Exploring Mathematical Concepts

Students of AI: Stop Getting Lost in Equations. I've been there—drowning in formulas, missing the deeper intuition. Then, I discovered a better way: focusing on the story behind the math. Let's walk through this with one of the hardest concepts I learned last year- The Natural Gradient. Typically textbooks give you the formula. Then they might walk you through the symbols and get deep into the derivations. Unless you love this stuff, there's a good chance that this is where you mentally check out and start scrolling through your phone (for me it was the football highlights on IG). Here's how I would suggest learning it instead. First we ask what’s Really Happening- 1. Standard gradient descent blindly follows steepest slopes. 2. The Natural Gradient introduces the Fisher Information Matrix (FIM), incorporating the geometry of your solution space. 3. This is interesting. Why Does the Fisher Information Matrix Matter? Looking more into this we see- High Fisher Value = Big fluctuations, tread lightly. Low Fisher Value = Smooth terrain, move boldly. Once we understand the function, the two bog blocks of math functions make a lot more sense- 1. Squared log-likelihood emphasizes magnitude regardless of direction—capturing change clearly. 2. Expectation (E[…]) filters out noise, spotlighting reliable, stable patterns. When I stopped fixating on derivations and asked “WHY this approach?”, the pieces fell into place. Don’t just memorize math. Understand the narrative behind it. In fact I'd go far enough and say that unless you're a researcher (or you LOVE) math, you're much better off ignoring all the math details and simply understanding the story. The math intuition can be built over time, as you come across the ideas repeatedly.

Are you ready to learn faster? People see things often as too complex. Sometimes the easier way is the best way. The Feynman Technique is a powerful learning method developed by physicist Richard Feynman. It's a simple yet effective way to deepen your understanding of complex concepts. Here's how it works: First, choose a concept you want to learn. Then, explain it as simply as possible, as if you were teaching it to a child. This process helps you identify gaps in your understanding. Next, review your explanation and simplify it further. Break down complex ideas into smaller, more manageable parts. Then, teach the concept again, refining your explanation as needed. Repeat this process until you can explain the concept confidently and clearly. The Feynman Technique helps solidify your understanding by forcing you to confront any areas of confusion and fill in the gaps in your knowledge. #personaldevelopment #humanresources #growth #people

