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Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
scalability, interoperability, and functionality compared to earlier iterations. Evolution of Blockchain Technology Blockchain technology has evolved significantly since its inception, with each iteration bringing new features and capabilities. Blockchain 1.0: The first generation of blockchain | medium | 2,488 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
technology was primarily focused on digital currencies, with Bitcoin being the most notable example. It introduced the concept of a decentralized, immutable ledger. Blockchain 2.0: The second generation expanded the use cases of blockchain beyond digital currencies. Ethereum, with its smart | medium | 2,489 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
contract functionality, was a key player in this phase, enabling the development of decentralized applications (dApps) and tokens. Blockchain 3.0: This phase saw improvements in scalability, interoperability, and functionality. Projects like Polkadot and Cosmos focused on enabling different | medium | 2,490 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
blockchains to communicate with each other, while also addressing scalability issues. Blockchain 4.0: The current phase is characterized by even greater scalability, interoperability, and sustainability. It aims to make blockchain technology more accessible and environmentally friendly, while also | medium | 2,491 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
enabling more complex decentralized applications and use cases. Overall, the evolution of blockchain technology has been driven by the need to address scalability, interoperability, and functionality, with each generation building upon the successes and challenges of the previous ones. Key Features | medium | 2,492 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
of Blockchain 4.0 Blockchain 4.0 represents the next evolution of blockchain technology, building on the advancements of previous versions. Here are some key features of Blockchain 4.0: ☛ Scalability: Blockchain 4.0 aims to address the scalability issues of earlier versions by implementing | medium | 2,493 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
solutions such as sharding, sidechains, and off-chain computations. These techniques enable the network to handle a significantly higher number of transactions per second, making it more suitable for mass adoption. ☛ Interoperability: Interoperability is a crucial feature of Blockchain 4.0, | medium | 2,494 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
allowing different blockchain networks to communicate and interact with each other seamlessly. This enables the transfer of assets and data across multiple chains, increasing the overall efficiency and usability of blockchain technology. ☛ Security: Security is a top priority in Blockchain 4.0, | medium | 2,495 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
with enhanced protocols and mechanisms to protect against various attacks and vulnerabilities. Advanced cryptographic techniques and consensus algorithms ensure the integrity and immutability of the data stored on the blockchain. ☛ Sustainability: Blockchain 4.0 focuses on sustainability by | medium | 2,496 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
implementing energy-efficient consensus mechanisms and reducing the environmental impact of blockchain operations. This makes it more environmentally friendly and aligned with global sustainability goals. ☛ Governance: Governance in Blockchain 4.0 is more decentralized and community-driven, with | medium | 2,497 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
transparent decision-making processes and mechanisms for resolving disputes. This ensures that the network remains democratic and resilient against centralization. ☛ Smart Contracts: Smart contracts in Blockchain 4.0 are more advanced, with support for complex logic and interactions. This enables | medium | 2,498 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
the creation of sophisticated decentralized applications (dApps) that can automate a wide range of processes without the need for intermediaries. ☛ Privacy: Privacy is enhanced in Blockchain 4.0 through techniques such as zero-knowledge proofs and homomorphic encryption, allowing users to transact | medium | 2,499 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
and interact with the blockchain without revealing sensitive information. ☛ Tokenization: Blockchain 4.0 supports the tokenization of a wide range of assets, including real estate, stocks, and commodities. This enables the creation of digital representations of these assets, making them more | medium | 2,500 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
accessible and tradable on the blockchain. Overall, Blockchain 4.0 represents a significant advancement in blockchain technology, offering improved scalability, interoperability, security, sustainability, governance, smart contract capabilities, privacy, and asset tokenization. Impact of Blockchain | medium | 2,501 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
4.0 on Web 3.0 Blockchain 4.0 is poised to have a profound impact on the development and evolution of Web 3.0. Here are some key ways in which Blockchain 4.0 could influence Web 3.0: ➥ Decentralization: Blockchain 4.0’s emphasis on decentralization aligns with the core principles of Web 3.0, which | medium | 2,502 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
aims to create a more decentralized and user-centric internet. By leveraging advanced consensus mechanisms and governance models, Blockchain 4.0 can help enhance the decentralization of Web 3.0 applications and services. ➥ Data Ownership and Privacy: One of the key features of Web 3.0 is the | medium | 2,503 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
