Hello, I'm Steve Anderlees from the Vaughan Company. My co-presenter is Peter Brink from Underwriter Labs. The title of our talk is Debating Linux in Aerospace, Objections and Paths Forward. In this talk, we'll give an introduction, then talk about some of the assurance concepts that are important in aerospace for assuring software, and then look at a few of the standards. We'll talk briefly about the benefits of using open source, and then walk through in sort of a debate style, a number of objections to the ability to use Linux in safety critical avionics. Hi, my name is Pete Brink. I am an automotive functional safety engineer working for Underwriter Laboratories. And when Steve proposed this idea of having a friendly debate, I was interested because I had actually started my career working on jet engine control systems, and I've been familiar with how avionics software works, especially in a safety critical context. So it was important to me to have this discussion because I am active in the enabling Linux and safety applications community working with the open source engineering process team. My motivation for this talk is thinking through how we might use Linux in higher assurance safety critical software in aerospace, but there are some challenges to doing so, so thinking through how we might address those challenges we thought would be a helpful presentation. Before we get into that back and forth of objections and possible answers, we wanted to go through a few of the concepts that are important in aerospace for safety critical software development, and some of the standards that form the regulatory environment that we must work under. First, let's go through two terms, reliability or reliable software means a system that does the right thing, that it's doing what you specify, it's acting according to the behavior that you require. A safe system is the complement, it doesn't do the things you did not want. That is, there's no unexpected behaviors, no unanticipated actions that are coming out of the software. In aerospace, we have several, the idea of several different levels or design insurance levels, software levels, from the most critical is level A. If something goes wrong with this software, fatalities could occur, and so this requires the most rigorous analysis demonstration through evidence the software is correct. The least critical is level E. Errors in this software don't result in any disruption of the crew and therefore doesn't require any formal evidence. Level A software might be, for example, software that forms the autopilot of the plane. Level E might be an example of software running the coffee maker or the entertainment system where it doesn't impact the safety of the system. Although maybe not giving coffee to the pilot, you consider it an issue, but generally yes. Level A most rigorous and then down through B, C, D, all the way to E the least rigorous. One of the key things that we look for when we are doing validation and verification of a safety critical system is specifically that we're following the appropriate processes that will minimize or eliminate the systematic error that we have the opportunity to build into the system. So specifically, there are two terms that we look for here. The first of them is validation, which is basically the mechanism by which we demonstrate that we built what we set out to build and verification. And verification is or are the set of processes that we go through to demonstrate that each of the different steps that we follow throughout the safety critical development life cycle are in fact being performed and are being performed in accordance with either the quality or the safety framework that we've established. And we do all of those things specifically so that at the end, we can demonstrate that the requirements that the things that define what we set out that what we are going to set out to build are in fact in place. And so we have a what's called a requirements traceability matrix where each phase of the development life cycle, we have we start with the system requirements, we do the software requirements based on those and in the context of the system requirements. And then those software requirements were expected to develop an architecture, a design and code that reflect and demonstrate that those requirements have been fulfilled with the ultimate goal that we have a set of tests that demonstrate that the requirements can be complete. And of course, then the requirements themselves must have a what's the best way to say it, the there must have the requirements themselves must be testable specifically. Another part of making sure software is safe and reliable is checking that our tests have exercised the software. Level D does require testing and testing against your requirements, but it doesn't require that we demonstrate coverage. All the other levels above that do. At level C, we have to show statement coverage. That is, our sets of tests must have executed every line of code. If there's any code there that was not exercised by the test, some justification analysis is required for that. For example, you might have some defensive coding that that can't normally be tested. And then that would be the justification for the test not covering it. At level B, we not only have to cover every statement, but we have to show that every branch has been taken for any decisions. And at level A, we have to show that every combination of those possible decisions at a branch have been exercised. Next, let's look at some of the standards in aerospace. There are a number of them that form the regulatory environment that we must certify software or a system under that has that software on it. On the U.S. side, the Federal Aviation Administration, FAA, has published an advisory circular which specifies the different means of compliance for showing how one can assure software. That advisory circular points to the U.S. and probably the key document with an aerospace is DO-178C. In Europe, the parallel standard is ED-12C. So it's titled Software Considerations in Airborne Systems and Equipment Certification. This document is the standard that guides how we develop software for aerospace for systems that will be flight certified. It was published about 10 years ago, a little over a decade ago, and it has four other documents that are appendices to it. DO-330 through DO-333 that talk about specific topics. For example, 330 talks about tools that we use to develop software such as a compiler or automated test framework and whether those also need to be verified or in this case that we call them qualified. DO-297 talks about how to approach modular systems, and so this is sometimes referred to as partitioning. And on the U.S. military side, Milhand Book 516C is one of the documents that guide how one would write software, particularly Chapter 15 of this standard. And then on the European side, there's a standard that just recently came out, AMC 20-193, that guides how we use multi-core processors. We expect a U.S. version of this document to come out fairly soon from the FAA as of this recording had not yet been published. So on the U.S. side, we would probably refer to the CAS-32A paper that talks about multi-core. There's also some system-level documents from SAE, ARP 4754A, and 4761, which talk about safety at a system level, but then also mention how this gets allocated hardware and software. And so this would flow down into software development under DO-178C. So with that background for assurance concepts and a quick overview of the different standards that are applicable, why might we want to try to use an open-source product like Linux for aerospace? Well, there are quite a few benefits. We'll just mention a few of them here. First, in safety-critical domains, it's fairly important to have peer review, expert review of the software to ensure it is suited to the purpose and is safe, it's correct, it's assured. So using an open-source licensed software like Linux gives us better visibility, and so you get much broader review. Review by experts, review by regulatory authorities is easier because it's open-source. Another benefit is crowdsourcing. When new technology comes along, when better ways to approach something comes along, when a fix to an existing problem comes along, those get filled in more quickly. You just have access to a broader array of experts and perspectives that are contributing into the software, the crowdsourcing effect. And then lastly, because Linux is so ubiquitous, it's a very well-known API. Use of Linux is across many, many different industries. It's really one of the most commonly used OSes in the world. So that gives us a couple of benefits. First, we should be able to find competent developers more easily because most students studying computer engineering or computer science will have used Linux already. They're familiar with it. And then it'll be easier for others to understand the code that we're developing if it's within the environment and uses Linux as an operating system. So we've heard from Steve about the specific advantages and opportunities that open-source represents in terms of how we might use it in aerospace. We have in the safety community some specific objections, and those are listed here on the screen, that Linux doesn't have certification artifacts. It doesn't protect the code. The design of the kernel of Linux doesn't specifically permit safety. And the culture, the development culture around open-source is not where it doesn't fit within the bounds of what we expect in a safety culture. So the first objection, and let me be very clear up front, the mechanism by which Linux is developed and works, there is nothing inherently wrong with that. It's just that there are specific issues when we look at it from a safety and or quality context in terms of why this objection is here. So specifically, the first of them is that Linux was not designed. Linux was grown organically, and there was a specific idea of what it was supposed to do and the job or role it was expected to fulfill. From a quality and a safety context, we're really looking to try and define that up front in order so that when we go through the development process, we can demonstrate that the architecture and the design specifically are there to fulfill or match or fulfill those requirements that we set out in the first place. There are, do we lack a set of requirements? And that's a key problem because when we talked previously about validation, one of the key things that we're looking for is a mechanism whereby we can demonstrate that the demand that the product is done that we want. We want to have a set of tests that demonstrate that all of the requirements have been fulfilled. It lacks a specific architecture, which is another key piece and another key problem, because what we're looking for here is a mechanism whereby we can perform a safety analysis against the architecture as presented. So in order for us to do that, we kind of have to have that architecture in place for us to do that analysis against. And the last of these is a unit design. So when we do the architecture, we're looking to demonstrate that we have a set of functional blocks and their interactions, and then the unit design is specifically there to demonstrate how those individual blocks are designed and how they fulfill the interfaces that were specified in the architecture. Although Linux was not designed according to a single set of requirements and processes and according to rigorous standard to which every developer adhered, it does have design and architecture is just more emergent, really a crowd source effect that has produced over decades a fairly well-despected and well-designed system. However, for critical domains, we need to demonstrate that that design is suited to the purpose that it's correct and safe. So how can we address some of these challenges? The path forward probably involves reverse engineering the certification artifacts that is the evidence from the source code and also foreign engineering from particular system requirements for particular aircraft to show that Linux provides the functionality required and does in a way that's safe and reliable. And so weaving these together in a way that shows that the system objectives have been satisfied that the software is safe and reliable is a way to start providing that evidence that's required in regulatory environments