Imagine: Parallel Neural Networks with Shaders 🤔 (GLSL instead... of CUDA) Parallelization by encoding the weights and activations as Buffer Objects or Textures, using fs to perform matmul and activation functions in parallel. Each fragm shader instance parproc 1..n neuron.show more

The First Omni-chain Restaking L2 now powered by the... Pyth Network 🔮? Indeed, the Pyth Price Feeds are now available on Alex T. Plumley and already secure the Parallel Super App Learn more below: ℹ️ About Parallel Network Parallel Network is the first omni-chain restaking L2 that focuses on decentralized, secure, restaking with EigenLayer. Backed by $30 million in investment from Sequoia, Polychain, Founders Fund, and more, Parallel Network aims to enhance the omni-chain experience through efficient restaking, lending, and liquidity. ℹ️ About Parallel Super App The Parallel Super App, stands as an integrated platform that offers a comprehensive suite of DeFi products including lending, staking, trading, and more across diverse networks. The Parallel Super App boasts $82m+ in TVL across its EVM and Polkadot products. Combining its Super App and L2, Parallel will significantly enhance the omni-chain liquidity and restaking experience. 🔮 Being powered by the Pyth Network on Parallel With the 450+ Pyth Price Feeds now live and permissionlessly usable on the Parallel Network, DeFi is ready to bloom. And as a matter of fact, the Pyth oracle is already securing the Parallel Super App. 🗣️ Quotes "We're excited to work with Pyth to secure our price feeds”, said Yubo Ruan, Founder and CEO, Parallel Network. “As the largest and most trusted oracle network, we will work closely with Pyth to ensure secure and accurate market data for Parallel and its products.”show more

How are developers scaling their work with AI? Some... Cline users are moving beyond 1:1 pairing, experimenting with orchestrating multiple instances simultaneously. Imagine managing several Cline workstreams, each on a different feature branch, tackling tasks in parallel. Integration happens via standard Git PRs, sometimes even reviewed/merged by another dedicated Cline instance. This points to a potential future where developers orchestrate AI-driven workstreams, leveraging familiar Git practices to manage parallel development. The video below shows one example of this approach in action. It's fascinating seeing the community explore this, and it gives us ideas...show more

Not now, s*ftware engineer, I'm writing firmware with my... left hand while soldering with the right and controlling CAD with a neural headset to create gears Go to your room and play with your LLM-electron-webslop.js thingy Let The Man do The Jobshow more

Here are more results from #RigidFormer: predicting physical dynamics... with purely neural simulators — an attempt to learn physical dynamics in a scalable manner. 🤖 1) Controllable Articulated Body Simulation — More Results Additional Unitree G1 humanoid rollouts under controlled motion. Each sample uses a different initial state and control signal (direction and velocity). 🏺 2) Object Fragmentation Simulating the cracking and fragmentation process of objects. Thanks Žiga Kovačič for suggesting this experiment! 🎬 3) Combining Rigidformer with Diffusion-as-Shader for controllable video generation. Note: the meshes shown here are only for visualization — the network takes point clouds as input and predicts the updated state of each point.show more

Added context to my tiny diffusion model to enable... sequential generation of longer outputs! Currently the context is a quarter of the sequence length (seq_len=256, context_len=64). I have a theory that the less semantic-value-per-token, the worse the “curse of parallel decoding” is. With parallel decoding, we independently predict multiple tokens in one step. With the sentence “My poker hand was a ___ ___”, two valid predictions are “two pair” and “straight flush”. Because each token prediction is independent though, we can end up with a nonsensical output like “two flush”. This seems to be exacerbated with low semantic-value-per-token, as now you need more tokens to express the same concept. Instead of needing to independently predict two tokens, we might need to predict 10 instead (which is of course much harder). The model currently has noticeably worse output compared to nanogpt (similar size) and I believe this is a main reason. I’ll try adding confidence-aware parallel decoding (from NVIDIA’s Fast-dLLM paper) and other tricks and see how much they improve generation quality.show more

