Hi everyone, my name is Lauren Zabrick and I am the executive director of the cyber project at the Harvard Kennedy School's Buffer Center. First, I just want to thank Rice and Bort and the entire ICS village for and team for inviting me here to discuss our paper entitled toward a collaborative cyber defense enhanced threat intelligence structure. I want to start with a story just to paint the picture, you know what how this is really informing my thinking. I spent over a decade in the United States intelligence community and a big chunk of that, where I was an analyst working counter terrorism. The group of analysts that I was a part of developed an entirely new methodology that at its core suggested that all data, no matter when it was collected, or from what sensor was collected is important to identify malign activity from the this mindset combined with leadership who removed your product burdens such as arbitrary reporting requirements. Let us be analysts and figure out the hard problems and you know isn't that what we all want as analysts who want to be empowered to do that. Our work allowed us to become very entrepreneurial in our work. We made relationships across the community, offering our cutting edge analysis for access to more data, and even creating our own forward deployed spots to make good on those relationships, and to continue the mission. In this setting, I deployed overseas multiple times sitting side by side with analysts and operators in joint operation centers or jocks where everyone was free to share information with each other, and where we all had awareness of the battle space, because the nightly operations and intelligence briefing. This is where I first became acquainted with this model that I described in the paper, which is best highlighted by the general stand in the crystals motto where he says, it takes a network to defeat a network and that's the whole crux behind this paper. So say what you will about our overall strategy in Iraq and Afghanistan from a tactical intelligence and organizational standpoint, I think we're innovative, and there's some really important organizational lessons learned here. When I came to the cyber world in January of 2016, I was fresh from government and fresh from this whole environment, and I watched organizations try to build up their own cyber capabilities for intelligence capabilities, operating on slim budgets and really trying to figure out how to defend themselves and during a mid rising cyber threats, largely without government involvement. And I just remember thinking, there has to be a better way. Now, I want to be very clear here. I know there was recent pushback over conflating cyber and CT, and while there are some distinct differences and some areas of similarity. You know, it's, it's the intelligence and the organizational concepts that I'm really trying to carry forward here. So the cyber threat environment demands the need for more collaboration increased collaboration, we're witnessing increasing ransomware attacks and intrusions into our critical infrastructure by both state and non state actors with increasingly sophisticated capabilities and sharpened intent. So the strategic vision set forth by our president and his senior cyber leaders is truly critical to helping us become more secure and resilient. But to operationalize that vision, we have to ensure that the structures and the policies to facilitate that collaboration are in place and as we know there they've been lacking. So this is my idea to create those structures and policies kind of like, if you build it they will come. From there, you obviously need strong relationships between people and an even better ways to operationalize intelligence. And this paper doesn't exactly address that, but I will make a quick one for share the mic and cyber which in October we're going to be really focusing on those human to human tactical relationships for public private partnerships. So is this a perfect answer. I don't know but it's a conversation starter. And there's new ideas in here building on some existing ideas from other people. So I'm really excited to talk to you about this today. So first I want to go over how we did the research and what we know about the current state and then I'll go into our five recommendations and wrap up and I'll wait for feedback. We conducted research and interviews, starting back in November of 2019. We wanted to talk to people across the entire ecosystem we did. We spoke with people at state fusion centers in national labs. People who were still in the federal government or who just left people from CISA, the Department of Energy, the intelligence community, the FBI. We talked to people at ISACs and ISOWs. We talked to people in various private sector companies, especially threat intelligence analysts of large and small and some even in the critical infrastructure sectors. We wanted to map the current state. And an earlier version of this paper has 20 pages dedicated to the subject but we cut most of that out for length because we want people to actually read the paper. But our thesis was that our current environment across the domestic landscape is that it's very stovepiped and uncoordinated. And as you know, it leaves the analysts and the operators over extended.  And it's not getting much better. And of course our critical infrastructure remains vulnerable, which was born out of our research. We know that for defense there has to be a layered approach between the diplomatic community, the military and intelligence community, of course, regulations. And this paper doesn't address that it's really just focused on the nature of our collaboration and threat intelligence sharing. So take that for what it is. We looked at the landscape through four different lenses, cultural, organizational, legal and technological. And it was part of my hypothesis from the outset that these were where some of those barriers were to effective collaboration. And as we know, domestic cybersecurity is really hampered by limited resources, talent data funding. There's a siloed approach and even especially at the state and local levels and emergency management law enforcement approach, which isn't necessarily that thing but doesn't really lend itself to strategic whole of nation approaches to the problem. The private sector is responsible for their own networks and their own data, as you know, with varying levels of maturity and resources. And similarly, the federal government suffers from swim lanes, budget limitations, paperwork requirements to actually work with the private sector classification issues there's a lot of trust issues between public and private and the state this thought of well, if I'm going to work with you what am I going to get out of it so a lot of entities thought they weren't really getting much out of that collaboration. Not one person has been able to really hammer this out between the interagency and you know engaging with the private sector on from a holistic standpoint. And so, to us the fundamental challenge is that the structures the policies and the incentives are lacking in the relationships that do exist are largely ad hoc, and their point to point. There's no clear operational picture of the entire threat landscape, or, again, a coordinated sustained strategic approach to really address the issues. And I lack a comprehensive understanding because we aren't collecting processing and sharing that data that's out there and in a coordinated manner. So before I jump into our five recommendations, I want to read you a quote from team of teams by Stanley with us on where he says organizations must be networked, not siloed in order to succeed. Particularly we restructured our force from the ground up on principles of extremely transparent information sharing and decentralized decision making authority. We dubbed this goal the state of emergent adaptive organizational intelligence shared consciousness, and it became the cornerstone of our transformation that quote right there is really just the driving vision for the structures and the policies that I described here so restructuring transparent information sharing and you know at the different level decentralized lowest levels and making those decisions but still having that that common thread across the whole landscape. So, our first recommendation to actually create those structures is to take something that exists now and transform it. And what we're recommending is taking the current CISA regional office structure. They have 10 regional offices right now and they are staffed by from what we could derive a handful of people they're more advisory posts right now. But what we are recommending is to take each of those offices and just build them out. And creating making them bigger outfitting them with all, you know, a number of workstations and common communications across across all these nodes. And we're calling them, or we want to come the collaborative defensive and analysis centers for CDACs so mirroring the jock. Sorry, the jock construct, but you know we don't want it to be over militarized, but in the setting. Analysts and operators would sit side by side analyzing sharing cyber threat intelligence. Providing early warning across the ecosystem and coordinating defensive actions with stakeholder organizations and mentioning the stakeholder organizations people may think well who has a seat in the CDAC. Obviously, not everyone can have a seat in the organization and of course it is incumbent upon the different entities staffing levels whether they can send somebody or not but at a minimum. We're thinking people like representatives from the FBI representatives from CISA representatives from the state fusion center so the states that are operating in that region. The different ISACs and ISOs especially ones that have a heavy presence in the region. Different private sectors companies, both big and small but especially ones in the critical infrastructure sector. So bringing all these analysts together and also giving me the ability to have reach back capability to their own organizations for additional analytic support and any sort of defensive actions. A lot of people have asked well why regional offices we think that they're key to this vision because they offer that physical breath to the mission and a field office touch point for businesses and states operating in that region. We think a structure like this would ensure a sustained government led coordinated presence in all regions of the country but to combat the threat on a local level, something that I've heard called reaching communities and highly distributed and and having intrinsic understanding of entities in that region I think is really important because there's just different needs and different realities on the ground. And then of course every day or night, we think there should be an enterprise wide operations and intelligence briefing that everyone in the CDAC could observe and gain situational awareness. You know the analysts would share strategic intelligence operators could debrief the incidents and you know no any major reflections from those activities and we think this would be the thread that holds all the different nodes in place and continues to build that connective tissue. Our second recommendation is to scale voluntary data collection and processing. We can't really have this structure without the data to to go through and to really provide that foundation for enhanced analysis. So collecting more threat data and processing it, and by processing I'm talking indexing applying analytics, minimizing it so stripping out all of that personally identifiable organizationally identifiable information to create that common operational picture and then to ensure that it is shareable at both speed and scale. We have the information out there you know that it's it's in all these different networks it's with companies it's with government organizations cloud service providers ISPs etc. But we need to get this data again this anonymized data into the hands of analysts to really drive those