Oedd ein chryfwyr chyfridwyr bwysigol iawn i'r tu'r cyffredinydd, i'n bwysigol i'r ffobl Stephen Van Colla, CEO of Corporate Investment Banking Barclays Africa, Vikas Potter, chief executive of the Varky Foundation based in the UAE. I'm going to just keep my comments as always to a minimum and let Stephen just jump right in there. Stephen, leaders discuss technology unlocking growth and driving entrepreneurship, but what's the role of the bank in doing this? If you have a look at what's driving change in banking, especially around payments, it's all about technology. Banks are having to being forced to just put this highly on the agenda. The beauty of it is that as you get more and more technology into your banking system, the costs go down. That creates ability for more financial inclusion to get more people involved in the banking system. It also allows you to use data on a far more dynamic basis, which allows you to forward-looking dynamically credit-score people as opposed to what we've done in the past. You could only credit-score them on a historical basis, and that makes it very difficult to go down the sort of a food chain in terms of giving people banking. I think it's absolutely critical. If you just have a look at Elly Barber, they're charging what, 0.18% per transaction, we have to get on that bandwagon, otherwise maybe they'll be the next bank. Financial inclusion is one thing, but what about elsewhere? Capital market is getting markets moving. One of the things we talk about alongside inclusion at the consumer level is enabling Africa's markets. In fact, it's one of the pillars of this meeting. If you have a look at what happened in a South Africa post-departates, a post-1994, if you go back just 20 years, in that time, the bond markets have increased tenfold, the equity markets eightfold, and it's that type of liquidity you need to bring into all these financial markets. Working with government to link both the institutions and the corporates and the government institutions is critical. One of the key roles, peristatals and government pension funds can play is they are the cornerstones of financial markets and trying to make sure that their mandates allow them to trade in government bonds or trade in corporate bonds or be market makers is going to be absolutely critical. So we're doing a lot of work with that, with those type of institutions at the moment to try and make sure these liquidity pools increase. Just moving on to skills. What kind of work can you do? We talk about public-private cooperation, but what kind of worker is Barcu's Africa doing in with regards to developing skills on the continent? There's a few things we do. Clearly, as you go first into South Africa, then from South Africa out, you're trying to take your global skills and put them in these markets so that you can actually develop them. So what we do is we take a lot of our staff, move them out of the continent for weeks, months, years and then bring them back again. So there's a continual flow because over time, as you're building a business, you want it to be 99% local because they understand the market, they understand the government, they understand the clients, but as you move in, you want to bring the skills in, you want to do the training, you want to move them out. So we have an expat policy which sort of limits expats to a three-year stint and then they must replace themselves with a local. So there's that kind of stuff we do. And then also with our entrepreneurship and SME development, we actually provide financial skills, business skills alongside the lending we do. And that's just really a risk mitigation, if you want to call it that and try and build the market by supporting them as you learn to them. Vikas, you've been working in education for a very long time. You talk a lot about the quality of education when I read your recent work and also you focus on one of the programs you're focusing on in Ghana, I believe, is about educating women. What are your views on education and the other technology can play in driving skills growth and making African workers and students more competitive through technology? Thank you for the opportunity. You know, like I said in my article that appeared yesterday, that if South Africa, for example, or Ghana provided a basic level of education to all of its children, students in schools, you'll see in South Africa at least something like a 2,500 percentage growth in your economic activity. And that's quite significant. So when we see people in the opening plenary, for example, the president of South Africa or of Ghana speak about economic growth, the question that I pose to people is, well, how much focus is there on education systems? And one of the key things I think that needs to happen is the involvement of the private sector. And that's why I attend the World Economic Forum and the different summits to talk to business leaders like Stephen and others to see how we can engage more meaningfully with education systems. I think that's a really important part of making sure that skills development just doesn't happen in the workplace, but also there's a focus at school level as to how this happens. A question to both of you. How joined up is the public and private sector and the foundations that we're a multi-stakeholder universe? How well are we working together? I mean, I think people in this room and on our orders online get the fact that there needs to be greater public-private cooperation. Problems and challenges are too large to be solved by any one particular actor. So there's definitely a realisation that what needed to happen is more cooperation. But how joined up is that cooperation? Are we there yet? Are we seeing good working models for public-private cooperation? Sure. I think actually not to embarrass him, but I think Barclays do a phenomenal job in this space in their life skills programme, which I'm familiar with from the UK. But however, when you look at the case of the private sector engaging with the education sector, I think it's woeful. When you come to the World Economic Forum, I think business leaders stand up and say that, you know, skills, jobs, these are very important. We can't find the human capital to drive our businesses further. And yet, can I ask you how many education ministers attended the World Economic Forum? Probably none. And I know this is a slight criticism of you, in that sense. But I think it's a very important point. We actually conducted a study called Business Packs Education. It's a campaign that we led. And what we found, based upon Brookings research from 2011, the private sector, just from its corporate social responsibility pool, spends 16 times more on global healthcare than it does on global education. And so the question is in terms of why is that happening? And when you think about what the likes of Bill Gates have been doing, you know, it's about leadership that's been provided to healthcare, global healthcare. And so then the next question that arises is who is the Bill Gates of Education? And you immediately realise that that's really a dumb question to us because we need a hundred, you need a thousand Bill Gates out there to actually push the conversation on. And that's what the campaign is about. We want business leaders to come on board and say, yes, we back education. And the one commitment and the one ask that we have is from the corporate social responsibility programmes that they have, ringfans 20% for global education in the areas that is needed the most. Now, if companies were to do that, we'd raise another $2.5 billion, just from the Fortune 500. Just imagine if that applied to all the companies in the world. I think it would be a great campaign. And I know that Barclays have supported us, which I'm thankful for. Stephen, yourself, any of you views that your own efforts or your bank services have just been lauded by Vickers. You've got off when that happens. Indeed. But from your perspective, could you see any better potential for greater co-operation or better more effective models for working to make sure that the gaps, the right gaps are addressed in the right order? I agree with Vickers, there's a lot more we can do. And it's becoming more and more important because as digitalisation happens and robotics and new technology comes in, we really have to upskill the masses into the next level. Otherwise, they're going to find it more and more difficult to be productive and actually have a job. And so I think it's incumbent on every corporate and every Government to really put this in a centre focus. And it's something we recognise as banks digitalise and that we will need to upskill our staff. And so we have those viewers that if you come in as a teller, you must leave with a degree. And if you come in with a degree, you must leave with postgrad, et cetera. And that's a key focus on us. What's interesting though now, and this is where I really think we can make a difference, is that we do a lot of these training courses for our staff. And in the past, it all had to be classroom based, which wasn't scalable, but with digital technology now, there's nothing to stop us putting that on a platform and just putting it out there for people to use. And so that's where we're heading now. And hopefully over time, we'll really make a difference just allowing people to use what we've done for our staff just generally. And this makes good business sense. In terms of public-private cooperation, it doesn't make sense just to do this on your own, to just look at the shortcomings for what's a better word of public education and then add on according to your needs. Or is there a model or a need, you think, for actually closer working together, because maybe? Well, I think the biggest thing that the private sector can bring is its disruptive nature to education. And I don't say that lightly. If you look at everything else in the world over the past 200 or two years has changed, apart from education systems, we still address classrooms in exactly the same way that they have been done. And so what the private sector has demonstrated time and time again is its ability to disrupt. And I think that's a really important thing that they can bring. I tell you one practical example that when we go around the world speaking to school leaders in particular saying, what would meaningful engagement with a private sector look like for you? And they often say to us, actually, we would like businesses to come into schools and speak to our children about careers advice. When I was growing up in London, we had a woeful careers advisory service, and we still do, apparently.  And so if there's one thing that we can talk about is when you talk about the future of jobs, actually go into schools and tell them that actually these vocations that you're thinking of will not exist when you graduate. So let's not focus on those. But have a look at these different competencies, these different areas of growth, and you should think absolutely about how you prepare for the future in that way. Interesting point. There won't be time for questions, but please allow me to indulge myself one more time. We had a gender breathing earlier on today in midday, and the conversation centered around the fact that Africa was actually punching above its weight, as it were, compared to other regions in economic empowerment for women. But what it was failing to do was the basics, getting health and education rights. And Sadiad Zahidi, my colleague who heads our gender parity efforts, made the point that we're doing a further piece of research on the future of jobs. And in fact, whereas Africa is in a relatively strong position in terms of economic empowerment for women, I'm taking the gender angle here, there was a very real fear that a new gap could emerge, as we see the need for more STEM training and higher value qualifications to cope with the changing global environment. Would you agree, Stephen? Well, if you have a look at the nature of it across Africa, and certainly what I've seen across our countries is what's coming out of university is very gender equal, and that needs to move further and further down, because if we don't harness the whole workforce, you're never going to get the productivity and the change you need. So for me, it's actually a maths thing, if I can put it that way, that we have to get right because otherwise we're not really going to move into the new productive world. So I think a world where you have got gender inequality, I think you've got a problem. I mean, my personal view is I've seen it in my own business is that once you get that gender parity across your business, and especially in your management structures, the diversity of that makes your business far more sustainable, and you get different views on how to do things, you get a diversity of ideas, which I think just going back to Vickers' point, is today that disruption is very important, because if you're not on the forefront of that, someone else will be. So I think it's an absolute business imperative at the moment. And Vickers, how are you helping driving leapfrogging in terms of using technologies for education, STEM, et cetera? Shall we hear so much about it? Yeah, so our foundation, for example, has been undertaking a project in Ghana called Making Ghanaian Girls Great, which is, when you think about technology, people think about the whizbang and the latest gadgets and everything. Actually, we use what is very steady, sturdy technology, it's called VSAT technology, in Ghana, where we have set up a classroom in a studio where a teacher broadcasts to 70 schools in rural Ghana. And at the school end, there are facilitators, experts facilitators that engage the students and make sure it's interactive in terms of the learning that they receive. And it's a pilot project funded by the Department for International Development from the UK that we are seeing promising signs of working. We're about to expand that program and that method to actually train teachers also in Ghana itself. So technology, we can get carried away speaking about the latest stuff, but let's not forget also in much of Africa, paper and pencil is still the most up-to-date technology you can use, which is missing in classrooms. So if we were to do a better job at actually providing classroom resources like that, again, this whole argument about providing a basic education, you can't really achieve that without having the infrastructure required to succeed. And so I'm cautious when it comes to speaking about technology and just meaning iPads or things like that. Do we have any questions? Okay. Before we close, I'll just ask one more. This is the meeting for public-private cooperation after all. We're nearing the end of the meeting. We're not quite there yet. But what has been your key takeaway or your key action point coming from this meeting? It's a question to both of you.  