brought to you by Private Property. My name is Malinuoco, your host every Tuesdays and Thursdays. This is another special edition of the Farming Podcast and we're speaking to a farmer who farms quite exotic plants. Some of them I've never heard of them before and his farm's name is called Thadmire Riverstel which apparently means farm more. So I think he's got such an interesting story to tell for us especially just what he does in his farming operations and the various commodities that he incorporates into his business and we're speaking to none other than Henry Finnemore. If you have any questions for Henry please feel free to drop your questions on the comment below and just comment about what you think regenerative agriculture is, what your thoughts on regenerative agriculture is and how we could diversify, you know, in this industry. We'd like to hear your engagements and continue to subscribe to our YouTube channel which is Private Property and for this specific show please go under the Farming Podcast playlist. Let's get straight into it. Henry, thank you so much for joining us. How are you doing? I'm Mbali Yesam. Well, thank you. Hi to you and your subscribers. Thank you. Thank you. It's always great to speak to a fellow farmer because you know, people say farmers are always in their small corners in rural areas in very outskirts areas as well. So it's great to speak to another farmer and get another farmer onto this Farming Podcast. So Henry, you are the owner and farmer of Farm Favmere Riverstel.  Mbali Yesam, my great grandfather started off with dairy farming. My grandfather continued and my father. But then the line was cut there. I think dairy farming, my dad always used to say it's a prison without walls. So I got another prison without walls. Protea farming is a bit like a jealous lover. It takes all of your attention, especially if you have a nursery, then things can become complicated, especially when you have fun times when FCOM switches off the lights and you've got to make plans to wet your plants. But I think you as a farmer will understand those hardships as well or difficulties. I think just to put you in the picture, Favmere actually means free range in France, which, you know, that's when me and my wife decided about seven years ago, we're going to say goodbye to Johannesburg and we moved down to Riverstel.  Yes, that's correct. I'm a vegetable farmer. Yes, well, then you must speak to my daughter. Well, I started a business in Benoni. It's called Field Fresh about 2006 and we started supplying cut vegetables, soups, processed beans, butternuts, anything you can think of with over 100 lines that we were supplying to checkers and spa. So possibly you can talk to her and you can do some business together with my daughter. She's still there. I'm not there. I sold my interests in the business and I then moved down to Riverstel. So just in short, regenerative agriculture came across my road for I think for lack of anything else, just the costs of the input costs. When I studied agriculture in 1984, we were all we were taught about fertilizers and even where they were made, you know, the different fertilizers in which countries they were made, etc. And when you start, when I was, say, long out of farming, coming back to farming seven years ago and just seeing how expensive input costs have become and also how the pest and disease list has increased, you know, for plants and for animals. In the time when I used to farm with my father, you know, just looking what he was doing, most probably the amount of pest and diseases were 20% of what they are today. So things have just become more difficult in terms of chemical requirements. So, you know, in general, everybody's farming chemicals nowadays and if you look at documentaries like Kiss the Earth and others, then you will see that, you know, there are only so many crops left before the elements in the soils are depleted completely. And you know, we keep on adding that just chemically and there are other solutions. You know, it was the farming the old way. Nowadays, you've got factory farming and you've even got big commercial form farmers. And I think some of them are also looking at alternatives, you know, like for example, no till has come into agronomy where they do that with grains, where you try and keep the stubble on the soil so that you can conserve the moisture in the soil and also build up your soil.  And without I think the old people knowing that these things were working hand in hand, you know, it's the way agriculture was intended was that, you know, you must have different animals on your farm so that they interact with each other. And then you get the benefits of that as the grower, whether it be the animal or the plant, because that interaction just means that you get healthier plants and animals. I think the more we have moved towards chemicals, the more dependency we have on chemicals, the more sick the system has become and also people in general, if you look at just the the the increase in cancer, you know, in people, it's just horrifying. So I think looking that's why a lot of people are looking and wanting to turn back to the old ways. But the problem is that you don't always get the remuneration for that effort you put in for growing things organically or more healthy. People are so bound by the commercial farming side. And even if you look at the price spread, you know, from your farm gate to your retailer, it is only economy of scale that keeps a lot of farmers in business. So you've got to produce a lot of something to eventually make money with it. Because if you look at a broiler farm, you know, you've got five in houses, each in house can take up to 10,000 chickens or fowls. So you know, you as a small farmer, if you're trying to farm with eggs or with meat broilers, you've got to compete with that scale of production. And that makes things difficult because then you need a client base that's actually willing to pay you for your product so that you can get the benefit of that economy of scale because you are producing a healthy product. And I think the closer you are to the cities, the more fortunate you are to actually have those LSM buyers that have got a higher living standard and that will be and they're looking at their life expectancy, they're looking after their life and their health.   You know, everything is developed for the benefit of health. I will support. So just in terms getting back to my farm, when I started, I started off with cattle. We went through a slight drought period. The farm was too small. I then looked at alternatives and then there were not farmers in the area. And I had started a macadamia nut nursery. And that was producing macadamia nut trees and then eventually I planted trees for myself. And then eventually I planted pecan nuts and almonds. And my idea with that is if you look at the regenerative agriculture practices and the guy I follow the most is Richard Perkins from Switzerland.  But you can only do as much as what the climate offers you. So if the higher the rainfall in your area, and then I must also add, it must not be too high because then you're going to sub tropical conditions, but you also can't have too little rain. So you must sort of sit there in between, you know, in terms of rainfall, or you must be able to irrigate your fields from a irrigation system, you know, where you, where there's a dam that's providing you water and you are listed in an irrigation scheme where you get a daily or a weekly allotment of water and you can farm with that. Because that's the only way really where I can see regenerative agriculture working well. Because in your drier parts or your very wet parts, you know, there's other farming conditions you are limited because of the the the weather conditions, like for example, the Karoo, you can you'll only get niche spots where you can apply regenerative agriculture because you've got water. So in general, you need a good supply of water. You need good soil. And obviously the sunlight is for free.  The animals you are using or the plants you're using in your system is beneficial to the soil.  I mean, we all talk we all know about the the the big carbon problem in the world that there's too much carbon and what it's doing to the world that's heating up the world. So nobody taught me adversity that there's actually carbon in the atmosphere that you cannot generally you can harvest and put back into the soil. You must only allow grass to grow on your soil on your soil. So if you allow grass, grass is the biggest sink source for carbon can actually bind a lot of carbon. So in the past where I used to use roundup in my fields to spray the macadamia trees to keep the grass off the roots and out of the fields, I don't do that anymore. So what I do now is I do a chop and drop. So I would take a weed eater and I would cut, I would mow the grass, I would mow the weeds, and I would also I started to learn all the different kinds of weeds on the farm. So and I looked at the beneficial ones, you get soft weeds and you get hard weeds. And some of the hard weeds you they they irritate you but they put down the root systems that actually break up the soil. So that's beneficial for you. Soft weeds gives you a lot of leaf so you can if you chop it and it falls onto the soil, it forms that organic mulch that you need where you can actually build up the ideal conditions for worms to come in, funnily to start growing and your micro your microresa and you can also build up carbon in your soil. Now the scientists have proven for every 1% you build up the carbon in your soil, you're actually taking carbon out of the atmosphere putting it in the soil. For every 1% you increase the carbon in your soil, you increase the carry the water carrying capacity of the soil by 250,000 liters per hectare. So that means if you can build up your soil if you start off with a soil that's got 1% carbon in it and you can build it up to 5%, you've got an extra storage capacity of a million liters in your soil. That carbon actually makes the texture of your soil much better, the structure of your soil. So that means that the water can infiltrate much better and it gets stored for a longer period of time. So when your neighbor is experiencing drought during a drought period you will still have the benefit of that moisture in your soil because you've got extra carbon in your soil. So yes, that's things that I'm looking at you know like for example if you plant alfalfa, Lucerne in your fields amongst your trees, yes they are competing with your trees but they've also but they've got rhizomes on their roots that are actually collecting nitrogen for you and making it available to the other plants. So now we know about this web, this web of interaction in the soil between plants where they actually exchange nutrients with each other to survive. So there's this whole network in the soil where certain plants will have more selenium or zinc which are micro elements which other plants don't have and they will exchange it for a favor from the other plant, the other plant might give it some phosphorus or calcium. So there's this whole interaction in the soil that has got my interest and since I've been applying these methods of farming, bringing in chickens, I eventually want to bring in sheep as well. I haven't got sheep now at the moment because my trees are small so I don't let the sheep to damage the trees. So my trees must get a bit bigger then I will bring in the sheep then I don't have to mow the grass anymore the sheep can can utilize the the grass. So it's just it's that whole interaction between we we've got chickens on the farm at the moment they are you cannot believe once you start watching them in nature how much they how much soil they can move around, how much they can dig and how much they can add value to the soil by the by adding fertilizer to the soil. Yes it is raw fertilizer yes you must compost it and but just moving