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This paper introduces a new numerical scheme for a system that includes evolution equations describing a perfect plasticity model with a time-dependent yield surface. We demonstrate that the solution to the proposed scheme is stable under suitable norms. Moreover, the stability leads to the existence of an exact solution, and we also prove that the solution to the proposed scheme converges strongly to the exact solution under suitable norms.

The main contribution of the present paper is the introduction of a simple yet expressive hybrid-dynamic logic for describing quantum programs. This version of quantum logic can express quantum measurements and unitary evolutions of states in a natural way based on concepts advanced in (hybrid and dynamic) modal logics. We then study Horn clauses in the hybrid-dynamic quantum logic proposed, and develop a series of results that lead to an initial semantics theorem for sets of clauses that are satisfiable. This shows that a significant fragment of hybrid-dynamic quantum logic has good computational properties, and can serve as a basis for defining executable languages for specifying quantum programs. We set the foundations of logic programming in this fragment by proving a variant of Herbrand's theorem, which reduces the semantic entailment of a logic-programming query by a program to the search of a suitable substitution.

We propose a new simple and explicit numerical scheme for time-homogeneous stochastic differential equations. The scheme is based on sampling increments at each time step from a skew-symmetric probability distribution, with the level of skewness determined by the drift and volatility of the underlying process. We show that as the step-size decreases the scheme converges weakly to the diffusion of interest. We then consider the problem of simulating from the limiting distribution of an ergodic diffusion process using the numerical scheme with a fixed step-size. We establish conditions under which the numerical scheme converges to equilibrium at a geometric rate, and quantify the bias between the equilibrium distributions of the scheme and of the true diffusion process. Notably, our results do not require a global Lipschitz assumption on the drift, in contrast to those required for the Euler--Maruyama scheme for long-time simulation at fixed step-sizes. Our weak convergence result relies on an extension of the theory of Milstein \& Tretyakov to stochastic differential equations with non-Lipschitz drift, which could also be of independent interest. We support our theoretical results with numerical simulations.

Based on the mathematical-physical model of pavement mechanics, a multilayer elastic system with interlayer friction conditions is constructed. Given the complex boundary conditions, the corresponding variational inequalities of the partial differential equations are derived, so that the problem can be analyzed under the variational framework. First, the existence and uniqueness of the solution of the variational inequality is proved; then the approximation error of the numerical solution based on the finite element method is analyzed, and when the finite element space satisfies certain approximation conditions, the convergence of the numerical solution is proved; finally, in the trivial finite element space, the convergence order of the numerical solution is derived. The above conclusions provide basic theoretical support for solving the displacement-strain problem of multilayer elastic systems under the framework of variational inequalities.

Out-of-distribution (OOD) generalization in the graph domain is challenging due to complex distribution shifts and a lack of environmental contexts. Recent methods attempt to enhance graph OOD generalization by generating flat environments. However, such flat environments come with inherent limitations to capture more complex data distributions. Considering the DrugOOD dataset, which contains diverse training environments (e.g., scaffold, size, etc.), flat contexts cannot sufficiently address its high heterogeneity. Thus, a new challenge is posed to generate more semantically enriched environments to enhance graph invariant learning for handling distribution shifts. In this paper, we propose a novel approach to generate hierarchical semantic environments for each graph. Firstly, given an input graph, we explicitly extract variant subgraphs from the input graph to generate proxy predictions on local environments. Then, stochastic attention mechanisms are employed to re-extract the subgraphs for regenerating global environments in a hierarchical manner. In addition, we introduce a new learning objective that guides our model to learn the diversity of environments within the same hierarchy while maintaining consistency across different hierarchies. This approach enables our model to consider the relationships between environments and facilitates robust graph invariant learning. Extensive experiments on real-world graph data have demonstrated the effectiveness of our framework. Particularly, in the challenging dataset DrugOOD, our method achieves up to 1.29% and 2.83% improvement over the best baselines on IC50 and EC50 prediction tasks, respectively.