Well, what I've found very encouraging is my meetings with the government officials, the ministers, the central bank governors, is there's a much bigger cooperation agenda. So for example, in South Africa we spent seven years working out a model for private-public partnership in renewable energy. And it's now accepted by the international community, the financiers, the operators, the generators, as working. And we're getting a lot of inward inquiry from the other countries as how do they leapfrog that, which in the past never really happened. Everyone wanted to boil the ocean and do their own bit. And the levels of cooperation across the countries I think has increased exponentially over the last four sessions I've been to. So that's been very, very encouraging for me. The second thing as well is the robustness with which the governments are looking to manage their economies. When you sit and have discussions with them and say, you know, what are you doing about the low oil price or the low commodity prices, what does that mean going forward, their planning and their strategic ability around that is so, so much more advanced than it was five years ago. And that gives me a lot of confidence and a lot of hope that we can actually get through this and continue this growth spurt we've got. So I'm actually very excited post this conference. The last thing, if I may, is that just having a look at the quality and the number of attendees that came this year. The global interest in Africa has really started to take shape and there's a lot of energy and that itself is going to drive a lot of change in the region and that for me is very exciting. To go back to your second point, I believe it's Annabelle Gonzales from the World Bank said yesterday that the oil price volatility that has hit so many African economies is actually going to be a good thing. It's going to force policymakers to look at structural reform, whereas they may not have if the markets had remained high. Do you agree? I agree absolutely. The interesting thing going back to my first point was, is that when you have the discussions, there's a real debate and you can see there's been a lot of thinking about this, where to next. There's a real term plan coming in, whereas before it would be, okay, we're going to sort ourselves out. You even saw in Zambia, for example, they've changed the royalty level on copper mining just because the copper price has gone on. So they realise this industry is important. So the government made a decision and that's fantastic, because they're really having a look at the long-term viability of everything and trying to partner, especially around power and infrastructure, with as many private sector partners as they can. That is a huge step forward. I really do think the next five years you're going to see a lot of change and there's a lot of opportunity for private sector to operate with government because they're embracing it and that's fantastic. Vikas, your action points from this week's meeting. I think there are three main points that actually stand out for me from this summit. Firstly, I want to congratulate the World Economy Forum for Organising in Cape Town again and it has been very dynamic and it's had a huge turnout and we found it very productive. The vibrancy that you get in Cape Town in South Africa actually is unmatched in other Africa summits. So I think it's a really important thing to do it here and you do it very well. There are three things that I take away from my discussions on education that, when I spoke with various public figures or captains of industry, there's that recognition that the private sector, whilst doing their own training programmes in their own companies, I think which are critical and important for their supply chain and for their businesses, actually need to engage further down and much earlier on. I think that recognition, that acceptance I think is coming through. The second thing that I wanted to mention was everyone speaks about the skills gap that exists in Africa and there's an acceptance that we need to look at innovative models of finance when it comes to actually paying for these kind of programmes and so whether you're looking at government-based schemes or whether you're looking at private sector corporate training, we have to think in terms of how we can make sure that at the lower level, so artisan level, how we can promote skills and greater skills base. The third thing is that on the question of leadership, it's actually something that Grachael Michelle said, was that we accept that leadership is important, but yet we do not see teachers as leaders and I think that's a really important statement to make in that if we expect reform to happen in education systems, you cannot do them without teachers and so when it's a very lazy argument to say that all teachers are not up to scratch, they're not doing their jobs, actually it's a massive systemic issue that we need to address and there's a greater acceptance of that but the reason I come back to the teacher point is that we should be celebrating and we should be fighting for greater respect of teachers because they deliver so much at school level so those are my three main points. Thank you very much back to you, thank you for joining us here and thank you for our audience online, this session is now closed.


Alright in this video I'm going to go through and review my contributor earnings over the years on deposit photos. I'm going to take a look at some of my best selling photos on deposit photos as well as some tips and tricks on how to use the site. Now if you're looking for a video about how to sign up for a contributor account on deposit photos I'm not going to get into that in this video but I do have another video on that and I'll put a link to it up here. So let's jump into the stats. So far I've earned about $1500 from my photos on deposit photos. Now that's a lot less than I've learned on sites like Shutterstock and Adobe stock so as a result I haven't spent as much time uploading photos on this site. I pretty much just uploaded a bunch of photos seven years ago and left them. So in this next chart I'm going to show you how much I've made each year and also on the chart it shows how many photos I uploaded each year with the red line. So the blue bars show the earnings each year and again the red line shows the number of uploads in that year. So notice that in 2012 I uploaded about 50 photos but I only made 30 cents. 2013 uploaded 141 photos and made under $40. Now I haven't uploaded a single photos since 2014 but in 2019 I made over $300 from those photos. I uploaded six years earlier. Now I recently uploaded 100 more photos in 2020 to see if I can raise my earnings. Now looking at the chart 2017 stands out. I hadn't uploaded a photo in over two years but it more than doubled in revenue. Let's dig into what happened here. So looking at my stats on the site on June 23rd 2017 I made over $60 with 36 on demand sales and on July 1st 2017 I made over $120 on 72 on demand sales. Those two days accounted for almost 40% of my yearly revenue. That's just crazy. Now on deposit photos you can see all your sales so I can go back to see what sold on those days and if I take a look at it all of these sales were for just one photo. So there's no way for me to figure out exactly what happened so I didn't really notice it until after the fact but I figure what happened is deposit photos maybe included my photo in one of their email campaigns to all their buyers or maybe they added one of them added that photo to the homepage and it just resulted in a lot more sales for that particular day but then after they removed it from the homepage maybe you know it went down to the regular sales. Now let's go in and take a look at my sales by photo. Now one of the cool things about deposit photos is that it does show you all the stats it shows you how many times your photos were viewed as well as how many sales you get. One thing that's not so cool is you can't actually sort by revenue. So what I had to do was I went and cut and paste out of the website and put it into a Google sheet and so then I could just sort it and I'm here I've sorted it by my most popular selling photos. Now as you can see the vast majority of sales just come from that one photo and I uploaded it back in 2012 and I'm still getting almost daily sales with it. Now the reason that is is it ranks really high in a lot of the search results. Buyers searching for just Canada in deposit photos are going to see it on the first page and if you type in Canada landscape then it's currently the top search result. Now these search results are going to change over time as a chance that my photo will go down but it will typically happen fairly slowly. Now I show you this not to brag but just to give you an idea of the type of sales that you can get from just one photo if it makes its way up in the search results on deposit photos. Now my second highest revenue photo is on the first page if you search for Toronto. Now searching for Toronto is not as popular as searching for Canada or Canada landscape but it still gets me enough sales to put it in my second best photo for revenue. And the third best photo for revenue is on the first page if somebody goes on deposit photos and searches for Vancouver. Now if you're watching this video and you've never uploaded any of your photos to stock and this is motivating you to sign up for deposit photos I'd probably recommend against that. I make a lot more money with the same photos on Adobe Stock and Shutterstock and there's other videos on this channel where I talk about that. So if you're looking to get into stock photography Adobe Stock or Shutterstock are typically a better place to start than deposit photos. However if you already upload your photos to Adobe Stock, Shutterstock or maybe some other stock sites deposit photos can be a great additional site to put your photos on to just get a little bit extra revenue each month. However you have seen some success on the other stock sites then you know adding another site to your workflow just adds additional time it takes to get every single photo online and it can get quite time-consuming if you want to do a lot of different sites. That's why we created a website called Fortaloo. What Fortaloo does is you upload your photo once to Fortaloo. Fortaloo will suggest the keywords and then take that photo and upload it to all of your different stock sites just saving you a ton of time getting your photos online. Now if you don't have a Fortaloo account yet we do have a free version and I'll put a link down in the description where you can go sign up for free try it out and then let me know what you think. If you already have a Fortaloo account and you just need to link your deposit photos account all you need to do is just go into options and then click on link deposit photos enter in your username and password and you're good to go. You can upload your photos to deposit photos the same time as you do the rest of your stock sites. Now if you want to sign up for a free Fortaloo account click the link down in the description or also put an icon on the screen. I also have another video about how to sign up for a deposit photos account and I'll put a link on the screen here. Best of luck selling your photos online.