the hen houses through the orchard we can already see the difference just in that the trees in the areas where we are moving the hens just react much better they are healthier and they they just look happier than the other trees. So we are we are slowly increasing the amount of animals on the farm as we have increased the the nut trees. So production. Yes I don't know if you are familiar with a silver pasture silver pasture methods where they grow trees and then they have pastures underneath the trees but you thin out the trees so that you have the benefit of the grass and you've got the shading effect of the trees. The trees have got all the fungal growth on them normally when a tree breaks down it's got fungus associated with it. The grasses have the the rhizomes correlated to it so you've got the interaction of the bacteria the rhizomes of the bacteria and the fungal spores on the trees so you get this interaction that is all beneficial to the plants and eventually as I said if you've got healthier plants you've got healthier animals if you've got healthier animals and you slaughter them you're selling a healthier product to people and eventually you have healthier people you know if you look at and you have a healthy environment I think and you have a healthy environment at the end of the day because you know all the different elements are working within each other and just supporting each other as an ecosystem. Before you go into further technical info I Henry you know you've really unpacked regenerative agriculture quite well with different methods and obviously brought certain case studies of what you've done in the farm incorporating chickens and soon to be sheep. You know for a farmer who's thinking how do I apply these practices and I want to start you know just to to farm for a better tomorrow etc etc where does a farmer begin to get such information you know you seem like you've got such an extensive technical knowledge it's like you know you've you've really put a microscope in every level of your farming production how can another farmer adopt these regenerative regenerative practices do they have to do just research do they have to speak to institutions or do they have to use the services of an agronomist soil scientists or any other scientists as well you know how can one start adopting these practices by themselves and their farms. I think it's a combination of what you said if you look at the internet as a wealth of knowledge some of these world-renowned speakers on on the topics like for example Gay Brown who features in Kiss the Soil I don't know if you've watched Kiss the Soil yes I have yes I have okay Gay Brown is the farmer in America that shows how his neighbors land is flooded with water and where his land is not because he's applying his binding carbon in the soil so Gay Brown was in South Africa he was in Cape Town he gave a talk their different Richard Perkins has been in South Africa he gave a talk in Cape Town I must be honest with you I missed both he talks as well I only saw the programs after the fact you know after I but I've I watched their videos I listen to what they say and I apply what they say I also do a lot of reading of my own I do research I do a lot of research on the internet because most you know to go to your local library you're going to get such outdated information yes there will be good information but it will all be based on the past the commercial you know I'm not saying modern libraries are not modern but most of them have got computers as well now you just go and you you surf the internet so there's so much good information on the internet I think just read up about it and then get to know your own farm because every farm will be different the weeds the weeds on my farm is not going to be the weeds on your farm it might be if we in the same district but if we are in different districts different areas regions it's going to be different and then different weeds come at different times of the year so you must devise a strategy for every weed on your farm like for example I have double keys you know I just I just hate them because apart from that you you cannot walk in the field they just you know they they sting you wherever you go I still have to use chemicals to kill them off I haven't got any other solution for them or you must just chop them when they're very young but they such they so hardy they just regrow and regrow so you know you spend a lot of time with your weed eater in the field so you've got to learn the different weeds you've got to learn your soil you know your specific soil if you've got a clay soil a sandy soil or a loam soil I think always a farmer so lucky to have a loam soil because that's in between a clay and a and a sandy soil you know it's much better than than a pure sandy soil and a and a very clay soil those are the two difficult ones to work with so if you've got a loam soil which has got a mixture of sand and clay then you are very fortunate and then you can do a lot of things on your farm yeah so you know get to know your farm get to know your own conditions get to know your weather conditions I mean I use YR as an app a weather app I don't you can use any app weather app that matter but what I do is I always look at next week's weather forecasts so that I know when it's going to rain what you expect so that I can you know do my planning according to that as well yes yes and so you've mentioned that you've got nut trees chickens as well so the concept of having to bring the the protea plants was that also all forming part of regenerative agriculture or did you just see an opportunity with protea plants approach your flowers that you could start farming and trading with I think you know you spoke about COVID when we started off our affected everybody yes it was bad yes for the flower industry you know it was terrible you know and the thing is if you think about it logically you know people don't eat flowers they they they eat protein they want nuts maybe maybe they want meat first and eggs and milk and bread and then they eat nuts you know so flowers start becoming as a nice to have so it's it is it is a tricky business and it is also economy of scale so you know I think I think the flowers came across my my path as an opportunity because I just I love plants in general and I mean if you if you if you've seen a king protea you know you just fall in love with him it's just such a beautiful flower and I just wanted to to plant them so I've planted I've planted you know a hectare of king proteas and I'm going to expand further with the flowers but I always I want to keep a balance between the flowers and the nuts you know you know you never know what's the next COVID going going to do so I don't want to I don't want to get knee deep in into the flowers but the opportunity has come and yes we are making cuttings we are exporting cuttings to to Ecuador so you know there are opportunities if you look for them and to get back to to pin cushions the the scientific is name is a lucos spurnum so you can you can look them up on the internet but it's a it's a beautiful color it's got a variety of colors in it's it's you get them in deep red you get them in yellow you get them in in yellow with with with red little stamens at the top so you know it's it's different colors that you get and they are very exotic and they're one of our as I said one of our biggest export flowers uh is um uh pin cushions you know apart from the proteas um if even if you look at you know if if you look at the variety within the plant kingdom it is it is so wonderful I mean you know sometimes I'm just gobsmacked by the variety that you know God's creation has given us if if you look at you know some people will talk about proteas and think yeah proteas okay but they are you know there's there's almost commercial there's most probably 30 30 commercial varieties and then you still get many more you know pin cushions the same there there are so many commercial varieties and then there are so many more that you find in nature so um the diversity the diversity is tremendous and then I'm not even talking about you get the lu luca dendrons the cone bushes they are ornamental flowers that they're also using arrangement they make little balls um they've got little balls on them um and and beautiful to put into an arrangement so I think I've I've given you next time you go to the flower market you must go and go and check out all these new things I'm talking about yeah the pin cushions etc but Henry would you think that would you say that regenerative agriculture is quite an expensive farming process to adopt you know because um you know when you go through very very harsh climatic conditions and you can't spray chemicals and maybe you just don't have the budget to buy all these different things that could help you know the soil etc like invest in nut trees invest in sheep invest in chicken invest in far in flowers um you know if you don't have all that money and capital to invest in all these resources what do you do so is could one say that regenerative agriculture does come at a price for the farmer specifically uh I think you know now I've answered the question why I've got a nursery because that's cash flow so yeah okay you know everything I've produced in the nursery I can I can sell within a year so that gives me a yearly income so that allows me to extend on the other farming activities and yes we do eggs locally so we provide all the local restaurants and the road stalls with eggs my wife does that there was actually she did a um interview with one of the with another I can't remember who but but um uh yes I think I think yes you are correct um it's not as if we I said if I don't spray any chemicals so for example rust is still a big problem on nut trees so I will I will use copper and um sulfur in my in my arsenal um which are chemicals but it's more the old school chemicals it's not got the um the chemical components that you know we are trying to go away from I'm talking about the copper and the and the diethane he's almost a contact chemical it's not as if it goes systemic it does not it's not systemic so it does not go into the plant um and and eventually into the fruit or the nuts that you eat so they say roundup isn't everything is now say you can you can go and if every person analyzes a bit of his fat his stomach fat you will find a roundup in it yeah so that's quite bad as we say as we some of the show Henry what is just you've mentioned quite a number of tips as well and things that farms could do just maybe for a farmer that's just you know been farming for many years who's so used to their their farming practices and methods and maybe that just want to adopt more the organic approach or regenerative agriculture approach what are just some of the top short key three things or tips rather that you could advise to a farmer just to try and going out and experiment regenerative agriculture to see if you know they could save some money back at the end of the day so what are some of the top three um uh tips that you could give to farmers around regenerative agriculture or exploring regenerative agriculture as a farming option I think um you know like everything in life don't don't jump head head over heels into something new you know um adopt it slowly you know do it do it on a small scale I always say you know and I learned this lesson with when I started in macadamia nursery is don't do things on a big scale you know don't do something so big that you know it overwhelms you so um just just start of um small you know take one egg day on your farm take one egg day on your farm or ten egg days depending on the size of your farm and slowly start changing your um uh your practices there I think the I think the most the the most important key was something a professor told us at university said that and this is something that's becoming a saying in agriculture is you know the best fertilizer is footsteps in