𝗪𝗵𝘆 𝗟𝗶𝗻𝗲𝗮𝗿 𝗔𝗹𝗴𝗲𝗯𝗿𝗮 & 𝗖𝗮𝗹𝗰𝘂𝗹𝘂𝘀 𝗔𝗿𝗲 𝗡𝗼𝗻-𝗡𝗲𝗴𝗼𝘁𝗶𝗮𝗯𝗹𝗲 𝗶𝗻 𝗗𝗮𝘁𝗮 𝗦𝗰𝗶𝗲𝗻𝗰𝗲? It’s tempting to think that Python libraries (Pandas, Scikit-learn, TensorFlow) are enough… but if you want to go beyond “plug-and-play,” you need math. 𝟭. 𝗟𝗶𝗻𝗲𝗮𝗿 𝗔𝗹𝗴𝗲𝗯𝗿𝗮 𝗪𝗵𝘆 𝗶𝘁’𝘀 𝗿𝗲𝗾𝘂𝗶𝗿𝗲𝗱:  • 𝗗𝗮𝘁𝗮 𝗶𝗻 𝗺𝗮𝗰𝗵𝗶𝗻𝗲 𝗹𝗲𝗮𝗿𝗻𝗶𝗻𝗴 is usually 𝗿𝗲𝗽𝗿𝗲𝘀𝗲𝗻𝘁𝗲𝗱 𝗮𝘀 𝘃𝗲𝗰𝘁𝗼𝗿𝘀 (features of a sample) or 𝗺𝗮𝘁𝗿𝗶𝗰𝗲𝘀 (datasets, images, transformations).  • Operations like 𝗱𝗼𝘁 𝗽𝗿𝗼𝗱𝘂𝗰𝘁𝘀, 𝗺𝗮𝘁𝗿𝗶𝘅 𝗺𝘂𝗹𝘁𝗶𝗽𝗹𝗶𝗰𝗮𝘁𝗶𝗼𝗻𝘀, 𝗲𝗶𝗴𝗲𝗻𝘃𝗮𝗹𝘂𝗲𝘀, 𝗮𝗻𝗱 𝗱𝗲𝗰𝗼𝗺𝗽𝗼𝘀𝗶𝘁𝗶𝗼𝗻𝘀 are the backbone of ML algorithms. 𝗪𝗵𝗲𝗿𝗲 𝘄𝗲 𝘂𝘀𝗲 𝗶𝘁 𝗶𝗻 𝗗𝗮𝘁𝗮 𝗦𝗰𝗶𝗲𝗻𝗰𝗲:  • 𝗠𝗮𝗰𝗵𝗶𝗻𝗲 𝗹𝗲𝗮𝗿𝗻𝗶𝗻𝗴 𝗺𝗼𝗱𝗲𝗹𝘀: Linear regression, logistic regression, SVMs, PCA, neural networks all rely on matrix/vector math.  • 𝗗𝗶𝗺𝗲𝗻𝘀𝗶𝗼𝗻𝗮𝗹𝗶𝘁𝘆 𝗿𝗲𝗱𝘂𝗰𝘁𝗶𝗼𝗻 (PCA, SVD): Helps reduce high-dimensional datasets into smaller representations while keeping important info.  • 𝗗𝗲𝗲𝗽 𝗹𝗲𝗮𝗿𝗻𝗶𝗻𝗴: Forward pass, backpropagation, weight updates are essentially matrix operations. Computer vision & NLP: Images are stored as matrices; embeddings in NLP are vectors. 👉 𝗪𝗶𝘁𝗵𝗼𝘂𝘁 𝗹𝗶𝗻𝗲𝗮𝗿 𝗮𝗹𝗴𝗲𝗯𝗿𝗮, 𝗶𝘁’𝘀 𝗶𝗺𝗽𝗼𝘀𝘀𝗶𝗯𝗹𝗲 𝘁𝗼 𝘂𝗻𝗱𝗲𝗿𝘀𝘁𝗮𝗻𝗱 𝗵𝗼𝘄 𝗺𝗼𝗱𝗲𝗹𝘀 𝗮𝗿𝗲 𝘁𝗿𝗮𝗶𝗻𝗲𝗱 𝗮𝗻𝗱 𝗼𝗽𝘁𝗶𝗺𝗶𝘇𝗲𝗱 𝘂𝗻𝗱𝗲𝗿 𝘁𝗵𝗲 𝗵𝗼𝗼𝗱. 2. 𝗖𝗮𝗹𝗰𝘂𝗹𝘂𝘀 𝗪𝗵𝘆 𝗶𝘁’𝘀 𝗿𝗲𝗾𝘂𝗶𝗿𝗲𝗱:  • Calculus is about rates of change and optimization.  • Most ML algorithms need optimization (finding best parameters to minimize error/loss). 𝗪𝗵𝗲𝗿𝗲 𝘄𝗲 𝘂𝘀𝗲 𝗶𝘁 𝗶𝗻 𝗗𝗮𝘁𝗮 𝗦𝗰𝗶𝗲𝗻𝗰𝗲:  • 𝗢𝗽𝘁𝗶𝗺𝗶𝘇𝗮𝘁𝗶𝗼𝗻 𝗼𝗳 𝗺𝗼𝗱𝗲𝗹𝘀: Gradient Descent (used in almost every ML/DL model) comes from derivatives.  • 𝗕𝗮𝗰𝗸𝗽𝗿𝗼𝗽𝗮𝗴𝗮𝘁𝗶𝗼𝗻 𝗶𝗻 𝗡𝗲𝘂𝗿𝗮𝗹 𝗡𝗲𝘁𝘄𝗼𝗿𝗸𝘀: Uses chain rule of derivatives to update weights.  • 𝗣𝗿𝗼𝗯𝗮𝗯𝗶𝗹𝗶𝘁𝘆 & 𝗦𝘁𝗮𝘁𝗶𝘀𝘁𝗶𝗰𝘀 𝗺𝗼𝗱𝗲𝗹𝘀: Involves continuous probability distributions, which require integration/differentiation. 👉 Calculus allows us to move from “just running code” to understanding why models learn and how to make them better. 👉 Without math, you’re just using models. 👉 With math, you can understand, debug & innovate. 💡 Math isn’t a barrier to AI — it’s the foundation.

