The royal's race day wardrobe is unmatched as she enjoys the Magic Millions on the Gold Coast with husband Mike Tindall

Zara Tindall stuns in stripes as she dons bold silhouette-skimming dress in Australia | HELLO!

Using titles of several Bollywood movies released in 2023 and an upcoming web series, the social media account of Delhi Police issued an advisory in its own style., Delhi News, Times Now

Delhi: ‘Zara Hatke Zara Bachke’: Delhi Police’s New Year’s Eve Advisory Has A Bollywood Twist | Delhi News, Times Now

The Waleses and the Tindalls always appear to be very close

Zara Tindall shows strong bond with Kate Middleton as Princess of Wales appears to borrow special accessory

Prince William's wife and Zara Tindall appear to share a close bond

Zara Tindall comes to Princess Kate's rescue ahead of family Christmas gathering | HELLO!

The Prince and Princess of Wales' children have developed a loving friendship with Zara and Mike Tindall's kids.

Adorable friendship formed between Prince William and Zara Tindall's kids - Daily Record

Prince George's bond with Zara and Mike Tindall was on display during the royal family's Christmas walk, as well as other sweet interactions with the families.

See Prince George's Sweet Moments on Christmas with Zara and Mike Tindall

Prince William and Kate's children have developed a sweet friendship with Zara and Mike Tindall's kids, following in the footsteps of their parents who have a close bond

Adorable friendship between Zara Tindall and Prince William's children since Meghan and Harry drift - Mirror Online

Members of the Royal Family, including the Prince and Princess of Wales, their children, and Mike and Zara Tindall, attended the Sandringham church service yesterday.

Touching friendship between Zara Tindall and Prince William's children and Mike's bond with Prince George prove how the Tindalls and Waleses have grown ever closer since Harry and Meghan drifted away | Daily Mail Online

Zara Tindall is becoming one of the royal family’s best dressed, wearing a Laura Green coat and Stuart Weitzman boots on Christmas Day at Sandringham.

Zara Tindall looks amazing in waist-cinching coat alongside daughter Lena, 5 | HELLO!

A customer at the popular fast fashion chain Zara had to remove hangers, magnetic strips at checkout herself. She's done.

Customer Slams Zara’s Self-Checkout, Asks 'Am I Clocked In?'

Zark

zark

Prisoners on the Electron

Prisioneros en el electrón

Prisonniers de l'électron

इलेक्ट्रॉन पर कैदी

電子の囚人

Prisioneiros do Elétron

Tù nhân trên Electron

电子囚徒

This is the official HackerNoon wiki for Cosmos. As part of our , this living wiki displays on . If you’d like to suggest edits to our official Cosmos wiki, please submit them in comment field below, a real human editor will read and review your suggestions for accuracy and publication. coin price pages Cosmos crypto price page Cosmos is a network of that communicate and work together to create a better experience for developers and users of the . ATOM is the that helps keep the Cosmos network secure. interconnected blockchains blockchain governance token History of Cosmos Jae Kwon and Ethan Buchman cofounded Tendermint in 2014. Tendermint was the consensus algorithm that went on to power Cosmos. Kwon and Buchman co-authored the Cosmos white paper and released the software in 2016. A Swiss non-profit known as The Interchain Foundation is the organization that helped in the development and launch of Cosmos. In 2017, The Interchain Foundation carried out a two-week offering of the ATOM token, raising over $17 million. Tendermint later raised $9 million to continue developing the project in 2019. Cosmos Biggest Claim to Fame Cosmos offers developers an opportunity to build application-specific blockchains which can interconnect easily. This gives Cosmos a leg up from the other leading blockchains which force developers to exist as on their blockchains. Cosmos allows for interconnection between blockchains instead of several blockchain applications competing on a single base layer. smart contracts Several well-known and well-respected apps and services are running on the Cosmos network. Several well-known projects have been built using Cosmos including the Binance Chain (BNB), (LUNA), and Crypto.com Coin (CRO). cryptocurrency Terra Cosmos offers the solutions to a problem all crypto has faced. The problem of slow transactions and bottlenecks, whenever there is a large number of transactions, has not been solved. But with the technology Cosmos offers, transaction speeds within the blockchain and the problem of bottlenecks can be solved. The technology will allow for faster and more efficient transactions. The cost of Cosmos and its currency has increased with the launch of its interface. The price has also increased with the company's integration with , which hosts most of the , and also with the between Cosmos and . decentralized finance Ethereum decentralized finance applications interoperability Bitcoin Criticism of Cosmos Investor confidence is low due to the lack of official regulation. The lack of regulation means that the environment to invest in Cosmos is considered unstable and undependable, which is something investors would not like to risk. The regulators worry that crypto products such as Cosmos work as traditional banks do but without the same consumer protections that traditional banks have. There are other platforms that offer the same blockchain advantages as Cosmos but have higher returns. Crypto is a competitive market and the idea behind Cosmos is not entirely unique so investors consider giving other better-performing markets their investments. The problem of interoperability, which is the potential future gain-making plan for Cosmos if they can get it fully operational, could be solved by another crypto before Cosmos fully solves it. This will inevitably cause the price of Cosmos to go down since that is the main idea behind Cosmos. is growing fast and the newer cryptocurrency may phase out the current cryptocurrency with newer and more integrated technology. Blockchain technology The Cosmos Team Jae Kwon is the cofounder of Tendermint and the president of the Interchain Foundation. He worked as a developer in Silicon Valley after his graduation in 2005, working on apps such as Alexa and Yelp. He together with Ethan Buchaman and Zarko Milosevik created the Tendermint Consensus algorithm and implementation which aimed to liberate the blockchain from the drawbacks of proof of work mining. Ethan Buchman is the co-founder of Tendermint and Cosmos. He is the chief technical director of the Interchain Foundation. He started his career as a core developer for Eris Industries in 2014. He joined the Interchain Foundation in 2017 and has been the technical director since 2019. In Conclusion Blockchain technology needs networks that can handle a large number of transactions simultaneously without compromising the speed of the transaction. Cosmos allows for blockchain projects to be created that can handle high transaction volumes through its development process. Read more cosmos stories on HackerNoon This is the official HackerNoon wiki for Cosmos. As part of our , this living wiki displays on . If you’d like to suggest edits to our official Cosmos wiki, please submit them in comment field below, a real human editor will read and review your suggestions for accuracy and publication. coin price pages Cosmos crypto price page

Cosmos $ATOM Wiki: Coin History, Techstack, and Crypto News

As part of my continuous work to see how much I can do with just CSS (see other work such as the ), I decided to try and recreate Windows 98 using nothing else apart from CSS and HTML. Did anyone ask for this? Not really? Is it fun to try and see what you can accomplish with just CSS? Yes, sort of. Was it time-consuming? Unfortunately yes. CSS-only Minecraft Chicken As a quick note - this is a desktop Windows recreation, so it is of course optimized for desktop viewing. Ironically, though, it probably won't work on Windows 98, since you would have to use a very old version of Internet Explorer to view it. since it is better viewed in full-screen mode. In this demo the things which I thought were cool included: The link to the demo can be found here with just CSS - although no scorekeeping. Minesweeper and logout with the memory of who is logged in using the brand new CSS parent selector. Login Animated . update process . Minimizing, maximizing, and closing windows It is however worth noting that the things that are hard or just downright impossible to do with only CSS, such as: Drag and drop - not possible at all with CSS. Multiple conditions - for example, hard to say that a window should be both maximized and put on top at the same time. Click anywhere to close - you have to click the start button to close and open it. You can't just click anywhere else to close it. Multiple conditions in general - CSS doesn't really have an AND operator. How to build Windows 98 in CSS So, the first thing I wanted to get right about this version of Windows 98 was the look and feel. I'm using some pretty cool Windows 98 icons (which I do think should make a comeback), as well as the standard Windows 98 color scheme. To get the indented and out-dented feel I used quite a complicated box-shadow, as you can see here: .windows-box-shadow, .minesweeper .content > label {
    box-shadow: -2px -2px #e0dede, -2px 0 #e0dede, 0 -2px #e0dede, -4px -4px white, -4px 0 white, 0 -4px white, 2px 2px #818181, 0 2px #818181, 2px 0 #818181,  2px -2px #e0dede, -2px 2px #818181, -4px 2px white, -4px 4px black, 4px 4px black, 4px 0 black, 0 4px black, 2px -4px white, 4px -4px black;
}
.inverse-windows-box-shadow, .minesweeper .content > label:active {
    box-shadow: -2px -2px #818181, -2px 0 #818181, 0 -2px #818181, -4px -4px black, -4px 0 black, 0 -4px black, 2px 2px #e0dede, 0 2px #e0dede, 2px 0 #e0dede,  2px -2px #818181, -2px 2px #e0dede, -4px 2px black, -4px 4px white, 4px 4px white, 4px 0 white, 0 4px white, 2px -4px black, 4px -4px white;
} Everything else was relatively straightforward look and feel-wise. The key to making all of this work is checkboxes and radio buttons. Using Checkboxes and Radio buttons as a store of information in CSS Checkboxes and radios are the only way to store information in CSS. We can then use them to implement style changes. Checkboxes, when checked, allow us to enable or disable a single feature (like showing a window, maximizing a window, or clicking a minesweeper square). For things where there is only one option allowed to be active at a time (for example, which window should be on top) - we can use radio buttons. Both follow the same syntax in CSS, where we use the selector: :checked #windows-11:checked ~ .windows-11 .text {
    /* -- CSS here -- */
} Here, when the input is checked, it will affect its sibling's child - so we can apply some custom CSS. Importantly, since we can't easily style an HTML input, we use s to model the different features of Windows 98. #windows-11 .text label For example: <form id="windows">
    
