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To obtain the highest confidence on the correction of numerical simulation programs for the resolution of Partial Differential Equations (PDEs), one has to formalize the mathematical notions and results that allow to establish the soundness of the approach. The finite element method is one of the popular tools for the numerical resolution of a wide range of PDEs. The purpose of this document is to provide the formal proof community with very detailed pen-and-paper proofs for the construction of the Lagrange finite elements of any degree on simplices in positive dimension.

The aim of this study was to compare vastus lateralis (VL) and rectus femoris (RF) muscles for their nervous and mechanical adaptations during two bouts of downhill walking (DW) with load carriage performed two weeks apart. Moreover, we investigated cardio-metabolic and perceived exertion responses during both DW bouts. Methods Seventeen participants performed two 45-min sessions of loaded DW (30% of body mass; slope: -25 %; speed: 4.5 km$\times$h -1 ) separated by two weeks. Rating of perceived exertion (RPE), cost of walking (C w ), heart rate (HR), and EMG activity of thigh muscles were assessed during the DW. Muscle shear elastic modulus ($\mu$) of RF and VL were assessed before each exercise bout. Maximal voluntary contraction (MVC) torque was assessed before (PRE), immediately after (POST), 24 and 48 h after the two exercise bouts. Results MVC torque decreased from POST (-23.7 $\pm$ 9.2%) to 48 h (-19.2 $\pm$ 11.9%) after the first exercise (Ex1), whereas it was significantly reduced only at POST (-14.6 $\pm$ 11.0%) after the second exercise (Ex2) (p < 0.001). RPE (Ex1: 12.3 $\pm$ 1.9; Ex2: 10.8 $\pm$ 2.0), HR (Ex1: 156 $\pm$ 23 bpm; Ex2: 145 $\pm$ 25 bpm), C w (Ex1: 4.5 $\pm$ 0.9 J$\times$m -1 $\times$kg -1 ; Ex2: 4.1 $\pm$ 0.7 J$\times$m -1 $\times$kg -1 ) and RF EMG activity (Ex1: 0.071 $\pm$ 0.028 mV; Ex2: 0.041 $\pm$ 0.014 mV) were significantly decreased during Ex2 compared to Ex1 (p < 0.01). RF $\mu$ was significantly greater in Ex2 (0.44 $\pm$ 0.18) compared to Ex1 (0.56 $\pm$ 0.27; p < 0.001). Conclusions The RF muscle displayed specific mechanical and nervous adaptations to repeated DW bouts as compared to VL. Moreover, the muscle adaptations conferred by the first bout of DW could have induced greater exercise efficiency, inducing lesser perceived exertion and cardio-metabolic demand when the same exercise was repeated two weeks later., Comment: Medicine and Science in Sports and Exercise, 2023

We propose a reservoir design, composed of fixed dissipation operators acting each on few local subsystems, to stabilize an approximate GHZ state on n qubits. The main idea is to work out how a previously proposed sequence of two stabilization steps can be applied instead in appropriate (probabilistic) superposition. We examine alternatives to synchronize the superposition using local couplings only, thanks to a chain of "clock" ancillas or to additional levels on the data subsystems. The practical value of these alternatives depends on experimental constraints. They all feature a design tradeoff between approximate stabilization fidelity and protection against perturbations. These proposals illustrate how simple autonomous automata can be implemented in quantum reservoir engineering to replace sequential state preparation procedures. Encoding automaton actions via additional data levels only, appears particularly efficient in this context. Our analysis method, reducing the Lindblad master equation to a Markov chain on virtual output signals, may be of independent interest.

This paper reviews the state-of-the-art of language models architectures and strategies for "complex" question-answering (QA, CQA, CPS) with a focus on hybridization. Large Language Models (LLM) are good at leveraging public data on standard problems but once you want to tackle more specific complex questions or problems (e.g. How does the concept of personal freedom vary between different cultures ? What is the best mix of power generation methods to reduce climate change ?) you may need specific architecture, knowledge, skills, methods, sensitive data protection, explainability, human approval and versatile feedback... Recent projects like ChatGPT and GALACTICA have allowed non-specialists to grasp the great potential as well as the equally strong limitations of LLM in complex QA. In this paper, we start by reviewing required skills and evaluation techniques. We integrate findings from the robust community edited research papers BIG, BLOOM and HELM which open source, benchmark and analyze limits and challenges of LLM in terms of tasks complexity and strict evaluation on accuracy (e.g. fairness, robustness, toxicity, ...) as a baseline. We discuss some challenges associated with complex QA, including domain adaptation, decomposition and efficient multi-step QA, long form and non-factoid QA, safety and multi-sensitivity data protection, multimodal search, hallucinations, explainability and truthfulness, temporal reasoning. We analyze current solutions and promising research trends, using elements such as: hybrid LLM architectural patterns, training and prompting strategies, active human reinforcement learning supervised with AI, neuro-symbolic and structured knowledge grounding, program synthesis, iterated decomposition and others.

Lebesgue integration is a well-known mathematical tool, used for instance in probability theory, real analysis, and numerical mathematics. Thus its formalization in a proof assistant is to be designed to fit different goals and projects. Once Lebesgue integral is formally defined and the first lemmas are proved, the question of the convenience of the formalization naturally arises. To check it, a useful extension is the Tonelli theorem, stating that the (double) integral of a nonnegative measurable function of two variables can be computed by iterated integrals, and allowing to switch the order of integration. Therefore, we need to define and prove results on product spaces, hoping that they can easily derive from the existing ones on a single space. This article describes the formal definition and proof in Coq of product $\sigma$-algebras, product measures and their uniqueness, the construction of iterated integrals, up to the Tonelli theorem. We also advertise the \emph{Lebesgue induction principle} provided by an inductive type for {\nonnegative} measurable functions.