Metabolic syndrome, also known as syndrome X, is a medical disorder characterized by the so-called deadly quartet, abdominal obesity, high-fasting sugars, high triglycerides, and high blood pressure. And it sets people up for a non-alcoholic fatty liver disease, stroke, diabetes, cancer, and heart disease. It's been estimated to afflict about a quarter of the American population. How do we stop it? How do we prevent it? Well, if it's all about obesity, the level of fat in our blood, and high blood pressure, then that seems like a job for plant-based nutrition. We didn't know for sure, though, until recently. Even after adjusting for lifestyle factors such as smoking and exercise, risk was highest in those eating non-vegetarian, intermediate risk for those eating semi-vegetarian, and lowest in those eating vegetarian, cutting the odds of having metabolic syndrome by more than half.


This is the VOA Special English Economics Report. Are you a saver or a spender? A penny saved is a penny earned. This old saying calls attention to the wisdom of saving money. Putting money away for a rainy day is another way to talk about saving for the future. People who hate to spend money are known as tight wads. While those who like to get the most value for their money are called thrifty. A thrifty person is different from a spend thrift. A spend thrift is someone who spends wastefully. People like that are often said to spend money like a drunken sailor or like there's no tomorrow. In the United States, people who want to start a savings account have different choices of where to put their money. These include banks and credit unions. Credit unions are cooperatives for people who have some kind of connection. For example, the members might work for a university or a government agency. Most credit unions are non-profit organizations. Credit unions, banks and other financial institutions pay interest on savings accounts. But the interest rates are low. Certificates of deposit pay higher returns. For a certificate of deposit or CD, a person agrees not to withdraw the money for a certain period of time. This term could be anywhere from a few months to several years. Longer terms and larger amounts pay higher interest. People can withdraw their money early, but they have to pay a penalty. Another way to save is through a money market fund. This is a kind of mutual fund. Mutual funds invest money from many people. The money is sometimes placed in short-term government securities. Money market funds, however, may not be federally guaranteed like other kinds of savings. The Federal Deposit Insurance Corporation guarantees savings up to $250,000. In a number of countries, people have been saving less over the years. The Organization for Economic Cooperation and Development is a group of 34 countries. The OECD says in 1990 Americans had a household savings rate of 7%. This year that rate is expected to be a little more than 4%. Many European countries have higher rates, but Americans save more than families in countries like Japan and South Korea. For VOA Special English, I'm Mario Ritter. To read, listen, and learn English, go to voaspecialenglish.com. You can also find our captioned videos at the VOA Learning English Channel on YouTube.


What's up everybody? Paul Garrett, filmforfood.tv here to show you how I single-handedly did the projection for a community theater production of In the Heights, a play by Lin-Manuel Miranda. Using Adobe's Premiere Pro, After Effects, an Element 3D to create the content and Resolume during the live performance. First step as in any multimedia project is pre-production. I picked up the In the Heights Libertto and read it front to back three times. Once to enjoy it, twice to see how projection would fit into it creatively, and a third time to see it in a technical light. Then I listened to the music to really get a feel for the audio side of the production. I would have liked to have seen the play live, but since it wasn't being performed anywhere near my area, I went online and found a high school production of the play, which was an ideal but still gave me some insight into the overall look and movement of the production. I went to the venue to look at the space. The venue has a 3,000 person capacity and a giant stage for them to create the barrio set. It was my main objective to create the Washington Bridge that appears in the background between the buildings. Now that I had the stage plans, I was able to do the math for my projector placement and for my cable runs from control to those projectors. When my preliminary research was done, I sat down with the director and went over his ideas and how I could help him bring them to life. Projection mapping in a live play setting is magical in a real cool new way to bring a stage design to life, but you've got to be careful not to turn your play into a movie. My main objective was to not steal the show, but to be a complementary element to the overall production. With a list of ideas in hand, it's time to animate. The first thing I needed to do was get a 3D model of the Washington Bridge and then make sure its size and angle was in tune with the overall stage design. I got stock images of the skyline and I threw those in the background and then animated 3D model cars driving across the bridge. I used 15 different cars in a whole bunch of different colors so that you couldn't tell it was the same cars driving across the bridge over and over. In the Heights takes place during the day, the night, and during golden hour. So I had to make three different versions of the Washington Bridge background, one for each coinciding part of the day. The final touch was to add moving clouds in the background and voila! The Washington Bridge background was complete. I put together some samples for the rest of the animations and then finished up around the same time that the set construction was ready for me to bring in my screens and projectors. We built a frame for a 200 inch screen using 45 feet of triangle truss and then placed it 12 feet from the back of the stage to allow my short throw projector enough room to create an image that size. We assembled our hardware and then climbed up onto the truss above the stage to hang the two projectors that were going to be mapped onto the buildings specifically for the club scene. My computer is a Windows based with a Rampage V-Extreme motherboard, 64 gigs of DDR4 RAM, an Intel i7-6 core chip, a GTX 970 graphics card, and a 1TB solid state hard drive. For my work monitor I used the display port and plugged that into a 4K monitor for the Resolume interface. I used my HDMI port and converted that to SDI. Then ran a 300 foot 3G SDI cord from my workstation to the backstage and converted that back into HDMI. Plugged that in to an Optima 1080 short throw projector which threw 3000 lumens onto the 200 inch screen at the back of the stage. For the DVI ports on the back of my computer I converted those to VGA and then from VGA into CAT 6 and ran two 400 foot CAT 6 cables to the projectors that were placed in the truss above the stage. I converted that back into VGA and plugged that into two 2400 mp projectors by Dell that were shooting 3000 lumens each at the buildings on the stage that were sitting about 17 feet away. Here's a tip. Make sure you work closely with the place lighting designer in pre-production. You've got to ensure that their lights don't flood out your projections. This defeats the entire purpose of having a projector set up in the first place. Now with the AV setup complete and your animation's done it's time to load everything into Resolume and see what it looks like on stage. In Resolume I created three decks. One for Act 1. Two specifically for the club scene. And three for Act 2. I created three layers inside each of the decks. One layer was dedicated to the projector at the back of the stage. The other two were going to the projectors hanging from the truss that were pointed at the buildings on the stage. In the advanced output settings I had three screens. One for each projector with one slice inside of each screen assigned to the coinciding deck layer. I added a vignette to each animation inside of Resolume and it seemed to help with their projections on stage by getting rid of that boxy squared off projector look. I labeled each individual clip inside of all three decks according to my projector cues and then I midi mapped my faders onto an Akai midi mix controller. My first cue was for pre-show. Something to put on the screen while everybody found their seats. This was just a still image of the bridge on the 200 inch screen with the in the heights logo over it. My second cue was triggered right when the play started. I faded out the logo and let the cars start driving across the bridge. Cue 3 was in scene 4. During the song it won't be long Vanessa is singing about a plane that flies through the sky. I created the animations in such a way to where I can trigger it in sync with the musical performance. Starting with the animation of the cars driving across the bridge then going up into the sky with the airplane flying through and then back to the bridge loop. We also had an audio person cue the sound of an airplane flying by all at the same time. All of this choreographed together made for a great little moment on stage. Cue 4. In the heights also has scenes that take place in doors. So for those scenes the lighting design would only light a few things on stage and my screen would go black. Cue 5 was at the beginning of scene 7. Now we're outdoors again. We bring back the bridge this time with moving clouds in it. During the song we decided to put twinkling stars in the background and have them slowly get brighter during the musical performance. Cue 7. For the song we see the golden hour backdrop appear on stage for the first time. Cue 8 again we go black for an indoor part of the play for the song the following scene and then we go up from black as we move into the club scene. For this scene we use all three projectors so that we could transform the barrio into a club. This scene has its own deck inside a resolute with 32 different visuals. I practiced this part over and over at home before we went live so that I could learn the musical cues and put the proper visuals in the right part of the song that coincided with both the actors and the musical performances. Coming out of the club the city is blacked out and then we transition from that to intermission with the logo fading back over the blacked out bridge. Coming out of the break I fade out the logo leaving just the blacked out bridge with the cars driving over it. For the song Sunrise I created an eight-minute long animation that goes from the blacked out bridge up through golden hour because there's a sunrise and then into daytime with moving clouds and made sure that it moves in sync with the lighting design and with the actor's performances. It also plays long enough to take me into my next cube which is going black for another indoor scene. For the song Alabanza we bring back the twinkling stars which pay homage to Abuela who has just passed away and we wanted to show the same visual that we showed during her performance earlier in the play. For the song Everything I Know we used actual pictures of our actors as kids and created an animated slideshow that went along in sync with the actor's lyrics as they were sung. The scene ends and we go back to the daytime backdrop with moving clouds. We bring back the golden hour backdrop for the song Piragua and then let that play through the performance of when the sun goes down. The final cue is the same eight-minute animation that I played at the beginning of act two and let that roll through the end of the play. After the play the stage goes dark and I fade up the logo for the curtain call and that's it. Remember to take your time in pre-production with the creative side. Dry run all of your equipment at home and make sure to work out any bugs that it has before you put it up on stage. Make sure you're up and running for at least three rehearsals so you can time out your cues and work out any technical or animation issues you have before opening night. Once everything is set all you have to do is make sure you hit all of your cues in the play have fun and watch the crowd react as you create technical magic on stage with projection during a live play. Thank you very much for watching. I'm Paul Geerit and if you have any questions or if you'd like to hire me for your next video production you can find my contact information on my website filmforfood.tv


Tezal Toronto If you choose to attend our TEFL Tezal course in Toronto, you will complete your training in a genuine language school environment. The school itself is centrally located close to public transport links that provide easy access to all the main attractions across the city, as well as major transport hubs including Pearson International Airport and Union Station. Within the school building you will find modern classrooms that are perfectly suited to language learning. We also ensure that you have all the things you need to successfully complete your training including a wide selection of teaching resources, Wi-Fi access and the use of photocopying equipment. The number of trainees on our courses is deliberately kept low to make sure that everyone receives the maximum benefit of our trainer's knowledge and experience. During your stay in Toronto, the in-class hours for our course are usually 5.30pm to 10pm Wednesday to Friday and 9am to 5pm on Saturday and Sunday. Between lessons you can take advantage of the wide selection of shops, cafes and other amenities in the local neighborhood, as well as our complimentary drink station.