operations and you may be asking ourselves well don't we have systems like this in place with say scissors automated indicator sharing system or do these crisp. Yes, however, while they can be or they should be very foundational and help to provide early warning across the system. They're just not, or at least ACE isn't to a point where it can really do that yet. So, they, both of those programs need to be upgraded as far as the technology and increased in scale so for instance, in September of 2020. The inspector general release to report that noted, at least in 2018 only 219 private sector organizations were members of a IS, and those who did participate found that the information that they were getting just really lacked usable context which as you know is just not, you know as helpful as that could be. And this also was understaffed which were really limited outreach to the private sector and the whole, you know where DHS is really seen as a regulatory agency, I think a lot of organizations are very reticent to participate that in that and so we need to change some of those policies incentives. So, to get wide, wide scale voluntary participation we think a number of things should happen so first, ensuring that anonymization or minimization so again stripping out that information that we don't need that is personally identifiable or organizationally identifiable but stripping that out making that automated but but even further than that is to put that burden on the solution. So, instead of having every single entity who participates to have that as their responsibility has to be on the solution. And that's not only from a legal standpoint but a technological standpoint to access controls based on authorities. Applying analytics and indexing this data again to be instantly shareable. And, you know, maybe it's through a data lake or maybe it's through the notional joint collaborative environment that the cyber slaying Commission recommends either way, but you know to create these analytics on top of the data to help analysts to free them up to do the higher order analysis and to essentially triage. And, you know, couple I mentioned the legal aspects of the cybersecurity information security or infrastructure security act of 2015 should be amended to help facilitate the sharing. So again, taking the minimization burden and putting it on the solution and really the government to do that. And then, removing the limited liability clause that limits private sector protection so essentially adding more protection to the private sector entities that participate on there so it doesn't have to be just sharing with DHS for instance. So, and ensuring that the incentives are there the business cases are there but making it easy for people to say yes to do this. So that I will turn to the next recommendation to create a culture shift. And again, I will read you another quote, I think from team of teams and maybe from my share of the task but my crystal said that in the world of intelligence information was power, leading people at each stage to ask themselves a set of questions, should we pass this intelligence and so how much will we get in trouble for it. And those doubts cost us speed and often diluted the intelligence making it less likely to lead to targets. We widely distributed without preconditions intelligence that we captured for analysis that we've conducted actual information shared was important but more valuable was the trust built up through voluntarily sharing with others and that I think is so key. Because as we've seen the, the trust really needs to be built up between the public and private sectors trust as well as again was the structures and policies. So, much like the task force in the early 2000s. I think that we need to make a major cultural shift in our domestic cybersecurity posture. A lot of interviews told us that there's a big disconnect between the field and Washington DC. So, they may ask questions to certain representatives and it would just take forever because they would have to root those questions back to DC so really, we have to flip that mindset on its head for both this expeditionary culture, in which a lot, you know the major activities happen in the field and DC is the reach back office for the federal government and in fact, Chris clubs, the former says a director has said the future of says that is in the field so along that that line we think this could really be an interesting solution. And then to incentivize people to buy into that, then we recommend tying the ability to go out into the field if you will, to promotions bonuses and raises for instance. And then we need to really create a culture in which sharing and collaboration is the norm, and not the exception, and the way to do that is for the leadership to set the mission and priorities, as well as the conditions, and the infrastructures technologies, etc to do so, but also to build that connected tissue between the nodes and ensure that people are executing in line with those missions to help do that. So in my example, we need to move from same metrics as measures of effectiveness metrics and numbers say reporting reports created to analytic outcomes. So, in my experience organizations that mandate a certain number of reports. And then collaboration and information sharing then becomes not the goal, but an inhibitor to creating those reports. So, where the leadership removes those burdens of production from analysts and instead focus on those hard analytical outcomes and subsequently operational successes then become more entrepreneurial and innovative. And most vital to realizing this is personnel from leadership on down to the most junior levels. Everyone has to be welcome diversity is key, focusing on those people, but ensuring that they're, they're welcomed and they're supported and really focusing on diversity and demographics background and experience. So my recommendation is to unravel the interagency. In our research we discovered that the interagency cyber environment was really characterized by competing equities and priorities and up until the creation of the national cyber director. Every person was really only are able to unravel that yarn to determine those priorities and who arbitrates among the interagency. You know the equities to classification battles and then as also as they are coordinating with the private sector. So, with the creation of the new national cyber director role. Notice that the deputy national security advisor for cyber and learning technologies under and new burger will handle title 10 and title 50 cyber issues so more on the offensive side and the NCD will be responsible for the rest of the interagency and engagement with the private sector so on the defensive side from a strategic standpoint, while in continuous coordination with each other and this distinction is important. And that the president really in view the NCD with that authority to ensure that the director can determine those priorities and hammer out those conflicts because when Congress set this position they really gave the president some some wiggle room on on how to actually take this role so we're really hoping that the president can really give that role the authorities that needs on par with the counterpart at the NSC so in the program. So, where a new burger and Chris English set that strategy, the other agencies and especially CISA will operationalize it. Given the agency's importance, and it's relatively small budget and the recent politicized political can't say it politicization, we echo the call for scissors independence from DHS. We believe that it would give greater authority, bigger budget, more operational flexibility to an agency that just really needs it right now. We think it could provide greater flexibility and hiring practices as it looks to scale up and out and informally we've heard you know some frustrations there so we believe that it would also allow for reform of the sector risk management agency construct that I'll get into in a second. But with CISA as its own freestanding agency, we're recommending that it establish its own intelligence arm becoming the newest member of the intelligence community. That way, the office can work with the different seed acts to inform the intelligence collection requirements and priorities. At this point there's no real institutional mechanism for to inform the collection framework, despite a lot of those threats really focusing on the homeland. On the SRMAs, we think first that PPD 21 should be revised to enhance collaboration and sharing across all sectors and entities so transforming the focus from a sectoral approach to a cross sectoral mission focus and collaborative one and then second, if you could take those missions to collaborate and share information with the private sector that are currently on each of the SRMA so think DOE think TSA for pipelines for instance. Instead I think you could transfer those missions to CISA but really operationalized at the CDAC so again at the CDAC level then they would be the ones taking care of sharing that information and working with their constituents basically in the private sector. Our final recommendation really focuses on personnel. We often read about the gap between the number of open jobs and cybersecurity and the number of quote unquote qualified personnel. According to CyberSeq there were over half a million open jobs in the US and we currently lack the personnel to fill them. The pipeline is always going to be an issue. There are a lot of people who are looking to get into this career field they're working towards this employment, but for a variety of reasons they're finding it very difficult to make that jump in. And one of the biggest reasons and there are a lot we won't go into them right now but one of the biggest reasons is that a lot of jobs, even entry level jobs want their candidates to have experience. So how can you get experience if you can't get in. And a lot of people are, you know, even if they're able to overcome those high costs of training and certification to make that jump has been very prohibitive. To bridge that gap we're proposing a service here in which a person interested in this field could say receive training and support for certification in exchange for at least a year of service at one or more of this is a regional CDACs for instance, but really getting that critical boots on the ground experience. Now, this is not the military you're not going to have to get up and do PT you're not going to have to deploy overseas. It's just this is your national service in exchange for training certification you're getting that experience. And then, importantly too I think it would really help to fuel civic renewal, which is something I think we desperately need right now. You know, to make this happen. The President actually signed a law the American Rescue Plan that included $1 billion for national service. And so we recommend that the White House establish this interagency service core between now DHS but if says it becomes its own regional freestanding agency and then something like that for instance, and you know eventually maybe include other agencies but in cooperation with the National Community Service organizations to create those opportunities to create those pathways in, because as we know we desperately need people. And then on the other end of the spectrum, you know to raise the level analytic capability across all federal entities, and to get people who are very interested in cyber but who haven't really found those pathways over yet to create those opportunities for analytic exchanges between the agencies but especially between CISA FBI and NSA. And so we think that it would sort of raise the level of capability, it would provide bio experience and training, but also build that connective tissue and that trust across the different interagency elements. So, thank you so much for hanging with me I know that was a lot to digest, but I'm really thrilled to be here and I really look forward to questions and your comments. So thank you very much.