the in the field you know so um go into your field go and look at your individual plants look at their needs see how you can make life easier for them you know obviously if you have big trees I I see that by me on the farm as soon as you get bigger trees protecting smaller trees you get a micro climate so now the scientists are talking about they're talking about the the the macro rain and also the micro rain so you can actually create the atmosphere on your farm that's beneficial for rainfall because you are increasing the amount of foliage you're increasing the amount of humidity so um plants start protecting each other so you you'll also find in a garden try and start a garden or just with fine plants you know plants that are soft now the sun will take them on the third day so you know what you must do is first establish your tree and then you get go to other plants but to get back to your your question I think it's it's take it slowly see how you can improve your soil do an analysis of your soil see what the carbon content of your soil is talk to an agronomist find out how you can carbon is for free you don't have to buy it in a fertilizer format you can actually pull it out of the atmosphere and put it in your soil nitrogen is for free you know so look at planting those beneficial plants clover for example or or Lucerne you plant in your fields try and plant them so they don't compete with your main crop but then um I think a guy brown from America with a kiss the soil and they've got mixes of up to 12 to 14 different plants that they seed at one time so that they can establish a cover crop so that that gives the benefit to the main main crop plus it gives the benefit to the soil you know that that green green fodder that that you're looking for so I think yes look at the plant look how you can improve that plant's condition and as I said in the beginning we started off it's not for everybody because you you will find farmers that are farming in in difficult climate climatic conditions he's got a maize farm for example you know what can you do and you can't plant trees amongst your maize farm now to make it absolutely a better place for the maize yeah so you know you but maybe you can look at cover crops you know to improve your soil's health so you don't have to put in everything chemically but you can slowly start weaning yourself from chemical products and I've seen that you have on me if you get plants that grow in natural conditions without having too much chemicals they are just healthier they just work naturally yes they are healthier so but that it's it's it is a bit of a patience game and as you rightfully said is you know if you want to if you're a small scale farmer what I'm talking about just becomes more difficult because nothing in South Africa caters for small farmers yes the the implements are scarce the advice is scarce the competition is strong you know you've got to compete against big commercial growth so for a for somebody on a small scale trying to produce maize you know it is it's it's you're competing with a maize price that's that's being done you know on on a large scale so it just becomes more and more difficult but it doesn't mean you can't it doesn't mean you can't do it if you want to be self sustainable yeah but thank you so much for your time Henry I think all the tips that you've given are quite valuable and I agree with you that the agricultural industry in South Africa does cater especially more so for the commercial farmer and it's very important to also note that not all the regenerative agricultural practices or farming practices can be adopted in every single type of farmer you know because like you said if you're farming maize you can't plant trees and if you're farming maize the last thing you want today is cattle you know going in and about your maize and and eating your carbs etc so yeah I think that's quite detrimental but I agree with you I think research is the most fundamental thing maybe speaking to other farmers who also have adopted those processes but thank you so much for your time for your expertise and I wish you all the best with your farming venture and I will continue to maybe yeah I will actually reach out to your daughter because we're in the same vicinity so I know about your company but yeah thank you for the tips that you've given and shared on the show tonight okay thank you for having me on the show it's a pleasure it's a pleasure that was Henry Finnemore who is a farmer and we were speaking about regenerative agriculture I think the most biggest tips that I learned from him in terms of how we can adopt these practices basically starting small trial and error I think research is also quite important and you know just just listening to some some talks online on youtube listening to the farming podcast because Henry definitely dropped some serious gems on how he's adopted regenerative agriculture in his farm and also just some other tips like harvesting carbon which is for free who would have thought so I hope you really found this conversation quite valuable and keep subscribing to our youtube channel just to get more content around the farming industry and how you could improve your farming practices as a farmer once again go to private property youtube channel under the farming podcast playlist this is where you'll find this episode with Henry and yeah that's it for me and I wish you a very good evening and I will see you soon take care


So Stephen, I should buy and I'm going to take you through how to do some electronic structure calculations on atoms in Spartan. So I've gone to this V desk interface and once I'm there, I'm going to click here on the lower right, you'll probably have to enter some credentials, which I've already done. And eventually it'll take us to this Spartan screen, which I'll double click on and it'll launch Spartan in just a moment. Now I've already set up these icons so they're ready to go. If you want to do that, you can go to preferences like this and set that up. But I'm going to go ahead and build helium and for that, I'm going to go to the inorganic menu, going to choose helium and which is there. And I'm going to choose this one stick coming off it because I'm actually going to delete even that one. And so now I have helium and we go to this, this calculations icon, which you would be able to also get through through the setup down, pop down. Now we can take all the defaults on this, so I'm just going to submit and we can save it however we want. And and it tells me that it's completed. So next what we're going to do is I'm going to choose this icon and which again is on here. It's under the display pop down and it will show you a couple of options here on the left. It's an energy ladder. I kind of like to show these orbitals in a mesh view. And I've let the model default to ball and spoke, which is fine. And there's an orbital. It's a 1S orbital and there's another orbital. It's actually the 2S orbital. So you can see the 2S is bigger than the 1S. It's got a little bit of structure inside. And after that, you do the same thing for other noble gases to begin with. I would say next you should go for neon, then argon and and and so on.


This retrospective study evaluated the effectiveness of socket preservation techniques used after tooth extraction. It found that there were no statistically significant differences between the use of autologous particulate bone versus deprotonized bovine bone in socket preservation. This article was authored by Glauco Chisai, Arje de Hacia, Eletra Chisai, and others.


Today, we are going to solve some problems based on numerical quadrature and after that we will consider numerical integration. So, some of simple problems, first problem is we are given a function f x which is piecewise linear. So, f x is defined as equal to x for 0 less than or equal to x less than or equal to half and 1 minus x if 1 by 2 less than or equal to x less than or equal to 1. We want to find an approximation to integral 0 to 1 f x dx by using trapezoidal rule, then composite rule based on partition 0 less than half less than 1. That means, we will be applying trapezoidal rule to interval 0 to half and trapezoidal rule to interval half to 1 and adding it up, then Simpson rule, so basic Simpson's rule and corrected trapezoidal rule. In all these rules, what will come into picture will be value of function f at two end points 0 and 1 and at the midpoint half. In addition for corrected trapezoidal rule, we will need f dash at 0 and f dash at 1. Our function f is such that it vanishes at the two end points, so f of 0 is 0, f of 1 is 0 and f at half is going to be equal to half. f dash at 0 is going to be equal to 1, f dash of 1 is going to be equal to minus 1 because on the interval half to 1, our function is defined as 1 minus x. So, when you take its derivative, so at 1 we are taking the left handed derivative. So, now we are going to substitute in these values in various formulae to obtain approximation to integral 0 to 1 f x dx. So, let us look at first the exact value. So, integral 0 to half x dx plus half to 1 1 minus x dx because that is how our function f is defined. When you integrate, you get the value to be equal to 1 by 4. The first rule is trapezoidal rule.  In our case a is 0, b is 1, f at 0 is 0, f at 1 is 0. So, the approximation using trapezoidal rule which we get is equal to 0. Now, we want to apply composite trapezoidal rule. We are going to apply it on the partition 0, half and 1. That means on the interval 0 to half, we apply trapezoidal rule. On the interval 0 to half, our function is f x is equal to x. That means it is a linear function and the error is 0 for linear functions for trapezoidal rule. So, using our trapezoidal rule on the interval 0 to half, we are going to get exact value of the integral. Similarly, on the interval half to 1, our function is 1 minus x. So, it is linear. We approximate our integral over half to 1 by trapezoidal rule. So, there again the error is going to be 0. So, composite trapezoidal rule is going to give us exact value. So, it is the using composite trapezoidal rule, you get 1 by 4 into half, 1 by 4 because b minus a. On the interval 0 to half, it is half. So, 1 by 2 and then these two. So, it is 1 by 4. Then we will be looking at value of f of 0 plus value at f of half. At 0, the value is 0. At half, the value is half. So, that is why 1 by 2 plus 1 by 2 to 1. So, again the length of the interval is half divided by 2. That gives us 1 by 4. Now, we will be considering f at half plus f at 1. So, that is going to be equal to 1 by 2 and f at 1 is 0. So, it is 1 by 4. So, for the composite trapezoidal rule, you get the value to be equal to 1 by 4. Next, for Simpson's rule, this is the formula f a is 0, f b is 0, f at a plus b by 2. At the midpoint, it is value is half. So, when you simplify, you get it to be equal to 1 by 3. And in the corrected trapezoidal rule, you have trapezoidal rule plus this term. Trapezoidal rule gives us 0.  So, it is going to be equal to minus 1 by 6. Thus, the best result which we obtain in this example is when you are using composite trapezoidal rule. And that is because our function is piecewise linear. And in the composite trapezoidal rule, we have used our partition such that on each sub interval, our function is linear. If instead of applying composite trapezoidal rule to interval 0 to half and half to 1, if I would have used say interval 0 to 1 by 3, then 1 by 3 to 2 by 3 and 2 by 3 to 1, then our error would not have been equal to 0. Because on the interval 0 to 1 by 3, our function will be linear. On 2 by 3 to 1 also it will be linear, but on 1 by 3 to 2 by 3 it will be piecewise linear. And if you compare the result which you obtained using trapezoidal and Simpson, then Simpson's approximation is a better approximation than the trapezoidal.  