This study presents a comprehensive numerical investigation of magnetohydrodynamic (MHD) convection in a conductive fluid subjected to a rotating magnetic field within a rectangular cavity. The model incorporates a convective flow induced by differential heating of opposing vertical walls under adiabatic conditions. The governing equations are derived based on Maxwell’s equations and the incompressible Navier–Stokes equations, with the magnetic forcing term, averaged over time under low magnetic Reynolds number conditions. A high-resolution numerical algorithm is employed to analyze the stability and transition to turbulence as the magnetic Taylor number (Ta) and Rayleigh number (Ra) increase. The results are consistent with prior experimental observations of flow destabilization at critical values of Ta. Furthermore, the study investigates the emergence of large-scale, nonstationary structures in the turbulent regime, quantifying the finite-time blow-up of solutions as a function of Pr, Ra, and Ta. Attractor formation in velocity space is examined to distinguish deterministic non-periodic solutions from fully developed turbulence. By computing over 104 parameter points, phase diagrams are constructed to illustrate regions of flow stability, deterministic chaos, and turbulence. These results offer novel insights into the interplay between electromagnetic forcing and convective instability, with potential applications in metallurgy, electrochemistry, and crystal growth processes. Continue reading at the publisher's website.

Wolfram has developed several features for integrating our technology with LLMs. The major products are chat notebooks, Notebook Assistant, LLM functions, remote MCP server, the MCPServer paclet and AgentOne. With all these offerings, it is confusing to know which one to use. This notebook should help you figure it out. Continue reading at the publisher's website.

A new paper arrived on the ArXiv last week which has generated a lot of attention across social media: "Single-minus gluon tree amplitudes are nonzero." Within the paper, the authors present various single-minus tree-level n-gluon stripped amplitudes before generalizing the result to arbitrary n > 2. In particular, the authors present the n = 3, 4, 5, and 6 stripped amplitudes in Eqs. 29-32: Continue reading at the publisher's website.

When making pancakes, the first one is always tricky. Is the oil in the pan hot enough? Or too hot? If you start too soon, that first pancake is pale and greasy instead of golden brown and toasty. Wait too long and the temperature may reach the oil’s smoke point, leaving you with a burnt […] Continue reading at the publisher's website.

LLMs don’t—and can’t—do everything. What they do is very impressive—and useful. It’s broad. And in many ways it’s human-like. But it’s not precise. And in the end it’s not about deep computation.So how can we supplement LLM foundation models? We need a foundation tool: a tool that’s broad and general and does what LLMs themselves don’t: provides deep computation and precise knowledge.And, conveniently enough, that’s exactly what I’ve been building for the past 40 years! My goal with Wolfram Language has always been to make everything we can about the world computable. To bring together in a coherent and unified way the algorithms, the methods and the data to do precise computation whenever it’s possible. It’s been a huge undertaking, but I think it’s fair to say it’s been a hugely successful one—that’s fueled countless discoveries and inventions (including my own) across a remarkable range of areas of science, technology and beyond. Continue reading at the publisher's website.

This paper introduces a framework for finite non-cooperative games where each player faces a globally uncertain parameter with no common prior. Every player chooses both a mixed strategy and projects an emergent subjective prior to the uncertain parameters. We define an “Extended Equilibrium” by requiring that no player can improve her expected utility via a […] Continue reading at the publisher's website.

On January 15, 2026, Wikipedia turned 25, and that birthday demonstrates a simple, radical fact: a vast, volunteer-built reference work that stays free to read has become a foundational record of human knowledge and an infrastructure for how the internet answers questions, quietly propping up organic learning, search engines, voice assistants and generative AI. In […] Continue reading at the publisher's website.

This book provides a concise and practical introduction to quantum computing, emphasizing an interactive, hands-on approach. It introduces and explores the essential concepts, principles and foundational quantum algorithms through guided modeling and simulation exercises. Each topic is developed computationally, allowing readers to build both intuition and technical proficiency by directly engaging with the computational mechanics of quantum systems. The approach taken here is computation first, meaning that understanding arises through the act of calculation, echoing David Mermin’s well-known slogan, “shut up and calculate.” Continue reading at the publisher's website.