Hi, and welcome to the session. I am Shashi and I am going to help you to solve the following question. Question is, using elementary transformations, find the inverse of each of the matrices. If it exists, the given matrices 2, minus 6, 1, minus 2. Let us start with the solution now. Let A is equal to matrix 2, minus 6, 1, minus 2. Now, to find the inverse by elementary transformation, row transformation method, we will write A is equal to I A, where I is the identity matrix or we can write 2, minus 6, 1, minus 2 is equal to matrix 1, 0, 0, 1 multiplied by A. Now, we will apply a sequence of row operations simultaneously on the left-hand side and the matrix I, that is the identity matrix on the right-hand side. The row operations will take place till we obtain identity matrix on the left-hand side. Now, to make this element equal to 1, we will apply an R1 row operation 1 upon 2, R1. So, we can write applying R1 row operation 1 upon 2, R1. We get 1, minus 3, 1, minus 2, matrix is equal to 1 upon 2, 0, 0, 1 matrix multiplied by A. Now, to make this element equal to 0, we will apply on R2 row operation R2 minus R1. So, we can write applying on R2 row operation R2 minus R1, we get matrix 1 minus 3, 0, 1 is equal to matrix 1 upon 2, 0, minus 1 upon 2, 1 multiplied by A. Now, to make this element equal to 0, we will apply on R1 row operation R1 plus 3 R2. So, we can write applying on R1 row operation R1 plus 3 R2, we get matrix 1, 0, 0, 1 is equal to matrix minus 1, 3, minus 1 upon 2, 1 multiplied by A. We know identity matrix is equal to A inverse multiplied by A. Now, comparing these two expressions we get A inverse is equal to matrix minus 1, 3, minus 1 upon 2, 1. So, our required inverse is given by the matrix minus 1, 3, minus 1 upon 2, 1. This completes the session. Hope you enjoyed the session. Take care and goodbye.


Hello, and welcome to this session in which we will discuss the variable interest entity or VIE. Well, to understand VIE, we have to understand when do we consolidate because VIE is related to consolidation. So variable interest entity. Well, this is what we learned so far. We learned so far that you have to consolidate when you have what's called controlling interests. That's when you consolidate. What does it mean controlling interest? Well, controlling interests usually it's presented through a percentage. This is what we learned so far. If you own between zero to 20% of some company's stock, equity stock, you account for this investment using the fair value method. In other words, you have no saying in the company. You have no significant control. Therefore, you will account for it fair value. What does that mean? It means if the stock of that company goes up, the value of your investment goes up, and if the stock value of that company of your investing goes down, the value of your investment goes down. Now, if we go above 20, up to 50, then you have to account for this investment using the equity method. And the equity method means what? It means your investment is proportionally directly related to the net income, net losses, and dividend from that investment. And we covered the equity method in details. What happened if you own more than 20, more than 50%, here you have what we called control and you consolidate. So this is called the voting model. Voting model means what? It means if you own more than 50%, guess what? You are in control of the company. You have a controlling interest. That's the assumption. Unless we know otherwise, you have a controlling interest. This is the voting model you consolidate. So what is controlling interest? Let's say this one more time. It means you have direct or indirect ability to influence or determine the direction of management and policies. In other words, you are in control of the company. And an easy way to find out whether you have control or not if you have more than 50% interest. Unless, again, there is some sort of an agreement on the side. Now, how do we have control? Again, equity interests more than 50%. However, sometimes you could have control by contract or some sort of a side agreement. So although you may not have 50% plus, you might have 0, 10, 15, any percentage or no percentage at all for that matter and direct equity ownership. But you have some sort of a side agreement or a contract that could give you controlling interest. Under those circumstances, we will disregard the 50% plus requirement. Now, what we have is we have a variable interest model. And really, what happens in the real world is first, if there's no percentage, you would look at the variable interest model. First, you have to see, do we have any contract or side agreement that's given us controlling interest? What is controlling interest? It means you influence management. You can tell the company what to do. If you can tell the company what to do, you are in control. It doesn't matter what the percentage is. So how do we use this variable interest model? Well, guess what? When controlling interests exist through a method other than equity ownership, then you have a variable interest model. In other words, although you don't own more than 50%, but somehow you control the company. Well, if that's the case, we have a variable interest model. Simply put, when you look at an investing, you ask yourself, do I have a variable interest model? If not, then I will go to the voting model. If I have the more than 50%, then I consolidate. Now, the best way to illustrate this is to learn about the company called Enron. So we will use Enron as an excellent case to illustrate the concept of this variable interest model. So after we look at Enron, we're gonna look at the characteristic of the VIE. How do we know we have a VIE? But once we look at Enron, it will make more sense. And how do we know that we have an EIE? We look at the characteristic of a VIE. So let's take a look at Enron. But before we look at Enron, most likely you're watching this because you are either an accounting student or a CPA candidate. Either or, I'm glad you are here. That's why you are watching. You are looking for some additional help. I'm here to help you, farhatlectures.com. My website will have additional resources, lectures, notes, PowerPoint, multiple choice through false. That's gonna help you in your accounting courses. This is a partial list. Or your CPA review course, which is my material is aligned with your Becker, Roger, Wiley, Gleam. I don't replace your CPA review course. I'm just gonna give you additional explanation, additional resources, which in turn will help you with your preparation. If you have not connected with me on LinkedIn, please do so. Take a look at my LinkedIn recommendation. If you're watching, please like this recording. If you like it, share it with other. Connect with me on Instagram, Facebook, Twitter, Reddit. And if you're a CPA candidate, I have a group me account called CPA Exam Support Group. You can join there as well. So the best is to look at Enron. Let's take a look at Enron. Well, here's Enron. Enron is a company that existed. Now it's no longer exist. What Enron used to do is something like this. Enron will create what's called, used to be called SPE, Special Purpose Company, which is now we're gonna call it Variable Interest Entity. So they will create some sort of a partnership, some sort of a company, some sort of a joint venture on the side. And that venture is for a well-defined limited purpose. So for example, if they wanted to buy barges or transportation boat from Nigeria, so they will set up a company on the side to do so. The sponsor, which is Enron, provide most of the resources and all the activities of the SPE. When I say SPE, Special Purpose Entity, it's the same thing as Variable Interest Entity, just because I'm used to saying SPE rather than Variable Interest Entity. So the activities of the Variable Interest Entity is governed largely by contract or agreement that rests with the sponsor. So now, so we created this joint venture or this partnership. What would this joint venture or partnership do? They will go to Bank One or Bank of America, Bank of America, and they will ask for a loan.  