Hey y'all, it's Tammy and Chris. It is lunch time at home and I've got two pots heating up on the stove. We are going to fry up a little bit of chicken, not fried chicken, but just sauteed chicken. And we're gonna, we're going to fry up these potatoes kind of like hash. So I am just chopping them kind of like a hash would be. And I'm just going to fry them in some butter. So I'm just going to throw some olive oil in it. And it's good and hot. And I'm going to put some butter in it as well. But we'll go ahead and throw our potatoes in here. Let it cool them down before I throw the butter in there or it burns the butter. So we're going to throw some butter in here. You guys hear that? I want to put a little bit of spice in here, but first salt and pepper, salt, pepper. And I guess that's all we need in there for now. Now for our chicken. The first thing we're going to do is cut up a little bit of onion and pepper that I went here. So let me turn this down on low. But it's good and hot now, which is a good thing. So all I'm going to do is take this knife and slice up some of this onion to go with our chicken. It's over there popping and fizzing good, ain't it? This is a new rate of knife. It's really nice and sharp. And then we're going to cut up a couple of little peppers. And we're going to throw it in there and let it start sauteing. So I like to cook my chicken really good and seared. You know, I like to sear it. Now in here, I'm going to put a little bit of, I'm going to use the heat of spices. All right. So if y'all have no way, I'm going to reach up here and grab my the heat of seasoning. This is my favorite. The heat of seasoning in the world. Badea. We'll put those in there. Add a hint of butter one on. Make it delicious. Mix that around good. And we're going to let these cook until they're nice and soft. And while we're letting these simmer and our potatoes are getting, you're going to want them to cook a little bit first, you know, get kind of done. I didn't have any rice made. A lot of people probably think, why didn't you make rice to go with this? Because I didn't have time and I'm in a hurry and I'm hungry. All right. So we've got chicken. The only thing is, and I got some fresh cilantro too, will you? The only thing about this chicken, it has these little tag things in it.  These were actual tenders. So I'm just cutting those out of the tender. Sometimes it's down deeper. How's that looking over there, Chris? Good. Get ready. I'll turn those potatoes up and get them nice and brown once we start browning the chicken. And this is all we're having for lunch. All right. We're just going to turn these a little bit. Those look good. They look good. They smell good. I'll stir these up a little bit. While we're waiting on that, I'm going to chop up a little cilantro rinsed off and chop it up a little, get it ready to use. And then we'll also chop up an avocado. Looking good. Looking good. Looking good. This is not too. Well, I'm afraid it'll get too brown if I leave it in here while I'm trying to get that chicken dark, because I like my chicken to get nice and kind of brown on the outside. So we'll go ahead and plate this on here for now and add the chicken. I like turning it all the way up on high. Get it good and hot and then add my chicken. Well, those potatoes are cooking up nice in this cookware. Looking good. Looking good. I just look in a butter in there to make them taste good. Let's see if this is going to be. This one seems pretty dog on soft and may turn out to be too soft. So we'll see how it looks. Of course, I'm going to, I barely cut it in half, so I think I want to come out. And we're going to slice our avocado with this in just a minute. Just a little bit of this. Is that the, the heat of? Yes. Did y'all get a good look at that? Well, if you didn't, here it is. Yeah, that's it. Or they should have been. Now, you could put some in here if you want to. Or you can just let them be cheap potatoes like what we're doing. Can see potatoes. All right. I got this up on high. It's going to take it a couple of minutes. So we're going to turn on this for three minutes. And then I'm going to flip it over and probably cook it for a couple of minutes. Add a little more oil to it. And a lot of y'all have a hard time. And want to know how to cook for two people. And really, you just grab it and go. Just like I did today. You know, you just don't cook in large quantities. And you don't have to cook in large quantities. You don't have to use per se a recipe. Just grab your meat. Throw in a pan. Throw some good stuff on it. Grab your vegetables. Throw them in a pan. Put something on them. And cut up as many as you think will be observing. So that's all there is to it. And that's how I cook most of the time, instead of by using recipes. I just cook whatever I've got in the quantities that I know will seem to be increased. And that's how we do it. How I do it. How we do it. To it. We should need to on Kato. No. Sometimes for lunch, I'll take an avocado when Chris is fishing. And I cut it up. I'll put salt pepper on it. And then I'll put a little bit of, I used the sodobe seasoning the other day. And it's a spice blend from that gourmet collection that I find at Marshall's or off to wherever you're at. And I put a little bit on it and I have it for lunch with a few chips. And that's all I have for lunch. But it's still pretty healthy. Better than eating just the chips. So Cheetos for lunch, right? Yeah. Better than Cheetos. Sure. Y'all haven't seen the way they're nice that we're using. They're really nice and very affordable. So you should check them out on the website. Just go to shop now and you'll find them. Will you hand me those little tongs over there for us? Okay. Thank you. About 16 seconds left. So I think I'm safe to turn these. Now at this point, if you want to go three more minutes, you can. In that last minute, I'm going to add just a little bit of butter. I am going to go ahead. We don't need to turn these. You just let me get them brown, aren't you, Chris? It's getting all up. Perfect. Now see how good these are doing in this stainless steel? You just got to make sure you got some oil in it and it'll do fine. And this little one is going to be done quicker. But hey, we're going to go ahead and add these peppers back in. Because we got this side as done as we need to. This is my home-bought cooking. And while it's just got about a minute left, we're going to put a little butter in it. Yes, we could do without the butter, but hey, a lot of y'all eat out all the time and y'all get plenty of butter. Trust me.  Good fat. Not good fat, but taste good fat. Tasty fat. Tasty fat. Don't that look good, y'all? Now see how simple that was?  I said, baby, you ready to eat? And he said, yeah. I said, well, come on in here. Make us something. And quick, easy lunch. All right, let's put some, uh, I'm going to turn that off. You're going to put a little cheese on it. We just turned off that chicken. We got to get it out where we get to done. We'll let that melt. Put a lid on it. Let it melt. And then we're going to get chicken out, or it's going to get to done quickly. Make sure this is good. All right. The thing that we're not doing that a lot of you would do if you were in a restaurant is having the bread or having the, um, you know, the burrito shell or the chips and the salsa. And of course you're not going to get cheesy potatoes there, but you know, we wouldn't be from the South if we didn't make some cheesy potatoes. And that's what it's all about. What it's all about here at the nickel's house. Good old, now that's a good lunch. I guarantee it. Don't y'all want to buy it? And, um, it works really good. I'll give you that one, Chris. The extra potatoes real quick. All right, let's get our good old avocado. Now this is a really soft one. So it may not be the prettiest in the world to slice. Let's just put your little cilantro on the plate, but really our plate looks good already. There's cheesy potatoes. I didn't put a lot of salt on it because the fajita seasoning is not real salty. It's so good. You sprinkle a little salt and pepper on your avocado to make it even better. I think I'll do that right before we eat. I hope y'all enjoyed watching Colored Belly Cooks, where we cook like mama did. Bye y'all. Love ya.


U.S. Senate leader Ukraine runs out of more soldiers ammunition because of lack of American funding. Ukraine is failing on the battlefield because of a lack of American funding. Senate Majority Leader Chuck Schumer said he demanded the House Republicans pass the $61 billion aid bill as soon as possible. Schumer, a New York Democrat, brought up Ukraine in a speech about the upcoming U.S. legislative agenda. The Democratic Majority Senate passed the funding proposal in mid-February, but the Republican Majority House has not voted on it yet. The situation in Ukraine is desperate, Schumer said, claiming that the funding bill has been collecting dust for 55 days while our friends in Ukraine fight and die on the battlefield with no support. With each passing day, Ukraine is running out of more soldiers, ammunition and hope, he added. Let's be blunt. The reason Ukraine is losing the war is because the hard right in the Congress has paralyzed the U.S. from acting. That's it. That's the reason, Schumer said. By passing the bill, House Speaker Mike Johnson would do the right thing for Ukraine, for America and for democracy. The New York Democrat argued. Otherwise, he claimed the Republicans would hand a large victory to Russian President Vladimir Putin. Schumer made much the same argument while visiting Ukraine in late February, claiming that if Kiev gets the money, they will win the war and beat Russia. Democrats have sought to split the GOP into moderates and MAGA Republicans, referring to former U.S. President Donald Trump's campaign slogan Make America Great Again to get support for their legislative agenda in both chambers of Congress. The tactic has paid off in the Senate, where 22 Republicans voted in favor of funding Ukraine more than offsetting three Democrats who were opposed. Ukrainian fighters have targeted a large number of military equipment in the territory of the gas station used as a Russian Army's warehouse in the occupied Bolshe-Kopani village of Kursin region. The military equipment detected by the shark drone were hit by the Himmars missile launcher and destroyed. Pol-21 radio-electronic warfare system is among the destroyed equipment. As the warehouse was being used as a repair base, the equipment under repair were blown up as a result of the strike. The number of casualties among personnel is not reported.