Gaming + Generative AI This is a scene from... GTA: San Andreas reimagined with Runway Gen-3 Jensen Huang, the CEO of Nvidia spoke of the future of DLSS where he mentioned DLSS to generate in-game assets, such as textures and objects and DLSS 10 hypothetically Delivering Full Neural Rendering. This may look bad and un-playable by our today's standards but we all know how good Generative AI Video has gotten over a few short years. This is the future of Gaming combined with Generative AI to create photorealism Video Credit: Niccyanshow more

[Backpropagation] by Hand✍️ [1] Forward Pass ↳ Given a... multi layer perceptron (3 levels), an input vector X, predictions Y^{Pred} = [0.5, 0.5, 0], and ground truth label Y^{Target} = [0, 1, 0]. [2] Backpropagation ↳ Insert cells to hold our calculations. [3] Layer 3 - Softmax (blue) ↳ Calculate ∂L / ∂z3 directly using the simple equation: Y^{Pred} - Y^{Target} = [0.5, -0.5, 0]. ↳ This simple equation is the benefit of using Softmax and Cross Entropy Loss together. [4] Layer 3 - Weights (orange) & Biases (black) ↳ Calculate ∂L / ∂W3 and ∂L / ∂b3 by multiplying ∂L / ∂z3 and [ a2 | 1 ]. [5] Layer 2 - Activations (green) ↳ Calculate ∂L / ∂a2 by multiplying ∂L / ∂z3 and W3. [6] Layer 2 - ReLU (blue) ↳ Calculate ∂L / ∂z2 by multiplying ∂L / ∂a2 with 1 for positive values and 0 otherwise. [7] Layer 2 - Weights (orange) & Biases (black) ↳ Calculate ∂L / ∂W2 and ∂L / ∂b2 by multiplying ∂L / ∂z2 and [ a1 | 1 ]. [8] Layer 1 - Activations (green) ↳ Calculate ∂L / ∂a1 by multiplying ∂L / ∂z2 and W2. [9] Layer 1 - ReLU (blue) ↳ Calculate ∂L / ∂z1 by multiplying ∂L / ∂a1 with 1 for positive values and 0 otherwise. [10] Layer 1 - Weights (orange) & Biases (black) ↳ Calculate ∂L / ∂W1 and ∂L / ∂b1 by multiplying ∂L / ∂z1 and [ x | 1 ]. [11] Gradient Descent ↳ Update weights and biases (typically a learning rate is applied here). 💡 Matrix Multiplication is All You Need: Just like in the forward pass, backpropagation is all about matrix multiplications. You can definitely do everything by hand as I demonstrated in this exercise, albeit slow and imperfect. This is why GPU's ability to multiply matrices efficiently plays such an important role in the deep learning evolution. This is why NVIDIA is now close to $1 trillion in valuation. 💡Exploding Gradients: We can already see the gradients are getting larger as we back-propagate up, even in this simple 3-layer network. This motivates using methods like skip connections to handle exploding (or diminishing) gradients as in the ResNet. I did the calculations entirely by hand. Please let me know if you spot any error or have any questions!show more

NVIDIA just made AI detect objects 10x faster by... deleting one step. It's called LocateAnything, and it removes the biggest bottleneck no one else was fixing in vision-language models. Normally a model builds each bounding box one coordinate token at a time. 100 objects means thousands of tokens before an answer. NVIDIA scrapped that: their Parallel Box Decoding predicts the whole box in a single forward pass, as one atomic unit. → 12.7 boxes/sec on one H100 → 10x faster than Qwen3-VL → +3.8% F1 on LVIS, accuracy up, not down → 3B params, runs on one consumer GPU Treating the box as one unit keeps its coordinates tied together, which is why accuracy climbed instead of falling. One model handles detection, GUI grounding, OCR, and document understanding, ready for computer-use agents, robotics, and document pipelines. 100% open source, weights, code, demo, and paper all live.show more