What is the purpose of your business to give you a loan? They would say, okay, no problem. They would say, we have actually a guarantor. Enron will guarantee our loan. Well, oh, we know who Enron is, sure. So how much money do you want to borrow for this project? We want to borrow $50 million. Bank of America will lend this SPE $100 million. So that's fine, now they will borrow the money. And Bank of America is not concerned about this loan. Why? Because Enron is guaranteeing this transaction. So if anything happened, Enron will guarantee the transaction. Now what would Enron do? Enron will provide services. For example, Enron, they will do many things, but let's keep it simple for now. They will provide services like management services, consulting services for this SPE. And the SPE will pay Enron $100 million in fees. So simply put, the money will transfer from the SPE books to Enron books. Enron will debit cash $100 million, credit revenue $100 million. This is just to kind of show you the example. Now, you're saying, hold on a second. If this SPE was created by Enron, it doesn't make a difference because this SPE will have to consolidate with Enron. Therefore, the loan will be showing on the Enron books. Well, based on the old rules, and notice I highlighted and I put it in red. Based on the old rules, Enron would not have to consolidate if they can bring 3% equity to this SPE. Simply put, if they can bring 3% equity, independent interest, then this SPE will be considered independent. And as a result, they don't have to consolidate. They don't have to consolidate. So what did Enron do? Enron will create those fictitious third party, which I'm going to show you on the next slide, to make the SPE and Enron as independent companies. So they don't have to consolidate. They would use the executive real estate agent and other individuals. So what is the purpose? So why did Enron do this? Well, you create SPE, the Enron style to do three things. One is to avoid losses. So this is what Enron used to do. Enron will make investments. They will buy investments. And by the way, they were starting a company like Netflix. Netflix, the Netflix that you know now, but with the blockbuster, then the deal did not go through. But simply put, they will buy investments. They did a lot of things, but part of it is they will invest in other companies. And if those investments were bad, it means they didn't do well on those investments. What Enron would do, they would fictitiously sell them to those SPEs, not only one, they did many SPEs, you will see in a moment. So they will sell them to this SPE for a profit to avoid the losses, thus creating illusionary gains. So it's like basically, they will sell it to them. Hold on a second. If they sell it to them, how would the SPE pay for this? Remember, the SPE will go to different banks and get the money, so they will buy the investment from Enron. And Enron will show that they are doing well. They are making good investments, but really they're buying and selling from themselves. Also what Enron would do, they will keep that off the books, off the books, which is called off-balance sheet financing. Simply put, remember that loan that SPE got for 100 million. It's supposed to be on Enron books. We'll turn the loan into revenue and no one will see the loan. Enron committed all three violations by using those SPEs. Now it's worth looking at it, the actual case of Enron. And this is what it looks like. Sorry about the quality of the picture. I looked everywhere and I could not find really a good quality, but I'm gonna try to explain this. So this is Enron. Enron will have a relationship with this Chukko investment LP. Now, for example, Enron would use this Chukko to buy barges from Nigeria for transportation, for freight. Now who owns really Chukko? Well, let's see. Chukko is owned by Michael Copper and William Dotson. So Michael Copper owns 96.5 and Bill Dotson who works at Chukko who's the domestic partner of Michael Copper owns 2.5. So hold on a second. Who are these two individuals? First, Michael Copper is a former SEO of Enron, but he left. Well, he's like now an outsider and Bill Dotson, his domestic partner. In that state, domestic partners were not considered related parties. So what happened is this? Enron will ask them because they control Chukko and they're considered independent and notice they don't control Chukko directly. They control this partnership and this partnership control Chukko. So what they would do, they will deal amongst themselves. They will buy stuff from Enron and show that Enron is making a profit. And again, how do they buy it? They go to Barclay Banks and they will ask Barclay Banks for loans. And this is how they booked illusionary gain by using Michael Copper and Bill Dotson. Simply put, here what we're talking about is this, substance over form. Although Chukko is considered independent, but Chukko is controlled by outside parties. And by the way, Enron executives will pass the money to Michael Copper indirectly and they will ask them to invest in this partnership. So these partnerships are in essence, and not in essence, artificially independent, but they're not really independent. So that's the case. The case is to not allow Chukko to be an independent party from Enron. So let's now dive into the characteristic of VIE. When do we have a VIE, Variable Interest Entity? Well, of course we have to disregard the 3% ownership level, okay? Actually, we have to disregard any percentage ownership level. You don't have to own anything. If you have any of the following four, you have a VIE. So those are the important criteria on looking whether there's a VIE or not. One, the entity cannot finance its activities without additional financial help. What does that mean? It means the company is thinly capitalized. There's not enough equity in the company. Therefore, they cannot really borrow. So shareholders cannot borrow without a guarantor. Someone is guaranteeing them. And remember, Enron guaranteed the loan for those SPEs. That's one. So if that's the case, you have a VIE. So you have to know if this is a VIE, who really controls the VIE? Who really owns that VIE? Definitely not the shareholders. Although on the books, there are shareholders. So Michael Copper is not really the true owner of Chukko, nor Bill Datsun, although they own the stocks, but they're not really the true owners. Two, shareholders lack any of the following. And what I mean by any is any. First, voting power. Sometimes they might give you stocks, but those stocks don't have voting power. They have a certain characteristic. Then you're not really a true owner if you cannot vote. You cannot make a decision. Well, although you are the owner, but you are not really in control of the company, management trust somewhere else.  Okay? You cannot enjoy the game. You don't enjoy the game. Even if the company make the profit, the game go to somewhere else and you don't absorb the losses. Simply put, you have no skin in the game. You're just simply put artificially an owner in the company. If any of those exist, we have a VIE. Also, if the expected losses exceeds the equity investment at risk, if this partnership is expected to incur losses and those losses are going to be larger than the amount of money invested. Well, guess what? This company is not really a true company because the losses are large enough to wipe out all the shareholders. Also, if there's any disproportionate large voting power relative to the investment, simply put, for example, one party invested 20%, but that party has 90% of the voting power. Well, guess what? This is not really a true ownership. Why? Because if you invested 20%, you're supposed to have 20% of the voting power. Here, what we're saying is the remaining, the party that invested 80% only will have a 10% voting power. Well, that's a disproportionate relationship. Well, if that exists too in any relationship, you have a VIE. Again, any of those four will give you a VIE. They don't have to be all four, any of those four. So what are the forms of VIE? Usually it's either a partnership, joint venture, legal trust, or a corporation. Enron relied heavily on partnership and some joint ventures. VIE cannot be a person. You cannot consolidate with a person. It cannot be a registered investment company. It cannot be an employee benefit plan. It cannot be a government organization. Why am I mentioning all of those? Because on the CPA exam, you might see a multiple choice. All the following are form of VIEs accept. So you wanna make sure you know what accept or they will give you three of those and a joint venture. So you know the answer is joint venture. So simply put, what are we talking about here? We're talking about who consolidate? Who would consolidate those VIEs? Well, the primary beneficiary or the sponsor, okay? So how do we know who's the sponsor? Who's the primary beneficiary? Only we can have one primary beneficiary. So that's the party that have skin in the game. That's the party of the SBE failed. They're gonna have to carry the losses. They're responsible for the losses. So they must meet two criteria. Two criteria is one of them is called the power criteria. What is the power criteria? It means if they direct the activities of the VIE, then they are the primary beneficiary. That's what it is. Enron controlled everything. Also Enron did not really control the ownership percentage, but they controlled everything that the SBE is conducted. Then they meet the power criteria two, the risk reward. Basically the same thing. If they're gonna enjoy the game or absorb the losses or get the residual value of those entities, then they are in control. They're the primary beneficiary. So Enron really controlled all of those. And that's why they were considered the primary beneficiary. Therefore those SBEs should have consolidated with Enron. And when they consolidated, well, Enron would not show profit. They will show losses and they could not hide the debt. And that was the purpose of those SBEs. So what we did is change the rules. So rather than the 3% outside ownership set, forget about any percentages. Let's look at substance over form. Who controls this entity? And whoever's control the entity is the primary beneficiary. And the primary beneficiary will consolidate. At the end of this recording, I'm gonna remind you whether to go to farhaplectures.com and look at additional resources, multiple choice through false. Don't shortchange yourself, invest in yourself, invest in your career. The reason you're watching me is because you need some additional help. I can help you more.  The CPA exam is worth it. And good luck.


