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Linear complementary pairs (LCPs) of codes have been studied since they were introduced in the context of discussing mitigation measures against possible hardware attacks to integrated circuits. Since the security parameters for LCPs of codes are defined from the (Hamming) distance and the dual distance of the codes in the pair, and the additional algebraic structure of skew constacyclic codes provides tools for studying the the dual and the distance of a code, we study the properties of LCPs of skew constacyclic codes. As a result, we give a characterization for those pairs, as well as multiple results that lead to constructing pairs with designed security parameters. We extend skew BCH codes to a constacyclic context and show that an LCP of codes can be immediately constructed from a skew BCH constacyclic code. Additionally, we describe a Hamming weight-preserving automorphism group in the set of skew constacyclic codes, which can be used for constructing LCPs of codes.

The impact of outliers and anomalies on model estimation and data processing is of paramount importance, as evidenced by the extensive body of research spanning various fields over several decades: thousands of research papers have been published on the subject. As a consequence, numerous reviews, surveys, and textbooks have sought to summarize the existing literature, encompassing a wide range of methods from both the statistical and data mining communities. While these endeavors to organize and summarize the research are invaluable, they face inherent challenges due to the pervasive nature of outliers and anomalies in all data-intensive applications, irrespective of the specific application field or scientific discipline. As a result, the resulting collection of papers remains voluminous and somewhat heterogeneous. To address the need for knowledge organization in this domain, this paper implements the first systematic meta-survey of general surveys and reviews on outlier and anomaly detection. Employing a classical systematic survey approach, the study collects nearly 500 papers using two specialized scientific search engines. From this comprehensive collection, a subset of 56 papers that claim to be general surveys on outlier detection is selected using a snowball search technique to enhance field coverage. A meticulous quality assessment phase further refines the selection to a subset of 25 high-quality general surveys. Using this curated collection, the paper investigates the evolution of the outlier detection field over a 20-year period, revealing emerging themes and methods. Furthermore, an analysis of the surveys sheds light on the survey writing practices adopted by scholars from different communities who have contributed to this field. Finally, the paper delves into several topics where consensus has emerged from the literature. These include taxonomies of outlier types, challenges posed by high-dimensional data, the importance of anomaly scores, the impact of learning conditions, difficulties in benchmarking, and the significance of neural networks. Non-consensual aspects are also discussed, particularly the distinction between local and global outliers and the challenges in organizing detection methods into meaningful taxonomies.

Motivated by the need for the rigorous analysis of the numerical stability of variational least-squares kernel-based methods for solving second-order elliptic partial differential equations, we provide previously lacking stability inequalities. This fills a significant theoretical gap in the previous work [Comput. Math. Appl. 103 (2021) 1-11], which provided error estimates based on a conjecture on the stability. With the stability estimate now rigorously proven, we complete the theoretical foundations and compare the convergence behavior to the proven rates. Furthermore, we establish another stability inequality involving weighted-discrete norms, and provide a theoretical proof demonstrating that the exact quadrature weights are not necessary for the weighted least-squares kernel-based collocation method to converge. Our novel theoretical insights are validated by numerical examples, which showcase the relative efficiency and accuracy of these methods on data sets with large mesh ratios. The results confirm our theoretical predictions regarding the performance of variational least-squares kernel-based method, least-squares kernel-based collocation method, and our new weighted least-squares kernel-based collocation method. Most importantly, our results demonstrate that all methods converge at the same rate, validating the convergence theory of weighted least-squares in our proven theories.

Model reduction is the construction of simple yet predictive descriptions of the dynamics of many-body systems in terms of a few relevant variables. A prerequisite to model reduction is the identification of these relevant variables, a task for which no general method exists. Here, we develop a systematic approach based on the information bottleneck to identify the relevant variables, defined as those most predictive of the future. We elucidate analytically the relation between these relevant variables and the eigenfunctions of the transfer operator describing the dynamics. Further, we show that in the limit of high compression, the relevant variables are directly determined by the slowest-decaying eigenfunctions. Our information-based approach indicates when to optimally stop increasing the complexity of the reduced model. Further, it provides a firm foundation to construct interpretable deep learning tools that perform model reduction. We illustrate how these tools work on benchmark dynamical systems and deploy them on uncurated datasets, such as satellite movies of atmospheric flows downloaded directly from YouTube.

