I welcome everyone to this presentation of the paper Beyond Security and Efficiency on Demand Ratcheting with Security Awareness. My name is Andrea Coforio and this is a joint work with Betil Durak from Boss Research and Serge Wodne from the LASAG Laboratory at EPFL. Although messaging applications permeate our everyday life basically everywhere around the globe, the cryptographic sub-discipline of secure messaging has only been gaining traction over the past few years with the emergence of several ratcheting protocols with varying security properties. The overall goal, however, for all of these constructions remains identical. How to secure an asynchronous challenge between two participants that arbitrarily switch their rules in the presence of state exposures? In this context, asynchronous means that both users do not need to be online in order to communicate, hence descending and reception of messages may be randomly interleaved. Nevertheless, this does not mean that the order of messages from a user can be mixed or that messages can be dropped. We captured the security of a messaging protocol through two notions. The first one is forward security. We want to prevent the decryption of past messages by deleting old states through one-way functions. And secondly, post-compromise security. We want to prevent the decryption of future communication by introducing some form of randomness into the user states. This is commonly called ratcheting. OTR, short for off the record, was the earliest ratcheting protocol that found real-world application. It is now superseded by the omnipresent signal construction released in 2013 and among others found in WhatsApp. The security of signal was only formally analyzed in 2017. The first ratcheting protocol that found its way on the academic stage was a construction by Corn Gordon et al that however required synchronized rules. A year later, Belarie et al proposed a unidirectional protocol with no forward security. At Crypto 2018, Petering and Rösler came forward with a protocol with optimal security in the sense that no better model is known. It is proven in the random oracle model but does not consider leakage of random coins. The construction is instantiated with a hierarchical identity-based encryption scheme that requires bilinear pairings and this does very inefficient. At the same conference, Jagger and Stepanov proposed a similar optimal protocol in the random oracle model also relying on HIDEs with random coin leakage before usage. In 2019, Durak and Vodane put forward a highly efficient ratcheting protocol with slightly lower security in the standard model that only requires ordinary cryptosystems. It is based on a publicly cryptosystem, a symmetric key system, a digital signature scheme and a hash function. They further introduced the notion of R-recover security, meaning that a participant should not be able anymore to receive genuine messages after having received the forgery, hence detecting the intrusion. Additionally, the authors proved that post-compromise security implies publicly cryptography, hence this security level cannot be achieved by symmetric primitives only. In the same year, Yoast, Maurer and Malarczyk proposed another protocol in the random oracle model with a security level somewhere between DV and PR with coin leakage resilience after their usage. As DV, it is only based on ordinary primitives but not as efficient as we will see later. Alvent, Corelli and Dodes introduced a reinterpretation of the single protocol in the standard model featuring immediate decryption, the ability to decrypt out-of-order messages. However, their construction only relies on symmetric crypto during periods when the direction of communication does not change, hence there is no post-compromise security for those messages. ACDPK is a tweak that incorporates public key cryptography into those unidirectional periods making it post-compromise secure. And finally, in 2020, Jan and Wodehne designed the so far most efficient protocol called encrypted and hash that only relies on symmetric primitives and thus offers no post-compromise security. Having reviewed state-of-the-art of secure messaging protocols, what is the crux? So in all proposed constructions, the users are oblivious to the actual protocols. Active attacks may occur undetected as communication progresses. Additionally, there is no way to modulate the security level. In some scenarios and applications, better performance is warranted at the cost of reduced security guarantees. The known protocols are either strongly secure but impractical or the other way around. In this paper, we propose the following contributions. First, a formal definition of the security awareness notion in which users are able to detect active attacks by noticing a communication breakdown, meaning the inability to further send and receive messages. Consequently, every acknowledged message is deemed secure. Secondly, users can deduce from incoming messages which of the outgoing ones were actually delivered. We call this concept an acknowledgement extractor. Additionally, given a transcript of sent and received messages alongside potential state exposures, we want to pinpoint which messages remain private. We call this a cleanest extractor. In practical terms, we give a generic toolbox that allows the composition of any two protocols with different security levels. When a strongly secured protocol is combined with a weaker but more efficient protocol, we obtain the notion of ratchet on demand. On a similar note, a hybrid system of two identical protocols allows to re-instantiate broken communication. Finally, we give a comprehensive implementation benchmark of all discussed scenes. This includes the Prithering and Rösler, Jäger and Stepanov's protocol, the construction by Durak and Vodane, the protocol by Joost, Maurer and Malarczyk, both Albin, Korty and Doris constructions, and the recent encrypt and hash protocol by Jan and Vodane. Note that some of those constructions are technically key agreement protocols, however from a security standpoint we consider them equivalent. Before jumping into the meat of the paper, we need to revisit some general concepts. An asynchronous-ratcheted communication protocol with associated data is a five-tuple of a PPT algorithms. A setup algorithm that creates the public parameters according to some security parameter, a key generation algorithm that creates a public secret key pair, an initialization algorithm that creates the initial state of a user, a center team that produces a ciphertext and an updated state, and a reception procedure that, given a ciphertext produces the plaintext, an updated state and a flag indicating whether the decryption succeeded. In terms of correctness, we demand that any untempered sequence of wretched send and wretched receive calls be permitted as long as no messages are dropped or a sequence of messages from a user are received in a different order. In the form of a correctness game, this means that we have two oracles, a sent and received one that populate a sent and received list for each user. Then for any schedule of sent and receive calls, we demand that the received list of each user is prefixed of the sent list of its counterpart. More formally, we say that a user P is in a matching status at time t if at any moment of the game before time t, the received list of the user P is a prefix of the sent list of the user P bar, and at any moment of the game before the time t bar for the user P bar, we demand that his received list is a prefix of the sent list of the user P. In order to simplify the analysis of various security properties, we would like to talk away trivial attacks, for example, trivial forgeries after a state exposure, behind a cleanest predicate cclean, such that the games for the other security notions become easier to pause. Specifically, we are interested in forge security, resistance against forgeries, or recover security, detecting a forgery through the inability to receive further genuine messages, predict security, the inability for an adversary to guess the ciphertext, and most importantly in CTA security for confidentiality. Making such a predicate simplifies our game significantly. For example, in the forge game, the adversary has access to a sent and received oracle, a state exposure oracle, and a plaintext exposure oracle, and his goal is to make a message accept by some user according to some cleanest predicate. In the R-recovered game, the goal of the adversary is to create divergence in the sent and received list of the users that goes unnoticed. In this NCCA game, there is an additional oracle called challenge where either a random plaintext is encrypted or the given one. As already indicated, R-recovered security averts that a user P continues to accept genuine ciphertext from its counterpart after having received a forgery. However, in this scenario, the user P-bar is still able to receive messages from P. However, we demand a complete breakdown of communication. Communication breakdown should reveal any forgery and receiving genuine messages should indicate that no forteries took place. We capture this with the S-recovered security notion whose game or in whose game the adversary attempts to make a user accept the message from another user that has received a forgery. The combination of R-recovered, S-recovered and predicted security yields the following very useful results. If an R-cat is R-recovered, S-recovered and predicted secure, whenever P receives a genuine message from P-bar, P is in a matching status except with negligible probability. In addition to R-recovered security, we want to give a user the power to verify whether a message has been accepted by its counterpart and check whether some cleanness predicate has been violated. For this, let T of P be the chronological partial transcript of the up-to-time T of send, receive, exposure and challenge calls involving the user P alongside the ciphertext pairs corresponding to each send and receive invocation. Furthermore, let T-ratch of P be the transcript that only contains the send, receive and challenge call. In addition to R-recovered security, we want to give a user the power to verify whether a message has been accepted by its counterpart and check whether some cleanness predicate has been violated. For this, let T of P be the chronological partial transcript of the up-to-time T of send, receive, exposure and challenge calls involving P alongside message pairs corresponding to each send and receive invocation. Let T-ratch of P be the transcript that only contains send, receive and challenge calls. Given such a transcript, it is rather simple to define an acknowledgment extractor. So given a transcript T of P and a message ADCT successfully received by P at time T that are sent by P-bar at time T-bar, let AD-prime and CD-prime be the last message received by P-bar before time T-bar. An acknowledgment extractor is an efficient function F such that F of T-ratch of P returns AD-prime and CD-prime for any time T when P is in a matching status. Furthermore, we can also define a cleanness extractor. So