Topology optimization is used to systematically design contact-aided thermo-mechanical regulators, i.e. components whose effective thermal conductivity is tunable by mechanical deformation and contact. The thermo-mechanical interactions are modeled using a fully coupled non-linear thermo-mechanical finite element framework. To obtain the intricate heat transfer response, the components leverage self-contact, which is modeled using a third medium contact method. The effective heat transfer properties of the regulators are tuned by solving a topology optimization problem using a traditional gradient based algorithm. Several designs of thermo-mechanical regulators in the form of switches, diodes and triodes are presented.

Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take any position in given uncertainty sets. Then, the cost function to be minimized is the sum of the distances for the worst positions of the vertices in their uncertainty sets. We propose two types of polynomial-time approximation algorithms. The first one relies on solving a deterministic counterpart of the problem where the uncertain distances are replaced with maximum pairwise distances. We study in details the resulting approximation ratio, which depends on the structure of the feasible subgraphs and whether the metric space is Ptolemaic or not. The second algorithm is a fully-polynomial time approximation scheme for the special case of $s-t$ paths.

Symplectic integrators are widely implemented numerical integrators for Hamiltonian mechanics, which preserve the Hamiltonian structure (symplecticity) of the system. Although the symplectic integrator does not conserve the energy of the system, it is well known that there exists a conserving modified Hamiltonian, called the shadow Hamiltonian. For the Nambu mechanics, which is a kind of generalized Hamiltonian mechanics, we can also construct structure-preserving integrators by the same procedure used to construct the symplectic integrators. In the structure-preserving integrator, however, the existence of shadow Hamiltonians is nontrivial. This is because the Nambu mechanics is driven by multiple Hamiltonians and it is nontrivial whether the time evolution by the integrator can be cast into the Nambu mechanical time evolution driven by multiple shadow Hamiltonians. In this paper we present a general procedure to calculate the shadow Hamiltonians of structure-preserving integrators for Nambu mechanics, and give an example where the shadow Hamiltonians exist. This is the first attempt to determine the concrete forms of the shadow Hamiltonians for a Nambu mechanical system. We show that the fundamental identity, which corresponds to the Jacobi identity in Hamiltonian mechanics, plays an important role in calculating the shadow Hamiltonians using the Baker-Campbell-Hausdorff formula. It turns out that the resulting shadow Hamiltonians have indefinite forms depending on how the fundamental identities are used. This is not a technical artifact, because the exact shadow Hamiltonians obtained independently have the same indefiniteness.

Linear non-Gaussian causal models postulate that each random variable is a linear function of parent variables and non-Gaussian exogenous error terms. We study identification of the linear coefficients when such models contain latent variables. Our focus is on the commonly studied acyclic setting, where each model corresponds to a directed acyclic graph (DAG). For this case, prior literature has demonstrated that connections to overcomplete independent component analysis yield effective criteria to decide parameter identifiability in latent variable models. However, this connection is based on the assumption that the observed variables linearly depend on the latent variables. Departing from this assumption, we treat models that allow for arbitrary non-linear latent confounding. Our main result is a graphical criterion that is necessary and sufficient for deciding the generic identifiability of direct causal effects. Moreover, we provide an algorithmic implementation of the criterion with a run time that is polynomial in the number of observed variables. Finally, we report on estimation heuristics based on the identification result, explore a generalization to models with feedback loops, and provide new results on the identifiability of the causal graph.

Extremiles provide a generalization of quantiles which are not only robust, but also have an intrinsic link with extreme value theory. This paper introduces an extremile regression model tailored for functional covariate spaces. The estimation procedure turns out to be a weighted version of local linear scalar-on-function regression, where now a double kernel approach plays a crucial role. Asymptotic expressions for the bias and variance are established, applicable to both decreasing bandwidth sequences and automatically selected bandwidths. The methodology is then investigated in detail through a simulation study. Furthermore, we highlight the applicability of the model through the analysis of data sourced from the CH2018 Swiss climate scenarios project, offering insights into its ability to serve as a modern tool to quantify climate behaviour.