Every sound you hear in a movie, a song or a phone call, has been shaped by audio effects—filters that cut unwanted noise, reverb that simulates the acoustics of a concert hall, pitch shifts that raise or lower a voice. In this post, we’ll build all of these from scratch using Wolfram Language, turning a […]

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.

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.

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:

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 […]

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.”