I hope you all are doing fine with the grace of Allah subhanahu wa ta'ala. Welcome to Aliyah's Kitchen. There are only three or four days left until Ramadan. May Allah subhanahu wa ta'ala allow us to reach Ramadan and cause our sins to be forgiven. Ameen. Today is the last recipe of this series and we are making beef meatballs or kofta inshaAllah. I would like to tell you a secret of delicious cooking today. And that is always make Pudu before starting cooking and send salawat on Prophet Muhammad sallallahu alaihi wa sallam inshaAllah. Then there would be lots of barakah, khair and taste in your food inshaAllah. Please like, subscribe and follow Aliyah's cooking channel. It's time to start our recipe ingredients now. So, here are our ingredients. This is one and a half pound minced beef. One cup of water for cooking. One tablespoon of ginger paste. One tablespoon of garlic paste. One tablespoon of Kashmiri chilli powder. One tablespoon of roasted coriander powder. Half tablespoon of salt and half tablespoon of turmeric powder. This is one tablespoon of fenugreek leaves. I took two medium sized onions and then I just deep fried them. So, it would be approximately one cup of deep fried onions. This is one cup of yogurt. And here are our some spices for making curry. One teaspoon of turmeric powder. One teaspoon of roasted coriander powder. One teaspoon of salt. One teaspoon of Kashmiri chilli powder. As I always mentioned that I use Kashmiri chilli powder because it is not that spicy. But it gives a very rich color to your food. If you are using the regular chilli powder, so then use less amount of chilli powder. Half teaspoon of cumin seeds. Two tablespoons of oil. Two long green chillies. I took two bread slices and then I soaked in water and just squeezed all the water. Half cup of mint. One large size onions. One teaspoon of ginger paste. One teaspoon of garlic paste for making curry. Half cup of oil. And one to two tablespoons of chopped cilantro. Yes, so this is our first step of this recipe and that is making meatballs. So, first we are adding chillies in the food processor and we are adding this one large size onion. Bismillah. That's why I did not chop it earlier because I know in food processor everything will be mixed inshallah. And we are adding this mint. Mint leaves. So, Bismillah. Now we will... Thing is chopped now. Now we are adding meat in it. Bismillah. This is one and a half pound. Now we are adding just half of it right now. Bismillah. And then we will add all the spices. This is one tablespoon of ginger paste and one tablespoon of garlic paste. Bismillah. This is one tablespoon of Kashmiri chili powder, half tablespoon of salt, half tablespoon of turmeric powder and one tablespoon of coriander powder. Roasted one. Bismillah. Now we are adding two tablespoons of oil and this is two slices of bread which I soaked in water and squeezed all the excess water. Now we are again mixing it. Bismillah. So this beef is mixed and now we are adding the leftover beef in it. Bismillah. All the spices, bread slices are mixed and now we are adding this. And then we will mix it again. Now we are mixing it. Bismillah. So you can see this minced meat is mixed up with all the spices and the ingredients. Now we can make meatballs of this mixture inshallah. Now we are making meatballs from this mixture inshallah. So Bismillahir Rahmanir Rahim. First wash your hands and then we will start making meatballs. So I am just using this spoon so our all meatballs will be in equal size inshallah. And first some one or two drops of oil on your pan and then just rub it up and then we are using this much Bismillah. See, so this is where we smooth. First make a firm ball and then just roll it on your pan. So we make meatballs from all the mixture and these are 26 meatballs now and we will cook 13 meatballs and making curry and we will freeze 13 meatballs for Ramadan inshallah. So this is the thing. We just took the baking pan and put the wax paper on it and then make meatballs and just put on top of it and then we will freeze it as it is and once it will get freeze. So then we will put in a Ziploc or any airtight container and write it down that these are beef meatballs. Now we are mixing yogurt and deep fried onions. So Bismillahir Rahmanir Rahim. This is one cup of yogurt. Now we are adding deep fried onions which is around one cup Bismillah. We did not add any water because it is mixing with yogurt and yogurt already have some water in it. So Bismillah. We will use it after frying the meatballs inshallah. So I just turned on the burner and we are adding half cup of oil. Bismillahir Rahmanir Rahim. Now we are frying meatballs in it Bismillahir Rahmanir Rahim. My hands are washed. That's why I'm just using my hands Bismillah. We will use half of the meatballs right now and we will freeze half of them for Ramadan. This is on medium heat and now it is fried from one side and now we are flipping it. Fry them from all sides. It will take just five minutes and then we will add spices in it and here's a garlic paste. Now these all meatballs are fried and we are adding one teaspoon of ginger paste and one teaspoon of garlic paste. Bismillahir Rahmanir Rahim. And now we are sautéing it. So we will just fry this ginger garlic little bit for few seconds and we are making meatballs curry. Whenever you want to make meatballs curry just take out the frozen ones and then just this is the step. So now you can directly fry them and then start making curry. You can see that all the ginger garlic paste are fried and now we are adding the spice, turmeric powder, salt, kashmiri chilli powder, half teaspoon of cumin seeds and one teaspoon of roasted coriander powder. Bismillah. Mix it. Bismillah. Now we are adding the paste which we made it earlier, the yogurt and fried onions one. Bismillah. Now there is some remaining in the jar. So now we are adding one cup of water in this jar. So we will use all the paste inshallah. So we just shake it up and now we are adding this water. So we did not waste any paste. Alhamdulillah. Now we are mixing it. Now we are cooking this for 20 to 25 minutes till the meatballs will get tender inshallah. Now covering it up. So this much water would be enough inshallah. Just cook it on the medium heat inshallah. Now these meatballs are ready. Alhamdulillah. Bismillah. As you can see, all the oil is on the top surface. All the water of yogurt is dried. And now at this point, we will add this dried fenugreek leaves inshallah. So just crush it with your hands. Bismillah. These are kusuri methi you can say or dried fenugreek leaves. Bismillah.  Bismillah. We will just put it on very low. Flame for just one minute and then we will take it out inshallah. So we are covering it for just one minute. Bismillah. See, it is all done after one to two minutes of adding kusuri methi or dried fenugreek leaves. Now we are taking it out in the dish inshallah. I can't explain it. What an awesome aroma is coming from this meatballs. Alhamdulillah. Now we are garnishing it with chopped cilantro. Bismillah. Curry is ready now. You can serve it with naan or with rice pilaf or with any plain rice. Make it at home and I'm sure that you will love it inshallah. Please remember me in your prayers you win Ramadan. Take care. Stay safe and stay blessed. Assalamu alaikum from Aliyah's kitchen.