    <input type="checkbox" id="login-screen-input" name="login-screen-input" />

    
    <label for="login-screen-input">Log Off</label>
</form> Here, the shown is associated with the checkbox . That means when you click the label, it will check the checkbox. This basically gives us free reign to track a user's clicks, and then use the checkbox status to show certain windows, in certain forms. The difficulty is you can only have label associated with input. label #login-screen-input :checked one one That means in a scenario where a button is supposed to open the window and place it on top of all other windows, you'd have to use Javascript since this will require tracking two states - the of the window, and whether it is open or closed. This is a major blocker in implementing CSS-only versions of complex UIs. z-index Using the parent selector to track who is logged in Since we have a login screen in a for when a user logs out, we can't use sibling selectors to easily track who is logged in. We can still use statuses to track this, but the inputs are too deep in our DOM to affect their parent's sibling's CSS. Fortunately, we can use the new CSS parent selector for just this task: div :checked #login-screen:has(#login-window .select-box #zark-muckerberg:checked) ~ #start-bar .zark-muckerberg,
#login-screen:has(#login-window .select-box #donald-trump:checked) ~ #start-bar .donald-trump,
#login-screen:has(#login-window .select-box #spiderman:checked) ~ #start-bar .spiderman {
    display: inline;
    padding-left: 0.5rem;
} Here, if has a div, we can use it to display the user name in the start bar, despite these checkboxes being deep within the DOM. This is pretty neat, and a useful way to use parent selectors if you ever wish to accomplish recreating a CSS-only version of a Windows operating system. #login-screen :checked There is no CSS AND Selector Much to my dismay, there was no way for me to create a CSS selector using chained checked boxes. For example, consider this situation where we apply some CSS based on a state: AND :checked #minesweeper-box-1-1:checked ~ .content > .minesweeper-box-1-1 {
    
} This works fine, but what if we want to check if two minesweeper boxes next to each other are checked, before applying CSS? I thought, logically, that the selector would only continue if both were checked - so tried this: #minesweeper-box-1-1:checked + #minesweeper-box-2-1:checked ~ .content > .minesweeper-box-1-1 {
    
} But unfortunately, that doesn't work. So while we have a way to track states in CSS, . That's disappointing, but it doesn't limit us too much for our Windows 98 implementation. it's quite hard to track multi-conditioned checkbox states to create logical statements and styles based on that Achieving Windows 98 Text Windows 98 text is anti-aliased. To remove anti-aliasing (at least for some browsers), and achieve that classic, crisp, Windows 98 finish, I used the following CSS: not body {
    -webkit-font-smoothing: none;
    -moz-osx-font-smoothing: grayscale;
} Recreating Minesweeper So one of the major undertakings in this project was recreating Minesweeper. I kept the grid relatively small (to maintain my sanity) - but I had to make my own Minesweeper map created out of labels. Each of these labels mapped to an input, which tracked if a cell had been clicked or not. If a mine is clicked, it's game over and you can't interact with the board anymore. As there were ~56 Minesweeper cells, we needed an equivalent of ~56 Minesweeper inputs. Tracking that all in CSS required of CSS, but the overall result is pretty cool looking. a lot Overall, this follows the same logic as the previously mentioned checkbox and radio trick - so conceptually it's not any more complicated than anything else we've done. Conclusion I hope you've enjoyed this guide. Doing this reminded me of how web development was about 10 years ago when things were a lot harder to accomplish and required a lot of manual creation of DOM elements. It's fun to see what can be achieved in CSS all these years on (including the parent selector). Is this a realistic way to create web applications? Not really in terms of speed, and not yet in terms of functionality, but CSS did a lot more than I thought it would be able to, and I'm pretty happy with the results. If you enjoyed this, . please consider following me on Twitter Also published . here

Recreate Windows 98 With CSS

Introduction Who are all those people on social media? Those people who randomly like, share, comment on posts, harass and praise other people and events. The anonymous users who influence public opinion, dictate the social discourse, and even have the power to decide the outcome of elections. They could have Ph.D.’s or be children. Or their personas could be entirely fabricated by AI. It can be concluded that they are unidentifiable. Here is a spooky thought experiment for you. Imagine that you are texting back and forth with someone or reading an article online. Naturally, you’d assume that there is a human, made of flesh and bones at the other end. In actuality, however, the communication or content you receive is solely generated by an AI. Just like the fake profile photo. How could you know? To take the thought experiment one step further: How can you prove that friends you have in your network on social media are really who they appear to be? When you meet them, their faces serve as proof of identity. But in the online space, there is hardly any direct way for users to verify if a profile is in fact linked to the person it represents. Even Instagram stories, TikTok videos, or the most heartfelt Facebook postings can in theory be deepfaked. Essentially, information online cannot be trusted. Sure, right now this dilemma is more philosophical than pragmatic. However, technologies that can distort reality in online spaces are getting stronger by the day. Just imagine what they can do in 10 or 50 years. Without a bulletproof method for users to confirm people’s identity, the internet may run into a situation that resonates with “dead internet theory” — an underground, conspiracy theory that proposes the internet is “fake”, basically a gaslighting project that is almost entirely controlled by AI. [1] Bizarre as it sounds, I do believe there is a disturbing ring of truth to it. In this post, I will go over what the dead internet theory is, and how a platform-agnostic identify system built on the “trustless” web 3.0 could resolve the current problem of digital identities. Dead Internet Theory — Main Premise “Dead internet theory” had a breakthrough into mainstream media when a on the forum caught the attention of The Atlantic writer Kaitlynn Tiffany. widely read post Agora Road Macintosh Cafe The post was written by a forum member called “IlluminatiPirate” who boasts about being an “ ” (meaning an old-time member) of imageboards such as 4chan who has “ ”. In the midst of his seemingly paranoid ramblings, there are some nuggets of truth that many people can relate to. oldfag seen it all The pseudonymous forum member compiled evidence to the theory which was according to him originally written by anonymous users on 4chan’s paranormal section and Wizchan. The latter is an imageboard dedicated to male virgins past the age of 30 or “wizards” as they are called in the community. As The Atlantic writer Kaitlyn Tiffany aptly wrote, dead internet is “ ”, yet it “ ”. patently ridiculous feels true [2] The TLDR-version (meaning the abstract) of the theory is perhaps a testament to that: [3] “Large proportions of the supposedly human-produced content on the internet are actually generated by artificial intelligence networks in conjunction with paid secret media influencers in order to manufacture consumers for an increasing range of newly-normalised cultural products.” Or when IlluminatPirate describes a strange hunch he has that something is very wrong about the internet: [4] “The Internet feels empty and devoid of people. It is also devoid of content. Compared to the Internet of say 2007 (and beyond) the Internet of today is entirely sterile. There is nowhere to go and nothing to do, see, read or experience anymore. It all imploded into a handful of normalfag [5] sites and these empty husks we inhabit. Yes, the Internet may seem gigantic, but it’s like a hot air balloon with nothing inside.” The main premise of the theory I believe is somewhat accurate. A report shows that bots make up almost two-thirds of all internet traffic anno 2021. [6] Additionally, curate and prioritize the content it shows to users based on the principle of profit-maximization. In other words, social media algorithms and search engines push content that entices users to buy more products and services, and neglect content that is controversial. social media algorithms Additionally, the unfathomable amount of data processed on the internet every minute, kind of makes it all blend together. The human mind was never designed to take in, or even less sort out, in such quantities of information. Dead Internet Theory — Valid Points In the most proficient imageboard slang, the author goes on to describe why dead internet theory is real and why the internet isn’t. Some of his points are hard to follow, passages are incoherent, and at times he is clearly delusional. However, some of the arguments coincide with my own research and personal beliefs. IlluminatPirate points to the superficial and impersonal nature of online interactions: “I