Singularly perturbed boundary value problems pose a significant challenge for their numerical approximations because of the presence of sharp boundary layers. These sharp boundary layers are responsible for the stiffness of solutions, which leads to large computational errors, if not properly handled. It is well-known that the classical numerical methods as well as the Physics-Informed Neural Networks (PINNs) require some special treatments near the boundary, e.g., using extensive mesh refinements or finer collocation points, in order to obtain an accurate approximate solution especially inside of the stiff boundary layer. In this article, we modify the PINNs and construct our new semi-analytic SL-PINNs suitable for singularly perturbed boundary value problems. Performing the boundary layer analysis, we first find the corrector functions describing the singular behavior of the stiff solutions inside boundary layers. Then we obtain the SL-PINN approximations of the singularly perturbed problems by embedding the explicit correctors in the structure of PINNs or by training the correctors together with the PINN approximations. Our numerical experiments confirm that our new SL-PINN methods produce stable and accurate approximations for stiff solutions.

In this work we consider the two dimensional instationary Navier-Stokes equations with homogeneous Dirichlet/no-slip boundary conditions. We show error estimates for the fully discrete problem, where a discontinuous Galerkin method in time and inf-sup stable finite elements in space are used. Recently, best approximation type error estimates for the Stokes problem in the $L^\infty(I;L^2(\Omega))$, $L^2(I;H^1(\Omega))$ and $L^2(I;L^2(\Omega))$ norms have been shown. The main result of the present work extends the error estimate in the $L^\infty(I;L^2(\Omega))$ norm to the Navier-Stokes equations, by pursuing an error splitting approach and an appropriate duality argument. In order to discuss the stability of solutions to the discrete primal and dual equations, a specially tailored discrete Gronwall lemma is presented. The techniques developed towards showing the $L^\infty(I;L^2(\Omega))$ error estimate, also allow us to show best approximation type error estimates in the $L^2(I;H^1(\Omega))$ and $L^2(I;L^2(\Omega))$ norms, which complement this work.

The accurate and efficient evaluation of Newtonian potentials over general 2-D domains is important for the numerical solution of Poisson's equation and volume integral equations. In this paper, we present a simple and efficient high-order algorithm for computing the Newtonian potential over a planar domain discretized by an unstructured mesh. The algorithm is based on the use of Green's third identity for transforming the Newtonian potential into a collection of layer potentials over the boundaries of the mesh elements, which can be easily evaluated by the Helsing-Ojala method. One important component of our algorithm is the use of high-order (up to order 20) bivariate polynomial interpolation in the monomial basis, for which we provide extensive justification. The performance of our algorithm is illustrated through several numerical experiments.

We present a new stability and error analysis of fully discrete approximation schemes for the transient Stokes equation. For the spatial discretization, we consider a wide class of Galerkin finite element methods which includes both inf-sup stable spaces and symmetric pressure stabilized formulations. We extend the results from Burman and Fern\'andez [\textit{SIAM J. Numer. Anal.}, 47 (2009), pp. 409-439] and provide a unified theoretical analysis of backward difference formulae (BDF methods) of order 1 to 6. The main novelty of our approach lies in the use of Dahlquist's G-stability concept together with multiplier techniques introduced by Nevannlina-Odeh and recently by Akrivis et al. [\textit{SIAM J. Numer. Anal.}, 59 (2021), pp. 2449-2472] to derive optimal stability and error estimates for both the velocity and the pressure. When combined with a method dependent Ritz projection for the initial data, unconditional stability can be shown while for arbitrary interpolation, pressure stability is subordinate to the fulfillment of a mild inverse CFL-type condition between space and time discretizations.

We consider the numerical approximation of a continuum model of antiferromagnetic and ferrimagnetic materials. The state of the material is described in terms of two unit-length vector fields, which can be interpreted as the magnetizations averaging the spins of two sublattices. For the static setting, which requires the solution of a constrained energy minimization problem, we introduce a discretization based on first-order finite elements and prove its $\Gamma$-convergence. Then, we propose and analyze two iterative algorithms for the computation of low-energy stationary points. The algorithms are obtained from (semi-)implicit time discretizations of gradient flows of the energy. Finally, we extend the algorithms to the dynamic setting, which consists of a nonlinear system of two Landau-Lifshitz-Gilbert equations solved by the two fields, and we prove unconditional stability and convergence of the finite element approximations toward a weak solution of the problem. Numerical experiments assess the performance of the algorithms and demonstrate their applicability for the simulation of physical processes involving antiferromagnetic and ferrimagnetic materials.