Thank you very much for the invitation to speak at this wonderful seminar, which I very much enjoyed going to. So this talk will be about the alternative and hypothesis and point processes. And this first slide is that this talk will be reporting on on work in two joint papers with Brad Rogers. There are correlations on the alternative hypothesis 2020 and limited mimicry of point processes by point processes supported on a lattice 2021 results in the first paper were obtained independently by terrorist Terry and his blog post. I'll say a little about that a little bit about that later on. The topics I would like to cover our history of the GUI hypothesis and the alternative hypothesis. And then I will discuss point processes models. And then the third part of the talk. We'll discuss the point process that behaves like the alternative hypothesis and can't can't be discriminated from the GUI hypothesis on the known data. And then the, there's a trade off on on how much on a band limited functions which I will describe briefly at the end of the talk. I've prepared slides for more but when I tried to do them. I ran out of time. So the GUI hypothesis, the Gaussian Unitary Ensemble concerns the distribution of normalize zeta zeros up to high T as T goes to infinity. The alternative hypothesis concerns an alternative distribution of normalize zeta zeros up to high T, which we've been unable to rule out, and it's believed to be false. I mean totally false and and and the number theory goal is to roll it out. And this hypothesis is related to the possible existence of exceptional zeros for Dirichlet L functions with real quadratic characters, which, which show up and estimating class numbers for quadratic fields and a goal. Since he goes paper in 1935 is to show there aren't any exceptional zeros. So this talk will be concerned. The number theory is concerned with the, the, the spacings of the zeta zeros. And the number of zeros of zeta of s in the in the critical strip from real part of between zero and one on an imaginary part between zero and T as a well known formula titch marsh T over to pi log T over to pi minus T over to pi plus a small error log T. But we are only concerned here with the main term there, which is the one over to pi T log T plus so of T. So it says the average spacing of a zeta zero up to high T is to pi over at around high T is to pi over log T. So there are two ways to handle normalized basings. So if you just in the bottom if you just take the ensemble of all the zeros up below T, you can rescale them all, we can rescale all the zeros. One at a time, you can rescale the spacings by multiplying by log T over to pi uniformly on that ensemble of zeros, or else you can rescale them separately one at a time. Yeah, gamma J tilde where you instead of T use the current value log gamma J to pi and then you do it once and for all. And then the with that definition the normalized spacing will be gamma J plus one minus gamma J, which achieves average spacing one. Why one is interested in if you take random Dirichlet series if you even go a little bit to the right of a half and you look at the spacings of where the real parts are equal to zero or where the where the thing is zero it's oscillating you'll you'll get a density that's the same density but the zeros will be spaced completely evenly like a clock. And there were they won't show any fluctuations. But on a critical line they apparently they do and, at least on Riemann hypothesis. So Conrini-Weinrich, if, if you can find a large fraction of zeta zeros having spacing, a little bit less than a half the average spacing, then you can affectivize Siegel's theorem and you can prove that for a sufficiently large T there's a real Dirichlet character. For every real Dirichlet character L, one chi is bounded by log q to the minus to some explicit constant. And this is generalized by the Dirichlet class number formula that the class number of a real of an imaginary quadratic field will be bigger than a square root of q divided by a fixed power of log q which then would allow you. Well, to, for example, to prove the prove the effective finiteness of the genus quadratic forms the idonial numbers, an open problem, long standing open problem due to Euler. Okay. So I remind you on the on the history of what's known about the normalized spacing of zeros and so the breakthrough result of Montgomery showing in 1973 shows that indeed there are fluctuations and he proved a positive proportion of the zeros have spacings less than 0.68 of the average spacing. The methods been that method has been refined and the record is current arrow Chandee Littman and Malinovich who showed and on our age, got it the number down to 0.60. And there's an averaging method due to Montgomery and odalisco that puts in weight functions. So you don't necessarily and allows you to show zero small infinitely many zeros with smaller spacings and they, they originally but without getting positive proportion, and they got down to 0.5179. And the record is 0.5154 and the Riemann hypothesis and a preprint put up earlier this year. Analyzes thing and says that the Montgomery odalisco method cannot in principle get below a smidgen above 0.5. So we don't we don't get below a half. We hypothesis. So this is it's evolved through many forms this is the form most convenient here. You take the sync function sine pi x over pi x, which is a nice smooth function and it interpolates to be one at x equals zero. It has an improper integral that's equal to one. And the hypothesis is, if you take in any test function that's a Schwartz function on on n dimensional space and you average it from t to 2t sampling against all the ordinance of the zeroes distinct ordinance of the zeros in the range from to 2t, then the limit as t goes to infinity will converge to an integral the test function against the sine kernel determinant, which is the determinant of this n by n matrix, which will be as it says the endpoint correlation function of a well for what we would do here a point process, which is the sine kernel process. I remind you of one of the many famous plots of odalisco, which is the normalized consecutive zero spacings histogram of them. Maybe this is the one for 10 to the 20th, the, the curve plotting through the points is the GUE prediction giving great agreement and a positive proportion of the normalized spacings are below 0.5 so that that would, according to the Conry and Bonyek's theorem would give an effective version, if we if we could prove it. So that's what the data says. The known results. Montgomery in 1973 pair correlation, a large calculation of Headshaw doing the triple correlation. And then the seminal work of Rudnick and Sarnac doing endpoint correlations. There's, there's a misprint here it's for fixed n and band limited test functions data on S of R n, whose Fourier trend, well, the Fourier transform of the test function is required to be in a bounded domain, given there. The fact that the Fourier coefficient of the Fourier transform is in a bounded domain makes the is equivalent to the function being band limited. Kind of the definition. And then we get the conversion I said earlier you, you take a integrated test function and as T goes to infinity it will, it will converge to the correct limit of the sine kernel. So, of the GUE distribution. So what this means is, we don't see the GUE distribution directly we we see it when we probe it with a test function, and we have a limited class of test functions for with which we probe it this way. And then do is verified against these test functions. The alternative hypothesis. So let's take those normalize a to zeros, and the alternative hypothesis says you get a very different distribution for all sufficiently large j, the differences, the consecutive differences look like they are half of an integer with it with an error going to zero is you go to infinity this is a very strong form of it. So if the, almost, and this is almost all of the pairs are doing this. So then, if you, if you take things that are far away they, they still likely will have a half integer difference so it's saying that the differences are mostly half integers. This is a straw man it's discussed in Conry's Riemann hypothesis survey and the notices of the AMS in 2004. I'm not sure if that's the first place it appeared in print. There's a there are very in the, in the last 10 years there have been a number of variations staring, studying the consequences of this hypothesis in detail. So we would assume our age, we allow the value zero which allows multiple zeros. But we would, we would expect simple zeros. I'd like to mention some conventions on the Fourier transforms that are in this talk. So the Fourier transform in this talk will be the, the integral against exponential minus two pi i times they product X Xi Xi. And then minus two pi i for the inverse Fourier transform. The important thing is with this definition the sync function as I gave it is the Fourier transform and the characteristic function of the interval minus a half to a half. It's a band limited function with that frequencies in the band minus a half to a half. The signal analysis, people use the usual Fourier transform without the two pi i. Again, the definitions of band limited and how wide the bands are might differ from this by a factor of two pi. Two pi sorry. More statistics the GUE two point correlation function. Do you allow the coincidences the diagonal that it would be the temperature distribution the delta function and zero plus one minus sine pi X over pi X squared. The form factor introduced by the physicist is essentially the Fourier transform of this. The transform of sine pi X over pi X squared is a convolution of the box car with itself so it's a triangular function supported on minus one to one. Okay, that's so I can put up the one plot where you can tell the GUE apart from the alternative hypothesis for two point correlation functions. It has a triangular gap near zero. It has a spike at the origin which is the delta function and it has it's flat outside one and the form factor for the alternative function is the same between minus one to one, which is where you can apply those test functions and then it repeats periodically with period two. It's easier to draw that form factor because if you drew the histogram the histogram for the alternative functions will just have spikes at the half integers. Now I would like to flash to the past and discuss the pre history of the alternative hypothesis. One of which appears in a talk of Heath Brown given in the aim and our age conference in 1996, which a few of us here were at. This talk is available on a video recording, which you can find on the web posted by him. Heath Brown never wrote the paper. So what what we have is the talk but anyway what the talk does looking at it is that find something like the alternative hypothesis produced by the exceptional zero. And the heart, the starting point in that talk is to assume you have a. You have an L L value it should be L one guys less than Q it's misprint Q to the minus one quarter minus delta. This violates the seagull, the seagull ineffective lower bound with which that that band proves there only finally many such But we are going to see the consequences of assuming you had such an L value, which would have a zero very close to one. And you want to produce the ideas to study things and eventually go back and get a contradiction. But this analysis has some true true things in it. He takes the zeros of the dedicated Zeta function of the imaginary quadratic field which is Zeta of SL of us Kai. He takes the function f of s which is a normalized version putting in the gamma factor. So that has the functional equation f of s is f of one minus s. So approximate it with a function that replaces L of s Kai was Zeta of two s and then fixes all the Euler factors up to height q where it differs from L of s Kai. So there's a difference that the primes dividing the conductor q finite set and then, then you have to switch the signs of some of the Euler factors where Kai piece plus one up for P less than q. h of s plus h of one minus s is supposed to approximate f of s, you've symmetrized it with the functional equation. This is a factor of a half somewhere. Not not sure about that but anyway, the sequel zero would imply most small primes have Kai p equals minus one so this last product will push things. Small zero. The first theorem he announces that if you had such a thing, then the mean value of f of s minus h of s plus h of one minus s will be very small with this factor q, q to the minus delta will give you and he said it applied for q to the a not for some large a not from T up to a very large value, but we're going to be interested in what happens where it's at this bottom M near cube a not. And he stated that they that the h of s plus h of one minus satisfies the Riemann hypothesis. And then he can prove it way we really needed to prove it for a range of heights. In this in this range of heights where he's going to do