let T-P and T-P-bar be two partial transcripts. There is a cleanness extractor for C-clean. If there is an efficient function G such that G given those two transcripts has the following properties. There is one challenge in T-P and either P received AD-test and CD-test or there is a roundtrip P to P-bar to P starting with P sending AD-test and CT-test. Then G given those two transcripts tells us whether the cleanness predicate has been violated. The reason that we need such a roundtrip for an outgoing challenge message is that there is no way to know whether the challenge message has actually been correctly received by P-bar. In the following, we give a generic construction that elevates any secure R-card 0 into a security-aware R-card 1 denoted by chain of R-card 0. Intuitively, a blockchain-like structure that contains a hash chain of sent ciphertexts that is sent alongside each message suffices to associate each message to the digest of the chain. For this, we define some auxiliary variables. H sent is the hash of all sent ciphertexts computed by the sender and sent along each message and updated with a hashing key HK. H received is the hash of all received ciphertexts, also updated with the hashing key upon every reception. It should be easy to see that H sent and H received are enough to ensure our recover security since those two variables only need to be compared upon every message reception. However, more work is required for S recover security. We define more auxiliary variables. H received is a counter of received messages that need to be reported in the next sent call. The latest H received is attached to the message, to acknowledge received messages, and then resets to 0. And then A sent is a list of hashes of sent ciphertexts that are waiting for an acknowledgement. With those two variables, any impersonation of a participant leads to an immediate cut in communication as the intrusion is detected. So how does such a protocol actually look like in practice? So we have a set-up routine that generates the public parameters and the hashing key and an initialization routine that initializes the states of the two users alongside the auxiliary variables. And then in the send function, we attach the H sent to each message as well as the H received and update the H sent variable with the hash function. And then in the receive call, we compare the received H sent to the H received and decrypt the message. And if it succeeds, we update our H received variable with the hashing function and increment the H received counter. We can now show that if H is a collision-resistant hash function, our chain RCATs construction is recover secure for both R recover and S recover security. It's also easy to see that our construction has an acknowledgement extractor. Furthermore, we can also show that our chain RCAT scheme has a clean extractor for a group of cleanest predicates. For the full definition of those predicates, I refer the listener to the full paper. And finally, we can show that if RCAT zero is predict secure, then our chain RCAT construction is also predict secure. We move on to the notion of on-demand ratcheting, where we combine a strongly secure protocol using public key cryptography with a weaker protocol that is only based on symmetric primitives that is more efficient. The weaker protocol is used for frequent exchanges, where no post-compromise security is guaranteed. And the strong protocol is engaged periodically. For example, ratcheting could be administered at the application level or even by the users themselves. For this, we denote by RCAT main the strong protocol and by RCAT sub the weak protocol. The composition of RCAT main and RCAT sub we call a hybrid. And we instigate a ratcheting execution with a flag that is sent alongside the associated data. Our protocol proceeds in so-called epochs, a concept that first appeared in the Patrick and Rosier protocol. Intuitively, an epoch designates a batch of sent and received messages until the direction of communication changes. For this, each user stores two counters, eSent and eReceive, indicating the epoch of the last sent and received message. An epoch counter is also sent alongside each message. So here an epoch counter e of m is equal to eSent if eReceived is smaller than eSent or equal to eReceive plus one otherwise. How does this look like in an example? So here in this example, user A starts off with a batch of messages that are sent to B and B responds with an incremented message epoch. A responds again, but this time the message of A crosses an earlier message of B. So in the following response to this message by A, the epoch counter is not incremented. In addition to the epoch counters, each user keeps count of the number of messages within one epoch. So from a high level perspective, each ARCAT main call creates a new ARCAT sub send and receive state. The sender stores the send state in an array and transmits the receive state in the ciphertext. So how does our protocol actually look like? The setup and initialization routines are very similar to the security or construction from before. In the initialization function, we create an ARCAT main state for each user and initialize their epoch counters and arrays. In the send routine, we first check if you wanna actually instigate a ratcheting call. If so, we first create a new ARCAT sub send and receive state. We keep the send state, but pack the receive state into the ciphertext and encrypt it with ARCAT main. Otherwise, if you don't want to ratchet, we simply use ARCAT sub for the ciphertext creation. Note that the very first message of a communication is necessarily a ratcheting call since no ARCAT sub states are created in the initialization function. Then the corresponding receive function, we first check if an ratcheting call has been demanded. If so, we use ARCAT main for the decryption. Otherwise, we simply use ARCAT sub. We can show that we can derive the security of our hybrid construction directly from the security properties of the sub-protocols, mainly ARCAT main and ARCAT sub. Furthermore, if the two protocols are actually identical, so if ARCAT main is equal to ARCAT sub, then we can use ARCAT main to generate a new ARCAT sub session whenever we want. So making it possible to restore a broken communication in ARCAT sub. For the last part of this presentation, we move on to the benchmarks. So we implemented the discussed protocols on a comparable, on a machine comparable to a high-end smartphone in three different scenarios. First looked at alternating traffic. So A and the users A and B take alternating turns in sending and receiving messages. Then we looked at unidirectional traffic. So first user A sends N messages that are received by user B and then user B proceeds to send his share of N messages. And finally, we looked at something similar called a deferred and unidirectional traffic which starts off by A sending N messages. But before B receives those messages, he sends his share of messages. What can we conclude from those three messages or traffic scenarios? First, we look at alternating traffic. So it is dominated, of course, by the ETH construction by Jan and Baudenay which doesn't use any public key cryptography. We also see that the two schemes which are relying on HIVEs are significantly less efficient than the other schemes relying on symmetric and public key cryptography. A similar thing is true for unidirectional traffic. However, if you look at deferred unidirectional traffic, we see that the JMM protocol, so the protocol by ULST, MAUDER and MULARCHIC actually has a hidden complexity when it comes to this type of traffic. And with this slide, I conclude my presentation.


Every citizen who lives in this country must be made to understand that today we go to our office and come back, today our children go to school and come back, today we are doing whatever we wish to do in this country only because certain number of men and women are standing, God, and taking care of this. Otherwise, you must look back at medieval times when raiders came and raped and looted and did what they did in the past and the same will happen even today. So it is very important in this country every citizen understands that we owe this to the soldiers, we owe this to the defense forces, we owe this to variety of security forces who are providing this to us.


Welcome to the Rich TV Lab podcast. I'm your host, Richard D'Souza. And today we're going to talk about the hottest stocks and cryptos in the market. And we are pre-market. The market does open in 14 minutes. And we're going to let you know what is popping. First and foremost, Bitcoin continues higher. And this is very bullish for all Bitcoin related stocks. Bitcoin been flirting around $71,000. We're sitting right now at $70,847. So very, very bullish price action for Bitcoin. About Reddit's RDDT on an epic tear since the IPO last week. Excuse me, since their IPO last week, they're now sitting at $67 pre-market and up 12% pre-market is Reddit at $67. So unbelievable price action here for Reddit. And that's kind of what they've done since their IPO. So they've done extremely well. And they are now up a large pre-market currently at $66.97 up 11.99%. Micro strategies. This is a big one. And the holder of Bitcoin, they're up 2.5% pre-market at $1900. Keep your eyes on micro strategies. All right, STOK. This is one of the big pre-market movers up 74% pre-market STOK. See that yellow line? That is where they are trading pre-market. So it's going to be a nice gap up here to start a trading day. As you can see here, right to the yellow line is where STOK is now trading pre-market gone from $6.50 to $11.38 up 75% pre-market. So big pre-market move here for STOK. They were also given a price target of $22. So that's the reason why the stock is moving today. All right. What else is moving in pre-market to get our attention? Let's see here. OPGN. Here's OPGN. They are up pre-market as well. As you can see here, let's get the trend line. So you can see they've gone from $0.46 to $0.95 pre-market. So big pre-market move here for OPGN. So be aware of them. Put them on your radar and put them on your watch list almost at a dollar and up 102% is OPGN. The biggest pre-market mover right now on my radar and CHEK. CHEK currently up 31% pre-market. Those are our pre-market gainers right now. And everything that is popping up on my radar that is worth mentioning. Oh, here's another one. ABVC up 38% pre-market. You can see where the yellow orange line is trading. That is where it is pre-market. So you can see it's doing this nice move here pre-market is ABVC. And that's also up on news and currently up 38%. All right. So those are all the pre-market movers that I'm watching now. I'm also going to be watching anything that is large cap. Here's meta. They're up slightly pre-market Uber on my list. Up slightly pre-market AMD. They're up slightly pre-market. The Dow Jones looks like it's slightly red. Google up slightly pre-market. The Nasdaq is actually up slightly pre-market as well. S&P 500 also looking a little bit slightly red. Ripple. Ripple's up pre-market. Amazon is up