to Community Matters here on Think Tech, the three o'clock clock on a given Monday. And we're talking about failure, solution, and fallout. We're from here. I'm by two veterans. But we're talking about a commentary they wrote, namely Vic Kraft and John Reese. Welcome to the show, you guys. Thank you. So we got that commentary. I thought it was very interesting. And important that you wrote it. And we posted it on our Think Tech blog, and we thought we'd have a show about it too. And I think this is very valuable to discuss your views and where you get that from. So the first thing you do, Vic, is tell us a little about your background in life. Hold it down to six hours, wouldn't you? Okay. Let me stop the start while I share it. Okay. Take it from the brief bio that I sent you guys and what you know of me from years past. John and I were both in the Air Force at Station Williams Air Force Base many, many moons ago. I raised Jim from a second lieutenant. And as me as a staff sergeant at the time, I was interested in meteorology and wanted to do it as a hobby. And I got some master sergeant pushing him on me. And we've known each other now for almost 50 years.  What brought you to collaborate on the piece? We've been writing together for a long time. We wrote a book together a long time ago called Pacific Cauldron. It was about what would happen to Hawaii should there no longer be a United States. And we've done some other things together as far as writing. We've been practicing this for years, writing about defense issues, space, the environment, various other issues around. Well, how important is this piece? I guess it's like maybe 1500 or 2000 words. And it goes into pretty lofty thoughts and concerns about the future of the country. But what brought you to do this? And how important is this for you guys? For me, when I saw the Afghanis running along the side of the aircraft, trying to grab ahold of it, thinking that they were going to be blown away to safety, it kind of motivated me to think about giving service and also what you're willing to commit. These guys just wanted to get the heck out of there. They didn't want to do anything. It seemed as though they didn't want to commit anything other than their own safety. And I thought of the sacrifice that we had made in the 20-some hundred lives of American Armed Forces people in Afghanistan. And what was it all for? And how did we get there? This has made me ask the questions. Okay, we'll go into the piece in a minute. But, John, how much of what Vick said you agree with? You're coming from the same place? Well, we're both operating out of a sense of frustration more than anything else because we're both futureists. We both have an abiding interest in science fiction and looking at the future or whatever. And we both know how good things could be. And we're frustrated at the fact that despite all of our technical sophistication and economic power and everything else that we still seem to both basically shoot ourselves in the foot as Americans or whatever. Americans have a great, great talent for writing a really sophisticated, tightly knit plot for a one-hour show or a two-hour movie. And it's like, why can't we do that? Why can't we do at least some of that in real life?  We've done things right in the past, World War II. I mean, not only did we contribute to winning that war, but we had a plan afterwards called the Marshall Plan. And all that's on the shelf. It's there. It's not some corporate proprietary information that's available to use. So it's frustrating for us to see us go into a situation like Afghanistan. And we have all these ready-made friends sitting on the shelf or whatever. And nobody seems willing to make the effort or make the effort to put them into practice. What's the lens that you see this from? I mean, both of you guys had long careers in and around the military. So I guess maybe this is hard for you to answer. But what's the lens? From what point of view do you look at these issues? I mean, the 30,000-foot level or? No, personally. Personally? Personally. Yeah. Well, you know, Vic was an actual combat veteran in Vietnam. And I listed in 73.  I didn't happen to end up in Vietnam. My background was in and when I entered the Air Force, I already had my bachelor's degree in applied mathematics. And the Air Force wanted to make use of that. They decided I should be a computer programmer. But still, you know, my career went from 73 to 93. So I went from Vietnam to Desert Storm. Well, a lot of people in the service who have been in the service, maybe I'm thinking of recent years, they treat themselves as apolitical. And some of them treat themselves as, you know, kind of conservative. Yeah. A lot of them support Trump right now, today. Yes, I mean, very troubling, troubling to my generation anyway. But it's very true. It's very troubling to me, too. I have to think of myself as apolitical, you know, but, you know, I've been through the Carter years, I've been through the Reagan years, I've been through the Clinton years and everything else. But the idea is that, well, it's good to be found, it's good and bad to be found in both sides.  And Trump is anathema.  And I can find virtually zero worthwhile in him in particular and his enablers within the Republican Party. That's not to say that I'm a fully committed Democrat, you know, because I think they have their issues also. But let me come to his defense in this regard. I think both of us we view ourselves as Americans first in terms of what we raised our right hand for in terms of supporting the Constitution. That's, I mean, the principles and ideals that were set up in the Constitution and what that's what we we put a uniform on for. And when I see people storming the Capitol, and I see people trying to grab onto an airplane to get their freedom, recognizing that they're both giving up. The people who stormed the Capitol gave up on the processes in the system. They were willing to abandon the law and take the law into their own hand. The people in Kabul who were trying to get on board these aircraft had abandoned whatever they were they were living in for fear of losing their freedoms or whatever. We fought for those freedoms. 22 or 2400 people died for those freedoms. And thousands upon thousands of Afghani soldiers also died. And these guys are thinking, I'm going to grab onto an airplane or I'm going to crawl over these other people to get my freedom and get away from them. I have no sympathy for them. And I have no sympathy for the people on January the 6th who gave up on America. They should be looking at themselves as being Americans and not just Trumpers or anything.  Okay, well let's go to your piece, your commentary. It covers three things, which is why we fashioned the name of the show as failure, solution, and fallout. So you spend a bit of time writing about the failures. And I guess it's driven, Vic, it's driven off the Afghanistan experience. But can you summarize what you're saying about the failures in your paper? I think John said it earlier, it's kind of like a proper pre-planning prevents this poor performance. If you're going to go invade somebody, you've got a plan. And if you're going to have a plan, you're going to have some means of what defines what your goals are. I don't think we really had any goals in Afghanistan. It was interesting, one of my mentors called me up the day after 9-11 occurred. And he was an army intelligence officer.  We're going to go down the military industrial path to try to fight these people in Afghanistan and in Iraq. No, no, the way to do this was to follow the money. It was also to be clandestine and fighting this thing. And we did not take any kind of lesson out of Vietnam in that we were going to put boots on the ground and this is how we're going to solve the problem. It doesn't. And that's one of the things that we wrote about in the piece is our foreign policy has been driven by defense and paranoia as opposed to the other way around in that our foreign policy should be driving defense policy. I constantly hear the business of American interests. What are those interests? Protecting American interests. What are they? And delineated where? And you read the defense posture plan that comes out about every five years for the decennial report.  We have the largest armed forces in the world in terms of carriers, other weaponry before. What are we using? Are we concerned about banana republics invading the United States? Okay, so these are failures. What would you add to that, John, in terms of the treatment by your joint paper here about these profound failures? Failures about which you and Vic are concerned? Well, it's, you know, Vic summed it up pretty well. It's the lack of setting clear objectives that, you know, okay, ostensibly, we went into Afghanistan because the Taliban government that was in place at the time was sheltering Al Qaeda. And Al Qaeda launched the 9-11 attack. So we wanted to not only eliminate Al Qaeda, we wanted to eliminate the group that was supporting them. But we accomplished that. We chased Al Qaeda, you know, Al Qaeda out of, out of Afghanistan. It took us several more years to finally track down Osama bin Laden in Pakistan. But, you know, we accomplished the first, first goal. And we, we knocked the Taliban, you know, out of, out of government there or whatever. But we essentially went in with no plan of what to replace it with. And not only that, we went in with no plan for an Afghan government. We went in basically with wishful thinking that once we eliminated the Taliban, the Afghans would rise up and embrace the principles stated in the U.S. Constitution and set up a, you know, set up their own republic, you know, or democracy in the shape of the United States government. Well, they have no experience with doing that. We needed, we needed to set up a, you know, we needed a plan to set up an Afghan government, not a copy of the U.S. government. Because they had, you know, they had different structures there. They had, you know, village chiefs. They had, you know, various other, you know, ethnic minorities and things that we just ignored. And it's like, you know, we went in with a plan across your fingers and hope that they form a government after we, after we take care of the Taliban. And we finished the first two girls with a matter of one or two years. And then what, what did we spend the remaining 18 years doing? Trying to prop up a government that wasn't working. Yeah. Jake, there was some accommodation of