The value of the integral is positive and the value which we obtain is negative. So, it is a simple example to compare various our numerical quadrature rules. Now, the next example, we want to construct a rule of the type a 0 f of minus half plus a 1 f of 0 plus a 2 f of half which is exact for polynomials of degree less than or equal to 2. So, that means we want to determine a 0 a 1 a 2 in such a manner that there is no error when you use this formula for f to be a polynomial of degree less than or equal to 2. One way of doing this problem is we have got our points minus half, 0 and half. So, you fit a parabola that means a polynomial of degree less than or equal to 2, then you integrate and then you are going to get values of a 0 a 1 a 2. Now, our interpolating polynomial p 2 itself will be equal to function f if f is a polynomial of degree less than or equal to 2 and that is why our numerical integration also will be exact if f is a polynomial of degree less than or equal to 2. We will do it slightly differently, we want that there should not be any error for polynomials of degree less than or equal to 2. So, if we guarantee that there is no error for three functions f x is equal to 1, f x is equal to x and f x is equal to x square, then there will not be any error for a general quadratic polynomial of the form a 0 plus a 1 x plus a 2 x square. Now, when we take these three functions f x is equal to 1, f x is equal to x and f x is equal to x square, we will get three equations in three unknowns a 0 a 1 a 2 and then we determine a 0 a 1 a 2. So, f x is equal to 1, then the left hand side is going to be equal to 2, f x is equal to integral minus 1 to 1 1 d x. So, that will be 2 is equal to f x is a constant function.  So, this is our first equation. Next, look at f x is equal to x, when you integrate you are going to get x square by 2, evaluation between 1 and minus 1 that will give us to be 0 and this will be equal to now f x is equal to x.  So, you get minus a 0 by 2 plus a 2 by 2. So, this equation gives us a 2 has to be equal to a 0. So, now, this is for function f x is equal to x, if f x is equal to x square, then the integration is going to be equal to 2 by 3 and here f of minus half will be minus half. So, f of minus half will be 1 by 4, f of 0 will be 0, f of half also will be equal to 1 by 4 and hence we will get a 0 plus a 2 divided by 4. From this equation, we have got a 2 is equal to a 0. So, using that fact, we get a 0 is equal to a 2 is equal to 4 by 3, then we have not used this equation. So, use this equation to deduce that a 1 is equal to minus 2 by 3 and thus the coefficients a 0, a 1, a 2, they are uniquely determined by the condition that the formula should be exact for polynomials of degree less than or equal to 2. Now, in the next example, we are given a rule that means, we are given the weights, we are given the interpolation points and we are asked to determine the degree of precision of this formula. That means, we want to find the highest degree of the polynomial for which the rule is exact that means, there is no error. Now, when we want to determine degree of precision, what we have to do is we have to proceed in an orderly manner that means check whether there is no error for f x is equal to 1, then look at f x is equal to x. If there is no error for f x is equal to x, then go to f x is equal to x square, but if there is error for f x is equal to x, then that means, the degree of precision is going to be only constant polynomials. It can happen that there is no error for the constant functions, there is no error for function f x is equal to x square, but in that case if there is a error for f x is equal to x, then we say that our degree of precision is going to be only 0. That means, it is going to be exact for only constant polynomial. So, here in this example or in this problem, we are given the interpolation points, we are given the nodes, we are given the weights and we want to determine the degree of precision. So, it is integral x 0 to x 3 f x d x is approximately equal to 3 h by 8 f x 0 plus 3 f x 1 plus 3 f x 2 plus f x 3, where x 0 is any real number and x k are given by x 0 plus k h, k is equal to 1, 2, 3 that means, x 0 x 1 x 2 x 3, they are going to be they form a equidistant partition, which means x 1 minus x 0 is h, x 2 minus x 1 is h, x 3 minus x 2 is equal to h. So, we want to know the degree of precision of this rule that means, the highest degree polynomial for which there is no error. What we are going to do is, we are going to assume that without loss of generality, let x 0 be equal to 0. So, we are just changing the position of our point to start with x 0 is any real number, but we do not lose any generality of the problem, if I can if I assume x 0 to be equal to 0. So, if I assume x 0 is equal to 0, my x 1 is going to be equal to h, x 2 is equal to 2 h, x 3 is equal to 3 h, this is just for the sake of convenience. And then, we will write the formula.  And then, we will try to find out the formula. So, calculate for f x is equal to 1, integral 0 to 3 h f x d x and the formula using formula, we are going to get a value. So, check whether these two are equal, if they are equal then we will go to f x is equal to x and we will continue till we get a function or we get a power of x for which the two values are not equal that will determine the degree of precision of our rule. So, this is by assuming x 0 is equal to 0, if I assume x 0 is equal to 0, integral x 0 to x 3 f x d x becomes integral 0 to 3 h f x d x, which is approximately equal to 3 h by 8 f of 0 plus 3 f h plus 3 f of 2 h plus f of 3 when you put f x is equal to 1, this is our left hand side L h s. So, L h s is 3 h, R h s will be 3 h by 8, f x is constant 1. So, it will be 1 plus 3 plus 3 plus 1. So, that is 8. So, you get 3 a. So, there is no error for f x is equal to 1. Next, for f x is equal to x, the integral is going to be x square by 2 evaluated between 0 and 3 h. So, that gives us 9 h square by 2, R h s will be 3 h by 8 f of 0. Now, it is going to be 0, then it will be 3 h plus 6 h plus 3 h. So, that again is going to be 9 h square by 2. Consider f x is equal to x square, the exact integral is going to be 27 h cube by 3. That means, it is going to be 9 h cube. The right hand side will be 3 h by 8, no contribution from this term. Here, it will be 3 h square, this will be 4 h square multiplied by 3. So, that will be 12 h square plus here it is going to be 3 h square. So, that is 9 h square. So, when you simplify, you get it to be 9 h cube. So, again no error and f x is equal to x cube. So, the exact integral will be 81 h raise to 4 by 4 and the right hand side also is the same. Next, f x is equal to x raise to 4. So, this is the first time when the left hand side is not equal to right hand side and hence the degree of precision is going to be equal to 3. So, here this is the first time when the left hand side is not equal to right hand side. So, this is an example of a Newton Quartz formula. We have considered Newton Quartz formula for like trapezoidal rule, when we had considered two equidistant points in the interval a to b. Then, we had Simpson's rule. So, we had the two end points and the mid point and then we fitted a polynomial of degree less than or equal to 2. Now, if you consider interval a b and look at four equidistant points including the two end points.  So, because you are fitting a cubic polynomial, if the function itself is a cubic polynomial, then there is no error and hence no error in the integration formula and if you now here what was given to us, it was given to us the formula and then we decided what is the degree of precision. So, this is Simpson's 38th rule, a special case of a Newton Quartz formula. If we were given Simpson's rule, the direct formula and asked to determine the degree of precision, then we would have again obtained no error for 1 x x square x cube and as we have noticed before that this is something unexpected. You are fitting a polynomial of degree 2. So, there should not be any error for quadratic polynomial, but we get a result that no error for cubic polynomial as well. So, this is where as here for the we are fitting cubic polynomial and no error for cubic polynomial. Now, we are going to look at composite trapezoidal rule. So, we will take a special example where we will apply composite trapezoidal rule to find an approximate value of our integral a to b f x d a. We have got a formula for the error in the composite trapezoidal rule. It involves second derivative of the function. Now, the function which we are going to integrate is f x is equal to 1 by x on the interval 1 to 7. So, we choose our interval 1 to 7 because our function f x is equal to 1 by x. It has got a singularity at 0. So, whatever interval you are choosing for integrating that should not include 0. For this function f x is equal to 1 by x, we can calculate its second derivative and then in the error formula we have got second derivative and then we have got power of h and some constant. So, we will try to determine the length of the sub interval so that we achieve the desired accuracy. Now, once we determine h, h is going to be equal to b minus a by n. So, it will also tell us equivalently the number of sub intervals. Next, we will consider the same example for composite Simpson's rule and then find the step size h and the number of sub intervals to achieve the same degree of precision. So, our function is d x by x integral 1 to 7 and we want to determine the step size h such that the error is less than 4 into 10 raised to minus 8 in the case of composite trapezoidal rule. In the case of composite trapezoidal rule error is given by minus f double dash c divided by 12 h square into b minus a where h is b minus a divided by n. Point c is going to be some point in the interval a to b. In this example, b minus a is going to be 7 minus 1. So, it is 6 f x is 1 by x f dash x is minus 1 by x square and f double dash x is 2 by x cube. Point c, we know that it exists, but we do not know what that point c is. So, one dominates f double dash c by norm f double dash infinity. 1 upon x cube is going to be a decreasing function on the interval 1 to 7. So, the maximum will be attained at the left hand point and that gives us norm f double dash infinity to be less than or equal to 2. So, our error, take the modulus. So, modulus of the error will be less than or equal to 2 by 12 b minus a is going to be 6. So, you are going to have 6 into 2 12. So, that will get cancelled. So, you will get h square should be less than 4 into 10 raise to minus 8 and h is going to be equal to nothing but 6 by n. So, modulus of error is less than or equal to h square, which is less than, which we want to be less than 4 into 10 raise to minus 8. So, that is what we want to get. So, h is equal to minus 8 and hence h should be less than 2 times 2 into 10 raise to minus 4, h is equal to 6 by n. So, that gives you 6 by n should be less than 2 into 10 raise to minus 4 or n should be bigger than 3 into 10 raise to 4. So, that is 30,000. So, if you choose n to be bigger than 30,000, then the error will be less than 4 into 10 raise to minus 8 in the composite trapezoidal rule. In the case of composite Simpson's rule, you have fourth derivative of the function h by 2 raise to 4 b minus a by 180. So, calculate the fourth derivative, that is 24 by x raise to 5. Again, the maximum will be attained at the left end point. So, norm f 4 infinity will be 24, h by 2 raise to 4. So, I write it as h raise to 4 and 2 raise to 4 is 16, 180 from here and b minus a is 16. So, that 6 is here. So, this is going to be equal to h raise to 4 divided by 20. Suppose we want this to be less than again the same number 4 into 10 raise to minus 8, then modulus of the error will be less than 4 into 10 raise to minus 8 provided h raise to 4 is less than 80 into 10 raise to minus 8, which gives us h to be less than 80 raise to 1 by 4 10 raise to minus 2. So, we have got 3 raise to 4 is 81. So, that is why 80 raise to 1 by 4 will be less than 3 into 10 raise to minus 2 and that will mean that h is 6 by n. So, our n should be