things. Anyway, that that that that result is interesting and I'd like to see more on it. The second resulting announces is that if we're now looking at n of T which is the counting the zeros of Zeta of s times L of s Kai so it's going to grow like one over pi t log T. And he says the the g function has the Riemann hypothesis so it's got it zeros one one half plus it and and presumably his proof will show they are simple zeros. And for each zero of this he's going to find unique simple zeros a of s L of s Kai very close to it. Okay, and so he's getting the full set of zeros, because n of t, because this thumb of the zeros will grow like one of our pilot keys. So almost all the zeros and then his assertion is. If you're down at the bottom end of the range you get TM minus TN all of all the, all the differences will differ by a half integer, plus a plus a small error. Which would be what this alternate hypothesis asserts in this equation I've divided his equation by two pi which is the density of the zeros of either is eight or L s Kai so you can see the half integer on the right. And he also states at the end of the talk something like the Fourier transform of the pair correlation is close to that of this alternate hypothesis form factor and namely that exists, it exhibits the period to periodicity outside for some finite range of one which increases up to two. So the ingredients of the alternate hypothesis are already in this talk. Now what has happened, what has happened since of course is he assumed a very strong hypothesis which we know is was wrong. But if you replace end of the one quarter with a much weaker hypothesis that says that the sequel zero is at a distance log log q to a power which which we have not ruled out. Then some weaker version of the same stuff will be true. And would would follow and so you would still see something like the alternative hypothesis. Okay, on the point processes. Well, I mean, the, the GUE and the CUE they've been studied in great detail with with the finite matrix models and we look at the GUE limit from the viewpoint of infinite finite processes rather than finite random matrix theory. So the random matrix I advise statistics involve taking a limit where the size of the matrices go to infinity but in this point process viewpoint we're going directly to n equals plus infinity. And we will we will look at infinite point processes on on the line with average space thing one. And finally, there's all this machinery to define point processes I mean, but informally we're throwing down a random set, possibly infinite points on to a space X so the actual measure space is a configuration space of the allowed set of configurations got a topology Braille sets and measure on it. Here I'm going to consider only the real line. Or, or a lattice wells, a space of lattice points on the real line. And then we are throwing down points on to these regions. The points may have finite multiplicity. And then when you sample you get a sample thing which will be a set of points a process will be simple if all of its configurations. You never always have more publicity one. These are sometimes called fermionic point processes. And then the information describing a point process for for a measure would be the moments. So this is analogy the endpoint correlation functions or intensity vent measures. The question functions then, if it's a continuous density boy, we row van times DX one through DX and they, they, they encode information on the expected weighted counts of endpoint configurations in these spaces when you integrate over some finite region. Some compact region for example, but we're going to create them against test functions. So if the if the test function is one inside a region and zero outside. Then that would correspond to getting correlations in that finite region. And the correlation functions will encode the expected values of these test functions in the regions. We can either use compactly supported smooth test functions. And then with some under some circumstances for its functions. And if you have the discrete function on the lattice we replace the integral with a sum. You'll notice the sum has a form resembling that which goes into the Poisson summation formula which is certainly what's used. As under some restrictions the correlation functions uniquely determine the point process. That's, let's say that's the standard state of affairs, but in the moment problem for measures it's known their their bad example or measures are not uniquely determined by the moments. If the moments don't grow too fast this doesn't happen. And in the papers we impose assumptions which put you in the case where you uniquely determine the point process and all the point process we're going to discuss will satisfy those conditions and therefore the correlation functions determine everything. They uniquely determine the process. The conditions are uniform local moments and some extra bounds exponential bounds in the uniformity. It is known there exist examples where the correlation functions don't uniquely determine the point process. A really big issue with point processes is if you're given a putative set of correlation measures. There's a lot of necessary conditions they have to satisfy but does there actually set does there actually exist a point process. Satisfying having these correlation functions. The answer is sometimes yes and sometimes no even when all the reasonable conditions satisfied there can be extra reasons that stop it from existing. So, the next two slides are just about how one can think about how you could produce a point process from Zeta zeros, you can consider the normalized basings of Zeta zeros at a particular high T. The finite ensemble of these normalized basings they give you a single a single set of points, but we will now allow random translations of these points. We will shift them all to the origin and then we allow random translation say up to T to the fourth, and then snip it off at this interval minus T to the four and then you get a then you get a configuration space of sample point sets on a finite interval. Then you, then you can run TJ off to infinity you'll get bigger and bigger intervals and you can hope that these sample spaces which are behaving like discrete point processes can converge in some limiting process to to a translation of the finite point process on the whole real line. Or you could ask for limit points of this set of zeros at height, G say that go to a limit translation invariant point process. For instance in the mathematics as you would form empirical correlation functions for the finite ones and then hope that your empirical correlation functions all all converge in the limit to the correlation functions of a point process that would be the kind of mathematics you would do. Anyway, a very strong form of the GUI hypothesis from this viewpoint might assert that all sequences of normalized data zeros converging in the limit to the sign process. So this limiting GUI processes called the sign process in the, in the point process literature, in the sense that all the correlation measures converge. We're not doing any of that in this talk. However, there is things like this have been studied in the literature and Shahibi Najnudo and Niccabale for example, try taking random characteristic polynomials of the CUE distribution and then they can go to a scaling limit and infinity and produce a model of random functions whose zeros then would function as a point process. And so they succeed to do this, but the issue is, so our question could be, could there be such a limit of normalized data zeros, if there were for example infinitely many single zeros and you sampled at a high TJ corresponding to that behavior in the talk of Heath Brown, then maybe you could extract a point process that's the alternative hypothesis in some scaling limit. Okay, well, if you're going to do that, then you would like to know that there actually exists a point process that could fill the bill. So you have to solve the existence problem for point processes. And that's what this talk is about is to exhibit a specific point process that does have the alternative hypothesis type statistics, and which will agree with the Rudnick Sarnac formulas on all the endpoint correlations that are in the in the Rudnick Sarnac theorem so that you will be unable to tell it apart from the GUI prediction. I did disturb you sadly. Yes. I just said very sadly. That you can't distinguish. I'm just lamenting the fact that you can't get rid of it. It's Peter. Yeah. Thanks Peter. I was about I was expecting you to say that I made a huge mistake. We're just. Okay. So, let's go back to the translation and variance. A point process and R will said to be will said to be translation and variant if you if you if you if you translate the process you get exactly the same distribution as before you get the identical distribution. In particular that would force all of the endpoint correlation functions to also be translation and variant. And so then we say it's translation and variant in the correlation sense. In our case the correlation functions determine everything. So the two notions of translation and variant are equivalent, but not in the other case. And when we took these limits, we looked only at, we were shifting around to get translating invariant point processes in the limit. So let me just say the Poisson processes are all translation and variant and the sign process that we will come to is translation and variant. The sign process is a determinant until point process. So these are point processes that have level repulsion near the near the origin. A process is determinant all if there exists a function. Okay, a kernel function are cross R to see on the whole real line such that has the property that if you restrict it to a compact regions, then this, this function will be an L2. And it's such that the correlation functions are given by the determinant. From I J to N of K X I X J. And, and if it's going to be translation then that translation and variant that kernel will have the form K X I X J is K of X I minus X J, I believe. I would say that there is a complete characterization of which kernel functions K give you a give you a realizable determinant all process first in Machi and then detailed in proof and Shashnikov. These conditions are that on a finite region it's a symmetric operator and it's got trace class, eigenvalues, and all the eigenvalues are between zero and one, or it's something like that. It's known that determinant all point processes are uniquely determined by their correlation functions. So in particular, there's only one correlation function for such processes and if you, if you have a set of correlation functions you can decide whether or not it could be the if it's a correlation function of a determinant all process. You know it's determinant all for sure. There isn't some non determinant all process that matches it. Okay, so the first existence theorem is there exists a determinant all point process the sign process whose kernel is this translated sync function. And this process is a simple point process and it is translation and variant under our the. There are several places in the literature where you can find proofs. They have to have to go by very verifying Machi conditions. No. Okay, so what we're going to need is a discrete version of the sign process. This is a version of the sign process that sits on the lattice AZ. And for several papers have discussed this. This process are used it, but we couldn't find a proof of existence. So we included one checking Machi's theorem in the in the paper. And the proposition is that for for each parameter a between zero and one there exists a process the a discrete sign processes whose configurations will obey that expected value formula except on the right hand side they give you a discretized form of the sign kernel on this lattice AZ AZ to the end for the endpoint correlation function. This this process is a simple point process, and it is AZ translation and variant. Okay, but it doesn't exist for a bigger than one. So that's a case where there's some there's some constraint to this. Okay. So now I want to discuss the construction of an alternative hypothesis point process. So it will start with the discrete a sign process that it will show that its correlation functions integrated against this range of brain band limited test functions agree exactly with those of the sign kernel process. This will apply only a phase between zero and a half. Okay, so, although the process itself is defined all the way up to a equals one these processes. The end bigger than one and any shorts function in in this class that's supported on the, the box minus one over to a to two a to end. So the box never gets bigger than one, because a is less than a half, then the claim is that the integral of the expected value of this thing. The left hand side is the discretized thing but the right hand side is now the continuous thing integrating that that function