used to be in perpetual contact with a solid number of people across multiple sites. Across the years each and every one of them vanished without a trace.” He points out how the same content keeps reappearing over and over again: “I’ve seen the same threads, the same pics and the same replies reposted over and over across the years to the point of me seeing it as unremarkable.” I also share his belief that truly original content is suffering because large players in the entertainment industry rely on algorithms and big data analysis of consumer habits to “feed the customers”. That is often the feeling I have when I watch a new movie from Netflix Originals: “Algorithm fiction. Do you like capeshit, Anon? How about other Hollywood stuff? Music perhaps? Have you noticed how sterile fiction has become? How it caters to the lowest common denominator and follows the same template over and over again? How music is just autotunes and basic blandness? The writer’s strike never ended. Algorithms and computer programs are manufacturing modern fiction. No human being is behind these things. This is why anime looms so large — even a simple moe anime has heart because there’s actual people behind it, and we all intuitively feel this.” IlluminatiPirate also points to the dangers of deepfakes. As implied at the beginning of this post, the technology could potentially take off within the next couple of years: “Fake people. No, not NPC’s. Youtube people who talk about this or that, and quite possibly many politicians, actors and so forth may not actually exist.” Social media algorithms and search engines alike are known to customize news feeds and search results so it fits with the individual user’s likes and preferences. This customization feature results in so-called “ ” where users are constantly reaffirmed in their own beliefs and opinions. epistemic bubbles Everything the users see confirms what they already know, and people they interact with tend to wholeheartedly agree with them. People are thus exposed to a very one-sided view of the world. It creates a potentially dangerous “us vs. them”, “they are wrong, we are right”-dynamic in society. “The internet is a fast way to get info, and info is what moves the mind, and the thing is, the mind likes recognition. When the “likes” were introduced without negative feedback they created a copy-feedback subconscious, they made it so only “positive” opinions be propagated (also accepted), and in it’s way negative opinions to be obsolete. Now everyone is too cowardly to have an opinion so they copy others they like, they are more likely to follow trends and say what others said, you can also see it with the paranoia of always wanting to listen to experts.” Finally, I think one of the most interesting points to be drawn from the compilation post on dead internet theory, is the distinction between “the old web” and “the new web”. If cut to the bone, IlluminatiPirate’s basic point is that he misses the old days of the internet, and dreads what the internet has become. The same notion is shared even by the founder of the world wide web, Tim Bernes-Lee, who says that the system is failing. [7] “Creation of original content is how the internet used to work. Anonymous people were willing to express their opinions and try radical or experimental things. More truly original content, uninfluenced by bots or paid influencers, was created due to anonymity as protection against negative feedback. On the old internet, you could start anew every time you posted something.” There is indeed a conventionally accepted distinction between an “old” internet (web 1.0) and a new internet (web 2.0). The first version of the internet was read-only, while the second version allow us to upload our everyday lives to various platforms. The problem with web 2.0 is that four companies control 67% of the world’s cloud infrastructure. The large tech monopolies claim ownership of our data, and indeed of our digital identities. [8] [9] With all this in mind — how can we prove that the internet is not fake? Or to turn the question around: how can we make it real? Building a Trustless Internet Satoshi Nakamoto, the anonymous creator of Bitcoin, had an extraordinarily deep understanding of human nature. He understood that transactions between humans are based on trust. Within good reason, trusting others is usually considered to be a positive trait. However, human beings are fallible. Therefore, Satoshi Nakamoto founded Bitcoin, a monetary system based on cryptographic proof instead of trust. [10] The transactions and interactions we have with other people on the web should be based on Bitcoin’s principle of “trustlessness”. What we need is a reliable and transparent method to link people’s digital avatars to the people they are supposed to represent in real life. That would mitigate a wide array of issues such as fake spam accounts, the deliberate spread of false information, deceivable media manipulation, fraud, catfishing, cyberbullying, and trolling because people can be held directly accountable for their actions in the online space. Importantly, online identity systems have to be (a) open and (b) decentralized. The contemporary solution where you upload your personal ID to the service provider for verification does not fulfill either of those criteria. If the identity system is closed and centralized, the system owner would be able to track and monitor people’s behavior much like China’s social credit system. Therefore, online identity systems have to be built on the infrastructure of web 3.0. The key characteristic of web 3.0 is that users are in charge of their data and digital identity. Plenty of companies are already working towards this shift by building platform-agnostic identity solutions for the web that can be scaled. Notably, recognizes the importance of being able to prove your online identity in a secure way that helps you to keep your credentials and information private. They currently offer a decentralized, identity system called for organizations. Microsoft [11] Azure AD Verifiable Credentials Examples of other companies that work on decentralized, blockchain-based identity systems are , , , , and . ONTology XSL Labs Cheqd Sovrin Decentralized Identity Foundation When digital identity solutions such as these become widely used, authorization with username and password or with social identity via accounts on Google, Facebook, or Twitter will be ancient history. Instead, you will own a unique digital passport that can be authorized by a peer-to-peer network in the place of an intermediary every time you register for a new online service. You can also decide what pieces of data to make available, to whom, and under what conditions. For example, if an application is age-restricted, you can prove with your digital passport that you are above a certain age without providing any further information such as your birthday. Or if an agency is interested in using your consumer preferences for commercial or statistical purposes you can set a price for giving them access to your data. [12] [13] That is only fair. Everything we do online leaves a digital footprint. These digital footprints are picked up by monopolistic technology companies and used as lubrication for algorithms and AI. Although they have no rightful claim to extract our subtle pieces of information for free. If nothing changes, the dystopian vision of dead internet theory could eventually become true. The internet may become a twilight zone controlled by powerful, proprietary-owned AI that will continue to make great profits from our personalities, beliefs, and human qualities. Luckily, the explosion in blockchain innovation and the hype surrounding web 3.0 are raising awareness of the digital identity problem and the current lack of data ownership for internet users. Kaitlyn Tiffany (Aug 31, 2021), Maybe You Missed It, but the Internet “Died” Five Years Ago -> (13–02–2022). [1] https://www.theatlantic.com/technology/archive/2021/08/dead-internet-theory-wrong-but-feels-true/619937/ Kaitlyn Tiffany (2021), “Maybe You Missed It, but the Internet ‘Died’ Five Years Ago” -> . [2] https://www.theatlantic.com/technology/archive/2021/08/dead-internet-theory-wrong-but-feels-true/619937/ (18–02–2022). [3] https://forum.agoraroad.com/index.php?threads/dead-internet-theory-most-of-the-internet-is-fake.3011/ Ibid. [4] Normalfag: slang for average internet use [5] Barracuda (Sep 2021), Bot Attacks: Top Threats and Trends -> [6] https://assets.barracuda.com/assets/docs/dms/Bot_Attacks_report_vol1_EN.pdf (12–02–2021). [7] https://www.theguardian.com/technology/2017/nov/15/tim-berners-lee-world-wide-web-net-neutrality Emily Nicolle (12–02–2022), Andreessen’s Dixon Spies Riches in Web3. Others See ‘Rubbish’ -> . [8] https://www.bloomberg.com/news/articles/2022-02-12/andreessen-horowitz-a16z-s-chris-dixon-sees-web3-riches-others-see-rubbish?srnd=cryptocurrencies (12–02–2022). [9] https://uk.pcmag.com/old-cloud-infrastructure/131713/four-companies-control-67-of-the-worlds-cloud-infrastructure See (18–02–2022)- [10] https://i.redd.it/njml2ng9j7h81.png Quote from Azure Friday (16. August 2019), An introduction to [11] https://docs.microsoft.com/da-dk/shows/Azure-Friday/An-introduction-to-decentralized-identities George Zarkadakis, The Internet Is Dead: Long Live the Internet, pg. 49. From Werthner et. al Perspective on Digital Humanism (2022), Springer. [12] Ibid. [13] Co-published here.