pre-market. Nvidia getting close to that $1,000 mark again and $958 looks like $1,000 is inevitable at this point. They are also up pre-market at $958. Tesla also up pre-market 3% at $178. I've been saying anything under 200 for Tesla is a steal and they are starting to prove me right. Ethereum starting to look really strong here. Look at the all-time highs on Ethereum and you can see they're approaching those all-time highs and I believe we will see them gap up and get back to that 4,600 mark all-time highs pretty soon around the corner here and Bitcoin at $70,894. Officially trading right now higher than it's ever traded in its history. We are now trading higher than its previous all-time high which was $69,000. We have hit a new all-time high of $73,000 and we're right around the corner from there at $70,900. This is historic for Bitcoin. Never has it ever really truly traded at these levels so the fact that it's trading at these levels is very bullish as far as I'm concerned and leads me to believe that we are going higher. Cardano at $0.67 up 2.56%. Solana. I am a holder of Solana. Full disclosure. Big fan of Solana. I told you they're going to $200. They're sitting at $194 so it seems like $200 right now is not even a matter of if more like a matter of when. Here's Disney. I'm a big fan of Disney. I've been talking about Disney since it was well-own under 100. It's now starting to make its move and you can see this move that it's made here from $80 to $119 where it's gone up about 50% since October of last year so it's starting to make this big move here quietly but don't worry we don't miss on moves. We've been talking about Disney since it was at the bottom. We told you guys we thought it was at the bottom and it is showing that clearly we were right and it is finding a bottom. Here's Arm. This is a AI IPO that has done extremely well. You can see it's gone from $40 range to where it trades now at about $140 up 1.58% pre-market. This is one of those really strong plays this year that you need to be aware of. It's an AI play that is on the move. We'll quickly just look at Bitcoin here as we get close to that $71,000 mark. We might hit it while we are live. Very, very bullish price action for Bitcoin as we approach that $71,000 mark. Very bullish. All right, STOK. One of the big pre-market movers is still up 76%. Let's actually delete this and let's use a trend line. That would be better. That's the pre-market move right there. So that's the gap up that we're going to see pre-market and then we'll see where we go from there. But with the $22 price target I wouldn't be surprised to see this one even go higher once the market starts to trade. Here's Robinhood. It was one of my top 10 plays for March. It has not disappointed. It's done extremely well. Dogecoin also continues higher. Dogecoin up 4.64%. And they've also had news recently that they may be used for Tesla purchases in the future. So I think that that is really big news for Doge. Lucid Motors, they're up slightly pre-market and Ethereum, ETHE, Grayscale Trust, also up pre-market slightly. Coinbase at $282, also up pre-market. GBTC up 11% yesterday. I expect them to climb higher. And IBIT, after a huge day yesterday, this is the BlackRock ETF, they're actually down slightly. I expect that this will probably go green as Bitcoin is looking bullish. MSTR maintaining in this green and Reddit new IPO up 30% yesterday, up another 11% today. Looks like there are a lot of people that believe in Reddit at the moment. So definitely be aware of Reddit. They have been one of the biggest movers since they did their IPO last week. And you need to be aware of all of the movers and shakers in the market. If you want to make money in these markets, you need to be aware of every single player that is out there. And right now Reddit is one of the biggest movers, period. They went out on their first day and traded just under a 50% move and then kind of retraced a little bit. And this week, they've come out really strong. Like I said, 30% yesterday and now 11% already in the pre-market. So definitely one to watch is Reddit. You can see this is what they've done. So this is their trading history. And you can see they made this big move pre-market on the day that they did their IPO, then they came down consolidated and then consolidated further, where the lowest they've really ever been was $44. When they've traded, the lowest that you could have bought in the open market is 44. And now they're sitting at 66. So they're on their way to having a monster day today. Be aware of Reddit, RDDT. And they are going to definitely be one of our hottest stocks to watch for April as they're a new IPO. They have momentum. And it just seems like people right now are getting behind this company. So it's definitely one that I think everyone should be aware of. Alright, let me just quickly search and scan for anything else pre-market that's on the move. Let's see here. Is anything worth our attention? Always searching the entire market every day looking for the biggest stocks and the biggest cryptos on the move. And if you'd like to join our ecosystem and our community, you can at richtv.io, an ecosystem built by investors, foreign investors, where we talk about stocks and cryptos 24 hours a day, seven days a week. So we'd love to have you guys join us there. Very simple richtv.io. Alright. So pre-market movers, OPGN, the big one right now up 97% looking really, really strong, really strong. And there's still the biggest mover of the day on my radar. And then CHEK also up big 26% right now. ABVC also up 32% right now on my radar. Alright. Bitcoin was approaching 71,000, but it looks like they're having a little bit of a pullback right now. We were almost at 71,000, but we're getting a little bit of a pullback at 70,800. That will affect the market. And STOK currently up 73%. And you can see that's what they look like pre-market. As we watch Robinhood, we've talked about will benefit from the price of Bitcoin and cryptos moving. And that's exactly what they're doing. Doge up 4% looking strong, lucid motors up 2% pre-market also looking strong. Coinbase still looking good and positive pre-market at $280. Micro strategies after being up really big pre-market, losing a little bit of steam, but still at $1891 and up 1.94% pre-market. And Reddit, one of our big movers that we're watching today up 11.54% as the market is about to open up any minute. We got one minute left until market opens. Hope you guys are ready. Should be an interesting day. We are in a short week due to the Easter holiday. So definitely a week where if you see profits, in my opinion, take them because it could be a very volatile week. A lot of different things that could happen this week. And once again, it is a short week. So remember that a lot of trading will not be happening on Friday in different places all over the world due to the holiday. Some places will be affected on Friday. Some people will be affected on Monday due to the holiday, the Easter holiday as we head towards Easter. All right, markets about to open any moment. We're going to continue to watch all of these picks that we've been watching very closely. Coinbase almost in the red now after being in the green. A little bit of tricky move there. ETHE, Lucid, and the market is open. Here we go. The market is now officially open. All right, OPGN still up and hot up 101%. You can see there's the gap up. That's the huge gap up that they're having. OPGN, STOK up 82%. So it gapped up with this open space. And now it continues to go higher. Reddit, strong up 12%. ABVC gapping up 30% and CHEK gapping up 30%. Hold on. Let's just quickly look at this. Okay, so Reddit now sitting in fifth spot today for us, up 11%. ETHE up nicely today. Doge up nicely. Tesla with a nice pre-market gap up. As I told you guys, under 200 would be a goodbye. Disney staying green. Solana, a crypto that is really doing well at $193, but losing some steam right now. ARM IPO now at $142 and gaining momentum. Cardano up 2%. Be aware of Cardano. Micro strategies was up. They are losing a little steam but still in the green at 1,870. Lucid motors right around $2.96. Hood up 2% on the day. Bitcoin up a little bit. Ethereum up a little bit on the day. 1% each. Uber slightly green. NVIDIA up a little bit. Meta slightly in the green as well. So large caps, mostly green that we're looking at. AMD also slightly green. The NASDAQ opening up green. S&P 500 opening up green. Google slightly in the green. I like Google at these prices. And Amazon slightly in the green. Dow Jones also in the green. So we're seeing a green day around. And XRP slightly red. IBIT, the BlackRock Bitcoin ETF slightly red. Outside of that, everything else is green across the board. STOK still up 91%, the biggest mover of the day. Right now, let me just scan to see if there's anything else we're missing. All right. What could we be missing? All right. So STOK up 92%. OPGN up 91%. Those are the two biggest movers of the day. CHEK up 30. Oh, 24% right now. So they've lost a little bit of steam. ABVC up 23% right now. And Reddit maintaining a strong bullish trend up 10% right now. ETHE, DOGE, Tesla really climbing right now at $179. And this is due to the fact that EV giant BYD posted its slowest quarterly profit growth in two years. That's really helping Tesla today. ARM IPO that has done extremely well. AI IPO up 3% and gaining steam. As you can see, these candles are growing, very bullish. Solana at $193, up 2% hood approaching that $20 mark, looking strong. Cardano staying up 2% right now, looking strong. Lucid Motors up 1% right now. Bitcoin struggling to get that 71,000 mark. Seems like there's a lot of sellers there. We're currently at 70,729 on Bitcoin. Ethereum also looking strong at $3,629. Meta at $508 in the green. NVIDIA 958 in the green. Micro strategies back over $1,906 now. Looking very bullish. Uber back over $80 looking strong. AMD pushing for $180. NASDAQ looking very strong today. That's very good for tech stocks. Google nicely in the green. S&P 500 climbing nicely. Disney also looking slightly green. The Dow Jones in the green. So all major markets are green today. Amazon slightly green. Ripple a little bit red. IBIT, the BlackRock ETF in the red, but it's starting to trend up. GBTC slightly red and Coinbase after being up over the last few days, giving back some of its gains on news that Cathie Wood arc takes profits in rallying crypto stock. So I'm sure she sold and took profits in Coinbase as she's seen enormous gains there. All right. So those are the movers and shakers of the day. Remember, if you like these videos, smash the like button, comment down below, share the video everywhere and subscribe. I'm your boy Rich from Rich TV. If you're not winning, you're probably not watching. We're bringing the winners, CEO interviews, breaking news, trending topics, and we bring it to you first. We're always watching the world of stocks and cryptos. And right now, the biggest mover of the day is STOK. We'll keep you guys updated on everything that is happening. We'll see you tomorrow. I'm your boy Rich from Rich TV and I'm out. Peace.