over all of our end. So that is the. So this is. So this is the key, the key lemon. Okay, so there is a proof of this lemon on this page. And of course, no, no one can follow proofs that are quickly going through. We want, we want to verify that to prove this equivalence, we want to verify for all test functions with Fourier transform supported in there that the equation star holds that the discretized thing on the left hand side, summed over the lattice exactly equals the continuous thing integrated over our RN. And the idea to do it is to use pause on summation on the left hand side of the equation. Which will turn in a sum over F half of L on the dual lattice so the, so the what the, the original lattice has side at most a half and the, the dual side will then have side at least two. Okay. And the, the key thing in this proof is to check that the sign determinant kernel has all its Fourier transform frequencies and minus one to one to the end, and that that that is a detailed calculation the original thing is on minus a half to a half but you, you have the I assume you know that that is true. Then the function eight of s debt si minus sj will have Fourier transform vanishing outside the closed box minus one minus one over two a to one plus one over two a of n, and since a is less than a half that box is inside minus two over two. But since we have Schwartz functions the, the Fourier transform is continuous so it, it, it must vanish on the boundary of the box as well so we get the open box minus one minus one over two a one plus one over two a to n. Okay, and this has the following feature with that Fourier transform, it says all the Fourier coefficients vanish except the zero Fourier coefficient, and the zero Fourier coefficient is exactly the integral on the right hand side. So that's the proof. Okay, now, now we have this discrete a sign process that's perfectly matching the correlation functions but it but it's supported on that lattice so it is, it doesn't have the translation and variance property. Okay, what it does happen is it's, it's a very another lattice but it's not invariant under our so to fix the process we fixed this make a new point process where we take the point configurations and we shift them by a random number drawn uniformly from the fundamental domain and the lattice which is a zero to a. Okay. So that won't do anything to the to the correlation functions we're interested in because they always, they always treat the difference of points in the process and the differences of points in the processes are completely unchanged so all those correlation things from n equals two on up will not be affected in the in the sign kernel because it's translation of variant. Okay. So the, the previous construction allowed us to do this from a from zero to a half. So we will choose the best point a equals a half, and we will call the process you get this way the alternative hypothesis point process. And now it's, it's correlation functions will be supported on our. They will differ from the one half discrete sign process when you did that, that averaging and the main result will say that this process when integrated against all they band limited test functions and the redneck Sonic result and additional ones when n is bigger than or equal to three will be satisfied so this is the statement of the result. The one half discrete sign process and you take a random variable drawn from zero to a half independently of this process, then the new point process you get a simple translation of variant and satisfies the matching condition for all band limited test functions, having support in cyber and psi plus one sign equals zero or sign one plus sign less than or equal to two. And more over any pair of points and any realized configuration will be separated by positive half integer distances a half one three half, so on so this thing is a completely limiting thing, and it, and it does exist. Now, I'd like to say a Terry towel in 2019 in a blog post. While we while we were writing the stuff up constructed an alternative hypothesis distribution for the CUE ensemble and fix finite and so you have an image on the unit circle so the average spacing is one over N, and he said that you can find a distribution where of matrices where all the matrices have eigenvalues with spacings between their eigenvalues integer multiples of one over and that this new distribution would agree in all of its low moments exactly with the moments of the CUE distribution. And with the, the moments growing the loud set of moments growing as n gets large, and he also observed at the end of the post about GUE that there would be an alternative construction which essentially matches what we've described above namely, discrete ties it on a lattice and then integrate over a fundamental domain of the lattice to smooth out. And I want to say that that finite construction and the very night is is is is it's it's just a very nice thing, because there are now finite versions of this. And the, the, how you handle the moment statistics is very elegant. Another remark I want to make is that actually this. When you, when you look at the higher correlation statistics with this domain you don't, you, they are not enough to uniquely reconstruct the higher order correlation function so any anything that be that behaves like alternative hypothesis must have the same two point point correlation function is what was on the previous slide but for the higher correlation things the, the whole cube is not covered and there are some regions where we don't know the correlation so they're put potentially are other alternative hypothesis like processes that do something different there and could be consistent with the Rudnick Sarnac test function data. This is a very specific one we constructed where it's, it's correlation functions are determined completely. And for example for this, for this distribution this alternative hypothesis process you could actually in principle you compute all you can compute all the, all the statistics for the probabilities of configuration so for example if you want to calculate the probability that given a point, what's the gap size to the next point then gaps size one half to the next point as probability one half minus two over pi squared, and you get a specific calculable distribution and these are the first few values for the gaps of a half or one or three halves or two between two, two points. Okay, that's the end of the alternative hypothesis part. Now I would like to, there's an underlying trade off there and that's the probability paper, which is something we would call band limited mimicry of point processes. Okay. We had, we found a point process on our and another point process, which was the sine kernel and another point process which was the discrete sine kernel supported on a lattice AZ that had the property that their endpoint correlation functions were not distinguishable using band limited test functions, having a restricted bandwidth bandwidth. In this case, we will say that the, the, the point process on the lattice is mimicking the point process on our at the bandwidth be. And then you can ask for what are the allowable trade offs between the parameters A and B for for a fixed process, where you could have this mimicry phenomenon where you the discrete ties thing matches exactly. By the way, we don't know whether there could be whether this can be done at all we we looked at two cases in detail the Poisson point process of any intensity and the sine process where you can figure out what's happened and it and it and it differs in the two cases and Okay, so the, the most basic point process is the Poisson point process is specified by its endpoint correlations being constants. The constants lambda and are the intensity of the process which are basically the expected number of points you can get in a unit interval. So if you want to compare with GUE you take lambda equals one then you'll get an average of one one point in an interval. The distribution of the number of points in an interval is given by Poisson distribution with mean lambda times the length of the interval. So they, they don't have level repulsion. One say that they, they would, I didn't put on this slide, because I don't know. I don't know how to say it. Things are as independent as possible for the Poisson process. Okay, well you can do the same thing you can make a discrete discrete Poisson process of intensity lambda which is the point process that's the number of points that each, each side are independent and identically distributed random variables which are Poisson random variables would mean alpha lambda. These processes exist for all a and for all lambda. They're uniquely determined by their correlation functions for all and and all lambda. The thing is, they're never just they're never a simple point process there's always a chance of getting, you know you're putting balls and boxes when you put down this process and the boxes are allowed to contain more than one ball that this is a, this is like. Okay. So, I, I, I thought I would only have time. I meant many, many slides and these things but I removed them from the talk. I'm just going to state the result. If, if you're, if you're, if you're given alpha a greater than zero and you're given a bandwidth, then the Poisson process can be mimicked at on a lattice AZ for minus B to be for any value of B that's less than one over a for for a between zero and infinity. And it cannot be mimicked by AZ whenever B is bigger than one over a for a for zero day to infinity, and the mimicking process is exactly a discrete Poisson process. The paper proves a general theorem which says if you have any translation and variant process. Any any translation and variant process at all you can never mimic at rate above one over a and this the Poisson process saturates that bound. And that that proves the upper bound in the picture so the upper bound in the picture this has a the the the lattice AZ on the on the x axis and it has the bandwidth be on the y axis and the red region for those of you who are not color blind. Is the region you can't achieve and the green region. For those of you aren't color blind is the. The region to the left. Okay, I say this because I am color blind. Red green color blind. Okay, and now I come to the interesting case of the sign process. So there again is a it as there is for any processes it's a band limited mimicry trade off. And the answer is different. This, the sign process can be mimicked on a AZ for bandwidth minus B to be whatever B is less than or equal to one minus a over a. And a can never go above one in this result. And the sign process cannot be mimicked for bandwidth one minus a over when B is greater than one minus a over a and that applies for a between zero and a half. And also, it cannot be mimicked when the bandwidth is bigger than one over to a, which applies when a is bigger than a half. I can't understand it and I can't understand what I said so here's here is the picture of what we know about this process at the moment there is a red region where we know you cannot mimic it. There's a green region where we know you can mimic it and there's a white region where we don't know what happens. The actual process we used in the construction for the alternative hypothesis is right there at the point where everything where we it's right at the lab the triple point in this in this picture. It's an extremal thing. So, what we can observe from what I said here is that the the band limited trade mimicry trade off differs between processes. It's not completely resolved for the sign process. And the alternative hypothesis process that we presented earlier was constructed using the process at the triple point in that mimicry plot above. I want to conclude with some open questions about this mimicry phenomenon is, first of all, is it generic or is it rare. Are there processes that you just can't mimic at all when you when you put them on a lattice that you just can't agreement. I don't know. And I also don't know whether this this phenomenon is is common or it's rare, maybe it only occurs in in in sort of discreet in in in frameworks where the process is extremely nice. For example, or the sign sign beta processes where beta varies over the random matrix parameter and beta equals two is the usual sign process. We don't know. And so to conclude the alternative my process shows the alternative hypothesis can't be ruled out current using the current results on the band limited test functions. We have an open problem of determining the exact trade off for the sign kernel process on when you can or cannot band limit, and I would say one main point is that this age point process that does exist even even whatever the status of the alternative process in this process may show up in other mathematical situations. Certainly the a discrete sign process has showed up in other mathematical situations and therefore it may prove interesting in in the future. Even even in number three. Thank you.