Explore the Problem of Digital Identities and the Impending Doom of the Dead Internet 

Starters For definitions, read the by . Primer Chjango Unchained For an overview of the Cosmos/Tendermint software stack, read by . The Value Proposition of Cosmos Gautier MARIN Watch the by . Cosmos Network Topological Overview Sunny Aggarwal Intro Itcan feel like information overload for the newly initiated to get a crash course about Tendermint, the Cosmos Network, and where the two intersect. at a high level, but in this post, we’ll dive a bit deeper into Tendermint Core’s consensus and networking layers. The design of said software stack is composed of an application layer (Cosmos-SDK), a consensus layer, and a networking layer (Tendermint Core). Tendermint marries consensus with a p2p gossip protocol in a completely novel way. Using a monolithic architecture is typically bad practice in computer science, as doing so makes it difficult to reuse components of the code, and attempts to do so . We’ve written about the Cosmos/Tendermint stack result in complex maintenance procedures for forks of the codebase Top Down: Application Layer (Cosmos-SDK), Application Blockchain Interface (ABCI), Tendermint Core stack (consensus algorithm + p2p networking protocol). First things first: semantics. Tendermint Core, the piece of software, is used synonymously with Tendermint, the organization that’s implementing Tendermint Core and the Cosmos Network software stack. Technically, Tendermint Core is a low-level protocol which is actually composed of two protocols in one: a consensus algorithm and a peer-to-peer networking protocol. Jae Kwon and Ethan Buchman, inspired by the design goal behind Raft and PBFT, specified Tendermint as an easy to understand, developer-friendly algorithm while doing algorithmically complex systems engineering. The new generation of BFT Proof-of-Stake (PoS) consensus algorithms draw elements from Tendermint’s BFT adaptation to the public blockchain domain. This adaption is what we refer to as Tendermint BFT, and is more generally classified as BFT-based Proof-of-Stake (as opposed to chain-based Proof-of-Stake). For a crash course of chain-based versus BFT-based PoS, see: . Casper vs. Tendermint Now that that’s out of the way, let’s bite into the meat of the introduction: What is Tendermint and how does it work at a high level? Tendermint BFT Stack Bitcoin is the intellectual ancestor of all blockchain-based cryptographic systems that we know and love today. The Tendermint protocol shares commonality with Bitcoin inasmuch as the two protocols record everything on a blockchain, yet they each provide their unique solutions to the Byzantine General’s Problem, also referred to as the consensus, or “agreement”, problem. Tendermint’s lineage is closely traced back to the world of distributed computing and Byzantine Fault Tolerance (BFT) literature in academia (e.g., see Ethan Buchman’s ). Whereas in the Bitcoin genesis story, after the many failed attempts of previous electronic cash systems — with the exception of PayPal — Bitcoin rose above the ashes as a censorship-resistant decentralized currency system. thesis The Bitcoin protocol optimizes for censorship-resistance, critically, due to its primary function as a payment system. Tendermint, on the other hand, optimizes for general Byzantine fault-tolerant distributed applications and data processing amidst a wide area network (WAN), e.g. hundreds of nodes (high node count). This distinction is nuanced and merits deeper investigation. For context, in the world of academia, there had been very little research done on BFT systems for WAN and only for a small number of nodes — 4 to 7 at maximum — and always for a single administrative domain. As for BFT systems for WAN with a large number of nodes and for multiple administrative domains, no significant body of work has been widely adopted in practice. Before 2009, when Bitcoin introduced a paradigm-shifting technology — the concept of a blockchain — to the world, the consensus problem in a WAN setting for high node counts was left largely unanswered. Despite solving the Two Generals’ problem, however, Bitcoin was not really an algorithm to solve consensus, in the purely theoretical sense in the research area of distributed systems. Furthering the work in the BFT domain was far from done. In 2014, Jae Kwon, coming from a computer science and systems engineering background, envisioned a purely BFT-based protocol which would scale to the tune of hundreds of nodes in the permissionless setting with Proof-of-Stake (PoS) as the underlying security mechanism. And so was created. Consequently, this system model designed with PoS as the primary security mechanism over a large number of validating nodes in a WAN turned out to be a significantly complex engineering endeavor, taking almost 4 years to implement in the public blockchain setting. That setting is Cosmos, slated to launch this summer in 2018. Tendermint The Model Tendermint is modeled as a deterministic protocol, live under partial synchrony, which achieves throughput within the bounds of the latency of the network and individual processes themselves. See where the purple is outlined in ’s triangle? Tendermint falls somewhere along the bottom-left corner. Vlad Zamfir ’s Tradeoff Triangle for fault tolerant consensus protocols Vlad Zamfir FLP Impossibility : “… ” As in, consensus cannot be solved in purely asynchronous networks with a deterministic protocol if at least a single process can fail. Theorem we show the surprising result that no completely asynchronous consensus protocol can tolerate even a single unannounced process death. Defined by Dwork, Lynch, and Stockmeyer in their paper, : “ ….” It was in attempting to solve this problem that Tendermint was born. of the aforementioned . Consensus in the Presence of Partial Synchrony Partial synchrony lies between the cases of a synchronous system and an asynchronous system. In a synchronous system, there is a known fixed upper bound…on the time required for a message to be sent from one processor to another and a known fixed upper bound…on the relative speeds of different processors. In an asynchronous system no fixed upper bounds…exist….The problem is to design protocols that work correctly in the partially synchronous system regardless of the actual values of the bounds Tendermint is thus a modified version DLS protocol Consensus Algorithm Partially Synchronous, Synchronous, and Asynchronous Communication Let’s explore the synchronous case, using a well-known protocol for reference: the Bitcoin protocol. In Bitcoin, there is a “known fixed upper bound”. This is referring to the 10 minute block time. In order for the Bitcoin network to move forward with creating blocks, the protocol artificially imposes a timing assumption, giving the entire network of nodes 10 whole minutes to listen for, collect, validate, and gossip transactions propagating across its peer network. Ethereum is another example of a protocol that makes synchrony assumptions with block times averaging 15 seconds. While 15 seconds is much faster than 10 minutes — giving the Ethereum network the benefit of higher throughput — this comes at the expense of its miners, as this results in more orphaned blocks; there is less time for transactions to propagate through its network. Tendermint belongs to a class of protocols which solve consensus under partially synchronous communication, wherein, a partially synchronous system model alternates between periods of synchrony and asynchrony; we sometimes refer to this model as “weakly synchronous”. What this means is, Tendermint does rely on timing assumptions in order to make progress. However, in contrast to synchronous systems, the speed of progress does not depend on system parameters, but instead depends on real network speed. Liveness & Termination Defined: “ ” The termination property is just that each correct processor should eventually make a decision. Those algorithms that of Nakamoto consensus, Peercoin, NXT, Snow White, Ouroboros, etc., in the synchronous system model, rely on synchrony assumptions not just for process termination but for safety. Those algorithms designed for synchronous systems always have fixed bounds which are known and always hold. And in the event that the synchrony bounds do not hold, consensus is broken, and the chain will fork. Therefore, Bitcoin’s 10 minute block time, for example, is duly conservative, as to preserve safety. By way of contrast, Tendermint never forks in the presence of asynchrony if less than 1/3 of processes are faulty. This property is what makes Tendermint a BFT-based PoS protocol, in which it strictly prefers safety over liveness (see: ). As a result, a Tendermint blockchain will halt momentarily until a supermajority, i.e. more than 2/3, of the validator set comes to consensus. CAP Theorem Now, there consensus protocols that work in purely asynchronous networks, but, adhering to the FLP Impossibility Theorem, they cannot simultaneously be deterministic. are Deterministic vs. Nondeterministic Protocol Nondeterministic protocols which solve consensus under the purely asynchronous case potentially rely on random oracles and generally incur high message complexity overhead, as they depend on reliable broadcasting for all communication. In the asynchronous environment, the overhead cost of a single reliable broadcast is about equivalent to . Protocols like HoneyBadger BFT fall into this class of nondeterministic protocols under asynchrony. Normally, they require three instances of reliable broadcast for a single round of communication. one [round](http://tendermint.readthedocs.io/projects/tools/en/master/introduction.html#consensus-overview) of Tendermint Tendermint instead is a totally deterministic protocol; there is no randomness in the protocol whatsoever. Leaders are elected deterministically, through a defined mathematical function in the implementation. As such, we are able to mathematically prove that the system is live and that the protocol is guaranteed to make decisions. Rotating Leader Election Tendermint rotates through the validator set, i.e. block proposers, in a weighted round-robin fashion. The more stake, i.e. voting power, that a validator has delegated to them, the more weight that they have, and the proportionally more times they will be elected as leaders. To illustrate, if one validator has the same amount of voting power as another validator, they will both be elected by the protocol an equal amount of times. The simplified explanation of how the algorithm works looks like this: Validator weight is established Validator is elected, their turn to propose a block Weight is recalculated, decreases some amount after round is complete As each round progresses, weight increases incrementally in proportion to voting power Validator is selected again Because , given that you know the validator set and each validator’s voting power, you could compute exactly who the next block proposers will be in rounds , ,..., . Because of this, critics argue that Tendermint isn't decentralized enough. When you can know predictably who the leaders will be, an attacker could target those leaders and launch a DDoS attack against them and potentially halt the chain from progressing. We mitigate this attack vector by implementing something called in Tendermint. the protocol selects block proposers deterministically x x + 1 x + n Sentry Architecture P2P Networking Protocol Sentry Node Architecture A properly set up validator will never expose the IP address of or accept incoming connections to their validator node. A correctly set up, well-defended validator will actively spawn , which act as full node proxies to the network, to obfuscate the real location of their validator node. IP addresses on the p2p layer are never exposed. sentry nodes That said, taking advantage of Sentry Node Architecture is opt-in; the onus is on the validator to maintain a fault-tolerant full node. This is where we make extra-protocol assumptions based on cryptoeconomic incentives. The assumption is, a validator will want to take all precautionary measures so as to maintain fault-tolerance, remain available, and ultimately play their part in keeping the network live. Because if they don’t, they get kicked out of the validator set for being offline for over a certain amount of time. Peer Exchange (PEX) Reactor Tendermint borrows from Bitcoin’s peer discovery protocol. More specifically, Tendermint adopts the p2p AddressBook from , the Bitcoin alternative implementation in Go. The by default. btcd PEX enables dynamic peer discovery Tendermint in Practice After all is said and done, beyond the fancy algorithmic design and academic jargon, what is Tendermint actually useful for? Bad news is, everyday people won’t find Tendermint useful. Good news is, application developers can bridge the gap between protocol and end-user. Tendermint is designed to be customizable and flexible enough to fit any setting, be it public or enterprise, where a consensus protocol is desired. Tendermint is ideal for a developer who wants to implement an application on top of its own blockchain. It comes pre-assembled, so if the developer chooses to go for a pure Proof-of-Stake, BFT-based consensus engine to power their , they can easily do so. dAppzone The fun part comes when users get to implement their high-level business logic with the Cosmos-SDK. To interface with the consensus/networking layers of Tendermint is then done through a Tendermint socket protocol we call the Application Blockchain Interface, or the . ABCI Application Layer: Cosmos-SDK (Blockchain Application Framework) Cosmos-SDK Alpha Release Tendermint ABCI Ongoing Research Currently, we are delving into BLS signature research which could potentially lead to a reduction in the size of Tendermint block headers from 3.2 KB (with ~100 validators) to 64 bytes. We also have a design for randomizing the round-robin proposer selection function in a secure way to make DDoS’ing the next proposer even more difficult, but the initial step is the Sentry Node Architecture. Cosmos Academy Inaugural Meetup! We’re hosting an in Berkeley, CA, with Zarko Milosevic, Senior BFT Distributed Systems Researcher, who will give a Q&A session about everything Tendermint. We will livestream this meetup and publish the broadcast on the . upcoming meetup Cosmos Network’s YouTube channel Register for the first Cosmos Academy Meetup on May 26th: here . Who Am I Chjango Unchained heads Communications and the Cosmos Academy at Tendermint. She is the chief editor of the Cosmos blog and brings the creative direction behind all the video and written content from Cosmos. Primarily, she writes articles breaking down technical subjects about protocols, distributed systems, and blockchain tech to broad audiences of all levels. Before Cosmos, she came from NASA’s Jet Propulsion Laboratory, having never left space. Additional Resources Read the Interview with Tendermint Core Senior Research Scientist, Zarko Milosevic: . here Kyle Kingsbury of showcases Tendermint BFT: Jepsen For the technical specification of Tendermint, see by . Readthedocs Zach Ramsay Read the original by . Tendermint whitepaper Jae Kwon Read the by . History of Distributed State Ethan Buchman Listen to the with on the Software Engineering Daily podcast. History of Distributed State Ethan Buchman Listen to on the Bitcoin Podcast Network with . The Blockchain Ecosystem in the Early Days Zaki Manian

