We propose a new second-order accurate lattice Boltzmann formulation for linear elastodynamics that is stable for arbitrary combinations of material parameters under a CFL-like condition. The construction of the numerical scheme uses an equivalent first-order hyperbolic system of equations as an intermediate step, for which a vectorial lattice Boltzmann formulation is introduced. The only difference to conventional lattice Boltzmann formulations is the usage of vector-valued populations, so that all computational benefits of the algorithm are preserved. Using the asymptotic expansion technique and the notion of pre-stability structures we further establish second-order consistency as well as analytical stability estimates. Lastly, we introduce a second-order consistent initialization of the populations as well as a boundary formulation for Dirichlet boundary conditions on 2D rectangular domains. All theoretical derivations are numerically verified by convergence studies using manufactured solutions and long-term stability tests.
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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.
Generalized linear models are a popular tool in applied statistics, with their maximum likelihood estimators enjoying asymptotic Gaussianity and efficiency. As all models are wrong, it is desirable to understand these estimators' behaviours under model misspecification. We study semiparametric multilevel generalized linear models, where only the conditional mean of the response is taken to follow a specific parametric form. Pre-existing estimators from mixed effects models and generalized estimating equations require specificaiton of a conditional covariance, which when misspecified can result in inefficient estimates of fixed effects parameters. It is nevertheless often computationally attractive to consider a restricted, finite dimensional class of estimators, as these models naturally imply. We introduce sandwich regression, that selects the estimator of minimal variance within a parametric class of estimators over all distributions in the full semiparametric model. We demonstrate numerically on simulated and real data the attractive improvements our sandwich regression approach enjoys over classical mixed effects models and generalized estimating equations.
Generating series are crucial in enumerative combinatorics, analytic combinatorics, and combinatorics on words. Though it might seem at first view that generating Dirichlet series are less used in these fields than ordinary and exponential generating series, there are many notable papers where they play a fundamental role, as can be seen in particular in the work of Flajolet and several of his co-authors. In this paper, we study Dirichlet series of integers with missing digits or blocks of digits in some integer base $b$; i.e., where the summation ranges over the integers whose expansions form some language strictly included in the set of all words over the alphabet $\{0, 1, \dots, b-1\}$ that do not begin with a $0$. We show how to unify and extend results proved by Nathanson in 2021 and by K\"ohler and Spilker in 2009. En route, we encounter several sequences from Sloane's On-Line Encyclopedia of Integer Sequences, as well as some famous $b$-automatic sequences or $b$-regular sequences. We also consider a specific sequence that is not $b$-regular.
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.
It is well known that Kleisli categories provide a natural language to model side effects. For instance, in the theory of coalgebras, behavioural equivalence coincides with language equivalence (instead of bisimilarity) when nondeterministic automata are modelled as coalgebras living in the Kleisli category of the powerset monad. In this paper, our aim is to establish decorated trace semantics based on language and ready equivalences for conditional transition systems (CTSs) with/without upgrades. To this end, we model CTSs as coalgebras living in the Kleisli category of a relative monad. Our results are twofold. First, we reduce the problem of defining a Kleisli lifting for the machine endofunctor in the context of a relative monad to the classical notion of Kleisli lifting. Second, we provide a recipe based on indexed categories to construct a Kleisli lifting for general endofunctors.
This paper presents a loss-based generalized Bayesian methodology for high-dimensional robust regression with serially correlated errors and predictors. The proposed framework employs a novel scaled pseudo-Huber (SPH) loss function, which smooths the well-known Huber loss, achieving a balance between quadratic and absolute linear loss behaviors. This flexibility enables the framework to accommodate both thin-tailed and heavy-tailed data effectively. The generalized Bayesian approach constructs a working likelihood utilizing the SPH loss that facilitates efficient and stable estimation while providing rigorous estimation uncertainty quantification for all model parameters. Notably, this allows formal statistical inference without requiring ad hoc tuning parameter selection while adaptively addressing a wide range of tail behavior in the errors. By specifying appropriate prior distributions for the regression coefficients -- e.g., ridge priors for small or moderate-dimensional settings and spike-and-slab priors for high-dimensional settings -- the framework ensures principled inference. We establish rigorous theoretical guarantees for the accurate estimation of underlying model parameters and the correct selection of predictor variables under sparsity assumptions for a wide range of data generating setups. Extensive simulation studies demonstrate the superiority of our approach compared to traditional quadratic and absolute linear loss-based Bayesian regression methods, highlighting its flexibility and robustness in high-dimensional and challenging data contexts.