In America, people want to tell you, oh, it's fine. We got a high standard of living and this and that. But they're totally ignoring the police brutality, the criminal government, the executions of black people. They just blocked that out. One brother's getting shot 20 times. And the police are saying it was an accident. Now, that's impossible to get shot 20 times in that beer accident, right? Wouldn't y'all agree? So I had to get out of there. I don't miss it at all. I don't understand why everybody else don't leave. Hey, family. You're at Black Eggers of the Gamma. My name is Rick. And this is my beautiful wife. Cynthia? Or a.k.a. Sweet Juicy Mama. We're back, guys. Yeah, we're back. Had to adjust the camera and stuff. Anyway, guys, for those of y'all that might not be familiar with us, if this your first time tuning in, go ahead and subscribe. Hit the notification button. You know the little bell. Little bell. Now prepare to be overwhelmed. You see the dynamic duo, guys? So anyway, we want to welcome them to Black Eggers of the Gambia. And we're going to talk a little bit. And then we'll give them a tour around the place and stuff. All right? Stay tuned. I just want to say that it's a pleasure meeting you in person. Thank you. I mean, I've always been seeing you on camera. I interviewed you on a live stream. Right. And we're going to do this in real life. Oh, OK. Got you. Thank you. What am I right here? The one and only annoying village boy. I love farms. That's why I didn't want to interview them in their own houses. I wanted to come to the farm. Let me see what's going on. What have you been showing us? Is it really real? Yeah, man. And I'm seeing the vegetable. Can I take some of them home? I have some cucumbers in this game. Later. You can take some. OK. Next time. Next time. Welcome to the YouTube channel. Thank you. Tell me more. You know, I really want to know more. All right. Back. What does back mean? OK. Bagged. Black anchors of the Gambia. So that's what back stands for abbreviated. Black anchors of the Gambia. And basically what we've done is we are sitting on right now about 12 hectares of land. And basically what we want to do is make the transition for people in the diaspora coming back home to the continent easier. Because when we came here in 2015, there was nobody ahead of us to say, hey, here's a smooth transition. So we was running into this and that.  We found that to be distasteful. And so what we want is a smooth transition for anybody coming to the African continent. You might not want to stay in the Gambia. You might want to go to Ghana. You might want to go to Tanzania or Egypt. But here is a smoother transition for you coming into the motherland and stuff, all right? And so what we do, we have organic crops, fruits, vegetables. So when we'll be getting livestock and hotels and restaurants, and we want to hold a wouldn't buy a Miss Trudy Day where all the YouTubers can come in. You're building all of that in the tour makers? That's why we're looking for 24 hectares. This is a city, a chocolate city. Where are you from, generally? Mentally or physically? Physically. Physically I'm from America. USA. African continent, baby. You know what I'm saying? What brought you to the Gambia in the first place? I did six months of research because in 1991, we were going to go to Burundi. But I'm an ex-battern and was fighting it.  So after that, I wanted to go to East Africa. But Sweet Juicy Mama kept having kids and stuff. So we didn't leave. So 2014, we started looking at leaving again. And so I did research. Six months came up with Gambia number one, Tanzania number two, and Zimbabwe number three. And I like those countries because the presidents were talking pan-African.  So I said, I'll take it. And so anyway, we came here and visited first and we just stayed here. We were like, this is it. Which hour is that? 2016. Yeah, 2016. June. You've been here since 2016. Yeah, been here ever since 2016. No, we went back to get another daughter. And more containers. And more containers. Because like I said, we ain't got what in my money. We can't just put it. Give me that, give me that, give me that, you know. So we just. Went back and put your dollar, store it. Yeah, we came back. That's right. Back family. Hey, back family. Back with another video. And today's topic, we want to talk about what makes a good night's sleep in the Gambia. Sound like something you might want to sleep on? Check out this video. All this life being since we moved back here, it's been amazing. Yes, amazing. I will change it for the better. You don't regret coming to Africa? No. One of my biggest regrets I will have to say is not getting here sooner. That's lovely. I almost talked Rick out of coming here. I'm sure he probably would have left me and came by himself. But yes, I believe that was one of my biggest regrets. It's not getting here sooner. I wish we'd have got here a little earlier. And maybe learn the language. And you know what? It's raining on us right now. What are we going to do? You want to go to another area, or we just stick it on the rain? Because you'll get wet. Because as I take you around, now I'm going to show you. I don't know if you can see a house. There's a house back there that's 4,000 square feet. And I told you this is sitting on about 12 hectares. In America, our house was the size of our bedroom. So do I regret coming to Africa?  But that's the same. Was it expensive getting this land? This land? Short answer is no. Well, we're using multi-crops. As you can see, we've got pineapples growing, and lettuce, and... This is collard greens. Collard greens? OK. Thank you, sweet mama. What's that? That's a pineapple. This is actually a weed. In America, people use this for stroke. For strokes. So I didn't want to get rid of it, so I kept it. You can use it, juice it. You can put it in your salad. It's good for a lot of diseases. Now whatever the land gives us, we'll take it. So we're not rejecting anything. Now, come rainy season. Certain parts hold water. And then when you come in here, you have tilapia there. You have catfish, just out here. So this rainy season, I'm going to grab it and make it a palm. Which is the people working here? Are people living around the village? Right in front of the village? The most we've ever had at one time was 20. And so now we've downsized a little bit. And so right now, we've got about, like, maybe 8 to 10 guys that are working for us from the village. How do you get water in here? Huh? You hear the generator pumping? It's a series of lines that is pumping through those hoses. And we do it that way until we can get the drip irrigation system in place. Well, you just hit a button and a pump. And there's a pump coming. That's generator number two. Generator number one. There's another one back there by the 4,000 square foot house. And we're going to put another one there starting Monday. What has been the major challenges you have most to offer? Oh my god, now. It is relating to our brothers. Now, we come in, we say, hey, look, we're going to pay you above average pay for acceptable work. And guys want to come in and do substandard work. And so I got to chase them to the gate to get them out of here. And that has been my biggest struggle. Getting the bang for the buck. You know, our pay quality, our we pay quality, our above average prices for the labor. And guys are not wanting to show up to work or complete the project that they started on.  And I can understand the death in the family, but not every day. So, you know, at the end of the day, somebody has to do the work. So something is generated income-wise to pay your salary. That's been our biggest struggle. So let me start you over here because this is my pride and joy right here. This is me and Sweet Juicy Mama's bedroom, master bedroom. This right here is our master bathroom. We'll put a jacuzzi, bathtub right here where we both can get in there. Like, then we got an enclosed shower here. Come on. Here's her dual vanity, my side, her side. The whole family are going to move in here? Only the ones that are here now. This is a closet. This where I do our live videos that we're doing now. That's just to go outside. And this is our bedroom here. Like I said, this here is 2,000 square feet. This is bigger than our house. Just me and Mom's bedroom is bigger than our house in America. So when people say, what's your biggest struggle? What struggle? Do you think you ever go back? Why? I got the food out here. It's free. I got the water. It's free. The rent is free. We're on the land. And this right here is Sweet Juicy Mama's kitchen. So next time you and Miss Trudy come down here, we'll treat y'all in the kitchen and stuff. And you know, we can go ahead and have a good old continental or African style breakfast. But is it going to be the first house you ever built since you moved to the Gambia? Yes, this is the first house in our life. In our life? In your life? Yes. In America, you just buy the house, everything done. You just buy the house, everything done. But over here, you build from the ground up. So would you say life in Gambia has really changed or transformed your life? Absolutely. I feel free now. You know what I'm saying? How free do you feel now? I feel free. How free? How free? Yeah. Peace! That's how free I feel. In America, you know what would happen to me if I would have said that? So free to say it. Huh?  Pow! So you ask me if I feel free? I feel free. Nobody's bothering me now. But I feel like you're so hyped in Africa. Because you're here. Ah! No, I think that's how I feel. No, no, no, no.  