bigger than 2 into 10 raise to 2, that means 200. So, here we have like for that composite trapezoidal rule, you will be evaluating your function if there are n intervals, then you will be evaluating it at n plus 1 points. In case of composite Simpson, we evaluate the function at the n plus 1 partition points and in addition at the midpoint, that means 2 n plus 1. Now, in order to achieve the desired accuracy in case of trapezoidal rule, we need number of subintervals to be 30,000, that means we will be evaluating our function 30,000 plus 1 times. For the Simpson's rule, we have got 200. So, we will be evaluating it 200 into 2 plus 1. So, that means it is going to be 401. So, it is much less than in case of the composite trapezoidal rule. This is about the function evaluation and then you evaluate the function, you multiply by weight and then you are going to add all these numbers. So, this illustrates that the higher the order of H, the formula it should be preferred that we had composite trapezoidal composite Simpson. If your number of intervals is the same, then in the case of composite Simpson, we had double the computation, but if you fix a desired accuracy, that will be achieved with much less effort in case of composite Simpson's rule as compared to composite trapezoidal rule provided your function f is sufficiently differentiable. You need your function to be 4 times differentiable. If your function is only twice differentiable, then we will not get H raise to 4 term in the composite Simpson's rule, then the order of convergence gets reduced to H square. So, for smooth function, the higher the power of H will be available and that formula will be more efficient as compared to formula with lower power of H. So, these were some of the simple examples to illustrate our theory. Now, we are going to consider what is known as Romberg integration. If we remember the corrected composite trapezoidal rule, so corrected trapezoidal rule was obtained by considering cubic hermite polynomial. That means, we looked at a cubic polynomial which interpolates our function f at point a and point b and also the derivatives at point a and point b.  Now, the term which contains the derivative, it is of the form constant times f dash a minus f dash b. So, if we consider composite corrected trapezoidal rule, that means our interval a b, it is divided into n equal parts on each sub interval you apply corrected trapezoidal rule.  So, corrected composite trapezoidal rule is trapezoidal rule plus a term which contains the derivatives at the two end points. So, it is of the form integral a to b f x d x is equal to the trapezoidal rule plus h square by 12 f dash a minus f dash b plus term of the order of h raise to 4. Here, the error is of the order of h raise to 4, but one needs to know what is f dash a and what is f dash b. So, you have order of convergence h square in the trapezoidal rule. In the correction trapezoidal rule, you have got order of convergence h raise to 4 with the rider that you should know what is f dash a and what is f dash b. Now, one wants to know whether one can do something and get a formula, which will have order of convergence or which will have error to be of the order of h raise to 4, but what should involve only function values because the derivative values they are not available. So, we look at this corrected trapezoidal rule more carefully. You have integral a to b f x d x is equal to t n plus c 1 h square plus term of the order of h raise to 4. What is c 1? c 1 is f dash a minus f dash b divided by 12. That means, our c 1 is independent of the partition. That means, it does not matter the what partition I am looking at. What is c 1 is?  If I change my partition, the term will remain the same, c 1 will remain the same. So, why do not I do like this? That I look at a partition with n intervals. So, I will have such integral a to b f x d x to be equal to trapezoidal rule plus a term plus a term of the order of h raise to 4. Now, instead of n intervals, I will consider a partition with n by 2 intervals. That means, earlier our sub intervals they had length to be equal to h. Now, I will look at length to be 2 h. So, then I will have integral a to b f x d x is equal to t to n. So, that means, that is the trapezoidal rule based on partition with n by 2 interval plus I will have c 1 the same constant and instead of h square, I will have 2 h square because now the length of the partition is 2 h plus 2 h. So, this is the term of the order of h raise to 4. So, I have got t n plus c 1 h square t n by 2 plus c 1 2 h square. So, now, I will take a combination of t n and t n by 2, so as to get rid of the term which contains h square. So, let me explain. So, we have integral a to b f x d x is t n plus c 1 h square plus term of the order of h raise to 4. Integral a to b f x d x is t n by 2 and denotes the number of intervals in the sub partition plus c 1 2 h square plus term of the order of h raise to 4. Now, what one can do is multiply the first equation by 4 and subtract the two equations.  So, that is going to be 3 times integral a to b f x d x. Here, it will be 4 t n minus t n by 2 the numerator here. Look at this term. It is going to be c 1 times 4 h square minus c 1 times 4 h square. So, this will get cancelled and then term of the order of h raise to 4. Big o of h raise to 4 that means, it is less than or equal to constant times h raise to 4. So, this is and then I divide by 3 throughout. So, I get integral a to b f x d x is equal to 4 t n minus t n by 2 divided by 3 plus term of the order of h raise to 4. So, I have got t n 1. So, integral a to b f x d x is equal to t n 1 plus a term of the order of h raise to 4. Now, what we are going to do is we are considering this partition say t 0 then t 1 t 2 t n minus 2 t n minus 1 t n and this is going to be of length h. So, t n is equal to t n minus 1 t is going to be h by 2 f a plus f b plus h times summation f t i, i goes from 1 to n minus 1. When we look at t n by 2 then it is going to be h times f of f f plus f b because wherever there is h now it is going to be 2 h plus 2 h times summation i goes from 2 to n minus 2 i even f of t i. So, here t n by 2 and t n they have some points they are in common like for the t n by 2 what comes into picture is a b and all even order t i's. When you want to calculate t n then to these points we add points t 1 t 3 up to t n minus 1. So, that means, whatever was the work done for t n by 2, what we are going to do is we are going to do calculation of the function values at t i's for i even that work one uses and then one looks at the combination 4 t n minus t n by 2 divided by 3 whatever formula you get that is going to give you the error to be of the order of h raise to 4. So, this is the first step of Romberg integration. So, if your function f is going to be 4 times differentiable then we obtain a formula which involves only function values no derivative values which gives us the error to be less than or equal to constant times h raise to 4. Now, whether suppose my function f is 6 times differentiable is it possible to obtain error obtain a formula in which case error is less than or equal to constant times h raise to 6. So, such a thing is possible. So, we have got a asymptotic series expansion integral a to b f x d x is equal to t n plus c 1 h square plus c 2 h raise to 4 plus c k h raise to 2 k plus term of the order of h raise to 2 k plus 2. So, we want the function to be sufficiently differentiable to k plus 2 times differentiable. These c i's the coefficients they are going to be independent of h and hence independent of h n and that means it is independent of the partition. In fact, the c i's are some constants alpha i's multiplied by the k minus first derivative at a minus k minus first derivative at point b. Now, this asymptotic expansion it is known as Euler Maclaurin series expansion and the coefficients alpha i's. So, for that what comes into picture are Bernoulli polynomial, but I do not want to get into details of those things. It is just I want to tell you that if the function is sufficiently differentiable then we have got a asymptotic series expansion. Now, using composite trapezoidal rule we got rid of the term h square and then obtained a result which has got error to be less than or equal to constant times h raise to 4. So, this asymptotic series expansion it tells us that it is possible now to get rid of the term h raise to 4 and obtain a error to be less than or equal to h raise to 6 and so on. So, this is known as Romburg integration. So, look at this expansion T n plus c 1 h square plus c 2 h raise to 4 plus term of the order of h raise to 6 when instead of considering n intervals I look at n by 2 intervals I have T n by 2 plus c 1. Now, our intervals are of length 2 h. So, it is 2 h square plus c 2 2 h raise to 4 plus c 2 h raise to 4 plus term of the order of h raise to 6. Now, what we are doing is we are considering 4 times this minus this and divided by 3. So, that was our T n 1.  So, nothing here this term is going to have c 2 h raise to 4 common 4 minus this term. So, 4 minus 16 divided by 3 plus term of the order of h raise to 6 and thus you have T n 1 plus c 2 dash h raise to 4 plus term of the order of h raise to 6 where now c 2 dash is minus 9 c 2. So, you have integral a to b f x d x is equal to T n 1 plus c 2 dash h raise to 4 plus term of the order of h raise to 6 c 2 dash independent of h. So, integral a to b f x d x is equal to T n 1 plus c 2 dash will be equal to T n by 2 1 consider partition with n by 2 intervals c 2 dash will not change what will change will be instead of h you will have 2 h raise to 4 plus term of the order of h raise to 6. Now, we want to get rid of these two terms. So, we will multiply this equation by 16 subtract these two equations and divide by 15. So, you will have integral a to b f x d x is equal to 16 T n 1 minus T n by 2 1 divided by 15 this term will go away and you are left with term of the order of h raise to 6. So, this is going to be your T n 2 plus term of the order of h raise to 6 T n 1 what it involves is 1 plus c 2 dash T n T n by 2 these are the composite trapezoidal rules T n by 2 1 will involve T n by 2 and T n by 4. So, T n 2 it is going to be based on trapezoidal rule based on n intervals trapezoidal rule based on n by 2 intervals and trapezoidal rule based on n by 4 intervals and their combinations that is going to give us a formula which has error to be less than or equal to h raise to 6. So, integral a to b f x d x is 16 T n 1 minus T n by 2 1 divided by 15 plus term of the order of h raise to 6. So, diagrammatically here you have T n here you have T n by 2 T n by 4 these are all composite trapezoidal rules then consider 4 times this minus 1 times this divided by 3 that will give you T n 1 next consider 4 times this minus 1 times T n by 4 T n by 4 divided by 3 that will give you T n by 2 1 after obtaining this consider 16 times this minus this term divided by 15 that is going to give you T n 2. More general case will be T n by 2 T n 0 is equal to T n T n is composite trapezoidal rule define T n m by this formula that T n m is equal to 4 raise to m T n m minus 1 minus T n by 2 m minus 1 upon 4 raise to m minus 1 m is equal to 1 2 up to k. So, the final thing T n k is going to give you the error to be less than or equal to constant times h raise to 2 k plus 2 or integral a to b f x d x is equal to T n m plus term of the order of h raise to 2 m plus 2. If m is equal to 0 that means the composite trapezoidal rule we know that it is of the order of h square then m is equal to 1 order of h raise to 4 and so on. So, thus if our function f is sufficiently differentiable then based on composite trapezoidal rule associated with different partitions we can construct approximation to integral a to b f x d x with the error of the order of h raise to 2 k plus 2 where k is equal to 0 1 2 and so on. So, this is the result of the next time we are going to consider numerical differentiation. Thank you.