You would also like to know what was your journey and what was your schedule, while you became a player, because everybody feels the glamour, the clapping, the waves going on in the stadium, or the people just going for the autographs. But what is the rigor so that people don't think that you've made six sixes, it's easy for me to hit six sixes. He has got this glamour, he has got so much advertisement, etc. What is your take on that, what is the schedule normally for any player who has to come to an Indian team, how many hours he has to devote? Sir, I think cricket in our country has a lot of passion, it involves a lot of passion and people are crazy about the game. In our country, if India wins, people feel that they have won. If India loses, people feel that they have lost. So they take the wins and losses very personally, which is good and the bad side. So eventually when young kids come to the block and they perform, the media and the people, they make them superstars. But eventually when they go through a bad patch, media scrutinizes them and makes them villains. That's the part of our culture and that's the part of certain media and certain people you have to realize. I think growing up for me was, I was into a lot in skating and tennis. I wanted to be a professional tennis player or a champion skater. I was at the age of 10 and 11 where I didn't know what my direction was in life, but I was never keen on playing cricket. I remember I was in YPS. I started to play cricket because my father was forcing me to play cricket. I went to YPS school in Mohali and I remember the cricket coach was not there and I started to play tennis. He actually came from nowhere and he said to me that Yuvraj, your father is not going to like this if he sees you playing tennis. So it was disturbing those times, but eventually when I got the hang of cricket, I think my father knew that I had cricket in me. There were times when I was woken up at 6 in the morning, probably had to go for 6-10 miles of running. My father would be coach of D.A.V. College Chandigarh where Mr. Bali is also studied and has been a victim of his coaching. So I'm sure he can relate to that and probably we would run a lot and I would have to bat from 9 o'clock to 12 o'clock, have a 2 hour break, then go and sleep in the afternoon and probably have to come back and again bat, feeling, bowling for the next 3 hours.  At the age of 16 when I went to England tour, I felt the fast bowlers were really quick and I came and though I did really well without a helmet but I came and requested my father that I should be allowed to wear a helmet and so on, went to play under 19s and then play the under 19 World Cup.


Welcome back to the Trade Hacker Mindset. In this episode, we're gonna talk about why trading stress is not necessarily a bad thing. Trading the markets can be difficult to master and seemingly just out of reach. Professional traders have a secret.  It requires the Trade Hacker Mindset. All right, so let's jump into our discussion of why trading stress is not a bad thing. And this really comes from, this topic comes from my interaction with traders over the last 20 years. And what I find is that a lot of struggling traders or a lot of new traders or someone who has not really reached the level of success that they want in trading, the conversation goes something like this. You know, if I could reduce the amount of stress that I have while I trade, then my trading would go a lot better. Or if I could eliminate the stress that I have when I'm trading, then I would be much more successful and much more profitable. But let me ask you this, do you think that's true? Because think about this. The fact that we are traders and the fact that we have daily encounters with risk and uncertainty, don't you think that having stress is part of the equation? Because I wanna give you an example of how stress can be a good thing. People typically, when they think of stress, they always think of stress as a bad thing. But I wanna clarify the difference between two different things. One is stress and the other is distress. Because if you can learn to manage your stress, if you can learn how to make your stress work for you in your favor, then stress during trading actually becomes a good thing. And if you ever think that you're going to get to a point where you are going to eliminate risk, then you're absolutely wrong. And in fact, if that's your ultimate goal, then trading may not be for you. And so let's get back to this stress versus distress because stress can be a good thing. Distress is not a good thing. So let's talk about first how stress can help you in your actual performance. I wanna give you an example that's not trading related and then we'll tie it back to trading. So I live in the Midwest, I'm in Kansas City. And in Kansas City, we sometimes get snow, but a lot of times the worst weather, especially for driving is we have these ice storms where it starts to rain and then it freezes and creates this layer of ice on the trees, on the roads, on your houses, on everything. Everything's just kind of a sheet of ice. So about 10 years ago, I was driving back from Indianapolis. A group of us went to a Kansas City chiefs game in Indianapolis and we were driving home. Now this is an eight hour drive. And so as you can imagine, once you're driving for a little bit, you start to feel a bit bored, right? But then all of a sudden it started raining and it started raining a little bit and then it started raining a little bit more and then it started coming down in sheets. I mean, it was coming down pretty hard and then the temperature dropped and all of a sudden this rain now turned into sleet. Okay, this was about 10 years ago or maybe more, but I can still visually see this.  My visibility started getting reduced. I can start to feel the road become slippery under my tires. And before I knew it, I would have to loosen my grip on the steering wheel because I would look over and you get the whole white knuckle thing where you're gripping something so hard that your knuckles are turning white and I had to kind of relax and readjust and get back into a normal state of mind. I had to reduce my speed and what was boredom before where we're just kind of cruising along and things were a little bit boring. Now my boredom had quickly turned into a high alert situation. Okay, I'm no longer operating on autopilot. My auto, whatever you call it, the auto speed thing. I'm definitely off of that. Like I've got my foot in control of the brake and the gas at this point. So in this situation, this would be considered a psychological stress. This is a heightened physical and cognitive state and this prepares us for dealing with challenges in life. As humans, we have this flight or fight response because it basically it mobilizes our mind, it mobilizes our body to avoid challenging situations or face them head on. So as I'm gripping the steering wheel and I have this muscle tension, I have this alertness and it's kind of like a flow of adrenaline into your bloodstream, right? So those are a few of the physical attributes that you can actually feel as you enter into a state of stress. If I just kept my cruise control on, I may not have slowed down my speed. I may not have taken measures that I needed to to avoid an accident or slide off the road. This new state of stress has actually mobilized my energy and it shifted me from being bored to being able to cope with the immediate situation. Now, if somebody told me, you know what, you really need to minimize or try to eliminate your stress in this situation, I would have laughed at them because if I'm in a situation like this where I need to be alert, where I need to have this stress to take care of the situation, then that's what it should be. I should be in a state of stress. I should not be in a state of boredom or relaxation.  I mean, these are not abnormal. I've driven in them many times. Now, this example, driving from Indianapolis to Kansas City, that was a pretty severe example, but I've driven when it's sleeting and icing and snowing many times. So that's not something that's only happened to me once in my entire life. But think about this. What if you have someone who's from Florida, who's rarely ever seen snow or they're from the southern tip of Texas and they've barely even seen ice or snow while they're driving? How do you think they would adapt to this situation differently than I did? They would probably feel a little bit more threatened, right? I mean, I know how to handle it. I know how to stay safe in those situations, but somebody who's never experienced it, their stress might actually become distress. I mean, you hear these stories about when it just starts to rain a little bit in California, all of a sudden, you have these big pile ups on the highways because people aren't used to driving in those conditions and they kind of freak out and don't know how to handle things. No offense if you're from California. So this simple example of drying in icy conditions, it kind of illustrates that your perception and experience can make the difference between stress and distress. All right, so how does this relate to trading? Well, when you put your money at risk, when you put your capital at risk, you're like the driver on the icy road. You will be alert. You'll be processing information in real time to make quick decisions and decisive decisions and you're sometimes going to need to make course corrections if needed in real time. And so if you view losing as part of the natural part of trading or if you've experienced comebacks from drawdowns and bounced back out of losses and you have specific rules and mechanisms in place to limit your losses, then that stress of trading is unlikely to become distress. A losing trade to me is kind of just like a little annoyance.  It's kind of a little annoyance. That's what a losing trade is to me. If a losing trade to you creates this massive situation of distress, then that's a problem. And that distress comes because you probably don't have position size limits.  Because if you don't have those parameters in place, if you don't have that stress that's needed to manage those situations, those become distress and then you don't have an ability to focus and make those rapid course corrections that you need to and you become impaired just like the driver on the icy road from Florida who's never experienced it before. You see, when you have position size rules, when you have a trading plan, when you journal your trades, when you document, when you have stop loss levels based on your position, those are kind of like snow tires on your vehicle, okay? They may not seem to do a lot for you when things are going perfect, but they definitely help you deal with adverse conditions.  And the same with trading. If you're a panicky or a nervous or fearful trader and you're unable to control your losses, the experienced trader knows that losses can always be limited. So your challenge as a trader is to embrace stress and always ensure that it does not become distress. One of the things, and I've talked about this on several podcasts in the past, is one way to do that is before you start your trading day, you should be writing down what your trading plan is for that day. What is your position size guidelines? What is your per trade loss limit? What is your per trade price targets? What's your daily loss limit? And how are you feeling? How did you sleep last night? Is there anything outside of your trading business that is gonna cause distraction? Or is there anything in your family life that is creating additional stress that could turn into distress in your trading?  Is there some kind of financial stress that is putting additional stress or distress on you while you're trading? If you become self-aware of this, if you document this, if you write it down, it's gonna make you become more and more self-aware and help you manage that risk and not allow it to turn into the distress that we don't want. You see, if you are prepared for adversity, you're going to respond with normal stress and not distress. Trading will always be stressful because we're always putting capital at risk. We are always dealing with uncertainty, but managing stress and using it in your benefit is a good thing. Remember, the idea around the trade hacker mindset is to maintain this mindset that keeps you confident, keeps your motivation high, and under those conditions, you'll work harder, you'll learn more, you'll grow your trading account over time. So it doesn't mean eliminating stress, it means creating this wall between stress and distress. Risk management, position size is one of the best psychological walls that you can put up between these two things. So I hope that was helpful. If you wanna be part of our trade hacker community, just go to community.navigationtrading.com. We've got hundreds of traders interacting on a daily basis, not only about the mindset stuff, but also sharing trade ideas with the sole purpose of helping each other become better traders. Just go to community.navigationtrading.com. I'll see you on the inside or I'll see you in the next episode.


