Matching Complexes of $3 \times n$ Grid Graphs

The matching complex of a graph $G$ is a simplicial complex whose simplices are matchings in $G$. In the last few years the matching complexes of grid graphs have gained much attention among the topological combinatorists. In 2017, Braun and Hough obtained homological results related to the matching complexes of $2 \times n$ grid graphs. Further in 2019, Matsushita showed that the matching complexes of $2 \times n$ grid graphs are homotopy equivalent to a wedge of spheres. In this article we prove that the matching complexes of $3\times n$ grid graphs are homotopy equivalent to a wedge of spheres. We also give the comprehensive list of the dimensions of spheres appearing in the wedge.

History Matching Complex 3D Systems Using Deep-Learning-Based Surrogate Flow Modeling and CNN-PCA Geological Parameterization

The use of deep-learning-based procedures for geological parameterization and fast surrogate flow modeling may enable the application of rigorous history matching algorithms that were previously considered impractical. In this study we incorporate such methods – specifically a geological parameterization that entails principal component analysis combined with a convolutional neural network (CNN-PCA) and a flow surrogate that uses a recurrent residual-U-Net procedure – into three different history matching procedures. The history matching algorithms considered are rejection sampling (RS), randomized maximum likelihood with mesh adaptive direct search optimization (MADS-RML), and ensemble smoother with multiple data assimilation (ES-MDA). RS is a rigorous sampler used here to provide reference results (though it can become intractable in cases with large amounts of observed data). History matching is performed for a channelized geomodel defined on a grid containing 128,000 cells. The CNN-PCA representation of geological realizations involves 400 parameters, and these are the variables determined through history matching. All flow evaluations (after training) are performed using the recurrent residual-U-Net surrogate model. Two cases, involving different amounts of historical data, are considered. We show that both MADS-RML and ES-MDA provide history matching results in general agreement with those from RS. MADS-RML is more accurate, however, and ES-MDA can display significant error in some quantities. ES-MDA requires many fewer function evaluations than MADS-RML, however, so there is a tradeoff between computational demand and accuracy. The framework developed here could be used to evaluate and tune a range of history matching procedures beyond those considered in this work.

Under Water Wireless PowerTransfer System

The wireless power shiftmodel mammoth potential in the hope for its protection and ease. self-directedsubmarineautomobile is single of the vitalmachinery to realize and build up oceans. In this manuscript, we increase and explore a wireless power transfer bargain based on magnetic resonance recipe to incriminate the autonomous vehicle under-water. We have adopted anexhaustiveprojected way that energy reachbase the vehicle and the coils in the wireless power transportarrangement were designed to enlarge a higher mutual inductance based on the sketch of the automobile in this paper. In estimate, we explore the correspondent circuit model and offer a kind of impedance matching complex to preserve system shield. The coils used in the planned system are 20 turns and radii are 70 mm.

Matching Complexes of Small Grids

The matching complex $M(G)$ of a simple graph $G$ is the simplicial complex consisting of the matchings on $G$. The matching complex $M(G)$ is isomorphic to the independence complex of the line graph $L(G)$. Braun and Hough introduced a family of graphs $\Delta^m_n$, which is a generalization of the line graph of the $(n \times 2)$-grid graph. In this paper, we show that the independence complex of $\Delta^m_n$ is a wedge of spheres. This gives an answer to a problem suggested by Braun and Hough.

Matching and Independence Complexes Related to Small Grids

The topology of the matching complex for the $2\times n$ grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes $\mathrm{Ind}(\Delta_n^m)$ that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain $\mathrm{Ind}(\Delta_n^m)$. Further, we determine the Euler characteristic of $\mathrm{Ind}(\Delta_n^m)$ and prove that several homology groups of $\mathrm{Ind}(\Delta_n^m)$ are non-zero.