like aerospace. We might also look to find what an appropriate coding standard would be and then apply that to the code being used from Linux for the aircraft. And perhaps identify whether there's some issues that the community would want to improve and address problems. Identifying the architecture and perhaps encouraging the open source community, Linux community to start coalescing around what that architecture looks like and simply just documenting what's been emergent over the decades. And then finally looking at different safety analysis to identify are there any potential flaws or areas where we want to make improvements to improve the safety of the system. The next objection is specifically that in terms of the mechanism of how Linux is developed that we don't that Linux doesn't protect the code. What we look for in a traditional safety development lifecycle or inequality development lifecycle is that there's a specific mechanism in place whereby we can go through and do an analysis on the code itself, not just the code, but the architecture and the requirements to determine the scope of what a particular change is going to do. And so as a consequence, we have this mechanism where the expectation if there's an architecture in place, we can determine what the scope of the change is going to be and what the impact and that's why it's specifically called an impact analysis on that architecture, that code, so that we can determine what the scope of the change actually is. And the fact remains that there are also specific competency requirements most of the time when we talk in a safety critical context. So the software engineers that work there are expected to be trained and knowledgeable on the safety standards and then follow the specific coding guidelines and design guidelines that define how we are expected to put the overall architecture or the system together. In this instance, anyone can write a driver. So we're limited to whoever is out there and there essentially isn't any confidence or at least no demonstrated confidence that the driver has produced is going to comply with what all the safety standards might expect. Well, it is true that anyone can propose in addition to the Linux kernel. It is not true that anyone can just put their code in. There is a system of maintenance that checks the quality and sufficiency of the code as it's pulled in and examined, considered, and eventually if it's shown to be tested and incorporated into the main line of Linux. For the aerospace industry, one way to formalize this is to provide a curated and baseline profile that is configure Linux for particular use, including only the code that's necessary through configuration and then maintain that in the company's own configuration management system, for example, within a Git repo. That configuration would include the compiler directives for how to build the system, the different configuration of which drivers are included or not. And since that now is a snapshot of the particular config Linux on the aircraft, that can be maintained and managed or curated. Objection three is specifically around the design of Linux, the nature of what the Linux kernel actually looks like. And this actually goes back a little bit to the idea around that anybody can build a driver and anybody can make a modification to the kernel. So specifically, the kernel is monolithic, which means that the kernel and all the drivers all draw, all execute in the same and the same privileged space. So on an Intel architecture, that would be ring zero. So specifically, what happens or what would happen to the kernel if a driver misbehaved? So the driver misbehavior, we have a wild pointer and it can overwrite memory pretty much anywhere, not just in the kernel, but in a in some nominal application or function that's executing in user space as well, because the kernel has indeed access to all of memory from a physical context. So if we look at the typical way that safety operating systems actually are organized, they usually have a kernel that is isolated unto itself and it's the only thing that can execute in a privileged space. And in fact, all of the user space, all of the drivers are expected to execute in user space. And only then the kernel during initialization or other things can give that the kernel can then give access to drivers or applications in order for them to get access to system resources. And that can be anything from memory. It can be CPU time. It can be peripherals or other communication media for in terms of how they're able to interact with either other applications or the outside world. Well, it is true that Linux is monolithic in that the kernel is relatively large, much the functionality is within the higher privileged operating system level. It is not necessarily the case that every driver must be included in the kernel. There is the ability to put drivers in user space and then the kernel provides a memory mapping so that the driver can get to those addresses related to the device it's managing and no others, providing some separation. The other path to take forward is to not include code that's not necessary. Generally at higher Dell levels we want less code that has to be proved and so we leave out through configuration the code that isn't needed for the particular function on an aircraft. And then for very high levels of assurance one can also consider partitioning. For example through a hypervisor where Linux is a desktop operating system within one virtual machine or partition and separating out functionality into different virtual machines. Understand this last objection is not a condemnation of how Linux works because it's clear that it does work and it's a strong mechanism for being able to produce an operating system that's used around the world. But we don't really have the confidence in it that we need to have when we're dealing with safety critical systems. And I'm talking about the software that operates a jet engine control system and keeps it flying or keeps it executing. The software that executes your brake controller on a vehicle. That's the kind of thing that's the kind of confidence that we want to have as part of that production. So when we look at these different items Linux doesn't have a safety culture or a quality culture. Well both of those are actually based specifically around what when when I say software engineering I mean something very specific. I mean that I mean it's the engineering of software. It's more like systems engineering than it is anything else. Because we follow a specific development lifecycle process. We follow requirements, architecture, design, implementation and then testing at all the different levels. Now it doesn't mean that it cannot be iterative and it doesn't mean that you can't skip specific steps based upon the complexity or the difficulty of doing those things. But ultimately what we want to have at the end of this whole thing is a mechanism whereby we can prove everything is in the code that we expect to be there and that there isn't anything else and there isn't going to be anything in there that doesn't work the way we expect it to. One way to handle the variety of code contributions coming into Linux and thinking about that coming from a variety of different developers that aren't necessarily working to a single standard or a single approach is to use a number of tools. First there are now available automated scanning tools that look for security vulnerabilities and this is something then that crowdsourcing can help us with to identify what are those vulnerabilities and what are the appropriate fixes to them. As I mentioned earlier in the previous objection, curation that is managing a particular baseline can help to ensure that the version flying on the aircraft has been carefully evaluated and so that can be done by a team at a narrow space company that is working within a rigorous safety process. So that then is a particular distribution. Think of this, for example, produced by Yachto as a set of build recipes that have been checked and managed to ensure that that is correct. Building trust at the particular configuration on a particular aircraft is correct and safe. Once that baseline is established, one is not forced to take a new driver or a new update of software. It can be evaluated and you can wait. You can see to just prove out its correctness over time and only adopt those features after they have been deemed safe. In conclusion, to support what Steve has said with regards to the possible paths forward, I agree completely and in fact I am part of an organization within Elyssa called the Open Source Engineering Process Group where we're looking specifically at what we as an organization can do in order to include the engineering processes that are necessary in order for us to be able to make any safety or quality claims about the development of the code. And with apologies to one of the ring series, one does not simply walk into aerospace using Linux. That is, there are multiple challenges have to be addressed. We've mentioned some of them here, not necessarily a complete list, but hopefully gives you a flavor for the sorts of things one might need to do to use Linux and aerospace. If this is an interesting topic to you, we'd like to encourage you to check out the Elyssa Aerospace Working Group. This is a group of industry, government and academic professionals that are gathering under the Elyssa project to think about how one could use Linux and Aerospace and tackle some of these challenges. I also encourage you to look for two papers in the upcoming IEEE Digital Avionics Systems Conference that will be on use of Linux and Aerospace that comes up in October. Thank you for listening. We hope that you will contact with the authors with your questions and try to respond to those in a timely manner.


Hi and welcome to lesson three of the linear theory curriculum for classical guitar and piano here at NCSSM this lesson deals with key signatures and It's the first introduction that you will have to a really useful tool in music theory called the circle of fifths So let's get right to work First of all, let's take a look at the staff and today. We're going to talk about key signatures now Key signatures are used to indicate sharps and flats that form a given scale quite honestly It makes scales a lot easier when you know how to do Sharps and flats in a key signature first. Let's start with the sharp key signature Now a lot of kids don't realize this But it's really important to know that whenever you see one sharp in a key signature that one sharp is always F-sharp Okay, so if you see one sharp in that key signature, it's always F-sharp and what that really indicates is no matter what what notes are written in on the staff every time you see a note written you'll play it as F-sharp the second sharp in the series is C-sharp and it is always second So if you see two sharps in the key signature, they are always F-sharp and C-sharp Okay now and then it goes on from there. The order is F C G D a E and B So if there are four sharps in a key signature those four sharps are always FC G and D so in this case it would be FC G and D and in this case it would be FC G and D If there are five sharps in the key signature, it's FC G D and a Hopefully you follow that and you understand what we're doing there if there's six FC G D a e and So then we would go ahead and add that e in the base and in the treble And then finally if there are seven we go ahead and add the B here And we go ahead and add the B here and that gives you your sharp key signatures Now Let me get rid of those sharps and let's go to flats and I should say that a lot of students like to remember that order of sharps by Using a mnemonic device. Let's just put a little reminder here FC G D a e B is the order now I'm not a mnemonic device guy But some people are and the mnemonic device people use for this is frogs can go dancing and eat Bananas so if you want to remember that little sentence frogs can go dancing and eat bananas that will always give you the order of sharps Flats have a different order the order for flats is B e a D G C F and the mnemonic device that some people use for that is Be ad good clean fun or bead good clean fun Again, not really the way I do it, but I totally understand if that helps you So if you see one flat in a key signature that flat is always be flat to B and e 3 B e and a Four