Here's what's new this month with Passport. Welcome to the Hotel Portofino. How do you find Italy? You mean how does Italy find me? What's the plan? If the threat's genuine, give him a love. Get him out of there. We need your help. All we want is to choose, and we will get to it. These and all your favorite shows are available with Passport on the PBS app. Download it today.


So today I'm going to talk about moments of class group distributions. So we were interested, last time we were talking about averages of functions of the P-CLO subgroup of class groups of, say for example, imaginary quadratic fields. And we also talked about the real quadratic case, so that would be the same thing here. So we're taking imaginary quadratic fields and we're saying over however many there are, if we average some function of the CLO P-subgroups of their class group, what does that look like at an odd prime P? And I'm going to write this with this sort of curly E-like expected value or average of this function of the class groups of P to the infinity. And the reason I made the curly E and not just the normal blackboard bold E for expected value is because this is a little funky because of this limit. It's not actually an expected value over some distribution. It's only the limit of expected values over these distributions. And of course for this to make sense you have to know that I'm taking, what case I'm taking and how I'm ordering them. But in any case. So this would be this sort of empirical average over class group distributions. And so far last time we were talking a lot about what proportion of the time which you get a certain CLO P-subgroup of your class group. So that would be taking the average of a characteristic function of a finite abelian P-group. The function that was one when your group is that group and zero otherwise. And if we simply had a distribution, a measure on finite abelian P-groups, then knowing the averages of the characteristic functions would determine the averages of other functions. Because there is this limit here, there are some funky analytic issues and even in fact if we knew the averages of the characteristics functions, it doesn't determine the averages of other functions because of this limit. And there are some exercises coming up with counter examples to this sort of thing in the notes. So that's like one, you know, you might think well do you really need to average other functions besides just the characteristic functions once you know what proportion of time you get each group, then you should know the anything you want but that's not quite true because of this limit. And so we're going to talk today about another very important class of functions to average over CLP subgroups of class groups and these are so for a fixed group B the function asks how many surjective group homomorphisms do you have to be? So these are surjective homomorphisms and the function F is an integer value function that just counts those surjections. And these averages are important enough that we are going to call such an average the B moment of the distribution of groups. So you can talk about that for a, you know, a literal probability measure on the space of finding the BMP groups or also in this class group case where we take a limit of distributions, we'll still talk about its B moment and this, these moments are in analogy to the classical moments of distributions of real numbers or of random variables valued in the reals and in that case the moments are indexed by natural numbers k and the k moment is the average of x to the k where x is from the distribution or the expected value of x to the k if x is a random real number and so these in the probability theory of real numbers these moments play a very important role and there are two big reasons for that. One is that they're often more accessible in computation essentially because of the linearity of expectation or of averaging often one can get a hold of moments of distributions and secondly because it turns out that if you know the moments of a distribution of real numbers and those moments don't go too fast they actually uniquely determine the distribution that gave those averages and so there's a little bit more about that in the notes and in this case now we're not talking about distributions of numbers but distributions of finite abelian p groups and these averages of the number of subjective homomorphisms to a fixed group play a similar role so here is an example a theorem so I said this in the language of random groups so let's say x and y be random finite abelian groups so by that I mean random variables valued in the set of groups not like in elements of a group but in the in the set of groups so if for every finite abelian group B we have that the expected so this is the average the expected value number of subjective group homomorphisms from x to B is the same as from y to B and if you're not as familiar with the language of of random variables and you are just thinking about a distribution of groups this average would be you know the integral of x you know it would still just be this average where x comes from from your distribution so if you were thinking about a measure of groups this would be what this average is I've just said it in the probabilistic language all right so if these averages are the same for every B and those averages don't grow too fast so say not more than a constant times the size of the second wedge power of B so that just is the normal wedge product as Z modules if that's the case if x and y have the same moments and those moments don't go too quickly then for every finite abelian group the probability that x is a is the same as probably as y is a so x and y have the same distribution so this shows that these averages have this this critical property of moments of distributions on the real numbers which is in fact they sort of contain all the information of the distribution at first it might just be like well they obviously have some information their averages over it but you might you know you might not have first guess that they in fact have all of the all of the information of the distribution now as I said before we have this kind of limit as x goes to infinity and so we're interested in limits of random variables valued in groups or distributions and so we need to maybe be we'd like to be able to recover a limiting distribution from limiting moments and in fact that's one of the the reasons that we well for simplicity stick to the C LOP subgroups so an analog I should say of this theorem for finite abelian groups is not true if you sort of introduce the additional layer of limits as we're about to and there's some exercises in the notes around around that but if you restrict to abelian p groups so we have p a prime and let's say we have one random abelian p group and then we have a sequence of of random abelian p groups x1 x2 now this says if for every abelian p group b the limit of the the limiting moments in my sequence are equal to the moment of some fixed y and they don't grow too fast then for for every finite abelian p group the limiting distribution the limiting probability that you get some group a is the same as as the probability that y had a so this is is like the last theorem but it lets you work through a limit which is important in this context because we when we're look talking about these distributions of class groups of number fields they're always limits of distributions that not you know an honest probability measure on the on the space of groups so all right so all right so the moral of that is just that the the averages of these these number of subjective homomorphisms to be for every b actually they actually determine the averages of the characteristic characteristic functions all right so that was sort of the second property that I said that moments from the probability theory of random real numbers had that was useful was that they actually determine the distribution and then the first the first thing I said was that they're useful because they're they're often accessible they're more accessible and it turns out that that is that that is true here as well and one of the main reasons is that the moments are actually very closely related to field counting so the question that we were talking about in the first lecture all right so I want to explain the relationship of these moments to field counting so we'll start with a number field K and we let H be the Hilbert class field of K which we were calling before also K on add the maximal unramified abelian extension of K and the reason we're talking about that is because that is a particular field that we know it's Gawa group over K is the class group of K so oh I wrote K I wrote that it was the I'm gonna do the quadratic case here I should I will change this to this should be degree 2 we'll take a quadratic number field okay so we're gonna take a quadratic number field and I just know that hanging around somewhere it's got its Hilbert class field whose Gawa group is the class group of K and now what these moments they cared about surjections from say the class group of K to B all right well what what you know once we have this this particular field whose Gawa group is the class group of K via Gawa theory such a surjection exactly produces for us so Phi exactly gives us some field here L where L is inside the Hilbert class field and L over K is a B extension right so this this via Gawa theory I mean we you had to use class I mean we sort of use class field theory to produce this this extension here but after that the this surjection just via Gawa theory gives us some field L all right and related to the computation we talked about in the case of genus theory class field theory it will tell us that L over Q is actually Gawa I mean that doesn't have to be the case when you say take a quadratic extension and so L over K is a is a Gawa B extension and but not every not every B extension of K will be Gawa over Q but class field theory because it's coming from this from this age over K does tell us that L over Q is Gawa and that the Gawa group is a particular group it is the semi-direct product of B with the Z mod 2 Z where the non-trivial element of Z mod 2 Z acts on B by multiplication by minus one so that's this minus one down here I'm telling you how the Z mod 2 Z acts so what happened we went from one of these surjections and we got we got out of it a field a field that was Gawa over Q with a particular Gawa group and conversely if I have an L over Q that's Gawa where the Gawa group I'll sort of make the Gawa group diagram over here is the semi-direct product of B and Z mod 2 Z you know because of you know the the the subgroups of this group you know inside of here I can take a you know an index index to to