Quick tip number two for today when you're not able to be consistent with the habit and You get that anxiety feeling for example You want to do you know 60 minutes of workout every single day? And then you start doing that and eventually it gets hard for you And now you stop doing it and every time you try you have that anxious feeling. Oh, it's too much I don't know if I can do it and then you just don't do it or you experience apathy Which is basically the opposite side of it and you're like, uh, I don't know I don't want to do it blah blah blah and just forget about doing that doing it But it's still on your goals list. What do you want to do is lower? The bar lower the bar as much as needed until you're able to do it without That anxiety without effort without thinking about it. You want to lower it to a point where it's like Okay, so not 60 minutes. So it's like, oh five minutes of working out. Yeah, sure. I can do that. That's no problem Okay, bam, like that's your new criteria because it's more important to be consistent Than it is to just have the habit again It's the cycles of action completing cycles of action I talk about it a lot and it just made a video about it I called it the reason you don't have the motivation you always lose motivation when you try things and you start things So make sure your complete cycles of action again if something is too hard lower the criteria and then do that and you can always go more but lower the bar the Minimum for how much you need to do now will make sure that you continue the cycle and you actually keep growing and growing and You love it you'll get accomplishments and it will only grow and get better from that point But don't lose this the consistency don't lose the streak do that and you'll be successful


O yo, se sente meio. Si, má, no, qua se sente, oya. Se sente, non se sente. Si, se sente. Ok. So, the last talk of this afternoon session will be given by Giuseppe Minjone in Spanish, Rosario, who has been working for years in non-linear elliptic and parabolic equations in Calderon Sigmund theory e he will explain here the relation of these tools with fractional operators. Ok, thanks a lot Juan Luis. It's a pleasure to be here and I'd like to thank the organizers for inviting me. As you notice, the title has lightly changed because there's now something added in parentheses, that is, and vice-versa. Actually because what I'm planning to do in this talk is first, ok, it's actually describing a double-playing interaction because the usual interaction we have seen over the last years where there was somehow an explosion of fractional problems has been taking non-linear and even linear methods from local problems and we have very often seen translation and far-reaching extension to the non-local setting and sometimes things can be quite different as we are going to see today but also I'd like to discover a wide range of methods that are actually non-local methods methods that can be actually applied to local classical problems. Classical, non-linear local problems. So what I'm planning to do is to show a double-interplay to see how non-linear classical methods can be applied to the fractional setting but also how let's say, some fractional methods can be applied to non-linear methods. Ok, as far as fractional results are concerned I'll be presenting some results with Tomo Coussi and Yannick Sear, fractional results and these are essentially in the first part where we extend non-linear methods and non-linear ideas to the fractional setting to solve some fractional problems while I will also present some other results where fractional methods where actually non-local methods can be applied to do some non-local problems and ok, as far as the first part is concerned let me start from non-linear potential theory potential theory ok, what is non-linear potential theory about is actually a far-reaching extension of the classical linear potential theory the classical linear potential theory is dealing with solutions of these problems in Rn larger than 2 and it's essentially dealing it also in some how incorporates classical Calderon-Sigmund theory and the idea is that we want to study as much as possible the regularity properties of solutions and of their derivatives in terms of the regularity of the assigned data this is given ok, that's a classical way to do this and the classical way is going the representation formulas representation formulas are telling you that you can represent this via convolution with the classical Green's function and the classical Green's function is the following this is 1 over x minus y and minus 2 if n is larger than 2 otherwise it's minus log x if n is equal to 2 and this is up to a constant up to a universal constant once you have this then essentially everything is known concerning u and its derivatives as far as u is concerned you get the first estimate via risk potential I1 of x which is defined in the following way and this is the classical risk potential this is for n larger than 2 otherwise you always have this actually this representation and actually this estimate up to a constant of course via the I2 risk potential this is I2 via the I1 risk potential which is the following one now since the behavior the behavior of these operators is known in essentially all reasonable function spaces then you can reconstruct via this inequality at least for instance all the integrability properties in any reasonable function space like for instance rearrangement in variant function spaces the big spaces all each law ends whatever you like ok, this is the classical fact this is the classical world but the classical world rests on this representation formula which is very much linked to the to the fact that you have this equation and actually you have a linear equation this is the classical approach and if you want to push it further up to the secondary derivatives then this becomes the classical calderon signal theory where let's say young's type in equality are not sufficient anymore and then you go to cancellations and to singular integrals but anyway as I was telling before this is the linear world so you have a fundamental solution and this motivated for instance in the 60s or in the 70s a great deal of studies with all the things related to fundamental solutions and microlocal analysis and blah blah now what is no linear potential theory about is actually the idea that you want to extend as much as possible the validity of this approach without having fundamental solutions of course to the non-linear setting and in the non-linear setting we are considering quasi linear equations of this type because to me this isn't in the most general case a measure but let's say that we are just talking about apriori estimates because defining solutions for measure data it's a bit problematic you have to go to approximation methods but anyway the idea is that we talk about we treat this as a measure but let's imagine is a same thing in function and then we get suitable apriori estimates which is just depending on the fact that on the total variation of the measure that for simplicity we imagine to be fine ok so the first idea we want to extend this to the linear setting and the linear setting means that we are considering no linear Poisson type equations this means that this is a linear growth and the ellipticity is prescribed by saying that you have this control from below on the eigenvalues for every lambda in our n and furthermore you also prescribe that you have this control on the largest eigenvalue so this means that the ratio between the highest and the lowest eigenvalue of this of the matrix a of x is L o and u and this is ellipticity furthermore you want to go on and you want to consider possibly the generator operator so the Poisson type where the model case is the following one and you get this operator which is still uniformly elliptic because the ratio between the highest and the lowest eigenvalues is uniformly bounded but this time can be possibly degenerate and more generally we shall consider we shall consider the classical equations on the equations like this that are modeled on this case that are modeled on this case this means that now when we consider the growth conditions these ones these will be bounded by z to the p minus 1 which certainly happens for this special operator and the ellipticity is described in the following way here it's lambda square because you have to have something elliptic but now you have a degeneration factor which is this one of course these assumptions reduce to these for p equals to 2 as well as the Laplacian reduces to the classical as the P Laplacian reduces to the classical Laplacian for p equals to 2 no yes, sure thanks ok ok, this is the general setting so in the first part of the talk I will first give a summary of how we surprisingly can extend these two estimates to the general setting which is quite non-trivial there are some fundamental solutions that are the first side appears to be an absolutely an absolutely crucial way to get such bounds and then we see how this can be extended to the classical to the new fractional setting and this is the first of the papers that I'm talking about here written with Tuomo and then we will see how other classical non-linear tools can be extended to the fractional setting now I'm going to switch to slides ok, as I told you I'm going to switch to slides ok, first now the new the new goal the first goal is to extend these two equal this to the non-linear setting where I'm considering well you can keep in mind this special operator but all the results are actually holding for this most general class of ok, this is the first part first let me rewrite the risk potential in a way which is suitable to localize the estimates there is no cheating here, this is a total triviality I'm introducing here what is called the truncated risk potential which is just another way of writing the classical risk potential you make a usual you can do this in very many ways maybe the composition in annual life would work anyway, from now on this is just the risk potential we talk about risk potential and we think of this of the truncated or localized risk potential ok now the point is that if you look at this operator let's say the pilaplash and type operator you immediately see that when p is different than 2 for instance these estimates just cannot hold because they do not respect the scaling of the equation if you get an estimate that doesn't scale as the equation then the estimate is wrong because you do a scaling and you see that all solutions are trivial this means that the estimate cannot be true so in when p is different than 2 the standard orthodoxy in a linear potential theory is to use this both type potential this is nothing but the classical risk potential that now incorporates the deficit scaling of this equation because you see that when this case is like p-1 then you see that this case is like p-1 this case is like 1 so introduce so you want to introduce the scaling and then you create this new potential this for p equals to 2 not surprisingly reduces to the classical risk potential so the orthodoxy in a linear potential theory is that whenever you have a degenerate of radar and p is different than 2 then you go to this both potential and you try to adapt the results and the first very nice result is due to Kilpelainen e Mali then they proved that for this operator and actually for any operator satisfying these assumptions but even further actually then you can point wise estimate you by the both potential w1p plus this object the real point is this one because this is the localization object which is there which must be there when mu is equal to 0 which is there otherwise you let capital R goes to 0 this goes to 0 because it's an integral and then you get u less than or equal than u otherwise if you want to be on the whole Rn then you let capital goes to plus infinity and this object disappears and then you get the both potential this is done by a clever application of a variation of truncation methods of the geology in the setting of measure data problems ok now what about and this is