ReLU vs Leaky ReLU 👉 = ReLU = ReLU... is the default activation in modern deep learning — cheap to compute, and stable enough to train networks hundreds of layers deep. To see what it does, picture five boba tea shops on the same block — 𝚊, 𝚋, 𝚌, 𝚍, 𝚎 — each running their own books. Each value is a shop's monthly profit — receipts minus rent, ingredients, and wages. When profit is positive, the shop stays open and the owner pockets every dollar. When profit turns negative, the shop runs out of cash and shutters — the lights go off, the books are wiped to zero. ReLU is exactly that rule, applied one shop at a time. Read the diagram left to right. The first column is the raw value x — each shop's profit at month's end. The second column is the gate: 1 if the shop is open (x > 0), 0 if it has shuttered. The last column is the ReLU output: open shops pass their profit through untouched, while shuttered ones are zeroed out. Five rows means five parallel shops on the same block, each evaluated independently. That's why ReLU is called an element-wise activation: every neuron decides its own fate. = LeakyRelu = Plain ReLU wipes negative values to zero — clean, but a shop that shutters can never recover, since both its output and its gradient stay pinned at zero. This is the dying ReLU problem, and in deep networks it can quietly kill a meaningful fraction of the units. Leaky ReLU is the one-line fix: instead of shuttering, the shop files for Chapter 11 protection and keeps the lights on at reduced capacity. Its debt is restructured down to a fraction α (typically 0.1) — the rest is forgiven, and the shop is wounded, not killed. A small negative signal still flows through, so the gradient survives, and the shop can crawl back to life if a TikTok goes viral. Read the diagram left to right. The first column is the raw value x — each shop's profit at month's end. The second column is the leakage α — the fraction of the loss held over after restructuring (default 0.1, editable). The third column is the gate: 1 for shops still in the black, α for those operating under bankruptcy protection. The last column is the Leaky ReLU output: y = x · gate. Profitable shops pass through untouched; struggling ones shrink by a factor of α but still carry a sign. Five rows means five parallel shops, each evaluated independently. Like ReLU, this is an element-wise activation: every neuron's fate is decided on its own merits. #aibyhahdshow more

Did you know about 21 million people worldwide are... diagnosed with #epilepsy in 2021? Epilepsy’s pathophysiology involves the aberrant electrical activity of neurons in the brain. This typically stems from an imbalance between excitatory and inhibitory neurotransmitters, with increased excitation or reduced inhibition leading to hyperactive neural networks. These abnormal electrical discharges manifest as seizures, which can vary in intensity and impact depending on the specific regions of the brain affected by this pathophysiological imbalance.🧠 . Video credit silvan_haber_ #scientissimum #neurology #neurologia #MedTwitter #Worlds2023show more

Predicting the next word "only" is sufficient for language... models to learn a large body of knowledge that enables then to code, answer questions, understand many topics, chat, and so on. This is clear to many researchers now, and there are nice tutorials on why this works by Ilya Sutskever resorting to compression ( ) and by Geoffrey Hinton ( ). However, the emergence of types of understanding is not unique to language models. In by Misha Denil and Brandon Amos the authors trained models to predict the next few time stems of over a hundred robot hand sensors (Touch, Gyro, Accelerometer, Joint Info, Actuator Info, etc.). They ten found out that they could regress the shape of the thing the hand was touching from the activations of the neural networks using probes. That is, the model developed an internal representation of shapes even though it was simply used to predict "only" the next few senses. Awareness follows from simple predictions and interaction with the world.show more

As announced in partnership with NVIDIA at CES, we’re... excited to introduce Stable Point Aware 3D (SPAR3D), setting a new standard in 3D generation. Ideal for running on NVIDIA RTX AI PCs, SPAR3D enables real-time editing and complete structure generation of 3D objects from a single image in under a second. You can download the weights on Hugging Face and code on GitHub, or access the model through the Stability AI API. Learn more here: (1/3)show more

When Deri V4? Now! Time to dive into the... world’s first-ever cross-chain derivatives trading today! 👉 Why Deri V4?👇 ✅ Flexibility of Network ✅Consolidated Liquidity for Enhanced Trading ✅Faster Execution and Reduced Gas Costs ✅Expanded Symbol Offerings and More Option Strikes Learn more 👉 Next steps Expand to more Layer 1 & 2 networks, driven by governance votes. In parallel, the Deri team is dedicated to evolving with traders' needs, continuously developing new features ✅Limit and stop orders ✅DERI token as base token ✅Optimized Gamma Swap ✅… Stay tuned! #DeriV4 #crosschain #SailToAmbitionshow more