While hunters registered almost 69,000 deer during the first weekend of the firearm season, the 68,958 deer registered during the first two days of the season is about 3% less than a year ago. DNR officials say these are preliminary numbers and it is too soon to say if the unusual weather had any impact on the harvest. The DNR is projecting the total 2016 deer harvest to be between 165 and 185,000 deer. The 2015 total harvest, as a reference point, was a little more than 159,000 deer. If you've enjoyed this segment of Lakeland News, please consider making a tax-deductible contribution to Lakeland Public Television.


Ladies and gentlemen, we now have a panel discussion. The topic of the discussion will be realizing the Army's vision for the future. The panel will be moderated by Mr. Patrick Tucker from Defense One. Please welcome our panel members, Professor Dieter Fox from NVIDIA, Professor Marshall Ebert, CMU, Lieutenant Colonel Christopher Lawrence, Army AI Task Force, Dr. Tony Stence, Uber ATG, Dr. Ethan Stump and Dr. Stuart Young, Army Research Laboratory, and Lieutenant Colonel Jay Wishem, next-generation combat vehicle cross-functional team. Please join me in welcoming today's moderator and panel members. Okay, thank you very much. Good to be here. So, a quick question for the audience. I'm Patrick Tucker, Technology Editor at Defense One. Who here knows the etymological origin of the word robot? Raise your hand. Handful. Some Europeans, which does not surprise me. So, it's a Slavic word, and it means slave or service, and it enters modern usage in 1920 with a play by a Czechoslovakian playwright named Karl Kipik. And the play was called Rosem's Universal Robota. And so, the robota were this creation of the play's protagonist, a guy named Rosem, and he's kind of a Silicon Valley Elon Musk type. Act one, Rosem is created an entity that can take over all human labor. He becomes fantastically wealthy. Act two, the robots have destroyed the economy and created, you know, civil strife. Act three, guess who's killing all the humans, right? So, I bring this up. I think it's important to remember that all of our fears and all of our projections about robots existed decades before anyone actually tried to make them, right? So that hope and that fear, it's just sort of ingrained in the concept, but it's always projection. It's always just a notion that we have and that we have to conquer. And that's why I'm so glad to have this panel here today to help us talk about the actual difficult work of making things that can relieve the labor that particularly we're putting on soldiers is an incredibly dangerous and difficult line of work. And so with that, I would like to, let me start first with, do we have Lieutenant Colonel Lawrence here? It's you. Hi. Hi. So, when you look at this, all of these things here from an operator perspective, I wonder if you can, to start us off, tell us a little bit about what you want Battlefield Robotics to give you in the next five to ten years. And then we'll get into the difficult technical challenges of meeting those expectations. Jay, that might be your up your alley if you want to go take this one first. Yes, we're Jay. Please go ahead. But I'm welcome to chime in if you want. Why don't you take a stab at it real quick and I can help. Okay, sure. So, Battlefield Robots, essentially we need them to, you know, be an augmentation system for the soldier. We need them to give us enhanced situational awareness. These are some of the things that we've seen. So they need to have some context understanding, be integrated as part of a team. And then ultimately, you know, like I said, feed that situation awareness back to the soldier, the operator, right? So that you get this increased standoff capability, increased survivability for the soldier. These are all really good, I mean, overall themes that robots can offer in terms of capabilities, right? So reduced cognitive load for the soldier, especially if you have a good human soldier interface where you have one soldier that's interfacing with many robotic systems. But again, going back to the situational awareness of what is, what are they perceiving, right, in the battle space? And then with that, obviously that's inherently going to improve the survivability of the soldier, right? Yeah. To be able to apply facts much more rapidly. Yeah, I think that's well said, Chris. So fundamentally, kind of the way I look at it, I'm very much like Kevin at. A lot of the CFT members, we, you know, the Army made a decision not to put technical experts or acquisition experts in the CFTs. We're all war fighters. So we take a very different sort of approach. I kind of boil it down to a very simple set of things. As a commander, I would like to be able to choose when I make first contact with an enemy force with a human or not. I would like to have that flexibility and option so that I can elect to put an unmanned system, whether that's a UAV or an unmanned ground system out, an unmanned system out to make that initial contact, you know, that very tip of the spearpiece. I want to be able to elect to do that on my terms. And that's where robotic systems really give you some interesting capability. What it also allows you to do, and Chris kind of alluded to this, I start changing what the decision, you know, processes for the human. How can I make better informed decisions faster than my competitor or my opponent? And again, a lot of this comes down to how do I make these choices in such a way that I'm preserving my capability and my forces and then putting the enemy in a very bad position. That doesn't always mean killing them necessarily, although, you know, the plight of the reality is solve more problems in human history than pretty much any other method, rightly or wrongly. But it's that ability to make those decisions faster and then choose when I make human contact. And sometimes you want to make contact with a human first because of the complexity of the task. Perhaps it's going to be in a very populated environment and there's going to be a sifting and a sorting of who the threats are and what's going on. Or perhaps your intent is not to have a kinetic engagement. I mean, there's reasons that you want to do that. But if you believe contact with an enemy threat is imminent, I would really like to have a robotic system out there that is increasing my situational awareness and then allowing me to make a choice on what is the most efficient way to deal with that threat that puts my soldiers in the best position to win. Okay, so this gets at a lot of what we're talking about here today and a lot of what you see around this room because core to that hope in terms of what you can have that system do is perceiving, right? I mean, that thing has to make a lot of decisions about what it is experiencing and then relay that information to you in a way that's helpful. And this gets at the problem of robotic perception. I don't know who is familiar with Rodney Brooks, the roboticist that invented is behind iRobot. But he said he has this very good quote about the evolution of robotic perception. He said, you know, in the 1970s, we thought that teaching a robot to see was going to be a summer program for a master's student. And now it's taken us, well, we're not done yet. We're not done yet. So let me turn to Marshall, Tony, and Dieter. What does machine vision look like in the next five years and in the next 20? I guess I'm the victim in the middle here. Yeah, I think where we really have to go is towards a much more robust, general, semantic understanding also of the physical world around the robot so that they can, for example, interact with it, as Sid also mentioned. So just to put this in contrast, we humans have this really good understanding, intuitive understanding of the world around us and how things work. For example, nobody here would be surprised if I opened my hand and suddenly the microphone would fall down, even though you might not have the exact equations in your mind but you know how this works. Or for example, I might not be able to tell you exactly how many ounces that microphone weighs or how long it is, but at the same time, I can interact with it in a very robust way. And if I bump on it, nobody is surprised that it makes that sound. So we have very good understanding and predictive models of the world that we then can use in order to interact with it. And in robotics, currently there's still a lot of focus on having exact models, kind of really physical understanding of these models, but the problem with these physical models is that it's very difficult to train them, to adapt them and to teach them. So I think what we need much more is how can we teach perception system over time and train them in the real world through experience. And as an extension of the real world, of course, I think one thing that's going to be important is how can we train them also in simulation just because it's very hard and very expensive to train robots only in the real world. So let me mention another aspect of perception in the future, which is this idea of embedded perception, let's call it that. And this is the following idea. I showed this example of semantic perception, semantic segmentation, which is a critical tool in autonomy. And the idea there is to basically label pixels in images as to the object that this pixel represents. And the way that works is that you have typically a neural network type of structure that tries to do as well as it can in labeling every pixel in the image correctly. So that sounds great. Now, can you think of any application in the world where you actually want to label every pixel correctly? It simply does not exist. One of the major limitations that we have now is that the perception systems that we have now solve the problem that is... solve the wrong problem in a way. I don't need to label every pixel correctly. I need to understand the scene only at a level that is sufficient for me to carry out the task. And worse than that, it is solving a problem that is harder than the problem that we need to actually solve. So what we need to do is to have a perception system that are actually developed and trained in the context of a task. A task-driven perception. That is something that we don't know how to do at the moment. We know how to do this for simple tasks, like grasping, for example. We don't know how to do this for more complex tasks like navigation. Yeah, I think in order for perception to be truly robust, we're going to need an immense amount of data and has been pointed out the labeling problem is a real bottleneck for that. And so I think we're going to need to leverage approaches that automatically label, automatically train models. For instance, we can have a human driver that can drive a vehicle around and the system can determine that, well, it's a road that they're driving on and when they, for instance, steal around something, it must be an obstacle. And so get very efficient classifications in that way. Additionally, the robots can go out themselves in the environment and encounter it and automatically understand the environment that way. So for instance, if a robot sees something in the environment and then drives over it, it gets much more information about what was actually there and can use that to automatically classify the data. So this labeling problem is a big one and if you look at some of the posters in the wall, they remind me that one of the big potential solutions to that that's emerged over the last couple of years is deep learning. Because in a deep learning system, the system finds its own labels to things in a way that conforms somewhat to your predictions, hopefully. So why, Marshall, I'm going to stick with you for a second. Why hasn't deep learning solved this perception problem? What's needed for it to solve this perception problem? Well, I guess three things. First of all, deep learning has been very successful precisely because of the availability of data and the availability of labeled data. So step one is, as Tony said and as we said earlier, is the ability to train the system with minimal amount of data and minimal supervision. And that is critical and in fact this is something that completely separates autonomy from other