essentially the analog of the first result then after this it remained an open problem to get the analog of the second result the analog of the second result is actually much more delicate just for instance for one reason that you can immediately see how do you get gradient estimator and a question like this you differentiate the equation you differentiate the equation this means that you have to handle with some quantities that are second derivatives now let's consider just the case p equal to 2 and let me consider the case when you have Poisson equation equal to mu then second derivatives do not exist because this is the classical failure of Calderon-Sigmund theory in the limiting case just for instance already when this is in R1 so therefore this means that the classical techniques cannot be applied because you do not have even a starting point you cannot differentiate the equation so this problem remained open for 20 years and just let me say that that the first result appeared in a paper of mine and essentially this tells you that there is no difference from the linear case to the non-linear case because here the real point is the jump from the linear case where you can use fundamental solutions to the non-linear case where fundamental solutions are not available or at least they are available but of course you cannot get general solutions via convolution you can always define fundamental solutions formally by solving something when the right hand side is a direct but then you don't do anything else with that ok actually I'll try to come back to this later on essentially when you are on the whole Rn when you have a suitable decayed infinity of the gradient which is satisfied then you get the usual estimate and this is a quite satisfying result because tells that there is no difference between the Laplacian and any other equation no linear operator ok what about p different and 2 the case p different and 2 then following the standard orthodox you look for a wolf potential wolf potentials are good because they can be anyway controlled via this inequality by iterations of risk potentials once you have this actually this is a point wise equivalence at least for p larger than 2 minus 1 over n the right hand side guy is called having magia potential and this is proved in a classical paper by having a magia from 71 anyway this is telling you that whenever you have wolf potentials your job is done because this can be controlled by risk potentials and the behavior of risk potentials is well known ok so let me go back to the case p different and 2 and the first result was achieved by Fran Duzer and myself and this tells you exactly what you would expect following the standard orthodoxy following the standard orthodoxy now you can point wise bound the gradient via the wolf potential and this is actually quite satisfying estimate because this allows you to get all the classical estimates that you get via fundamental solutions by wolf potentials this is this guy and it as you can see for p equals 2 to 2 this reduces to i1 so you get back this estimate and the previous estimate one but now let me make a twist because when you find the estimate you do not want to follow the standard orthodoxy that is what the society is telling you but you want to follow just the equation it's the equation that is telling you what you're talking about so what you're doing when you're doing Calderon-Sigmund theory what you're doing when you're doing Calderon-Sigmund theory you know that by Calderon-Sigmund theory and this is actually the Laplacian well when you have such an equation when you have such an equation and mu for instance belongs to LQ for Q larger than 1 then this implies that the second derivatives are in LQ as long as Q is not equal to plus infinity so this means that you are making a trait between the divergence operator and the derivative so you're making this trait and this belongs to LQ like this one now let's make, let's believe in the power of regularity and let's discuss what happens when you deal with the pilaplacian equation when you're dealing with the pilaplacian equation then you're dealing with such an equation and now let's make a trait let's treat this as it would be something which is let's say a derivative so let me decompose this as this and this is equal to this now the point is that you can always this is not an elliptic operator so you cannot get up your estimates but you can solve this and the classical solution tells you that you can bound let's say the derivatives of B via mu and the formula if you apply I1 to both sides I1 is an integration so it should cancel the divergence and this means that this is this is giving you that this should be less than or equal than the I1 risk potential for the equation you are just applying I1 to both sides I1 cancels the divergence and then you get formally this so the equation is telling you that also that okay the equation is telling you well I'm non-linear I'm a non-linear equation in the gradient but I'm a linear equation in the flux in the whole tensor so the equation is telling you that this is the estimate that should hold so actually also when you see different and two you should be able to bound the gradient point-wise via risk potential rather than wolf potential this appears at the first side as a crazy as a crazy heuristic argument but it's actually true because what the equation is telling you is exactly this and this was proved by Tomacoos and myself a few years ago so this result that tells that tells that actually holds for general operators and for more general than the pilaplashon for instance having any elliptic operator under the divergence form and this is a non-trivial work of Baroni well this equation this result allows you to in a certain sense linearize the whole theory of the pilaplashon because now whenever you want to prove on the gradient of the pilaplashon it can be proved exactly as it were the laplashon and this holds for general operators so this is a general result that in other words incorporates all the results all the regularity results given by the previous theories for instance the theories of bocado Galue for measure that a problems is implied by this the classical estimates by the Benedetto Manfredi and Ibanez are implied by this even the classical estimates by Atlantic can be recovered by this this holds for pilaplashon these holds for pilaplashon than 2-1 over n at the moment but there could be improvements for that yeah but let me concentrate just on the case pilaplashon or equal than 2 because yeah ok and essentially when you are on the whole rn you have the classical estimate and now holds for any general possible solution to this equation ok now let me go to the non-local non-linear potential theory and let me present a few extensions to this ok, the extensions to the non-local problem let me start step by step by the classical fractional aplashon which in a distribution way it's defined as this one so this is the analog of this equation so at the moment this is the analog of this equation and this is the nice equation that can be solved by a Fourier transform because essentially you have a kernel whose Fourier transform is perfectly known and then you can do whatever with this so this is the analog of this equation so you see the kernel is this object it's 1 over x minus yn plus 2 12 then we switch to a general kernel and now you still have something which is linear with respect to the solution but you lose the perfect shape of the kernel and you replace it by two bounds that replicates the growth this is the analog of this equation with measurable coefficients so essentially the first thing you would like to do is for instance extending the classical the Georgian-Ashmoser theory to such equations then we go to the quasi linear case we have now something which involves a measurable kernel as before but now incorporates a non-linearity with respect to the u-parable but this non-linearity is qualified because it's a monotone this is monotonicity essentially this is the ellipticity of the phi and these are the analog of these equations and then we go even further we take we go to the p-range and now we produce a ellipticity at the p-level and we take p-level and now this is the non-local analog of the classical p-laplation operator so we have produced the reproduction, the fractional version of the laplation then of linear equations with measurable coefficients non-linear equations with measurable coefficients and degenerate equations with measurable coefficients so these are the three cases essentially the prototype is the following one where the non-linearity is very much explicit in this way and this is essentially the operator on the left hand side is known as the fractional p-laplation and the fractional p-laplation essentially emerges when you want to minimize this normo in respect to this kernel so it's a natural object because you're minimizing the p-norm you're minimizing the p-norm so now you remember that the first estimate of Kitt-Pellang and Amalie tells you that you can locally bound this guy the w1 pu x r where w1p mu x r is equal to this plus the localization term the unavoidable localization term which is this one so this is the classical estimate of Kitt-Pellang and Amalie ok now we want to reproduce an estimate of this type for the fractional p-laplation and in particular we consider problems of this type and the natural assumptions that must be prescribed due to the non-locality of the operator because there are long-range interactions and essentially the boundary can be p-capacity zero in several cases right if p is too small and the natural space is the space where this object called the tail of the function is considered to be in this space so this is the essentially the right space because this is a quantity that is nothing fancy but it's the explicit quantity that emerges whenever you try to have energy estimates for this equation so you always have this quantity popping up everywhere so you prescribe that this quantity is fine otherwise you cannot deal with anything and this is the minimal condition for which you can handle your equation so this is the tail and essentially in fact there's a beautiful estimate by Dicaster, Guzi and Palatucci that tells you that if you have such a solution then you have the classical object plus the tail that takes into account the long-range interactions of the solution of all the points so it is obvious that our estimate if there's an estimate if there's a potential estimate so if there's a potential estimate the potential estimate must incorporate also the tail actually let me make a brief sketch of the regularity for this type of radar and the status for this non-linear operator there's a beautiful theory by Dicaster, Guzi and Palatucci that also extends previous theory by Casman where a full theory is presented and the superestimate, the helder continuity and even Arnaquia qualities are proved for solutions to these problems then there's a more reason a beautiful paper by Cotsi where he extends this in a variational setting so the good thing of this paper is that the paper is given for minimizes of functionals for minimizes of potential non-differentiable functionals because the functional f can be for instance just a helder continues so this means that you want to get regularity information not from the Ola Lagrange equation that could not exist at this stage but directly from minimality in other words, from minimality as in the classical case it is possible to derive a cacciopoli or energy type in equality so from these inequalities then you get regularity this is what Cotsi does and this is the reproduction of the classical Jacinta and Jussi's theory of regularity where they prove for instance that any minimizer of this function will just pick growth so without assuming that that there is that there is a Ola Lagrange equation our helder continues and locally helder continues involving also higher inequality and this is the classical approach and as far as the higher gradient theory is concerned this is very much open because differentiating something which is already fractional