Wikipedia took serendipity to another whole level. You just go and you always go in the end reading about the history of black metal in Sweden. I'm Andrea Zanni and I work as a volunteer in the Italian Wikisource. Wikisource is a digital library. It owns and hosts ancient books like old books from public domain ones and we put books for people to be read and we digitize some of them. We put them on Wikisource and then we transcribe them. We also try to link the books between each other because there is a possibility to link another to another, link a text to another and try to follow quotes and create a sort of, you know, hypertextual digital library. I really love that about Wikisource, the possibility of creating a connected integrated library. Serendipity is one of the best things in life and it's just serendipity is when you find something interesting but you are looking for something else. It always happens when you go in a bookstore. It always happens when you go in a library with an open shelf. As everyone, I was searching a word in Google. I came out on Wikipedia and then I started, okay, this is not possible. It can't be open to anyone. It can't be like this, like, editable for anyone. And it was. And then I discovered Wikiquote that was a project that I really enjoyed the first times. So I started working there and then discovered Wikisource and I said, okay, this is my place. And, you know, there were the books and there was the nerdy part with computers, software, all the stuff that was together and, okay, I belong here. Wikipedia, yeah, it's probably the best Serendipity of sync ever, right now.


I'm trying this again. Okay, guys, I'm trying this again. Let me know if you can hear me. I apologize for the distraction of it not working, but hopefully it is working now. Let me know if you can hear me in that chat section there. So let me know, yeah, if you can hear me, all right? Let me put something in there. Maybe I can do that. Let's see here. Can you hear me now? Loud and why? Loud and why? Hmm, loud and why? So all that stuff I said on the other live, you guys missed.  Now let me see if I can find the actual thing I'm doing now, the live. Let's see here, I go to my channel and I'm trying to hook into it to make sure that it is working. And it looks like I got like two of them going here. Let's see here. Clear and loud, you know, that's so weird because I didn't do anything any different. I didn't change any settings. I didn't mess with my microphone or anything. Isn't technology awesome? As a matter of fact, I'm doing a live on Saturday if you're interested in this on how almost everybody's job is in jeopardy right now. It's kind of scary, but you know, I do want to talk about that. Now I'm trying to find myself live on my screen down here because so far I haven't been able to do it. I think I did, I think I got it. So if you haven't subscribed, by the way, I forgot to tell you that, make sure you subscribe, make sure that you turn on all bell notifications so you're notified when I do go live. And today we're going to be talking about niches and picking a profitable niche. And it's probably not going to be what you think it is. It's going to be a lot different than what you think it is. And I know people want to pick niches. As a matter of fact, I got some comments in there. I think it was James that said something about wedding niche and then when I said it's seasonal, he said, well, how do I go about picking a year-round one? I will tell you this, at the end of this training today, I'm going to put a link in the description and it'll take me about an hour after this, but you're going to be able to go to that link and download the most profitable niches. Okay, so that's going to be kind of cool for you to be able to do that. And that's free, you know, it's going to be a list of all the top niches. But, you know, the conventional thinking is go find something that you really love and you'll be really good at it. That's not necessarily true because what you love may not even be popular, you know? So just finding what you love is not enough. And then the other people will say, well, just go find something that's selling really well and then you're going to do well at it. Well, that's not true because I started off selling cell phone cases. Now, cell phone cases were hugely popular in 2009 and I guess they are now too. I mean, most people have a cell phone case, right? But do they buy them online? I don't know, I don't know where they get them from. I failed at that. And then I got into survival products and is it something I love and it's something I'm really into? I mean, not really. I mean, it's interesting, but is that my life outside of, you know, selling survival food? No, it's not. I mean, I don't even really think about it on a daily basis. So a lot of those things that you hear about being successful online, just plain aren't true. They just aren't true. And finding a successful niche is more about finding out what the market wants more so than what you want to do or what you're interested in because you can have your hobby without selling online. And I don't know if I would want to destroy my hobby by, you know, doing it for a living. You know, I used to fly airplanes a lot. That was a really cool thing I liked doing. I flew all the time. I started when I was 16 years old. Flew all through college. Went out to California, ran in an air charter service. And I would not want to do that all day long. I mean, that would ruin it for me. And if you look at, you know, pilots that have to fly every day and they get stuck in cities and they're all cramped up in the cockpit for hours on end.  That was fun.  Not that it doesn't always work. My wife's in the dogs. She does dog training. So don't really go with the custom way of thinking, the ordinary way of thinking. Kind of break out of that a little bit. But the one thing I will tell you that's important is it's got to be popular. And that's why I want to provide you with a list of these niches. And I don't want to bore you by going through every single one. I will go through some of them. But, you know, there's a lot of niches where you can make money. Let's see here. Sort of read some of this. Paul says, USA Today reported one in five jobs is in jeopardy. It's more than that actually. It's really scary. And on Saturday, I'm going to get into a lot of depth on that. But man, I'm telling you, if you've got a job right now and you think it's secure, I'm telling you everything from lawyers, to doctors, to everybody's job is in jeopardy because of stuff that's coming down the pike. Isn't it better to sell high-priced products? Not necessarily. I mean, high-priced products, you're going to sell fewer of them.  Well, yeah, but they pay you a lower commission and there's fewer customers. It's really about how automated you have your system and how well you run that automation. Today, I was sending out some emails and I put a new thing in there. And it's a timer inside the email and it counts down to the date. And I'm going to try that out, see how it works. If that's something brand new, I hadn't been doing. But I'm always trying to automate new things. You know, trying to do new things. I think that's important. Let's see here. I think it's important to get started even if it's part-time, take action and revise later.  Yes, Paul, that's really good advice. I think that's one of the biggest obstacles is people keep looking for the perfect niche, for the perfect product. And when you do that, it kind of just stops you because there is no perfect niche. There is no perfect product. There never will be, okay? It's really about just getting going and getting automations going. Automations are super important. I will say that if you're not doing automations. JR, my Facebook account was hacked. Stole $250 in Facebook ads. Then they put an offensive photo which caused my account to be shut down. Facebook support is impossible to reach for me now. Ideas.   They, oh man, they did so much to my accounts and everything. And I'm not saying it's not a good platform to advertise and make money. You know, I bought their stock. What's their stock doing today? I'm kind of curious about that, and you brought that up. Because everybody's been really having a hard time with Facebook. What is their stock today? It has not been doing well. It's kind of flat right now. It's a 264. I do, I worry about the future of Facebook. Well, I worry about the future of my stock is what I really worry about. I don't care about Facebook. Let's see. Hey, what else we got in here? FYI, in Kissimmee, Florida, you can fly a jumpstart in a P-51. I think that would be cool. Yeah, I used to fly T-34s with a sliding canopy. I was a member of the Norfolk Navy Flying Club. My dad was retired Navy. So I had access to that flying club. So I flew some pretty cool airplanes. You know, Seneca twin engines, 152s, 172s, 172RGs. I flew a whole bunch of different airplanes. LOL, just reported to cyber crime. Yeah, good luck with that. Gary, Jerry, Jerry, I'm sorry. Looks like you're new in here. Is the affiliate marketing over? Never, ever, ever is it over? No, of course not. It's bigger than ever. Clickbank's doing great. If you look at my description after this video's over with, I've got all kinds of affiliate links. I make money every single month on it. I have affiliates that sell my product who share a sale. I sell other people's products. Oh no, no, affiliate marketing's awesome. The only thing I would tell you about affiliate marketing, I'm gonna get into all of the top niches in just a few of the top niches. What was I gonna say? Affiliate marketing. Oh yeah, yeah, yeah. If you're just doing affiliate marketing of other people's products, it's really hard to build a list unless you have a page between what your ad is and theirs. And then you run into, if it's a redirect or something like that, you can get your account shut down in Facebook, so it's a nightmare.  Facebook can just be a nightmare. And more and more people, I think, are getting off that platform. Is it gonna go away? I doubt it. It's probably always gonna be around in some form or fashion. But remember, we had was it MySpace? MySpace was huge. I mean, MySpace was the biggest thing out there. And where is MySpace now? I mean, it's gone. I think the guy sold it for $100,000. And at one point it was doing millions of dollars. But because of problems on there and the wrong kind of people on there, it crushed it. So I think they do worry about that stuff a lot. Maybe we'll sip. This is our saying for the day right there. You got that? Good. All right. So let's talk about some of the niches that are really popular and that probably will always be popular. Number one, I'm gonna say is weight loss. Weight loss is huge. And you know, weight loss is probably the simplest thing in the world to solve. You know, I could lose 15.20 pounds. I mean, I know that. I could lose some weight. But I know the answers to doing all this. Eat healthy, okay? Whatever healthy is, because every day that changes, right?  I mean, all that kind of stuff. But eat healthy, okay? Do your best to eat healthy. And I think the best thing you can do for eating healthy is to eat stuff that is less processed. Now you can take that to the extreme and you can go to a raw food diet, which I think is horrible for you because your body has to work really hard to break down all that food. Just my opinion, not a doctor.   