In recent years, two major breakthroughs have revolutionized the field of Large Language Models (LLMs): 1. 2017: The publication of Google’s seminal paper, (https://arxiv.org/abs/1706.03762) by Vaswani et al., which introduced the Transformer architecture – a neural network that fundamentally changed Natural Language Processing (NLP). 2. 2022: The launch of ChatGPT by OpenAI, a transformer-based chatbot…
Brief Historical Introduction to Laplace Transform The Laplace Transform is a cornerstone of mathematical physics and engineering, with deep historical roots dating back to the 18th century. Named after Pierre-Simon Laplace (1749–1827), a French mathematician and astronomer, this transformative tool emerged as part of his work on probability and celestial mechanics. Origins Laplace originally used…
Partial differential equations (PDEs) are classified into different types based on their characteristics, which determine the nature of their solutions and the appropriate solution methods. The three most important PDEs in mathematical physics are: Partial differential equations (PDEs) are used in machine learning (ML)—especially in advanced fields like: Physics-informed neural networks (PINNs): These use PDEs…
Welcome to our mathematically rigorous exploration of tensors. This guide provides precise definitions, theoretical foundations, and worked examples ranging from elementary to advanced levels. All concepts are presented using proper mathematical notation with no reliance on programming languages. The Story of Tensors: From Curved Surfaces to Cosmic Equations Long ago, in the 19th century, a…
At WWDC 2025, Apple unveiled its most ambitious software overhaul in years—Liquid Glass, a cohesive design language spanning all platforms, and Apple Intelligence, a privacy-first AI framework that enhances productivity, creativity, and communication. This year’s keynote wasn’t just about incremental updates; it was a foundational shift toward a more unified, intelligent, and visually refined ecosystem. From…
What Are LLMs? LLMs are machine learning models trained on vast amounts of text data. They use transformer architectures, a neural network design introduced in the paper “Attention Is All You Need”. Transformers excel at capturing context and relationships within data, making them ideal for natural language tasks. 1. Architectural Types of Language Models (Expanded with…
The Drama of the Cubic Equation: Rivalries, Betrayals, and Renaissance Mathematics The 16th century was a time of mathematical duels, secret solutions, and bitter rivalries. The quest to solve the cubic equation wasn’t just about algebra—it was about fame, survival, and revenge. 1. Tartaglia: The Stuttering Genius Who Outsmarted His Rivals Niccolò Tartaglia (1500–1557) was…
What is artificial intelligence? Artificial intelligence is a field of science concerned with building computers and machines that can reason, learn, and act in such a way that would normally require human intelligence or that involves data whose scale exceeds what humans can analyze. AI is a large field that includes many disciplines including computer…
Course Introduction: How Large Language Models (LLMs) Work What You Will Learn: The LLM Processing Pipeline In this course, you will learn how Large Language Models (LLMs) process text step by step, transforming raw input into intelligent predictions. Here’s a visual overview of the journey your words take through an LLM: Module Roadmap You will…
Introduction to MCP and Tiny Agents The Model Context Protocol (MCP) is revolutionizing how we build AI applications by standardizing tool integration for LLMs. In this guide, I’ll show you how to create a fully functional Python agent that leverages MCP to dynamically discover and use tools – all in under 100 lines of code….