empowerment of users to own and control their data. Blockchain 4.0’s advanced privacy features, such as zero-knowledge proofs and homomorphic encryption, can help ensure that users’ data remains private and secure, furthering the goals of Web 3.0. ➥ Interoperability: Blockchain 4.0’s focus on | medium | 2,504 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
interoperability enables different blockchain networks to communicate and transact with each other seamlessly. This interoperability can extend to Web 3.0, allowing for the seamless transfer of assets and data across various decentralized applications and platforms. ➥ Smart Contracts and | medium | 2,505 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
Automation: Blockchain 4.0’s advancements in smart contract capabilities enable the creation of more sophisticated and automated decentralized applications. This can enhance the functionality and user experience of Web 3.0 applications by enabling complex logic and interactions to be executed | medium | 2,506 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
without the need for intermediaries. ➥ Tokenization and Digital Assets: Blockchain 4.0’s support for asset tokenization can enable the creation of digital representations of real-world assets on the blockchain. This can unlock new opportunities for creating and trading digital assets within the Web | medium | 2,507 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
3.0 ecosystem. ➥ Governance and Community Participation: Blockchain 4.0’s decentralized governance models can be applied to Web 3.0 applications and platforms, enabling more democratic decision-making processes and greater community participation in the governance of the Internet. Overall, | medium | 2,508 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
Blockchain 4.0’s advancements in decentralization, data ownership, privacy, interoperability, smart contracts, tokenization, and governance can significantly impact the development and implementation of Web 3.0, helping to create a more decentralized, user-centric, and secure internet ecosystem. | medium | 2,509 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
Applications of Blockchain 4.0 in Web 3.0 Blockchain 4.0 introduces several innovative applications that can revolutionize Web 3.0. Here are some key applications: Decentralized Autonomous Organizations (DAOs): Blockchain 4.0 can enable the creation of more sophisticated DAOs in Web 3.0. These are | medium | 2,510 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
organizations governed by smart contracts and run by their members, with decisions made through transparent and decentralized voting mechanisms. Decentralized Storage: Blockchain 4.0 can improve decentralized storage solutions in Web 3.0 by ensuring data integrity and incentivizing users to | medium | 2,511 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
contribute their storage space. This can create a more secure and efficient way to store and access data. Cross-Chain Communication: Blockchain 4.0’s interoperability features can facilitate cross-chain communication in Web 3.0, allowing different blockchain networks to exchange information and | medium | 2,512 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
assets seamlessly. This can unlock new possibilities for decentralized applications (dApps) that require interactions across multiple chains. Tokenization of Real-World Assets: Blockchain 4.0 can enable the tokenization of a wide range of real-world assets in Web 3.0, including real estate, art, | medium | 2,513 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
and intellectual property. This can make these assets more liquid and accessible to a global audience. Decentralized Marketplaces: Blockchain 4.0 can enhance decentralized marketplaces in Web 3.0 by enabling more secure and transparent transactions. Users can buy and sell goods and services without | medium | 2,514 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
the need for intermediaries, reducing costs and increasing efficiency. Decentralized Identity: Blockchain 4.0 can improve decentralized identity solutions in Web 3.0 by providing users with more control over their identity data. This can enable secure and verifiable digital identities, reducing the | medium | 2,515 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
risk of identity theft and fraud. Immutable Content Publishing: Blockchain 4.0 can enable immutable content publishing platforms in Web 3.0, where content creators can publish their work in a tamper-proof manner. This can help protect intellectual property rights and ensure the integrity of | medium | 2,516 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
published content. Supply Chain Traceability: Blockchain 4.0 can enhance supply chain traceability in Web 3.0 by enabling transparent and immutable tracking of goods. This can help improve product quality, reduce fraud, and increase trust between supply chain partners. These applications | medium | 2,517 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
demonstrate the potential of Blockchain 4.0 to transform various aspects of Web 3.0, including governance, data management, security, and commerce. Use Cases of Blockchain 4.0 in Web 3.0 Blockchain 4.0 introduces several new use cases that can enhance the capabilities of Web 3.0. Here are some | medium | 2,518 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
examples: ➢ Decentralized Social Networks: Blockchain 4.0 can enable the creation of decentralized social networks that are resistant to censorship and offer users more control over their data. Users can own their data, decide how it is shared, and be rewarded for their contributions to the | medium | 2,519 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