failures there. Yes. Let me say something, Jake. This is something that has troubled me for the last 60 years. A guy by the name of William Letter wrote very interesting books back in the 50s. One was All the Ships at Sea, Nation of Sheep. He was also co-author of The Ugly American. And he pointed out the failures at that time of the American diplomatic policy in Asia. And I stopped to think about this. When we went into Afghanistan, I'm sure that there were people who were sitting on the desk with the specialists, the least subject matter experts within the State Department, who were probably feeding memos to their management saying, this is a bad idea. And it got distorted. Okay. So, yeah, it is say no, no proper pre-planning to go in there. And I had a conversation with a friend of mine who was a retired diplomat. And I said, he had been to Kabul and been talking with Hamid Karzai several times. And I asked him, I said, just exactly what is Afghanistan? Considering that it has had a history of a lot of warlords, you got a lot of drug use or drug production going on over there, how do they make a living? And I said, Afghanistan seems to me to be about a 60 mile radius around Kabul. And he said, oddly enough, Hamid Karzai had turned to him and said, yes, his writ does not extend beyond the city limits. So, exactly, we did not know what the heck to do. And didn't do our homework, went in there. That's a huge failure right there. We're going to. So, who's responsible? Is it the military? Is it the State Department? Is it Congress? What do you mean by us? Are you and I and John recharged with knowing all these things? We didn't have any authority to do anything. So, clearly, the people with the authority to do something, who made the mistakes? We're a Democratic Republic. We are a Democratic form of government. We are supposed to have our representatives represent our views. Given the feeling of the American populace today and its lack of advocacy in its own institutions, that sounds pretty far. I follow you, but it's the idea is that people are frustrated and they don't trust in their own system. How did we get out of Vietnam? It was the pressure of the public on its representatives. It took a while to do it, but at least it was a matter of numbers. And once the numbers grew and people or the representatives recognized their constituents were not happy, guess what? Things started to change. Okay, well, I suppose that people around the country were not fully aware of what was going on in Afghanistan and maybe they should have been out in the streets protesting and writing letters to Congress, which they didn't do. Most people went along with it. And let's assume for this discussion that there were huge failures, failures by all the government agencies involved, including presidents, the intelligence agencies, the military who never really learned how to speak the language, and of course the State Department and the coordination of all of them for 20 years. But let's go to the second area of your discussion. John, you talked about solutions. You and Vic came up with three or four solutions. This is really important because we're on a constructive track here. We want to find answers so that these failures are not repeated. What are your solutions, John? Well, first and foremost is, if we admit the fact that the failures occurred due to what amounts to poor leadership, the first solution is we need better leadership. It's easy to state, but then how do you go about doing that? And quite frankly, in today's current political environment, they talk about polarization. Well, that's a very apt description. It's divided between liberal Democrats and conservative Republicans and the definitions have been distorted to the point where a conservative from 50 years ago wouldn't recognize the conservatives, the contemporary conservatives. That be that as it may, our failure seems to be stemming from, quite frankly, our two-party system. Our election. Every election night, they talk about the swing voters, the independent voters, that each party has its own base that seems to be locked in, but the final decisions seem to be made by the swing voters, and yet the swing voters don't have any representation. Okay, so to me, as a mathematician, the solution seems to be we need a third party. Now, we need a third party to completely take over, dominate the politics. No. There's already been talk in the Republican party of Republicans dissatisfied with the Trump supporters. And there was some talk a few months ago, apparently not too serious, but some spoken speculation that maybe we ought to break off and farm our own party, which I was going to say yes, for the trigger. Good.  You know, they wouldn't have a threat of a majority in either house of Congress, you know, in either chamber of Congress. What about the other possible solutions that you included in the paper? Oh, we talked about national service. Vic and I, as members of the military, and oh, by the way, as an retired Air Force officer, retirement is not a discharge. I am still officially a member of the US Air Force, okay? My status is retired. When I retired, I did not receive a discharge. I wasn't separated from the service. I'm still part of it. And the thing is, and Vic has the same, his technical, his legal status might be, you know, not the same as mine, but his mental status and in his heart, it still is, okay? We recognize the value of national service. And we have other examples on the planet. Vic is always bringing up Switzerland, okay? Everybody in Switzerland ends up doing some national service at some point in their life. And whether that's in the military or some other function, you know, in the United States, we have the Peace Corps, we have, you know, other things like that. We could make up things like that. But, you know, you've got to get involved. You got to, as we put it in our paper, you got to get some skin in the game. You got to have some investment.  And, you know, the whole idea after Vietnam, we formed this all volunteer force.   Let me add to that, because this is one of the things Jay, I think that's important as far as the skin in the game aspect of this thing, how much debate has gone on since we left Afghanistan. I haven't seen anything in the press. People aren't talking about it. I don't know whether they're ashamed or what happened there, but there doesn't seem to be anything other than reprimanations against the current administration. That's all their fault. Now, we as a country are responsible for this. We are a Democratic Republic. We elect our representatives to go to Congress that are supposed to speak for us, or at least they are to speak to their district. And if their district is against a particular issue, then they need to raise that. And that's where the leadership comes in. Now, John mentioned the Republicans having some difficulties in keeping everybody in line. Democrats are having the same problem in Congress right now. And perhaps maybe we will get some leadership out of this. I don't know, but it's a matter of what you're willing to accept. One of the professional delegates to Washington from Hawaii has shown that kind of leadership. And bucking the system and saying, no, I don't want to do it your way. There is a better way. Vic, if I could jump in here, if 10% of the Republicans left the Republican Party and 10% of the Democrats joined them, forming a block, 20% of the current representation or whatever, that would have a moderating effect on the other two remaining parties. Because each party, with their 40% share, their plurality, whatever, would be trying to get that 20% swing vote or whatever enlisted in their various causes, which would cause them to moderate their policies to try and attract that middle 20% block. Well, I want to get to Jay's question regarding the getting to the how. And we started that at the beginning of the program. How do we go about doing this? Peter Adler wrote a very interesting piece in Civil Beat last week. He called it Gorilla Bridge Building. And basically, we know that Peter is a mediator. That's his specialty. But I recommend that you go read the article. It's very good. And basically, brings about the idea that we work at these things a little bit at a time, and we don't take issue with the other people as far as putting labels on it. You find collaboration. And that's what we need to do is find collaboration. How we go about doing it? I don't know, one step at a time, Jay. I don't know what else to do. But it's a matter of I've invested my life in the Constitution and the ideals of this country. I'm not willing to give it up. Hey, we're almost out of time, guys. So, John, let me ask you for a minute of closing remarks of how you want to leave this with our viewing audience. What would you like them to remember from this discussion? Well, we have to encourage the idea that collaboration and cooperation are not dirty words. That both sides of an issue can have some constructive thoughts. And compromise doesn't have to be a process of whittling away until the only thing that's left is mutually abhorrent to both parties. Compromise should be surveying the points of both sides and picking out the good things from both sides and then synthesizing those together to produce something greater than the original whole. Okay? Compromise should be a building effort, not a destructive effort. Hey, we've got to leave it there. What would you like to leave with our audience? The former Hawaiian legislature said to me one time, because politicians have such a bad reputation with the public, he said politics is the art of compromise. And that is one of the things that we need to work on. And the problem, as we've pointed out in our article and also in this interview, is the lack of people wanting to compromise. And that's why I get back to Peter Adler's business of derelict collaboration. There's got to be some people that are going to be willing to stand up and say, I'm not going to follow the dog. And I think of John and Kennedy and profiles and courage and the people that he wrote about and what they stood for. And there is no, as Kennedy said, no Republican answer or Democrat answer, just the right answer. That's what we need to look for. Big craft veteran, efforts veteran, and John Reece, another efforts veteran, been around the block and we're writing together for years and been thinking about our situation, both through patriots. Thank you very much, Vic. Thank you, John. Hello. Thank you, Jay.