🤯 𝗦𝘁𝘂𝗰𝗸 𝗶𝗻 𝘁𝗵𝗲 ‘𝗪𝗵𝘆’ 𝗼𝗳 𝗠𝗮𝘁𝗵𝘀? 𝗟𝗲𝘁’𝘀 𝗖𝗵𝗮𝗻𝗴𝗲 𝗧𝗵𝗮𝘁! 𝗪𝗵𝗲𝗿𝗲 𝗺𝗮𝗻𝘆 𝘀𝘁𝘂𝗱𝗲𝗻𝘁𝘀 𝗮𝗿𝗲: Sitting with a complex maths problem, staring at the paper, feeling frustrated. They’ve memorized formulas, but when it comes to breaking down a tricky question – they freeze. ❄️ 𝗧𝗵𝗲𝗶𝗿 𝗱𝗿𝗲𝗮𝗺 𝗼𝘂𝘁𝗰𝗼𝗺𝗲? They want to approach any problem with confidence. To see complexity and think, “I’ve got this.” To understand every step—not just apply it. 𝗪𝗵𝘆 𝗗𝗼 𝗠𝗼𝘀𝘁 𝗦𝘁𝘂𝗱𝗲𝗻𝘁𝘀 𝗦𝘁𝗿𝘂𝗴𝗴𝗹𝗲? Most students fail because they focus on what to do, not why they’re doing it. They memorize steps but miss the logic. So, when they face an unfamiliar problem, they get lost.  • They don’t question the question.  • They skip the ‘why.’ 𝗧𝗵𝗲 𝗦𝗼𝗹𝘂𝘁𝗶𝗼𝗻: 𝗕𝗿𝗲𝗮𝗸 𝗗𝗼𝘄𝗻 𝗣𝗿𝗼𝗯𝗹𝗲𝗺𝘀 𝘄𝗶𝘁𝗵 ‘𝗪𝗵𝘆’ 𝗧𝗵𝗶𝗻𝗸𝗶𝗻𝗴 🧐 Teaching students to ask “why” at each step transforms their understanding. Here’s how it works: 1️⃣ 𝗦𝘁𝗮𝗿𝘁 𝘄𝗶𝘁𝗵 𝘁𝗵𝗲 𝗣𝗿𝗼𝗯𝗹𝗲𝗺, 𝗡𝗼𝘁 𝘁𝗵𝗲 𝗦𝗼𝗹𝘂𝘁𝗶𝗼𝗻: Instead of rushing to plug in formulas, encourage them to ask: • “What is this problem actually asking?” • “What information do I have?” • “What do I need to find?” 🧠 𝗧𝗶𝗽: Break the question into smaller parts. Each part should be a mini-problem to solve. 2️⃣ 𝗤𝘂𝗲𝘀𝘁𝗶𝗼𝗻 𝗘𝗮𝗰𝗵 𝗦𝘁𝗲𝗽: When they apply a formula or make a calculation, they should ask: • “Why am I doing this step?” • “How does this help me get closer to the solution?” 🔍 𝗘𝘅𝗮𝗺𝗽𝗹𝗲: Solving an equation? • Why do we move variables to one side? • Why do we simplify terms first? 3️⃣ 𝗥𝗲𝗳𝗹𝗲𝗰𝘁 𝗔𝗳𝘁𝗲𝗿 𝗦𝗼𝗹𝘃𝗶𝗻𝗴: Once they reach an answer, teach them to look back and ask: • “Did every step make sense?” • “Why did this method work?” • “Could I explain this to someone else?” This reflection cements learning 𝗧𝗵𝗲 𝗧𝗿𝗮𝗻𝘀𝗳𝗼𝗿𝗺𝗮𝘁𝗶𝗼𝗻 🌟 When students adopt ‘why thinking,’ they gain confidence, reduce mistakes, and develop deeper mastery of concepts. This approach shifts them from memorizing to truly understanding, paving the way to top grades. Let’s help them question the question. The answers will follow! 👍 Like | 💬 Comment | 🔁 Repost | 👤 Follow me, Faisal Naqvi #MathsMastery #GrowthMindset #QEDTuitions #CriticalThinking #ProblemSolving #WhyThinking #ConfidentLearners

Applied Mathematics in Trading Most traders think math ends at moving averages and Fibonacci retracements. That’s surface-level. Real trading edge begins when you treat the market as a system of measurable relationships: • Price is a function of liquidity, volatility, and time. • Every candle is a data point — variance, distribution, skew. • Structure isn’t “random”; it’s a probabilistic map where liquidity pools act as attractors. Institutions don’t gamble. They model. They run probability distributions on order flow, volatility clustering, and expected ranges. That’s why they win over the long run. Applied mathematics isn’t about predicting the future it’s about narrowing the uncertainty so that risk becomes calculable. That’s the only way to stop trading stories and start trading numbers.