subgroup B and that will give me some that will give me some quadratic field so since this is an index to subgroup I get a quadratic field and since this is B by you know Gawa theory so this is this is just you know L fixed by B and then this is a B extension so this this was starting all right it's like the same you know like why is she drawing the same picture the point was this picture just started with a Gawa L over Q isomorphic to to B semi-direct product Z mod 2 Z extension and then it produced a quadratic extension and a B extension on top of that and now we had one other thing that our our L's that that came up here okay so the the this is the maximal unramified abelian extension so B is of course an abelian group here from out of a suggestion from the class group to it so the abelian part is taken care of but we also know that this that this L over K that we get here will be an unramified extension because it's sitting inside of the Hilbert class field there and so we can just see from the Gawa theory and the theory of the inertia group that L over K is unramified exactly if all the inertia in the Gawa group of L over Q intersects trivially with B all right because we want the B extension sitting there we want the B extension to be unramified so there should be no inertia no non-trivial inertia subgroups there okay so then that tells us that the average of the number of surjections from class group of K to B is actually the number these surjections correspond to these L's so it's the number of L over Q the G extensions for this group G such that dot dot dot and the dot dot dot here is basically just basically just this condition here because we wanted if we want the L over K to be unramified we need to put in this condition so that L over K is unramified now also if your average meant to be over imaginary quadratic fields the dot dot dot would also want to include the condition on the decomposition group at infinity so that you got that quadratic field is imaginary but so that you know that's that's the numerator that this average number of surjections and then the denominator is just the average number or sorry just the you know the number of quadratic fields and all of this is sort of up to discriminant x and limited x goes to infinity but basically this turns the numerator and the denominator of this average both into field counting questions now we already knew that the denominator was a field counting question and so the numerator is also a field counting question with a slight twist that we're going to maybe impose these conditions on inertia groups and um and decomposition groups now we haven't discussed that um very much except that we did we you know we have already we split our quadratic fields into imaginary quadratic fields and real quadratic fields so that's imposing some condition at infinity but a natural follow-up to these uh to these counting questions for county g extensions are to also count g extensions with various local conditions like oh the inertia should be here decomposition groups should be here um and i've given some references in the notes to to much more discussion about the variance of the counting questions where you're counting with local conditions uh but yeah so in any case this turns the question of finding the b moment into just two count two field counting questions um one in the numerator with some some additional conditions on on inertia groups and then one traditional one as we already had in the denominator and uh so this is not only a nice relationship between the class group questions and uh the field counting questions but also the actual only average that we know um i keep forgetting to put my infinities in here the only average that we that we know i mean i suppose you could average like the identity like the all ones function but only non-trivial average that we know on silo p subgroups fraud p of class groups of quadratic fields precisely uses uh this this relationship so um the only and uh the only thing we've been been able to access at all uh it via proof uh in this setting uh comes comes via this relationship between the moments and and field counting and so what is that so um that's the theorem of downpour and how brawn uh that says in this case this case that we were considering um today uh was that over k imaginary quadratic which says that this average remember this is like the limit as x goes to infinity of the average the average number of subjective um group homomorphisms from the classical of k to z mod 3z is 1 over k imaginary quadratic fields they also do it for k real quadratic fields and then you get one third but we're we'll just um focus in the in the lecture on the imaginary quadratic case and again in the notes the all the also all the details about the real quadratic case and so perhaps you recall i talked about um uh david we're a heilbron in my first lecture or very very briefly gave you a sketch of how they approached counting cubic fields and so in this case these these uh l that you need to count for the numerator just say get this z mod 3z moment right we have this similar product of z mod 3z with z mod 2z by minus one that's just the group s3 so they that so one just needs to count s3 gawa extensions but of course s3 gawa extensions there are sactic extensions but they correspond um to non gawa cubics they all have like one isomorphism type of non gawa cubic in them and every non gawa cubic you know has a gawa closure um and so uh we already talked about how they could count non gawa cubics and so they can count s3 gawa extensions which is almost what you need for the numerator here except um that they also were able to do this imposing conditions local conditions on inertia um which is what you need to to get this sort of you know maybe i should say inertia conditions maybe there's also a condition at infinity um here all right and so that counting uh so it turns out that you know they were remember they were counting lattice points in a fundamental domain and uh these inertia conditions are are mod p squared conditions on the coordinates of the lattice points for every p and so you know they just have these um sort of uh sub sub uh lat and well sub lattices or transites of sub lattices uh that that they have to count and there are some issues there are some some things that one has to do with the getting that all to work out for infinitely many p and the error bounds um but they do that and then they were able to prove um that this average number of subjections is one all right and so this uh is indeed as predicted by the Cohen-Linster heuristics so we um uh yeah so as you would hope especially because this theorem of Davenport and Hybron I mean I failed to put years in everywhere but this predates the Cohen-Linster heuristics so certainly the fact that this was already known uh was in in one case uh was a was a helpful helpful piece of evidence um and you know shortly uh of course uh just by uh sort of kind of meta thinking of course the Cohen-Linster heuristics must have predicted this uh since they wouldn't have put out conjectures contradicted like the one theorem which still remains the the one theorem all right but let's see um yeah so what does that mean that it was predicted by the Cohen-Linster heuristics if you remember um how I described that previously I said oh well the the empirical average over class groups uh is predicted to be that just this group theoretical average over a billion p groups of where every billion p group is weighted by one over its number of automorphisms so when I say that this theorem appears saying that this average or moment is one I was predicted by the Cohen-Linster heuristics I mean um uh that that that the average the group theoretic average is one for every um a billion p group b so that's a kind of cool thing also about the uh Cohen-Linster distribution for uh imaginary quadratic fields it's the distribution you know this is the distribution on a billion p groups whose whose all of whose moments are one um and one could actually um work that work that out in a number of different ways um and Cohen-Linster's original paper they they develop some great machinery with with generating series that can help average you know average all kinds of functions uh with this weight because of course once you make a conjecture like this you'd like to actually be able to to see what it it says in various cases so that's a very powerful machinery um that you could use to for example to to find to find this this average um now but I since we talked before about this matrix model I'll just give you maybe a quick way to think about uh to think about these averages being one so remember last time we took a hard random matrix uh from uh in by in matrices over the periodic numbers so let's think about that uh that matrix and then we took its co-kernel uh and and thought about that as a random a billion p group and so what what are the moments of of those co-kernels so if we want the the b-th moment the expected number of projections from the co-kernel to b so what is that co-kernel it's just zp to the n mod the image of zp to the n under n or otherwise this is like this is the you know this is the column space column space of n all right um and this is actually you know last time we very briefly talked about something that was where we were essentially uh computing the expected number of isomorphisms from this to b and so actually computing surjections is like just what we just stopped one step short uh of that that argument um so so every such surjection would come from a map from zp to the n to b so we can sum over those maps and we can say what's the probability that this column space is actually in the kernel um and as we discussed yesterday when you take har random uh piatic matrix in biometric matrix each column is independent from har measure on zp to the n and so the chances that each column lands in the kernel is one over the size of b and you have n of those independently so you get this this sum over the surjections of b to the minus n and so how many surjections are there from zp to the n to b you can actually write that down explicitly but especially when it is large it's practically the order of b to the n of them because you know most of the time when n is very large compared to b as you throw down um uh in elements of b there'll be enough to generate b so these moments of these co kernels of the har random matrices as n goes to infinity they go to one um so so the limit of the moments uh is one and now it this doesn't automatically give that the moment of the limit distribution is one um and why would we want that because we said last time the limiting distribution is the colon lindstra distribution um because to to to get between what we set up here and this