Hello everybody! Watch this video to find out how to recover data from hard disks within an x-ray array based on a Netgear NAS device. You will also learn how to enable FTP protocol and configure a shy folder in readyNAS. In our channel and blog, you will find solutions to any problem, from installing an operating system or configuring it to fixing possible bugs and errors or optimizing mobile gadgets. Our specialists will answer any questions you ask in your comments under the videos for articles. In this Netgear device known as readyNAS, the disks which are connected and used to store data are automatically combined into an x-ray disk array. This technology can automatically take all the disk capacity which is not currently in use and make it redundant regardless of the size and type of disk combined this way. All server functionality is managed by Linux's beta-operating system known as NASOS for Netgear. All Netgear devices can be classified into two groups. The first group comprises devices produced before 2017 and managed by NASOS version 5 or lower. In these devices, the data storage scheme looks like this. Devices are divided into partitions which are then combined into a RAID system with AMD technology. And then AMD arrays are combined into a unified LVM volume. As to the file system, such configurations tend to use EXT. The second group includes devices using NASOS 6 and produced in 2013 or later. These modern Netgear storage systems treat data differently. Yes, their hard disks are still combined into a RAID system with AMD technology, but the next level is the BTRFS file system instead of LVM. Just as any other storage device, such systems may suffer from breakdowns, hardware and software failures. That is why, when choosing a data recovery tool to restore data from disk arrays based on the RAIDing NAS, you should know it if the tool supports the file system used with a specific network drive, depending on the type of the storage device. Any changes you make to the RAID level of your RAIDing NAS volume will erase all data. That is why, before changing the RAID level on modifying other settings, which may potentially erase the disks, it is crucial to have a backup copy of important information and keep it on another hard disk to prevent data loss. If some information was accidentally deleted, hard disks were formatted or the storage device was damaged. In all these cases, you will need a specialized software product to recover your data. In today's video, I'll also show you how to choose a suitable tool for the job. Let's explore this XRAID technology to get a basic understanding of how it works before we choose a recovery tool and start using it. XRAID is an auto-expandable RAID technology that is available only on RAIDing NAS systems. With XRAID, you don't need to know intricate details about RAID to administer your system. XRAID allows you to add storage space without reformatting your drives or moving your data to another location. Because the expansion happens online, you can continue to use your RAIDing NAS system while the volume capacity increases. Because XRAID is a single-volume architecture, if you configure your hard disk drives to use XRAID, your storage system has only one volume that is made up of all installed hard disk drives. XRAID's single-volume architecture has two major advantages – easy system management and auto-expansion. If you replace existing disks with larger capacity disks, XRAID automatically accommodates the new disks. With XRAID, you can configure the system with one hard disk, add a second disk for data protection and add more disks later for additional storage capacity. Keep in mind, though, that XRAID requires a minimum of two hard disks to provide protection against disk failure. This RAID tag uses the capacity of one disk for data storage and reserves the capacity of a second disk for data protection, which allows the volume to recreate data if a disk fails. To disk system, the usable storage space is one disk. In S3 disk system, the usable storage space is two disks.  In this picture on the screen, you can see how XRAID uses new disks. The first disk that you install is used for initial, unprotected storage space. The second disk that you install is reserved for data protection, it's parity information. Installing additional disks increases your storage space. Although you can reuse the physical disk from another system to add capacity, any existing data on the disk must be removed before the disk can be added to the XRAID volume. If there is any important data on the disk, remember to copy it to another storage device before proceeding. Now that we know what XRAID is, the disk array is built automatically, so we can go on to find out how to configure network access to the drive. Start by opening the services tab. Standard file protocols. To enable FTP protocol, check the box next to the option and click to apply it. Now that FTP is active, we can connect to the server, add a shared folder and configure network access. To add a new shared folder, choose the Shared tab, click Add Shared Resources, give the folder name, add description and password if necessary, then click to apply the changes. Now that the shared folder is ready, let's configure network access to this directory. The directory is added with default access. To configure access to a shared resource, go to the page List of Shared Resources. Next to the new folder, click on the network protocol that needs to be enabled. After that, modify the default access time only for reading or read and write, then click apply. Network access is active, so now you can connect to the server by FTP and write some information to the disk. Unfortunately, this NAS device doesn't have a recycle bin feature for shared folders, so if something gets removed accidentally, there is no way to restore it, until you get to use Hetman Red Recovery. Modern software fillers, run configuration, accidental removal or formatting may cause a major data loss from the disk's storage inside your NAS device. In any of these situations, you'll need a reliable tool to get your information back. Not all recovery software products support disk rate time, so you have to keep this aspect in mind when choosing which tool to use. Also, most programs supporting x-rayed are designed in such a way that they don't find much data if one of the disks in the array was damaged. On the other hand, Hetman Red Recovery is a comprehensive solution for NAS data recovery to help you restore your files in many different scenarios, including the following. Hide-raying software fillers, run configuration, accidental removal files, disk formatting, and so on. This program supports the most popular file systems including those using NAS devices and all popular rate types including x-rayed. It can build an x-rayed system even if one disk is missing and still let you recover all the data you need. Bear in mind that before starting data recovery operations, you should prepare some free disk space with a capacity equal to the amount of data you are planning to recover from your disk array. Take the disks out of the NAS device and connect them to a Windows computer. Make sure that all disks are recognized by the Windows-integrated Disk Management utility. Right click on the Start Manage and select Disk Management. If you are suggested to initialize the disks, don't agree. Disk can erase the remaining data. If you are trying to recover data from a disk array and the motherboard of the computer where you want to connect the disks doesn't have enough setup words for all those hard disks, you can connect all but one disk, because this rate type remains operable even if one of the disks is missing. As remember, one of the disks is used to store parity information for others. Install and run Hetman Red Recovery. As I said before, here is how information is distributed across the disks. The first disk contains the operating system and data. The second one is a parity disk, and all other disks will expand the storage space for the first disk. At the beginning of the disk, you have service data, then the data partition, and these partitions are combined into a spent J-Bot volume, which is displayed in this program as a local hard disk. Here are the disks, each beginning with a dedicated space for service data, followed by data partitions, and some reserved free space. Here is the second disk containing parity data and two additional hard disks. And below, there is our network drive, a spent volume consisting of four disks. If the program manages to identify it properly, you will see its capacity and a file system type displayed here. To illustrate how the program works, I deliberately excluded with the first disk, as this array type is based on J-Bot, so all information is primarily written to the first disk until it's full and only then to other disks you have in this array. In the case with a missing disk, the program is supposed to read the parity disk and then build and display the restored disk array. To search for information, right-click on the disk and choose Open. Select the scan type and start the analysis. The program will analyze the disks and display their contents when the operation is complete. As you can see, it has found all the data stored on the network drive and in the case with x-rayed, the data is intact even though one disk is missing. If you are looking for files that have been removed, you can identify them by right crosses. Select the files you want to recover, click Recovery, select where you want to save them, and click Recovery again. When the recovery process is over, you'll find all of the files in the folder you have chosen. It is important to remember, however, that if such disk array suffers damage to more than one disk, you won't be able to recover all the data. Most files will either be damaged or unavailable for preview and recovery. And that is all for now. Hopefully, this video was useful. Remember to click the Like button and subscribe to our channel. Push the bell button to receive notifications and never miss new videos. Leave comments to ask questions. Thank you for watching. Good luck.


Josh Peterson shows us why the candidates feel confident and hope the success of the 2016 election continues into 2018. It's only been 24 hours since the 2018 campaign tickets have been finalized and has the GOP rallying in Bemidji for a fundraiser. Already top Republicans are out in force doing what they can to continue the success they experienced in 2016. The momentum for our president continues to remain really, really high and that trickles down to all of our candidates as they're going out and campaigning for their respective races. Coming off a strong primary win, gubernatorial candidate Jeff Johnson's running mate Donna Bergstrom says their ticket is ready to take on the Walls-Planigan campaign and stay on message. We've got too much government in our lives and I think we can make the case that Congressman Walts and Representative Planigan represent that. The Republicans feel they have the momentum and are looking to unseat long-term U.S. Senator Amy Klobuchar. She's against our President Trump and she wants to be in office for 18 years. Those two things right there really sum it up. A race that has received national attention is the contest for the 8th congressional district where Pete Stauber has received ongoing support from both President Trump and Vice President Mike Pence. The support for our president is as strong, if not stronger than an election day in Minnesota 8. And so we're feeling very good, we're very happy that the President and the Vice President has supported our campaign. Republicans are still reeling from the Trump train in 2016 and hope that the red wave can continue to carry them to victory come November. They're all confident President Trump in his tariff situation, namely they understand that that's a leverage for long-term gain and so I think they're ready to make a switch. In the race for Minnesota Attorney General, Republicans feel optimistic as they take on Congressman Keith Ellison. We need attorney generals to stand up for the rule of law and force all of the laws, keep our community safe and take the politics out of the attorney general's office. With 83 days remaining until election day, candidates from both parties will be out in force, vying for your vote. In Bemidji, Josh Peterson, Lakeland News. Over 250 people attended that event tonight. Republican candidate for Secretary of State John Howe and state auditor candidate Pam Myra were also in attendance. If you enjoyed this segment of Lakeland News, please consider making a tax-deductible contribution to Lakeland PBS.


In order for you to have a good, vulnerable, open dialogue between two people, it needs to be reciprocal. And part of that may be, if you're neurotypical, learning about autism, learning about how we work, how we think, the things that are different in our baseline level of communicating with people and the world. On the other side, autistic people, we do tend to struggle in a lot of key areas, particularly for relationships I've found was cognitive empathy and alexifimia. Alexifimia difficult to notice and categorise emotions you can see in a relationship, which is very, very emotionally based, or it should be, if it's love and it's a long, long lasting thing, then it's kind of important to know how you feel.