applications of computer vision where you have the luxury to train those networks offline with massive amount of data and so forth. The second thing is going back in time, going back to the time of the Rodney Brooks quote that you had here and to be able to incorporate more structural knowledge, additional knowledge in those systems. We went very far into the realm of pure data driven learning pure black box system that basically learned from scratch from data. That has gone very far, but we need to now step back and see how we can use additional knowledge, common sense knowledge, geometric information, knowledge about the task, et cetera to add to this data driven black box learning. The black box thing is an important one because neural networks, they've actually been around for like a little bit. In the 90s there was some very interesting work with neural networks and it was before there was like really big data sets and before there was really great compute, but they were around. I talked to a couple of folks that were trying to create an application for them in the public space, particularly like predictive policing actually here in Pittsburgh. A lot of people don't know about this. There was a neural network sort of like project that was based on predictive policing here in Pittsburgh and it worked great, but city planners and municipal folks were like, I don't know if I can want to adopt this because I don't understand how it made its decision. I don't know how it reached its decision because that opaqueness that was neural networking in the 90s and to a certain extent today was like a huge inhibitor. So I want to ask our operators here if you don't fully understand how a system arrived at the decision that it made, how much do you trust it? And I want to go back to you, Marshall, or anyone that feels like they have something to contribute to that. Is the black box opaqueness, is that still the best descriptor of the way neural networks are operating now or have they evolved from that? So first to the operators, how much do you trust a neural network that you can't understand its reasoning, but you do see the output being correct? You know, I think it comes down to the soldier interacting with the system early on in the process and start practicing kind of the, you know, the teaming concepts that we're working on with man and unmanned teaming. How do they see themselves interacting with the system so they can build trust over time and understand the system's limitations, what its capabilities are and what it can offer? And I think over time taking that feedback too, you know, from the soldiers saying that, you know what, this is throwing too many alerts. I don't trust it. These aren't accurate. And then we might have to go back to the drawing board a little bit. But I think, you know, there's systems we feel that before that, you know, maybe somewhat as a black box to the soldiers. So, I mean, we have means of, you know, testing and evaluation and through modeling and simulation. And we've already just discussed recently about putting some bounds and some of the, you know, around the systems in terms of interjecting, maybe physics models. You know, bounding necessarily what it can do in terms of its response. So I think there's means for us to tackle this problem. Yeah. I think I would say it kind of depends on what the task is. So I would just offer that everybody in this room, you put in an order in an amount of trust in anything, but from the time you woke up to the time you got here in something you can't actually describe how it works either. Unless you've like torn apart a car on your own and put it back together, you probably in theory understand how our car is working, particularly a new model car with the digital systems in it. But you probably really don't know. Like you have an idea. And by the way, most humans are not as explainable as we might like to think either if you've ever asked your 10-year-old why he did a thing. So from that aspect, a simple task, I think it comes down to demonstrative effect. Like if you demonstrate a soldier that the tool system that you're trying to enable them with generally functions relatively well and adds some capability to them, some minimal level of capability, they will grow trust very, very rapidly. And in more complex activities where you're trying to assist into actual decision support tools or higher cognitive processes where like multiple complex systems together, that is going to be a little bit more difficult. But again, these are all uniquely human activities that you chain series of humans together and then you have these arbitrators that are again humans that are doing this. These are not always explainable either. So I think that you can gain trust, but a lot is going to come down to what you demonstrate as a capability. Yeah, I mean I would echo that. You trust something because it works, not because you understand it. And so the way you show it works is you run many, many, many tests and you do a statistical analysis and build confidence that way. And that's true not only of machine learn systems but other systems as well if they're sufficiently complex. You're not going to prove them correct. You're going to similarly need to put them through a battery of tests and convince yourself that they meet the bar, that they work. Yeah, I fully agree also on the notion of maybe we have to let go a little bit of this idea that we can prove everything that it's correct and perfectly predict its behavior just because the kind of data that we're dealing with is so complicated, right, that you just can't explain it anymore at every level of detail. And as you pointed out, we humans aren't that great at explaining why we're coming to our decisions. Also we're not that good at that. And also I think in the machine learning community there's of course a lot of work going on exactly in this direction where techniques are being developed, how you can kind of introspect these deep networks and kind of see why the network might come up with a certain recognition solution or something like that. You can look at which part of an image led it to recognize a cow and things like that. And also other areas are going into how we can put this physical structure, specific constraints into these networks so that they learn within the confines of what we think should be physically okay. So I think it's a lot of good progress going on. Okay. So let me turn to you, Stuart, Chris and Gary. The army modernization priorities, we just heard a little bit about those. And there is a bit of a gap between what commercial industry is working on and what the army needs. So how are the army modernization priorities shaping right now AI research that you're doing? So the biggest thing for me is they're enabling us to focus our research efforts on capabilities that the army actually needs. And it also is an opportunity for us to inform them as they develop requirements, we can help them understand what's technically achievable and what's a little bit further out. And we use this interaction to help us focus our research and discover new things that are of interest to them. So instead of looking at, you know, a wider swath of problems which we're always inquiring about, it allows us to understand and get deep into the area of interest that they have. So the engagement with the NGCV, which is why I invited them to be here in such force, is because they've really adopted what we're trying to do in the CTA and they are really helping us understand how they would actually use them in the field. And that's informing the research experiments that we have to do to meet the requirements that they have. So on this for Tony with Uber, what is Uber doing right now in this space and what are the lessons that you learn from the RCTA that are applicable to your work in Uber and what can the military learn by partnering right now with Uber? Because I know that's something that they're talking about. Sure, yeah. So I would say there's a lot of technology that's in use not only at Uber but in the self-driving car industry that traces back to programs like CTA. Technologies that were either innovated on the program or matured or extended in some way or another. For starters, a lot of the perception technology, for instance, detecting people, detecting pedestrians is very important. The multimodal approaches and machine learning approaches have been very successful for doing that. Other perception problems as well, I think there's a big deal made about the fact that self-driving cars use maps, and they do, but there's an awful lot of stuff you encounter in addition to moving actors like cars, pedestrians, and bikes that aren't in a map, things like a construction site. And so when the car encounters that, it needs to have some understanding of what that is. So a lot of the semantic perception techniques have been applicable there. Prediction is a very important component for a self-driving car. The faster a car goes, the more time it needs to stop, and therefore the more important it is to predict what other actors in the environment are going to do for the next several seconds. And so the same sort of machine learning techniques that are used in perception are also used there. Planning is another area. Routing technologies for getting a car from A to B sourced in these programs. A lot of local navigation approaches where the car has to follow a lane, stay behind another car, or drive around something. Uber uses sampling-based planning approaches where in faster than real time we investigate different sequences of steering throttle and braking commands and evaluate them and pick something to execute. So that's been very successful. And then other technologies as well. It's very important that the car knows where it is at all times. And so a lot of the technology that was developed in the extensive slam research has made its way into that application in self-driving. Are you guys working on something that's GPS-independent because that's a big issue for Army right now? Sure, yeah. GPS, for instance, driving around in cities with tall buildings, GPS is not particularly good at localizing. So we've had to adopt multimodal approaches for getting the position of the car. Okay. Go ahead. Are you now exploring like a crater with the Army? Is that a thing? Certainly have looked into possibilities of teaming and we'll look into more in the future. Okay. So Ethan, what are the autonomy challenges that ARC has to focus on that industry won't focus on? And also sorry for neglecting you for so long. So I think probably the number one task, and it's kind of mentioned before, is that I think we have to deal with unstructured environments and there are certainly aspects where industry is interested in this, moving off-road terrain. I think the adversarial aspect is the key thing that the Army needs to face, that industry might not necessarily face. And I think it's adversarial not in that we're worried about people trying to goof with the cars or mess with them.  