🚨🇷🇺🇨🇳 Russia and China develop breakthrough AI-powered earthquake forecasting... method The new AI model predicts earthquakes by tracking changes in the Earth’s crust with superior accuracy—at a fraction of traditional computing costs. 🔸 Geoacoustic emissions are used — sounds produced as tectonic stress builds up underground — to pinpoint possible earthquake precursors 🔸The new system combines Physics-Informed Neural Networks (PINNs) with Kolmogorov-Arnold Networks (KANs) to process seismic data 🔸 The team proposes using human-made noise as seismic sonar to reveal underground details that natural signals miss 🔸 The approach slashes computing costs while delivering sharper 2D and 3D geological maps Beyond earthquake forecasting, the same technology could also aid mineral exploration and infrastructure planning—by revealing subsurface structures with unprecedented clarityshow more

I've been editing this article about "brain mapping" and... connectomics, and I'm just stunned by how quickly the cost estimates to map, say, a mouse brain have plummeted in just the last couple years. It actually seems feasible that we could map the entire human brain -- all 86 billion neurons, and their connections -- in this lifetime. In the 1970s, Sydney Brenner started mapping all the connections between neurons in C. elegans. His team sliced the worm into thin pieces, took photos using an electron microscope, and manually traced and reconstructed each synapse for 302 neurons total. This project took more than a decade of work, and it cost about $16,500 to reconstruct each neuron. Scaling this up to a human brain boggles the mind. Electron microscopy remained the norm in connectomics for decades, because it was the only option available to see synapses at a resolution high enough to be able to trace their paths. Each electron microscope costs several hundreds of thousands of dollars, though, and you need lots of them to map even a mouse brain in a reasonable timeframe. In 2023, the Wellcome Trust released a report estimating how long, and how expensive, it would be to map the mouse connectome (~70M neurons). They estimated that imaging alone would cost $200-300M, and that proofreading (or ensuring that traces between neurons are correct) would cost $7-21 BILLION. (A human can only manually trace about 1 mm of neuron per hour.) Also, the images would occupy about 500 petabytes of data, and getting those data would require 20 electron microscopes running in parallel for about 5 years, continuously. They estimated the whole project would take about 17 years of work. This is, understandably, insane. But now it seems like there's an actual path toward mapping the full mouse brain in about five years for ~$100M dollars. There have been three major breakthroughs in the last year or so: 1/ Expansion microscopy, first developed in 2015, showed that it's possible to "enlarge" the brain by about 5x using a swellable polymer. But an improved method increases this number to >20x expansion, meaning we can now expand brains and image neurons much more easily using cheap light microscopes, rather than expensive electron ones. 2/ E11 Bio (a nonprofit research org) developed protein barcodes that get delivered into brain tissue; each neuron gets a unique combination of barcodes. These cells are then stained with colorful antibodies, which stick to a matching protein barcode, causing each neuron to light up in a distinct color. This makes tracing neurons so much easier. 3/ Google Research released PATHFINDER this May, an AI-based neuron tracing tool that can proofread about 67,200 cubic microns of brain tissue per hour, with very high accuracy. It works on electron micrographs, but something similar could be presumably be developed for the E11 / colorful tag approach. This is an extremely exciting time for neuroscience. (C. elegans connectome below.)show more