Because I feel like your hypheness is overshadowing my hype. Oh, I'm going to stop then. I'm going to calm down. OK, I'll stop. But this is the kitchen. And this right here is the living room, dining room. And this is where we put the big screen TV to watch our videos. We only got four of seven children here. So we got two girls and two boys. Two girls share this bathroom. Two boys share this bathroom. And then over here, we got the dog house. We got about 20, 30 dogs. Today is Gambia's independence. In the days. That's right. Absolutely. It's Gambia really independent. If you're independent, you can control your finances. You can control your economy. You can control your politics. You can control your education. And we don't think that Gambia controls these things. As far as, let's say, I wanted to teach. And I want to teach that we are the superior people. Nothing racist about that. Nothing racist about it. You can do your research and find out who the gifted people are. The gifted people are the ones that have the best continent, the most prosperous people on the planet, and the smartest on the planet. Because we got nine strands of DNA, which means we are genius. We just don't know it. Where all the other races have six strands of DNA. So I'll let you come to your own determination. So anyway, if we use that, then we get back on top like we used to do before colonization and enslavement. And so if I wanted to teach that curriculum in the school system, I would have problems. Because they would say, well, that's not in conducive with, you know, we discovered y'all and y'all was primitive. You understand what I'm saying? So there would be some pushback on that. But still, we're going to push that anyway. Yeah, it's already bought for it. Paid with cash. Everything's bought and paid for. In Africa, we pay staff with cash. Yes, I like it. You usually pay with a lot of money. Me, no. We came to Africa with $6,000, US dollars. $6,000. $6,000. So how much do you worth right now? How much do you worth right now? If we can break the land down and sell it what they're selling it for now, we're multimillionaires. Do you miss America? No. No. There's nothing to miss. I mean police brutality. You know what I'm saying? I'm a realist. I will look and see what is actually there versus what you tell me. Because people can tell you now the sky is purple. And I would look and say, wait a minute now. I know what purple is. It's not purple. It's blue. You see what I'm saying? So in America, people want to tell you, oh, it's fine. We got a high standard of living and this and that. But they're totally ignoring the police brutality, the criminal government, the executions of black people. They just block that out. I don't block it out. I see it for what it is. One brother's getting shot 20 times and the police are saying it was an accident. That's impossible to get shot 20 times and that'd be an accident, right? Wouldn't y'all agree? So I had to get out of there. I don't miss it at all. I don't understand why everybody else don't leave. They block it out and say, oh. What are you doing to inspire more people to come join you guys? I show them what they can have. I'm not the smartest person in the world. I'm just the average person that wants to do above average things. I just showed you the house that we're building. This right here, after we're complete, it'll be 60 acres. So we're living 10 times the value of what we were living in America and peaceful, you know what I'm saying? You haven't jumped me yet? It's the same amount that you can spend to live in America. No, man. Oh, no. In America, if we didn't have $10,000 a month income, we were struggling. Over here, if we got $1,500, we good. Are you living or surviving? No, no. No, no, no. We're living. We're finally living. Now surviving, well, you just barely got your nose above water. We weigh above that. And when we get to the land, it's a done deal. Once we get to the land, this is crossing the finish line for us. Right now, we're still in the race. But once we get to the land, this is the finish line. Because again, with the beautiful sun, solar panels, no electricity bill. You hear the solar panel pump scoring? Free water. No rent. Everything is paid for, you know what I'm saying? So we were to cross the finish line and be independent of the system. Do you think there are opportunities in Africa? Yes. It's a bond. I don't understand why the youth don't see it. So somebody's misleading the youth. You know, there's too many opportunities here.  And I explained to them, if I was in, if I was born on an African continent, physically, the way I would come up is, well, first of all, I don't mind putting in the price to get what I want.  You can't put it off no more. Now you got to go back to work. That's the backwards way. So the way I would come up in the system on the African continent, if I was a youth, was I would not place any limits on what I would do, provide it. I'm not hurting anybody or doing anything illegal. And here's my story. I would not hesitate to work 12 hours a day. I would find a way on the African continent, what we found is if you could find an open piece of land, nobody would bother you as far as planting, sub-known it, waiting for rainy season. So once rainy season comes in, whatever you planted, you can harvest that. And once you harvest that, let's say peanuts, for example, you can sell those peanuts. And you do that for a couple of years. You should have enough profit to go and buy you some type of taxi. You work the heck out that taxi until you get two taxes, working 12-hour days if you got to. Work the heck out of it until you get three taxes. Once you get three taxes, now you can buy you a big bus of galley-galley. Now you got three taxes in the galley-galley. Just keep repeating that to where you can step out the picture. Now you got three. I think if I understand what you're saying, you know, when it comes to agriculture like this. Yes. So many of the youth of Africa don't see a value in it. Right. It's because of our education system. Exactly. Our minds were tuned just to sit in the office. Right, exactly right. But there's no money in that because in agriculture, if you look at the business opportunities on the African continent, always in the top 10 is agriculture. In fact, the African continent can feed the rest of the world. So don't believe the lies that you're being told that people are starving here in the place on the African continent, Africa can feed the rest of the world. That's why I'm not sure which one of your cameramen told me, but there's a foreigner here that's bought up a large piece of land and they're selling agriculture, but they're selling most of it to the UK and America. They're exporting it. So why ain't they supporting people here, see? Because sweet mama, please tell the story about the chicken and the eggs, how the price is going up. Oh yeah, that's where it has gone up. We actually, I believe last month, I went to look for eggs and I couldn't find them for four days. It used to be when we first moved to the continent, the age was 125, then it got up to 150. The last thing before it went up, I believe it was 190. The four days that I was looking for and I couldn't find them, the next fifth day I found some, it was 290. 290, see? 290 for the eggs. So yeah, they put it in perspective, the prices are doubling, but we know that the youth and the workforce here in the Gambia, the incomes isn't going up with inflation here. You know what I'm saying? So somebody's being left out and we say we want to try to offset that or we will offset that where our prices will be consistent. And just because there's no rain this season, we won't double our prices. You've been here a couple of years. What is the best advice you're gonna give to the best? Well, if somebody wants to come here and visit the Gambia, I would say that's exactly what you need to do. Most people want to say, hey, what's it like, tell me this and tell me that. No, you need to come and look at it with your own eyes. Come here, visit. And if you know some people, like for example, what am I, you're in Ghana, reach out to him or somebody that you know that you think is trustworthy. That's the key word, trustworthy. Reach out to him. They'll show you around, hopefully free of charge and help to acclimate you into the system. So definitely come here for a visit because the opportunities are endless. Endless opportunities. Karen, can I ask you a question? Now, why is it that people like myself from the diaspora can see the opportunities here, but youth like yourself on the continent, not exactly you, say there's no opportunity. No, it's more like exposure. Even me, I never saw the opportunities until I left the continent. Oh, okay. When I came back, I'm like, why are this here? That we can turn this around. Before I left, my mindset was all about going abroad, staying there forever, even without coming back. And I thought that is where the greener pastures. When I went there, when I started, I mean, moving around with Chinese people, seeing the pretty things that they do just to get money out, I'm like, you know what, four years time in China after five years, I'm like, I'm going back home. That's right. And when I came, I keep on, even as I'm talking today, I don't even have my own age mates as friends because when I talk, it's like, you know too much. When you talk to them, it's like, you know too much. So sometimes I don't blame them, but the little chance that I get through my videos, I keep on preaching on young people, all the opportunities are around. I mean, I met a young guy who has never left the country. He wanted to go abroad, so bad. Wow. At the end of the day, he realized that, you know what, he has money to buy a pig. Right, a pig. One pig. See?  Now he has over 200 pigs. See? Wow, see? He sees himself as a billionaire. Right, exactly. This is the kind of education that I think that when we meet people like you, we're able to transfer what you know from a younger generation in Africa. I believe that they are watching, they want to learn something. Because I even think that education system in Africa needs to be changed. They need to scrap all of it and all. Because that is what is affecting our families. And they're not getting it. Right. I encourage the African diaspora to come back. Right. Whatever they've learned there, even some of us go to school, we've learned a lot. We just have to come and help develop our culture. Right, exactly. We don't want to. Exactly. And we're not each other's enemy. There's this little trickery out there to say, we fight each other. No, we're not. We're brothers and sisters, you know what I'm saying? Check out DNA. It all comes here, you know what I'm saying? And so when people... Are you in here? Yeah, and so when people put this theory out there, oh, we don't like black Americans, or we don't like gun-ins, or we don't like gamblers, no man. Put that out your mind. Hey, we're one people. We're one people. How do people in here show you their love, making you feel like you are one? Absolutely. They say, welcome home. And I said thank you. You know, I feel at home. You know what I'm saying? Mentally, this is where I'm from. You know what I'm saying? Physically, I was just born in another place. But yeah, we get showed their love all the time.  Let me show you how to do this. And I said, thank you. I just learned something different today. You've been here for quite a long time. Yeah. Could you have one thing to change on the continent, or would it be? I would say, if I could make wooden miles of money, I could change a lot of things. No, no, no, no. No, I'm not sure. Sweet Mama's different. Mine is more of education. Yeah. That's what I would like. Yeah. Be taught more to the youth. Right. And that's the basis and stuff. But I would say there's no greater place than home. You know, people think the grass is greener on the other side. No, no, no. Like you just gave the story about the guy with the pig. This is the opportunity here on the African continent. If you don't see the opportunity, hold on for a little bit longer. It is coming. It is coming. I want to say thank you so much for taking me around. Thank you for coming. I appreciate your time. Thank you. And I'll be in Gambia for one more day. And I know that people watching us know that you have a YouTube channel. If you don't know, I think now you know. Yes, so the YouTube channel name is Black Echoes of the Gambia. Or Bag Family. Or it's Black Echoes of the Gambia, but they refer to us as the Bag Family. Can we all go and subscribe to the YouTube channel right now? All right. Thank you guys. Thank you so much. I appreciate you all. Thank you for visiting. I'll see you all in Ghana. Yeah, thank you. Thank you, thank you. We definitely will be there. We'll be in Ghana. No, no, never. We're coming. We're coming.