it turns out to be difficult there are a few results by Brasko and Lindgren but the higher order or the regularity is still an open issue ok let me just please you can also completely forget about this this slide because I'm just getting I'm just giving you the definition of solutions to measure data problems so essentially you have now the pilaplash and type operator on the left hand side you want to solve with a mu and you do it with approximation this is what you classically do in the local case in the local case when you have a measure data problem like for instance this one the classical issue is very simple you can first define very weak solutions or distributional solutions and these are just distributional solutions to this one you can always you can always define in the classical case and in the classical case you get that a solution a measure data problem a very weak solution solves in the weak form this identity if you are in omega so if you think of the pilaplash if you want to give sense to this where the distributional solution prescribes that this is sin infinity then you just require that this is L1 which is the same this is in LT1 so you are not in the natural energy space associated to the operator so these solutions are natural but these solutions are absolutely unminageable you cannot do anything with them because you cannot get energy estimates you cannot for instance test with something which is proportional to you so these are not good essentially no one is able to handle with them so what you do you define special classes of solutions so you do the most natural thing you smooth it out the measure in a way or in another so in a way that this is become for instance L infinity you consider approximating solutions and then you pass to the limit provided that you have good a priori estimates now you can have good a priori estimates because you can test this you can test this and then you can pass to the limit so the same thing is done in the local case in the no local case but you can just forget just forget and you can think that you have now the good definition in your pocket ok so the main result with Tuomo and Yannick is the following one you have exactly you have exactly the potential estimate that now incorporates exactly the term you are waiting for so you have the both potential bound which is now not n-p but n-p-alpha because it must respect the scaling of the equation you see that if alpha is very small then you have less regularity properties of your operator and in fact the potential becomes worse when alpha is very small you get a strong singularities and this reflects that you have less regularization properties of your operator when alpha gets larger you get better and better so you get the term you would expect the localization term plus the tail when mu is equal to zero this gives you back the superestimate before moreover this also involves a classical a criteria on fine regularities of solution because whenever you have that your boof potential is finite then you have a Lebesgue point this is classical in classical linear theory the real one of the first historical goals of linear potential theory was to study the fine properties of solutions of Poisson type equations via potentials now you know that potential is finite you have a precise representative and then you have the you can estimate the capacity of the singular sets and the fine properties and whatever essentially this is the result and this is the perfect analog of the result and you see the boof potential w alpha p is the follow ok so as far as the gradient theory this is still open so this is the first adaptation ok to the no local setting of a classical local no linear result so this follows the lines and extends in the way you expect something which is established in the no linear case now I will present by the way you can also for instance criterion that if you have that mu is in this special lowering space then the solution is continuous this is the borderline optimal space you would expect and so forth this is implied by our results for instance you have that if the boof potentials goes to zero uniformly then the solution is continuous this is essentially a by product of the previous proof also tells you that you have a precise representative if the boof potential is fine and if this goes to zero uniformly then this means that the measure is not spreading is spreading enough to guarantee the absence of singularities and the final continuity and so forth you can get estimates like this the definition of lowering space is this one they are interpolating spaces they are interpolating the big spaces and they are able to catch the final borderline results ok this was defining a first extension that let's say it's a first type of extension it replicates a local result now I would like to present a second fractional result that does not replicate a linear result essentially where there's a new phenomenon coming up and this is a non-local self-improving property the non-local self-improving property has something to do with the so-called estimate so my as estimates are the following one what can you tell about already for instance in the case p equals to two and the right hand side equal to zero what can you tell about the regularity of the gradient and the integrability of the gradient of an equation like this you take already when the right hand side is equal to zero and you start from an energy solutions now you are considering the case p equals to two in these assumptions in the assumptions before ok then the classical elca admires and Jacinta Modica's estimates are the following one these bootstraps into something which is better than L2 for some small delta depending only on the ellipticity constants of the equation this is the so-called higher integrability it's classically proved by now there's a better approach due to Gehring's lemma Gehring's lemma tells you the following assume that you have a so-called reverse order inequality reverse order inequality is the following one you take f just consider g to be zero at the moment the classical order inequality tells you that this inequality is true if q is larger than p now assume that q is less than p so you reverse the principle of elders inequality then this improves this self improves the integrability of the function getting you an other exponent for which you get better and better as long as the right hand side allows you which is a source term in this in this setting ok the point is that for these equations you have what is classically called the cacciopoli type inequality also called the reverse pointer inequality or energy estimate so you can bound for ellipticity the gradient by the solution itself so you can bound a higher-order object by a lower-order object and this is according to the principle that for instance for harmonic functions point-wise convergence puts trapping uniform convergence in all derivatives right so you can control higher-order objects with lower-order ones this you can prove under these assumptions and this implies the higher integrability of the gradient so let's see how this implies the higher integrability of the gradient this is a very simple inequality that you can easily obtain by testing with a simple possible choice of the equations so you take a cutoff and you multiply by u and you test it and then you get it in two lines ok so how do you get this one so you get the gradient can be estimated by this now apply point-carries inequality for the right hand side and you get what a reverse order inequality because now you can bound du squared by du squared to a lower power which is n over n plus 2 now you can conclude with the non-trivial Gering's Lemma getting you that you self-improve the gradient so it's a very simple fact so you get an elliptic equation you get a cachoppoli type inequality so you can control du by u but you can be controlled by a lower power of the gradient and then you self-improve concluding by Gering's Lemma ok now you can conclude the non-trivial Gering's Lemma ok so this is the improvement what can you tell about now the oscillations of the gradient you can actually tell nothing and this is the counter example take the simplest possible elliptic equation this is the solution take one over at to be measurable just measurable and you see that the gradient of the solution is one over at and so at can be as bad as you like and so there's no improvement whatsoever end of the story so in the local case you get an improvement of the integrability but no matter you can do you cannot get this and this happens where coefficients are bad bad coefficients ok now now we go to the now we want to extend this self-improving result to the non-local case and we go to this a very simple setting to the linear setting but with measurable coefficients so you just get a lower and upper bound on the kernel K and this doesn't allow you to use Fourier transform because you don't know exactly the specific shape of K so the analog ok what is the definition of energy solution you are in W alpha p alpha 2 I will not recall the definition so please allow me to not in this conference recall the definition so the natural definition you want to start with which is the natural analog of the energy solution which is in W1 2 in the local case is W alpha 2 and if you want to extend what you get in the linear case you are allowed to assume that you is in W alpha 2 plus delta for some delta so this is the Maya's property in this case and let me recall you that this is essentially the this norm ok this is essentially what you have because Bassenrand approved that this is true and essentially this relies on the classical scheme you get reversal, the inequality you get that this quantity is in L2 plus delta and this implies by a characterization of Stein and Strikert's that this is in W alpha 2 plus delta but the surprise is that now there is a new unexpected phenomenon is that what you can actually prove is that you do not only have a self-improvement of integral building but you have a self-improvement of differential building and this is one of the very few cases of a purely non-local of a purely non-local effect so this is not true in the local case so this is not the replication of the local result this is a new phenomenon and the proof of this is very adding on trivial it involves essentially 30 pages of hard harmonic analysis when I say hard harmonic analysis this means that you are not using the tools, the pre-made tools of harmonic analysis but you really have to do coverings and combinatorics and things like that so you have to do really things by hands so you have to use ideas from a harmonic analysis but not tools and there is no analogue of that later on Chicago was able to extend this by saying that if you prove if you go a bit below then you can go a bit further on there is a difference between this result essentially and you can get also this result by the methods by adapting some of our methods our methods start from energy in equality these methods require solutions in fact we do everything by hands Chicago in these very nice papers use delicate properties of commutators in fractional spaces all the things extend to the p to the p case ok, let me now present our approach and it is a fractional approach to Garin Glem the fractional approach has one advantage for instance if you want to consider minimizes or functionals for which you do not have Euler Lagrange equations then you can use this approach you cannot use the approach via testing because you do not have the equation for instance I think that this could follow in the realm of this paper of Kotzi ok so just to two slides, because the proof is ok ok, what is the starting point as I told you the starting point is this Cachoppoli type in equality in the local case then from this you get the reverse order in equality and then you conclude via Garin Glem ok, now the first our first approach is that we prove an energy type in equality you see on the left hand side you have the alpha derivative dimensionally on the right hand side you have the same thing before scaling proportionally and