[Graph Convolutional Network] by hand ✍️ Graph Convolutional Networks... (GCNs), introduced by Thomas Kipf and Max Welling in 2017, have emerged as a powerful tool in the analysis and interpretation of data structured as graphs. This exercise demonstrates how GCN works in a simple application: binary classification. -- Goal -- Predict if a node in a graph is X. -- Architecture -- 🟪 Graph Convolutional Network (GCN) 1. GCN1(4,3) 2. GCN2(3,3) 🟦 Fully Connected Network (FCN) 1. Linear1(3,5) 2. ReLU 3. Linear2(5,1) 4. Sigmoid Simplications: • Adjacent matrices are not normalized. • ReLU is applied to messages directly. -- Walkthrough -- [1] Given ↳ A graph with five nodes A, B, C, D, E [2] 🟩 Adjacency Matrix: Neighbors ↳ Add 1 for each edge to neighbors ↳ Repeat in both directions (e.g., A->C, C->A) ↳ Repeat for both GCN layers [3] 🟩 Adjacency Matrix: Self ↳ Add 1's for each self loop ↳ Equivalent to adding the identity matrix ↳ Repeat for both GCN layers [4] 🟪 GCN1: Messages ↳ Multiply the node embeddings 🟨 with weights and biases ↳ Apply ReLU (negatives → 0) ↳ The result is one message per node [5] 🟪 GCN1: Pooling ↳ Multiply the messages with the adjacent matrix ↳ The purpose is the pool messages from each node's neighbors as well as from the node itself. ↳ The result is a new feature per node [6] 🟪 GCN1: Visualize ↳ For node 1, visualize how messages are pooled to obtain a new feature for better understanding ↳ [3,0,1] + [1,0,0] = [4,0,1] [7] 🟪 GCN2: Messages ↳ Multiply the node features with weights and biases ↳ Apply ReLU (negatives → 0) ↳ The result is one message per node [8] 🟪 GCN2: Pooling ↳ Multiply the messages with the adjacent matrix ↳ The result is a new feature per node [9] 🟪 GCN2: Visualize ↳ For node 3, visualize how messages are pooled to obtain a new feature for better understanding ↳ [1,2,4] + [1,3,5] + [0,0,1] = [2,5,10] [10] 🟦 FCN: Linear 1 + ReLU ↳ Multiply node features with weights and biases ↳ Apply ReLU (negatives → 0) ↳ The result is a new feature per node ↳ Unlike in GCN layers, no messages from other nodes are included. [11] 🟦 FCN: Linear 2 ↳ Multiply node features with weights and biases [12] 🟦 FCN: Sigmoid ↳ Apply the Sigmoid activation function ↳ The purpose is to obtain a probability value for each node ↳ One way to calculate Sigmoid by hand ✍️ is to use the approximation below: • >= 3 → 1 • 0 → 0.5 • <= -3 → 0 -- Outputs -- A: 0 (Very unlikely) B: 1 (Very likely) C: 1 (Very likely) D: 1 (Very likely) E: 0.5 (Neutral)show more

