This is the Picard rank one analogue to Proposition 5.3 in the main text. The number in each dimension is given in Table 1.

Mixed-precision training has emerged as an indispensable tool for enhancing the efficiency of neural network training in recent years. Concurrently, JAX has grown in popularity as a versatile machine learning toolbox. However, it currently lacks robust support for mixed-precision training. We propose MPX, a mixed-precision training toolbox for JAX that simplifies and accelerates the training of large-scale neural networks while preserving model accuracy. MPX seamlessly integrates with popular toolboxes such as Equinox and Flax, allowing users to convert full-precision pipelines to mixed-precision versions with minimal modifications. By casting both inputs and outputs to half precision, and introducing a dynamic loss-scaling mechanism, MPX alleviates issues like gradient underflow and overflow that commonly arise in half precision computations. Its design inherits critical features from JAX's type-promotion behavior, ensuring that operations take place in the correct precision and allowing for selective enforcement of full precision where needed (e.g., sums, means, or softmax). MPX further provides wrappers for automatic creation and management of mixed-precision gradients and optimizers, enabling straightforward integration into existing JAX training pipelines. MPX's source code, documentation, and usage examples are available at github.com/Data-Science-in-Mechanical-Engineering/mixed_precision_for_JAX .

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This is the Picard rank one analogue to Proposition 5.3 in the main text. The number in each dimension is given in Table 1.

I didn't expect it to change my entire computing setup. But here I am now, using the 300 Beelink SER5 Mini PC as my daily workhorse, powering an ultrawide 1440p 100Hz monitor and smoothly handling any productivity task I've thrown at it. This little computer has been so delightful to use that I've relegated my full-sized desktop tower PC to the basement television, where it's now serving exclusively as a gaming rig. Consider this a lesson on technological overkill. Outside of some specialized use cases, the required compute power for getting things done might be a lot less than you think.

At HP Amplify 2025, the company's big annual conference for showcasing its latest products and services, HP unveiled nearly the entire set of its new PCs for the year. I lost count at some point, but HP claims over 60 new models of laptops and PCs. While technically true, it's a bit fudged--the company counts some variants of the same computer as separate. For example, if the same laptop comes in Intel, AMD, and Snapdragon options, then each one is a distinct "model" even if everything else is the same. That also goes for screen sizes, 2-in-1 variants, and so on.

Berlin's annual IFA electronics show isn't typically a hotspot of PC news, since it follows hot on the heels of the computer-centric Computex in June. Intel used IFA to launch its hotly anticipated "Lunar Lake" Core Series Ultra 2 processors, packing a radical new architecture and seriously improved graphics chops. Not to be outgunned, Qualcomm revealed new 8-core variants of its Snapdragon X PC chips, aiming to bring multi-day battery life to sub- 900 laptops. With so many notebook reveals crammed into such a short time span, however, the deluge felt a bit same-y at times – but not with the delightfully unorthodox systems below, which lean on surprisingly nifty gimmicks to stand out from the crowd. Without further ado, let's get weird.

MSI is hoping that you aren't yet tired of hearing about "AI" in consumer PCs, as the company's lineup of high-end laptops and monitors at the German trade show IFA is jam-packed with artificial alleged-intelligence. Most of it includes the usual bumps up to the latest processors, but a few products caught my eye -- like a wacky desktop PC with a huge built-in screen for its dedicated AI-powered chatbot. The MSI MEG Vision X AI is a desktop PC that's truly thicc. It's filled with the latest parts, of course, built around either an AMD Ryzen 9000 or Intel Arrow Lake CPU, with your choice of Nvidia or AMD graphics card. But none of that is what's giving this desktop PC its bulk.

I didn't expect it to change my entire computing setup. But here I am now, using the 300 Beelink SER5 Mini PC as my daily workhorse, powering an ultrawide 1440p 100Hz monitor and smoothly handling any productivity task I've thrown at it. This little computer has been so delightful to use that I've relegated my full-sized desktop tower PC to the basement television, where it's now serving exclusively as a gaming rig. Consider this a lesson on technological overkill. Outside of some specialized use cases, the required compute power for getting things done might be a lot less than you think.

I didn't expect it to change my entire computing setup. But here I am now, using the $300 Beelink SER5 Mini PC as my daily workhorse, powering an ultrawide 1440p 100Hz monitor and smoothly handling any productivity task I've thrown at it. This little computer has been so delightful to use that I've relegated my full-sized desktop tower PC to the basement television, where it's now serving exclusively as a gaming rig. Consider this a lesson on technological overkill. Outside of some specialized use cases, the required compute power for getting things done might be a lot less than you think.