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An extremely schematic model of the forces acting an a sailing yacht equipped with a system of foils is here presented and discussed. The role of the foils is to raise the hull from the water in order to reduce the total resistance and then increase the speed. Some CFD simulations are providing the total resistance of the bare hull at some values of speed and displacement, as well as the characteristics (drag and lift coefficients) of the 2D foil sections used for the appendages. A parametric study has been performed for the characterization of a foil of finite dimensions. The equilibrium of the vertical forces and longitudinal moments, as well as a reduced displacement, is obtained by controlling the pitch angle of the foils. The value of the total resistance of the yacht with foils is then compared with the case without foils, evidencing the speed regime where an advantage is obtained, if any.

We propose a generalized free energy potential for active systems, including both stochastic master equations and deterministic nonlinear chemical reaction networks. Our generalized free energy is defined variationally as the "most irreversible" state observable. This variational principle is motivated from several perspectives, including large deviations theory, thermodynamic uncertainty relations, Onsager theory, and information-theoretic optimal transport. In passive systems, the most irreversible observable is the usual free energy potential and its irreversibility is the entropy production rate (EPR). In active systems, the most irreversible observable is the generalized free energy and its irreversibility gives the excess EPR, the nonstationary contribution to dissipation. The remaining "housekeeping" EPR is a genuine nonequilibrium contribution that quantifies the nonconservative nature of the forces. We derive far-from-equilibrium thermodynamic speed limits for excess EPR, applicable to both linear and nonlinear systems. Our approach overcomes several limitations of the steady-state potential and the Hatano-Sasa (adiabatic/nonadiabatic) decomposition, as we demonstrate in several examples.

We consider the vorticity formulation of the Euler equations describing the flow of a two-dimensional incompressible ideal fluid on the sphere. Zeitlin's model provides a finite-dimensional approximation of the vorticity formulation that preserves the underlying geometric structure: it consists of an isospectral Lie--Poisson flow on the Lie algebra of skew-Hermitian matrices. We propose an approximation of Zeitlin's model based on a time-dependent low-rank factorization of the vorticity matrix and evolve a basis of eigenvectors according to the Euler equations. In particular, we show that the approximate flow remains isospectral and Lie--Poisson and that the error in the solution, in the approximation of the Hamiltonian and of the Casimir functions only depends on the approximation of the vorticity matrix at the initial time. The computational complexity of solving the approximate model is shown to scale quadratically with the order of the vorticity matrix and linearly if a further approximation of the stream function is introduced.

Weighting with the inverse probability of censoring is an approach to deal with censoring in regression analyses where the outcome may be missing due to right-censoring. In this paper, three separate approaches involving this idea in a setting where the Kaplan--Meier estimator is used for estimating the censoring probability are compared. In more detail, the three approaches involve weighted regression, regression with a weighted outcome, and regression of a jack-knife pseudo-observation based on a weighted estimator. Expressions of the asymptotic variances are given in each case and the expressions are compared to each other and to the uncensored case. In terms of low asymptotic variance, a clear winner cannot be found. Which approach will have the lowest asymptotic variance depends on the censoring distribution. Expressions of the limit of the standard sandwich variance estimator in the three cases are also provided, revealing an overestimation under the implied assumptions.

Identifying predictive covariates, which forecast individual treatment effectiveness, is crucial for decision-making across different disciplines such as personalized medicine. These covariates, referred to as biomarkers, are extracted from pre-treatment data, often within randomized controlled trials, and should be distinguished from prognostic biomarkers, which are independent of treatment assignment. Our study focuses on discovering predictive imaging biomarkers, specific image features, by leveraging pre-treatment images to uncover new causal relationships. Unlike labor-intensive approaches relying on handcrafted features prone to bias, we present a novel task of directly learning predictive features from images. We propose an evaluation protocol to assess a model's ability to identify predictive imaging biomarkers and differentiate them from purely prognostic ones by employing statistical testing and a comprehensive analysis of image feature attribution. We explore the suitability of deep learning models originally developed for estimating the conditional average treatment effect (CATE) for this task, which have been assessed primarily for their precision of CATE estimation while overlooking the evaluation of imaging biomarker discovery. Our proof-of-concept analysis demonstrates the feasibility and potential of our approach in discovering and validating predictive imaging biomarkers from synthetic outcomes and real-world image datasets. Our code is available at \url{//github.com/MIC-DKFZ/predictive_image_biomarker_analysis}.