network. ➢ Supply Chain Management: Blockchain 4.0’s enhanced scalability and interoperability make it well-suited for supply chain management in Web 3.0. Companies can use blockchain to track the provenance of goods, ensure authenticity, and streamline processes across multiple parties. ➢ | medium | 2,520 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
Decentralized Finance (DeFi): Blockchain 4.0 can further advance DeFi applications in Web 3.0 by offering increased scalability, security, and interoperability. This can enable more complex financial instruments, decentralized exchanges, and lending platforms to operate more efficiently and | medium | 2,521 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
securely. ➢ Digital Identity: Blockchain 4.0 can improve digital identity management in Web 3.0 by providing users with greater control over their identity data. Users can manage and share their identity information securely, reducing the risk of identity theft and fraud. ➢ Content Monetization: | medium | 2,522 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
Blockchain 4.0 can enable new models for content monetization in Web 3.0. Content creators can tokenize their work, allowing fans to support them directly through microtransactions or subscriptions, without the need for intermediaries. ➢ Gaming: Blockchain 4.0 can enhance gaming experiences in Web | medium | 2,523 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
3.0 by enabling true ownership of in-game assets. Players can buy, sell, and trade assets across different games, creating a new economy within the gaming ecosystem. ➢ Healthcare: Blockchain 4.0 can improve healthcare in Web 3.0 by enabling secure and interoperable sharing of medical records. | medium | 2,524 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
Patients can have greater control over their health data, and healthcare providers can access a more complete picture of a patient’s health history. ➢ Real Estate: Blockchain 4.0 can revolutionize the real estate industry in Web 3.0 by enabling fractional ownership and easier transfer of property | medium | 2,525 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
rights. This can make real estate investments more accessible and liquid. These use cases demonstrate the potential of Blockchain 4.0 to enhance various aspects of Web 3.0, including decentralization, data ownership, security, and efficiency. Challenges and Considerations Blockchain technology | medium | 2,526 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
faces several challenges and considerations that need to be addressed for its widespread adoption. One of the key challenges is scalability, as current blockchain networks struggle to handle a large number of transactions simultaneously. Interoperability is another issue, as different blockchain | medium | 2,527 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
platforms often operate in isolation, limiting their ability to communicate and share data. Additionally, there are security concerns, as blockchain networks are not immune to cyber-attacks and vulnerabilities. Regulatory uncertainty is also a consideration, as governments around the world grapple | medium | 2,528 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
with how to regulate this relatively new technology. Finally, there are environmental concerns, as the energy consumption of blockchain networks, particularly proof-of-work-based ones, has raised questions about their sustainability. Addressing these challenges and considerations will be crucial | medium | 2,529 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
for the continued development and adoption of blockchain technology. Predictions For The Future of Web 3.0 And Blockchain 4.0 The future of Web 3.0 and Blockchain 4.0 looks promising, with both technologies poised to revolutionize various industries. Web 3.0 is expected to offer a more | medium | 2,530 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
decentralized, user-centric internet experience, where individuals have more control over their data and digital interactions. Blockchain 4.0, on the other hand, is likely to bring advancements in scalability, interoperability, and sustainability, making blockchain technology more accessible and | medium | 2,531 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
environmentally friendly. These developments are expected to drive the widespread adoption of decentralized applications (dApps) and smart contracts, transforming industries such as finance, healthcare, supply chain, and more. Additionally, the integration of artificial intelligence (AI) and | medium | 2,532 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
Internet of Things (IoT) technologies with Web 3.0 and Blockchain 4.0 could further enhance their capabilities, leading to a more connected and intelligent digital ecosystem. Conclusion In conclusion, Blockchain 4.0 introduces transformative changes to Web 3.0, revolutionizing the decentralized | medium | 2,533 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
landscape. The advancements in scalability, interoperability, and sustainability are poised to address key challenges faced by earlier blockchain generations. By enhancing scalability, Blockchain 4.0 enables blockchain networks to process a significantly higher number of transactions, paving the | medium | 2,534 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
way for mainstream adoption. Interoperability improvements foster a more interconnected ecosystem, allowing seamless interaction between different blockchain platforms and assets. This interconnectedness unlocks new possibilities for decentralized applications and smart contracts, driving | medium | 2,535 |
Blockchain, Blockchain Technology, Blockchain Development, Web3, Web3 Development.