flats B e a and D if there are five be a D G If there are six be ad G C and then the last one of course is f and there are your key signatures for flats be ad Good clean fun Okay, so that's your key signatures now the next thing I want to introduce to you is the concept of using something called the circle of Fifths the circle of fifths is a magnificent tool, and I recommend that you use it just like you would use a Calculator in a math course the way it works is this is it tends to give you the name of a key signature in this case we have a C and The number of sharps or flats that are in that key signature You'll notice here at the very top for C. There are no sharps or flats So what that really is saying that if you have a C major scale C D E F G a B C There will be no sharps or flats to get your whole whole half Whole whole whole half you don't need to add any sharps or flats to that C major is basically all white keys If you go to the right The circle of fifths helps you with your sharp key signatures a G major scale has one sharp now if we go back to our Sharps that one sharp when there's one sharp in a key signature. It's always F So for a G major scale G a B C D E F sharp G and I promise you if you go through and you do your whole half pattern whole whole half whole whole Whole half feel free to check me on that that will work every single time D major has two sharps We go back to this the two sharps are always the first two here F and C So if we do our D major scale then D E F sharp G a B C sharp D Promise you whole whole half whole whole whole half It'll work for all of these it will work for all of these So and you've got a circle of fifths on your theory assignment same deal with the flats it works the exact same way and The other thing is if I asked you to write the key signature for the key of let's say for the key of B flat then really to write a key signature in B flat it gets Two flats and those flats are B flat and E flat or in bass clef those flats would be B flat and E flat Okay, so that circle of fifths gives you a ton of information that you can use Okay, now I have One other thing that I want to show you on this circle of fifths down here on the bottom These three key signatures D flat major F sharp major and B major Also have an N harmonic spelling remember N harmonic means another name for the same thing B major has five sharps but C flat major Has seven Flats F sharp major has six sharps, but the N harmonic key G flat major has six Flats and Finally D flat major has five flats What's the N harmonic spelling of D flat? If you said C sharp you were correct and it has seven sharps So these three key signatures D flat F sharp and B Each have an N harmonic key signature that you need to consider as well And that is spelled out on your worksheet as well, and that's going to be important to today's assignment Let's go to the assignment if you take a look at the first Assignment you'll see something that looks like this It gives you a scale D E F sharp G a B C sharp and D and then it gives you a little double bar like this And it says study the sharps and flats used in the scale and write the collect correct key signature for each I want you to write the correct key signature here at the end of the line Recreate the treble or bass clef depending on the line, and then what are the sharps that are used an F sharp and A C sharp Don't write that F sharp down here That would be incorrect Okay, you want to write it on the absolute correct line And you can always refer back to the front page of your theory to know exactly where that line is Now in part two, it's a little tricky to understand what they're doing here in part two So follow me here. They give you a key signature, and I think this first one in part two is in bass clef So there's a key signature that looks something like this six flats right B E a D G and C and Says identify the given key signature will you do that down here? this is the key of sort of to identify that you want to go back here and How many flats was it again? It's six flats So six flats is down here on the bottom the key of G flat So let's get to that and we're going to identify that as G flat major and then Look across the page, and it gives you another bass clef and it says key of and you got a blank and it asks for the N harmonic key Well the N harmonic key of G flat There's our G flat the N harmonic key is I got another color here just to remind you again It's F sharp so Here over on the right The N harmonic key is F sharp, and then we want you to write The key signature for F sharp so we'd come back here and It's six sharps F C G D a E and then F sharp C sharp G sharp D a And that would be the correct answer for the N harmonic key signature for G flat major Hope that all makes sense Note that on part two there is another Third example of that up on the top of page 14 and then finally on At the end on part three Simply for each key indicated write the approach appropriate signature in both clefs, and I think that should be fairly Easy to understand and so really what we're looking for is like for instance on number one they ask for The key signature for B flat major to get that I go over here Look at my circle of fifths B flat major has two flats those two flats are B flat and E flat and so B flat E flat B flat E flat and that would be the correct answer for Number one of part three you can do all of those. I hope that all makes sense if you have any questions Don't hesitate to come and ask


Hello and welcome to the session. The given question says evaluate the following definite integral, 15th is integral 2x plus 3 upon 5x square plus 1 dx and the lower limit of the integration is 0 and the upper limit of integration is 1. So let's start with the solution and first let us find the value of integral 2x plus 3 upon 5x square plus 1 with respect to x. So it can be written as integral 2 hangs of x upon 5x square plus 1 into dx plus 3 hangs integral dx upon 5x square plus 1. Now let us name this integral as i1 and this as i2. Now let us solve both these integrals one by one and then we will substitute the values here. Now the first integral i1 as integral x upon 5x square plus 1 into dx. So let us suppose that 5x square plus 1 is equal to t. So this implies 10x into dx is equal to dt or x into dx is equal to dt upon 10. So this integral can further be written as dt upon 10 and the value of 5x square plus 1 is t. Now taking the constant outside the integral sign we have dt upon t and this is equal to 1 upon 10 log mod t and the value of t is 5x square plus 1. So we further have 1 upon 10 log mod 5x square plus 1. So let us substitute the value here. We have 2 into 1 by 10 log mod 5x square plus 1. Now let us solve the second integral. It is i2 which is integral dx upon 5x square plus 1. It can further be written as integral dx upon root over 5x whole square plus 1 square. And now let us put 5 into x is equal to t. So this implies root 5 into dx is equal to dt or dx is equal to dt upon root 5. So here we have integral dt taking 1 by root 5 outside the integral sign and in the denominator we have t square plus 1 square and this is further equal to 1 upon root 5 tan inverse t since integral 1 upon 1 plus x square with respect to x is equal to tan inverse x. So this is further equal to 1 upon root 5 tan inverse t is root 5 into x. So let us substitute the value of i2 here. 3 we already have and the value of i2 is 1 upon root 5 tan inverse root 5 into x. So this is further equal to 1 by 5 log mod 5x square plus 1 plus 3 upon root 5 tan inverse root 5 into x. Now by the second fundamental theorem of integral calculus the value of the definite integral which is integral as 3 upon 5x square plus 1 into dx from 0 to 1 and the written as 1 upon 5 log mod 5x square plus 1 plus 3 upon root 5 tan inverse root 5x lower limit of the integral is 0 and the upper limit is 1. Since by the second fundamental theorem it says if we have integral fx dx and the lower limit of the integration is a and upper limit is b then it is equal to the value of the antiderivative of the given function at the point b minus the value of the antiderivative at the point a where f is a continuous function defined on the closed interval a to b. Now let us simplify this we have 1 upon 5 log mod 5x square plus 1 lower limit 0 and upper limit 1 plus 3 upon root 5 tan inverse root 5 into x from 0 to 1 and this is further equal to 1 by 5. Now putting upper limit in this function we have log mod 5 plus 1 minus log mod 1 plus 3 upon root 5 tan inverse root 5 minus tan inverse root 5 into 0 is 0. This is further equal to 1 by 5 log 6 minus log 1 plus 3 upon root 5 tan inverse root 5 minus tan 0 is 0 so tan inverse 0 is again 0 and this is further equal to 1 by 5 log 6 since log 1 is 0 plus 3 upon root 5 tan inverse root 5. Thus on evaluating the given integral our answer is 1 by 5 log 6 plus 3 upon root 5 tan inverse root 5. This completes the session by intake care.


The Bible says that all Israel will be saved, but when it says all of Israel will be saved, does it mean all of Israel will be saved? All of the Jews will be saved? Well, the answer is yes, all of the Jews, all of Israel will be saved. But the question is, which of all of Israel? Sounds like a strange way to kind of put it, but that needs to be understood because he's not saying that every single Jewish person, every single person that claims to be Israel will be saved. To get a better understanding, we need to look at the exact passage that's spoken of and then see if we can glean what Paul is speaking of. He says in Romans chapter 11, verse 26, he says, so all Israel will be saved. Just as written, the deliverer will come from Zion and he will remove ungodliness from Jacob. And that's what he says. This is my covenant with them when I take away their sins. So the question is, what is he speaking of? What is he referring to? He's referring to the book of Isaiah. Oftentimes you'll see Paul quote Isaiah all throughout Romans, but particularly in Romans 9 to 11. Because in Romans 9 to 11, he's speaking specifically about the salvation of Israel. And before we cover what he means by Israel and what he's getting at in Romans 9 to 11, as well as chapter two and three. And so let's go to Isaiah chapter 59, verse 20, where he says, a redeemer will come to Zion and those who turn from transgression in Jacob declares to Lord. Now notice what he says to those who turn from transgression in Jacob, meaning of Israel. And then the following verse he states in verse 21, he says, as for me, this is my covenant with them, says the Lord, my spirit which is upon you and my words which I have put in your mouth shall not depart from your mouth nor from the mouth of your offspring, nor from the mouth of your offspring's offspring, says the Lord from now and forever. Now this hasn't happened yet, but he's speaking what's going to happen. Remember even going to Isaiah 53, he's speaking of what he's going to do with Israel. At some point in time, Israel will say those words, looking back at what the Messiah did that for our transgression, he was bruised, he was run through for the chastisement, peace was laid upon him and he bore our sins. This is why Isaiah speaks in past tense about Israel because Israel will say that in the future about what has happened. Now let's also look at the passage that he brings up about this covenant that he makes with Israel in Jeremiah 31. And this kind of helps us get an understanding of what he's meaning in Romans nine when it says all of Israel. Behold, days are coming says the Lord when I will sow the house of Israel and the house of Judah with the seed of man and with the seed of beast, as I have watched over them to pluck up, to break down, to overthrow, to destroy and to bring disaster. So I will watch over them to build and to plant, declares the Lord. So he's speaking of what he's going to do. He says in those days, they will not say again, the fathers have eaten sour grapes and the children's teeth are set on edge. So in other words, their sins won't be attributed to someone else. You will be guilty and charged and the consequences of your own sins will be your own. He says, but everyone will die for his own iniquity. Each man who eats sour grapes his teeth will be set on edge. And here it is. Behold, days are coming declares the Lord when I will make a new covenant with the house of Israel and with the house of Judah. And he's specific that who the