statement you have to move the limit from the outside to the inside of this expected value uh which is an infinite sum and you can't just arbitrarily move limits uh however morally it's perhaps the i think the the most convenient way of thinking about uh the fact that you get that these moments one because you can really sort of see it in these roughly b to the n maps from zp to the n to b and then each of them has a order of b to the n uh chance of surviving and actually being a map from the co kernel um and indeed one can use this and this is uh done in the notes with a convergence theorem like dominated convergence theorem uh you you can interchange the limit and the expected value like i said one doesn't just have that this is a limit but you can write down this number of surjections very explicitly uh and then you can use that um with a convergence theorem to show that indeed uh the moments of this colon lindstra distribution are one any questions okay so um all right so just in summary if uh we knew overclass groups that the average number of surjections and man i really don't like writing these infinities in here from the co p subgroup of uh our class groups uh was one for every b then we would uh uh we would actually know this is from this moment moment determining the distribution theorem that i mentioned before we would actually know that the proportion with any particular co p subgroup of their class group was as predicted um but as i pointed out at the beginning but not vice versa uh so so and because of because of the appearance of the limit in the whole whole story uh so in that sense knowing these limiting moments is it's it's strictly stronger than than knowing the limiting distribution it implies uh the limiting distribution all right so in the next um lecture is it my final lecture tomorrow i'm going to talk about the generalization uh of these conjectures to class groups of higher degree extensions so so far it's been all class groups of quadratic extensions just because that was a good place to start um but we're interested in of course class groups of higher degree extensions as well now um um because of the fact uh that um uh the maybe i should say here a class group sort of class group elements elements of class groups of quadratic fields correspond to orbits of binary quadratic forms so i'll just say so this is a this is a correspondence as far as i understand that that's due to denokin so you often hear people talking about oh gauss conjectured this and that and this about class groups of quadratic fields gauss was really conjecturing uh about uh the orbits of of binary quadratic forms and now we just translate that into the language of class groups but because of this um because of this relationship and then us having a good reduction theory over here on the binary quadratic forms uh there are very large tables of class groups of quadratic fields so a lot of data available um and that uh you know data has played an important role as i mentioned before in both sort of uh inspiring you know creating a need for conjectures to sort of explain phenomena that were being observed empirically and also to provide evidence uh for those conjectures and these very large tables uh show that this cone lister conjectures for quadratic fields look really good i think there's every every reason to to believe them um exactly exactly as i've stated and um so in higher degree uh one does not have this same this same correspondence um and so the tables are much smaller uh at computations of are much slower and all of the kinds of statements that we've made so far about distributions of class groups of quadratic fields and the statements that we're going to make um tomorrow about higher degree fields are all just limiting statements their statements about limits but they don't include any any conjectural information on how quickly that limit is going to be approached or any kind of speed of convergence we say the limit of as x goes to infinity is supposed to say be one but we don't say how quickly were we planned to get there um and this introduces significant challenges uh in using empirical evidence for conjectures of class groups distributions of um of higher degree fields and and that is actually really important because unlike this situation where these uh these quadratic field conjectures look quite good there are there are various anomalies and issues with the higher uh degree conjectures some of which we will discuss tomorrow and so you would really like to so there's a much bigger space for for good empirical evidence uh to to play a role in understanding those conjectures so why am i talking about this um today uh uh before i i talk about all these other conjectures because i think that there's something important to be done in the quadratic case and um and that that is i think that in the case of the colon linscher heuristics for distributions of class groups of quadratic fields where we're pretty confident that the limits are are approaching the numbers that we we think they are it would be it would be excellent to have good heuristics predictions conjectures for things like the speed of convergence or error terms or secondary terms um um you know for these colon linscher conjectures here here i'm talking about four quadratic fields as a start um because this is a case where we're pretty confident in the in the actual limits and that makes it much easier to to be on solid ground to try to develop a theory or even just predictions from computations from tables of uh how quickly we're going to get there and then you know especially how uh does this sort of speed of convergence or error terms or secondary terms depend on which moment we're computing or maybe which group we're you know averaging how often it happens um and i think that this is especially interesting if it can be done in a way such that it would give insight into what we might imagine for higher degree fields um and so maybe it's a little bit uh you know confusing if all the real questions about what is is going on or not are for higher degree fields why would you start here um uh where the where where we believe very solidly in the conjectures it's because um it it's it's very hard to uh make guesses about the speed of convergence when you're not totally clear what limit you're actually even heading towards um so i think that having a good understanding um of uh of of what we should predict for these sorts of things uh will be in you know starting in the quadratic case maybe using that to build up to higher degree cases will be very important for future empirical investigation of the conjectures for class groups of higher degree fields um all right so i want to yeah i'll show a picture a little bit about what some of these um what some of these things look like so here is um a graph okay so there are four um lines uh uh in this graph and so these are um these the and they're three labeled here the three labeled are all um moments so these are moments of imaginary so these are of imaginary quadratic fields and so remember we said that the moments of the class groups uh you know of class groups of imaginary quadratic fields should all be one okay and so this this purple line is the z mod 5z moment the average number of surjections on the z mod 5z and the blue line uh is the z mod 3z moment average number of surjections to z mod 3z and the red line is the z mod 3z cross z mod 3z moment so it's uh and maybe i should explain this um axis here is saying if you take discriminant up to x and you see it's um plotted here the you know log of x over log of 2 so this is like you know up to discriminant of k is less than 2 to the 30 so this is as the discriminant increases you know up to you know 2 to the 5 2 to the 10 2 to the 15 2 to the 30 it looks like this goes up to 2 to the the 32 so as x is going to infinity the purple line and the blue line and the red line are all predicted um to to hit hit one here um and so um even from this picture one can observe many things uh uh one they they all seem to be coming up from below all right so it's not an asymptotic being being sort of approached like this or something they're all uh coming from below which really suggests some kind of secondary term um and that persists with with with you know this is just a sample but with other other moments this um uh hitting one from below persists so suggesting there's some sort of negative uh secondary term it's interesting uh that the here if you see the purple line the z mod 5z moment is a is a head it's getting there faster um which you might be kind of surprised just because it's a bigger group you might it feels like it should be harder to have uh subjections on to z mod 5z so that i really have no um explanation or understanding of of why uh the z mod 5z moment is ahead and then this of course this is this z mod 3z cross z mod 3z moment is is slower not even so clear uh maybe at this point um uh where it where it's going to level off and of course it theoretically this this you know red z mod 3z cross z mod 3z moment has to be behind the blue one because you can't start having subjections to z mod 3z cross z mod 3z until you have subjections to z mod 3z so um there's some theoretical uh restriction uh there uh so maybe yeah i'll mention um yes yes that's what i'm gonna say next good question okay um uh so uh yes so um this uh chart is is from a paper uh that was mostly interested in this other line this yellow and green line and these were being used as a reference to show the kind of what you would expect convergence uh uh to to be like so exactly in the uh in the um sort of uh uh way that i was suggesting uh for the class group distribution conjectures for higher degree fields however this is um this is a case of a generalization not two higher degree fields it's still of quadratic fields um but this is so remember we said that these uh you know z mod 5z moments is like the average number of unrammified z mod 5z extensions and these are the unrammified z mod 3z extensions and so this is about this green and yellow line is about counting unrammified um uh a four uh extensions so that's why this this a four of course um it's that isn't about uh surjections from the class group to to something it's surjections from the fundamental group of of spec okay or the gaol group of the maximal unrammified extension um so it's not directly a class group thing that's what you see here in the title of this paper not a billion coneland straw um uh moments uh and so uh the the context is that it was just it's very helpful when trying to uh to guess where this uh is going to land uh to have to have a sense of the growth of other analogous functions so just in case you're wondering we predict that this is going to land at two um and uh and