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Comparison of Contemporary Large Language Models

 This blog presents a concise structural comparison of four prominent large language models: GPT, Claude, Gemini, and xAI. Although all are built on  Transformer -based foundations, they differ markedly in mathematical design, alignment strategy, training dynamics, and multimodal architecture. GPT (OpenAI) follows a scaling-law paradigm using a Transformer backbone enhanced by  s parse Mixture-of-Experts  layers. Claude (Anthropic) preserves the same basic architecture but introduces  Constitutional  AI, an alignment method that incorporates explicit behavioral constraints. Gemini (Google) adopts a unified  multimodal  Transformer that represents text, images, audio, and video within a single token sequence. xAI's Grok retains the Transformer form but is trained on a non-stationary, continuously shifting real-time data stream, giving it  distinct temporal behavior.  As a disclaimer, I am not a practitioner in this area; my background is ...
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Effective PDE Coefficients for Electrical Tissue Property Imaging

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In this blog, I discuss how the effective (or homgenized) coefficient of the elliptic partial differential equation \(\sum_{i=1}^3 \partial_i \big( a_{ij}\,\partial_j u \big) = 0\) in a body arises in the context of electrical tissue property imaging, where \(u\) denotes the electrical potential. In brief, bioimpedance is directly linked to this coefficient, and several companies , such as  InBody  and  Sciospec , are actively developing bioimpedance-based devices. This blog is based on the book *Electromagnetic Tissue Properties MRI* (Imperial College Press) written by Jin Keun Seo, Eung Je Woo, Ulrich Katscher, and Yi Wang.  The mathematical model for electrical tissue property imaging is derived from an appropriate reduction of Maxwell’s equations. In the time-harmonic regime, the electric field \( \mathbf{E} \), current density \( \mathbf{J} \), magnetic field \( \mathbf{H} \), and magnetic flux density \( \mathbf{B} \) satisfy the following relations: ...
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Practical Vector Calculus for the AI Era

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1. Vectors Vectors  are mathematical entities characterized by both magnitude and direction. They are essential for describing physical quantities such as blood flow velocity, forces acting on joints, and features in image analysis for deep learning. A vector in three-dimensional space is written as: \[ \mathbf{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k}, \] where \(\hat{i} = (1,0,0)\), \(\hat{j} = (0,1,0)\), and \(\hat{k} = (0,0,1)\) are the unit vectors along the \(x\)-, \(y\)-, and \(z\)-axes, respectively. The components \(v_x\), \(v_y\), and \(v_z\) represent the magnitude of the vector in the \(x\)-, \(y\)-, and \(z\)-directions.  Vectors can describe the speed and direction of  blood flow  in vessels, facilitating the analysis of hemodynamics. They are also used to model  forces  acting on prosthetic joints or tissues, aiding in the study of biomechanics and prosthetic design. A vector in \(n\)-dimensional space is written as: \[ \mathbf{x} = (x_1, ...
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