Tendermint Explained — Bringing BFT-based PoS to the Public Blockchain Domain

Uruguay’s President Luis Lacalle Pou signed a bill in October to regulate cryptocurrency in the country. The new law, which was initially proposed over two years ago, recognizes cryptocurrency as virtual assets and gives the Uruguayan Central Bank the power to regulate the currency. The Central Bank will be responsible for overseeing virtual assets service providers (VASPs) and granting them permits, while it will be the job of the Superintendence of Financial Service (SSF) to identify exchanges, wallets, and miners that can be considered VASPs. While several Latin American countries have enacted crypto regulations — including some with volatile economic and political landscapes like El Salvador, Venezuela and Argentina — Uruguay’s stable democracy could set a precedent for crypto regulation in the rest of Latin America. Crypto regulation around LatAm Along with Uruguay, ten other Latin American and Caribbean countries have introduced frameworks regulating the use of cryptocurrency, including Brazil, Venezuela, and Argentina: the latter of which has been ranked the top country in Latin America for crypto value received. Between July 2023 and June 2024, Argentina received an estimated $91.1 billion worth of crypto, while Brazil received approximately $90.3 billion. In Mexico, a fintech regulation bill has been in place since March 2018. Cryptocurrencies are recognized as “virtual assets,” and there is no law in place prohibiting the exchange of cryptocurrency for services or operations. Cryptocurrencies can thus be used as payment in establishments that accept them as a form of payment, and there are machines that can buy bitcoin from or sell it to customers. In El Salvador, Bitcoin has been recognized as legal tender, with Bitcoin ATMs emerging across the country and the government announcing plans for a “Bitcoin City.” However, journalist Katarina Hall recently reported the significant difficulties she faced trying to spend it, with all establishments she visited refusing to accept Bitcoin as payment. Not all Latin American countries have been so open to the development of cryptocurrency. In Honduras, the government has expressed concern about the significant risks associated with unregulated currency and has prohibited banks from handling cryptocurrency in any capacity. Meanwhile, the Guatemalan government strongly advises against the use of cryptocurrencies; although they are technically permitted, they are not recognized in a legislative framework and they do not function as a legal means of payment. Uruguay can be considered the most politically and economically stable out of the Latin American countries that have established a legal framework for crypto thus far. The Economist’s 2023 Democracy Index found Uruguay to have a democracy score of 8.66 out of 10, ranking 14th in the world, and making it the most democratic country in the Americas. In 2023, Uruguay had a Gross Domestic Product (GDP) per capita of approximately $22,565, while Venezuela’s was estimated to be $3,740. Is Uruguay setting a precedent for crypto regulation in the region? The strong position Uruguay finds itself in could well set a precedent for other Latin American countries in similarly stable circumstances. By regulating cryptocurrency, Uruguay is not only legitimizing and lending credibility to virtual assets, but also increasing confidence among potential investors and users, including those in surrounding countries. Emiliano Zapata, Uruguayan Blockchain Developer and Professor at the ORT University in Montevideo, spoke to Latin America Reports about the future of crypto in Uruguay. He explained that, as things stand, crypto is not yet “widely adopted” in Uruguay despite assumptions that Latin American countries tend to use crypto “because it solves a lot of issues.” He noted that in neighboring Argentina there are significantly more “day-to-day people” who use crypto regularly. While Zapata does not believe that the regulation of crypto will necessarily lead to an increase in the quotidian use of crypto in everyday scenarios, he believes that the new law could “be extremely useful for business development” in the country. Zapata elaborated on the conditions which make Uruguay a favorable location for the expansion of crypto, explaining that Uruguay tends to be perceived as a constantly stable country without any “heavily polarized landscape,” even in spite of the recent election period. He explained, “This kind of stability fosters a good environment to develop any kind of commercial activity here in Uruguay,” something which he believes “separates us a little bit from the rest of the majority of Latin America.” Although Zapata noted that Uruguay’s tradition of being “conservative” may risk limiting the full potential of crypto, he said that the country has “all the conditions” necessary to become one of the Latin American leaders in the industry. While Zapata acknowledged that the power assigned to the Central Bank and the recognition of crypto in a legal framework will “add layers of bureaucracy and regulation”, he said that this is necessary to provide the “security and the safety” needed to “develop either local or foreign projects” in Uruguay. Martín Benítez Aramendía, Vice President of the Uruguayan Chamber of Fintech and Head of Business Development at financial payments company Ripio, echoed Zarpata’s observations. He told Latin America Reports that Uruguay functions as “a hub for business,” given that Uruguay “is the most economically and legally stable country in the region,” which creates “a natural and organic interest in cryptocurrency in Uruguay.” Benítez explains that this interest has manifested itself in investments and payments between Uruguay and Asian countries, including China. Benítez added that, due to the recent economic crisis, many Argentinians have based themselves in Uruguay, where they partake in cryptocurrency transactions. He explained, “There’s a natural use of cryptocurrencies in Uruguay, but it is also influenced by Argentina and Brazil, which have their operations here.” In most Latin American countries, he said, cryptocurrency “is primarily used for investments, not as an investment asset itself,” but noted that Argentina is an exception from this rule, where “it is used daily, especially for commerce and access to dollars.” For countries like Argentina, “Cryptocurrencies have provided a solution that the market wasn’t offering.” He described the new law as a “very positive” move, pointing out that there had long been concerns that it might not get passed at all. Now, the Uruguayan Fintech Chamber’s hope is that the Central Bank will be prompt in clarifying the regulations it will impose on crypto companies, so that Uruguay can “consolidate the advantage” it currently has over other countries which are still developing their own laws. Elizabeth Bratton, The Sociable Uruguay’s President Luis Lacalle Pou signed a bill in October to regulate cryptocurrency in the country. The new law , which was initially proposed over two years ago, recognizes cryptocurrency as virtual assets and gives the Uruguayan Central Bank the power to regulate the currency. new law The Central Bank will be responsible for overseeing virtual assets service providers (VASPs) and granting them permits, while it will be the job of the Superintendence of Financial Service (SSF) to identify exchanges, wallets, and miners that can be considered VASPs. responsible While several Latin American countries have enacted crypto regulations — including some with volatile economic and political landscapes like El Salvador, Venezuela and Argentina — Uruguay’s stable democracy could set a precedent for crypto regulation in the rest of Latin America. Crypto regulation around LatAm Crypto regulation around LatAm Along with Uruguay, ten other Latin American and Caribbean countries have introduced frameworks regulating the use of cryptocurrency, including Brazil, Venezuela, and Argentina: the latter of which has been ranked the top country in Latin America for crypto value received. Between July 2023 and June 2024, Argentina received an estimated $91.1 billion worth of crypto, while Brazil received approximately $90.3 billion. ranked In Mexico , a fintech regulation bill has been in place since March 2018. Cryptocurrencies are recognized as “virtual assets,” and there is no law in place prohibiting the exchange of cryptocurrency for services or operations. Cryptocurrencies can thus be used as payment in establishments that accept them as a form of payment, and there are machines that can buy bitcoin from or sell it to customers. Mexico bitcoin In El Salvador, Bitcoin has been recognized as legal tender, with Bitcoin ATMs emerging across the country and the government announcing plans for a “Bitcoin City.” However, journalist Katarina Hall recently reported the significant difficulties she faced trying to spend it, with all establishments she visited refusing to accept Bitcoin as payment. reported Not all Latin American countries have been so open to the development of cryptocurrency. In Honduras , the government has expressed concern about the significant risks associated with unregulated currency and has prohibited banks from handling cryptocurrency in any capacity. Meanwhile, the Guatemalan government strongly advises against the use of cryptocurrencies; although they are technically permitted, they are not recognized in a legislative framework and they do not function as a legal means of payment. Honduras Guatemalan Uruguay can be considered the most politically and economically stable out of the Latin American countries that have established a legal framework for crypto thus far. The Economist’s 2023 Democracy Index found Uruguay to have a democracy score of 8.66 out of 10, ranking 14th in the world, and making it the most democratic country in the Americas. In 2023, Uruguay had a Gross Domestic Product (GDP) per capita of approximately $22,565, while Venezuela’s was estimated to be $3,740. The Economist’s 2023 Democracy Index Uruguay Is Uruguay setting a precedent for crypto regulation in the region? Is Uruguay setting a precedent for crypto regulation in the region? The strong position Uruguay finds itself in could well set a precedent for other Latin American countries in similarly stable circumstances. By regulating cryptocurrency, Uruguay is not only legitimizing and lending credibility to virtual assets, but also increasing confidence among potential investors and users, including those in surrounding countries. Emiliano Zapata, Uruguayan Blockchain Developer and Professor at the ORT University in Montevideo, spoke to Latin America Reports about the future of crypto in Uruguay. He explained that, as things stand, crypto is not yet “widely adopted” in Uruguay despite assumptions that Latin American countries tend to use crypto “because it solves a lot of issues.” Latin America Reports He noted that in neighboring Argentina there are significantly more “day-to-day people” who use crypto regularly. While Zapata does not believe that the regulation of crypto will necessarily lead to an increase in the quotidian use of crypto in everyday scenarios, he believes that the new law could “be extremely useful for business development” in the country. Zapata elaborated on the conditions which make Uruguay a favorable location for the expansion of crypto, explaining that Uruguay tends to be perceived as a constantly stable country without any “heavily polarized landscape,” even in spite of the recent election period. He explained, “This kind of stability fosters a good environment to develop any kind of commercial activity here in Uruguay,” something which he believes “separates us a little bit from the rest of the majority of Latin America.” Although Zapata noted that Uruguay’s tradition of being “conservative” may risk limiting the full potential of crypto, he said that the country has “all the conditions” necessary to become one of the Latin American leaders in the industry. While Zapata acknowledged that the power assigned to the Central Bank and the recognition of crypto in a legal framework will “add layers of bureaucracy and regulation”, he said that this is necessary to provide the “security and the safety” needed to “develop either local or foreign projects” in Uruguay. Martín Benítez Aramendía, Vice President of the Uruguayan Chamber of Fintech and Head of Business Development at financial payments company Ripio, echoed Zarpata’s observations. He told Latin America Reports that Uruguay functions as “a hub for business,” given that Uruguay “is the most economically and legally stable country in the region,” which creates “a natural and organic interest in cryptocurrency in Uruguay.” Benítez explains that this interest has manifested itself in investments and payments between Uruguay and Asian countries, including China. Latin America Reports Benítez added that, due to the recent economic crisis, many Argentinians have based themselves in Uruguay, where they partake in cryptocurrency transactions. He explained, “There’s a natural use of cryptocurrencies in Uruguay, but it is also influenced by Argentina and Brazil, which have their operations here.” In most Latin American countries, he said, cryptocurrency “is primarily used for investments, not as an investment asset itself,” but noted that Argentina is an exception from this rule, where “it is used daily, especially for commerce and access to dollars.” For countries like Argentina, “Cryptocurrencies have provided a solution that the market wasn’t offering.” He described the new law as a “very positive” move, pointing out that there had long been concerns that it might not get passed at all. Now, the Uruguayan Fintech Chamber’s hope is that the Central Bank will be prompt in clarifying the regulations it will impose on crypto companies, so that Uruguay can “consolidate the advantage” it currently has over other countries which are still developing their own laws. Elizabeth Bratton, The Sociable Elizabeth Bratton , The Sociable Elizabeth Bratton Elizabeth Bratton

Can Uruguay's New Crypto Regulation Set a Precedent for Latin America?