Clustering and outlier detection are two important tasks in data mining. Outliers frequently interfere with clustering algorithms to determine the similarity between objects, resulting in unreliable clustering results. Currently, only a few clustering algorithms (e.g., DBSCAN) have the ability to detect outliers to eliminate interference. For other clustering algorithms, it is tedious to introduce another outlier detection task to eliminate outliers before each clustering process. Obviously, how to equip more clustering algorithms with outlier detection ability is very meaningful. Although a common strategy allows clustering algorithms to detect outliers based on the distance between objects and clusters, it is contradictory to improving the performance of clustering algorithms on the datasets with outliers. In this paper, we propose a novel outlier detection approach, called ODAR, for clustering. ODAR maps outliers and normal objects into two separated clusters by feature transformation. As a result, any clustering algorithm can detect outliers by identifying clusters. Experiments show that ODAR is robust to diverse datasets. Compared with baseline methods, the clustering algorithms achieve the best on 7 out of 10 datasets with the help of ODAR, with at least 5% improvement in accuracy.
Mass lumping techniques are commonly employed in explicit time integration schemes for problems in structural dynamics and both avoid solving costly linear systems with the consistent mass matrix and increase the critical time step. In isogeometric analysis, the critical time step is constrained by so-called "outlier" frequencies, representing the inaccurate high frequency part of the spectrum. Removing or dampening these high frequencies is paramount for fast explicit solution techniques. In this work, we propose mass lumping and outlier removal techniques for nontrivial geometries, including multipatch and trimmed geometries. Our lumping strategies provably do not deteriorate (and often improve) the CFL condition of the original problem and are combined with deflation techniques to remove persistent outlier frequencies. Numerical experiments reveal the advantages of the method, especially for simulations covering large time spans where they may halve the number of iterations with little or no effect on the numerical solution.
Multidimensional quaternion arrays (often referred to as "quaternion tensors") and their decompositions have recently gained increasing attention in various fields such as color and polarimetric imaging or video processing. Despite this growing interest, the theoretical development of quaternion tensors remains limited. This paper introduces a novel multilinear framework for quaternion arrays, which extends the classical tensor analysis to multidimensional quaternion data in a rigorous manner. Specifically, we propose a new definition of quaternion tensors as $\mathbb{H}\mathbb{R}$-multilinear forms, addressing the challenges posed by the non-commutativity of quaternion multiplication. Within this framework, we establish the Tucker decomposition for quaternion tensors and develop a quaternion Canonical Polyadic Decomposition (Q-CPD). We thoroughly investigate the properties of the Q-CPD, including trivial ambiguities, complex equivalent models, and sufficient conditions for uniqueness. Additionally, we present two algorithms for computing the Q-CPD and demonstrate their effectiveness through numerical experiments. Our results provide a solid theoretical foundation for further research on quaternion tensor decompositions and offer new computational tools for practitioners working with quaternion multiway data.
Logistic regression is a classical model for describing the probabilistic dependence of binary responses to multivariate covariates. We consider the predictive performance of the maximum likelihood estimator (MLE) for logistic regression, assessed in terms of logistic risk. We consider two questions: first, that of the existence of the MLE (which occurs when the dataset is not linearly separated), and second that of its accuracy when it exists. These properties depend on both the dimension of covariates and on the signal strength. In the case of Gaussian covariates and a well-specified logistic model, we obtain sharp non-asymptotic guarantees for the existence and excess logistic risk of the MLE. We then generalize these results in two ways: first, to non-Gaussian covariates satisfying a certain two-dimensional margin condition, and second to the general case of statistical learning with a possibly misspecified logistic model. Finally, we consider the case of a Bernoulli design, where the behavior of the MLE is highly sensitive to the parameter direction.
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