okay. So this is lecture number 35 it is a continuation of the previous lecture uhhh. So what we what we were discussing was the uhhh the proof of the following uhhh statement that the automorphisms the holomorphic automorphisms of the unit disc. So delta is unit disc and uhhh we were trying to show that uhhh set of holomorphic automorphisms of the unit disc uhhh are exactly maps of the form. They are Mobius transformations they are given by maps of the form z going to e to the i alpha uhhh z-z0 by 1-z0 bar z where alpha is of course uhhh an angle uhhh and uhhh and z0 is an element in the unit disc okay and what we uhhh what we saw in the towards the end of the last lecture was that uhhh any uhhh we saw uhhh uhhh map of the form z going to e to the i alpha into z-z0 by 1-z0 bar z is uhhh is uhhh indeed a Mobius transformation. So this Mobius is this o with 2 dots on top of it it is it is called as oum lot and it is supposed to be uhhh the German symbol for oe put together okay. So this is m oe oe uhhh half o and half e okay. So this is Mobius transformation it is of course also called as you know bilinear transformation or linear fractional transformation okay uhhh so it is indeed a Mobius transformation that uhhh maps uhhh the unit disc on to uhhh the unit disc okay. So so the right hand side is of course contained in the left hand side. So an element here is here what I will have to show is a conversely any holomorphic automorphism of the unit disc is of that form okay. So what I am going to do is uhhh so I will conversely let f be a holomorphic automorphism of unit disc. So you know what is happening is see you have you have f here uhhh uhhh all all I know about f is that it is an isomorphism it is holomorphic its inverse is also holomorphic and it goes from the unit disc uhhh to the unit disc and you know uhhh and I have to show that uhhh f is of this form okay. So I have to find an alpha uhhh and uhhh z not such that uhhh f of z is actually e to the alpha into z minus z not by 1 minus z not bar z right. Now what you must understand is that see if f takes 0 to 0 then we are already done because if f takes 0 to 0 then it will be an automorphism of the unit disc which fixes origin and we already seen as a corollary to Schwartz's lemma in earlier lectures that it has to be a rotation. So uhhh you so f will be of the form e power i alpha times z okay which is just this map with z not equal to 0 okay. Therefore uhhh the problem is when 0 does not go to the origin right that is that is the issue. So what we do is uhhh uhhh you know we just take uhhh see we we uhhh well let us look at what 0 goes to alright. So for for so for that matter you know I I can fix any point z not okay and uhhh uhhh suppose z not goes to the point w not. So my map is w equal to f of z I take a point z not it goes to the point w not alright. Now what I am going to do is I am going to write on this side a holomorphic automorphism of the unit disc which takes uhhh 0 to z not okay I am going to do that and you know what I am going to do I am going to use the first statement I am going to use this statement. So what I am going to do is it is it is very very simple you know if I take this map to be z going to uhhh uhhh z minus z not by 1 minus z not bar z suppose I take this map okay. Then I already told you that this map is a mobius transformation that maps the uhhh interior of the unit disc to the unit disc okay and it will take z not to 0. So the point z not will uhhh go to the go to the origin and what I am going to do is this is this map this is going to be the inverse of that map okay. So you know so I am going to call this as g okay and and this is my g inverse and I know g inverse uhhh uhhh is is is a map that takes uhhh the unit disc to the unit disc and it will map z not to 0 because you know in this if you substitute z equal to z not you will get 0 and this is a mobius transformation and I am going to take its inverse. So the inverse map will take 0 to z not okay and then this f will take z not to w not okay because w not is f of z not right and then uhhh you know I am going to then again apply another map here that will move w not to 0 alright and what is that map you know what that map is I will call that is h. So and this map is going to be so I am going to just map w w not to 0 and you know what that map is it is just w going to same formula w minus w not by 1 minus w not bar w okay this map will map w not to 0 and it will be an automorphism unit disc okay. So both for this side and for this side I am using this this fact that I have already proved in the last lecture right and now now watch if I if I take uhhh so this is h okay if I take the composition what will I get see this is also an automorphism unit disc f is also an automorphism unit disc and h is also an automorphism unit disc therefore if you compose you will get the composition of all the three will give an automorphism unit disc unit disc and it will fix the origin because it will take 0 to 0 that is the whole idea of putting this g on this side and h on that side. So it will be an automorphism unit disc which fixes the origin which by an earlier corollary of the Schwarz lemma we have seen is a rotation. So what you will get is you will get that uhhh should take uhhh first apply g then apply f then apply h is uhhh an automorphism a uhhh a holomorphic automorphism of the unit disc which fixes the origin okay. But you know any automorphism holomorphic automorphism the unit disc which fixes the origin is rotation that is the first corollary one of the corollaries of the that is the corollary of the Schwarz lemma that we have seen okay. So uhhh by a corollary to the Schwarz lemma what you will get is that h circle f circle g of uhhh uhhh w is just going to be a rotation and therefore it is of the firm e power i alpha w okay. Since you know this is something that we have already seen the set of automorphisms holomorphic automorphisms of the unit disc which preserve the origin. So this is this is a subset of this set these are all holomorphic automorphisms bijective holomorphic maps of the unit disc onto itself and these are the subset of maps which which send 0 to 0 and this is this is the same as this is identified with uhhh this is equal to rotations the set of all z going to e to the i alpha z where alpha is varying from 0 to up to 2 pi okay and this is identified with you know S1 which is uhhh the boundary of the unit disc we have already seen this and this is what I am using here okay.  So it is just the variable multiplied by uhhh an e power i alpha for suitable for some alpha okay for some uhhh for some alpha 0 less than or equal to alpha less than 2 pi of course alpha is just an angle and you have to read it mod 2 pi okay. So I think I will have to worry about my variables here. So here this is the z plane this is z plane and this is omega plane and uhhh let me call this as eta plane. So this is eta so eta is h of w okay and uhhh w is f z and here let me use zeta and uhhh zeta yeah z is g of zeta. So this is zeta uhhh zeta goes to z under z goes to zeta is g inverse and zeta goes to z is g okay. So so here you have variables so if I if I use these variables then I should let me change this variable to eta okay because uhhh uhhh not eta it should be zeta. So you know uhhh uhhh you will get uhhh uhhh so g zeta g of zeta is z okay. So I will get h of uhhh f of z see h of f of g of zeta is h of f of g zeta but g zeta is z so I will get f of z. So I will get h of f of z is equal to uhhh e power i alpha into zeta and what is the formula for zeta? The formula for zeta is z minus z naught by 1 minus z naught bar uhhh z okay. So I will get so f of z will be h inverse of this okay and uhhh uhhh and what is what is the formula for h inverse? It is going to be uhhh it is going to be the map in this direction uhhh. So it will be h inverse of eta will be what uhhh uhhh you have to solve this for w omega okay. So you have to solve this for omega and what will you get? Well if I make that calculation somewhere uhhh let me let me make it somewhere here say I will get uhhh neta is equal to w minus w naught by 1 minus w naught bar w you cross multiply you get neta minus w naught bar w neta will w minus w naught. So I am solving for uhhh uhhh so it is neta going to w I want write w in terms of neta that is h inverse h inverse of neta is w once I have a formula for h inverse neta I will put I will apply it to this okay. So uhhh so I have to calculate uhhh w from this or omega from this. So I will get omega into uhhh so I will get neta plus w naught is equal to omega plus uhhh w plus w naught bar w neta which is w into 1 plus w naught bar neta. So you will get w equal to neta plus w naught by 1 plus w naught bar neta this is the formula for w. So this is neta plus w naught by 1 plus w naught bar w of w naught bar neta. So this is the formula for h inverse of neta right and apply it to this apply it to this and you will get again an expression of this form okay you uhhh I mean if you are you must have probably done this such exercises when you have uhhh taken a first course in complex analysis but it is very easy to do it even if you have not done it. So I will I just have to apply so this is my neta okay. So I substitute it in here I will get e to the i alpha into z minus z naught uhhh by 1 minus z naught bar z uhhh plus w naught by 1 plus w naught bar into neta which is this e to the i alpha into z minus z naught by 1 minus z naught bar z this is what I get okay. In in this whole calculation I have chosen z naught I have chosen some z naught in the unit disc and I have taken w naught to be the image of z naught okay but you know I can make special choices for example I can take w naught equal to 0 if you want I I can reduce the calculation by I can make the calculation simple by simply taking w naught equal to 0. W naught equal to 0 means I am taking z naught so I am taking the unique point z naught that goes to 0 under f okay. So you make that assumption uhhh choose choose the unique z naught in delta with f of z naught is equal to w naught is equal to 0 this has to happen because after all f is a f is given to be a holomorphic automorphism of delta therefore uhhh there is something is mapped to 0 okay whatever is point is mapped to 0 you call that as z naught okay. So if you do this then you are actually putting w naught equal to 0 and if you put w naught equal to 0 you get f of z in the form that you want okay. So w naught equal to 0 gives f of z to be simply you put uhhh w naught equal to 0 you see this is gone this is gone you will simply get e power i alpha z minus z naught by 1 minus z naught bar z okay immediately and that is a proof that is the end of the proof. So what you have done is you started with an element here f here and showed that it is of this form okay. So these two are one and the same okay and uhhh so in fact there is uhhh uhhh uhhh so in fact the truth is that this is also a group the holomorphic automorphisms of the unit disc that is also a group because this is subgroup this is also uhhh certain subgroup of mobius transformations mobius transformations mind you form a group because you can compose mobius the the the group law the group multiplication is just composition okay. So under composition mobius transformations form a group and this is a subgroup of mobius transformations okay these are the uhhh these are certainly uhhh I mean these are mobius transformations of this particular form and they are all they are exactly the automorphism unit disc okay and it