innovation and collaboration across the decentralized space. Moreover, the focus on sustainability underscores the importance of eco-friendly practices in blockchain development, mitigating concerns about the environmental impact of blockchain technology. Overall, Blockchain 4.0 marks a pivotal | medium | 2,536 |
Self Driving Cars, Automation, Technology, Ford.
By Venky Krishnan, Autonomous Vehicle Systems Core Supervisor, Ford Motor Company When was the last time you thought about how insects affect your driving? Chances are you never have, since they don’t make much of an impact unless a particularly large one gets smeared across the windshield of your | medium | 2,538 |
Self Driving Cars, Automation, Technology, Ford.
car. Well I do think about it — a lot. I find myself dedicating an unbelievable amount of time to thinking about these critters as my team at Ford helps make progress on its self-driving vehicle business. It turns out that insects pose a significant challenge to self-driving vehicles. Over the last | medium | 2,539 |
Self Driving Cars, Automation, Technology, Ford.
few years, Ford has been conducting some serious research into making sure our self-driving vehicles can always see the world around them, no matter what may try to get in the way. We’ve sprayed dirt and dust onto our self-driving vehicle sensors. We’ve showered LiDAR sensors with water to simulate | medium | 2,540 |
Self Driving Cars, Automation, Technology, Ford.
rainfall. We created our own synthetic bird droppings and smeared it on camera lenses. When it came to bugs, we sat down for discussions with the author of “That Gunk on Your Car,” zoologist Mark Hostetler, to gain insight into the various insects that are regularly making contact with vehicles — | medium | 2,541 |
Self Driving Cars, Automation, Technology, Ford.
and how often they’re doing so. We even built a makeshift “bug launcher” that lets us shoot insects at vehicle sensors at high speeds, so we could really understand the best way to clean them off. All the various sensors on these cars are, after all, constantly working to deliver the best picture | medium | 2,542 |
Self Driving Cars, Automation, Technology, Ford.
of the world they possibly can, but an untimely splat could seriously interfere with their ability to do that. That’s where my team comes in. All this research sparked a question: Wouldn’t it be a lot easier if we just kept our self-driving sensors from getting hit with bugs in the first place? To | medium | 2,543 |
Self Driving Cars, Automation, Technology, Ford.
do that, we decided to take maximum advantage of the “tiara,” the structure that sits on top of all our self-driving vehicles and holds the collection of cameras, LiDAR and radar that helps the car “see” where it’s going. Our team has already submitted around 50 patents related to self-driving | medium | 2,544 |
Self Driving Cars, Automation, Technology, Ford.
cleaning and structural systems, demonstrating that there’s a lot of innovation happening outside the world of self-driving software. As you’ll see, we’ve designed our tiara to do a lot more than just house cameras — it’s actually the first line of defense for our sensors. Here’s how it works: As | medium | 2,545 |
Self Driving Cars, Automation, Technology, Ford.
the car is driving, the tiara funnels air out through different slots near the camera lens. This creates an “air curtain” that actually deflects bugs away from the sensor itself. So anytime bugs are making a bee-line for a camera lens, the air flowing out of the tiara pushes it aside so it doesn’t | medium | 2,546 |
Self Driving Cars, Automation, Technology, Ford.
make contact with the lens. It’s like changing the course of an asteroid on a crash-course with Earth. This method was remarkably successful. With bugs, for example, our tests showed that the air curtain successfully diverted the vast majority of them away from our self-driving sensors. We were | medium | 2,547 |
Self Driving Cars, Automation, Technology, Ford.