And we need to think about how we do this in a way that we have as little sort of preconceptions as possible. We don't need to, we need to be able to adapt if the plans change. We need to be able to do this without a lot of lead time and be able to adapt as the environment changes and the mission changes. So I just came from AUSA, it was a wonderful big show. General Murray was there, the head of Army Futures Command, and as well as a wonderful assortment of new semi, or not very much so, but robotic tank prototypes for the next generation combat vehicle. Industry's getting out there and making this stuff that looks like it just rolled off the set from like Mad Max. But if you talk to General Murray, one of his biggest concerns, and I think this relates a little bit to the work you're doing, but also to the future applications of these in a battlefield environment, Murray's big concern was the network. And whether or not the comms are sufficient to create a situation where an aerial drone, a ground drone, a semi-autonomous self-driving vehicle can all share enough data with both the soldier and with the folks in back of the soldier to allow for a successful operation. So that bandwidth is a huge concern of his. I wonder if you can talk a little bit about how big for the operators, how big a concern that is for you. And also, what role does intercommunication among systems play in your work? I know we're talking about autonomous systems that are supposed to be thinking on their own, but until we get there, there's going to be a lot of need for the transfer of video imagery and other imagery and just a big network that the Army wants to have behind these things. So talk a little bit about the challenge there. Sure. So I think ultimately we got to get, as you pointed out, it's a great question. We know we're going to be operating in a contested environment, and we can't necessarily rely on our networks to transmit high bandwidth data, like full motion video. So we need to continue to push and develop our autonomy ability for these systems to, if you will, get minimal guidance initially, and then they move out, right? But then, of course, we know they need to adapt, and that's going to be one of the biggest challenges of these systems. Yes, we're pretty good now in terms of adding waypoints to these systems and then autonomously navigating out to those waypoints, avoiding obstacles, and so on, and reporting back. But in terms of situational awareness, how do they add value to the team, and as Jay talked about, an increased standoff capability is making contact first. They essentially need to report back a salute report, right? A situation, what's the size, what kind of class of the threat am I looking at, what's my confidence level, and then maybe a chip image, right? A still image of that being relayed back to the operator in a common operational interface where you're getting this common picture, right, of that. And then also, obviously, we need a geolocation, right, at that threat as well. But we need to continue to push that in terms of research that we can't necessarily rely on and we want to move away from the tele-operation full-motion video. Sure, there might be times where we could have that option. We want the full-motion video, or we want another still image, and we can do that at times, but we need those systems to be able to function without it. I can augment this a little bit and talk about sort of what the current thoughts are about how network interfaces with autonomy and the use of robotics and autonomous systems. You know, now the military operates drones and we basically assume a full-motion video link, and we know that that's not a reality. We don't have the bandwidth for that. But what we're doing is we're thinking about how the networking actually becomes just another aspect of the entire planning process. You know, you don't need a lot of bandwidth if all you need to do is just, you know, recognize one small thing in the image. Most of the image is actually useless for that. So that task actually doesn't require as much bandwidth. And if you're going to be using, if you're going to be doing a control task that requires a lot of sort of high-speed, reactive control, and you want human space to be doing the tele-operation, you need more bandwidth. But if it's something that's low tempo, low impact, you know, you can get away with less bandwidth. And so there's a relationship between the perception and the control and the networking required. And so you actually want to live in the space where you're balancing those three things. It points to understanding better what the task is and tailoring the communications to the task. And then we can also look further and think about how the networking is not just a passive thing that we actually are actively shaping the network that we have. I mean, the Army already does this. You know, every unit from Italian and above, they're going to be equipped with a retrans station. So, you know, they'll put their operating posts down and the radio is going to reach so far. But you have a unit that can go out and put up an extra set of antennas so they can retransmit the comms out to get a wider field across the battlefield. You know, we can use robots for that task. And moreover, we can have the robots be able to adapt to how the network situation is changing and replace and move around the retransmit capability based on what the needs of the mission are and how the mission is evolving. Yeah, I was going to basically build on the point that Ethan was just talking about. So, ARL, you know, we've looked at this autonomy and networking communications as a dual that the autonomy enables the communications through the active approaches that Ethan was talking about. And some of our recent visits out to NTC, they actually highlighted this. The bedan commander was like, hey, I'd love to have a robot just be able to do the retrans for me so that they can have the connections. So I think, you know, this network, and then of course, as was already alluded to, you know, the autonomy is what I believe is so critically important to reducing the requirements on the network, which we know will be huge and contested. So the more we can do on autonomous fashion, the more, you know, it frees up, gives us maneuver space in that network domain. I think there's one thing that you don't want to lose sight of, and it sounds very simple, but the things that you desire to network in terms of these capabilities and this information you want to push to the, you know, the three or four, let's say three to like 12 human beings that are onboard that platform, and that's your raw numbers, right? You know, between the crew and then like a dismounted squad in the back, for example. The information is critically valuable to them, whether or not that information is going off-board that platform to some other node, for example. So a lot of these things that you want to apply some autonomy to or different functions and applications to, to the human beings onboard that individual platform, even if that node becomes isolated from the network, can be critical to them. So we talked a lot about, you know, the computer vision problems that we have in the frame of artificial intelligence assisted target destination recognition and tracking, one of my favorite acronyms ever, by the way. That is critically important to those human souls on that platform. The information you would get then one of you would want to pipe out to some other thing to provide this sort of collective understanding. That's critical, too. But I think that's something that we lose sight of is that there's a minimal threshold for capability that we can strive to. We all seek to go into this high, you know, XY axis of granular solution and complex environment. You all want to go in the top right because that's your bent from a research standpoint. That's really the goal. But at a real visceral level, there's real minimal capability that you can draw in. So network is a critical concern, absolutely. But if I can solve some problems for that crew, that's better than what I have now. And now, as I have more assurance in my network, as I build a more robust network, or we become much more dynamic in how we treat the network, generally through some very innovative AI applications that I've seen recently in terms of how you monitor a network and then flow information to the right actor at the right time, I'm just gaining more capability over time. But don't lose sight of the fact that network is a challenge, but to the crew on that vehicle, what you can enable them with, that's life and death. So think about questions that you might have because very shortly we're going to go to the audience for questions. And if you start coming up with them, raise your hand and I'll find you. And I think that that would be good. But I want to bring up something that you mentioned a moment ago, Ethan, which was this challenge of dealing with all of this stuff in an adversarial environment, in a data adversarial environment. Who here knows what again is a generative adversarial network? So you're all like roboticists, so you do. Okay, that's great. For folks at home that are watching, it's basically a means to, it can be a means to spoof a data set, right? You can use it to test whether or not a data set is true, or you can also use it potentially to inject faulty data into a data set and create a impression that is wrong. The NGA right now, the National Geospatial Agency, is very worried about some research that's come out of China that applies that methodology to spoofing pictures of the world, right? Spoofing like geolocation data, like satellite photos of this place or that place. And so they're trying to get ahead of that. But I wonder if you can talk a little bit about, first to Tony, the data integrity problem at Uber in the commercial space. And how lessons there, but also the data integrity problem in the military space, particularly with this, you know, this issue of autonomy in these environments. And that's kind of the second part of that is kind of a jump ball to whoever feels like they can tackle it. I guess there are lots of different types of data at Uber, but one of the most important kind is to get representative data of all of the situations that the car can encounter as it drives around. So that you come up with models that properly represent everything that the car will see and are not over-fitting to any particular situation so that you would do very well in some areas, but then struggle in others. So certainly data integrity in that sense is very important. Yeah. Like adversarial attacks on data integrity. How is the military looking at that? What are your worries and what are your solutions? Particularly in terms of autonomy and vision. Yes, it's clearly a really important question, right? We see this one more fake, even generated videos out there that when you look at them, you feel like, yeah, that's correct, and it was someone who was actually saying some very different words and you just can generate videos that look like someone said, something that they actually didn't say. I think one way forward will also be just to make sure that the data that is being collected that we can keep track of that data. Like what is the origin of an image? When was it taken? How was it taken? And then make sure as it goes through these different computing processes that we keep track that nobody really tempered around with them. And then at the same time, yeah, you mentioned GAN and a part of GAN is of course the discriminator network which is the job of that network is exactly to distinguish between fake generated or artificially generated data and real data and maybe these kind of lines of discriminators trained specifically for these purposes can help with that problem as well. If I can just chime in here for, I think the military, we've already had the structures in place in terms of protecting our data and I think as long as we know that we have to have the proper security measures for this data because if we want to protect against things like GANs, you have to have either the data of the neural network because you have to have the adversarial network pitted against the discriminator in order to basically find the vulnerability. So hence you have to protect your network and you have to protect your data. So this is something of interest obviously to the military but I think we have the right structures in place to be able to do that. So we talked a lot about corrupting the data set basically but there's another way to adversarily corrupt the performance of the system which is to generate input that will adversarily change the output of the neural network. This is something that is being studied extensively in the context of face recognition where you can present an image that is not necessarily a fully realistic image but will produce a recognition output that is basically what you, the adversary, want. This is an even more concerning problem because then you don't have to mess around with the data set basically. All you have to do is to figure out how the network operates on input output and be able to fabricate inputs that are not necessarily realistic images necessarily but that will produce the output that you desire. There is extensive research on that topic both on how to extract, how to identify those vulnerabilities and how to build networks that are immune to those vulnerabilities. Just