is a subgroup under composition of mappings alright and there is a much smaller subgroup among these you have the smaller subgroup which are further those which fix the origin and they are the rotations okay these are not rotations there is this part is a rotation there is something more here okay. So the fact is just like the this subgroup of mobius transformations that fix the origin or the rotations it it can be identified with the boundary okay that also can be identified in a nice way okay it can be identified with delta it can be identified with delta cross the boundary okay. So you see what will happen is that if you take the the holomorphic automorphisms of delta uhhh that is that I have written down a set of all maps of the form z going to e to the i alpha into z minus z not by 1 minus z not bar z where alpha is an angle trade mod 2 pi and uhhh z not is a point with uhhh point in the unit disc okay. So from here I can uhhh you know I can give you a map into delta cross dole delta which is delta cross s1 s1 is the unit circle okay set of all complex numbers with model s1. So it is numbers of the form e power i alpha okay and you know what is this map this map is you know you send the map you send the mobius transformation z going to e to the i alpha z minus z not by 1 minus z not bar z you simply send this to a pair where this first member is this z not because you see z not is an element of the unit disc and the second member is that e power i alpha and because e power i alpha will be on the boundary of the unit disc which is unit circle okay and what will happen is that this will be also an isomorphism of groups this will be an isomorphism of groups alright. So and what will happen is that this uhhh this will contain uhhh properly the automorphic holomorphisms I mean holomorphic automorphisms of uhhh uhhh of the unit disc which preserve the origin okay which are only the rotations and only rotations means that it will correspond to z not equal to 0. So only rotations okay and what you will get is the same map will give you this isomorphism it will you are going to send z going to e to the i alpha z to simply s1 you are going to send this to the element e e power i alpha of s1 okay and this is identified as uhhh uhhh 0 cross s1 which is sitting inside s1 okay. So the the holomorphic automorphism unit disc are identified with delta cross s1 okay whereas the subgroup of holomorphic automorphisms which fix the origin are simply identified with s1. So s1 so you have an identification of uhhh uhhh as a group uhhh of both the automorphisms of delta and the automorphisms of delta which fix the origin alright. So this is a remark now uhhh of course this diagram commutes okay and these vertical arrows are both group homomorphisms okay they are not just bijective maps they are group homomorphisms right. So uhhh now you see we come to uhhh uhhh uhhh uhhh so you know see what we have just looked at are the automorphisms of the unit disc okay the question is what can you say about a general analytic map of the unit disc onto itself what are what are the most general statements you can make. So I mean what can you say about uhhh maps which are not automorphisms of the unit disc but which are maps analytic maps of the unit disc into itself okay what can you say about such maps. So I will tell you what the main result is the main result is that on the unit disc okay there is a there is a metric called the hyperbolic metric okay it is a way uhhh of defining the distance between two points in the unit disc of course you know metric is something that gives you the distance between two points in a space okay. So on the unit disc there is a special metric it is called the hyperbolic metric and we will we will see what that metric is later but the point is what uhhh the Schwarz lemma actually says is that if you take an analytic function from the unit disc to itself which is not an automorphism then with respect to this hyperbolic metric it is a contraction mapping okay. So this is the this is the this is the philosophical statement I mean this is the ideological statement the ideological statement is I mean this is the statement that we need this is this statement is needed for proceeding with the proof of the Riemann mapping theorem okay you see last in the last lecture what we did was we proved that if you give me a simply connected domain which is not the whole complex plane then I can conformally map it to a subdomain of the unit disc that is what we proved okay but then somehow you have to a alter the map so that the image fills out the whole unit disc because finally the Riemann mapping theorem requires I mean the proof of the theorem requires you to find an isomorphism holomorphic isomorphism of the given simply connected domain which is not the whole complex plane with the whole unit disc okay. So that is the reason we have to study the hyperbolic geometry on the unit disc right and the point is that the Schwarz lemma actually in a in a disguised avatar of itself actually tells you that with respect to this hyperbolic metric on the unit disc if you take an arbitrary analytic function which is not an isomorphism of the unit disc onto itself then it has to only contract so the moral of the story is if either it is an isomorphism of the unit disc onto itself okay if it is not an isomorphism unit disc onto itself it will be a contraction mapping which means you take two points okay and you take their images the distance between the images will be smaller in the hyperbolic metric than the distance between the original points that is what a contraction mapping is a contraction mapping between two metric spaces is a map that reduces the distance okay. The distance between two source points is greater than the distance between the image points that is what a contraction mapping is and what the statement of the result that we need is that if you take an analytic map of the unit disc into itself which is not an isomorphism which is not a member here then it is certainly a contraction map for the hyperbolic metric alright so for that I will have to introduce the hyperbolic metric okay and then of course there is also the question that what will happen to these maps what will happen to automorphic I mean holomorphic automorphisms of the unit disc under the hyperbolic metric the answer is that they will preserve the metric they will be isometrics okay a map between two metric spaces is called an isometry if it preserves distances that is you take two points in the source metric space and you take their images under this map in the distance between the images is the same as the distance between the source points if the distance is preserved the mapping is called isometric okay. So what happens is whenever you take a map from the unit disc to itself which is analytic either it is an isomorphism and it preserves the hyperbolic metric or it is a contraction so it will shrink the image of the domain okay so this is a geometric fact in hyperbolic geometry which we need okay and that is what I am trying to explain alright so you know everything comes from Schwarz lemma you know so the fact is that Schwarz lemma gives you a description of the holomorphic automorphisms of the unit disc which fix the origin and also the general holomorphic automorphism of the unit disc not only that it also gives you information about any other analytic map of the unit disc onto itself okay so the first thing is let us recall the Schwarz lemma so here is the Schwarz lemma the Schwarz lemma is well uhhh uhhh f is defined from the unit disc and taking values in the closed unit disc uhhh analytic f takes 0 to 0 okay then mod f z is less than equal to mod z for all z in delta this is the Schwarz lemma which already tells you that you know any analytic mapping of the unit disc is is a contraction okay in the sense that if you if you measure the distance from the origin then it is a contraction okay the the mod f z is just the distance from the origin of the image f z of the point z under f and the distance from the origin of the image cannot exceed the distance from the origin of the source point that is what it says. So the fact that there is a contraction uhhh is already there in Schwarz lemma okay but what we are going to do is we are going to uhhh improve upon this okay and of course what is the other statement in Schwarz lemma if you get equality here for even a single z not then it is a rotation okay equality occurs for some z not in delta z not not equal to 0 if and only if f is a rotation okay. So this is the uhhh this is the Schwarz lemma from this version of the Schwarz lemma I am going to I am going to get another version of the Schwarz lemma which is called the differential version of the Schwarz lemma okay. So let me write this uhhh differential version of the Schwarz lemma so what is the differential version of the Schwarz lemma uhhh you see again differential version is that uhhh you have uhhh the the result in terms of derivative at the origin okay. So again you take f uhhh defined of the unit disc and taking values from the closed unit disc analytic and f of 0 is equal to 0 that is it takes the origin to the origin okay. Then the differential version says you take the derivative at the origin of f that cannot exceed 1 the derivative at the origin of any analytic map of the unit disc into the closed unit disc the derivative cannot exceed 1 okay and again equality occurs if and only if it is a rotation if and only if f is a rotation okay. So this is the differential version of uhhh Schwarz lemma or infinitesimal version of Schwarz lemma because it involves the original statement in uhhh the original Schwarz lemma involves uhhh f okay whereas the differential version will involve its derivative it gives you a bound for its derivative. The derivative of uhhh any analytic map from the unit disc to the unit disc at the origin cannot exceed 1 and it will be 1 only if it is a rotation if it is not a rotation then it will be strictly less than 1 that is what it says okay. Now uhhh so what is the proof? Proof of the differential version you can get the proof of the differential version from the original version okay and uhhh what is the proof for that? So you know you see uhhh see you we have uhhh mod f of z is less than or equal to mod z this is already there by the Schwarz lemma alright. So what you do is for z not equal to 0 you divide and you take a limit and you will get this because after all limit z tends to 0 f of z by z is just f dash because f of 0 equal to 0 okay. So you so you get it so you know uhhh so so so this implies so mod modulus of f dash of 0 is modulus of limit z tends to 0 uhhh uhhh f of z minus f of 0 by z minus 0 which is modulus of limit z tends to 0 f of z by z but you know if the limit of some expression exists okay then uhhh modulus is a continuous function so I can remove the limit outside okay. So I can write this as limit z tends to 0 mod f z by z but mod f z by z uhhh of course when I say limit z tends to 0 I am assuming z is not equal to 0 okay. So if z is not equal to 0 mod f z by z is less than or equal to 1 and this is this inequality is because of the original Schwarz lemma okay. Therefore so this so this is this is the quantity less than or equal to 1 and you are taking the limit so this is also going to be less than or equal to 1 okay. So you get this this is a differential form of infinitesimal version of Schwarz lemma okay then we have to say one more thing we we we must say that the derivative is 1 if and only if it is