Uni-OVSeg: A Step Towards Efficient and Bias-Resilient Vision Systems

Authors:
(1) Halil Yigit Oksuz, Control Systems Group at Technische Universitat Berlin, Germany and Exzellenzcluster Science of Intelligence, Technische Universitat Berlin, Marchstr. 23, 10587, Berlin, Germany;
(2) Fabio Molinari, Control Systems Group at Technische Universitat Berlin, Germany;
(3) Henning Sprekeler, Exzellenzcluster Science of Intelligence, Technische Universit¨at Berlin, Marchstr. 23, 10587, Berlin, Germany and Modelling Cognitive Processes Group at Technische Universit¨at Berlin, Germany;
(4) Jorg Raisch, Control Systems Group at Technische Universitat Berlin, Germany and Exzellenzcluster Science of Intelligence, Technische Universitat Berlin, Marchstr. 23, 10587, Berlin, Germany. Table of Links Abstract and Introduction Problem Setup Federated fair over-the-air learning (FedAir) Algorithm Convergence Properties Numerical Example Conclusion and References VI. CONCLUSION In this paper, we have introduced the FedFAir algorithm which uses Over-the-Air Computation to carry out efficient decentralized learning while providing fairness and improved performance. We have shown that the FedFAir algorithm converges to an optimal solution of the minimax problem. Furthermore, we have also illustrated our theoretical findings with a numerical example. Future research will include the development of resilient federated learning algorithms when there are malicious agents in the system. REFERENCES [1] B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas, “Communication-efficient learning of deep networks from decentralized data,” in Artificial intelligence and statistics. PMLR, 2017, pp. 1273–1282. [2] V. Smith, C.-K. Chiang, M. Sanjabi, and A. S. Talwalkar, “Federated multi-task learning,” Advances in neural information processing systems, vol. 30, 2017. [3] K. Bonawitz, H. Eichner, W. Grieskamp, D. Huba, A. Ingerman, V. Ivanov, C. Kiddon, J. Konecny, S. Mazzocchi, B. McMahan et al., “Towards federated learning at scale: System design,” Proceedings of Machine Learning and Systems, vol. 1, pp. 374–388, 2019. [4] Q. Yang, Y. Liu, T. Chen, and Y. Tong, “Federated machine learning: Concept and applications,” ACM Transactions on Intelligent Systems and Technology (TIST), vol. 10, no. 2, pp. 1–19, 2019. [5] A. Nedic, “Distributed gradient methods for convex machine learning problems in networks: Distributed optimization,” IEEE Signal Processing Magazine, vol. 37, no. 3, pp. 92–101, 2020. [6] V. P. Chellapandi, A. Upadhyay, A. 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Bjelakovi´c, and S. Sta´nczak, “Over-the-air computation in correlated channels,” IEEE Transactions on Signal Processing, vol. 69, pp. 5739–5755, 2021. [17] K. Yang, T. Jiang, Y. Shi, and Z. Ding, “Federated learning via overthe-air computation,” IEEE Transactions on Wireless Communications, vol. 19, no. 3, pp. 2022–2035, 2020. [18] F. Molinari and J. Raisch, “Exploiting wireless interference for distributively solving linear equations,” IFAC-PapersOnLine, vol. 53, no. 2, pp. 2999–3006, 2020. [19] D. Bertsekas, A. Nedic, and A. Ozdaglar, Convex analysis and optimization. Athena Scientific, 2003, vol. 1. [20] M. Mohri, G. Sivek, and A. T. Suresh, “Agnostic federated learning,” in International Conference on Machine Learning. PMLR, 2019, pp. 4615–4625. [21] Z. Hu, K. Shaloudegi, G. Zhang, and Y. Yu, “Federated learning meets multi-objective optimization,” IEEE Transactions on Network Science and Engineering, vol. 9, no. 4, pp. 2039–2051, 2022. [22] D. P. Bertsekas, “Necessary and sufficient conditions for a penalty method to be exact,” Mathematical programming, vol. 9, no. 1, pp. 87–99, 1975. [23] K. Srivastava, A. Nedi´c, and D. Stipanovi´c, “Distributed min-max optimization in networks,” in 2011 17th International Conference on Digital Signal Processing (DSP). IEEE, 2011, pp. 1–8. [24] R. Ahlswede, “Multi-way communication channels,” in Second International Symposium on Information Theory: Tsahkadsor, Armenia, USSR, Sept. 2-8, 1971, 1973. [25] A. Giridhar and P. Kumar, “Toward a theory of in-network computation in wireless sensor networks,” IEEE Communications Magazine, vol. 44, no. 4, pp. 98–107, apr 2006. [26] F. Molinari, N. Agrawal, S. Sta´nczak, and J. Raisch, “Over-the-air max-consensus in clustered networks adopting half-duplex communication technology,” IEEE Transactions on Control of Network Systems, 2022. [27] W. Rudin et al., Principles of mathematical analysis. McGraw-hill New York, 1976, vol. 3. [28] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge University Press, 2012. [29] A. F. Molisch, Wireless communications. John Wiley & Sons, 2012. [30] B. Sklar, “Rayleigh fading channels in mobile digital communication systems. i. characterization,” IEEE Communications magazine, vol. 35, no. 7, pp. 90–100, 1997. [31] R. G. Bartle and D. R. Sherbert, Introduction to real analysis. Wiley New York, 2000. [32] B. T. Polyak, Introduction to optimization. optimization software. Inc., Publications Division, New York, 1987. [33] Y. I. Alber, A. N. Iusem, and M. V. Solodov, “On the projected subgradient method for nonsmooth convex optimization in a hilbert space,” Mathematical Programming, vol. 81, pp. 23–35, 1998. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Halil Yigit Oksuz, Control Systems Group at Technische Universitat Berlin, Germany and Exzellenzcluster Science of Intelligence, Technische Universitat Berlin, Marchstr. 23, 10587, Berlin, Germany; (2) Fabio Molinari, Control Systems Group at Technische Universitat Berlin, Germany; (3) Henning Sprekeler, Exzellenzcluster Science of Intelligence, Technische Universit¨at Berlin, Marchstr. 23, 10587, Berlin, Germany and Modelling Cognitive Processes Group at Technische Universit¨at Berlin, Germany; (4) Jorg Raisch, Control Systems Group at Technische Universitat Berlin, Germany and Exzellenzcluster Science of Intelligence, Technische Universitat Berlin, Marchstr. 23, 10587, Berlin, Germany. Authors: Authors: (1) Halil Yigit Oksuz, Control Systems Group at Technische Universitat Berlin, Germany and Exzellenzcluster Science of Intelligence, Technische Universitat Berlin, Marchstr. 23, 10587, Berlin, Germany; (2) Fabio Molinari, Control Systems Group at Technische Universitat Berlin, Germany; (3) Henning Sprekeler, Exzellenzcluster Science of Intelligence, Technische Universit¨at Berlin, Marchstr. 23, 10587, Berlin, Germany and Modelling Cognitive Processes Group at Technische Universit¨at Berlin, Germany; (4) Jorg Raisch, Control Systems Group at Technische Universitat Berlin, Germany and Exzellenzcluster Science of Intelligence, Technische Universitat Berlin, Marchstr. 23, 10587, Berlin, Germany. Table of Links Abstract and Introduction Abstract and Introduction Problem Setup Problem Setup Federated fair over-the-air learning (FedAir) Algorithm Federated fair over-the-air learning (FedAir) Algorithm Convergence Properties Convergence Properties Numerical Example Numerical Example Conclusion and References Conclusion and References VI. CONCLUSION In this paper, we have introduced the FedFAir algorithm which uses Over-the-Air Computation to carry out efficient decentralized learning while providing fairness and improved performance. We have shown that the FedFAir algorithm converges to an optimal solution of the minimax problem. Furthermore, we have also illustrated our theoretical findings with a numerical example. Future research will include the development of resilient federated learning algorithms when there are malicious agents in the system. REFERENCES [1] B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas, “Communication-efficient learning of deep networks from decentralized data,” in Artificial intelligence and statistics. PMLR, 2017, pp. 1273–1282. [2] V. Smith, C.-K. Chiang, M. Sanjabi, and A. S. Talwalkar, “Federated multi-task learning,” Advances in neural information processing systems, vol. 30, 2017. [3] K. Bonawitz, H. Eichner, W. Grieskamp, D. Huba, A. Ingerman, V. Ivanov, C. Kiddon, J. Konecny, S. Mazzocchi, B. McMahan et al., “Towards federated learning at scale: System design,” Proceedings of Machine Learning and Systems, vol. 1, pp. 374–388, 2019. [4] Q. Yang, Y. Liu, T. Chen, and Y. Tong, “Federated machine learning: Concept and applications,” ACM Transactions on Intelligent Systems and Technology (TIST), vol. 10, no. 2, pp. 1–19, 2019. [5] A. Nedic, “Distributed gradient methods for convex machine learning problems in networks: Distributed optimization,” IEEE Signal Processing Magazine, vol. 37, no. 3, pp. 92–101, 2020. [6] V. P. Chellapandi, A. Upadhyay, A. Hashemi, and S. H. Zak, “On ˙ the convergence of decentralized federated learning under imperfect information sharing,” IEEE Control Systems Letters, 2023. [7] T. Omori and K. Kashima, “Combinatorial optimization approach to client scheduling for federated learning,” IEEE Control Systems Letters, 2023. [8] T. Sery, N. Shlezinger, K. Cohen, and Y. C. Eldar, “Over-the-air federated learning from heterogeneous data,” IEEE Transactions on Signal Processing, vol. 69, pp. 3796–3811, 2021. [9] T. Gafni, N. Shlezinger, K. Cohen, Y. C. Eldar, and H. V. Poor, “Federated learning: A signal processing perspective,” IEEE Signal Processing Magazine, vol. 39, no. 3, pp. 14–41, 2022. [10] M. Ye, X. Fang, B. Du, P. C. Yuen, and D. Tao, “Heterogeneous federated learning: State-of-the-art and research challenges,” ACM Computing Surveys, vol. 56, no. 3, pp. 1–44, 2023. [11] J. Koneˇcn `y, H. B. McMahan, F. X. Yu, P. Richt´arik, A. T. Suresh, and D. Bacon, “Federated learning: Strategies for improving communication efficiency,” arXiv preprint arXiv:1610.05492, 2016. [12] T. Li, A. K. Sahu, A. Talwalkar, and V. Smith, “Federated learning: Challenges, methods, and future directions,” IEEE Signal Processing Magazine, vol. 37, no. 3, pp. 50–60, 2020. [13] P. Kairouz, H. B. McMahan, B. Avent, A. Bellet, M. Bennis, A. N. Bhagoji, K. Bonawitz, Z. Charles, G. Cormode, R. Cummings et al., “Advances and open problems in federated learning,” Foundations and Trends® in Machine Learning, vol. 14, no. 1–2, pp. 1–210, 2021. [14] H. Y. Oksuz, F. Molinari, H. Sprekeler, and J. Raisch, “Federated learning in wireless networks via over-the-air computations,” in 2023 62nd IEEE Conference on Decision and Control (CDC), 2023, pp. 4379–4386. [15] F. Molinari, N. Agrawal, S. Sta´nczak, and J. Raisch, “Max-consensus over fading wireless channels,” IEEE Transactions on Control of Network Systems, vol. 8, no. 2, pp. 791–802, 2021. [16] M. Frey, I. Bjelakovi´c, and S. Sta´nczak, “Over-the-air computation in correlated channels,” IEEE Transactions on Signal Processing, vol. 69, pp. 5739–5755, 2021. [17] K. Yang, T. Jiang, Y. Shi, and Z. Ding, “Federated learning via overthe-air computation,” IEEE Transactions on Wireless Communications, vol. 19, no. 3, pp. 2022–2035, 2020. [18] F. Molinari and J. Raisch, “Exploiting wireless interference for distributively solving linear equations,” IFAC-PapersOnLine, vol. 53, no. 2, pp. 2999–3006, 2020. [19] D. Bertsekas, A. Nedic, and A. Ozdaglar, Convex analysis and optimization. Athena Scientific, 2003, vol. 1. [20] M. Mohri, G. Sivek, and A. T. Suresh, “Agnostic federated learning,” in International Conference on Machine Learning. PMLR, 2019, pp. 4615–4625. [21] Z. Hu, K. Shaloudegi, G. Zhang, and Y. Yu, “Federated learning meets multi-objective optimization,” IEEE Transactions on Network Science and Engineering, vol. 9, no. 4, pp. 2039–2051, 2022. [22] D. P. Bertsekas, “Necessary and sufficient conditions for a penalty method to be exact,” Mathematical programming, vol. 9, no. 1, pp. 87–99, 1975. [23] K. Srivastava, A. Nedi´c, and D. Stipanovi´c, “Distributed min-max optimization in networks,” in 2011 17th International Conference on Digital Signal Processing (DSP). IEEE, 2011, pp. 1–8. [24] R. Ahlswede, “Multi-way communication channels,” in Second International Symposium on Information Theory: Tsahkadsor, Armenia, USSR, Sept. 2-8, 1971, 1973. [25] A. Giridhar and P. Kumar, “Toward a theory of in-network computation in wireless sensor networks,” IEEE Communications Magazine, vol. 44, no. 4, pp. 98–107, apr 2006. [26] F. Molinari, N. Agrawal, S. Sta´nczak, and J. Raisch, “Over-the-air max-consensus in clustered networks adopting half-duplex communication technology,” IEEE Transactions on Control of Network Systems, 2022. [27] W. Rudin et al., Principles of mathematical analysis. McGraw-hill New York, 1976, vol. 3. [28] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge University Press, 2012. [29] A. F. Molisch, Wireless communications. John Wiley & Sons, 2012. [30] B. Sklar, “Rayleigh fading channels in mobile digital communication systems. i. characterization,” IEEE Communications magazine, vol. 35, no. 7, pp. 90–100, 1997. [31] R. G. Bartle and D. R. Sherbert, Introduction to real analysis. Wiley New York, 2000. [32] B. T. Polyak, Introduction to optimization. optimization software. Inc., Publications Division, New York, 1987. [33] Y. I. Alber, A. N. Iusem, and M. V. Solodov, “On the projected subgradient method for nonsmooth convex optimization in a hilbert space,” Mathematical Programming, vol. 81, pp. 23–35, 1998. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Boosting Fairness and Robustness in Over-the-Air Federated Learning: Conclusion and References