then you take a tail type term which takes into account the fact that the problem means it's on the whole Rn and involves the long range interactions between points and functions and then we prove that if you have this in equality then you have this so just this fractional Cachoppoli type in equality implies the higher differential build actually it is sufficient to prove higher differential build because then by embedding you know that from higher differential build you also get intermediate differential build and integrability ok, so what does it mean that you is in W12 being W12 means that trivially du is integrable with respect to a finite measure so the idea is that where is this additional differential build coming from now what does it mean that you belongs to Walpha2 this means that this object which is dimensionally speaking analysis to the gradient but you know that there is no local effect here well this is integrable against an infinite set function so this should improve this should tell you that this object is in a way better than the other so this is actually the source of this extra differential build because this object this object must be smaller than what you believe because it must be integrable against something which is non-finally integrable how this this connects from L1 this is a Marchinkiewicz factor it's a log factor so this means that if this is finite against something that blows up as a log this should be better of a log type but you whenever you get a log in higher integrability then you can improve and bootstrap to a power this is the first proof the first unknown proof of Gehring's lemma the first unknown proof was getting first a log integrability then you bootstrap to a power so this is the first suspect that should be something so the idea is the following one we make this this measure of finite but making this measure of finite means that we do n-epsilon and we do something here now this is finite against this so we lift the problem into rn times rn and we say that this function is finite against this measure so the measure is is more regular as the object too because you trade the blow up of the measure you trade it in the denominator but trading in the denominator means improving the French bill so we translate the Cachoppoli typing a quoting a reversal the inequality for the lifted function capital U we prove a version of Gehring's lemma for dual pairs for these new dual pairs we call them dual pairs in this sense then the higher integrability of capital U translate into a higher differentiability of U let's see why assume that U is in L2 plus log what does this mean? this, this, this means this so this is a way to exploit the thing and in fact what we prove is the following one the previous trend is that this, for this dual pair this inequality then this inequality itself improves now this at the first site might appear as a one of the many generalizations of the of the Gehring's lemma but in fact it is not why? because the tilde b which are the product of both this means that this is Rn times Rn while Gehring's lemma would require an information on every possible bowl here you only have bowls around the diagonal a complete loss of information outside so the idea is that you have a very difficult trade between the distance from the diagonal where you lose information and when you get information so the idea is that you have to quantify the distance from the diagonal if the distance is far enough then you are good because the kernel is not bad if you are closing up you have information and this makes the combinatorics very very delicate ok and now part 4 briefly, very briefly I'll go now in the other direction of the talk let's say how fractional methods and this is probably less known to the audience how fractional methods can be extended to local that can be used to get a local results so first of all a first example limiting Calderon-Sigmund theory as I told you if you have a measured data equation and now this is a local fact for instance you have this model equation what do you have the classical theories that tells you that the gradient is in this space and this is sharp this is already sharp for the case p equals to 2 otherwise it can be seen to be sharp if you see the classical non-linear grains function which is this one minus, ah sorry ok, this is the classical non-linear grains function ok now there's a gap because you know you cannot get that the gradient is in W11 this is once again the failure of Calderon-Sigmund theory in the limiting case so you get that there is a lack of theory why because you have a second order equation a second order equation is something that formally prescribes the value of second derivatives but now you have all the information about first derivatives so you would like to lift it up ok this was done for the case p equals to 2 by me several years ago where I proved that if the gradient cannot be in W11 this is anyway every sub of space before W11 so you almost lose nothing this is a a special technique that allows to get a sort of analogue of local little palette decomposition so this is for the gradient what happens when p is different in 2 well analyzing the behavior of the fundamental solution this would lead to the idea that the gradient must be in this space and in fact this is also true this is sharp because this embeds into a sub aller space if you take epsilon equal to 0 this would embed in a lebeck space where the fundamental solution is not doesn't belong to and essentially this paper lifted the whole theory up to the differentiability level so this is the first case of limiting Calderon-Sigmund theory but now essentially as I explained before as I explained before the whole essence of Calderon-Sigmund theory is that when you have a regularizing operator you can trade for instance in this case the more general case provided the regularity assumption on the vector field considered you can trade this with the divergence divergence with the gradient so this is formally telling you that the derivative of the whole the derivatives of the whole object should belong to a good sub aller space as in the case of the Laplacian so in the Laplacian you have that you can trade the divergence with the derivatives and this is what has been done in a recent paper by Abdelin Kuzin myself so we can essentially trading the divergence with the derivatives getting the full limiting regularity of let's say getting the limiting case of Calderon-Sigmund theory in the non-linear case this is a delicate result that couples with the previous result the previous result that is the potential estimate states that you can control essentially A of the U with I1 of mu X plus A of the U so it's all intrinsic so in the previous case you apply I1 to both sides and you get this estimate in this case you get the Calderon-Sigmund analog so this is the potential estimate analog while this is the Calderon-Sigmund analog ok and this comes along with the suitable catch-up polytype inequality which is explicit observed that in the case mu equal to 0 you get actually the derivatives of A and you can estimate ok and let me give the final example and this is probably surprising because actually we know from the initial paper of Caffarelli and Vassar and Kassmann that the fractional de Georgi technique has been usado to prove there's a fractional de Georgi technique used to prove regularity for solutions to fractional operators actually there's another paper where I introduced a similar fractional de Georgi technique to prove essentially regularity for solutions to local operators what is the difference the difference is the following one so I'll give the first proof of the gradient potential estimate for p equals to 2 I'll give the proof of this estimate and so this shows essentially the interplay I was talking about at the beginning of the talk so the interplay between fractional and local between not only results but also between methods ok this is the first proof of this result and is the following one ok how do you get a gradient estimate when mu is equal to 0 so how do you usually get the usual gradient estimate for a solution which is the following one the u is less than or equal than its average this is classical for harmonic functions ok you take a solution of this guy you differentiate the equation as was telling before so the gradient becomes the solution to a linearized equation with measurable coefficients and actually this is the core of the de Georgi's theory and then you get the du is bounded how do you get in turn a bound for du by writing a fractional catch up on the equality on level sets the second derivatives because now it's the gradient to be a solution can be bounded by first derivatives and then you can use subal efe embedding theorem to get a linear a non linear iteration eventually leading to the bound for the gradient so this is the classical approach now in the classical fractional problems for instance if we follow these these papers that I mentioned before you what you get you cannot get the higher differentiability you cannot get such a fractional catch up on inequality to three minutes it's almost done ok now the point is the following one so in the classical when you deal with fractional problems you don't have local derivatives so you cannot even write them here you have derivatives but you cannot once again write them not because the problem is no local but because the problem involves a measure if the problem involves a measure then this program fails from the very beginning because if this is a measure or even if this is in L1 then the second derivatives are not in L1 so how do you replace this method by recalling that this is the Gallardo norm of the function D what you get is the following you recover the theorem I was talking about before and actually what you do you prove a fractional catch up polytype inequality so you don't have so the idea is that although the problem is local there are no second derivatives because the right hand side is bad but there are fractional derivatives so you can write a fractional catch up polytype inequality where you bound not the second derivatives of the solutions but the fractional derivative of the solution plus the remaining part which is due to a non-trivial right hand side this is essentially the same approach that comes from the papers that I mentioned before by Caffarelli and Vaser and Kasman and was dying also independently here the heavy fact comes later on because you have the tail terms to control while the fractional catch up polytype inequality comes for freeing one line just by testing here the very delicate point is deriving a non-local fractional catch up poly inequality because the problem is local and forces things to be local here you want to go down and this is essentially the little pale decomposition type method that I was mentioning before anyway you can prove this where you get some sigma you don't care how large sigma is because the iteration in the georgis nashmosa theory are geometric so whenever sigma is larger than zero you are finally converging so this is the first result I proved in this paper which was actually written in 2007 and then the second result you see compare the classical catch up polytype inequality with the fractional one you get fractional derivatives full derivatives you get L1 here because solutions are not in L2 when you deal with measure data problems L2 here so this is the first line so any solution satisfies this and then the final stage is that any function satisfying this fractional catch up polytype inequality satisfies this bound according to the orthodoxy that everything must come from standard energy estimates and therefore if you combine these two methods then you are done and I think that this is a good point to stop thanks a lot for your attention