a quick point from an operational perspective we, the chart that Kevin put up earlier where you have the kind of a climb of what a machine would do, what a human would do and then really where's the sweet spot of what you do together. And we view this problem as how do you queue a human to help assist, augment and interrogate an object to kind of make your way through some of those problems. So basically that would be your next generation camouflage if you will. You're not camouflaging necessarily from a visual perspective like you would in traditional military sense but it could be something as simple as pixels or scrambled images on board of a platform that didn't produce a result you don't want from a system. But a human if queued to look at it can assist in the classification of that object and that's what we think is one of the again it's what's the threshold of right and it's probably where's that human in the loop for that teamed activity between a human and a machine. So let's go to the audience for some questions. We've got a little bit of time left and you've got a fantastic panel here to interrogate in terms of your hopes and fears and expectations of the future of military and their adaptation and adoption of machine vision. So shy. Okay. A little bit shy. Oh wait here's one right here. Yes they do. They are coming up right there. Thank you. Good afternoon. And please name your affiliation as well. Yes I'm Lieutenant Colonel Matt Cooke. I'm with Future Vertical Lift. I'm the modular open systems approach lead. So although our problems are very similar on the battlefield the solutions may be very unique. One of the things with MOSA that we all have to do for open systems architecture is make sure that we're standardized so that we can get the rapid adaptability and the high systematic reuse of our components for hardware and software. That being said a lot of people here working on unique solutions. Are you guys using as essentially a baseline Stain Ag 4586, Air Inc. How will the U.S. military be able to pull these from 6162 all the way into S&T and then beyond that into production? So I can speak what our plans are for this program specifically. So the pathway that we've outlined is we've been working with the Ground Vehicle System Center. So these are the folks that own the Ground Vehicle Robotics portfolio for the Army. And what they've done is they have put forth that the Army should standardize for ground vehicles around Ross. Everybody has at least heard of Ross. They should be familiar with it. And they put forward what they call RossM, which is Ross Military, to kind of play off the idea of Ross Industrial, where a large industrial consortium got together and said what are the important aspects of Ross that we need to focus on? What are the capabilities we need? And how do we sort of decide which packages and standardize the interfaces between them? So RossM is an effort to essentially build a business model which is around modular systems. So GVSC has... I think of it as a reference architecture which essentially says here are the basic building blocks of a system and this is built upon a host of programs they've done over the last decade starting... Stewart showed the chart of the flow of things. These things are all interconnected when it comes to ground vehicle robotics. So we're actually in the process of moving things from our CTA into that and we're proposing modifications to that because that's very much architecture that was kind of what came out of our CTA-1 and we've learned a lot of things since then. We have this entire intelligence architecture which goes from language and reasoning and ties in perception and so we see those as things that are augmenting that and so we're going to be folding those into the reference architecture that then for ground vehicle robotics at least we'll be able to build upon. So that's the strategy that we see forward as a path forward for ourselves. Right there. Yep, so Shad Reese, senior EOD robotics technologist out of Indian Maryland, question on standards. Does industry have standards and are they bleeding over into the DOD for us to leverage for testing evaluation of all these technologies? So we're tracking on the same topic. Yeah, I guess we certainly have standards but I would say that we're using our own architecture that is proprietary and home grown. I'll just for the camera. What are the names of the standards? So there's been work with what was called JAWS and I don't really know where that stands. A lot of the work that we're showing here in the CTA is pretty foundational and related to your question on the testing side of things, and the systems become more and more autonomous. There's I wouldn't say debate but there's definitely recognition of the need for doing a better job of understanding how can we test these systems. So we're working with ATEC and the different testing agencies to help inform them on how you might go about doing these even to the point where what is the definition of developmental testing and operational testing even mean when you're talking about systems that continuously evolve and learn and reason. So as was discussed earlier in the understanding of, you know, is a system good or trust, you know, there's approaches that we worked on in the autonomy COI across DoD to basically say, hey look we're going to have to think about this problem differently.  But they demonstrate certain capabilities and they demonstrate enough for you to convince the DMV to give you a license. So it's not a guarantee but it is evidence-based so you have some level of trust and so Tony was talking about this earlier. So I think we're going to have to get past perfect testing of every edge corner case because that's a fallacy and time and on earth doesn't permit that. So we're going to have to think about that related to the standards. This is something that Chris and I are talking a lot about trying to figure out how can we do this across the AI Task Force and ARL and informing this ROS and GVSC community to standardize on these interfaces, what are the right interfaces so that we can get maximum effort. So the RCT is doing a lot to help inform what they should look like and then we'll push that along to our colleagues in GVSC and working with the AI Task Force to solidify what those standards need to be so that we can get maximum industry involvement in that. Yeah, I guess one additional thing is there are a lot of conceptual and architectural similarities between systems but there's not a formalized standard in what I think you mean by that. So the industry has standards but they're not standard. Exactly. I think one of the problems is that many of the algorithms we're talking about here like the perceptual algorithms and these kind of pieces are really still at the research stage and as we know research groups, they use whichever tool is out there that they can build on top that's kind of the easiest first step and ROS is one of the ones that's being used a lot sometimes people just because there's no great alternative out there, for example in NVIDIA is developing an SDK that's more geared toward high speed GPU computation but again I think at this point there's nothing that the whole community fully agrees upon. In an attempt to make this discussion even more depressing I want to point out that standards are not only about software and software architectures, they're also about data and how to represent data, how to share data, how to transfer data how to reuse it, issues of privacy in particular in the self-driving area. The government I believe needs to issue more information a couple of months ago exactly on that topic so the fact that there is an RFI from NIST is evidence that this is still a very much a work in progress. Yeah, you don't request information on something that already exists so other questions? Must be one or two, I think I saw a hand okay well I guess did you have a follow-up? Yeah okay So this question is on I guess the interactive piece it's not clear to me where we are in regards to the algorithms generating a model and then the machine learning system knowing what's missing in that model and then tell the human what's missing in the model so you know kind of have these systems figure out what they're missing and kind of ask for more information from the human so I don't know if there's research being done in that area or someone can elaborate on that. Yeah that is a huge problem in fact this is related to a deeper mention in introspection that is the system being able to understand its own performance and understand what is missing in its performance and there is research in this area that is launched a program called competency aware system part of it is I mean the system means self-aware of its own performance and if you will its own weaknesses so that's a step in that direction Yeah it's a difficult one because humans are terribly bad at understanding the information that they're missing so right back there yes Larry Matthews JPL I've been a participant in the program and it's an opportunity to ask Tony since robots for the military ultimately need to work day and night all weather all season if you can give us any insight into where industry is at with that range of environmental conditions. Yeah it's certainly very difficult to field a self-driving car that can operate in all weather all conditions and so really the approach we're taking is since we control the vehicles in other words it would be part of our network we can selectively deploy them so we can start by deploying them only in good weather conditions only in easy environments and work our way up from there that allows us to realize the potential of self-driving vehicles long before we solve the entire problem but the entire problem itself meaning a self-driving car that can operate anywhere anytime all conditions is a ways off Can you specify or elaborate a little bit on a ways off could you like 10, 20 years I know people hate to do that and yet that is useful Yeah it's very difficult to estimate that and to solve the entire problem I would say is years off and I've made estimates in the past and I've been wrong so perhaps I shouldn't make another guess What was the wrong one? Which one? Other questions? I think we've got time for maybe one more so if you were holding back but oh we do we have one here in front thank you wait for the mic please and affiliation helpful thank you Don Rego from the Army I have a comment hopefully lead to a question about the issue of trust and black box mentality I've looked at the outputs of these things for as long as anybody and the thing that's most troubling about them is they change their mind from frame to frame ostensibly you've got the same image and from frame to frame the output will differ and I think the thing that's going to erode confidence the most is something that appears to change its mind in a indiscriminate way so is there something we can do in algorithm development to make those outputs more consistent? Yeah I fully agree that that's of course a big problem right if you see a video for example or something like that and it just doesn't know what the previous frame how it analyzed that anymore or at least seemingly I think on the one hand side yeah we can do a lot of course on how we train these systems so for example one common thread is that we see a lot of what called domain randomization which means we train these systems so that they are robust to changes in lighting conditions and to other kind of changes in the coloring and in the scene and I think as we train them on a higher variety of scenes then they will become more robust and also incorporating techniques that that we've known from the more traditional settings kind of temporal reasoning so that they learn to be more consistent over time I think those kind of things will of course increase the quality of these and in the end yeah we might always face this problem that a human looks at the output and says like that's not intuitive that doesn't make sense to me and that's a general problem we'll have to face yeah. Well folks I by my clock it's been an hour and I want you to please join me in thanking these panelists for their insight today thank you so much.


