A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines $\ell_1$ and $\ell_2$, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In the the bipartite permutation vertex deletion problem we ask for a given $n$-vertex graph, whether we can remove at most $k$ vertices to obtain a bipartite permutation graph. This problem is NP-complete but it does admit an FPT algorithm parameterized by $k$. In this paper we study the kernelization of this problem and show that it admits a polynomial kernel with $O(k^{62})$ vertices.
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In this paper, we examine the relationship between the stability of the dynamical system $x^{\prime}=f(x)$ and the computability of its basins of attraction. We present a computable $C^{\infty}$ system $x^{\prime}=f(x)$ that possesses a computable and stable equilibrium point, yet whose basin of attraction is robustly non-computable in a neighborhood of $f$ in the sense that both the equilibrium point and the non-computability of its associated basin of attraction persist when $f$ is slightly perturbed. This indicates that local stability near a stable equilibrium point alone is insufficient to guarantee the computability of its basin of attraction. However, we also demonstrate that the basins of attraction associated with a structurally stable - globally stable - planar system defined on a compact set are computable. Our findings suggest that the global stability of a system and the compactness of the domain play a pivotal role in determining the computability of its basins of attraction.
This paper studies the complexity of classical modal logics and of their extension with fixed-point operators, using translations to transfer results across logics. In particular, we show several complexity results for multi-agent logics via translations to and from the $\mu$-calculus and modal logic, which allow us to transfer known upper and lower bounds. We also use these translations to introduce a terminating tableau system for the logics we study, based on Kozen's tableau for the $\mu$-calculus, and the one of Fitting and Massacci for modal logic. Finally, we show how to encode the tableaux we introduced into $\mu$-calculus formulas. This encoding provides upper bounds for the satisfiability checking of the few logics we previously did not have algorithms for.
Non-Hermitian topological phases can produce some remarkable properties, compared with their Hermitian counterpart, such as the breakdown of conventional bulk-boundary correspondence and the non-Hermitian topological edge mode. Here, we introduce several algorithms with multi-layer perceptron (MLP), and convolutional neural network (CNN) in the field of deep learning, to predict the winding of eigenvalues non-Hermitian Hamiltonians. Subsequently, we use the smallest module of the periodic circuit as one unit to construct high-dimensional circuit data features. Further, we use the Dense Convolutional Network (DenseNet), a type of convolutional neural network that utilizes dense connections between layers to design a non-Hermitian topolectrical Chern circuit, as the DenseNet algorithm is more suitable for processing high-dimensional data. Our results demonstrate the effectiveness of the deep learning network in capturing the global topological characteristics of a non-Hermitian system based on training data.
By interpreting planar polynomial curves as complex-valued functions of a real parameter, an inner product, norm, metric function, and the notion of orthogonality may be defined for such curves. This approach is applied to the complex pre-image polynomials that generate planar Pythagorean-hodograph (PH) curves, to facilitate the implementation of bounded modifications of them that preserve their PH nature. The problems of bounded modifications under the constraint of fixed curve end points and end tangent directions, and of increasing the arc length of a PH curve by a prescribed amount, are also addressed.
The broad class of multivariate unified skew-normal (SUN) distributions has been recently shown to possess fundamental conjugacy properties. When used as priors for the vector of parameters in general probit, tobit, and multinomial probit models, these distributions yield posteriors that still belong to the SUN family. Although such a core result has led to important advancements in Bayesian inference and computation, its applicability beyond likelihoods associated with fully-observed, discretized, or censored realizations from multivariate Gaussian models remains yet unexplored. This article covers such an important gap by proving that the wider family of multivariate unified skew-elliptical (SUE) distributions, which extends SUNs to more general perturbations of elliptical densities, guarantees conjugacy for broader classes of models, beyond those relying on fully-observed, discretized or censored Gaussians. Such a result leverages the closure under linear combinations, conditioning and marginalization of SUE to prove that such a family is conjugate to the likelihood induced by general multivariate regression models for fully-observed, censored or dichotomized realizations from skew-elliptical distributions. This advancement substantially enlarges the set of models that enable conjugate Bayesian inference to general formulations arising from elliptical and skew-elliptical families, including the multivariate Student's t and skew-t, among others.
