MediMath Science

This blog presents a concise structural comparison of five prominent large language models: GPT, Claude, Gemini, LLaMA, 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.   LLaMA ( Meta AI ) emphasizes dense (non-MoE) Transformer scaling and data efficiency, prioritizing compute-optimal training and architectural simplicity. xAI's Grok retains the Transformer form but is trained on a non-stationary, con...

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: ...