Hello and welcome to the session. In this session we discussed the following question which says evaluate the following integral 0 to pi by 2 sin square x upon sin x plus cos x dx. Let us see how we can evaluate this integral. First of all we assume let i be equal to the given integral which is integral 0 to pi by 2 sin square x upon sin x plus cos x dx. We have a property which says that integral 0 to a fx dx is equal to integral 0 to a f of a minus x dx. So using this property in the above given i we get i is equal to integral 0 to pi by 2 sin square so in place of x we will put a minus x that is pi by 2 minus x. So in the numerator it would be sin square pi by 2 minus x upon sin pi by 2 minus x plus cos pi by 2 minus x dx. Now this becomes i is equal to integral 0 to pi by 2 cos square x upon cos x plus sin x dx. Since we know that sin pi by 2 minus x is equal to cos x and cos pi by 2 minus x is equal to sin x and so sin square pi by 2 minus x is equal to cos square x. So using these results we get i is equal to integral 0 to pi by 2 cos square x upon cos x plus sin x dx. Now we assume this as equation 1 and this as equation 2. Now next we shall add equations 1 and 2. So adding equations 1 and 2 we get i plus i that is 2i is equal to integral 0 to pi by 2 sin square x upon sin x plus cos x dx plus integral 0 to pi by 2 cos square x upon cos x plus sin x dx. So this becomes i is equal to integral 0 to pi by 2 sin square x plus cos x dx. Now we know that sin square x plus cos square x is equal to 1 therefore using this result in this 2i we get 2i is equal to integral 0 to pi by 2 dx upon sin x plus cos x since we have the sin square x plus cos square x is equal to 1. Now let's write sin x and cos x in terms of tan x upon 2.  Using these values of sin x we get 2i is equal to integral 0 to pi by 2 dx upon.  We will use this value for cos x that is 1 minus tan square x upon 2 upon 1 plus tan square x upon. Now we take let upon 2 be equal to t. Now differentiating both sides with respect to x we get 1 upon 2 into sin square x upon 2 dx is equal to dt which means that dx is equal to 2 dt upon sin square x upon 2 which is 1 plus tan square x upon 2 that is 1 plus t square since we are taken tan x upon 2 as t and we know that 1 plus tan square x is equal to 6 square x. So we have got the value for dx. Now the limits will also change accordingly like when we have x is equal to 0 then t would be equal to tan 0 since t is equal to tan x upon 2 and in place of x we put 0 and this is equal to 0. So when x is equal to 0 t is equal to 0. Now when x is equal to the upper limit that is pi by 2 then t would be equal to tan pi by 4 since t is equal to tan x upon 2 and in place of x we put pi by 2 and tan pi by 4 is 1. So we get t is equal to 1 then x is equal to pi by 2. So now the limits will change from 0 to pi by 2 to 0 to 1 thus we get 2i is equal to integral 0 to 1.  So this gives us 2i is equal to integral 0 to 1 2dt upon 2t plus 1 minus t square. This means we have 2i is equal to minus 2 integral 0 to 1 dt upon t square minus 2t minus 1 that is we take minus common from the denominator and 2 common from the numerator. So outside the integral we get minus 2. So this gives us 2i is equal to minus 2 integral 0 to 1 dt upon t square minus 2t the minus 1 can be written as minus 2 plus 1 so this further gives us 2i is equal to minus 2 integral 0 to 1 dt upon now t square minus 2t plus 1 is t minus 1 whole square minus 2 can be written as root 2 whole square. Now we can cancel this 2 with this 2 so we get i is equal to minus integral 0 to 1 dt upon t minus 1 whole square minus root 2 whole square. We have the formula integral dx upon x square minus a square is equal to 1 upon 2a log modulus x minus a upon x plus a. So using this formula in this value for i we get i is equal to minus 1 upon 2 into a now here a is root 2 log modulus x minus a now x in this case would be t minus 1 so t minus 1 minus root 2 upon x plus a that is t minus 1 plus root 2 and here the limits are from 0 to 1 this means we get i is equal to minus 1 upon 2 root 2 log modulus.


Good morning everybody.  I was sitting quietly on my couch minding my own business When she leapt up and hit my leg and broke my wound open didn't you? Yes, I did. I Talked to my prosthesis office and what that means is that I will not be able to get casted for my socket this week I will not be able to put pressure in it this week. I'll not be able to maybe take my first step This week because this little monster broke it open You can't put any pressure on an open wound So I just had my first real real setback now Here's the other issue now this all this little monster who is now trying to be real cute and she has real cute May have actually saved me in a fashion because when she broke it open there was this is gonna be delightful for those of you who Like pus a lot of pus And so I'm wondering if maybe there's some kind of small infection or something like that that I need to be watching out For and so maybe she saved me so I'll be seeing my doctor in two days I have a call into them and we will see what's going on I'll definitely be watching for signs of infection and I will be taking my temperature and all that good stuff and my sweet little puppy We'll have to go back to my parents house I'm gonna roll some footage of me visiting the dogs before I brought monkey back So dogs are like the love of my life next to my amazing husband I have three of them and two cats because we have too many animals and life is amazing with animals And so I got to go visit them At like three weeks after surgery because we had them outside the house because we didn't want this to happen And then I thought I could bring monkey home her name's monkey and I think I already told you that maybe I'm high on drugs still and I brought her home too early apparently because she broke it open. That's okay Roll the footage of adorable puppies So that's it guys I'll let you know how the healing process goes and hopefully I don't have a crazy infection Take care of yourselves. I'll talk to you soon


Okay, so let me explain the essence of the Oselege theorem, just the idea. Let me consider the trivial case, but which is the essence then the trivial case is the case multiplication of random number. We have a sequence of random number a1 a2 a n where this number these are positive and distributed according to a certain probability distribution, smooth enough. Okay. And then you consider this problem, you take X, Z, Z, and equal the product of this, this number. You know the probability distribution this time. Very independent and you wonder what happened when and go to infinite. Okay, this is the difference. What happened so if you, if you have to guess the behavior of this quantity when n is very large. One temptation is a this is the average of a to n. This is wrong. This is not true. Okay, the behavior, the most probable behavior, the typical behavior, which happened with probability one is that these go like exponential and the log, the average or log of a white is more formal way is a the limit of one over n Z and go to this. Go to this with the probability one. Why it's real because you just take this. Perform the logarithm, you have the logarithm of Z and is some of log of a and. Then, since these are independent, even log is dependent. And so, and if you divide by n, you invoke the law of the larger number and you have this approach to log of a. And that's it. Okay, this is just trivial, just a trivial application of just trivial remark. This is the consequence of the law of general larger now. Now, the Oceli deck theorem is the same kind of result, but I know trivial situation. The solution is the following you have now instead to have a number, you have matrix, let me call a one, a two, and so on. Okay. And let me define the be and as the product in order away. And then let me introduce the, the matrix is transposal matrix, which is a one. Doug a two. And you wonder, they said it is about this, you take this matrix, you multiply this by this in such a way the matrix is, is, is a symmetric. And then you wonder what happened when these go to infinite. Okay, now, just to do in a more precise way. Take this quantity. Okay. Which is a formally, this is the same, I think that is correspond to that and formally that and is the same. One over two and these go to log of a. Okay. Okay. Now, I consider this, this now this is a matrix. Now it is a matrix. And you wonder what happened when I go to infinite. So, in this case, this quantity go to this with probability one. It's absolutely not trivial but in the same spirit. The statement is when all go to infinite approach to a certain metrics, let me call the approach a certain metrics. Of course, this metrics that you can expect that the metrics depend of the exact sequence you have. Okay. But you see that in analogy with this, you added here the this quantity. Almost all the sequence approach to the same value. Even here you're the same all these things approach to the same, the same probably to the same metrics, which is independent from the specific sequence, this is the theorem. Which is in some sense is a generalization of the this is trivial. Okay, this is not trivial at all. Okay, so from this factor, if you translate this in term of the of the of the tangent vector, you have the reference point.  And these are positive, of course, because it is a real and then you're also some some eigenvalue, some eigenvector, let me call the one the two and so on. The fact that there exists that this is this is asymptotic metrics. You can translate this stuff for with for the Lyapunus for the opponent. Now you have that with the notation yesterday ZN is equal to be N at zero. Okay, now you take ZN model square. And this is ZN. This is B and that zero. You got this. Okay, now you see that appear appear this magic. Now.  And this is, this is one over two N. This quote. Okay, so now you see now you understand the why isn't that it's important that this is some some limit. Okay, because this means that, of course, this depend on that that zero. Okay, and so now depend on the, if that zero is perpendicular or not to the eigen eigenvalue eigenvector, all these metrics. So, and you have the Lyapunus point. Okay, this is the, this is the, there exists the Lyapunus point and the meaning, the meaning. Okay. This is the mathematical part without any proof just just just to mention the. The, the method by Benetton et al. Carlo Benetton is very wonderful mathematician, very kind person in Padua, he went in pension just few years ago. He's a great gentleman, not only a great scientist, but also a great gentleman. Benetton et al. Actually the paper, the first paper has been published on physical review, but is the, the first paper was say, not, not, not very rigorous. It was in the style of physical thesis, what there is a, some years later it was a rigorous paper published in a very obscure journal, and make the name is mechanical. And, in spite of the fact that this journal is completely unknown almost to everybody. It's very important. Okay. And the, the, the method is the following. So, I, I discussed yesterday, discussed for, for maps for the, okay, for maps or differential equation, there is no big difference. So the case on map, okay, like map and ordinary difference. So in the case of a map, we have something like that x. At time t plus one equal g of, of x time t differential equation we have x dot x dot equal f the, the, the, the, the, the engine vector is written this way. We are a is a j here is a set dot equal a, the same.  Imagine that you want to call the system is the dimension and n, n is the dimension of the system. So for example, in the case of a no market to in the case of the, the, the, the lower system is three and so on. Okay, now the, the, the, the, the, the algorithm is the following. At equal zero, you start with the zeta one to n zero, and these are orthonormal vectors. Okay. These are i zero j are mutually orthonormal. Then each of these evolve, each of these evolve according to these equations. Up to up to a certain time t equal tau, okay, this evolve this way. And after this evolution. I consider, let me call alpha one equal alpha one tau equal the modules of that one. And let me call alpha two tau, the following quantity. So this is that one is that two. And I look at this area. And this, let me call this up to down, let me call alpha, alpha three, the volume when they tell me by zeta one, zeta two, zeta three, and so on. Okay. Then after this.  