Boom! Grok Tasks Make It One Of The Most... POWERFUL Real-Time AI Systems In The World. — My How to Use Grok Tasks With Hidden Tools For Powerful Daily Output. Grok Tasks are customizable AI workflows that integrate a variety of tools to streamline daily activities, from research and analysis to creative planning and problem-solving. I have been using them for quite sometime and because of the vital heartbeat of news and first person data on X, it is the most powerful AI platform available. By combining Tasks with tools like web searches, X platform interactions, code execution, and media viewers, you can build efficient, automated processes. These tasks work by prompting Grok with a clear description of what you want to achieve, and Grok will intelligently call the necessary tools in sequence or parallel to deliver results. Here's a step-by-step guide to creating and using Grok Tasks: Step 1: Define Your Task Start by clearly outlining the daily activity or goal. Consider what inputs you have (e.g., a URL, a query, or an attachment) and what output you need (e.g., a summary, calculation, or visual analysis). Break it down into subtasks to identify tool needs. For example, if your task involves researching current events, note that you'll need search and browsing capabilities. Step 2: Review Available Tools Familiarize yourself with the tools Grok can access. Here's a quick overview: - Code Execution: Run Python code for calculations, data processing, or simulations using libraries like numpy, pandas, or sympy. - Browse Page: Fetch and summarize content from any website URL with custom instructions. - Web Search: Perform general internet searches, returning results with optional operators like site:. - Web Search With Snippets: Get quick, detailed excerpts from search results for fact-checking. - X Keyword Search: Advanced search for X posts using operators like from:, since:, or filter:. - X Semantic Search: Find semantically related X posts based on a query, with filters for dates or users. - X User Search: Locate X users by name or handle. - X Thread Fetch: Retrieve a full X post thread, including context like replies and parents. - View Image: Analyze an image from a URL or conversation ID. - View X Video: Extract frames and subtitles from an X-hosted video. - Search PDF Attachment: Query a PDF file for relevant pages using keyword or regex modes. - Browse PDF Attachment: View specific pages of a PDF with text and screenshots. Select tools that align with your task. Aim for a mix to handle data gathering, processing, and visualization. Step 3: Craft Your Prompt Write a detailed prompt to Grok describing the task. Include: - The overall goal. - Specific steps or subtasks. - References to tools if you want to guide the process (e.g., "Use web_search to find sources, then code_execution to analyze data"). - Any constraints, like dates or limits. Example prompt: "Create a Grok Task for my morning routine: Search recent X posts about tech news using x_keyword_search, fetch a key thread with x_thread_fetch, and summarize with browse_page on linked articles." Step 4: Submit and Interact Send your prompt to Grok. It will process the task by calling tools as needed, often in parallel for efficiency. Review the output and refine with follow-up prompts if required (e.g., "Expand on that using view_image for visuals"). Iterate to fine-tune the workflow for reuse. Step 5: Save and Reuse Once refined, note the prompt as a template for future use. You can adapt it for similar tasks, making Grok Tasks a habitual part of your day. Finding Grok Tasks To discover existing Grok Tasks or inspiration for new ones, use X searches with tools like x_keyword_search or x_semantic_search (e.g., query: "Grok Tasks examples" with mode: Latest). Browse community-shared threads via x_thread_fetch, or web_search for tutorials on xAI features. Prompt Grok directly: "Show me popular Grok Tasks for productivity." 1 of 3show more

