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Morse 2-functions are higher-dimensional analogs to morse functions. A morse 2-function is a mapping to 2-dimensional disk with its critical points satisfying certain generality conditions. The goal of defining morse 2-functions and trisections is to generalize methods of 3-dimensional manifolds to dimension 4. A morse function on a 3-dimensional manifold leads to a decomposition of that manifold to two manifolds with boundary with common boundary. This decomposition is called Heegaard splitting. This decomposition is unique up to stabilization. Trisections are 4-dimensional analogs of Heegaard splittings and decompose manifold into three parts. The intersection of each pair is a solid genus... 

Invariant manifolds are important objects in the theory of dynamical systems. The stable manifold theorem is a very important theorem about this concept which proves the existence of stable and unstable manifolds in a wide range of dynamical systems. The importance of invariant manifolds encourages us to view their pictures. It helps us to understand their bahavior. For this purpose, at first we need to approximate the invariant manifold we want to visualize. There are several algorithms designed to approximate invariant manifolds. Those algorithms approximate a set of points on an invariant manifold and then provide an approximation of the manifold by the calculated points. Visualizing an... 

With the advent of virtual memory machines and parallel processing, there has been a considerable shift toward computer-aided design and nonlinear analysis of structures. But increasing the size of computer memory or increasing the number of processing units are not the only ways to achieve a satisfactory solution to a large, complex problem. Another useful method is to reduce the size of the problem so that the reduced model is small enough to be solved at an appropriate processing level, and yet the important engineering behavior of the model is preserved in the reduced problem.Reduced models as an alternative, have gained considerable attention throughout the years in order to meet the...