Okay, so I rescale this. I take this, I rescale. Let me call W. W one. Z is just just Z one. Normalize the divide by alpha one for for W two is a vector which is perpendicular to this one. W three is perpendicular to W one and W two and so on the orthonormalization. Okay, and then I reconsider this and restart the game, a repeat again. Okay. So, I look at the evolution after a certain time tau, I perform a grass meat orthonormalization I compute this quantity. And then, and then go on. So, then I perform this, iterate this a long time. And what happened that from this quantity alpha one are computed time tau to tau and so on. So I have that today. So one is the limit and no N is M. M go to infinite one over M tau, some of log of alpha one alpha one. Jerry from M. Jay. So if you want to lambda two, I have that lambda one plus lambda two is the same, but here I have alpha two and so on. So if you want to lambda one plus lambda two plus lambda k. This is the same stuff with here I have just alpha k. And this means that they can compute all all all the all the characteristics on it. There is no problem. So it just it just long because the computer time increase if they want to one, practically a to double the time if you want to get factor three, and so on. And you see that it's not possible to compute the second without the computation on the first and you cannot compute the third without the computation on the first and the second and so on. And this is, this is the method. So this method was in the history of the dynamic assistant was extremely important was really a turning point of the history of dynamic assistant, because before this method. What means chaotic is really very generic is something which is irregular looking at the spectrum, looking at the correlation function. So, this is this is the borderline between between the pre history and the history of the dynamic assistant before this dynamic assistant was a matter for only for mathematician. After this it was possible even for us physicists entering the game, studying in a not just some very abstract T or M just looking at what type just look at some, it's a number. Okay. Okay, this is the for the part of the app. Okay, of the of the Japanese point. Now, the other aspect I mentioned yesterday is the characterization of the tractor. And I said yesterday that a way to characterize the tractor is so before before to go on before to go on. There are a list of the list of the property of the leopon exponent which I give you without any, any proof. Okay. So, the conservative system, the sum of the leopon exponent is zero. And in simplicity system, I mean Tony I'm sorry, it's in practice, in simplicity system. We have that the, the, the last one, let me call lambda to N is minus lambda one. And this is a, this is a specular situation. Okay. And in the C but a system. Okay, this is generally the sum of the leopon exponent is equal to the divergence. So it's negative. In the law in the Lawrence model, so in the Lawrence model in the, in the, okay, this is for the, sorry, this is for the differential equation otherwise is the determinant of the matrix. For example, in the, in the case of the another nice property, which in the case of ordinary differential equation, a part of the stupid case with fixed point. Okay, if the system is just before this if you have a stable fixed point, you have that the, all the leopon exponent are negative. This is real. Okay. If you have a stable stable limit cycle, you have that one leopon exponent is one leopon exponent is zero. And the, and the, and the other and the other are negative. And the entire situation is when at least one is positive. So in the case of ordinary differential equation at least a part of the trivial case of stable fixed point at least at least one leopon exponent is exactly zero. Okay, this help in the computation, for example, with this result, when you have to, you can use this result for the pain in the end on map, you have to the opponent, the first and the second, but the second is fixed by this condition. In the, in the law and system, you have three leopon exponent, but only one is independent, because the first one, because the second must be zero. And the third is fixed by this condition. Okay, so there's a lot in long, in long dimensional system, this, this kind means that practically just one leopon exponent, because the other are truly determined by, by this general Okay, now, the, the, the, the structure of the track before to go on. Let me, yesterday I discussed one of you in a private form, an important, important fact that why, what is the, the, the intuition why you expect that in the dissipative in the dissipative quality system you have some strange things that are, the, the, the, the reason is the following. These are to say to the stretching a max, stretching, stretching, stretching, stretching, no, stretching and folding mechanism. What is the stretcher for the image in that the just to fix the idea the enormous. Okay, so you have a disability you have a disability system, which is chaotic. and live in a bounded domain. Okay, then you start with the set of initial condition like this. Okay, so whatever. So because it is dissipative, these evolve and this area must decrease. Okay, into this area volume agent must decrease. Okay, but but can decrease in different way for example one one possibility just decrease in a trivial way just decrease. But this is not possible. No, this is possible way to decrease, but this is possible only in stable situation because in a stable situation in counting situation this is not possible because if you take two points here. These two point the distance must increase. So this is not possible. So what is what happens that this must become something like that. So if you take here some some point the point and so on the distance must must must increase. So this means that you are you are something like that. Okay, so because the system is dissipative, the area must decrease but because it's chaotic, the shape is necessary some location. Okay, and then this mechanism must go on. So since there must must be bounded in a finite region, in a finite domain, the unique way, the unique possibility at that not only you have this stretching but there's also a folding you have to fold this stuff in order to remain inside this. So it's necessary something like that. Okay. It must be repeated all the time you want. So at the end, so you start from something like that. And then the end that you is necessary to have something like that. Okay. This infinite time. So this means that this cannot be as much stuff can be something so you can hope that this is self similar for example you can imagine that if I take this and I perform a zoom. I obtained the same. This is not the proof of the course but so you can, you can expect something like this and there is always this this mechanism. So it means that putting together the fact that this is put together the fact this is secret is chaotic this is bounded these three things together. You have the intuition that you expect something like that. Okay. So this is the reason why you expect factor. This is not the proof. Nobody has been able to prove that some attractor is really fragile. Maybe there are some specific case but there's some. But it is what it is what you expect and you find that in my guess so I guess you are familiar with fractal is not necessary to define the fractal dimension. I get to find fractal dimension. Yes or not. I have to. Okay, so what what what means factor. What is fractal, which, if I say fractal, you expect to say that the father of fractal was Mandelbrot wrong. The father of mandel fractal was not Mandelbrot but was another gentleman. I suspect that you don't know the name of this, this guy. You know what this guy. You know, so every day you see the weather forecasting program. This way, this guy, this gentleman invented the modern modern methodology to perform the weather forecasting among the many among the other things. And so and he had the first was the idea that the famous question how long is the cost of Great Britain is from him, not from Mandelbrot. Okay, apart of the historical remark. What the means factor. The real factor is the general dimension. Yeah, measurement. What they mean dimension. So usually I say why why you say that these is dimension one, you say this dimension one because a point here is determined by one number. The longitudinal curvilinear coordinate. This L1, this L2, and that's it. So if you have a sphere on the sphere, a point is identified by the number latitude and longitude. So this is, you have, you say this is dimension one. This is dimension two. The volume is dimension three because you need the X, Y, Z. Okay, and so on. So you associate the idea of dimension with discrete number. In terms of the, how many, how many coordinate I need to identify a point. Okay, this is an idea. Another idea, which is a empirical is due to Richard's is the following. You have a line. I have a line. And you have a compass. Okay. There is a second. You have an instrument. Okay, you have an instrument. Okay, you have an instrument with a certain precision.  This and this. So you approximate the, you approximate this as something with a segment of size delta X. Okay. And then you count how many, how many, how many, how many segment I need to cover the stuff. Okay, I count this and then I multiply by delta X. And I take the limit delta X go to zero. And this is what you call length. Right. This is called what you call length, the length is one. So if you repeat the same game with the, so with a sphere. So you count, you count how many, how many square. You have the edge delta X, you need to cover the answer this stuff this and this approach to what you call surface. Okay, in the case of volume, you have n delta X delta X cube, and this go to the volume. Okay, this is another way to define. So you see that here you have a data. In this case, you have delta X go like delta X minus one. In this case delta X minus two. Here you have delta X to minus three. This is another way to define dimension because instead to say how many variable I need blah blah just just from empirical point of view. I count how many, how many boxes quantities, how many elements as necessary to approximate the stuff in the limit of delta X go to zero. I have that delta n delta X go goes like delta X to minus D. I define the dimension in this way. I found I have to count and counting looking at the scaling. Okay, fine. But, okay, this way I obtained what I know 123 up to now nothing tell us now the next question is, is it possible to have an object such that this quantity is not the one or two three. This is, this was the smart idea of Richard. There was also some in the prehistory of the stuff that I guess the richer son is the father but the grandfather probably was by a stress. He was in his work on, on, on Fourier series and on differential. So, apart the, the, the historical aspect, so you can go on the if there exists, if there exists the situation where this number is something for that between one or two or two or three and so. So, so what is important is that not only is possible to show some, some mathematical logic, such that this D is not integer. And these, maybe you can say this is a mathematical curiosity, okay. Now, the next step is this that exists in nature, or not this important from physical point of view. So, the first step in mathematics, and this is easy. In 20 seconds, two minutes. The, the, the, the, the following step is that there is this stuff in a trivial way, for example, in the electric attractor. Because a chaotic attractor is not very physical, it's just mathematics, okay, you can wonder if you look outside the window, I observe the stuff. Yes, in turbulence, for example, in turbulence, in turbulence, the, the, the, there is the energy distribution distribution of turbulence is an audible object. Which is not a factor, but it's more complicated. It's called multi factor, but don't worry. So, so now, now, let me just convince show you a situation where this is not, is not, is not the one. Oh, so in the, in the, in the paper. How, how is along the cost of a great beaten the gentleman take perform this performer explicitly this exercise. Okay, we are on the maps of course not not not working along the, the cost and the plotting log of N log of delta X, put the number. You said that this law, but you said that the was not the one was not the one but was something like I don't remember 1.2 1.3 so but something for sure different from one. So, why, so this means that this means that this means that the in some sense that the, the, the, the, the curve is not differentiable because it's differentiable this is one. At least it's not differentiable at a certain scale. Okay. No, now, let me just show an example of, of fractal. The phone cock. So the phone cock is the following is very simple, you can construct in an iterative way. So, at the, at the steps, zero, you have a segment of size one. Okay.   