Some more personal advice for those learning Mathematics: (1) Not all concepts are equal, so allocate your time accordingly. When learning new material, spend considerable time on the building blocks of the topic—see how to express them in different ways, look at many examples, and visualise them. For Riemannian geometry, you must know what the Riemannian metric tensor is; spend ample time trying to understand it. It is a worthy investment, given every geometric quantity derives from it (geodesics, angles, volumes, connections, curvature tensors, etc.)—these will be far more intuitive and easier to remember when you understand their foundation. Similarly, in analysis and group theory, if you don't fully understand injections and surjections, none of the morphisms (and there are many of them) will be clear or memorable. There should be an asymmetry in the time you spend on different concepts. (2) Step back and (subliminally) ponder. As one delves deeper into abstract mathematics, theorems and techniques become difficult to understand and apply, and forcing clarity in one sitting seldom works. At first passing, go through concepts without working through every detail, then step away. Let your mind coalesce them in the background—some theorems take weeks or months to process, and that's fine. If you continue this over a prolonged period, especially with a visual understanding of the foundations (from point one), you will be able to connect ideas from different areas of mathematics and (possibly) find creative ways to apply them. Just those two pieces of advice for now. The photo is of my desk at Oxford Mathematics (I ran out of space for my other books so I have now begun using my friends' shelves).

🔢 ENGINEERING CALCULATIONS – The Backbone of Process Control & Operations No matter the industry — power, chemicals, utilities, polymer, oil & gas — strong fundamentals in engineering calculations separate reactive operators from confident decision-makers. Recently reviewed a detailed technical module covering core engineering calculations used in field operations. Here are some powerful reminders 👇 📏 1️⃣ Unit Systems & Conversions Matter Field Units vs Metric System: • Pressure → psi, bar, kPa • Density → kg/m³, g/cm³, lb/gal • Volume → m³, liters, barrels • Length → m, ft, inches One wrong conversion factor can distort: ❌ Pressure calculations ❌ Pump performance ❌ Density control ❌ Volume estimation Precision starts with correct units. 🏗️ 2️⃣ Tank & System Volume Calculations Rectangular tanks → V = L × W × H Cylindrical tanks → πD²H / 4 Understanding system volume helps in: ✔️ Chemical dosing ✔️ Batch preparation ✔️ Inventory control ✔️ Level monitoring Volume per meter and per centimeter calculations are extremely useful in real operations. 🚰 3️⃣ Pump Capacity & Circulation Time Pump Output depends on: • Liner diameter • Stroke length • Efficiency • Stroke rate From pump output, we determine: ✔️ Flow rate (m³/min) ✔️ Circulation time ✔️ System turnover time This directly impacts: • Process stability • Flushing efficiency • Chemical mixing • Cleaning cycles ⚖️ 4️⃣ Density Control & Additive Calculations Engineering formulas allow us to: 🔹 Increase density using high SG materials 🔹 Reduce density using dilution 🔹 Predict final volume after mixing 🔹 Perform mass balance calculations This is critical in: • Chemical preparation • Utility systems • Polymer solutions • Slurry handling • Cooling & treatment systems Engineering is applied mathematics in motion. 🌊 5️⃣ Hydrostatic Pressure & Fluid Behavior Hydrostatic Pressure = Density × Height × Conversion Factor This concept applies in: ✔️ Storage tanks ✔️ Columns ✔️ Pipelines ✔️ Vertical vessels ✔️ Closed loop systems Understanding static head prevents: ❌ Pump cavitation ❌ Overpressure ❌ Structural stress 🎯 Final Thought Engineering calculations are not just exam formulas. They are daily decision-making tools in operations. Strong fundamentals reduce: • Operational errors • Chemical wastage • Energy losses • Safety risks Before automation and DCS — there was engineering logic. And it still matters. #Engineering #ProcessEngineering #Operations #FluidMechanics #IndustrialEngineering #Utilities #MassBalance #TechnicalLearning