a rotation okay and why is that true uhhh if uhhh the derivative is equal to 1 okay then uhhh the function the analytic function g of z is equal to f z by z okay. So uhhh if you remember the uhhh in the last the last lecture uhhh or the lecture before that where we proved Schwarz lemma what we did was we defined this function g of z to be f z by z and applied the maximum principle to it okay because it was analytic it was not defined at z equal to 0 but then at z equal to 0 it had only a removable singularity okay which you can see because the z will cancel off if you write out a Taylor expansion of f at 0 namely if you write a McLauren expansion of f at 0 the first uhhh constant term will be 0 that is because f of 0 equal to 0 alright and the z will cancel out and therefore g of z will be represented by a convergent power series at the origin therefore it is also analytic at the origin therefore this is actually though I am writing f z by z it extends to an analytic function at z equal to 0 it is just like sine z by z if I take limit z tends to 0 I will just get g of 0 is equal to f dash of uhhh uhhh 0 okay and uhhh what this will tell you uhhh so so modulus of g dash of modulus of g of 0 will be uhhh will be 1 if mod f dash of 0 is 1 then mod g 0 will be 1 okay and you see uhhh now go back to the uhhh again to the proof of the Schwartz lemma in the Schwartz lemma what we did was we applied the maximum principle to g which is f by f of z by z and showed that uhhh uhhh the maximum value of uhhh uhhh the maximum value of mod g is always less than or equal to 1 I mean actually we proved mod g is less than or equal to 1 which translated to mod f z less than or equal to z which is the conclusion of the Schwartz lemma okay. So and what is the maximum principle what does it say if the maximum modulus if the maximum value is uhhh is it occurs at an interior point then the function is a constant okay so by the maximum principle for g principle for g g is a constant on the unit disc okay you you will you will get g is a constant for g is a constant on the unit disc and uhhh uhhh and that constant is uhhh and and uhhh mod g will be mod of that constant on delta which will be which will be mod g0 which will be 1. So this implies that this constant has uhhh modulus 1 okay so mod g is mod of that constant and mod g is mod of that constant it has to be equal to mod g0 but mod g0 is 1 so modulus of that constant is 1 okay it is a complex number with constant modulus 1 therefore it is of the form e power i alpha. So g of z is of the form e power i alpha it is a constant function but what is g of z g of z is f of z by z. So this will tell you that f of z is a rotation so in other words you know you have proved the differential form of Schwartz lemma the derivative the derivative at the origin of an analytic map of the unit disc on to itself cannot exceed in modulus 1 okay its modulus cannot exceed 1 and it is equal to 1 if and only if it is an automorphism and that automorphism has to be a rotation. So if it is not a rotation the derivative of the origin is strictly less than 1 that is the infinitesimal version of the Schwartz lemma okay so that finishes uhhh the proof of this. So this is the differential version of the Schwartz lemma and now uhhh we come to uhhh you know uhhh this question of uhhh trying to generalize this differential version of the Schwartz lemma. So you can ask what will happen if I put instead of 0 is z here okay. So take an analytic function from the unit disc to the unit disc okay I know f dash 0 in modulus is less than or equal to 1 but my question is what will happen if you take f dash of z for z an arbitrary point on the unit disc not necessarily the point 0 okay. Then also you get a bound and that is uhhh grander version of the Schwartz lemma and that is called Picks lemma okay and which is a which is a fundamental lemma that is required for hyperbolic geometry okay. So I will I will just state uhhh Picks lemma uhhh somewhere I think uhhh so here is Picks lemma. So again same assumptions f from delta to delta bar analytic uhhh f takes 0 to 0 okay uhhh no I do not think I need f takes 0 to 0 I do not need that this is very general f is any analytic map f is any analytic map from the unit disc taking values in the closed unit disc okay and uhhh then you see mod f dash of z okay is less than or equal to uhhh 1 minus uhhh mod f z squared uhhh by 1 minus mod z squared for all z in delta. You see if you uhhh in this you know if I take in particular an analytic function which takes 0 to 0 okay then if I substitute z equal to 0 what I will get is mod f dash of 0 is less than or equal to 1 minus 0 by 1 minus 0 because I am putting z equal to 0 and f of 0 is also 0 I must if I assume 0 goes to 0 then I will get mod f dash of 0 is less than or equal to 1 which is the differential form of the uhhh Schwartz lemma. So this is a this Picks lemma is a generalization of the Schwartz lemma okay the only point in Schwartz lemma is that uhhh you are looking at the derivative at the origin the modulus of the derivative at the origin and you are also assuming that the map takes 0 to 0. Now you are relaxing both the things you are taking any analytic map of the unit disc into itself and you are saying that the derivative modulus of the derivative at any point has a bound and this is the bound okay and again just like that case you will get equality if and only if f is a automorphism okay otherwise you will have strict inequality right. So uhhh further equality occurs uhhh for a point of uhhh delta if and only if f is a holomorphic automorphism of delta so this is Picks lemma okay. So you you see uhhh uhhh you you can see the flavour of our arguments we are more and more worried about the mappings of the unit disc into itself. The Schwartz lemma itself is a statement about the mapping of the unit disc into itself. So we are worried about the mappings of the unit disc into itself we are worried about which of these mappings are automorphisms and we are worried about properties of such mappings okay and this con and this focusing on the unit disc is actually uhhh leading us to studying hyperbolic geometry on the unit disc which is uhhh essentially given by defining the so called hyperbolic metric on the unit disc okay. So so this is Picks lemma uhhh and what do you uhhh what do you do to prove Picks lemma uhhh the nice I mean I it is a pity I erase this diagram here I have to use the same diagram to prove Picks lemma but I will have to finally apply the infinitesimal or differential version of Schwartz's lemma. So let me remove I I do not want it to take values on the unit circle alright. So let me remove this let me just assume that I am taking the map is from the unit disc to the unit disc okay. Let me not think the let me not take make the assumption that f takes a boundary value alright. So map from the unit disc into the unit disc right. So uhhh so you know the diagram is the same so I have I have this uhhh unit disc I have the map f and uhhh it goes from the unit disc to the unit disc okay and this is w equal to f z this is z plane and this is the w plane okay and well if you take a point z0 it will go to a point w0 okay and I want this w0 to lie inside the unit disc that is the reason why I did not take f to be a mapping from the unit disc to its closure okay. So uhhh so I want w0 to lie inside the unit disc right and uhhh well if you take any point in the unit disc it is going to go to a point inside the unit disc because I have I have I have removed the bar there okay and what I am going to do is as I did before I am going to put here this map which uhhh which is an automorphism unit disc which takes z0 to 0 okay and you know what is that map. So you know if you remember I call this the zeta plane earlier in this lecture and then I I I wrote a map like this I wrote a map like this which is g which is an isomorphism it is an automorphism unit disc which takes 0 to z0 and its inverse is an automorphism unit disc which takes z0 to 0 and you know what that map is it is simply z going to z-z0 by 1-z0 bar z this is the map and this is this is this is our zeta okay and this is g inverse okay and I am taking g to be this map so that it takes 0 to z0 because g inverse takes z0 to 0 g will take 0 to z0 alright and I have f here mind you f is not f is not an automorphism okay f is not an isomorphism it is not even given to be 1 to 1 it is only given to be analytic nothing more is given about f okay and then here what I do is I take this map from here a neta is equal to h of w this goes lands in the neta plane and I will take an automorphism unit disc that maps w0 to 0 okay and you know what that map is what is that map it is w going to w-w0 by 1-w0 bar w which is your neta okay. So, I have this set of maps alright this is an isomorphism the last one is an isomorphism the central one is just an analytic map the central one is analytic I do not know it is 1 to 1 I do not know it is 1 to I do not know anything about it now what I do is I compose if I compose what I get is g followed by f followed by h if I compose I get h circle f circle g it is a map from the unit disc to the unit disc okay I cannot again say anything about this map except that I can only say 0 goes to 0 okay because 0 goes to z0 z0 goes to w0 w0 goes to 0 so 0 goes to 0 but that is good enough for me to apply the Schwarz's lemma the inferential version of the Schwarz's lemma which will essentially give me the proof as you as you as you will see. So, apply differential version version of Schwarz's lemma this composite function h circle f circle g what will you get you will get h circle f circle g derivative at 0 modulus is less than or equal to 1 that is what the infinitesimal version of Schwarz's lemma says infinitesimal version of the Schwarz's lemma says whenever you have an analytic function unit disc to unit disc which takes the origin to the origin okay which this is the origin then the derivative of the origin has modulus less than or equal to 1. So, h circle f circle g is an analytic function from the unit disc to unit disc which takes 0 to 0 so its derivative modulus of the derivative at the origin cannot exceed 1 okay but what is this this is if you see this is g dash you see see I will have g dash of 0 okay. So, if I differentiate it using chain rule I will get h dash of f of g of 0 into f dash of g of 0 into g dash of 0 this is what I will get I will get this because you know if I differentiate this by chain rule I differentiate the outermost function then the next inner function and then the next inner function this is just the chain rule okay and you know but this is what you see this is mod g dash of 0 mod f dash of g of 0 but what is g of what is g of 0 it is z not so g of 0 is z not and what is and I will have mod h dash of f of g of 0 which is w not this is h not equal to 1 I will get this alright so I will tell you what to do what you do is you calculate g write out g alright g will be just zeta going to zeta plus z not by 1 plus z not bar zeta you just have to change minus to plus that will give you the inverse map okay you differentiate it and substitute 0 okay you literally calculate it and you will get mod f dash of z not is less than or equal to 1 minus f of z not the whole squared by 1 minus z not the whole square you will directly get it it is direct calculation ordinary calculation nothing to do nothing complicated and since this is true for any z not you get that estimate alright it is just a straight calculation ordinary differentiation right.