Also, youths were in the streets of Ibadon, the oil state capital, to peacefully protest against bad governance, public corruption and other forms of maladministration and the worst in state of insecurity across the country. The youth were carrying different placards, banners and calling on the federal government to prefer solutions, battling the country or step aside. Heavy armed security men were also at the venue of the protests and on all major streets and junctions in Ibadon. Some of the youth who spoke to plus TV Africa demanded the government to do more for the people. Nigeria cannot continue to endure poverty. Nigeria cannot continue to endure the condition of unemployment. There's a need for Nigerians to be transformed. A country whereby Nigeria will be able to proud of the country we have. That is the situation. I mean what Nigeria wanted and that is why Nigeria's a house side today. Let me speak for myself. I'm tired. We can't keep keeping us in fear. We are afraid to go out. We are afraid to sleep. We are afraid even to go about our businesses. People are not at peace with their mind. People are not free to do what they want to do. People are not even encouraged even in this country. People don't believe in this country because of the insecurity, because of the inflation, because of the nepotism, because of killings all around.


So you see what we did last time was to try to define when the when an analytic function is having infinity as a point of analyticity okay and so as I told you if you recall one of the standard ways of defining analyticity at a point is to say that the function is differentiable at that point and also in a neighbourhood at every point in a neighbourhood of the given point okay. So this is how you define analyticity at a point in the complex plane okay but since we are worried about analyticity at infinity okay what we do is that we tend to look at the point at infinity as a singularity, a singular point of an analytic function which is defined in a deleted neighbourhood of infinity which means that the function is analytic for mod z greater than r for r sufficiently large outside a circle of sufficiently large radius the function you are given a function which is analytic and then infinity the point at infinity becomes an isolated singular point and then you want to say the function is analytic at infinity so it would not help to say that the function is differentiable at infinity because if you write the differential limit it does not make sense at infinity. So what is the way out? The way out is to actually get the draw inspiration from Riemann's removable singularity theorem. Riemann's removable singularity theorem tells you that if you are looking at an isolated singularity of an analytic function at a point in the complex plane then saying that the function is analytic at that point namely that the which is essentially saying that the function can be extended to an analytic function at that point including that point okay which is one of the definitions of what a removable singularity is okay is equivalent to requiring that the function has a limit at that point it is which is equivalent to the continuity of the function at that point in fact it is also equivalent to the function being bounded in a neighbourhood of that point bounded in modulus of course. So this is so the so the model of the story is that you can use these conditions to define the function to be analytic at infinity okay what you can say is that though it will not it does not make sense it will not work to say that the function is differentiable at infinity you can always say that the function has a removable singularity at infinity in the sense that the function either has a limited infinity so the limit as z tends to infinity f z exists okay that is one condition the other condition is the function is bounded in bounded at infinity that means there is a deleted neighbourhood of infinity where the function the modulus of the function can be made less than a positive constant okay and these conditions are all one and the same okay and why these conditions are one and the same is because of this other important philosophy that the studying the function at infinity studying f of w at infinity is the same as studying f of 1 by w at 0 okay. So and I told you that the rational or the justification for that is that z going to w equal to 1 by z is actually a homium of some of the Riemann extended complex plane onto the extended complex plane which interchange is 0 and infinity okay and further if you throw out the point 0 and infinity then you get the punctured complex plane the complex plane punctured at the origin this map is it going to 1 over z which is equal to w is going to be an analytic holomorphic isomorphism of the punctured complex plane onto itself okay and under such an analytic isomorphism the nature of the singularity at 0 and the nature of the singularity at infinity which is the image of 0 they should correspond okay this is the philosophy that we use now what I want to say is that I want to go ahead with go ahead with this and so now that we have defined a function being analytic at infinity okay and the weakest definition for a function being analytic at infinity is that it is bounded at infinity namely there is a point I mean there is a small neighbourhood of infinity okay which should be thought of as all z such that mod z greater than r for r sufficiently large then mod fz should be made you should be able to make mod fz less than m okay for some positive constant m okay so that is exactly what you want to do so what I want to do next is I want to worry about how I want to worry about what it means to say that a function has a removable singularity at infinity okay so we want to analyse this right so let us take the let us take the so we will look at the case of an entire function alright but even before that I wanted to say that you know this one more aspect that we need to actually look into okay and this aspect is about this aspect is about the Laurent series okay so you see you know the in some sense how do you study the isolated singularity of a function at a point in the complex plane at a point z not in the complex plane one of the ways of studying this is by looking at the Laurent expansion of the function centred at z not that means you expand the function in positive and negative powers of z minus z not that is the Laurent expansion you get the the coefficients are the Laurent coefficients and the Laurent theorem says that there is such a Laurent expansion okay which is in general valid in an annulus alright and then you know that the nature of the Laurent expansion to be more specific the nature of the principal part or the singular part of the Laurent expansion will tell you what kind of singularity z not is so you know that if the Laurent expansion has only negative powers I mean it has only finitely many negative powers of z minus z not you know it is a pole if it has no that is if the principal part has only finitely many terms then it is a pole okay if the principal part does does not exist namely if the principal part is 0 then it is a removable singularity okay and if the principal part has infinitely many negative powers of z minus z not then it is an essential singularity okay this is this is the trying to classify singularities based on the Laurent expansion okay which is something that you know now the question is what is the analog this of this when z not is a point at infinity not a not a when z not is a point in the complex plane but when z not is a point at infinity so how do you get a handle on this how do you get a handle on this so you see so let me write that down so my point is what is this Laurent series Laurent series at infinity which is rather funny thing again you know the you must remember that the way you deal with infinity is you have to be very clever there are certain things you can do with infinity there are certain things you cannot do with infinity so for example when you want to define a function to be analytic at infinity you do not take the root of trying to define it differentiable at infinity because derivative at infinity is not defined it is not easy you cannot define it so easily right of course you can go to another level of abstraction called what is called of what is called a Riemann surface okay and you can make the you can make the complex plane along with a point at infinity namely the external complex plane into Riemann surface and once you have a Riemann surface and you have a holomorphic function then you can talk about derivative at any point by using local coordinates okay but then we do not want to get into that amount of generality you can look up that point of reasoning if you look at my video course on Riemann surfaces which is available on the web okay but then let us not go to that level of abstraction okay at this moment at this level of our exposition we do not try to define derivative at infinity but trying to say it is analytic infinity we get away by using the drawing inspiration from Riemann's removable singularity theorem okay by simply for example just requiring the function is continuous at infinity okay so in the same way if you look at the Laurent series at infinity how do you define this so you know that see the idea of so you have to think like this the idea of what is a Laurent series see the whole point about Laurent series Laurent series is a generalization of the Taylor series okay and what is the Taylor series a Taylor series is trying to express an analytic function in terms of simple analytic functions functions is a simplest possible analytic functions so you know if when I say Taylor series of an analytic function at a point Z not in the complex plane I am simply expanding the function in powers of Z minus Z not and I know Z the powers of Z minus Z not they are this they are simple functions they are simple polynomials okay and I am trying to expand I am trying to write the function the given analytic function as a limit of polynomials after all a power series for that matter any functional series by definition when it converges is just the limit of partial sums okay and if you take a Taylor series or a power series then the partial sums are all polynomials okay and it is the limit of these polynomials that gives you the given function okay which is the limit of the series. So the purpose of a Taylor series is to expand a function as a series in terms of simple analytic functions that is the idea okay and this is this is at a point where the function is analytic okay but suppose a point your in question is a point not a point of analyticity suppose it is a point where the function has an isolated singularity then what comes in is the Laurent expansion. The Laurent expansion gives says that well you will all you can still get a expansion of the function in the form of a series but then now you will have to allow also negative powers okay now so in general we think of philosophically we think of the Laurent series as a as a generalization of Taylor series and the guiding philosophy is that it number one is that it allows you to expand the function in terms of simple functions that is that is point number one. Point number two is that the Laurent series can be broken up in two pieces okay there is one part of the Laurent series which consists of positive and zero powers of z minus z0 where z0 is a point where you are looking at the centre of the series that is called the analytic part of the Laurent series okay and then there is also the part of the series that involves the negative powers of z minus z0 which you call as the singular part or the principle part of the Laurent series. So you see the general idea of the Laurent series that it breaks the function into two functions it breaks the functions into two pieces as a it expresses the function as a sum of two pieces one