# Latest News on Linear Problem April-21

[1] Linear Problem Kernels for NP-Hard Problems on Planar Graphs

We develop a generic framework for deriving linear-size problem kernels for NP-hard problems on planar graphs. We demonstrate the usefulness of our framework in several concrete case studies, giving new kernelization results for Connected Vertex Cover, Minimum Edge Dominating Set, Maximum Triangle Packing, and Efficient Dominating Set on planar graphs. On the route to these results, we present effective, problem-specific data reduction rules that are useful in any approach attacking the computational intractability of these problems.

[2] Development of NAVDAS-AR: formulation and initial tests of the linear problem

A 4-D implementation of an observation space variational data assimilation system is under development at the Marine Meteorology Division of the Naval Research Laboratory (NRL). The system is an extension of the current US Navy 3-D operational data assimilation system, the NRL Atmospheric Variational Data Assimilation System (NAVDAS). The new system, NAVDAS-AR, where AR stands for accelerated representer, is similar in many respects to the European Centre for Medium-Range Weather Forecasts (ECMWF) 4DVAR system. However, NAVDAS-AR is based on a weak constraint observation space, while the ECMWF system is based on a strong constraint model space. In this paper the formulation of NAVDAS-AR is described in detail and preliminary results with a perfect model assumption and comparisons with the operational NAVDAS are presented.

[3] On the three‐body linear problem with three‐body interaction

A three‐body potential is introduced for which Schrödinger’s equation of the three‐body linear problem with additional harmonic and inverse cube forces is solved exactly.

[4] The Constructivist Approach of Solving Word Problems Involving Algebraic Linear Equations: The Case Study of Mansoman Senior High School, Amansie West District of Ghana

This paper is an action research which involves a sample of forty (40) second year students of Mansoman Senior High School. The study was aimed at using the constructivist approach to enhance students’ competence in solving word problems involving algebraic linear equations. Prior to the study, it was observed that the students were not able to understand and solve word problems under algebraic linear equations. The constructivist approach of teaching and learning was employed as the intervention strategy and was carried in a series of activities. The pre – test and post – test scores obtained by the students were analyzed quantitatively based on the research questions that preceded the study. Comparatively, the results obtained from the pre – test and post – test showed a significant improvement on the students’ ability to translate word problems into algebraic linear equations and solve the equations as well. It was then concluded from the findings that the constructivist approach of teaching and learning employed during the intervention processes improved the students’ academic achievements. The constructivist approach of teaching promoted students participation in the teaching and learning process and environment, and it must be encouraged by all.

[5] Asymptotic Domain Decomposition and a Posteriori Estimates for a Semi Linear Problem

Coupling heterogeneous mathematical models is today commonly used, and effective solution methods for the resulting hybrid problem have recently become available for several systems. Even if in certain circumstances, asymptotic evaluations of the location of the interfaces are available, no strategy are proposed for locating the interfaces in numerical simulations. In this article, a semilinear elliptic problem is considered. By reformulating the problem in a mixed formulation context and by using an a posteriori error estimate, we propose an indicator of the error due to a wrong position of the junction. Minimizing this indicator allows us to determine accurately the location of the junction. By comparing this indicator with a mesh error indicator, this allows to decide if it is better to refine the mesh or to move the interface. Some numerical results are presented showing the efficiency of the proposed indicator.

Reference

[1] Guo, J. and Niedermeier, R., 2007, July. Linear problem kernels for NP-hard problems on planar graphs. In International Colloquium on Automata, Languages, and Programming (pp. 375-386). Springer, Berlin, Heidelberg.

[2] Xu, L., Rosmond, T. and Daley, R., 2005. Development of NAVDAS-AR: Formulation and initial tests of the linear problem. Tellus A: Dynamic Meteorology and Oceanography57(4), pp.546-559.

[3] Wolfes, J., 1974. On the three‐body linear problem with three‐body interaction. Journal of Mathematical Physics15(9), pp.1420-1424.

[4] Andam, E.A., Okpoti, C.A., Obeng–Denteh, W. and Atteh, E., 2015. The constructivist approach of solving word problems involving algebraic linear equations: The case study of Mansoman Senior High School, Amansie West District of Ghana. Advances in Research, pp.1-12.

[5] Pousin, J. and Slimani, K., 2012. Asymptotic Domain Decomposition and a Posteriori Estimates for a Semi Linear Problem. Journal of Advances in Mathematics and Computer Science, pp.226-241.