Derek, how nice is it to be home a place that you guys just simply haven't been from much of the last seven weeks? Yeah, we've been on the road for a little while, but it'd be nice to get back home and play in the stadium. It's been a little while since we've been playing in there, so it'd be fun to get back in front of the fans and play a home game. You've got to go for ice much from maybe being back in Tuscaloosa. How much do you pay attention to maybe four family guys? Yeah, I got to spend time with them last year on the house commercial. Got to know them a little bit and just talk to them. I'm happy for their success and hopefully they don't have much this Sunday. What's it like for those new guys to come in for the housing commercials? How does that kind of break them in? Well, I didn't get anybody this year, so I don't know how this year went. I talked to Nissan about that one, but yeah, it's always fun. You know, we just laugh and joke and spend time with one another and get to know one another if a guy's new. And really just have fun because it's some long days. Derek, this is the time of year where in your career you've taken a basically Thanksgiving on to be able to see that. What gives you confidence that this year, with this offense blind and this offense, it will be the same? Yeah, I mean, I'm just really focused on just trying to get a win. That's my main focus, and then let the other stuff happen when it happens. But my main focus is doing my job effectively enough to help us win in any way possible, whether it's me taking over or being effective in other ways. So it's just about winning. And then when you win, then those stuff will happen the way it's supposed to. How hard is the losing of the last year been on you? You came from Alabama. You used to win in the first five or six years you were here. You guys won a lot of games. How hard has that been on you from just your psyche in a personal standpoint? Yeah, I mean, it's tough on all of us. Nobody likes losing. Nobody goes in the game trying to lose, especially when you work so hard throughout the week to go out there and try to get a win. It's just not going the way you want it to. It's tough on everybody collectively. And you just got to keep working, keep focusing on improving, and just have an emphasis on it during the week, even more, even more, until that day comes where we get a victory. And I think that's just the main focus of guys is just trying to find any way possible for us to get a win. And just doing what we can to go out there and execute the way we know how to. Derek, and all the time at Nissan Stadium, it's yourself and Eddie George. Those are the only names that get chanted. When you hear them chanting, Henry, Henry, how much does that charge you? What do you think when you hear them? Well, it's been a little while, but it's very cool. I appreciate the fans that appreciate me and we love and we always need their support. It's always a surreal moment for me because I always dreamed of being able to play in this league and be effective and have a big impact. Eddie definitely had his impact. So my name to be mentioned here is always cool and something I cherish. Mike talked a little yesterday about wanting to make sure you get some more touches as the season goes on. Is there anything you feel like you can do personally to make that happen, or is it kind of more dependent on the other 10 guys? I just think the touches I get just be more effective, be more efficient in those plays when the ball is in my hands and trying to create explosives in any way possible. But I think efficiency has been my main focus this week, just trying to get the most out of a play that I can when the ball is in my hands. I don't really worry about the touches, but when I get the ball, it's being efficient. That's just my fault. Me executing and doing what I need to do to make the play effective, that's just all on me. You know, it's got to be better. How much has that been an overall theme, just the play volume for the offense after having so few plays in Jacksonville? Yeah, I mean, when you don't run a lot of plays, not many opportunities can happen for a lot of guys. So that's why you want to sustain drives, execute, have self-inflicted wounds that we've been talking about to be able to have drives with a lot of plays guys being able to get the ball in the hands to be able to make plays and score in touchdowns. What's your impact against the running game, maybe the whole offense when you've had so many different combinations on the offensive line, and how do you work through that? Yeah, I mean, I think this is the next guy up. I think guys have been trying to prepare like the starter throughout the whole season. And then when the time comes, you've got to go out there and play like it as well. And I think guys are just trying to focus on their job day by day. And then when Sunday comes, try to go and execute. We've got a lot of guys in there.  What kind of Panthers defense specifically, top 10 defense, when it comes to the rush defense, giving up a lot of yards on the ground, where do you see the breakdowns and the opportunities for the ground game? Yeah, I mean, I think the defense is pretty solid. They have game breakers over there, guys that play hard, guys that get to the ball, guys that are physical. But I think the main focus is just us, you know, making about us, executing an offense, sustaining drives, putting drives together, converting on third downs and being able to have more plays. I think, you know, they're probably trying to do what they can and making about them and get them a win as well. But I think it's mostly about us and what we're doing on offense and how we execute and how we be able to put drives together at the end in points. And I think that's just a big thing, focus on execution and what we need to do to be better on Sunday. Coach, how would you best describe the attitude in the locker amongst the players? Yeah, I think guys are motivated, guys are still hungry. In this time now, it's easy to make excuses and, you know, finger point, but I don't think no guy is doing that. Guys just focus on improving individually and collectively and trying to help one another just so we can spring forward and get it going in the right direction that we want it to.  I think it's just time to win. That's what guys want. Derek, Mike's message has been the same throughout his entire time. But now that message is what? 14 losses and 17 games. How does it keep resonating? How do you guys keep the confidence and belief that this week you're going to win? Because it's worked and, you know, we know it works and we know that as players, you know, it falls on us to go out there and execute. You know, they can call a play, put plays together, but we've got to go out there and execute. And, you know, this standard and this culture has worked for the time he's been here. You know, it's been tough right now because we haven't been winning. But I believe and truly believe eventually, you know, everything will turn in the way we want it to as long as we go out there and execute in all three phases and play how we want to play and not have, you know, this self-inflicted stuff that, you know, we know that we don't need to have to be able to win games and not hurt ourselves in games as well. Going against the team. What's the worst record in the NFL? Is there some sort of mindset of, hey, circle this one? We really have to have this. This is a perfect chance to kind of break the shit and see if we can start stringing something together. Well, you want to treat it as the next game and the next opponent, not because they're the worst team. Because, I mean, anybody can be beaten in his league. No matter who it is, the worst to the first. So, you know, you got to approach this game seriously. Be locked in on the details and what we need to do just so we go out there. We can play outside of football and be able to win the football game. Thank you, sir. We're doing this turning in the right direction. We just got to find a way to get a win. We're Bryce Younger. I've heard guys refer to going against like chasing a rapid. What kind of challenge does he bring, you know, the way he runs around back there? I mean, he tried to stand, please. I think the thing is when you're playing a guy like that, as we talk about, you know, each and every week when you play against a, playing against a Scrammer quarterback, we have to be on the same page. You know, the Rushland got to be coordinate. He just got to rush relentless. A guy like that who, you know, when he feel pressure, he like to move around. But I don't, I wouldn't consider him the same way I consider it as like, when I, when you say that phrase, I think of a guy like Colin Merrill. I don't think he that type of, like, scrambler like that. Like I said, I do think he, he have, sometimes when he's scrambling, try to get out the pocket. But I would consider him like Colin Merrill when you're playing against a guy like that, trying to chase around him. But, you know, when you get a guy like that, or where's Bryce Young, I think he easy, he's more easy to get to the ground than a guy like Colin Merrill.   Yeah, I mean, you know, we try to keep the streak alive and keep winning at home, you know, especially with our home crowd. As long as we out there making plays and, you know, hopefully get a win, you know, they in it. So that's the goal to try to keep our crowd in it.  So, you know, hopefully coming back home this weekend and, you know, we can get back on the right track and, you know, keep building off that.  I think you can say to spark guys around or you hope them maybe they watch. Man, you know, I just try to do my job. No matter if it's, you know, me out there talking or, you know, the way I carry myself on the field and out the field. You know, I don't think I need to do anything more. I wouldn't tend to want to go watch my film or whatever it be. Because I have so much I need to get better at. You know, I can play better. We all can play better. And at the end of the day, once we all get our best football and we all get on the same page and, you know, on all three phases we could get a win. And I think that's the message we just have to keep improving and, you know, put it together. Because as when we watch film, it's just one guy every play or, you know, two guys or whatever it may be. We just have to put it together. And I think, like I said, it's nothing, no better way to do it than playing at home in front of all crowd this weekend and, you know, trying to start a better second win. How bad does this team need a win just to improve around and get guys kind of a little more bounce in their style? That's what's going to help us out. I mean, win and take care of everything.  That's when there's really points and everything's more emphasis on. But I think once we get this win this weekend you know, that's the mindset of winning. You know, we could just keep stacking wins. I think when you win, like I said, when you win and then you keep winning that's build the confidence of God. So like I said, as long as we win, win and take care of everything. You're one of the most motivated players in this locker room, but how difficult has it been for you just to stay positive during the last couple of weeks when you are losing?   It sucks, you know, after the game I'm frustrated, but at the end of the day it's about, you know, being the leader. I can't come in here, you know, dragging. I can't go out there on the practice field dragging. It's about trying to come in. What can I do first to get better? How can I play better as an individual? And then try to translate that message to the team.  It's not going to wait on no one else. Not going to wait on the Titans or whoever it may be. So every day I come in here with a positive mindset that we're going to win. And like I said, start this weekend. Most definitely. So my foundation give them a reason. I decided to give out 500 turkeys.  And, you know, just being able to give back to our communities means so much to me.   I'm thankful to be able to give back to our community. And around this time, I know that, you know, maybe someone didn't have, you know, that money or whatever it may be to go buy a turkey. That's the things I think about.  Were you able to be out there? Were you able to distribute a few turkeys? Oh, most definitely. And what was it like, an expression from most people who were almost saving in? Most definitely.  I did the first portion of practice yesterday. I was able to get over there kind of towards the end a little bit. But, you know, when one of the ladies asked me, what's the cause?   Thank you for the community. And that's what it's all about, man. You know, I don't look for, you know, all the publicity on the outside.  I do it for, you know, our community.  You're going to give them away 500 turkeys? Well, I wouldn't say we gave that many out there. We did give out a lot out there. What we did with the extra, it's a lot of housing here. You know, even with the different shelters and stuff like that, you know, we found houses and everything to take them, drop them off too.