We consider the problem of sketching a set valuation function, which is defined as the expectation of a valuation function of independent random item values. We show that for monotone subadditive or submodular valuation functions satisfying a weak homogeneity condition, or certain other conditions, there exist discretized distributions of item values with $O(k\log(k))$ support sizes that yield a sketch valuation function which is a constant-factor approximation, for any value query for a set of items of cardinality less than or equal to $k$. The discretized distributions can be efficiently computed by an algorithm for each item's value distribution separately. Our results hold under conditions that accommodate a wide range of valuation functions arising in applications, such as the value of a team corresponding to the best performance of a team member, constant elasticity of substitution production functions exhibiting diminishing returns used in economics and consumer theory, and others. Sketch valuation functions are particularly valuable for finding approximate solutions to optimization problems such as best set selection and welfare maximization. They enable computationally efficient evaluation of approximate value oracle queries and provide an approximation guarantee for the underlying optimization problem.
Understanding the connection between complex structural features of RNA and biological function is a fundamental challenge in evolutionary studies and in RNA design. However, building datasets of RNA 3D structures and making appropriate modeling choices remains time-consuming and lacks standardization. In this chapter, we describe the use of rnaglib, to train supervised and unsupervised machine learning-based function prediction models on datasets of RNA 3D structures.
{We analyze a general Implicit-Explicit (IMEX) time discretization for the compressible Euler equations of gas dynamics, showing that they are asymptotic-preserving (AP) in the low Mach number limit. The analysis is carried out for a general equation of state (EOS). We consider both a single asymptotic length scale and two length scales. We then show that, when coupling these time discretizations with a Discontinuous Galerkin (DG) space discretization with appropriate fluxes, an all Mach number numerical method is obtained. A number of relevant benchmarks for ideal gases and their non-trivial extension to non-ideal EOS validate the performed analysis.
Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, with solutions over the spatial domain. However, obtaining these solutions are often prohibitively costly, limiting the feasibility of exploring parameters in PDEs. In this paper, we propose an efficient emulator that simultaneously predicts the solutions over the spatial domain, with theoretical justification of its uncertainty quantification. The novelty of the proposed method lies in the incorporation of the mesh node coordinates into the statistical model. In particular, the proposed method segments the mesh nodes into multiple clusters via a Dirichlet process prior and fits Gaussian process models with the same hyperparameters in each of them. Most importantly, by revealing the underlying clustering structures, the proposed method can provide valuable insights into qualitative features of the resulting dynamics that can be used to guide further investigations. Real examples are demonstrated to show that our proposed method has smaller prediction errors than its main competitors, with competitive computation time, and identifies interesting clusters of mesh nodes that possess physical significance, such as satisfying boundary conditions. An R package for the proposed methodology is provided in an open repository.
Mesh-based Graph Neural Networks (GNNs) have recently shown capabilities to simulate complex multiphysics problems with accelerated performance times. However, mesh-based GNNs require a large number of message-passing (MP) steps and suffer from over-smoothing for problems involving very fine mesh. In this work, we develop a multiscale mesh-based GNN framework mimicking a conventional iterative multigrid solver, coupled with adaptive mesh refinement (AMR), to mitigate challenges with conventional mesh-based GNNs. We use the framework to accelerate phase field (PF) fracture problems involving coupled partial differential equations with a near-singular operator due to near-zero modulus inside the crack. We define the initial graph representation using all mesh resolution levels. We perform a series of downsampling steps using Transformer MP GNNs to reach the coarsest graph followed by upsampling steps to reach the original graph. We use skip connectors from the generated embedding during coarsening to prevent over-smoothing. We use Transfer Learning (TL) to significantly reduce the size of training datasets needed to simulate different crack configurations and loading conditions. The trained framework showed accelerated simulation times, while maintaining high accuracy for all cases compared to physics-based PF fracture model. Finally, this work provides a new approach to accelerate a variety of mesh-based engineering multiphysics problems
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