covenant is going to be with, he's mentioning Israel and the house of Judah. Not like the covenant which I made with their fathers in the day that I took them out of the land to bring them out of the land of Egypt, my covenant which they broke, although I was a husband to them declares the Lord. So he said that I'm going to make a covenant with Israel. And so his point is that he is going to redeem. He's going to save Israel. The reason why this is important, even for those of us who are not Jewish who are not Israel is because if he makes a promise for Israel then we should also expect that he would keep that promise. Why is that important for us? Because if he makes a promise for us to us which he has, then we should also expect him to keep those promises. If he's not going to keep the promises with Israel that he made, then what's the expectation for us to think that he will keep the promises that he made for us? Why this is important also? Notice what it said in Psalm 89. Look what it says in verse 34. He says, my covenant I will not violate. Speaking of this new covenant that he's making with Israel, my covenant I will not violate, nor will I alter the utterance of my lips. In other words, he's not going to change. He's not going to deviate from that. That's what this word, alter, comes from. He's not going to differ. He's not going to make it any way to fit what we think. Well, if someone were to think that, yes, he's going to keep this covenant with Israel but we are Israel now, no, he said I'm not going to alter that. I'm not going to make it differ in any way, shape, form or fashion even more. So he says, once I have sworn by my holiness I will not lie to David. He says, his descendants shall endure forever and his throne as the son before me. It shall be established forever like the moon. In other words, he is not going to stop off David having any descendants that is of Israel, of the Jews. Notice what he also says here. Going back to Isaiah 9, 20, 70 says, Isaiah cries out concerning Israel, though the number of the sons of Israel be like the son of the sea. It is the remnant that will be saved and this is where we get the understanding of all of Israel. Who does he mean by all of Israel? Well, he says a remnant of who? A remnant of Israel that will be saved according to verse 27, verse 28, for the Lord will execute his word on the earth thoroughly and quickly. That's important thoroughly and quickly. He says, just as Isaiah foretold, unless the Lord of Saba had left us a posterity we would have become like Sodom and would have resembled Gomorrah. Here's what he means. Sodom and Gomorrah, there are no more descendants of that land. He says, we're not going to be like Sodom and Gomorrah. We will have a posterity. In other words, there will be Jews left. There will be some Jews that God will keep. Who will those Jews be that he will keep? It will be that remnant that he promised to keep, those whom he will change their hearts, the elect of Israel. And that's how we understand who all of Israel is. Now, he makes a statement also in chapter nine. Let's start in verse one so we can know what he's speaking of. Paul is speaking of salvation all throughout Romans. When he gets to nine, he's speaking about Israel in particular. He says, I am telling you the truth in Christ. I am not lying. My conscience testifies with me and the Holy Spirit that I have a great sorrow and an easing unceasing grief in my heart. Here's why. For I could wish that I myself were cursed separate from Christ for the sake of my brethren who is his brother. He says, my kinsmen according to the flesh who are Israelite. So these are ethnic Jews, physical Jews to whom belong the adoptions as the sons and the glory and the covenants and the giving of the laws and the temple service and the promises. Whose are the fathers and from whom is the Christ? According to the flesh who is over all God bless forever. Amen. That's what he says though. But it is not as though the word of God has failed for they are not all Israel who are descended from Israel. In other words, not all of those who literally physically come from Israel are the Israel that he's speaking of.  Just because you are of Israel he's not calling you true Israel. True Israel is not a Gentile who places his faith in Christ nor is it an Israelite who is not placing his faith in Christ. It is an Israelite who does place his faith in Christ. It's a Jew who has faith in Christ. Now does that mean that he's any better off or any worse already different than the Gentile who places himself in Christ, places faith in Christ? No, his only point is that he's speaking specifically of these Jews. What about them? Why is it that they are not coming to Christ? And so he says, has God's promise failed them? No, they will come, but God will save a remnant. But right now he is giving them a spirit of stupor and a partial hardening. But at some point in time in the future he is going to save the Jews of Israel, the true Jews of Israel. And so at some point in time he is going to do what he stated that he would do in Ezekiel 36, as well as Jeremiah 31. He is going to change the hearts of the people. He's going to provoke them to jealousy. He is going to bring wrath upon them and cause them to turn their hearts back to him. Now, how many of those Jews will there actually be? We don't know. He doesn't say. We can rest assure that whatever the number is he is at some point in time going to save Israel. Is that going to be in a year, in a hundred years, in 500, we have no idea.  No, it doesn't. As a matter of fact, if he didn't come back for another hundred years, 200 years, 500 years, then that necessarily means that it's not likely to be you. However, an individual Jew will place his faith in Christ from time to time on this planet as we live. But at some point in time he's going to keep his promise that he made with Abraham, Isaac and Jacob, Jacob being Israel. Now at the end of Romans chapter two, going into three, he's speaking about Jews and how there should be some sort of benefit in the fact that they were given the covenants and so forth. But other than that, nothing else. He says, for indeed circumcision is a value if you practice the law, which Israel was not doing. But if you are transgressive of the law, your circumcision has become uncircumcision as though why are you even circumcised in the first place? You're not keeping the law. You're not keeping this covenant, which is the sign of this circumcision is the sign of this covenant. And so you're not keeping it. Look what he says. So if the uncircumcised man keeps the requirements of the law that it's a Jew, will not his uncircumcision be regarded as circumcision? In other words, those who are uncircumcised are keeping the law. They'll be treated as though they are circumcised. The problem is though, and he's being hyperbolic because there's no uncircumcised person that's also keeping the law. Keep that in mind. Verse 27, and he who is physically uncircumcised if he keeps the law, will he not judge you though having the letter of the law and circumcision are a transgressor of the law? For he is not a Jew who is one outwardly nor is a circumcision that which is outward in the flesh. But he is a Jew who is one inwardly and circumcision that which is of the heart by the spirit not of the letter. So his point is the true Jew, the true circumcised person is the one who circumcised the flesh and in the heart. But ultimately it's the heart that must be circumcised. That's what he's speaking of in terms of salvation. That's where there is no difference, no delineation. And it's not to say that there are going to be two different people of God. There's one people of God, one flock. But they do have different tracks. The history of the Jews is vastly different than the history of the Gentile coming to Christ. Just like your history of coming to Christ is different than mine. Similarly he has a specific plan that's recorded with the people of Israel. And so his promises to them he's saying, do not worry, I'm going to keep my promises. Look what he says, what then are we better than they? Not at all. For if we have already charged that both Jews and Greeks are under all, you know what I'm saying? As it is written, there is none righteous, not even one. So all of us are unrighteous. All of us are right, none of us are good. None of us on our own have sought after God. However, that does not nullify, as he said, his plan for what he's going to do with Israel. And what is he going to do with Israel? At some point in time, he is going to touch the hearts of Israel. Those Jews right now, the nation that has turned their back on the Messiah, at some point in time, he is going to, one, inflict wrath upon them, which he says. Two, he is going to save those who are saved out of that. He is going to elect those, save a remnant out of them, change their heart. And those will be the ones that he considers to be true Jews. Not some Gentile who places his faith in Christ will be the Jew or a new Jew. No, nor some Jewish person who does not have his faith in Christ. He's not a true Jew either. The only true Jew, the only true Israel will be those who are of Israel, who are descended ethnically of Israel, who will then have their heart circumcised as he says that he will do in Deuteronomy 36. He says, I will circumcise your heart in the future. He will do so and they will look back on him and the Messiah and regard him as Messiah and place their faith in him. And so we come back to the very beginning. He says in chapter 11 verse 26, as a matter of fact, let's go to verse 25 first, for I do not want you brothers to be uninformed with this mystery so that you will not be unwise in your own estimation that a partial hardening has happened to Israel until now, until the fullness of the Gentiles has come in and then we can heal, say, and so all Israel will be saved just as it is written, which are the passages that we've gone over. And so he will save all of Israel, but not all of those who call themselves Israel, not all of them who call themselves Jews will be saved. Those that he chooses, those will be the true Jews, the true Israel of Israel and all of them will be saved. Amen.


Now, entertainment news, Chicago court has sentenced actor Joseph Spillett, one time star of the TV drama Empire, to 30 months probation and 150 days in jail for staging a hit crime against himself. During December, found Smollett, 39, guilty of five of the six felony disorderly conduct counts he faced. One for each time he was accused of lying to police. Cook County Second Court Judge James Lane also ordered Smollett to pay over $120,000 in resuscitation and find him $25,000 saying he didn't agree to damage to people that are genuine victims of hit crimes. I am innocent and I'm not suicidal. If I did this, then it means that I stuck my fist in the fears of black Americans in this country for over 400 years and the fears of the LGBTQ community, your honor, I respect you and I respect the jury, but I did not do this and I am not suicidal and if anything happens to me when I go in there, I did not do it to myself and you must all know that. It is the worst thing that can happen, especially in our country with all our history and all that we're going through now to try to get around some of these issues.  There are people that are actual, genuine victims of hate crimes that you did damage to. These are people that have a difficult time coming forward. They may be mistrustful. They may not want to bring it to the attention of the community or first responders. There may be some trepidation. I don't know for sure how much damage there was.  Emotional moment. My brother was sentenced to 150 days in prison, which while on its own might not sound like a long sentence to all of you, but if you could listen to the recital of the prosecutor's entire, entire prosecution at the mouth of the duly elected judge, James Lin. All right? I don't understand and I spoke specifically today to the judge and I asked for leniency. I did not expect him to be completely lenient and listen, but at the same time he shamed my brother. He spoke about arrogance. He doesn't know the struggles my brother is encountering. He doesn't know anything that he's dealing with. So, hope you enjoyed the news. Please do subscribe to our YouTube channel and don't forget to hit the notification button so you get notified about fresh news updates.