saving (insect) lives! Still, this solution wasn’t perfect. Insects could still break past the air curtain in some situations, so we needed a way to successfully clean our sensors when necessary. After all, trips to the car wash aren’t really practical with self-driving vehicles due to all the | medium | 2,548 |
Self Driving Cars, Automation, Technology, Ford.
sensitive technology packed inside. Instead of taking these cars to the cleaners, we had to the take the cleaners to them. Literally. Fully integrated into the tiara, our cleaning system features next generation nozzles next to each camera lens that can spray washer fluid as needed to clean the | medium | 2,549 |
Self Driving Cars, Automation, Technology, Ford.
sensors. Using advanced software algorithms that helps our self-driving vehicles determine when a sensor is dirty, our cleaning system can specifically hone in on dirty camera lenses (whether it’s just one dirty lens or several), efficiently cleaning each one individually without wasting washer | medium | 2,550 |
Self Driving Cars, Automation, Technology, Ford.
fluid on already-clean sensors. It’s like having a personal cleaner visit each sensor when it’s time for a bath. After a sensor has been sprayed down, our tiara has a clever way of drying itself off. It releases air through a slot which quickly “dries” the face of the lens. Needless to say, a great | medium | 2,551 |
Self Driving Cars, Automation, Technology, Ford.
deal of engineering work is being put into ensuring our systems are as safe, extremely quick, capable and precise as possible. Our team has gone to great lengths to test the effectiveness of our system, taking one of our test vehicles and driving it through the Huron-Manistee National Forests in | medium | 2,552 |
Self Driving Cars, Automation, Technology, Ford.
western Michigan too see how our cleaning system reacted to swarms of bugs. This system has also been equipped on the third generation of our self-driving test vehicles, which are now hitting the streets in Detroit, Pittsburgh, Miami-Dade County and Washington, D.C., so it’s going to see continuous | medium | 2,553 |
Self Driving Cars, Automation, Technology, Ford.
testing in very different environments. As fun as some of this development may sound, these are not features that would simply be nice to have when self-driving vehicles are ready to be deployed; they are critical functions that vehicles must be able to carry out on their own in order for safe | medium | 2,554 |
Self Driving Cars, Automation, Technology, Ford.
deployment to be possible. Just as we must equip self-driving vehicles with the brains to process what’s happening in their environment, we must also equip them with the tools to deal with that environment — no matter what kind of gunk it decides to throw at them. | medium | 2,555 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
An intuitive basis for understanding all things “eigen” Motivation We often want to transform our data to reduce the number of features while preserving as much variance (i.e., the differences among our samples) as we can. Often, you’ll hear folks refer to principal component analysis (PCA) and | medium | 2,556 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
singular value decomposition (SVD), but we can’t appreciate how these methods work without first understanding what eigenvectors and eigenvalues are. Etymology “Eigenvector” is a pretty weird word. As with many weird words (think kindergarten), we can blame the Germans. The most useful translation | medium | 2,557 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
I’ve heard comes from the Coursera course “Mathematics for Machine Learning Specialization”: “eigen” means “characteristic.” An “eigenvector” is a vector that “characterizes” a linear transform. Visual intuitions Let’s take a look at a couple vectors under arbitrary linear transforms like | medium | 2,558 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
translation, scaling, rotation, and shear. Most of our vectors are shifted around. Some, though, point in the same direction before and after a transform. Let’s take a look at a simple horizontal scaling! We can achieve this with the linear transform matrix [[2, 0], [0, 1]]. Figure 1: Visualizing | medium | 2,559 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
three vectors through a horizontal scaling. Image by author. If we plot three unit length vectors— one at 0°, one at 45°, and one at 90° — and visualize what happens after applying our transform matrix, we see that some vectors remain pointed in the same directions (0° and 90°) as before while | medium | 2,560 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
others do not (45°). There’s something interesting about the two vectors that remain pointed in the same direction before and after. Under the linear transform, these vectors are just scaled by a scalar term. Our unit vector at 90° is unchanged, i.e., scaled by 1 while our unit vector at 0° is | medium | 2,561 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