Table of Links Abstract and 1. Introduction 1.1 Option Pricing 1.2 Asymptotic Notation (Big O) 1.3 Finite Difference 1.4 The Black-Schole Model 1.5 Monte Carlo Simulation and Variance Reduction Techniques 1.6 Our Contribution Literature Review
Methodology 3.1 Model Assumption 3.2 Theorems and Model Discussion Result Analysis
Conclusion and References 5. CONCLUSION The research findings have demonstrated that finite difference methods offer a powerful tool for approximating option prices and hedging parameters, especially in cases where closed-form solutions are unavailable or impractical. The asymptotic analysis conducted in this study has provided valuable insights into the convergence properties and accuracy of finite difference approximations, shedding light on the optimal choice of discretization schemes and grid sizes for different option contracts. Furthermore, the combination of finite difference methods and Monte Carlo simulation with variance reduction techniques offers a comprehensive approach to hedge error analysis in option pricing. By leveraging the strengths of both methodologies, financial practitioners can obtain more robust and reliable estimates of option prices and hedge parameters, thereby enhancing their risk management capabilities and decision-making processes. It is important to acknowledge the limitations of this study and areas for future research. While the asymptotic analysis provides valuable theoretical insights, further empirical validation is warranted to assess the robustness of the findings across different market conditions and asset classes. Additionally, exploring alternative variance reduction techniques and incorporating more sophisticated models of market dynamics could yield further improvements in hedge error analysis and option pricing accuracy. this study has advanced our understanding of the factors influencing hedge errors and provided practical tools for mitigating their impact on derivative pricing and risk management. As financial markets continue to evolve, the insights gained from this research will remain valuable for academics, practitioners, and policymakers striving to enhance the efficiency and stability of global financial systems. REFERENCES [1] C. W. Smith, “Option pricing. A review,” J. financ. econ., vol. 3, no. 1–2, pp. 3–51, 1976, doi: 10.1016/0304-405X(76)90019-2. [2] S. Guernsey, “an Introduction To the Black-Scholes Pde Model,” 2013. [3] B. Peeters, C. L. Dert, and A. Lucas, “Black Scholes for Portfolios of Options in Discrete Time,” SSRN Electron. J., no. 03, 2005, doi: 10.2139/ssrn.469022. [4] J. He, “A Study on Analytical and Numerical Solutions of Three Types of Black-Scholes Models,” Int. J. Trade, Econ. Financ., vol. 13, no. 2, 2022, doi: 10.18178/ijtef.2022.13.2.720. [5] A. S. Shinde and K. C. Takale, “Study of BlackScholes model and its applications,” Procedia Eng., vol. 38, pp. 270–279, 2012, doi: 10.1016/j.proeng.2012.06.035. [6] Y. Zhang, “The value of Monte Carlo model-based variance reduction technology in the pricing of financial derivatives,” PLoS One, vol. 15, no. 2, pp. 1–13, 2020, doi: 10.1371/journal.pone.0229737. [7] P. L. Bonate, “A brief introduction to Monte Carlo simulation,” Clin. Pharmacokinet., vol. 40, no. 1, pp. 15–22, 2001, doi: 10.2165/00003088- 200140010-00002. [8] D. R. Tobergte and S. Curtis, “Importance sampling for high speed statistical Monte-Carlo simulations,” J. Chem. Inf. Model., vol. 53, no. 9, pp. 1689–1699, 2013. [9] E. C. Anderson and M. Stephens, “Importance Sampling and (MC)^3,” Sisg 2020, vol. 2, no. July, pp. 1–29, 2020. [10] I. N. Medvedev, “Vector estimators of the Monte Carlo method with a finite variance,” Russ. J. Numer. Anal. Math. Model., vol. 28, no. 3, pp. 231– 244, 2013, doi: 10.1515/rnam-2013-0014. [11] E. Saliby and R. J. Paul, “A farewell to the use of antithetic variates in monte carlo simulation,” J. Oper. Res. Soc., vol. 60, no. 7, pp. 1026–1035, 2009, doi: 10.1057/palgrave.jors.2602645. [12] P. W. Glynn and R. Szechtman, “Some New Perspectives on the Method of Control Variates,” Monte Carlo Quasi-Monte Carlo Methods 2000, pp. 27–49, 2002, doi: 10.1007/978-3-642-56046-0_3. [13] W. Gustafsson, “Evaluating the Longstaff-Schwartz method for pricing of American options,” 2015. [14] L. Andersen and M. Broadie, “Primal-dual simulation algorithm for pricing multidimensional American options,” Manage. Sci., vol. 50, no. 9, pp. 1222–1234, 2004, doi: 10.1287/mnsc.1040.0258. [15] E. Wipplinger, Jim Gatheral: The volatility surface, a practitioner’s guide, vol. 22, no. 1. 2008. doi: 10.1007/s11408-007-0072-4. [16] M. Avellaneda and S. Stoikov, “High-frequency trading in a limit order book,” Quant. Financ., vol. 8, no. 3, pp. 217–224, 2008, doi: 10.1080/14697680701381228. [17] S. Mayoral, D. Moreno, and A. Zareei, “Using a hedging network to minimize portfolio risk,” Financ. Res. Lett., vol. 44, no. April, p. 102044, 2022, doi: 10.1016/j.frl.2021.102044. [18] S. Gayathri Devi, K. Selvam, and S. P. Rajagopalan, “An abstract to calculate big o factors of time and space complexity of machine code,” IET Conf. Publ., vol. 2011, no. 583 CP, pp. 844–847, 2011, doi: 10.1049/cp.2011.0483. [19] M. S. H. Mojumder, M. N. Haque, and M. J. Alam, “Efficient Finite Difference Methods for the Numerical Analysis of One-Dimensional Heat Equation,” J. Appl. Math. Phys., vol. 11, no. 10, pp. 3099–3123, 2023, doi: 10.4236/jamp.2023.1110204. [20] I. R. Khan and R. Ohba, “Taylor series based finite difference approximations of higher-degree derivatives,” J. Comput. Appl. Math., vol. 154, no. 1, pp. 115–124, 2003, doi: 10.1016/S0377- 0427(02)00816-6. [21] S. Danho, “Pricing Financial Derivatives with the Finite Difference Method,” p. 237, 2017. [22] G. Connor and T. Lasarte, “An Introduction to Hedge Funds Strategies. Introductory Guide,” pp. 1–16, 2004, [Online]. Available: http://eprints.lse.ac.uk/24675/1/dp477.pdf [23] M. Nurul Anwar and L. Sazzad Andallah, “A Study on Numerical Solution of Black-Scholes Model,” J. Math. Financ., vol. 08, no. 02, pp. 372–381, 2018, doi: 10.4236/jmf.2018.82024. [24] J. Akahori, F. Barsotti, and Y. Imamura, “Hedging error as generalized timing risk,” Quant. Financ., vol. 23, no. 4, pp. 693–703, 2023, doi: 10.1080/14697688.2022.2154255. [25] R. C. A. Oomen and G. J. Jiang, “Hedging Derivatives Risks - A Simulation Study,” SSRN Electron. J., no. May, 2005, doi: 10.2139/ssrn.302525. [26] K. A. Froot, “Hedging Portfolios with Real Assets,” J. Portf. Manag., vol. 21, no. 4, pp. 60–77, 1995, doi: 10.3905/jpm.1995.409527. [27] H. E. LELAND, “Option Pricing and Replication with Transactions Costs,” J. Finance, vol. 40, no. 5, pp. 1283–1301, 1985, doi: 10.1111/j.1540- 6261.1985.tb02383.x. [28] T. Hayashi and P. A. Mykland, “Evaluating hedging errors: An asymptotic approach,” Math. Financ., vol. 15, no. 2, pp. 309–343, 2005, doi: 10.1111/j.0960-1627.2005.00221.x. [29] J.-L. Prigent, R. Hentati, and A. Kaffel, “Dynamic versus static optimization of hedge fund portfolios: the relevance of performance measures,” Int. J. Bus., vol. 15, no. 156, pp. 1–17, 2010. [30] R. A. Jarrow, D. Lando, and F. Yu, “Default Risk and Diversification : Theory and Empirical Implications * Default Risk and Diversification: Theory and Empirical Implications Abstract,” no. 607, 2003. [31] S. Kim, D. Jeong, C. Lee, and J. Kim, “Finite difference method for the multi-asset black-scholes equations,” Mathematics, vol. 8, no. 3, pp. 1–17, 2020, doi: 10.3390/math8030391. [32] D. Jeong, M. Yoo, and J. Kim, “Finite Difference Method for the Black–Scholes Equation Without Boundary Conditions,” Comput. Econ., vol. 51, no. 4, pp. 961–972, 2018, doi: 10.1007/s10614-017- 9653-0. [33] T. Guo, Black-Scholes Process and Monte Carlo Simulation-Based Options Pricing. Atlantis Press International BV, 2023. doi: 10.2991/978-94-6463- 010-7_75. [34] P. Glasserman, “Monte Carlo Methods in Financial Engineerring,” J. Chem. Inf. Model., vol. 53, no. 9, pp. 1689–1699, 2013. [35] Q. Jiang*, “Comparison of Black–Scholes Model and Monte-Carlo Simulation on Stock Price Modeling,” vol. 109, no. Icemci, pp. 135–137, 2020, doi: 10.2991/aebmr.k.191217.025. [36] E. Derman and E. K, “Stochastic Caclulus for Finance,” pp. 1–63, 2000. Authors:
(1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com);
(2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in);
(3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Abstract and 1. Introduction 1.1 Option Pricing 1.1 Option Pricing 1.2 Asymptotic Notation (Big O) 1.2 Asymptotic Notation (Big O) 1.3 Finite Difference 1.3 Finite Difference 1.4 The Black-Schole Model 1.4 The Black-Schole Model 1.5 Monte Carlo Simulation and Variance Reduction Techniques 1.5 Monte Carlo Simulation and Variance Reduction Techniques 1.6 Our Contribution 1.6 Our Contribution Literature Review Methodology Literature Review Literature Review Literature Review Methodology Methodology 3.1 Model Assumption 3.1 Model Assumption 3.2 Theorems and Model Discussion 3.2 Theorems and Model Discussion Result Analysis Conclusion and References Result Analysis Result Analysis Result Analysis Conclusion and References Conclusion and References Conclusion and References 5. CONCLUSION The research findings have demonstrated that finite difference methods offer a powerful tool for approximating option prices and hedging parameters, especially in cases where closed-form solutions are unavailable or impractical. The asymptotic analysis conducted in this study has provided valuable insights into the convergence properties and accuracy of finite difference approximations, shedding light on the optimal choice of discretization schemes and grid sizes for different option contracts. Furthermore, the combination of finite difference methods and Monte Carlo simulation with variance reduction techniques offers a comprehensive approach to hedge error analysis in option pricing. By leveraging the strengths of both methodologies, financial practitioners can obtain more robust and reliable estimates of option prices and hedge parameters, thereby enhancing their risk management capabilities and decision-making processes. It is important to acknowledge the limitations of this study and areas for future research. While the asymptotic analysis provides valuable theoretical insights, further empirical validation is warranted to assess the robustness of the findings across different market conditions and asset classes. Additionally, exploring alternative variance reduction techniques and incorporating more sophisticated models of market dynamics could yield further improvements in hedge error analysis and option pricing accuracy. this study has advanced our understanding of the factors influencing hedge errors and provided practical tools for mitigating their impact on derivative pricing and risk management. As financial markets continue to evolve, the insights gained from this research will remain valuable for academics, practitioners, and policymakers striving to enhance the efficiency and stability of global financial systems. REFERENCES [1] C. W. Smith, “Option pricing. A review,” J. financ. econ., vol. 3, no. 1–2, pp. 3–51, 1976, doi: 10.1016/0304-405X(76)90019-2. [2] S. Guernsey, “an Introduction To the Black-Scholes Pde Model,” 2013. [3] B. Peeters, C. L. Dert, and A. Lucas, “Black Scholes for Portfolios of Options in Discrete Time,” SSRN Electron. J., no. 03, 2005, doi: 10.2139/ssrn.469022. [4] J. He, “A Study on Analytical and Numerical Solutions of Three Types of Black-Scholes Models,” Int. J. Trade, Econ. Financ., vol. 13, no. 2, 2022, doi: 10.18178/ijtef.2022.13.2.720. [5] A. S. Shinde and K. C. Takale, “Study of BlackScholes model and its applications,” Procedia Eng., vol. 38, pp. 270–279, 2012, doi: 10.1016/j.proeng.2012.06.035. [6] Y. Zhang, “The value of Monte Carlo model-based variance reduction technology in the pricing of financial derivatives,” PLoS One, vol. 15, no. 2, pp. 1–13, 2020, doi: 10.1371/journal.pone.0229737. [7] P. L. Bonate, “A brief introduction to Monte Carlo simulation,” Clin. Pharmacokinet., vol. 40, no. 1, pp. 15–22, 2001, doi: 10.2165/00003088- 200140010-00002. [8] D. R. Tobergte and S. Curtis, “Importance sampling for high speed statistical Monte-Carlo simulations,” J. Chem. Inf. Model., vol. 53, no. 9, pp. 1689–1699, 2013. [9] E. C. Anderson and M. Stephens, “Importance Sampling and (MC)^3,” Sisg 2020, vol. 2, no. July, pp. 1–29, 2020. [10] I. N. Medvedev, “Vector estimators of the Monte Carlo method with a finite variance,” Russ. J. Numer. Anal. Math. Model., vol. 28, no. 3, pp. 231– 244, 2013, doi: 10.1515/rnam-2013-0014. [11] E. Saliby and R. J. Paul, “A farewell to the use of antithetic variates in monte carlo simulation,” J. Oper. Res. Soc., vol. 60, no. 7, pp. 1026–1035, 2009, doi: 10.1057/palgrave.jors.2602645. [12] P. W. Glynn and R. Szechtman, “Some New Perspectives on the Method of Control Variates,” Monte Carlo Quasi-Monte Carlo Methods 2000, pp. 27–49, 2002, doi: 10.1007/978-3-642-56046-0_3. [13] W. Gustafsson, “Evaluating the Longstaff-Schwartz method for pricing of American options,” 2015. [14] L. Andersen and M. Broadie, “Primal-dual simulation algorithm for pricing multidimensional American options,” Manage. Sci., vol. 50, no. 9, pp. 1222–1234, 2004, doi: 10.1287/mnsc.1040.0258. [15] E. Wipplinger, Jim Gatheral: The volatility surface, a practitioner’s guide, vol. 22, no. 1. 2008. doi: 10.1007/s11408-007-0072-4. [16] M. Avellaneda and S. Stoikov, “High-frequency trading in a limit order book,” Quant. Financ., vol. 8, no. 3, pp. 217–224, 2008, doi: 10.1080/14697680701381228. [17] S. Mayoral, D. Moreno, and A. Zareei, “Using a hedging network to minimize portfolio risk,” Financ. Res. Lett., vol. 44, no. April, p. 102044, 2022, doi: 10.1016/j.frl.2021.102044. [18] S. Gayathri Devi, K. Selvam, and S. P. Rajagopalan, “An abstract to calculate big o factors of time and space complexity of machine code,” IET Conf. 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Financ., vol. 08, no. 02, pp. 372–381, 2018, doi: 10.4236/jmf.2018.82024. [24] J. Akahori, F. Barsotti, and Y. Imamura, “Hedging error as generalized timing risk,” Quant. Financ., vol. 23, no. 4, pp. 693–703, 2023, doi: 10.1080/14697688.2022.2154255. [25] R. C. A. Oomen and G. J. Jiang, “Hedging Derivatives Risks - A Simulation Study,” SSRN Electron. J., no. May, 2005, doi: 10.2139/ssrn.302525. [26] K. A. Froot, “Hedging Portfolios with Real Assets,” J. Portf. Manag., vol. 21, no. 4, pp. 60–77, 1995, doi: 10.3905/jpm.1995.409527. [27] H. E. LELAND, “Option Pricing and Replication with Transactions Costs,” J. Finance, vol. 40, no. 5, pp. 1283–1301, 1985, doi: 10.1111/j.1540- 6261.1985.tb02383.x. [28] T. Hayashi and P. A. Mykland, “Evaluating hedging errors: An asymptotic approach,” Math. Financ., vol. 15, no. 2, pp. 309–343, 2005, doi: 10.1111/j.0960-1627.2005.00221.x. [29] J.-L. Prigent, R. Hentati, and A. 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Model., vol. 53, no. 9, pp. 1689–1699, 2013. [35] Q. Jiang*, “Comparison of Black–Scholes Model and Monte-Carlo Simulation on Stock Price Modeling,” vol. 109, no. Icemci, pp. 135–137, 2020, doi: 10.2991/aebmr.k.191217.025. [36] E. Derman and E. K, “Stochastic Caclulus for Finance,” pp. 1–63, 2000. Authors: (1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com); (2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in); (3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). Authors: Authors: (1) Agni Rakshit, Department of Mathematics, National Institute of Technology, Durgapur, Durgapur, India (spiritualagnimath.statml@gmail.com); (2) Gautam Bandyopadhyay, Department of Management Studies, National Institute of Technology, Durgapur, Durgapur, India (gbandyopadhyay.dms@nitdgp.ac.in); (3) Tanujit Chakraborty, Department of Science and Engineering & Sorbonne Center for AI, Sorbonne University, Abu Dhabi, United Arab Emirates (tanujit.chakraborty@sorbonne.ae). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

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