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We consider a class of linear Vlasov partial differential equations driven by Wiener noise. Different types of stochastic perturbations are treated: additive noise, multiplicative It\^o and Stratonovich noise, and transport noise. We propose to employ splitting integrators for the temporal discretization of these stochastic partial differential equations. These integrators are designed in order to preserve qualitative properties of the exact solutions depending on the stochastic perturbation, such as preservation of norms or positivity of the solutions. We provide numerical experiments in order to illustrate the properties of the proposed integrators and investigate mean-square rates of convergence.

Ensuring the accuracy of responses provided by large language models (LLMs) is crucial, particularly in clinical settings where incorrect information may directly impact patient health. To address this challenge, we construct K-QA, a dataset containing 1,212 patient questions originating from real-world conversations held on K Health (an AI-driven clinical platform). We employ a panel of in-house physicians to answer and manually decompose a subset of K-QA into self-contained statements. Additionally, we formulate two NLI-based evaluation metrics approximating recall and precision: (1) comprehensiveness, measuring the percentage of essential clinical information in the generated answer and (2) hallucination rate, measuring the number of statements from the physician-curated response contradicted by the LLM answer. Finally, we use K-QA along with these metrics to evaluate several state-of-the-art models, as well as the effect of in-context learning and medically-oriented augmented retrieval schemes developed by the authors. Our findings indicate that in-context learning improves the comprehensiveness of the models, and augmented retrieval is effective in reducing hallucinations. We make K-QA available to to the community to spur research into medically accurate NLP applications., Comment: The data and the evaluation script are available at https://github.com/Itaymanes/K-QA. Results and model comparisons can be viewed at https://huggingface.co/spaces/Itaykhealth/K-QA

We present some of the salient aspects of the scientific motivation for high resolution soft X-ray spectroscopy of early-type stars with the Line Emission Mapper X-ray Probe. The major strength of {\it LEM} for hot star physics is its large effective area, aided by the inherent energy resolution of its microcalorimeter that readily achieves resolving powers of 1000 and obviates the need for relatively inefficient dispersive optical elements. This increased sensitivity enables much fainter and more distant high mass stars to be observed than are accessible with present-day facilities, greatly increasing the pool of potential targets. For brighter sources, the sensitivity opens up time domain studies, wherein sufficient signal can be garnered in short order and exposure times, probing source variations on ks timescales. We argue that these capabilities of {\it LEM} will yield breakthroughs in all types of hot star systems, from understanding single OB and WR star winds and how they vary with metallicity, to probing the shocks of colliding wind systems and the magnetically channeled winds of magnetic OB stars. {\it LEM} will also study the energetics of WR star bubbles and feedback from their powerful pre-SN stellar winds., Comment: A Line Emission Mapper White Paper

We present analytical results for the Gini coefficient of economic inequality under the dynamics of a modified Yard-Sale Model of kinetic asset exchange. A variant of the Yard-Sale Model is introduced by modifying the underlying binary transaction of the classical system. It is shown that the Gini coefficient is monotone under the resulting dynamics but the approach to oligarchy, as measured by the Gini index, can be bounded by a first-order differential inequality used in conjunction with the differential Gronwall inequality. This result is in the spirit of entropy -- entropy production inequalities for diffusive PDE. The asymptotics of the modified system, with a redistributive tax, are derived and shown to agree with the original, taxed Yard-Sale Model, which implies the modified system is as suitable for matching real wealth distributions. The Gini -- Gini production inequality is shown to hold for a broader class of models.

The paper establishes the strong convergence rates of a spatio-temporal full discretization of the stochastic wave equation with nonlinear damping in dimension one and two. We discretize the SPDE by applying a spectral Galerkin method in space and a modified implicit exponential Euler scheme in time. The presence of the super-linearly growing damping in the underlying model brings challenges into the error analysis. To address these difficulties, we first achieve upper mean-square error bounds, and then obtain mean-square convergence rates of the considered numerical solution. This is done without requiring the moment bounds of the full approximations. The main result shows that, in dimension one, the scheme admits a convergence rate of order $\tfrac12$ in space and order $1$ in time. In dimension two, the error analysis is more subtle and can be done at the expense of an order reduction due to an infinitesimal factor. Numerical experiments are performed and confirm our theoretical findings., Comment: 30 pages, 2 figures

We introduce a positivity-preserving numerical scheme for a class of nonlinear stochastic heat equations driven by a purely time-dependent Brownian motion. The construction is inspired by a recent preprint by the authors where one-dimensional equations driven by space-time white noise are considered. The objective of this paper is to illustrate the properties of the proposed integrators in a different framework, by numerical experiments and by giving convergence results.