So, Dylan, there's a bunch of people who had small accounts here that built them up to six and seven figure accounts. So I think they would be much better to give you, if all I'm saying to give you any type of tips, I just don't know how to. No, it's not to do about the money, it's just the point of, I started out on prop, so capital was never an issue for us. If all I'm saying it was never an issue for us, that's the whole point, it was never an issue. Anybody who ever traded a prop, that's all they know is capital was never an issue. I think what Steve's saying makes a lot of sense, swinging the pivots. So Dylan, when there's a pivot on the table like a Tesla, let's just say above 881, right? You want to take the value trades with the highest probability of a measured potential. You know what I'm saying? Like you really want to wait for those big measured potential trades. You know what I'm saying? Like you really, really want to wait for those big measured, like you can't trade every day. That's for sure. You know what I mean? Like you definitely can't trade every day. That's for sure. The key is to embrace the process and just wait for these, like put it this way, like Apple over 38 in this lifetime of the next, Dylan, like you see what I'm saying? Like Apple, right? If Apple wakes up in this lifetime of the next above 138, then you can swing it, then you could swing it for like six, seven points, because here's your whole macro channel until possible move to the upside. But yeah, you want to wait for measured potential. Like not everything has measured potential. That's the whole point. That's why trading unfortunately is so difficult for a lot of traders who have smaller capital because they just want to, well, number one, you're constantly fixated with the idea of social media, press, social media, press, social media, press, press, press, press, social media, press, go home run diamond hands.   And the unfortunate part is when you embrace all that noise, right? When you embrace all that incredible unnecessary noise, you're not sticking to the big picture. And the big picture is you can't trade every day. Number one, and number two, you can't trade every single day when you have a small account. Okay, so you really have to wait for value. So when you're looking at a trade, for example, like a Zoom, or I'll even give you a better idea. Dylan, I'll even give you a better idea that will give you a better bang for your buck. Right, Dylan? You see how a lot of these small cap stocks have big aggressive option flow, right? Like OGI, take those traits, right? Like obviously not here, but we were a long OGI right here, right? I was a long OGI right here, and the reason why I started buying it, because buyers came in for the $5 calls, right? So you have option flow, correct? You have option flow coming into your name. So anytime you see a stock with really big option flow, right?  So even if you buy a thousand little shares, right? Even if you buy a thousand little shares, and the stock goes up, you know, $1.50, that's how you could build up your account. You know what I'm saying? Like that's how you can really, really build up your account very, very quickly. The problem is that type of market, right? That type of market is not around all the time. Like I tweeted out last week and I said, look, when this is all said and done, you better have, right? You better have a share, because correct me if I'm wrong, the last time we had a market like that, it was 20 years ago, right? It was 20 years ago. So the moral of the story is when you have a market like this, and again, we're in the market like this, right? It's very, very important to understand what you have in your disposal. So when you see big option flow coming into small cap stocks, I would definitely do those, right guys? For all you guys who traded OGI on either equity or either equity or options, right? Was that not the play to go, right? Option flow out of the money calls, that's a great recipe. It's putting peanut butter and jelly together, right? It's a great, great recipe. Yeah, absolutely. But other than that, man, I just don't know. Yeah, TRXC was another one. I just don't know. Even, Dylan, even a stock like UEC, right? Even a stock like UEC yesterday, right? Even a stock like UEC yesterday, for example, right? There was order flow they came in, order flow came in for the $3 calls. And it went from $3.16, $3.17 to $3.30 today, right?  So I think if you're very patient, you could do very, very well with pivots, right? Until you start building up your account. And we've had a lot of people with less than $10,000. I mean, there's tons of people here in the room that over a year or two years, they went to six, seven figures in their account. So it could be done. I just don't know how they did it, you know what I'm saying? And that's the complete honest truth. I just don't know how, right? Like Larry started with like five grand. He's like mid six figures now. It's not easy, but it's doable. It's definitely doable. It's just I don't know how they did it. You follow what I'm saying? Like I don't know, like I don't know how they did it on a day that, yeah, there's a lot of discipline involved, right? There's a lot of discipline involved, a lot. There's a lot of discipline involved. And we had the recent conversation I had with Richard, you know, a couple of months ago went from like three to five grand to over the pattern day trading rule, right? He didn't do it overnight. He did it with a lot of discipline. You know what I mean? It's a lot. It's so much harder taking your account from 5,000 to 30 than it is from 30,000 to 100. It's just the reality.  It's much easier to let go everything when you have $200,000 and you have $5,000. So yeah, I wish I can help you with that. There's plenty of people who are great people in here. That's the one thing that everybody here helps everybody out. Also, Kenyon, on the website, I think Larry, I think Larry and Matt did something, right? I think Larry and Matt did something on the Access to Trader website. It's on the website in the members area that explains how they did it via options. Yeah, so Dylan, if you go, Kenyon, can you just tell them how to get, how to access it? They'll tell you, because they both started with under, I think both started under $15,000. I think Matt had like, right, Matt, Matt had like 12 and Larry had like five or seven grand, whatever it was. Yeah, yeah, yeah. They'll definitely give you some good nuggets of how they did it. But it's definitely doable, because I get emails all the time, hey, Dan, I just turned five into blah, blah, blah, or I get it all the time, I just don't know how to do it. You know what I mean? Like again, somebody said a long time ago, hey, Dan, why don't you try to do, take five grand, I'm like, how, how would I do that? Like, how can I possibly take five grand and turn to the 25? Like, how would I do that? Or maybe I'd trade 11 shares of Tesla? Like, how am I personally gonna do that?  My diamonds, of course. Yeah, so, yeah, that's kind of the deal. It is, this is the best way of saying it. Unfortunately, this is a business that you do need capital. You really do. You do need capital. You need a lot of it, right? But as we've seen, people do have really good success in a longer period of time that they could accomplish that. Maybe not go from 5,000 to half a million, but maybe go from five to 50, right? Everything is in stages. Before you can make, you know, $500, you have to learn how to make 50. Before you learn how to make 50, you have to learn how to make 20, right? And on and on and on and on. So, yeah, it's tough. It's a very tough business. It's a very, very hard business.


Hi everyone, Joe for Jaspy's casewrakes.com coming at you with a box of 2022 clearly Daunris football 2022 edition we're Brock party hunting. We're also giving away a spot If you're one of the first five to pick up spots, you got a chance at an early bird spot So here's the first five right here And let's gather these names right here. Let's roll it and randomize it one into four five times for this group one two three four and five Wow after five times Michael all it takes is one Mike Sorry Tristan randomizer a little brutal this week But congrats to Michael you got an extra spot now, let's see what we can do here Let's roll and randomize it two into two for the hard way one two three and four and then once again four times for the divisions One two three and four South NFC south and to the NFC east All right, so here's how it shakes out for everybody Tristan you have the NFC south Aaron AFC south Tristan NFC west Michael AFC east Aaron AFC north Michael NFC north and Tristan AFC west NFC east So let's alphabetize by division and And we'll pause the video just for a little bit when we come back We're gonna see if there's any trades, then we'll have breaks took around BRB All right, welcome back everybody. No deals were done. So this list right here remains the same Here on Wednesday the 20th hump day. Thanks everybody for joining us and for making this part of your day It's a quick little one box break. So now let's see if we can find some some Brock purties in here among others From the 2022 draft class, but obviously with the way Brock purge has been playing we want to try to See if we can find some of him Even if it's not an autograph, maybe we find like a low numbered card or something like that rated rookie card would be sweet We'll see Hey there Duncan. Oh They started off with a little bit of flooding in the shot a little rain And a clogged drain outside, which we didn't realize was clogged up and We've been spending a little bit of the early part of the day dealing with that we're have to remove all the showcases and clear out the puddles of water underneath so Tyler Algear is the autograph not a bad auto Not Brock Purdy. This is for the NFC south that will be for Tristan With the Falcons running back Showed some moments some flashes. There's Nick Chubb 275 for the AFC north sir chance that you might come back FC north is Aaron Devo Drew Brees, Brian Robinson, Tremaine Johnson Got another numbered car to see he lamb 24 out of 49 for the Cowboys Chris Olave and a Jordan Davis Olava going to the NFC south and the CD lamb going to the NFC east That will be for Tristan as well. There you go gang The hunch continues for Brock Purdy autographs. Go to Jasper's case breaks calm for clearly donors. I'm Joe I'll see you next time for the next break. Bye. Bye


Howard Thurman always spoke of his relationship with the younger visionary Martin Luther King Jr. in personal and endearing terms. In the following selection, delivered over radio station KSFO in San Francisco on the evening of King's assassination, Thurman speaks to the ideal of community and the hope for a transformed American society symbolized in the life and witness of the fallen prophet. He writes, Martin Luther King Jr. is dead. This is the simple and utter fact. A brief few hours ago his voice could be heard in the land, from the ends of the earth, from the heart of our cities, from the far sides of the humble and the mighty, from the sails of a thousand prisons, from the deep central place in the soul of America, the cry of anguish can be heard. There are no words with which to eulogize this man. Martin Luther King was the living epitome of a way of life that rejected physical violence as the lifestyle of a morally responsible people. His assassination reveals the cleft deep in the psyche of the American people, the profound ambivalence and ambiguity of our way of life. Something deep within us rejects nonviolent direct action as a dependable procedure for effecting social change. Yet, against this rejection, something always struggles, pushing, pushing, always pushing with another imperative, another demand. It was King's fact that gave to this rejection flesh and blood, courage and vision, hope and enthusiasm, for indeed in him the informed conscience of the country became articulate. And tonight what many of us are feeling is that we, all of us, must be the conscience wherever we are living, functioning and behaving. Racial prejudices, segregation, discrimination were not regarded by him as merely un-American, undemocratic, but as mortal sins against God. For those who are religious, it awakens guilt. For those who are merely superstitious, it inspires fear. And it was this fear that pulled the trigger of the assassin's gun that took his life.