What should dating apps do to combat discrimination? Go back to Mexico.   Grindr has launched a new campaign aimed at combating the discrimination plaguing the app. It's set to tackle sexual racism, transphobia, body-shaming, film-shaming, and more. The move's positive, but is it enough? And what are the other apps doing?  I think if you ask most gay and bisexual men of colour whether they've had negative experiences through Grindr because of racial stereotyping and discriminatory language, most of them would say yes. When I used to use the app, I used to get things like you should f off back home even though I was born in the UK. As a trans woman, I also got fetishised so a lot of people would get in touch with me because they felt Asian people would do certain sexual acts or Asian girls are stereotypically more submissive or tighter down there. Thing is, many parts of the LGBTQ plus community have been vocalising issues with the app for a while. There's definitely a problem here and actually if you put this into context, people of colour that use apps like Grindr have been saying, four years, there's only to be done. And Grindr's existed for almost 10 years and we're only just now hearing about their solidarity and then wanting to do more work. I think it's a really great first step. Earlier this year, trans activist and model Munro Bergdorf tweeted a screenshot of some of the hate she'd received writing, After that, Grindr talked to Munro about how they were going to fight discrimination. Fast forward to now, we've got the Kinda Grindr campaign. The campaign is trying to create a more inclusive space for minorities within a minority. They're doing this through videos profiling users and their experiences with the app as well as changes to community guidelines. Ultimately, these are issues that have been pervasive in the queer community for quite some time and it is our hope that by elevating these experiences and driving awareness into discrimination that's taking place, not just on Grindr but on other apps overall, that we can engage in a more thoughtful, constructive conversation around how we can be more respectful, inclusive, tolerant and just generally kinder to each other as we interact both online and in person. Grindr's Zach Stafford tweeted, So I'm seeing some confusion around Kinda and wanted to clear up one big thing. Discriminatory language like no fats, no fems, no Asians is officially considered discriminatory and will now be banned on Grindr. He adds, A couple of years ago, Tinder also ran their own campaign promoting inclusivity called All Types, All Swipes. It was fronted by Munro Bergdorf and aligned with the app opening up over 40 gender options for people to choose from. These moves are good and necessary but how much will these types of campaigns really change things? I think it's a really great first step. The kind of work that they're doing now is putting them on the path towards understanding these issues a bit better and I definitely commend Zach Stafford at Grindr who is really pushing forward a lot of this kind of stuff. But what I would say is we do need to think about this in context. Has Grindr allowed people of colour to be the victims of racism through their platform before almost 10 years? The answer is yes. Are the changing of these community guidelines useful in a good first step? Yes. While it's a step in the right direction, there's more work to be done. People are calling for the removal of the filters on the app that literally let you pick and choose what ethnicity is you do and don't want to appear. So what else are people asking Grindr to do? There's an opportunity here for people who understand that racism exists within the LGBT community but aren't necessarily a victim of it themselves because I'm talking about white allies. If you're on Grindr and you're seeing people use racialized and discriminatory language call them out on it or use Grindr's new policy and block. Don't make people who are the victims of racism have to call it out all the time. You can work in solidarity and you can be a true ally. It's a step in the right direction for what Grindr are doing but I think it needs to be a bit more of a holistic approach and not just put on one app.  I don't think it's the responsibility of one little app to kind of tackle this ingrained discrimination that LGBTQ people have for people of color or foreigners or trans people or anything like that. Grindr and apps like Grindr and Tinder did something more to really pinpoint racism on social media and on online dating and it's about time that we put a stop to it. Ultimately it's always great to see anyone pushing back against hate and bigotry but unfortunately it's rife and there's no quick fix.  So if you see an egg call them out. Even if you're not the one targeted. Especially if you're not the one targeted.


I am Dr. Murli Raj. I am currently head of department of psychiatry here and I practice clinical psychiatry at the Manipal Hospital. Depression, all of us know, is a state of mind where you feel sad or unhappy. It also includes a variety of other things such as being unable to enjoy oneself, feeling whether either slowed or the thoughts have become very fast, feeling negative about everything, either the present, past or future, not sleeping well or sleeping too much, poor appetite and occasionally ideas and attempts at suicide. That's the danger in depression.


We have a seven hour travel day, seven hour travel day, maybe more, probably more. And we're gonna be in the city that we're traveling for seven hours, for 36 hours. It's 30%, don't quote my math, it's four a.m. in the morning. I did in New York City, my boy Brett's gonna meet us there. Our flight's to six, I might be getting sick, we're gonna make this happen. My secret is to turn off my cell phone. Now in 2017 that's your watch also, so I turned it off and that makes the time disappear. Because flights at 6.01, it's probably about 5.30, maybe 5.25, we're supposed to board about 5.35. And I guess I've never traveled in the holidays that much. Still got 80% chance I'm getting on this airplane.  Barely made the flight, but we made it. 6 a.m. flight, Charlotte, about four, four and a half hours. Half hour layover, nothing. Now what? Hour and a half here, two hours here in LaGuardia. It's Friday holiday season, so that's where I'm gonna take us another hour or two to get to the freaking Manhattan. Then hopefully pizza and a little bit of sightseeing. As the man, the magic man behind the camera has never been in New York, he's never seen anything, so we're gonna try to show him a little bit. I've never been in the Christmas season, so I want to see the tree in Tangs. So we're gonna go see the Christmas tree. Should we wait for Sandy? Buddy, just focus on your ankles. I'm gonna give this thing all to Sandy. And then, so I want to see the Christmas tree, Christmas tings at Rockefeller, and then show Connor the city as much as we can tonight, because we might have to leave 9 a.m. Sunday. Working on that so we can spend Sunday sightseeing, having fun. But as of right now, seminar tomorrow, sightseeing today, Boston Sunday, and break. Eastern time now. I don't know if I said why I'm here. We're doing a seminar. I saw this gym. My boy Kenny runs it. The homies at Reebok want us to come out here. I think it's like $25 to do the class. All for the seminar. All proceeds are going towards November for like men's health awareness and things. So that's why I'm here in New York. I'm gonna sleep. Try to shake this cold. I got that itch right here. Every time I try to breathe, it feels like that.   So there's Joe DeFranco, Bert Soren, Brandon Lilly, me and Tehran. Apologize to the others names who I aren't at the top of my head right now.  He trains Triple H. Brian Cushing.  He worked with Onnet for a while. He trains like the WWE. So like he's legit. He's from Jersey. I watched a lot of his videos going up. Bert Soren and I have had a good chat on the phone, but I've never met him. Obviously, Brandon Lilly, who I've known for some years now and learned a bunch from.  But we'll just have to sleep as deep and as much as I can and figure out in the morning. And then we leave 9 a.m. on Sunday to Boston. So give this tang a thumbs up seminar coming tomorrow. Flu or not, I'm gonna crush the seminar, but we'll figure out the rest of the day based on how I feel. Good night. Sleep tight. Don't let the bugbugs bite. And if they do, count to two and wake me up in the morning. And something that keeps me in the gym and keeps me kind of doing what I do, whatever the heck that is, is that the barbell doesn't really care where we're from. The barbell doesn't really care who we voted for. The barbell doesn't care how much money is in our bank account, who our parents are or anything once you get under it. So strength, like Taran kind of just mentioned, is the great equalizer. You lift it, you don't. Pounds don't change day to day. Kilos don't change day to day. And any opportunity I can get to get around like-minded individuals.  You know, 10 years ago it was a serious like feud.  And the more you hang out with people that are of that extreme nature, politically, financially, socially, the further we are going to be apart from each other. So if we can all just focus on, again, the barbell doesn't care what's in our bank account, the barbell doesn't care our skin color, our religion, anything of that nature. And we can start to draw more together. I think the better off we're going to be.  So shout out to Kenny and Solace, shout out to Reebok, shout out to Jen. Big breath, squeeze here. Yeah, you got a big breath in here. Squeeze this, take your time. Good. Squeeze this tighter, squeeze, squeeze, squeeze. Good, nice big breath, force those knees out. Good, drive into the bar. Good, one more, a little bit more aggressive, no pause. Drive into the bar. Yeah, that's good, bro. If I'm your coach, that's how we're starting. What if your talent is before the high bar? If I'm your coach, that's how we're starting. You can score however you like. You know more of a new elliptic, and I feel like I can do it as well. And you're a new elliptic, you don't know anything about your posterior hurting. You know what I mean? If you're a new elliptic, what your body can do, and the loads you're handling, are not fatiguing your low back. When you're squatting and deadlifting 700 pounds, multiple times a week, then you can be like, Mike, man, my back's having issues, I gotta figure this out. If you're not in, if you don't have an injury, and I'm your coach, I'd squat that way. How'd it go? I don't know, you tell me how it went.  You can. All right, killed it.  So we'll figure that out if I did good here, but he told me in the past, like, Mike, that seminar needs a little more feeling, or Mike, not your best work. He's actually, he's nice enough friend to cut it how it is, and still be nice, like, not your best work.  Level up, outside ways.  He's got his Gandalf staff and stability. Woo, that's pretty good. That sounds like it could be in Dungeons & Dragons, or what's it called? Zelda. Zelda, or what else do they play? What's the real nerdy one? Oh, World of Warcraft. That's a World of Warcraft weapon. The Gandalf staff has stability. We got this new gimbal.  Appreciate you guys. I think there's a party upstairs, which I'm going to try to avoid, and then we'll, I think we're gonna go to dinner, hopefully with these guys, as they're going to talk to. This is the 35th Hunger Games, cut to cool train scene. Probably about four years. But every time we come down here, they're at old HQ, is off in the suburbs. So it's probably about 30 minutes from downtown, but they just moved downtown. They're trying to be more part of the city and the culture. So my dad used to travel here a ton for work. He worked for John Hancock for a long time. And he always talked about how much he liked Boston. So this trip, I'm excited to explore downtown a little bit. It's freezing. I got a headache. My flu is getting worse. We're going to check in the hotel, and then, I don't know. I don't know what we're going to do. Find some food, maybe. We'll see. There are the mardits here. There are the potted. Fuck. Shit. Cunt. The potted. Shake Shack.  What is hair, dude? Hey, dude.  My dude? Huh? You talking about Boston?  There you go. Appreciate you. Podcasts come in soon. I know you're hyped. I'd love to give you the link, but I don't have it yet. Smash this thing with a thumbs up. Give it a like. Give it a subscribe. Next video will be in Boston. See you in Boston.