which is just l1 of your probability measure but in reality if you take a sufficiently big set of functions that is dense inside l1 in the suitable topology then that's enough. So often often enough to take f a holomorphic for instance f of z sorry f of t to be one over t minus z for z fixed or even take f to be polynomials. Another popular choice is to take f of t to be e to the 2 pi i t theta the Fourier transform right because if you take the expected value of this with respect to t you get the Fourier transform of my measure at theta. So there are various sets of functions that you may want to take as test functions and well the first two are enough always to characterize the law of your variable. This is enough to characterize the law of your variable if the measures say compact is supported or if you have some growth conditions. So you define two n-tuples of variables in this non-commutative space to have the same law if when you take the trace of an arbitrary function of them you get the same answer. And for all f here again can be qualified if x's and y's are bounded operators then it's enough to take polynomials this is like this compactly supported case or more generally you may have to take some kinds of resolvents or products of resolvents and so what's nice is that if you know that two such families are the same in law then actually you know that the algebras they generate are isomorphic namely if you simply take x's to y's this map extends to an isomorphism of algebras. So and we already talked about the case of a single variable if you take a single self-adjoint variable then what we mean by laws is nothing more than the usual classical probability measures right because this is just the spectral theorem it tells you how to evaluate the trace of an arbitrary function it will be integration against some probability measure okay all right well this sounded very abstract let me just give you some examples right away so the first example I want to give is that of a discrete group so let's take as an example gamma to be the integers z then I can look at l2 of gamma so this is the Hilbert space whose basis is my group gamma okay so in this case this is just the the orthonormal span of things like delta n n in z right I have a an orthonormal basis element for every element of the group now every group acts on its l2 space I have that's called the left regular representation from gamma to l2 of gamma it's defined by saying that the action on the delta function of h is just the delta function at gh so in this case if I look at lambda of 1 this is simply the shift that takes delta n to delta n plus 1 right the bilateral shift on l2 of the integers right yes yes to bounded operators and l2 of gamma thank you okay um so lambda of 3 is the shift by 3 right and so forth uh so now if you look at the algebra generated by all these things by all of these lambda of n n in z you can identify them if you if you take the Fourier transform map and identify it with l2 of the circle using Fourier transform then this algebra can be identified simply with trigonometric polynomials because this this shift right corresponds to multiplication to e to the i theta and so a linear combination of such shifts would correspond to a linear combination of e to the i n theta which is a trigonometric polynomial um now the thing that we often do here is to look at what's called the phenomenon algebra generated by all of these lambda n's n in z this is what's sometimes called the group phenomenon algebra l gamma and uh that algebra will correspond to a certain closure of these polynomials in fact it will correspond to all essentially bounded functions on the circle okay so you see why taking this bigger algebra is useful it allows you to play with more functional calculus right I mean you cannot apply an arbitrary function to compose a polynomial with an arbitrary function you won't get a polynomial but if you apply an arbitrary function to polynomials you get things like that and finally there is a a preferred linear functional there's a preferred vector in l2 which is the delta function at the neutral element of the group and so you define tau of x to be delta e x delta e and if you work out what is tau of lambda of g you see that well let's work it out it's delta e lambda of g acting on delta e so delta e gets shifted to delta g so we get delta e delta g which is one precisely when g is the neutral element so this linear functional just records whether you are a trivial element of g or a non-trivial element of g and in this isomorphism by the way the trace tau corresponds to integration against the uh harm measure on the circle right this is what it does if you take a function trigonometric polynomial such an integral gives you the trivial I mean the constant coefficient of this polynomial and all the other coefficients are killed now um this setup is is very often used actually in probability maybe not quite said this way but nonetheless let's do it suppose we take s inside g inside gamma a symmetric generating set the symmetric simply means that together with every element the inverse of that element is there so for instance we could take s to be plus minus one inside z these are two generators of z and i'm put plus one and minus one to make it symmetric and then we can look at this kind of random walk Laplacian it's called which is in this case going to be one half of lambda of one plus lambda of negative one right and so