Hello and welcome to the session. My name is Mansi and I'm going to help you with the following question. The question says check whether in the following AD is the bisector of angle A of triangle ABC given in figure 8.61 that is this figure. Now we have to tell whether AD is the bisector of this angle or not. So the measurements of the side given to us is first is AB is equal to 5 centimeter, AC is equal to 10 centimeter, BD equals to 1.5 centimeter and CD equals to 3.5 centimeter. The second part is AB is equal to 4 centimeter, AC is equal to 6 centimeter and BD equal to 1.6 centimeter and CD equals to 2.4 centimeter. The third part is AB is equal to 8 centimeter, AC equals to 24 centimeter and BD equal to 6 centimeter and BC equal to 24 centimeter. Now before starting with the solution let us see the key idea that we use in this question. The key idea behind this question is AD is the angle bisector of angle A in figure 8.61 that is this figure given to us only if AB by AC is equal to VD by CD that means we see that AD is the angle bisector of this angle only if AB by AC is equal to VD by CD. Now with this key idea in mind we'll start with the solution to this question. Let us start with the first part. Now here first of all we find AB by AC. Now it's given to us that AB is equal to 5 centimeter and AC is equal to 10 centimeter so AB by AC will be 5 by 10. Now if we divide the numerator and the denominator by 5 we get 1 by 2. Now we find BD by CD. Now BD is given to us as 1.5 centimeter and CD is given to us as 3.5 centimeter so BD over CD is 1.5 divided by 3.5 and that is equal to 3 by 7. Now AB over AC is not equal to BD over CD that is 1 by 2 is not equal to 3 by 7 hence AD is not the angle bisector of angle A. So for the first part we see that AD is not the angle bisector of this angle. Now let us see the solution to the second part. Again here we will find AB by AC. Now AB given to us is 4 centimeter and AC given to us is 6 centimeter so AB by AC is 4 by 6. If we divide the numerator denominator by 2 we get 2 by 3. Now again we find BD by CD. BD by CD will be 1.6 centimeter by 2.4 centimeter and that is equal to 2 by 3. Now because AB by AC is equal to BD by CD that is 2 by 3 is equal to 2 by 3 hence AD is the bisector of angle A. Now let us see the solution to the third part. Again we find AB by AC. Now AB by AC will be 8 centimeter by 24 centimeter so it is 8 by 24. Now if we divide the numerator and the denominator by 8 we get 1 by 3. Now we see that BC given to us is 24 centimeter that is this is 24 centimeter here we are given BD that is 6 centimeter. So we can find out CD. CD will be equal to BC minus BD. Now BC is given to us it is 24 centimeter and BD is given to us is 6 centimeter. So CD is 24 minus 6 that is 18 centimeter. Now we can find BC by now we can find BD by CD. So therefore BD by CD is equal to 6 by 18 again we divide the numerator denominator by 6 we get 1 by 3. So we see that AB by AC is equal to BD that is 1 by 3. So because AB by AC is equal to BD by CD hence we say that AD is the bisector of angle A and this is our answer to third part. So let us summarize our answer. Now to the first part we see that AD is not the angle bisector of angle A. So the first part answer is no to the second part and to the third part we see that AD is the angle bisector of angle A. So our answer is yes to the second part and yes to the third part. So this is our answer to the question. I hope that you understood the question and enjoyed the session. Have a good day.


Let me just remind you what's involved in a Riemann-Hilbert problem. There's a contour, sigma, which is oriented, and by convention, the plus sign that lies on your left as you follow the arrow.  And we say m, which is a function of z, solves the Riemann-Hilbert problem, or is a solution of the Riemann-Hilbert problem. If m of z is analytic, in c, take away the contour.  This is for z belonging to the contour. And in the case where m of z goes to the identity, as z goes to infinity, this is what we mean by saying that m solves. Can people hear me? Yeah. OK. Took it off. That's what we mean by saying that m solves the normalized Riemann-Hilbert problem. Now, there are many, many, many technical issues connected with a Riemann-Hilbert problem. I wrote on a list of nine or 10 of them. I just mentioned some of them. The most obvious ones is what you mean by m plus and m minus. I remind you that m plus means, if you have a point z over here, m plus of z is the boundary value of m of z. As you approach z from the plus side, m minus is the boundary value from the left. In what sense are those limits taken? When m goes to the identity, in what senses are taking place? Different problem is how do you know that there exists such a solution? Is the solution unique? What kind of contours are you going to look at? Many, many problems. I'm not going to address this in a systematic fashion at all. We're just going to go along and face some of the issues as they come up. Now, a key observation which has been apparent in many of the recent lectures is the role of the Cauchy operator, or sometimes people call it the Cauchy transform. That had the effect, for example, we saw in Los Los Talk that it draws the analysis from the upper half plane down to the axis. In exactly the same way, the Cauchy operator is going to take the problem from the complement of the contour onto the contours. So the most important operator in the whole business is the Cauchy operator. And it looks like this. We have a contour, an oriented contour. It acts in the following fashion, f of s, d bar s upon s minus z, so the sigma, whenever I d bar s, I mean ds over 2 pi i. And this is for z, allowing to the complex take away the contour. This is the Cauchy operator. And what we'll be studying in this lecture is the properties of some of these properties, some of the properties of this operator, because as I say, it's going to play the key role in understanding Riemann-Hilbert problems. So, just as an example, I wanted to just take the case where sigma is just the real line oriented from Meijers-Vietu plus infinity.  So this first part of this talk is just going to be that kind of review and to just move you on ahead, the remarkable thing is that these properties for the Cauchy operator just on the line are now going to be shown to be true in a much more general situation where we have a contour of that kind. So here, let me just write c of f of z as the integral of f of s d bar s upon s minus z, and now the integration takes place over R. If I end up writing too small, can somebody at the back just give me a word, okay? All right. Now, what we have, let's do a very simple computation. Let's take z equal to x plus i epsilon, epsilon positive, and x real.   And so let's call that 1 plus 2, okay? So now, if we take 1 and we change variables, this will take the form one-half of f of x plus epsilon u du over pi times u squared plus 1. So that's the first thing. So let me assume, first of all, that f belongs to the short space. So it immediately means we can let epsilon go down to zero, and this is going to go to one-half, one over two pi times f of x by dominated convergence du over u squared plus 1, which is just f of x over 2.  And the part where 2 is bigger than epsilon, I just put here x minus s is bigger than epsilon. Okay, now comes the key fact about these operators. And one of the reasons that these operators are so important in analysis is that they are a prime example of what are called pseudo-differential operators, or particularly here, called they're on Ziegman operators, and there's a fundamental cancellation which is built into them which you're about to see now.  I can put this in for free because this is an odd function and I'm integrating over an even interval, so I can just put that in for free. And this is this cancellation which is absolutely essential for what comes. We find that this then here tells us that pi less than epsilon, this quantity, is then less than or equal to 1 over 2 pi times f prime in the infinity norm times the integral of s minus x squared, s minus x squared plus epsilon squared ds, and this is over s minus x is less than epsilon. This number is less than 1, so this is less than 2 epsilon upon 2 pi f prime in the infinity and that goes to 0 as epsilon goes down to 0, essential thing.  Now, so 3 is, oh 4, okay, so 3 here is going to equal, going to be bounded, 1 over 2 pi times integral s minus x bigger than epsilon, and we've got epsilon squared upon s minus x squared epsilon squared f of s ds quantity, and we change variables, this is 1 over 2 pi, and we've got f of x plus epsilon u, d u left something off, there's an s minus x here.  And again, this is an odd function, so this integral is 0, so this limit here is 0, so what we left with is a part 4 which is just the integral, so this is equal to 1 over 2 pi times part 4, which is just the, okay, so the limit as epsilon goes down to 0 of 1 upon pi integral of f of s, we've got your x minus s is bigger than epsilon f of s upon s minus s ds. So that's what we left with that, let me put up what I want to say, and this is the key fact, c plus f at a point x, which is the limit, goes down to 0, cf on the real, it's 1 half f of x plus i over pi, cf of x is 1 over pi x minus s, yes, and this is what's called the Hilbert transform. So that is the key fact, if I come from the bottom, I let epsilon be negative and go up to 0, everything stays the same, instead of having plus here I have a minus, the big one so far, as I said I'm just sticking with f in short space. Okay, now the key things to observe is that c plus of f minus c minus of f is just f, and c plus of f plus c minus of f is just i times the Hilbert transform. Now one of the things is the fact is we looked at this limit, epsilon goes down to 0, you can replace it with what's called a non-tangential. So instead of just taking the limit as you come down there, you can, here's your point x, you can put any cone, say an opening angle to theta over here, and you can take the limit not just as you come down but anywhere as you come down. In general you can't go if this was a parabola, that wouldn't work. But so all the limits are going to be point-wise limits which are non-tangential. So far we're just working on short space functions. So now to get a handle on this as an operator we use the Fourier transform f hat of z, which is the limit in the mean, or the limit, so r f of s e to the minus i xz e to the i sz ds upon root 2 pi. And it's inverse transform which will be the same thing as this except you put e to the i sz ds over root 2 pi f of s, this was dz now. You can get this at s now. And again it's a limit in the mean from minus r to r, you just flip the sign. Now it's a straightforward calculation in residue theory to establish the following. Fix epsilon positive and define c plus, let me call it cx or c epsilon, f at x to be c, the transform at the point x plus i epsilon. So think of epsilon as being fixed now. It's a calculation, a straightforward calculation using residues and you get the following. That if you want to compute c epsilon f, you take your function, you take its Fourier transform, you multiply by the function chi bigger than zero e to the minus dot epsilon and you take the inverse Fourier transform and okay that's what it is. So what it is to evaluate this operator here which is now a function of x, epsilon