doubled. These vectors are our eigenvectors! The eigenvectors of a linear transform are those vectors that remain pointed in the same directions. For these vectors, the effect of the transform matrix is just scalar multiplication. For each eigenvector, the eigenvalue is the scalar that the vector | medium | 2,562 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
is scaled by under the transform. The mathematics Plotting a bunch of vectors and waiting for an animation to render isn’t a terribly efficient approach. Luckily, all we need to do is formalize the intuitions we’ve already built. Some of the the equations look a little intimidating, but they aren’t | medium | 2,563 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
so bad once we understand where they come from. While the mathematics extend to matrices of arbitrary dimensionality, we are going to stick to a 2x2 matrix for this demonstration. Let’s consider a linear transform matrix A. As we saw, the eigenvectors for a matrix are the vectors that are just | medium | 2,564 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
scaled. These scalars are eigenvalues, and we’ll call them λ. We’ll call our eigenvectors x. All together now… the eigenvectors x are scale by our eigenvalues λ by our matrix A. We can formalize this with the top equation in Figure 2. Figure 2: Basic formalization of eigenvectors and eigenvalues. | medium | 2,565 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
Image by author. Let’s move some terms around! Subtracting off λx and factoring out x gives us a nice zero-valued equality. To subtract a constant (λ) off of the matrix A, we need to multiply it by an identity matrix the same dimensionality of A. Let’s solve for our eigenvectors and eigenvalues! We | medium | 2,566 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
aren’t interested in the “trivial” solution to these equations where the x vector is zero-valued. Instead, we want to know when the term (A-λI) is equal to zero. We can achieve this by checking the determinant (Figure 3) is zero-valued! Figure 3: Definition of the determinant of a 2x2 matrix. Image | medium | 2,567 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
by author. Let’s expand our matrix A now, so that we can see each of the values in the matrix (Figure 4). Figure 4: Expanding the matrix A. Image by author. Piecing things together, we get equalities shown in Figure 5. Figure 5: Checking det(A-λI)=0. Image by author. Using our definition of the | medium | 2,568 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
determinant from Figure 3, we can substitute in our values. Figure 6: Substituting in the values of our matrix into the definition of a determinant. Image by author. Finally, multiplying out terms, we recover a form called the “characteristic polynomial.” Figure 7: Definition of the “characteristic | medium | 2,569 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
polynomial.” Image by author. That’s as far as we can go in the abstract! Now, let’s apply this “characteristic polynomial” and solve for our eigenvalues (λ). Figure 8: Sample matrix A. Image by author. Plugging in the values from our matrix into our characteristic polynomial… Figure 9: Solving the | medium | 2,570 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
characteristic polynomial for our matrix A. Image by author. We get our eigenvalues (λ). Now, we can substitute our eigenvalues back in to solve for our eigenvectors. Figure 10: Determining our eigenvectors based on eigenvalues. Image by author. This is an odd result… @λ = 2, [0, -x₂] = 0. Our x₁ | medium | 2,571 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
seems to have disappeared. What does this mean? It means that for λ = 2 as long as x₂ is zero, x₁ can equal anything. [5, 0], [1, 0], and [-3, 0] for instance Similarly, for λ = 1 as long as x₁ is zero, x₂ can equal anything. [0, 2], [0, -1], [0, 8] for instance We express this invariance by | medium | 2,572 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
substituting in a placeholder variable t for the terms that can take on any value. Figure 11: Defining the eigenvectors of our space. Image by author. This is exactly what our visual intuitions showed us! All the horizontal vectors of our space are eigenvectors and they are scaled by the eigenvalue | medium | 2,573 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
2. All the vertical vectors of our space are eigenvectors and they are scaled by the eigenvalue 1. Try plotting yourself! I found a lot of value in plotting things for myself. If you want to try, check out the source below! Acknowledgements It is important to give credit where it is most definitely | medium | 2,574 |
Python, Mathematics, Computer Science, Data Science, Machine Learning.
due! While the code in the article is mine, the package used for visualization (manim) is certainly not! The visualization library and the method of explanation are shamelessly stolen from 3blue1brown. I mentioned it earlier in the article, but I love this Coursera series: Mathematics for Machine | medium | 2,575 |
Space, Technology, Engineering, Research, Hardware.