Okay, this is one, this is one third. And then you repeat the game. Repeat the game is enough. This is one third. This is one. One third square. And so on. Okay, you see that here, yeah, delta, okay, yeah, delta x here is one is one third is one. And the N, and here is one. Yes for here is 16.  And so, and then if you apply this formula, what you have. You have that N, N of one third to N is equal to four to N. Okay, then you apply this formula. And you obtain that D is in this, this is log of four log three, which is something between two and one. Okay, is a number. Okay, so you see this is this is an example. Okay, you can say, oh, okay, this is the pathology is artificial example. Yes. You are all the right to say that this is just a joke. This is just a game. Okay. But if, if the factor remain on this level are not important. Fortunately, they're appearing in a trigger way. Okay. So, okay, this is the way to compute the dimension. And the problem is that, okay, but if I, for example, if I have, if I have an attack, if I have empirically an attract, imagine that now you have your computer, your computer you iterate the system for example, the, the, the, the, the new map or, or, or, or Lauren system. And then you want to assume that you assume because you expect you hope that the, the attractor is the fractal. The first question is, how is how you computed the fractal dimension, how you compute the fractal dimension. Okay, these things was very popular when I have your age in the 80. And the first approach, the first approach was very, very, very nice. What just the first approach is just to apply the definition. Of course, okay. The first approach is what is called both counting. So just to repeat this. So the idea is the following. So this is the, the first method of course the first method is not efficient as you always the first method was counting. You follow me. Imagine that you have your system, and you generate the X one X to X, and so this is the, the, the point of evolving time. Okay. Imagine that the time is this great. This is continuous you take it at a certain time is not important. This point, for example, in the, in the, and then you have your, your system. And you perform a, a partition of your system of size, so it's a epsilon. So you have, you have this point. Okay, this point, well, and it is very large of course and, and, and must be very large. And at a given, at a given epsilon, you count how many non empty boxes you have. So you count how many non, non empty box you have that you call this end of epsilon. Okay, then if this is a fractal. If this is a fractal means that this is scaling this way. Okay, so now you take this you put here log epsilon. You repeat the game with the different epsilon. Small, your point. So if this is fractal this point will be on the line. If it's on the line. Do a fit from this fit you have a diffusion.  The question is the method is efficient. No, of course, no, it's not efficient. So he's efficient for the normal map. For the map is efficient because for if you do this for the normal map for, for example, the normal map with the B equals zero three a equal one point four which is a typical the typical value just to play the dimension is is around one point. Okay.  Okay, but this is this is simple. It's okay if you zoom the stuff. If you zoom here you see that you are more. If you do this is normal map. Okay. So if you do the same for the Lawrence system. Are equal 28. Here you have the D is around two point zero five. This is the case of low dimensional system the method is okay but so you see the method is very stupid because they put for example in in today. This is not empty. This is an empty box, but the, the total number of boxes much larger is is epsilon two minus two. Okay, so it means when you do this. The dimensions of the box are empty. So you have an enormous memory consuming for nothing. Okay. And so it's clear that this method cannot be applied to large system. Even even something embedded in the same system with the dimension with the five, five, five variable. The method is not efficient. Okay, and so the, the, the, the, the, the, the, the work to solve the problem at least not completely but at least. This is the first step is the grass burger Procaccia. If I remember correctly was 83. Okay, so the progress burger progression method is to avoid the, the, the, to have a method, which is, which is efficient and is not limited by the fact that the, the, you have to use an enormous number of boxes and these boxes are empty. Okay. And the idea is the following. Actually, in this way, actually, the method is not the method doesn't have to compute the fractal dimension, but another dimension, which apparently is disturbing because you want to compute the fractal dimension but what actually what you compute is something different. And you can want you can you can complain on this but it's not true. This is the point to for a break. Oh, okay. They are the computation of another dimension. And then, after this, the people realize that the fractal dimension is just one dimension is not the dimension. So it's possible to use the many, many, many, infinite dimension. And, and the fact that dimension in some sense is not the most relevant. Even the dimension introduced by Gersberg approach is not the most relevant, but in any case, better than nothing. So, and the general generalizing this is the first step. And this is the correlation dimension. And then, so, generalize Gersberg progression method. We have as possible to compute the generalized dimension. And there exists a thermodynamic interpretation of this dimension is a thermodynamic. And some strange thermodynamics thermodynamics what we thermodynamics for the annual map is something ridiculous. No, but I show you that. Okay, so this is the, you see when you repeat you, you appear this, this. Okay, this is not the proof that is self familiar but convincing that you suspect that this is really said similar. Okay. And actually it's not exactly self similar. So in fact that the, the fact that the, the, the, the, when you try to generalize the idea you realize that there is no unique dimension and this dimension has been interpreted in terms of statistical mechanics formula. So apparently say, so you know the usual statistical thermodynamic formula, the free energy and so on. Partition function. Partition function and so on. What is the relation partition function free energy this stuff. No, there is a connection. There is a connection. I show you in the next, next, next hour. I stop you. Okay, we take a break. Questions. Oh, okay. Okay, thank you very much. Okay. The, the idea was very prokaccia is very, very simple but as usual when you know it's simple. Here is the following. So you started with your X one X and okay and very large. Okay.  Okay, and these are representative of the attract. And you want to understand something like a dimension. The idea is the following. I take one point X. And I take a ball of size epsilon. And I count how many of these stuff are inside this. So, and then I perform an average on all the time. This means this I think so, in order to count out I can count I take the data function epsilon minus X i minus X j. So this is zero. The distance is larger than epsilon. And these one is the distance is smaller than epsilon so discount how many couple as distance less than epsilon. Then I take the average I and j and divide by n square n square and my n minus one over two but that is okay. And so, and now you're then I take epsilon smaller and the statement is this when it's you know go to his mall, this is epsilon to something, and let me call something the two. And so you expect that this the two is some relation with the fractal dimension. Actually, it's not exactly, but why imagine that the way you expect this and why you expect this is the dimension. Imagine that you have that the pointer are on a line. If the pointer on a line and you perform this is clear that what you can obtain epsilon.  Okay, so for the same reasoning, you say this is an easy to make is a dimension. Okay, and you expect that this dimension some relation with the fractal dimension. But actually, it's possible to prove that the to is a lower bound is the dimension is surely smaller or equal to before to go on. Why this method is much a part that this is not exactly this but almost, for example, if you compute this this in for the, for the, for the animal map, instead of 1.26 say you take 1.24. It's almost impossible to distinguish but this is just a coincidence. Well, apart this, so this method is much powerful than the box counting method because it's not necessary to introduce a partition. You don't you don't introduce any partition and what is important at the end of the story that this dimension is not too large, but the original dimensions that we call x, I, in R to let me call the small D. So, this D here can be also larger, maybe five or six okay, what is important that this number here is not too big. Okay, so if this is large is larger, the bottom the boss county method is practically impossible. Okay, this is just at the computational level, so the computational this was a great advantage for the method, and then the people start to say that it's so important that this is smaller than what the means that is smaller, what is the relevance of the difference between the two and the and so on. Okay. Okay, and this is the story of the next step. The introduction of of the range dimension. And what is called multi fractal description, the multi fractal description of this object has been introduced in around 1984 in Rome and Chicago, and I present the, the Chicago version, just to be kind, because it was in the other group, whatever, and, okay, the, the, the group, the two leader was one was Parisi in Rome and in Chicago was kind of, and the, the, the idea is just posteriori so the approach is the following. In the mathematical term, you have the problem of the characterization of a singular measure. Okay, so mathematically you have this problem you have a measure. Let me go new day new bags, which is singular, singular means that this is, is, is not possible to write in this way, not possible to write in this way where this is a smooth function. And so you want to, you want to characterize how this is singular in which sense is single and so on. Okay, so the area is very simple. No, it's very simple, the area is the following. And you have your system, and you, I present a simplified version so it's possible to do something more sophisticated but so it's not very important. You introduce a partition epsilon regular or you can also also know regular partition but you introduce an epsilon partition, and you have this cell, you numerate the cell. And so this is, for example, the cell I, and you introduce, so you have a resolution epsilon. And you, let me call PI epsilon, the probability to say here. So the integral of this stuff on the cell. So this is lambda lambda I epsilon is the cell of size epsilon center in this part. So if the system is not is is regular. In the case of a regular, if it is irregular in the case regular regular means that you have the new equal raw days. If it is irregular this quantity is just just epsilon to the dimension to the dimension of your system, apart the pre factor 234 and that's it. Maybe there