[Discrete Fourier Transform] by Hand ✍️ In signal processing,... the Discrete Fourier Transform (DFT) is no doubt the most important method. But the math involved is extremely complex, literally, involving a summation over a complex number term e^(-iwt). I developed this exercise to demonstrate that underneath such complexity, DFT is just a series of matrix multiplications you can calculate by hand. ✍️ Once you see that, it should not surprise you that a deep neural network, which is also a series of matrix multiplications, with activation functions in-between, can learn to perform DFT to process and analyze signals so effectively. How does DFT work? [1] Given ↳ Signals A, B, and C in the 🟧 frequency domain: ◦ A = cos(w) + 2cos(2w) ◦ B = cos(w) + cos(3w) + cos(4w) ◦ C = -cos(2w) + cos(3w) ◦ Each signal is a weighed sum of four cosine waves at frequencies 1w, 2w, 3w, and 4w. ◦ We will apply Inverse DFT to convert the signals to time domain representations, and then demonstrate DFT can convert back to their original frequency domain representations. ↳ Signal X in the 🟩 time domain. X is sampled at 10 time points 1t, 2t, …, 10t: ◦ X = [-2.5, -1.8, 3, -0.7, -1.0, -0.7, 3, -1.8, -2.5, 5] ◦ Suppose X is also a weighted sum of the same four cosine waves, but we don’t already know their weights. We will apply DFT to discover them. [2] 🟧 Frequency Matrix (F) ↳ Write the coefficients of A, B, C as a matrix F. Each signal is a row. Each frequency is a column. ↳ A → [1, 2, 0, 0] ↳ B → [1, 0, 1, 1] ↳ C → [0, 1-, 1, 0] [3] Cosine → Discrete ↳ Sample from the continuous cosine waves at discrete time points 1t, 2t, 3t, to 10t. [4] Cosine Matrix (W) ↳ Write the samples as a matrix, Each frequency is a row. Each time point is a column. [5] Inverse DFT: 🟧 Frequency → 🟩 Time ↳ Multiply the frequency matrix F and the cosine matrix W. ↳ The meaning of this multiplication is to linearly combine the four cosine waves (rows in W) into time-domain signals (rows in T) using the weights specified in F. ↳ The result is matrix T, which are signals A, B, C converted to the time domain. Each signal is a row. Each time point is a column. [6] Transpose ↳ Transpose T, converting each signal’s time domain representation from a row to a column. [7] DFT: 🟩 Time → 🟧 Frequency ↳ Multiply the cosine matrix W with the transpose of matrix T. ↳ The purpose of this multiplication is to take a dot-product between each time-domain signal (columns in the transpose of T) and each cosine wave (rows in W), which has the effect of projecting the signal onto a cosine wave to determine how much they are correlated. Zero means not correlated at all. ↳ The result is an intermediate version of the “recovered” frequency matrix where each column corresponds to a signal and each row corresponds to a frequency. ↳ Compared to the original frequency matrix F, this intermediate matrix has non-zero weights in the correct places, but scaled up by a factor of 5 (n/2, n=10). For example, signal A, originally [1,2,0,0], is recovered at [5,10,0,0]. [8] Scale ↳ Multiply each value by 2/n = 1/5 to scale down the intermediate matrix to match the magnitude of the original frequency matrix F. [9] Transpose ↳ Transpose the recovered frequency matrix back to the same orientation of the original frequency matrix F. ↳ Like magic 🪄, the result is identical to the original F, which means DFT successfully recovered the frequency components of signals A, B, C. [10] Apply DFT to X: 🟩 Time → 🟧 Frequency ↳ Now that we have some confidence in DFT’s ability to recover frequency components, we apply DFT to X’s time-domain representation by multiplying W with X. ↳ The result is the an intermediate matrix. [11] Scale ↳ Similarly, we scale down by a factor of 5 to obtain the recovered frequency components of X (a column). [12] Transpose ↳ Similarly, we transpose the recovered column to row to match the orientation of the frequency matrix. ↳ Using the coefficients [0,0,3,2], we can write the equation of X as 3cos(3w) + 2cos(4w). Notes: I hope this by hand exercise helps you understand the essence of DFT. But there is more technical details, such as: • Sine: The complete DFT math also includes sine waves that follow a similar calculation process. • Phase: Here, we assume all the cosine waves are aligned at the origin, namely, phase is 0. If a phase p is added, for example, cos(w+p), we will need to calculate the sine component and use their ratio to figure out what p is. • Magnitude: If phase is not zero, the magnitude will need to be calculated by combining both cosine and sine terms.show more

Well Islamist groups are not shying away from describing... exactly how they will pull this off. They now openly describe how they plan to do it gradually by seizing control of Muslim-dominated neighborhoods first before they expand outward. It starts in concentrated areas like Tower Hamlets and Bradford. Sharia councils already rule on marriages and divorces there, often clashing with British law on women’s rights, while past patrols and “Sharia zone” signs tested informal enforcement. Authorities back off, and parallel norms take hold. Expansion follows through demographics and politics. Higher birth rates and migration grow these enclaves, voting blocs capture local power, and concessions multiply to keep the peace. Norms then spread to nearby areas as the parallel system strengthens. And the endgame of course is national. What begins as local Sharia in small pockets becomes institutional pressure for broader change, as many groups have openly pursued a caliphate for years. So Britain either enforces one secular law for all or watches these zones link up and redefine the country.show more

Math and the Mind. Dodecahedron. By Andrea Belloni, Water... flowing | anbello.eth | anbello.tez, Used with permission. In your personal opinion, do dodecahedrons exist, or are they abstract ideal geometries, Platonic solids that exist in some non-human realm? Perhaps they exist in both ways: as precise, abstract mathematical ideals and as real physical objects you can hold, study, or even find in ancient ruins and nature. The abstract, ideal version a convex polyhedron with 12 identical regular pentagonal faces, 20 vertices, and 30 edges, where the same number of faces meet at each vertex (all faces congruent, all angles equal, etc.). Platonists argue these forms inhabit an abstract realm independent of human minds or the physical universe. Others see them as human inventions or patterns inherent to logic and reality itself. [Humans have been building dodecahedrons for millennia, and close approximations appear, such as pyritohedrons in pyrite (fool's gold) crystals.]show more