is fixed, x is real. You take the Fourier transform of the given function, you multiply it by this function, zero of s e to the minus epsilon s, for s positive and zero otherwise and that's what you have. In particular it tells you that this operator, initially you do all these calculations on short space and you immediately see from here because of the properties of the Fourier transform, I remind you the Fourier transform is a bijection, an isomorphism between l2 and itself and the inverse Fourier transform is the inverse transform. So this takes a function f onto function f hat with the same l2 norm. You're multiplying by a bounded function so it stays in l2. Then f inverse brings it back to the original l2 with the same norm. So this is a bounded operator along to the bounded operators in l2. And moreover we see that as epsilon goes down to zero, c epsilon minus c plus in l2 goes to zero. Now c plus is clearly our original operator. It's the value of this quantity when epsilon goes down to zero. On the other hand we see if we look at this object here when epsilon goes down to zero, this goes just down to a particular operator. What I do with, so we've got that c epsilon goes down to the operator c plus which is just the inverse Fourier transform multiplied by this bigger than this and then the Fourier transform and this operator is necessarily the Cauchy transform limit as epsilon goes down and down from the top. So you've established a number of things over here which are very important in application. First of all you see that the Cauchy operator which we wrote down, bb4 for points above the line converges to the operator c plus. That operator c plus is necessarily a bounded operator. It's just multiplication by the characteristic function of the right hand line in Fourier space. Moreover, c plus is the limit in l2 of the functions defined on lines of height epsilon above the real axis. You just look at the definition. It's an integral only over the real line. The complex comes in only in the variable z. This is purely real. Fourier transform goes from the real line to the real line. So here, cf is defined, let me remind you, cf at z is defined for complex number z which is the integral of f of s. Have you got it? Yes. Okay. So let me just re-interpret these are absolutely critical facts. The operator c plus is a bounded operator from l2 to l2. That follows from the calculation it is on short space and the density of short space in l2. Moreover, if you look at the values of this function for fixed epsilon in l2 of x, it will converge in l2 to a function on the line which is given by c plus of f. Okay. Now, if we use this fact that c plus and similarly we would find that c of epsilon converges to c minus as epsilon goes up to zero in a similar calculation. And there, if you write out so a similar calculation will give you c minus but if you now write out c plus is one-half, c plus on f, plus i hf over 2 and you take the Fourier transform of both sides, so you get one-half fh plus plus i over 2 and then you take the inverse Fourier transform you'll see that this implies that hf is a Fourier transform times i of the signum function times the inverse Fourier transform. So instead of just being the characteristic function of the right-half interval, it's now the signum function which is plus one on the right and minus one on the left. So this is just a little piece of algebra where you plug in this fact over here, this fact that we just plug it in over here. Okay. Now, there's a beautiful argument due to Ries and he tells us the following. If you take c f at z, z belonging to the upper half plane which we know of f of s upon s minus z d by s on the real line. And let's assume first of all that f belongs to c zero infinity or complexly, compactly supported infinitely differentiable functions. Then this function here clearly has an analytic continuation or analytic in c plus and is of order one over z as z goes to infinity. This means we can integrate c f of z d z over a contour where mod z is equal to r. And we can raise this to any power we want. Let's raise it to the fourth power. Then because this decays like one over z, this goes like one over z to the fourth, so it looks like one over r to the fourth. This will give you the length of the contours r. So this behaves like one over r cubed, which goes to zero as zero as you go. But then by Cauchy's theorem, the integral around here is just the integral around here. So this tells you that the integral of c plus of f to the power four dx minus infinity to infinity is zero. This is great because you take this formula here and you just plug it in there. So let's also assume for simplicity that f is real. So you got this integral here. Now you write out the real and imaginary parts and you get an f to the fourth minus, I think it is four, f squared f hf squared plus hf to the fourth integral must be zero, taking the real part of that. But now you see what's going on. You've got an h to the fourth here. You've got an f to the fourth here and you've got this thing over here, which by Schwartz's inequality is controlled by this object and by that object. So a little application of Schwartz tells you hf to the fourth. So you immediately see a beautiful argument that not only is the Hilbert transform bounded in L2, it's bounded in L4. And I can repeat this argument for any even integer just using Schwartz over here. So then you can do it for every even integer. So by interpolation you then have it for any positive integer bigger than two and then by duality you get it for all p bigger than one. So this implies together with interpolation and duality to the p. So that's the first big piece of information that the Hilbert transform is a bounded operator in Lp for any p bigger than one and less than infinity. It's not a bounded operator in L1 as you can see easily because suppose I've got some function of s minus z.


far too long since I've talked about Star Wars on this channel, so let me ray mind you of why I love it so much. I think you're gonna be pleasantly surprised by some of the ray violations. Let's get started. Believe me, listen to what I ray. I'm a geek. If you haven't seen The Rise of Skywalker yet, there is going to be spoilers in this video, so just walk away. But everybody's seen this movie because it's wonderful. It's the best. It came after the last Jedi, which was even better. The beginning of the film. We see Kylo Ren searching for answers in the form of a wavefinder on a planet we're not really sure the name of. It's kind of gross. There's a lot of reds and crusts and bad guys getting killed or maybe the good guys. We don't know because the movie doesn't tell us anything. But now we do know because a recent novel came out and explained everything the movie didn't, which was everything. Grab on to your sacks, boys, because the reveal of this planet's name is going to blow a lot of you out of your drawers. It is called Oostafar. Oostafar is where Sir Darth Vader was born and bred, a volcano planet filled with ash and molten lava, magma, and of course one of the most powerful Siths on the planet. Anakin Skywalker, Darth Vader. That's right. Kylo Ren was there for approximately one minute of film time to retrieve the wavefinder so he can find a secret hidden thing or base or world or something. I honestly don't know. There's a lot of running and jumping and moving and then planet hopping in this film. It's hard to make heads or tails of anything. Actually we do find out who these guys are that Kylo Ren takes out through the Alzheimer's, which were a tribe of sorts that worshiped Vader and then stayed on this planet until, I don't know why, they're just there guarding it I guess, but he kills them so they're not guarding, they're doing a bad job guarding it. You're dead. They're on a volcano planet so I guess you could say, you fired. You fired. That's my president. Not my president, but that's the president. Later half of the film we're introduced to a new strong female lead named Janna and she's black. It's about time. This is important because there's only like three black people in all of Star Wars in all of the universe and one of them happens to be her father probably and that's Landau Calrissian who we do have a weird exchange with later where he's like, hey what are you doing? You know who your dad is and she's like, no I don't but you're black so odds are we're related somehow, right? Who's my mom?  That part's not true, but I bet it is because he definitely had a thing for that robot. It's also gleamed from the Navelle that Luke was training young Ben from the age of birth. That's right. At zero years old he was learning the ways of the force and for 13 wonderful years he was he was kind of becoming a douchebag and eventually he burned down the Jedi temple. That was Ben. Ben did that too. So just kind of all around a pile of garbage. This is a big one. I've been wondering this ever since I saw the movie 25 times in theaters and that is who are these people that are in Palpatine's audience at the end of the film when they're fighting in that coliseum? Who are these people? Austin Powers reference subscribe. Turns out this beautiful book has revealed to us that those are the younglings the children of fallen Siths. I didn't know if Palpatine was like making them show up with some magical Sith ability but no they were actually chilling in the auditorium there and Ray just like completely eviscerated all of them. So that was cool. There are a lot of kids, a lot of younger members she killed. There's a lightsaber in the last Jedi it's a very special important lightsaber with a blue crystal inside of it. Well that's destroyed presumably by Kylo Ren. I forgot what I just read but I'm gonna go with that and Ray uses her magical force healing ability to bring back the lightsaber to life. She fixes the blue crystal, it's back and that crystal can be seen again in all of its glory. Rise of Skywalker. Potentially the biggest revelation is the fact that old Palpy wasn't actually Palpy at all. He was a clone of Emperor Palpatine. That's right there was multiple clone processes going on. We saw that was Snoke in the tube. So what we can infer from this is there is a room full of lifeless husks of Palpatine lying around that just didn't get fully formed. Why go for the geriatric 900 year old look? I don't know. They kept it Sith.  You know I can barely walk. He's kind of limping around. That's the one they went with. And I remember hearing as a joke that Emperor Palpatine was gonna be back in Rise of Skywalker way back when Solo came out. There was a review or something that that showed it. Radish do we have that clip? We have a clip from that that guy that predicted that.  Palpatine fell down.  Gave him like a robotic Xavier chair. That's all I know for now but I'm sure Disney's gonna give us another heaping spoonful of more as time goes on. They're like J.K. Rollins. They just kind of rewrite everything that's been done. And since the movie is so vague it's really easy to just kind of make up whatever you want and just kind of you know put it right in there. But here's the thing. I don't watch Star Wars movies for these these big reveals and these gotcha moments. I'm in a foray. I'm only in a foray. Oh here she comes.  Oh here she comes. She's a Skywalker. Oh here she comes.  Oh here she comes. She's a Skywalker. Unless they change that too.