is a pre factor he had you do the same but the scaling is epsilon. So this is regular. Okay, is regular and you have always the same day, and this is the treaty. Now, the problem is that to, to put in evidence that they, the regular, the singularity, and the idea is the following and to perform a sort of a partition function. The terminology is actually the statistical mechanics argument. It's not a coincidence that the Parisian codon of some knowledge of this. And so the, the z, z of z epsilon q. Okay. They find this way, if I had some, I, of P, I, epsilon q, where q is not necessary integer. So, and then you look at the scaling of this prop scaling of this quantity for small laptop. And this is the definition of this epsilon go to q minus one dq you define, you define this number, you define this number. And then you look at the range dimension range dimension is no no Alfred Rennie was a great Hungarian mathematician. I guess he never introduced this stuff about so in honor of all these are cool. He introduced something similar for information to you but the idea is the same. Now, these are the, the, the, these general. It's, it's, it's simple to see that if I take q equals zero. If I take q equal zero. Okay, this quantity is zero is this empty or one is not empty. So if I take q equal zero, I have that the deal zero is the fractal dimension. Okay, then okay it's possible to prove that don't worry so it's possible that they take q equal to. The same to two is the grass burger dimension, the grass burger procachio dimension. Okay. And so it's possible to prove that this quantity this quantity must be a non increasing quantity. This is just a standard inequality in probability of it's a convexity argument. Okay, this the q is a monotonic. And so, now, what, what, what happened that the, the shape of the q is important to understand the, how, how this structure, how this fractal is trivial or not. Okay. Imagine that you have a, a, a, a, a simple fractal simple means that is the, the, the, the, the density of the point are homogeneous on the factor. Okay. So in the, in the homogeneous factor. In the homogeneous factor you have the view q is equal is equal to the fractal dimension. So this is a flat stuff. Okay. Something like a uniformly distributed in some sense, the fact that the if this is not true this means that you have a hierarchical you have a homogeneity on the, on the, on the, on the measure. So you see that we start to define a fractal as an object, but in mathematical term the fractal is not an object is something related to a measure. Okay, so in order to characterize in an interesting way you have to look in terms of the measure. And then you can wonder with this definition you, you are forced to use the, the partition, don't worry there is a trick to avoid the partition at numerical level is possible to avoid the, the, this is just a definition but so it's possible to, to, to, to introduce a trick to bring the idea of the grass burger Procaccia for generic, for a generic, for a generic. Oh, let me write the trick, the trick is the following so just, just for to be complete. So the idea in generalization of the grass burger Procaccia method you define the, this quantity you, you can sort of define this, this local density. So this way you avoid the, the, you avoid the box counting, and then you perform the, if you compute the moment of the stuff, if you compute the one over N sum of the, and I have, you know, to Q is not difficult to prove that this is K as, as what is Q, the. So, this is the definition and this is the, the practical computational method. Okay, so now you can wonder, so you can wonder, but there are all these dimension are all this dimension at the same level are. conceptually, technically, the same relevance. No, there is one dimension which is the somehow the, the dimension, the most important, the most important dimension is not the two is not the zero is the one. Why is this one and fact that this one, this is called information dimension. It's associated to the entropy. Okay. Why. So the statement, the important dimension is the one is this one, and this is this is called information dimension. Why. Okay, from this stuff. Let me take Q equal one plus epsilon. Absolutely. Absolutely. I use epsilon for the action to the small. Okay, and then from this, I, I take this stuff. And I, I have that Q minus one, your Q is limit epsilon go to zero, a lot of some PI epsilon to Q, divide by Q. No, the right dog. Okay, this is there. Okay, then let me take Q equal one plus absolutely there. And so I have that PI epsilon to Q is equal to PI epsilon PI epsilon to the. Let me remind the trick. It's a simple trick, but I have to remind the trick. Okay, okay. So, using the, the, the standard trick in statistical mechanics, let me hide this as exponential epsilon to the log of P, which is sure is correct.  And then let me see I use the fact that this is smaller. So you have PI one plus epsilon to the log. Okay, so. So you have that sum of PI Q is equal to some stuff, some PI plus epsilon to the sum PI log PI. This is one definition. One plus epsilon to the sum PI PI log PI. So we are something interesting. Okay, now, I have to take the logarithm of the stuff. The logarithm of this, I have to take the logarithm of this, but the logarithm of this by definition, so the logarithm one plus something small. Okay, so you have that the log of some PI one plus epsilon tilde is equal to one. And then I have to add to this stuff to epsilon tilde. Some of PI log PI and I can go on and so if you want something more, but at this step at this level is enough. So I have this. And then, and then I am. I am, I am okay, because now, now I have here. This is equal to one plus epsilon tilde. Okay, so this, and so this is epsilon tilde. The one plus epsilon tilde equal to what to limit it to zero. Oh, yeah, I have. I have epsilon tilde. Some I PI log PI over log epsilon. Okay, and I take the limit epsilon till they go to zero. And they are the two one, I have the definition, a way to define the one is necessary to do this because otherwise. This is defined this way at zero as the sum I PI log PI epsilon log. Okay. Now, if you put here, minus. This is a sort of entropy. This is an entropy. This is an entropy of the, let me put minus here. I have that minus sum of PI epsilon log PI epsilon is minus is minus the one log of epsilon. No, up now. You don't, you know that to see the, the, the, let me write another way. This is the one log one over epsilon or log one over epsilon. The one, let me write this way. So if you like this way, you can appreciate so this is what is this this is this is the entropy. This is the solution epsilon. Yeah, the epsilon. The solution is equal to log one over epsilon to the one. So now you understand why this is the dimension. No, because you have a system informally distributed. You have that this is the log of the volume and the volume is the size to the dimension. Okay, so this is one. This is in the scary unit. Okay. So you see that the one is associated to to an entropy. Okay, and this is possible to prove possible to prove believe me that if you pick a point on the attractor at random with the major natural measure. If you look, look at the ball around this point and you wonder how, how, what is the probability to say they are outskate the probability. You realize that the probability escape as epsilon to the one. So this is the most, the most probable singularity to have in free singularity but at the end of the story this is most. This is the most. So, and this is can be computed because you can can be computed, you know, using this formula here you repeat the game and you take the logarithm blah, blah, blah. So, using the. Okay, with the same idea you perform the average of the log of an you can compute this stuff. So, so this can be computed. It's a bit longer. Not only can be computed but if you go on here in the if you go on in the computation, considering also the next order, what, what you have. If you go on, you have that the of q or q around one is equal to the one minus something, let me call sigma, sigma square, minus one. So I am considering the correction to this one. And this sigma one is really a fluctuation of something. Okay. And so you see that is true that the view q is something like that because he's an on decreasing functions on something like that. Okay, but what is really important is, is this value. And this look, so this look, which is given by this sigma must be negative because this will give you an idea how you know more genius is the factor. So if this is more you are in. If this is in almost almost homogeneous we have something like that. Something like this is very, very flat. Okay, so this is, so now I promise you that there is, there is a thermodynamic interpretation of all this stuff. The one question so so there are other ways in which you could define a dimension say for example by looking, you could define a dimension. Okay, so for example there is a spectral dimension. So if you look at the spectrum dimension is another story the spectrum. No, no, no, no, no, the spectrum dimension is another story because this is this is about the static stuff. The fractal dimension is associated with the diffusion property is a concept is different because all the stuff you have to work in a finite domain. Why the fractal dimension is associated to the diffusion property which is an open space and open domain. No, there is no, in general, there is no, no, no, no, no, as far as I know there is no connection. Yeah, it's true that also in different case that are different dimension but here the infinite dimension. Okay, it continues better. So now they, I promise you they interpret the thermodynamic interpretation of this stuff. The potential interpretation is a. I never, I never discussed with the, with the Kaldanov and the Parisi, how they arrive to this, now it's impossible to discuss with Kaldanov because he is no more with us, but also with Parisi is not simple to speak. And I saw this is my personal reconstruction of the origin of the idea is personal reconstruction. I don't know if they had this idea or not my personal reconstruction is the following. Imagine that you have a day standard. You compute the system with n particle. You put the partition function is something like that. You have to compute this. Okay. So, and the standard trick is instead to compute this stuff you connect with the micro canonical property. So you define omega. Okay, you have n go back very large. So, you, so you, you, you expect that energy, you write the energy as n times the energy for particle, and you write that the and the micro canonical entropy right as n time the micro canonical entropy for particle something like that. Okay. And so you write this, you write this way. This is just a change of variable introducing the density of states. Okay, so you remember this is, this is exponential of, okay, forget the, forget the Boltzmann constant just just for simplicity. Okay, so this is n s. Yeah, this stuff. Okay, so you, so this is exponential and s, x, and here you have minor beta. Okay, this stuff. Okay, but this is just a shift shift the problem. Okay, so of course it is not easy to compute but imagine that you. So if you do this, then, so then, okay, this is the like exponential of minus beta and F. This is the free energy for particle. Okay. Immediately realize that, no, let me put it here. Let me put here. Okay, so. Check. So you see that the these you compare this with this. So if you are interested in the limit, very large n, you can perform a Laplace methods and plus the last computation, and what you have you have that that you have that the beta F of t is the minimum on a of beta. Minus minus. Okay, so you see that you can, you can write the, the, the, the free energy in terms of religion transform of. Microphone. Now. Now know what is important here is the entropy, the entropy is the counter counter how many states you have such. Now repeat the game. Yeah. Now, if you were to repeat the game, using this district. So in this week, so you put together all the, the part of the face space such that you have a certain energy. You count how many you have. Now the idea is the following I, I look at the, at the point such that you have that this go as absolute alpha. This is some sense is similar that you look at the region where this you have a certain energy. And the analogous of this, you have to count how many boxes you have, you have such that you have alpha. So I count how many box or size x, you know, present the scaling. So, and these. This is water. So here this is an entropy. The, this is, is associated. So if I count how many box such that this I am introducing a fractal dimension. This means that they have epsilon to mine something. This something is called f of alpha. Okay, it's not this effort. Sorry. Chicago they call it. Okay. And so I introduce. So this is not in some sense you see that alpha is some some our analogous of energy. And the f of alpha is the analogous of entropy. And the, the fact that and go to infinite is the analogous of epsilon to zero, which are low to performance at the point. Okay, this is the analogy. The exotic correspond to small epsilon and so with this stuff. So you have that now you repeat the game. Yeah, this is integral of what of epsilon to q alpha minus f of alpha. The alpha. Okay. A person decoration. Okay, so now you repeat the game. So you have that you are interested in the limit of small epsilon. So in the little small laps, you know, this means that the dominant is given up by the minimum.  