The line between science fact and science fiction is often now blurred, and nowhere is that more evident in the way our computing is changing. However throughout the last decade we’ve seen the line blurred in other places and the landing of a rocket on its tail, followed by the flight of the SpaceX | medium | 2,577 |
Space, Technology, Engineering, Research, Hardware.
Falcon Heavy with its ballet landing of its side boosters, alongside the race to win the smallsat launcher war, and a serious push to make cubesats more useful and open source, means that space is now one of those places. Over the last couple of years we’ve seen a Raspberry Pi on the space station, | medium | 2,578 |
Space, Technology, Engineering, Research, Hardware.
plans to 3D print a satellite, and hardware sent to the space station via a single email. Now Ariel Ekblaw — a student at MIT’s Media Lab and founder and lead of the Space Exploration Initiative at the lab — is looking to biology for inspiration. Sensors embedded in the TESSERAE prototype. (📷: | medium | 2,579 |
Space, Technology, Engineering, Research, Hardware.
Ariel Ekblaw) The TESSERAE project is a design experiment in imagining how to bring the concept of geodesic domes—based on the structure of Carbon-60 that occurs in nature, known as a “Buckyball”—from Earth into orbit to help create habitable structures in space. Intended to be multi-use and | medium | 2,580 |
Space, Technology, Engineering, Research, Hardware.
low-cost, the orbiting modules should be self-assembling and reconfigurable. “Unlike large-scale habitats proposed for entire space colonies, the TESSERAE should be thought of as flexible and reconfigurable modules to aid in agile mission operations. Our mission concept focuses on supporting Mars | medium | 2,581 |
Space, Technology, Engineering, Research, Hardware.
surface operations, with multiple, interlocking TESSERAE acting as an orbiting base, in addition to supporting the coming waves of space tourists in Low Earth Orbit.” The prototype employs 3D-printed magnetic plates that snap together, with solar panels and sensors—including a 9-axis IMU—added. | medium | 2,582 |
Space, Technology, Engineering, Research, Hardware.
While the proof-of-concept prototypes made use of traditional PCBs, the next generation of prototypes will embed flexible circuitry into the tiles themselves, and make use of 3D-printed traces between them. Proof of concept circuit design. (📷: Ariel Ekblaw) TESSERAE was one of 14 experiments to fly | medium | 2,583 |
Space, Technology, Engineering, Research, Hardware.
on a MIT-chartered ZERO-G flight in November last year. The flight was the initiative’s first research deployment. Other projects ranged across the disciplines from design, to architecture, engineering, biology, and robotics, through to music and art. Space Exploration Initiative in Zero Gravity | medium | 2,584 |
Space, Technology, Engineering, Research, Hardware.
Flight. (📷: MIT Media Lab) The flight validated the TESSERAE mechanical structure and self-assembly protocol. Another follow up experiment is scheduled to fly on an upcoming Blue Origin New Shepard sub-orbital flight to test embedded sensor network, communication architecture between tiles, and | medium | 2,585 |
Space, Technology, Engineering, Research, Hardware.
additional self-assembly. You can read more about the TESSERAE project in Ekblaw’s paper on “Self-assembling Space Structures: Buckminsterfullerene Sensor Nodes,” published in the proceedings of the AIAA/AHS Adaptive Structures Conference, which was held in Kissimmee, Florida back in January. | medium | 2,586 |
Connectivity, Silicon, Photonics, Technical Analysis, Technews.
Technology Light in an optical fiber covers much greater distance without repeaters (and associated repeater delay) and loses far less energy in route than do electrical signals in even the best cable. Three attributes — reach, bandwidth density, and energy consumption — are critical to the future | medium | 2,587 |
Connectivity, Silicon, Photonics, Technical Analysis, Technews.
of large-scale data centers. Consequently, optical fiber, driven by plug-in interface cards, has rapidly replaced electrical connections in the top layers — the spine and leaf networks — of the largest data centers. But major advances in silicon photonics — implementing optical components on | medium | 2,588 |
Connectivity, Silicon, Photonics, Technical Analysis, Technews.
silicon wafers using processes derived from standard IC production techniques — are poised to spread this transformation to applications which require much shorter distances. The near future will see rapid evolutionary change as silicon photonics shrink the size and power consumption of optical | medium | 2,589 |
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