since you believe that this is going to land at one uh then maybe you can believe this is going to land at two um the the other actually the other i'll just note another thing here in this picture there's the sort of green and the yellow um so uh the green are is a computation of actually all fields up to that discriminant and then the yellow uh is a computation of sampled fields so instead of all fields of you know uh discriminant we randomly sample uh some and if you are actually able to take a good um a good sample you expect that to be to be very good uh now for quadratic fields you probably aren't ever going to i mean it's going to be a while before you need to do that because it's so fast to compute uh class groups and some you know but here um one has to make much more expensive computations and so this is something that i discussed at several places in the notes um that especially uh especially once these computation the computations you want to do um are more expensive per field when they're doing things like computing class groups of higher degree fields you it's probably very much quicker and just as useful to have sampled data where you don't do all the fields up uh to uh discriminate two to the thirty but just have a a good good random sample of them all right so um all right so we do actually understand so one thing theoretically um about this picture which is actually this this z mod three z this blue line this convergence um i said oh it really looks like there's a secondary term there um in actually both cases but we know we we do know one thing um so we're you know the moral of these several talks is but we can count cubic fields that's the thing we could do all right so um there is a known secondary term uh so again it comes from from counting cubic fields really really well so um uh from work of uh barge of a shaker at zimmerman and taniguchi and thorn and then actually a more recent paper of barge of a taniguchi and thorn um we and this is a little bit of abuse of this notation but say for k either imaginary or real quadratic fields this z mod three z moment and the reason this is abusive notation is i said this was a limit but if you forget about the limit with this this average without the limit they prove is one minus and i wrote this minus here so this c is a is a positive constant so this is why it's coming up from below um x to the minus one sixth all right so this is so i was talking about secondary terms and error terms so this is precisely what i mean by a secondary term they give c um you know so c greater than zero is given explicitly and so that is that is explaining this um you know this gap here that it's negative and you see by the time you get to here it's sort of approaching at a very regular rate all right and that's the that's the the x to the minus one sixth rate uh and then beyond that um they have you know a further error term at x to the at x to the minus one third um uh so um yes yes oh except oh sorry for real quadratics um for real quadratics it's one the the value of the moment it's one third and the and the constant this next constant is different too but it's the oh so this this is this is the real yeah the real case all right so we'll say that that is for imaginary quadratics and this is for real quadratics so i haven't talked about you know we didn't talk about the values of the moments those are those are for the real quadratic case they're one over the size of the group and that's that's worked out in detail um in the notes but they have the same uh you know the same shape secondary term also negative just with a different constant um then the error term okay uh yes and so uh just the final thing i want to say is since i'm encouraging a more computational investigation uh in to um you know into the speed of convergence and sort of secondary terms and error terms of these i want to point out um uh one uh one paper so this is of um lewis and williams sort of who did some numerical investigation into secondary terms and this was a um an undergraduate research project it has has some nice data and like if you want to think about this i think you know you you should start and look and and see um and i just pulled up one uh one chart from their paper to highlight so this is the the difference um that they're seeing between a predicted a value and an actual value and these axes here start at zero and so the fact that uh um the fact that the differences are always positive you see even in the five seven equals eleven twenty the fact that these differences are always positive and they're not going um below this line is just more cases of this claim that i i i told you seems to always empirically happen um that um these averages seem to be approached from below uh and also i think it's a very um very interesting question to understand sort of heuristically like um why why that is is so strongly the case um all right that is it for today okay we have


Hello, and welcome to the Prosperity Alchemist channel. I'm David Wong, and we're going to start this video with testicles. So according to Professor Matthew Walker, he's the director of UC Berkeley's lab for sleep and neuroimaging. He discovered that men who sleep five to six hours regularly per night have significantly smaller testicles than men who sleep regularly eight hours or more per night. And by measuring the testosterone levels of the men who sleep only six hours a night, they have the same testosterone level of men who are 10 years senior than them. They also found that the same goes for reproductive health in females. Well, that's the good news. So you want to know the bad news? Okay, well, the bad news is that not only does it affect your testosterone levels or hormones, it affects your brain function too. So what it is, it's an experiment with two groups of people. One group of people had eight hours of sleep. The other group of people didn't have any sleep. They monitored them. It was very miserable for people who did not have sleep. And the next day, they put them into an MRI machine to measure all the brain activity and they gave them a list of things and facts to memorize. And from the test results afterwards, they found that the people who had only six hours of sleep had 40% less cognitive ability and memory compared to people who had eight hours of sleep.   And they also found that you need sleep before you learn to in order to clean out your memory circuits. And if you don't sleep before, your circuits are already overloaded and cannot absorb more information. It's a fact that one out of two American adults sleep less than six hours a day. Studies found that the sleep gets worse with age. They found that the older people get, they lower the quality of sleep. So what does it mean to have quality sleep? So what they discovered was that during deep sleep, your brain would have these electrical spikes. And during those spikes were when the data in your brain, so to speak, was being transferred from the short-term memory into the long-term memory in the cortex of your brain. With age, people are having a harder time or the brain is having a harder time to get into those really deep, high-quality sleep states. And now they're associating the disruption of deep sleep with cognitive decline and diseases like Alzheimer's. So they found that there was something called genetic distortion or DNA distortion when you had lack of sleep. So there's a group of genes that were heightened in their abilities. And there were a group of genes that were lessened. The ones that were lessened were the genes that were associated with immune function. So their immune function would be lessened. And the ones that were heightened were the genes that were responsible for increased tumors, inflammation in your body, stress and cardiovascular disease. So when we lack sleep, we are actually genetically modifying ourselves and programming ourselves in the DNA level for sickness and for disease. So you're probably asking, how can I get deep sleep? I don't have enough time to sleep eight hours a day. Is there a way for me to get really high quality sleep during those even six or even less hours a day? Well, there are some things that you can do in order to increase the quality of your sleep so that your brain will get into those states. So what are those states? Those states are actually called delta and theta brain waves.  And this brings us to the technology that I've been developing with some scientists and some holistic practitioners right now. So we have the technology using electromagnetic fields in order to induce these states in the brain in a matter of minutes. So what we did is we have clients who come to us with insomnia, with anxiety who cannot sleep. So for example, some students that have been worried with exams and have huge anxiety problems and they haven't been able to sleep in months. We're talking about probably three to four hours of sleep per day that's gone for months. They're totally remissurable and their health is totally decline. And they come to us and we use our technology and within minutes they fall asleep and in minutes they are asleep on the table right in front of us and they stay like that for the whole session. So this is just completely amazing. We have this new technology using electromagnetic frequencies. Totally non-invasive, doesn't even have to use any electrodes. So with people who are having difficulty getting into those deep sleep states, we are able to get them into delta and theta sleep brain wave states in a matter of minutes. So you will feel more recharged, you will feel more clarity, you'll feel more awake, more aware. Your memory will increase and your cognitive ability will increase. And also you will feel a lot happier and you'll feel a lot more optimistic about the future too. So just imagine by experiencing the rejuvenation and revitalization by experiencing these deep brain wave states that you only get through deep sleep. So imagine being able to program your DNA for health and also make yourself smarter, have better memory, have more focus and more clarity. So you want to learn more about the frequencies and waveforms that we discovered that help to induce these deep, deep relaxation and deep, deep sleep states. And all the technology and research that's behind it, make sure that you attend my webinar that's coming up. It's free and you can learn more about this technology. So I hope you enjoyed this video. If you like it, make sure you click the like button, comment if you have questions or you have some suggestions. And make sure you share this with your friends and subscribe to this channel. So make sure you click the notification bell. You get instant notification of all the new videos coming out. So until next time, use the force and prosper.