piece is the analytic part of the function at that point in the neighbourhood of the point the other piece is the principle part of the function at that point which is not analytic at that point okay now using these two guiding philosophies you can also define what a Laurent series at infinity means okay and well the point is that the point is as follows so you suppose f of z is analytic at z equal to infinity or let me not even start with analytic at infinity let me just say let me say it is analytic in a neighbourhood of infinity suppose f of z is analytic in a neighbourhood of infinity say in and you know for obvious reasons let me do the following thing let me not use z let me use w okay say in mod w greater than r okay. So I am using w as a variable because I will always you know when I want to study w at infinity I will rather study you know there is one of the tactics that we have been using is that you study w equal to 1 by z at 0 so that is why I want to reserve w for 1 by z okay so suppose f is analytic in a neighbourhood of infinity say in mod w greater than r so of course by this I do not mean that the function is analytic at infinity mind you okay so you know you have to be a little careful when I say a function is analytic in a neighbourhood of a point in the complex plane it is understood that the function is also analytic at that point okay but then when I am saying f of w is analytic in a neighbourhood of infinity I am not necessarily meaning that it is also analytic at the point at infinity the point at infinity could is a singularity okay it is an isolated singularity I do not know whether it is analytic or not okay. So let me state that we do not know we do not know if f is analytic at infinity okay now what do you do with this of course you know the let us go by the philosophy that you to study f of w at infinity you study f of 1 by z at 0 okay the behaviour of f of z f of w at w equal to infinity is the same as the behaviour of f of 1 by z which is g of z mind you f of 1 by z is the same as w f of w where w is equal to 1 by z at at z equal to 0 okay note that g g of z is defined in defined and analytic defined and analytic or holomorphic in a deleted neighbourhood of the origin which is just given by 0 less than mod z less than 1 by r which is actually writing this mod w greater than r in terms of z okay putting w equal to 1 by z okay. So and 0 less than 0 strictly less than mod z strictly less than 1 by r is a deleted neighbourhood of the origin which is circle I mean this is the interior of a circle with the origin to remove radius 1 by r okay. Now but then you know now you are looking at the point 0 in the complex plane and we have Laurent's theorem since now 0 is an isolated singularity of g of z okay and the idea is that the nature of the singularity of g at 0 should be the same as the nature of the singularity of f at infinity okay that is the idea okay and of course you know why that is correct because z going to w which is z going to 1 by z is an isomorphic okay of deleted neighbourhoods alright. So now so g has a Laurent expansion okay so g has a Laurent expansion has a Laurent expansion so what is the Laurent expansion it is g of z equal to sigma n equal to minus infinity to infinity a n a z power n this is the Laurent expansion of g okay and mind you I am the centre of the expansion is the origin normally if the centre is the point z not then you have to use powers of z minus z not but here z not is 0 so use powers of z and the point that it is Laurent expansion is that there are negative powers of z included as well that is why this summation is running from minus infinity to plus infinity and well so this is the this is g as it is now you know what you must understand is that you know if you look at if you look at this what does it mean for f so what this will tell you is that see this is valid for mod z less than 1 by r z not equal to 0 okay and if I plug in see of course instead of z I can put 1 by w okay instead of z I can put 1 by w and g of 1 by w is just f of w okay so this is the same as writing this is equivalent writing f of w is equal to sigma you know n equal to minus infinity to infinity a n w power minus n okay so I can simply replace z by 1 by w so z power n becomes w power minus n and summation will run from again minus infinity plus infinity the only thing is that because I have changed my variable the powers the power of the nth power of the variable has a negative subscript okay right so this is well this is the Laurent expansion but now so you know the point is that the Laurent expansion so this gives you a clue as to what you should call the Laurent expansion of f at infinity you can very well call this expression that you have written for f as a Laurent expansion at infinity okay because it is in line with the philosophy that it is expressing f as a series okay in terms of simple functions the functions are just powers of z okay so you can very well call this thing on the right this expression for f w as a Laurent expansion of f at infinity that is fair enough but then there is a little bit more to be seen you see if you write if you look at g of z the Laurent expansion splits into 2 pieces as I told you the Laurent expansion splits into a principal part which is a singular part plus an analytic part okay and the principal part of the singular part consists of negative powers of z so you know let me write this like this so this is sigma n equal to minus infinity to minus 1 a n z power n plus sigma n equal to 0 to infinity a n z power n okay there are 2 pieces and this fellow here is the so what I have written here is the principal part is the principal or the singular part at the at the origin okay and this part is the analytic part at the origin okay so you know if you want let me give symbols to these things so this is so if you want this is gs of z gs of z is the singular part okay and this is ga of z ga of z is the analytic part alright now let us make this change of variable which is z going to w which is equal to 1 by z okay and you know now watch carefully what is our philosophy our philosophy is that if you change variable from z to 1 over z okay then the behaviour at 0 at z equal to 0 should correspond to the behaviour of w equal to 1 by z at infinity therefore if you go by this if I transform gs of z to w which is 1 by z what I should get should be the singular part at infinity okay because gs of z is a singular part at z equal to 0 if I transform gs of z by putting z equal to 1 by w what I should get is a singular part at infinity okay and similarly if I transform ga of z by putting z equal to 1 by w I should get the analytic part at infinity so what do I get see basically what I get is I get f of w you see is so I will get sigma n equal to minus infinity to minus 1 An so I will get w to the minus n plus sigma here I will get n equal to 0 to infinity An w to the minus n okay and if you watch carefully so let me use a different colour at this point if you watch carefully now you see this guy here this corresponds to what is this this is just gs of 1 by this is gs of 1 by w okay because I put z equal to 1 by w and but gs was a singular part and therefore gs of 1 by w should also be a singular part so this should be in principle this must be equal to the singular part of f at f of w and what are you and watch carefully the singular part of f at w has what powers of w it has positive powers of w okay it has positive powers of w because n is negative so w to the n minus n is positive therefore the moral of the story is that if you write a if you write a Taylor series if you write a positive power series in a variable that at infinity corresponds to a singular part okay and that is that is very believable because as you go to infinity okay the partial sums which are polynomials are going to go to infinity so infinity is a pole actually the for the parties for the partial sums okay it is not bounded at infinity so it is correct okay so the whole point is that when you look at the so the Taylor series at infinity should be thought of I mean the when you look at the Laurent series at infinity the principle part should it will look like a Taylor series at the origin it is because it is exactly that by the transformation is it going to 1 over z okay so this is this is a singular part of f and then and then this guy here this is f a of w which is g a of 1 by w and this is the analytic part at infinity okay so you see f of w is also split into a singular part f s w plus f a of w which is an analytic part at infinity okay and mind you the analytic part at infinity contains all the negative powers including the constant because I have put n equal to 0 is also here so a not is here in the analytic part at infinity and so you see when you look at the variable at 0 okay then the analytic part consists of the non-negative terms and the singular part consists of the negative terms but when you look at the variable at infinity the analytic part consists of the negative terms and the including the constant and the principle part consists of the positive terms this is the this is exactly what happens and it is correct okay now why do you think that this analytic why do you think that the negative powers are the portion of the expansion which involves the negative powers is analytic at infinity that is correct because you see as a variable approaches infinity the negative powers approach 0 so you see that the it is bounded essentially okay so it is analytic and because our definition of analytic at infinity is either that it should be bounded or it should tend to a limit okay and no positive power of the of a variable will ever tend to will ever be bounded or will ever tend to 0 if you let the variable go to infinity okay so everything is fine so you know so here is the so here is our definition our definition is you take a function analytic you take a function is analytic in a deleted neighbourhood of infinity okay write out its laura expansion okay basically the laura expansion is the laura expansion of by gotten by changing the variable to its reciprocal okay and then what you do is you take the positive part of the laura expansion okay in the in the in the in the original variable and call that as a singular part okay and the negative part including the including the constant term is what is called the analytic part okay so now we have this clear definition of what a laura series at infinity should mean okay fine now once you have made this definition of laura series at infinity what you need to know is that whether this fits in well with our with with the theory that you do on the finite complex plane so for example you can ask this question suppose of you know for a point z0 in the finite complex plane a function is analytic at that point if you assume to begin with that point is an isolated singularity the function is analytic at that point if and only if you write the laura expansion about that point it has no singular part its principal part is 0 okay so if you go by that philosophy for a function f which is defined in a neighbourhood of infinity I mean for which infinity is an isolated singular point the function will be analytic at infinity if the laura expansion of at infinity has no singular part that is what should happen now does that happen it does because you see here is my function fw defined in neighbourhood of infinity its singular part is this is this part which consists of positive powers of w okay if that singular part is not there okay that means that I can express f only in terms of negative powers of say w then that is of course analytic at infinity because all these go to 0 as w goes to infinity okay so the moral of the story is that the our definition is correct so you know the point is that sometimes you may have to make definitions based on certain philosophy and then you have to check whether it matches with what what