I mean, that's pretty much it drink water. So there's every fitness program and they're not fitness, every diet program in the world right there in a nutshell. So that is a huge niche, weight loss. If you're into helping people with that, if you're good at that, I would say be careful with that because I do see people marketing weight loss stuff and they need to lose weight. So you kinda gotta be an example of what you're trying to sell so be careful there. Now an offshoot of that or something close to that is fitness, being in shape. There's a difference in losing weight in being in shape. If you try to go out there and say, I'm gonna help you lose weight and get in shape, you're not niche down enough. You gotta pick one, pick one, okay? Could you make it by doing it that way? Yeah, but you're gonna make it a whole lot better if you pick one and you're more niche down because people want specifics, okay? Somebody who wants to lose weight and get in shape, the first thing they wanna do is lose weight, okay? They're not thinking about having the six-pack abs. They're just not. I mean, yeah, they would like to have that, but they know they gotta lose weight first. Let me catch some of your comments in here. LOL, just three, okay, okay. I've been researching Khartra, super cool setup, just wondering if the starter package is actually enough to get started. It is, I mean, it is. It has to do with your bandwidth. You're not gonna get as much bandwidth. If you get past the starter package, you have unlimited everything, which is really cool that, you know, of course, I had that because my list is too big to do any of the starter stuff, but yeah, I would say do the starter package first. And if you don't have that, please use my link. That link will get you some discounts and will get you a $1 trial and use some really cool stuff. So if you don't have that link, it will be in the description after this video. Matter of fact, I had planned on talking to you after this too, right? We're gonna have a quick conversation. How much due diligence did you do when you went into food manufacturing? When I actually went into manufacturing, it was a ton. I mean, it was a ton. It took months of really getting stuff put together. And I'm actually going through some of that now because I'm doing some reformulations on our freeze-dried foods. So yeah, I'm still doing a lot of that. But so we've got fitness and we got weight loss, okay? What's the next one that's really big? Dating. Dating is huge. Now, dating is probably a little bit different now. And a lot of the dating stuff has been taken over by some of the apps that are out there now. And I don't know any of these. I think there's Tinder and I don't know what the other ones are. There's a whole bunch of different apps for that. So I don't know. But I don't know if those are really for dating or if they're for other things. I mean, they could just be to meet somebody very quickly and be intimate and be on your way. I don't know. I came along, I was married by the time all that stuff got really popular. So I don't know much about it. Haven't even looked into it. Had no need to. But that is a big niche, dating. And you know, if you're interested in one of these, you can go to something like ClickBank and type in dating and look at the top offers in there. Find out what is selling. Find out what is working by the program of that person. You know, and then get in their funnel and see what their emails are. I mean, it's really that easy to do that. And then you can do something, excuse me, similar. You don't want to copy what they're doing. You don't want to copy what they're doing. But dating is a big one. Next thing is relationships. So once you get people dating and they're together, then they have problems. So the next really big niche is relationships. And I'm just going to go over some of the huge ones right now. But in the download I'll give you, and I'll put a link in the description about an hour after this video, that will give you a whole lot more niches that you can pick from. So relationships and getting people to get along. And relationships can be between a couple, OK?  You can do relationships with your children. You can do relationships with your dog.  So that's another really big one, is relationships. It's a huge, huge, huge one. Let's see, is cartra effective for affiliate marketing and drop shipping? 100%. I do it. I do it right now. As a matter of fact, if you search in my channel, affiliate marketing, you're going to find a lot of videos on it. And I'll walk you through that. I'll walk you through share sale. I'll walk you through cartra. I'll walk you through click bank, a lot of those different things. How much due diligence did you do? OK, you've already answered that one there. OK, got that. Keep putting those questions in there, guys. These are awesome. I really appreciate you doing that. The more questions, the better. The more comments, the better. I really appreciate it if you put that stuff in there. It helps the whole session out. Let's me know you're interacting with me. So that's really cool. So what else? What else? What's another good niche? I will tell you, self-help is self-help, a huge niche. Now, this one can branch off in a lot of different directions. It can branch off into people who are depressed. It can branch off into people who want to be super successful. It can be branching off into people who want to have time management. There's a lot of self-improvement things that you can do. And that's just a huge market, self-improvement. And most of us do want to improve. I mean, we don't want to get up every day and just kind of get through life. I mean, what's the point? Why are you even here? I think the point of life is to help others and improve who you are. I mean, that's probably it. I don't know of anything more than that. Some of you may say, well, too amass tons of money. But yeah, then what are you going to do with it? Are you going to help other people? Are you going to improve yourself? Are you going to improve the world? What are you going to do with it? Because you can only buy so many cars. You can only drive one car at a time, unless you have self-driving cars.  So I guess you could do that. So put in that comment section. What do you think of these niches? Have you tried any of these niches? Are these things that interest you? Put that in there. Like I said, every day, I try to improve things just a little bit. I have a sequence that's going out this morning. If you're not on my Survival K food list, you should get on that. You should really go to that page and get on there. Let's see. James Brain is Shopify the best platform for drop shipping. Well, that's a good question. It is probably one of the biggest ones in the world as far as drop shipping. There's probably other good ones out there. I know Shopify better than most others. I have a course on Shopify called Ecom Road. And it walks you through setting up a Shopify store and how to do it. So if you ever want to look at that up, let me put that in the comments section here. For Shopify, check out Ecom Road. Ecom Road. And if you get on my list, of course, you'll see it. Siddharth, what's going on? He just showed up here. What niche would you stay away from starting out? Personally, I think the food business is the hardest thing I ever did. Yeah, ours is not easy. We're established now, which helps because we have customers that have been with us for 10 years, 11 years. So that's kind of nice. And it's not really a matter of what's hard is doing stuff you don't want to do. I don't mind doing, you know, our survival food company. You know, I want to get back to that question that James had about seasonal. He said the wedding thing. Now, weddings are seasonal. It's rare for somebody to get married in the last week of November. Okay, or any day during December. I'm not saying some people don't do it, but most people don't. Most people don't get married in January or February either. It's generally a springtime summer thing. They don't normally have weddings in the fall, but there's some people that do. Now, does that make it a bad niche? No, it doesn't because, you know, maybe you just want to work those months.  Maybe you work two niches, you know, or three, I don't know. So it doesn't make it bad because it's seasonal, but you've got to understand that since it is seasonal, you won't make money year-round. My survival food company is seasonal. It's like September through March. I mean, it's just what it is. Does that mean I won't sell stuff for a year-round? No, I do sell stuff year-round. However, the majority of my business is during that period. This last year wasn't like that. All through the summer, I was selling the heck out of food. And now it's a little bit slower this past fall, which makes no sense, right? But that's just the way it is. Maybe a lot of those people bought what they were gonna buy. I don't know. We still sell every single day. We do well. Can you give me a tech company name idea? You know, tech company names just don't really matter. If you look at most of them that are out there, I don't think the name means a darn thing. I really don't. I don't know. If I knew more about doing that, I would say something, but I just don't think it matters. Along with due diligence, check out trade magazines that are available, usually free for food. Check out Pizza Today or PMQ, Pizza Marketing Quarterly for food. Yep, you wanna do food? You can do that. Those are some really good suggestions. But be careful about doing research because we can over-research stuff very easily. We can over-research stuff. If you have any ideas on niches, put that in the chat section. If you've tried something, put that in the chat section. It's the brand and behind it.   I mean, if you think about what does Adidas mean? What does Nike mean? I mean, we don't even know what those things mean, but they're huge brands. What does McDonald's mean? McDonald's was just the name of the brothers that originally had the hamburgers. They were a croc, went into business with it. That's all that is. So I think we tend to sometimes get tied up on a name. Though for search engine purposes, if that's what you're trying to do, it does help, you know, the Dummies series, I don't know, there's a million of them, writing for Dummies or cooking for Dummies or whatever. I mean, they did pretty good with branding on that. You know, that worked out well. And it basically said, anybody can do this if you read my book. So it kinda opened up that world. But I don't really get hooked up on names. I mean, I have a company called Digital Enterprise Marketing. It sounds really fancy, it's really not. It's just a parent company for other courses and things I have. I have Survival Cave Food, which is just something I thought about sitting on the couch one day. I thought of Survival Cave. I don't know why I just did. I didn't do any research on that name. It's a Google search term now. So that's kinda cool. But you can overthink it. It's like Paul's saying there right now, don't get hung up with analysis paralysis, yeah. What about outdoor sports? Outdoor sports are great, good things. Keep in mind, they're seasonal. So that's another thing. Depending on where you live, right? I don't see Aldean in here today, but he's in Jamaica. So you're around in Jamaica, you know the weather's good, you can do outdoor sports. If you're in Alaska, you're not gonna do outdoor sports, you're around.  