What's going on everyone Sam here late in sports car driven one box of 22 Pini Chronicles draft basketball hobby for Joshua L. So good luck here Joshua. See what we got man Yeah, and Bob A is in the new prism tomorrow. I have autos in it too. Just pretty awesome So definitely something we're really excited for man Congren Wow Tyrese Halliburton For Iowa State nice one there. I'm now a pacer That's wild Thurin Shaden Sharp Jovic and Jason Tate. Yeah, that's tough. That's really tough Jalen Duran. There's a 1.99. We got Devin Booker Kentucky nice one there Wildcats There's a Palo power formulas Keegan Murray Jaden Hardy rise above Palo optic and Malachi Brenna or DeVoe Michael DeVoe. That's what it is For George attack nice one there on the foil not numbered contenders the yellow jackets. We got a red Quartz side coming up here. That's gonna be oh check Bajie. Nice 93 out of 149. Yeah, that's true too That's true too. Yeah, the imbop a caller is gonna be crazy dude Yeah, Admiral tomorrow morning forest is streaming at Uh Like 11 and then I come and take over at five, but yeah, we're doing prism pretty much all day We've got Johnny Davis for Wisconsin nice 93 out of 99 on the gold standard Congrats Joshua. That's nice Dyson Daniels Silver Jalen Duran And Nicola Jovic Shane save me on flag For what is that same Houston state? Nice one there Save me on flag. You don't know save me on flag. No Scotty Barnes For a state Jaden Ivy Quartz side and Keegan Murray Yes Jaden I've Johnny Davis check home grand power formulas. That's not numbered Walker Kessler and EJ Liddell That'll do it for your break there Joshua. Thanks again for the sport and jump it in we'll get them right out to you, buddy


Okay, so before we get to making up the fountain, we'll do a quick unboxing of the solar-powered fountain pump that I purchased off of Amazon. I will share the link below. So this cost me $21 plus taxes and was available through Amazon Prime. So there was free delivery and I received it in three days. So the cord according to the product description is 2.7 meters long, which is a nice size. I should be able to set this up in my bird bath and find a spot to hang the solar panel not too far away. So this comes with six different attachments to create different types of sprinkling. I am planning to just use something that is very low sprinkling. I basically just want to set this up so that it keeps the water moving to kind of keep the water from getting too stagnant and create a spot for mosquitoes to lay their eggs. I find that a lot of these when you get them set up and they it is you know a very sunny day that they shoot up the water pretty high and if you have a lot of wind which we do here all the time it basically just blows all the water out of your bird bath and you end up with no water at the end of the day. So we will see once we get this set up out in the sunshine what kind of water pressure it has and build our water source for the bees and the birds and the pollinators kind of around that. So let's take it outside and get her set up. I am here for this water source that I want to try and create for the birds and the butterflies the bees insects is to try and create some kind of a container that has a good spot for the birds to sit so they can bath and also you know have some wet rocks that you know the insects and the bees and the butterflies can come and have a quick drink. So I am just going to test out see how powerful this is to start with. So this is basically the solar panel and this piece which has some suction cups they don't work very good I can tell you that already but I'm just going to try and secure it under the water with some some rocks. So I'll just show you kind of what the flow looks like to start with. So this is without any of the attachments and this is kind of what I would prefer to use because it's not spraying too far or too high and it's shouldn't scare away anything. If you turn it away from the sun it just kind of loops a little bit so I'm assuming in on cloudy days it won't be such a high flow on it. We'll just try a couple attachments here I'll just show you how they work. So it comes with four different tops here that you can switch out. I've tried them and they all seem to kind of do the same thing but once you put it on here submerse it under the water put the solar panel in full direct sunlight you can see how high it shoots up. So if that's kind of feature you're looking for that's great. I can see that in on a windy day that this water will all get blown out of my bowl here and you know eventually dry it out. So I really don't want a high rising spray like this. I'm not going to use these features I just wanted to show you what they look like full sun and I'm assuming if it's a cloudy day they probably won't spray a whole lot out. Let's see if we can just tone down this spray a little bit with some rocks. I'm going to try and secure it for one so that it doesn't move. I think it's just a matter of playing with whatever kind of rocks you have. I don't have very good rocks here in my yard. I have to go foraging to find some good rocks. I kind of like to just block it off a little bit so it just kind of flows nicely over top of these rocks. It's very simple but I can see you know the bees and the butterflies could come and just sit on these rocks get something to drink and the birds can land on these bigger bricks. So I'm just going to leave it like this for now that was pretty simple. So when you're finding a spot to put your water source here for your pollinators and for the birds of course you want to put it somewhere in your garden and I've got it here in my potato patch and I've set it up near this post here that I have bird feeder hanging on so that I have something to secure the solar panel too. So you want to make sure that you put your solar panel somewhere that is going to get the most amount of sunlight all day long. This spot here may not get a the early morning sunlight but it should get for the most part a lot of sunlight. Something like some twisty ties like this should work good too as well so that's what I'm going to do right now. I just want to also show you a setup I have here with an electric pump in my bird bath that sits outside my living room window. So I get to see a lot of great bird baths going on here and because I have power close to the house here I have an electric pump set up so basically just the same kind of setup with an electric pump that I've covered up with a bunch of rocks just to kind of slow down the flow, keeps the water moving, keeps the mosquitoes from laying eggs in here and I get to see all sorts of wildlife come and take drinks and have baths here. So that's pretty much it on how to set up a water source for your bees, your birds, the hummingbirds hopefully, maybe some of the butterflies and dragonflies will come here as well. So it's not really much to look at but its purpose is solely to provide a good water source for all those pollinators so that they stick around in your garden and do their job and help you out with pollinating all your garden plants.


Hey everyone welcome to today's day trading update today's Thursday November 5th. I hope everybody's doing well today. Nice green day today plus $605 that's back-to-back days over 600 bucks so that's always nice.  Disney was a runner I was trying to catch an upside runner on Disney right let's see where was that oh yeah big push up started to pull back tried to get long here for continuation and just kept just kept grinding lower so I ended up just closing that out cutting my loss I found other positions I would rather be in better high quality high quality opportunities so I just closed out my Disney and jumped in something else at that time Microsoft small loser of 105 Microsoft I took as an early early morning runner so we had this initial push up above the pivot it started to consolidate this is a little bit earlier than I typically take runners it started to work out really nicely I actually got out of some and then it just rolled over and flushed me out of the rest so I took a small loss of 105 Mu Mu was a runner as well so after it pushed up here and started to consolidate I got long and was able to run some of that up it did roll over a little bit when it started to roll over here I just got out of the rest and ended up still booking a profit of 154 in Netflix I made 289 let's take a look at Netflix Netflix we did a an upside runner in here so Netflix pushed up came back down we got long in this area here and rode it up um and then it came back down I added to it again and rode it back up a little bit a little bit longer so nice winner netflix for 289 in video was my biggest loser minus 315 and this was a situation where it had this big push up I got long it didn't really stop and consolidate much I just I just wanted to get long at this pivot to see if it would make a quick bounce unfortunately it did not flush me out so I took the loss there peloton my biggest winner plus 460 peloton it was I was watching this one all morning and it it was consolidating consolidating finally I got long and had to hold it through all this choppy chop chop which is kind of excruciating in real time but ended up catching a late day run and finally at this point about one o'clock I just I just bailed on it and ended up taking a a nice winner had I held on a little bit longer could have gotten quite a bit more but that's okay tesla minus 270 so I had a couple trades in tesla one of which was a mighty 90 so on this initial push up as it was approaching the highs of yesterday I got short took some heat pushed higher I was almost at a point of wanting to add to it but I didn't and it flushed down so ended up just taking basically a scratch plus 10 bucks on that one and then I tried to get back in for a runner it was coming down and made a nice bounce and then just rolled over so I ended up taking a loss on that one so net net minus 270 on tesla uber this is one that I've been I was trying to trade yesterday and trying to trade again today so yesterday right out of the gate it hit a new all-time high a lot of times when that happens when it hits a or it's close to all-time highs it'll act like a magnet and want to come and take that out and so after this big initial push I was looking for an area to get long finally got long right in here added a little bit in here and just got out some here sat through this sat through this all day finally right at the end of the day here I read about one o'clock got out of my last piece of uber right here up at the pivot never did hit that all-time high but still booked a nice profit 250 on the trade and then lastly zoom zoom was a nice trade here zoom let me zoom zoom in on this a little bit when zoom came down and this initial push this is a mighty 90 trade I got long right here at yesterday's high it took a little heat for just a few minutes and then boom popped right up for a nice $310 profit in zoom so that's everything plus 605 for the day will be live streaming again next monday we are off friday which is tomorrow so no live stream tomorrow but we'll be back live streaming on monday at the market opens guys have any questions let me know otherwise we'll see you then