But this by definition, this by definition is q minus one. You have this the q minus the q is equal to one q minus one, the minimum on alpha alpha q minus f of alpha, which is in a general form. You see that this is the perfect parallelism of this. So this is the reason why the mathematician called this thermodynamic formalism. It's the thermodynamic form because at the formal level. So you have that q is some sense in analogous of the temperature of beta. No, and, and so on.  When you change the temperature at the value in the temperature you you explore different. So here is the same. So at the value in q, you are able to explore the singularity. So if you want to see the very strong singularity, which correspond to small value of q to small value of alpha, you have you have to, you have to look at that very large q, very large is q give you an idea of the terrible event. Okay, this is the idea. So, of course, this is f of alpha cannot be a completely arbitrary function as as the entropy is not an arbitrary function. So you know the entry must be convex for some reason, somewhere, even here for some very good reason, F or must be convex, the derivative must be negative. And so down, let me just, so the F of alpha must have some, some property, F alpha must be, okay, let me, if this is F or must be something like that. Okay. F of alpha, F of alpha must be surely smaller or equal than alpha, and that there exists a point. Alpha star. There exists a point alpha star such that F of alpha star is equal F star, and this is, this is the, the, the, the important dimension.  So, okay, you have a, you have a, a, a parabolic shape, a parabolic shape of the Gaussian approximation, translating the proper way. And, okay, you have all these must be convex, must be convex, F, second must be negative. Okay, of course, the maximum, the maximum, the maximum is given by the fractal dimension. The maximum is given by, by the fractal dimension. So this stuff can be measured. No, so simple, but true. And then.  So then this for the, this for, for the, for, for the attractor. So then you can wonder, okay, but this, the attractor of the law and system. We are physicists who care or the attractor or the system. So it's possible to have some property using this kind of formula in real physics. The answer is yes. So in turbulence, of course, I have no time to introduce you the problem of turbulence but believe me in turbulence you have in turbulence. What is the singular, what is the singular measure of turbulence in turbulence what is important for, for some good reason. It's called energy dissipation. So apart some decoration so you, you look at the, you look at the epsilon, the, the, the, I am not sure of the, the, the, but something like that. Some more. ij derivative ij plus square. So this is, this is by definition is a positive quantity. Let me call this epsilon. No. Okay. This energy dissipation. And especially in the point x, of course, must be a non-negative function. And so you can, you can, you can take, you can take as, as, as, as, as measure, this is the measure. Okay, this is the, this is the measure. This is the new, new of x, a part, a part, a constant to normalize this one. Okay. And so what you have that the structure of this is this stuff, which is why there are some very good reason to study this. Andrei Nicolai should come over to explain us that this is important stuff.  This story is called multi-fractal. Multi-fractal, why? Because it's not, because there are different fractals embedded in super-posted in, for, for, to explain that. So I have a question. So really interpret this construction in this way, in the sense that it's like you have, say, different fractals with different singular, so the different singularity are really placed. So in the sense that if you take a point. No. Then you ask yourself with, in some sense, but not, I can give you a trivial situation, but the, actually this is much more complicated. Okay. So the trigger solution is, as you are saying, okay, but for example, the simple situation is that what you call defractal. So imagine that they have that the tractor is a line, the point that distributed on a line, and then are also distributed on, on, on, on, on the surface. Okay. Then if I take here, I observe that the p scale like epsilon. If I take here, I have that p scale like epsilon square. Okay, so if I perform this, this computation, what you said by yourself, this or this depends on the value of, of, I have something like that and the, the, the IQ in the latest way, which is a bit stupid. Okay. So this is just to keep to have an intuition.  I don't know if this was fine. So this is just to develop intuition. So this is a big factor. So yeah, this and this. So the question is, is the converse now, whenever you find the fractal multi fractal structure, can you always interpret it as local singular. Yeah, in some sense, yes, but not not not not not too much because this is considered this as a probabilistic interpretation. So because if you look, if you wonder what happened if you look at a certain point with probability one you obtained the one. That's it. No, sorry. If you try to obtain something different from one is possible because zero mission. You have to consider in this way just perform me an average and so on if you just try to take a point and consider the quantity or the point you have in the in a certain ball. There is a general theory and say this you take the one that's it. This is the reason why the one is considered a day dimension, the dimension because if you take a point at random, at random in the sense of you take the one. You take something different because you perform an average, and you take the average and the average. You are waiting at the point in different way because there are these two, this is the reason. So in some sense that the correct that the correct average is when Q is close to one which correspond to entropy. So this is for me, my point is very nice because it is a perfect correspondence between the statistical mechanics formalities and the stuff. I understand and say, is there a way to tilt the probability such in such a way that you can take that you can sample rare points, whichever. You sample their point this way you sample their point this way so you know that the performing the study with different you do realize that there are these rare points my rare point you cannot serve a directive because if you try to serve a rare point systematically you take a typical point. This is the trick. Okay, I guess we can stop and so tomorrow I finish the. Tomorrow is the last day not part. Yes, tomorrow is the last lecture. Okay. So no, okay. I have to discuss before before to pass to change argument. I have to discuss the connection between the dimension and the opponent because apparently say, I have the opponent from the side and dimension on the other and apparently are to this disconnected chapter of the book. And no, it's not true. There is a connection. This is very important. Let me finish the problem is the following. If you. If you try to compute the dimension. Don't worry which dimension and because that's better dimension care. Of course, the method as a limit. But when the dimension is large, it's not possible to use them at large means the 45 or six. Why, because you are trying to, to feel a dimensional space. If you try to feel a dimensional space you need an astonishing number of points. Okay, so imagine that you want to feel a segment segment with 100 points enough. And then you need 10 to four in three to 10 to six and so on. If you want to feel a system with dimension 10 with the same precision use you need the 10 to 20, which is out of your possibility. So you can wonder how it's possible to compute the fractal dimension of an attractor in a system with the dimension is around 20 is not possible to do with this method. It's clear it's not possible to realize immediately, maybe to 10 is possible but 20 for sure it's not possible. So but there are a lot of system with the whose dimension is large is not two or three maybe is 100. So how I can compute the fractal dimension of the system with the one on the first point is why why you want to compute the fractal dimension of the system which is 100 why why is important for example in two bullets. So in a atmospheric science. So the reason is the following the dimension, a part of the integer part, the dimension attractor give you an idea of how many relevant variable you need. In principle, so imagine that you are a system with the 10 to five variable, but the dimension is only 22. Okay, then the conclusion is that they may be in the future some bart and mathematician will give us the proper 22 variable between 23. Okay, no, this is the idea. And this is the reason why so defective dimension is given by the fractal dimension attractor not by the dimension of the, the nominal dimension, but how you can compute not with this method with this method is impossible. Unfortunately, there is a formula that actually there is no proof that is only conjecture that put the relation between the upon exponent and the fractal dimension. The fact that the upon exponent that there is no problem to computational level if you want to compute the fact that the upon exponent of a system with 2000 variable, you can do. When I, when I was young also with the Stefan roof, we compute the system like system with the hundred and under the variable, and this was almost 40 years ago. It is, it is absolutely straightforward. So the upon exponent are not difficult to compute all the upon exponent you want, even some 1000. Now the point is, knowing the upon exponent is possible to understand something about the dimension at the level of theorem. No, but at the level of acceptable argument. There is a conjecture is called Capra and York conjecture which put relation between these, these two fields, and this is the reason why the upon exponent are important, not who carries the upon exponent is 1.2 or 1.3. Okay, it's important to have this one or 10. Okay, what is the relevance of the other the second and so on. Okay, the mathematically at this and this out close the area and so on. Okay, I would say that the physical level, what important is the first one, but then what's important, all the opponents opponent together to try to understand the, to try to understand the fractal dimension of the, of the attract. I don't approve of this but I guess everybody believe that this is the essence is true. In any case, it is not possible in the case where you can check the independently, it works. And so you hope that is correct. But in any case, this is the only possibility. In fact, the people invent an extra name called couple of York dimension. Assuming that this conjecture is true. Which is tautological. Okay, but everybody believe even great. Even very sophisticated mathematics, even see now I believe that is true. Okay. It's in I believe this rule, must be true. Okay, thank you very much. So we meet the