happens what you expect to happen as morally correct okay so if from this it is very clear that a function is analytic at infinity if and only if its singular part at infinity vanishes okay then if you take the laura expansion at infinity okay then its singular part vanishes so let me write a few things so this is the this is called the singular part of fw at infinity and this guy here is called the analytic part of fw at infinity and now you know if you look at it in a very simple way you know when you are looking at infinity the good functions are negative powers of w because they go to 0 and the bad functions are positive powers of w because they go to infinity so if you expect a function to be good at infinity it should be expressible only in terms of negative powers of w and that is why the negative powers of w along with the constant that part is the analytic part at infinity okay so the so the definition is very clear and with this definition you see that Rayman's removable singularity theorem is also valid in its various forms at the point at infinity a function namely a function which is which has infinity as an isolated singularity has that singularity as removable singularity if and only if it is bounded in a neighbourhood of infinity if and only if it tends to a limit at infinity and that is also equivalent to saying that the Lorentz series at infinity has no singular part they are all equivalents okay so you get the same version of the theorem as you would get in the case of a finite point a point in the usual complex plane okay so everything fits well the only thing that does not work is trying to define a derivative at infinity that does not work okay fine so now what I am going to do is I am going to ask this question as to what it means to having a removable singularity at infinity for example for a good function for example for a function like an entire function okay so what does it mean and you will see that that will throw up connections with Lieuwell's theorem and so on so see so let us analyze this suppose that f of w has w equal to infinity as a removable singularity okay so f w is f analytic so it has no singular part it has only the analytic part of its expansion and that is what this is writable as you know the analytic part of the expansion at infinity will involve negative powers of the variable and also the constant term so it will be n equal to 0 to infinity if you want I can call it as b n w power okay so this is analytic part at infinity now let us analyze what it means to say that the function is for example you know entire suppose so I am looking the following case suppose I have an entire function and suppose it is analytic at infinity what happens we will see that it will reduce to a constant okay and that is just another avatar of Lieuwell's theorem okay so how do you see that see suppose f has a removable singularity at w equal to infinity then f of w has this expansion which is analytic at infinity then you see g of z which is f of 1 by z where I put w equal to 1 by z what I will get is I will get sigma n equal to 0 to infinity I will get b n e z power n okay which is you can see that is clearly a Taylor series at the origin it is a positive it is a power series at the origin okay center at the origin so it has to represent an analytic function and that is correct because f is analytic at infinity f w is analytic at w equal to infinity if and only if g of z which is f of 1 by z that is analytic at z equal to 0 let us go back to Riemann's removable singularity theorem okay saying that f is analytic at infinity is same as saying that f is bounded at infinity okay so it means f is bounded in a deleted neighborhood of infinity alright so by if you want Riemann's removable singularity theorem f is bounded so I am using b dt as an abbreviation for bounded at w equal to infinity so f of w if you want. So there exists a positive constant m greater than 0 such that you know mod f of w is less than m for if mod w if mod w is greater than m okay so this is what bounded at infinity means in a neighborhood of infinity the function in modulus can be bounded by a positive constant okay and this is equivalent to infinity being a good point namely it is equivalent to infinity being a removable singularity okay now watch see for mod w less than or equal to R okay so mod w greater than R the modulus of the function is bounded by m okay and look at mod w less than or equal to R I am using the assumption that so I am putting this extra condition that f is entire and mind you I am saying f is entire as a function of w okay so I am trying to look at an entire function which is having a removable singularity at infinity so f of w itself is an entire function even for w finite okay so you see if f of w is entire okay then you know f is analytic at 0 so is analytic at 0 okay f is analytic at 0 because an entire function is supposed to be analytic at every point at every finite point so if I say f of w is entire as a function of w it should be analytic at for all values of w in the complex plane in particular it should be analytic at 0 and if it is an analytic at 0 then you know if you write out it should tend to a limit as w tends to 0 okay but then look at this expression look at this look at this expression this expression mind you normally this expression will be valid it is supposed to be a Laurent expansion at infinity so it should be valid only in a neighbourhood of infinity but since the function is entire it is valid everywhere it is valid on the whole complex plane so it is valid at 0 also okay and if it is valid at 0 you can see all the b n's for n non-0 they all should be 0 okay because the moment I get a negative power of w at w equal to 0 it is not going to give me a finite limit it is going to go to infinity because it is going to become like a pole okay so the moral of the story is that if you assume f is entire then f is analytic at 0 and this will imply that all the b n's are 0 for n not equal to 0 and this implies that f is a constant okay so that is very obvious so all I am trying to say is that if you have an entire function which has a removable singularity at infinity then it is a constant what is the counter positive of that the counter positive of that is suppose you have a non-constant entire function then infinity is certainly not a removable singularity for a non-constant entire function infinity cannot be a removable singularity because the only entire function entire functions which are analytic at infinity are the constants okay and you can the reason why I got into this stuff about modulus is because I wanted to say that this is actually another avatar of Lieuwell's theorem see because you see look at this stuff that I have written in between see f is analytic at infinity so outside the circle of sufficiently large radius mod w greater than r mod fw is bounded okay but if you look at the interior of that circle and the boundary of the circle I will get mod w less than or equal to r and you see mod w less than or equal to r is a compact set it is both closed and bounded and I have assumed f is entire so it is continuous you know a continuous function on a compact set is bounded because the image of a compact set under a continuous map is again compact and compact is will imply bounded so what this will tell you is that there is a bound for f even in mod w less than or equal to r that combined with the bound for mod w greater than r will tell you that f is an entire function which is bounded on the whole plane and then Lieuwell's theorem will tell you that f is a constant so that is the point I want to tell you I want to tell you that see this argument that an entire function which is analytic at infinity is constant is actually another avatar of Lieuwell's theorem okay it is actually another avatar of Lieuwell's theorem that is what you have to understand. So the moral of the story is that whenever you are looking at a non-constant entire function infinity is certainly a singularity it is a honus singularity it is not a removable singularity so it can either be a pole or it can be an essential singularity it cannot be a removable singularity okay and the only exemptions are constants which are very uninteresting okay so I will stop here.


I got into a fight with Stacey Dash one time. Sure. And we were, it was at spin class. Bob from the biggest loser was the teacher. Random as fuck, right? So Crunch is back in the day. I put my stuff on the spin bike and then I went to go do abs and then I went back and my stuff was off the spin bike on the ground over in front of another spin bike. And Stacey Dash is on my spin bike. Back in the day, Crunch only had like 15 spin bikes out of the 50 they had that were like still good high end.  It's astonishing. And how my face gets redder as I drink throughout this podcast. But also how you act like it's totally normal. Like I got in a fight with Stacey Dash. So Bob from the biggest losers, of course. Bob's a very inspirational dude and for real. Like the guy's, guy legit knows how to connect with people. There's a few people that do that. But he's a, he was a legit great coach, very great coach. So I go to Stacey Dash. I go, hey, my stuff was on that bike. And she's, Stacey Dash has never had to use her words to do, to get anything, you know? Other than, can I have a coffee please? And then everyone was like, sure. Yeah, what size, what size? You're gorgeous. So I said, that was my spin bike. She goes, no, it's mine. And I realized in that instant, I was dealing with someone that doesn't talk, that doesn't use words. Well, she doesn't have to persuade anybody to do anything. Right. Her thing is. When women say, I'm going to seduce him, you're just going to make yourself available. There's no seduction.   When they like, like your, your dog is probably at the age where they, if they think you guys cute, they just go to his job. Because that's all they have to do. Yeah.  I ran circles around her intellectually in a conversation about spin bikes. And she just started crying cause she had never used her words before. And she just, I watched it and I went, of course. And by the way, I was single at the time and I was super misogynistic. And, and I was like, of course that's what you do.  You're going to cry. And then Bob was like, yo. Did you, you said that? Yeah. And Bob goes, let it go. And I went, no, Bob, no. She took my spin bike and he goes. And this is in the middle of a spin class, by the way. There's still a spin class happening. Everyone's clipping in. Everyone's warming up. Everyone's watching. And you're going like, and another thing, Stacy Dash. But here's the interesting thing. Every one of these people in this group knew also there's only 15 good spin bikes. Everyone knew the currency I was dealing in was spin bikes. And that's what they built in every day. That's why they showed up early. We're talking to early, early people. And Bob only wrote a one spin bike, the one at the front of the class. That's going to be the best spin bike there is, of course. Right?  You don't, he called that bike because that's how everyone operated. And Bob was like, you get to pick the battles you want to fight. I think this isn't the one. And in a weird way, I kind of heard it like, Bob had a great way of saying things that I understood. One time I was about to go to Thanksgiving and he was like, he was like, do you, we're in this, he goes, everyone has a saddle. That means like stand up a pedal.  And I was like, in my head, I'm like, that's all I've ever wanted. And he was like, then stay out of that saddle. And I'm like, okay, stay out of the saddle. Stay out of the saddle. Do you want to walk into the bar that you grew up going to and have everyone at college see you and go, God damn, you look good. I slept on that guy, then stay out of the saddle. He spoke so sincerely to my heart that I stayed out of that saddle.