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.”

Discover how the Raspberry Pi 5 turns into a full STEM lab with free Mathematica. From real-time sensor dashboards to image processing, neural networks and stunning math art, this guide empowers students, teachers and parents to explore, invent and visualize science and engineering like never before.

If we regard interference and entanglement as the most distinctive features of the quantum world, then “which-way” experiments lie at the heart of quantum physics, vividly illustrating the contrast between classical and quantum conceptions of nature. The idea of “which-way” experiments can be traced back to Einstein’s famous proposal of a double-slit experiment with a […]

To quantify the immune response against a rapidly evolving virus, groups routinely measure antibody inhibition against many virus variants. Over time, the variants being studied change, and there is a need for methods that infer missing interactions and distinguish between confident predictions and hallucinations. Here, we develop a matrix completion framework that uses patterns in antibody-virus inhibition to infer the value and confidence of unmeasured interactions. This same approach can combine general datasets—from drug-cell interactions to user movie preferences—that have partially overlapping features.

Comet 3I/ATLAS is the third detected interstellar visitor, identified by its hyperbolic orbit; backward integration places its origin beyond the solar system. It poses no hazard to Earth, passing no closer than about 1.8 au (~170 million miles, ~270 million km). Perihelion occurs around October 30, 2025 at roughly 1.4 au (~130 million miles, ~210 million km), just inside Mars’s orbit. Its size and physical properties are under active study worldwide. It should remain observable to ground-based telescopes through September 2025, become unobservable while near the Sun, and reappear by early December 2025 for renewed observations.

Huge congratulations to John Clarke, Michel H. Devoret and John M. Martinis on the 2025 Nobel Prize in Physics “for the discovery of macroscopic quantum mechanical tunneling and energy quantization in an electric circuit.” Their superconducting Josephson-circuit experiments made quantum effects unmistakably visible at circuit scale, discrete, anharmonic energy levels and coherent tunneling between macroscopically distinct states, laying key groundwork for modern superconducting qubits. In this short computational essay, we’ll walk through compact simulations that reproduce those signatures: a spectroscopy-style level map for the Cooper-pair box/transmon, and time-domain tunneling dynamics with realistic decoherence to mirror the original observations.

The quantum Schur transform is a unitary change of basis from the computational product basis to the so-called Schur basis, a basis labeled by the irreducible representations of the symmetric and unitary groups, based on the Schur-Weyl duality in the group representation theory. The latter basis is called the Schur basis. The quantum Schur transform […]