This paper leverages various philosophical and ontological frameworks to explore the concept of embodied artificial general intelligence (AGI), its relationship to human consciousness, and the key role of the metaverse in facilitating this relationship. Several theoretical frameworks underpin this exploration, such as embodied cognition, Michael Levin's computational boundary of a "Self," Donald D. Hoffman's Interface Theory of Perception, and Bernardo Kastrup's analytical idealism, which lead to considering our perceived outer reality as a symbolic representation of alternate inner states of being, and where AGI could embody a different form of consciousness with a larger computational boundary. The paper further discusses the developmental stages of AGI, the requirements for the emergence of an embodied AGI, the importance of a calibrated symbolic interface for AGI, and the key role played by the metaverse, decentralized systems, open-source blockchain technology, as well as open-source AI research. It also explores the idea of a feedback loop between AGI and human users in metaverse spaces as a tool for AGI calibration, as well as the role of local homeostasis and decentralized governance as preconditions for achieving a stable embodied AGI. The paper concludes by emphasizing the importance of achieving a certain degree of harmony in human relations and recognizing the interconnectedness of humanity at a global level, as key prerequisites for the emergence of a stable embodied AGI.

The randomized singular value decomposition (SVD) has become a popular approach to computing cheap, yet accurate, low-rank approximations to matrices due to its efficiency and strong theoretical guarantees. Recent work by Boull\'e and Townsend (FoCM, 2023) presents an infinite-dimensional analog of the randomized SVD to approximate Hilbert-Schmidt operators. However, many applications involve computing low-rank approximations to symmetric positive semi-definite matrices. In this setting, it is well-established that the randomized Nystr\"om approximation is usually preferred over the randomized SVD. This paper explores an infinite-dimensional analog of the Nystr\"om approximation to compute low-rank approximations to non-negative self-adjoint trace-class operators. We present an analysis of the method and, along the way, improve the existing infinite-dimensional bounds for the randomized SVD. Our analysis yields bounds on the expected value and tail bounds for the Nystr\"om approximation error in the operator, trace, and Hilbert-Schmidt norms. Numerical experiments on integral operators arising from Gaussian process sampling and Bayesian inverse problems are used to validate the proposed infinite-dimensional Nystr\"om algorithm.

The study of linear codes over a finite field of odd cardinality, derived from determinantal varieties obtained from symmetric matrices of bounded rank, was initiated in a recent paper by the authors. There, one found the minimum distance of the code obtained from evaluating homogeneous linear functions at all symmetric matrices with rank, which is, at most, a given even number. Furthermore, a conjecture for the minimum distance of codes from symmetric matrices with ranks bounded by an odd number was given. In this article, we continue the study of codes from symmetric matrices of bounded rank. A connection between the weights of the codewords of this code and Q-numbers of the association scheme of symmetric matrices is established. Consequently, we get a concrete formula for the weight distribution of these codes. Finally, we determine the minimum distance of the code obtained from evaluating homogeneous linear functions at all symmetric matrices with rank at most a given number, both when this number is odd and when it is even.

We consider the problem of estimating the error when solving a system of differential algebraic equations. Richardson extrapolation is a classical technique that can be used to judge when computational errors are irrelevant and estimate the discretization error. We have simulated molecular dynamics with constraints using the GROMACS library and found that the output is not always amenable to Richardson extrapolation. We derive and illustrate Richardson extrapolation using a variety of numerical experiments. We identify two necessary conditions that are not always satisfied by the GROMACS library.

Traditional neural networks (multi-layer perceptrons) have become an important tool in data science due to their success across a wide range of tasks. However, their performance is sometimes unsatisfactory, and they often require a large number of parameters, primarily due to their reliance on the linear combination structure. Meanwhile, additive regression has been a popular alternative to linear regression in statistics. In this work, we introduce novel deep neural networks that incorporate the idea of additive regression. Our neural networks share architectural similarities with Kolmogorov-Arnold networks but are based on simpler yet flexible activation and basis functions. Additionally, we introduce several hybrid neural networks that combine this architecture with that of traditional neural networks. We derive their universal approximation properties and demonstrate their effectiveness through simulation studies and a real-data application. The numerical results indicate that our neural networks generally achieve better performance than traditional neural networks while using fewer parameters.