now it turns out that if i'm interested in things like tau of lambda to the m which is delta e lambda to the m delta e then this is something that is very natural has a probabilistic interpretation namely you start with uh the your z this is zero one two minus one minus two and so forth you start with some point let's say it's zero and then you toss a coin and depending on whether it comes out heads or tails you walk either forward or backward in other words you pick one of these elements of s and you either and you add that element okay so if you're here with probability of half you jump here probability of half you jump there having jumped here you do the exact same thing again you take another independent coin you flip it and depending on that you might jump forward or back so let's say you jump forward and then maybe you jump back and maybe you jump forward and maybe you jump back and jump back again so in fact this is going to be the probability of return to the neutral element after m jumps and what this general theory is telling you is that you can actually write down this probability uh the style l l m you can write it as the moment of some probability measure and one of the exercises that are in the notes is to actually compute what that measure is it's not very hard given all the identifications here but it's some exercise that you should do you know to understand how this works okay now there is a a notion of independence that you can talk about in this very general framework of non commutative probability spaces so here it is if i have two sub algebras a one a two in a probability space they're independent precisely when first of all they commute and secondly you have this funny law that the expected value of the product of two variables is zero and i wrote it in a very funny way provided that each variable has zero expectation and provided that they come from different algebras this condition here that i one is different from i two simply saying that a one comes from either a one or a two and a two comes from a different algebra than a one came from okay and you can check that this is exactly the same thing as the usual way of saying uh independence which is that the product of sorry the expect expectation of a product is the expectation of is the product of expectations the only thing that i've done is i've centered the variables but that can always be done because if you re-center the variables here in here you see uh that the general case can be reduced from from the centered case there's just a little algebraic manipulation that you can do so one example of such a situation would be to take a product of two groups let's say instead of taking a random walk in gamma z i could i look at z squared right then i will have generators corresponding to uh i will have things like lambda of one zero lambda of minus one zero lambda of zero one lambda of zero negative one and the point is that these guys are going to be independent from these as it's very easy to check okay and now what's amazing is that in this non-commutative framework there is a new parallel notion of independence which is called freely independent and the rule that this time is a little different so again i have two sub-algebras a1 a2 and i say that they are freely independent if no condition on commutation anymore so they do not commute but then i want to say that a product of a whole bunch of variables has expectation zero provided that first of all each one of them is centered so it has expectation zero and the consecutive ones come from different algebras so if the first one comes from the algebra i1 and the second one comes from the algebra i2 the third one comes with the algebra i3 then the first one comes from an algebra that's different than the next one and this one comes from the algebra that's different from the next one and so on so this looks very much like the previous one we had only there you could get by with just products of two because of commutation here you have to deal with products of just of n and one example which you will work out and written in the notes is the case of a free product of two groups so instead of z squared you could look at z free product z so that would be a free group on two generators and you can also look at generators of that you will have let's say four generators to corresponding the first copy of z to corresponding the second copy of z and these things are now going to be freely independent now this random walk for z times z will be a random walk on z squared so on the integer lattice the random walk here will be on a by on a on a tree on the calligraph of this group which will be a certain tree okay all right so there are there's a couple of main theorems that you can prove which are very simple the first one says that if i know that my two of my algebras a1 and a2 are independent so let's say a1 a2 are freely independent freely independent and i know the restriction of my i know how to take expectations in a1 and i know how to expect to take expectations on a2 then i i know how to take expectations on the algebra generated by a1 and a2 and here's the trick suppose i want to prove figure out what does tell of a1 a2 a3 and so forth an well the first thing i can do is i can assume that each one of those a's lives in different algebras why well let's say that a1 lives in a1 and let's say that a2 also lives in a1 well then their product a1 a2 also lives in a1 because it's an algebra