I promise you Julius Irving Dr. J and let me just tell you before we bring Julius in the nets and Marvel Present the Dr. J inspired by Black Panther bobblehead to the first 10,000 fans in attendance on Friday, March 6 When the next play the Spurs as part of Marvel Superhero night and here is the the bobblehead I'm showing you on your it really is beautiful and It's Dr. J in a black Nets Jersey and rip to always rip He's got to be very happy with that now He will be in attendance at the March 6 game watching his first ever Brooklyn Nets game in person at Barclay Center for those that don't remember and you should he played with the New York Nets of the ABA and He led them to two ABA titles The Nets are going to host an exclusive pregame meet and greet with the dock Which includes a lower level game ticket and a dr. J inspired by Black Panther bobblehead for 250 bucks So Brooklyn Nets comm is where you go for more information and tickets are still available for March 6 that Brooklyn Nets Dot-com and we bring in Julius Irving Dr. J. It's Michael Don and Peter. How are you? I'm doing great guys. How are you guys doing? We're doing great? Have you seen this bobblehead because it's pretty impressive? I'd show it a couple of days ago and And I compared it to one of my old bobbleheads. It was it was like night and day Now I've got to tell you this I play this game a lot and I wonder if you do as well What would have happened if they didn't deal you to Philadelphia? What I mean with the Nets still be in New York? That game yeah game that game. Well, you know that the sad irony of that is the Roy Bowell and Dave de busher Kevin Lockery and Rod Thorn.  There's the next year wasn't that good For the class so, you know, you have to make decisions though. They're only bad decision to notice it You know, in fact, you make these decisions that you want. I think they went out. They got tiny archibald Got George Johnson and he's a seven footer and good shot blocker And whatever and probably probably would have had a pretty good team there in New York in 7677 as opposed to, you know joining the new franchise and plan for the series Six was made the finals that year, but we were a very very Interesting study in basketball that that particular group you know, particularly moving from the camaraderie we have with the Nets and you know the success that we had to Move into another state another city another team and and really another journey and I and I ended up, you know, stand there the rest of my basketball career So it just goes to show that that probably was the right place to be because I was able to play 11 more seasons Whereas the Nets, you know moved over to the sky the way For a while played over and you know, the college gym In that hard floor and I remember I remember when these felt after playing as a visitor I would imagine if that was my home court, you know I probably would have already played five or six years as opposed to 11 just because of that so I look back on it maybe a little differently than a lot of other people and You know the fact that you know things do have a way of coming full cycle I mean, I always always pull for the four ABA teams, you know I pull for the sixes and I pull for the Nets and the Pacers and the Spurs and The Spurs of all the ABA teams have won all the championships But but I still pull for the other three and I feel like it's only a matter of time, you know When you look at the other sports, there was the alternative league the AFL in football the WHA in hockey How did you view yourself in the ABA? Did you view yourself as a Rebel League? I think when the ABA first started, I think that mantle probably would have been appropriate But as we moved on and you know, suddenly there were nine teams in the ABA There were 14 teams in the NBA and you know, guys are making decisions or what which way are you gonna go? ABA and NBA, you know, there was the feeling that you know, there was the experience was going to be comparable in either league and And you know, I played in the last five years of the ABA So I missed the first four, but I heard a lot of stories about the first four It was part of creating stories in the next five and Felt like You know when we got to the off-season and all you know The ball players were gonna play in the summer league, play in the Rucco League and played in DC and played in Chicago It's all the same guys, you know, I mean it's the same game I think just that the feeling there was probably maybe a little more depth on the NBA teams and You know, but the NBA was up there. It's kind of like the slow down brown ball league, you know Where's the ABA was, you know, red, white and blue America, you know forever and And I think when you got publicity in the ABA Everybody reveled in it, you know, we all enjoyed supporting one another in that capacity and that wasn't where the NBA was run Dr. Jay is our guest here on the Michael K. Show. I gotta ask you this I don't know if you realize but I did ten years of the Yankees with John Sterling and John was your play-by-play guy with the New York. Now. What was he like back then, Don? Absolutely. Absolutely, you know John was John was it was awesome. You know, we knew on channel nine and channel 11 Yeah in the day WPI X and Bobby Goldshaw worked with him and You know Steve Albert, I think too. Yeah So, you know, they're quite a few guys who are You know Hall of Fame quality Analysts and announcers and commentators because they always did their homework they always showed up for work and And and we got to know one another I mean your players in the and the media, you know Always had an amicable relationship because you know, that was what helped float the ship You know, it's one for all and all for one. Was he a good as good a tennis player as he says, doc? I Remember the tennis No, but he tells me these great stories that he was really good before his knees got shot Doc what are you? How do you feel about the league right now? How much are you enjoying this version of the NBA that we're getting in in 2020? Well, you know the league has been empowered by having an international audience having the economics go through the room for me, you know, it's 24-day dollar enterprise and And it's driven by partnerships and And I think the players are you know getting a fair share of the the equity With the way contracts are structured now, so I mean it's everybody's happy Nobody has anything to complain about and I think from a fan standpoint, you know, maybe the You know guys sitting out what do you call it? There's something over load or whatever. Yeah, low management The load management situation. Yeah, that's earth to few people and I probably if I was you know Buy a ticket to one game a year and I wanted to see Kawhi Leonard And he's not playing that day because he's he's sitting out. I'd be a little pissed too But you're gonna get a better version of that player in the playoffs if they don't play all 82 games So so leagues in good shape, I like watching it, you know, I enjoy watching my teams When I come into Brooklyn and the Spurs are gonna be there, you know it's gonna be a little nostalgia there because I Was a net when we you know had the battle the Spurs To get to the ABA title and that last game was against the Denver Nuggets actually the last championship year For the Nets it was the Nets and the Denver Nuggets and and you know, a lot of Hall of Famers were in that game Dan Essel was in the game and David Thompson was in the game, of course, I was in the game and You know, that's three that I can remember I think it was some other guys who were who were maybe worthy Ralph Simpson comes to mind and Maybe more of a Western did some good things in the ABA and on our team We had Brian Taylor who hopefully will be up as a nominee You know one of these years to be able to recognize how good a career he had Now Doc, I've got to ask you this. I was enthralled by the all-star game dunk contest this year But I don't understand how you and Dominique Wilkins are in the building and they have Common and Chadwick Boseman judging the dunks. Why weren't you guys sitting there judging the dunks? We're not from Chicago They wanted to make it feel, you know Have some home cooking there and You know that I think You know these decisions are made behind closed doors. Yeah But if Dr. Jay's in the house, he should be judging the dunk contest. That would be a natural thing But I've done so many of them. I mean, I've got probably more than 20 and Gone to you know, 30 plus all-star games or whatever. So it's nice to take a break Did the guy who won deserve to win or did The guy from Orlando, I think as it turned out his his duck was probably better than the one jumping over the seven footer Because that was a little, you know, it wasn't smooth. It was it was a little shaky. It was a two-handed. So it's very safe dunk And the other from inside the foul line with a little twist in the trail I mean, it wasn't it wasn't as good as some of the fifties that had happened before by both the guys But you know, it was a 48 and it beat out of 47 so Close yeah, but I think the thing you probably have a little angst about and I do too You know, the contest went so well and it was neck and neck for so long You hated to see somebody have to lose. Yep You know, it was almost like if they gave two trophies who would be mad in the building? Nobody And That's what I thought about it. All right, how about your nets? So they're hanging in there for a playoff spot pretty much without Kyrie in the next year You hope a healthy Kyrie and then you added to rant, you know, is the sky the limit for this next team Can we talk championship in Brooklyn next year? Well, they automatically have to be contended if you can get those guys It's just if they played 60 games I mean each of them because they're probably gonna still go through the load management situation next year It seems to be a fixture in the league now. There's an 82 game schedule But I think the benefit of the load management is now you see what you got in the in the rotation in terms of the next guy And then now you go nine deep or ten deep, you know instead of eight deep and That's going to help you play off time too because you're going to know what you're going to get out of these guys I mean, you know, they've compensated you got 12 guys on the side of the taxi squad So, you know, you might have 14 or 15 guys available and When the when the stars Get on the first rest You can see what you got other than just in practice you get to get to play real games So for so for some guys, it's an opportunity and and and we all know what happened in baseball When the star sat out and somebody took his spot and then 35 years later or 20 years later or whatever was he still had his spot and how the guy never got back to the lineup Julius what what were your thoughts when Kobe passed? I would I was saddened by it, you know, I was moved From my relationship with his dad and his mom You know his dad played with me and Philly my first year there. You got traded in the second year And Kobe was born during a decade when now my wife and I had five children. So So I kind of you know, look at it at him like another another child, you know, I kind of never My my my processing of this whole thing in Monday and Tuesday Sunday and Monday, you know when it was happening You know, so many people say well, you know, Kobe is what Michael Jordan used to be what used to be and you know Guys used to be in the league and I I never looked at the ball player Comparison piece because I never related to him That way it kind of we always related to him is, you know, Kelly being son and His greatness was associated with you know the titles and they and you know the relationship with Shaq and You know the things we heard and saw Over over the course of his career But you know, this was personal. I mean, you know, suddenly a life has been lost and And you know his daughter With him and this is not something that you know should happen. This is not something that happens every day to people who you have feelings for and Or even have a distant relationship with I was you know, I was taken back by it and You know, I mean, I think it's something that I mean, we're not going to get over Cuz there's gonna be more discoveries just like You know, we learned that the pilot had a history of being a risk-taker So that doesn't enhance the story, I mean it just makes it more pitiful in terms of you know saying Why did this have to happen because it was something that probably was avoidable Doc we thank you for coming on March 6th You're going to be in Brooklyn for the first time to watch the the Nets and the bobblehead is outstanding and we thank you for Giving us some time today Awesome guys nice talking to you and take care. All right. Good luck. All right So you just go to Brooklyn Nets comm for more information and you could actually be part of an exclusive meet and greet with Dr. J So there are tickets still available for that March 6 game at Brooklyn Nets comm


Hello and how are you all today? The question says three coins are tossed simultaneously 200 times with the following frequency of different outcomes. Now these are the outcomes and this is the frequency. If the three coins are tossed simultaneously, if the three coins are simultaneously tossed again, compute the probability of two heads coming up. Let us proceed on with our solution. Here the total number of times the coin was tossed is equal to 200. Now number of times is equal to, heads are coming up 72 times out of 200. So that is equal to 72. So the probability that two heads are coming when we are tossing three coins simultaneously again, that is equal to 72 times out of this 200 times. And on simplifying, we have the answer as 9 by 25. This is our required answer. Hope you enjoyed the session. Take care and have a good day.