We often get excited about being 𝘳𝘦𝘢𝘭𝘭𝘺 𝘨𝘰𝘰𝘥 at FEA. But ask any experienced practitioner, and they’ll tell you: the real foundation isn’t the software—it’s the skills we’ve been practicing for years: ↳ Math ↳ Science ↳ Mechanics ↳ Problem Solving ↳ Critical Thinking ↳ Communication No amount of training in FEA tools can overcome deficiencies in these areas. They are our bedrock. Without them, we can’t truly grasp the complexity—or the elegance—of applying FEA to real-world problems. You might be thinking: “𝘉𝘶𝘵 𝘐 𝘥𝘰𝘯’𝘵 𝘯𝘦𝘦𝘥 𝘮𝘢𝘵𝘩 𝘵𝘰 𝘳𝘶𝘯 𝘮𝘺 𝘍𝘌𝘈 𝘴𝘰𝘧𝘵𝘸𝘢𝘳𝘦… 𝘥𝘰 𝘐?” Actually, you do. • When you read a contour plot of stress, you’re using 𝗱𝗲𝗿𝗶𝘃𝗮𝘁𝗶𝘃𝗲𝘀 to judge how fast values change (gradients) near hot spots. • When you compute equivalent forces from stresses, you’re relying on 𝗶𝗻𝘁𝗲𝗴𝗿𝗮𝘁𝗶𝗼𝗻 over a surface. We’ve all had frustrating moments with math. But those same concepts quietly make our engineering decisions easier and sharper. That’s why it’s worth going back to our calculus roots. The 𝗙𝘂𝗻𝗱𝗮𝗺𝗲𝗻𝘁𝗮𝗹 𝗧𝗵𝗲𝗼𝗿𝗲𝗺 𝗼𝗳 𝗖𝗮𝗹𝗰𝘂𝗹𝘂𝘀—linking derivatives and integrals—isn’t just abstract math. It’s one part of our foundation of how we interpret results and build trust in our FEA models. To make this refresher less painful, let’s lean on one of the best teachers out there: Khan Academy’s Sal Khan explains FTC here on YouTube: https://lnkd.in/e4SWiDeY. His clear and approachable style makes even the toughest concepts feel manageable. And here’s a fun fact: Khan Academy recently published a one-page statement in the 𝘕𝘦𝘸 𝘠𝘰𝘳𝘬 𝘛𝘪𝘮𝘦𝘴 about their optimism for AI in education—backed by real case studies from school districts already benefiting from it. Inspiring stuff to contemplate for the future.

Statistics is a crucial tool for interpreting data and making evidence-based decisions, even for those without formal statistical training. Understanding basic statistical concepts allows professionals to analyze information critically and apply quantitative reasoning in their work. This document provides a practical introduction to statistics for non-statisticians, explaining key concepts in an accessible way without requiring advanced mathematical knowledge. The guide covers data types, descriptive statistics, probability fundamentals, and basic inferential techniques, helping users understand how to summarize, interpret, and draw conclusions from data. It introduces sampling methods, hypothesis testing, confidence intervals, and regression analysis, emphasizing practical applications over complex formulas. Special focus is placed on common statistical misconceptions and errors, ensuring users develop a solid foundation for accurate data interpretation. For professionals in business, social sciences, healthcare, and public policy, this document is a valuable resource for applying statistical thinking in real-world scenarios. It highlights how to assess data credibility, avoid misinterpretations, and make informed decisions. Whether analyzing survey results, tracking performance, or evaluating project outcomes, these insights help non-statisticians use statistics effectively for decision-making and problem-solving.

Will you ever be able to use math in real life? The answer is a definite “yes.” I am not talking about using simple arithmetic when counting cash (who uses cash these days anyway). Even the most abstract math concepts can help you understand our world better. Math, after all, is about real life (or maybe real life is about math?). Take, for example, algebraic topology—as abstract a field as they come. How on Earth can you use it? Let’s see. Every day I spend a nontrivial amount of time brushing my two Siberian cats. They are mostly fur, plus a strong commitment to not being “fixed.” No matter how carefully I brush—trying to make the coat look like one smooth, unanimous flow—there is always a cowlick. I can move it around, I can flatten it for a minute, but I can’t make it vanish everywhere at once. This is not a lack of technique. It’s a fundamental law. This particular law is known by a memorable name: the Hairy Ball Theorem. What it says is that if you cover a sphere with hairs lying flat along the surface and try to comb them so the direction varies continuously from point to point, you must fail somewhere. In mathematical terms, any continuous tangent vector field on the sphere has to be zero at at least one point. There will be a “bald spot” (a zero) or a little swirl where the direction can’t be defined smoothly without breaking continuity. The sphere does not permit a perfect global comb. The cool part is that it’s not about the material—fur, grass, magnetic arrows—it’s about the shape. Change the topology and the outcome changes. On a torus (a donut surface), you can comb perfectly: point everything along one of the circular directions, and you’re done. No mandatory cowlicks. The sphere is just built differently, and there is nothing you can do about it. You might think, of course, topology works for cats—the notorious shape-shifters—but it’s not only them. Consider the wind on Earth. At each point, the horizontal wind is a direction and speed: a tangent vector on an (almost) spherical surface. If the wind field changes continuously, the theorem implies that at any instant there must be at least one point with zero horizontal wind. That zero is the “eye” of a cyclone (or anticyclone), so there has to be at least one such swirl somewhere at any given time. So yes, abstract math can teach you something about everyday life—humility. No matter how hard you try, there are things you cannot achieve in life: you cannot change the topology of a sphere, you cannot stop a cyclone, you cannot fully brush a cat. Pictures. My unbrushable cat, Amber (left), and a cyclone over Iceland (right).