You can certainly do that. But that's my overview on it. I would suggest that you go into the description, give me about an hour because I gotta put the link in there. But I'll put a link for the top niches to market to and then you guys can have it. You can download it, it's free.  I'd say it's worth something. It's got some value. And I think there is a value on a hundred bucks or something like that. But I'll give that to you. One niches gifts and gift ideas along with gag gifts. Yep, that's another one. That's a good one. And that could be a year-round thing too, you know? But like I said, don't get hung up on just year-round things. I'm like, my courses are a little bit seasonal depending on the time of year. I mean, spring and summer, people are thinking about vacationing and going places. This past year they weren't. Okay, so there could always be a rich thrown into your plans. Nobody was going on vacation. This past year they just weren't doing it. I did. I went to Florida and I went to see my daughter in Tampa. Then I went to Key West and hung out down there. I did do a little vacation. I was cautious.  But I wasn't gonna end my world. I'll put it that way. Let me read some of these comments here. In sports, golf and fishing are a huge market. They are, but it also depends on how popular that sport is. When Tiger Woods was playing, golf was immensely popular. Right now it pretty much sucks. A lot of golf courses are closed around the country. If you research golf right now, it's way, way, way down since he wasn't involved. And when he was gonna get back involved, it was starting to tick up a little bit. Who knows what's happening with him now? I mean, we know he's been injured in a car accident and he just brought new life to the game. I don't know why. Everybody just liked him. He's a cool guy, I don't know. Yeah, a lot of girlfriends, I give them that. Fishing are huge markets and places where fans are fanatics. Yeah, fishing's a good one. My niche will be get S done bells. Where'd you get that from? I don't know where you would have gotten that from. See, I can't say it, but you do. Wanna ring the bell. Cause when you ring the bell, you turn all bell notifications to know what I upload videos. You'll know when I go live. So yeah, that's stuff you wanna do. How can we identify a long-term niche versus an overnight fad that is gone tomorrow? Oh, I think that's pretty simple. I think that's pretty simple. Fidget spinners, okay? Was everybody gonna do that forever? Mm-mm, pet rocks. Was everybody gonna have a pet rock forever? Mm-mm. So I think I could probably look at anything and tell you if it's a fad or not. You know, I think cupcake shops was a fad.  Really, how many cupcakes you gonna buy? I've seen some people, it looks like they buy a lot. But for the most part, you're gonna go, eh, yeah, tired of cupcakes. Same thing with frozen yogurt shops. I mean, you saw that come and go. The orange Julius thing, you saw that come and go. So those are all food related. But I can pretty much tell, I guess maybe because I've been in marketing, you know, my whole life, but I can pretty much tell if something's a fad just by looking at it. Is it applicable to everybody? Is everybody gonna do it? Or is it something that once you buy it, you're never gonna buy it again? You just apply some common sense stuff to it. Hula hoops, yeah, hula hoops was big when I was a kid. You know, that was like 60s, 70s.  It's like, you know, you've done it, you learned how to do it, so what, it's over with. So a lot of stuff like that. Travel bargains, niche is huge, along with credit card bargains. Commissions are big because an average person keeps a card for eight years. So companies pay big commissions. Yeah, but I will tell you, it's very expensive to do ads for credit cards. Very expensive. I mean like uber, uber expensive to do that. So where are you gonna get the leads from? Also for doing business loans, that's very profitable too. Uber expensive to market. Very, very hard to market business loans. It's a tough one.  Gelato is coming up, is a trend. Yeah, Gelato's been around for a long time though. I don't know if that's a new trend. Gelato's been around for years and years and years. I remember being in Puerto Rico, San Juan, Puerto Rico with my kids. And they were like 12 and eight, or 13 and nine, something like that. And they're now, gosh, my son is going to be 27 and my daughter 31 this year. So yeah, that was popular then. I don't know how popular it is now. And it's seasonal. You know, if you live in a city where there's seasons, I'm not buying Gelato in January. I'm just not. It's not gonna happen, right? So I've seen a lot of those come and go. So I hope that helped you guys a little bit today. Kinda get you in the right direction.  Don't feel like it has to be your all being. You know what I mean? I like flying and I haven't flown in years, but you know, I could start a niche in that, but I don't know if I'd wanna ruin, you know, what I like doing by selling, you know, aviation stuff. I don't know. I mean, it's probably a niche out there for that. So don't worry about it. Don't overthink it. That's the one thing I'll tell you. Mike Roberts says pet supplies is huge. Yeah, my wife actually, she does dog training and she had a pet supply site, but it is very, very competitive. It's a race to the bottom as far as price goes. So you're either gonna be trying to mass market against like Petco and places like that in Walmart, or you're gonna be selling only super high-end stuff that you can't find other places. And I think the high-end stuff is the way to go. The big thing in pets now is pet food, raw food diets, our dog is on the raw food diet. We buy it frozen, thawed out, feed our dog. So a pet nutrition is a big thing because it's been so crappy for so many years. YouTube cooking like Bobby does on Flake City. He demos products and then links in his Amazon store. Yeah, and you can do that. Don't overdo that. I do see some of these people overdoing it. And Bobby is kind of going that way where it's like every single video is sponsored. And that kind of wears on you after a while because I don't always want to see that. I'm tired of seeing VPNs and all the stocks and all these different things that people are pushing. And I don't do that much. I mean, I can, I could, I could every single one of these I can tell you about some product and make a little bit of money. But I don't think you come here to hear about these products. You know, I'll talk about cartridge because it's the best thing since sliced bread. And I'll have a link to cartridge in the description. If you are not watching this live, please don't be put off by the fact that you don't have that chat section. You do have a comment section. And in that comment section, you can put whatever you want. I'll come back, I'll answer you. Late last night I was answering comments in there. I love getting your comments, keep those coming guys. I really do appreciate those. Let's see here. All right, yeah. Do you guys got anything else? Do you want to put anything else in that comment section? I'll give you a drink because there's like a 10, 15 second delay. That you could put in there if you have questions. If you just want to tell me what you think of the training. If you want to ask me for different training, all those things you can put in that chat section there. So feel free to do so. I love, I love, really love your comments. You guys really do come up with some good stuff. You really make me think. And this is as educational for me as it probably is for you because it makes me think about stuff. And you guys have great ideas. I'm not all knowing. Believe me, I just make some money online. I mean, we've done millions of dollars of digital and physical products. But man, I learn stuff every day. Paul says, what trade magazines or sites do you follow? None.  I don't follow trade anything. I do my own research. I look at my own stuff. I think a lot of trade magazines are biased because they have advertisers. Anything that has advertisers is gonna be a little bit biased. What I do is I look at the top people in the industry and see what they're doing rather than having a trade magazine tell me what to do. I just don't. Things a bunch, JR. Training is great. Peace and blessings always. Thank you so much, James. I really appreciate it. And Mike, you can contact me on Facebook. I'll be happy to talk with you. I promised you I would do that, Mike Roberts. And guys, if I could help you in any way, anytime, please feel free to contact me. Training at JR Fisher. You can send me an email if you wanna do that. You can always comment on any of these videos. If you haven't subscribed, please do so. Give me a thumbs up. Everybody on here, just click that thumbs up if you would please do that. That really gets me a lot more distribution of my videos. Really trying to work on building this channel. And I wanna build it by helping you guys out, helping you make money online. Any of your success stories, I'd love to hear, put that in the chat, put that in the comments section, whatever's available to you, put your success stories. I'm really, really like seeing that. I like seeing that somebody made $1,000 in a day or $10,000 in a week or whatever. Even if it's $50, your first $50 or $10. My first sale was 10. Looking forward to Saturday. So what bank or company do you recommend for online marketing so they don't hold your money? I use Navy Fed. My dad was ex-military, so I use Navy Fed. I also have Chase. I also have Citibank. I also have Fidelity. So I've got several different ones I use, but for credit card processing, excuse me, I use Authorize.net and I use Stripe. Stripe is very good. I do highly recommend Stripe. Stripe's a little bit more expensive than Authorize.net, but you know what? It is what it is and they do a good job. Just don't bend any of the terms or conditions because if you lose one of these accounts, guys, it can cost you a ton. Do what they tell you to do. Always follow all the rules. Super important there. I want to thank you guys for being here today. I really do appreciate it. Like I said, if I could ever help you, let me know. And I'm just going to say goodbye now and go to work. And how many currently subscribers do you have? Let me look, Paul. Let me look and see what's going on with my subscribers. And I think it's just under 5,000 right now. And is it huge? No, but it's getting there. It's getting there. So I got 4,500 or 700. 4,570 right now. But I had like a year ago, I only had like 900 or 1,000. So it's gone up and it's going up every day. And is it going up as fast as I want? No, I went 4 million right now. But my training is not for everybody because I don't tell people that this stuff's easy. I tell them it's simple, but I never tell them it's easy. I don't tell them that they don't have to work much. You know, if I were to come on here and say, guys, I got this deal, man, and you only work four hours a week, you're going to make a million dollars a year. I would have a lot more subscribers than I do now.  And I tell you the truth. And the truth does not resonate with everybody. It just doesn't. They don't want to hear the truth. But I can't lie, so I mean, it is what it is. Have a blessed week ahead gang. I really appreciate you guys being here. I hope.