it doesn't matter if you have vision if you're too afraid to take that opportunity. So you may be able to see that pass and a lot of players will see the right pass but they're not confident in their ability to play that pass or they're too afraid to play that pass because they think they'll make a mistake and everyone will yell at them. So having vision is the first step, having the confidence that I will send the better in one day soccer blueprint to anyone who comments hashtag soccer on this video. Bravery the courage to actually try and play that pass to your teammate well that's the second step so play me a pass again please.  If you like this content check out the soccer success planner you can download it for free there's a link in the description below and for advanced training check out the online soccer academy.


Have you ever met a person who has strong opinions and convictions, unquestionable, and unproven things? But no matter how much you try to make sense talking to them, nothing works? Such conversations can be extremely irritating, yet becoming frustrated and yelling at them that they are wrong never works. This is where Peter Burgosian comes in. Peter these days is known as one of the best critical thinking experts in the world. Recently, him and his colleague James Lindsay wrote a book called How to Have Impossible Conversations, a very practical guide where they introduce scientifically proven methods of effective communication with people who have opposing views. In this video, I will introduce eight practical techniques that you can use when having conversations with people you can't stand. Sounds good? Then let's get started. Technique number one, what's your purpose? Often in the heat of a difficult conversation, we lose track of what we are really talking about to begin with. As Peter suggests, it is important to first identify your your conversational goal. Ask yourself, why am I having this discussion? What are my goals? What do I want to get out of this? Do you want to better understand whether the other person believes what he or she believes in? Do you want to prove them wrong? Will this goal help you in any way? Knowing your goal will already make the conversation much easier, since you will better understand what you're looking to get out of it. To say it in a different way, how will you get somewhere when you don't even know where you're going? Technique number two, partnership. During the 1970s, Peter's mentor, Portland State University psychology professor Dr. Frank Wesley, investigated why some U.S. prisoners of war, POVs, deflected to North Korea during the Korean War. His research showed that virtually all of the defectors came from a single U.S. training camp. As part of their training, they had been taught that the North Koreans were cruel, heartless barbarians, who would despise the United States and single-mindedly sought its destruction. But when those POVs were shown kindness by their captors, their initial indoctrination unraveled, they became far more likely to defect than those POVs who either hadn't been told anything about the North Koreans or had been given more neutral accounts of them. The way to change minds, influence people, build relationships, and maintain friendships is through kindness, compassion, empathy, treating individuals with dignity and respect. It comes naturally to all of us to respond favorably to someone who listens, shows kindness, treats us well, and appears respectful. Meanwhile, a sure way to entrench people in their existing beliefs is through adversarial relationships and threatening environments. It is easy to dislike someone who is mean-spirited, treats you poorly, doesn't listen, or disrespect you. The best way to create this partnership is to view yourself as a conversation partner. That is, treat others as if you're working together to have a fruitful conversation, because you are. Seeing your conversations as partnerships is the single biggest step you can take to ensuring conversations stay civil and to building relationships instead of damaging them. Technique number three, from winning to understanding. How do you switch from viewing people as opponents, moral degenerates, or even enemies to valued partners and collaborators? Answer, shift your goal from winning to understanding. Make understanding your conversation partner's reasoning your initial goal. Treating an individual as a partner in civil dialogue does not mean accepting their conclusions or buying into their reasoning. It means thinking along with someone so that you understand not just what they believe but also why they believe it. In that process, maybe you'll come to understand your reasoning or see that their reasoning is an error or maybe you'll even discover that you're harboring a false belief yourself. Technique number four, report. When you build rapport with your conversation partner, you experience a closeness where you both feel comfortable, get along, and mutually empathize. The more individuals diverge in their stances, the more important it is to build and maintain rapport. As this connection grows, the person with whom you're speaking is less likely to be defensive and more open to suggestion. Some tips on building rapport. Build rapport immediately. Do not start the conversation with a substantive issue, especially if it is controversial. Find common ground. Do not parallel talk, explanation of parallel talking. It is taking something someone says and using that to reference yourself or your experiences. For example, if someone tells you they just got back from Japan, don't start talking about the time you went to Japan. Ask them about their experience instead. Don't make their stories about your life. Parallel talk damages rapport. Engage in substantive conversation only if you're willing to make time. Be courteous. To build rapport, ask sincere questions, that is, for which you like to find answers, as opposed to asking questions as a tactic. Technique number five, listen. If you do not listen, you cannot understand. And if you cannot understand, there is no conversation. Listening is more difficult than it seems, so it requires practice. Do what you can to make listening the center of your approach to conversation. Many times when we talk with people we don't agree with, while they speak, instead of listening, we focus on our own internal explanations of why they are wrong, where we are preparing a counter argument to shoot as soon as the other person stops speaking or we interrupt them. Yet this way we not only hurt the rapport, which is essential in finding some sort of an agreement, but we may also miss important information that would help better understand the other person's viewpoint. Listen, and sometimes you will be surprised by what you will hear and learn yourself. Technique number six, shoot the messenger. Nobody likes to be lectured. The research literature on effective conversations shows that delivering messages does not work. Conversations are exchanges. Messages are information conveyed in one way transactions. Messengers espouse beliefs and assume their audience will listen and ultimately embrace their conclusions. Even when messages are not delivered across any sort of political or moral divide, they tend to be poorly received. In order to avoid throwing out messages at other people, Peter Bogosian offers these suggestions. Distinguish between delivering a message and authentic conversation. Approach every conversation with an awareness that your partner understands problems in a way that you don't currently comprehend. You'll be less likely to deliver messages if you're more focused on figuring out how someone knows what they know than if you presume to understand the reasoning behind someone's conclusions.  When you realize your partner is being a messenger, do not shoot the messenger.  Shooting the messenger applies only to yourselves. If your partner enters messenger mode, begin a listening and learning mode, centering on asking questions. Questions can be an effective way to nudge the conversation back on track.  Socrates once said that people do not knowingly desire bad things. Individuals act, believe, and desire based upon the information they have. If they had different information, they'd derive different conclusions. We all have an impulse for goodness. However, lacking a comprehensive picture contributes to the failure of arriving at correct conclusions. When you encounter a person with radically different beliefs, you might think they are ignorant, crazy, or malicious. This in turn may lead you to feel frustrated, resist the synchronization and instead consider that they view issues from a different perspective or that they are acting upon what they think is the best available information. Chances are far better that they mean to help but aren't great at communicating than that they're actually ignorant, crazy, or malicious. Assuming your partner has malicious intentions, stifles your conversation. It holds cooperation and undermines the possibility of using the conversation to arrive at truth. If you must make an assumption about your partner's intentions, make only one. Their intentions are better than you think. Technique number eight, walk away. Know when to walk away, even when the conversation is going well. Your open listening, authentic, and curious questions may naturally lead your conversation partner to start doubting their opinion. As this doubt starts to build, the person may feel uncomfortable. Feeling this doubt, you may feel the urge to keep pressing your viewpoint but here you should restrain yourself. Putting pressure on your partner to continue a discussion beyond their comfort level shuts down listening, encourages defensiveness, and turns the conversation into a frustrated rehearsal of why one of you is correct and the other is not. People need time to wrestle with doubt, incorporate new information, mull over challenges and different perspectives, and rethink their positions. And so do you. Changing one's mind happens slowly and in a way that suits one's individual psychology and habits. Over time, new beliefs and attitudes integrate with, or entirely replace, existing ones. Even the smallest doubts in your conversation partner may grow into greater realizations with time. Forcing a conversation beyond someone's comfort zone denies you and your partner an opportunity to reflect while placing a strain on the relationship. Politely leaving a conversation when all parties are getting along can be an opportunity for those involved to reflect on issues. Always try to end a discussion on a positive note. Talking with a person who holds different beliefs from you may be a very frustrating experience, yet it doesn't have to be. As Einstein once said, no problem can be solved from the same level of consciousness that created it. If you want to experience a different relationship with people you can't normally stand, waiting for them to change or trying to force them to change will not work. First, we need to change how we approach the conversation ourselves, and if we do, the results may surprise us. If you want to learn more about these techniques or discover advanced techniques, make sure to read How to Have Impossible Conversations. For more practical thinking tips and videos, subscribe to the Practical Thinkers YouTube channel now. This was Rokas, and I wish you to keep thinking practically and critically.