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Fundamentals to Geometric Modeling and Meshing
Published in Yongjie Jessica Zhang, Geometric Modeling and Mesh Generation from Scanned Images, 2018
Geometric models exist in images implicitly, and image visualization is basically a display of implicit functions or isocontouring. Consider the implicit function f (x, y) = 0. Given an x, in general we cannot directly find the corresponding y. However, given a pair of x and y, we can easily check if they lie on an isocontour or not. In many applications, we have the height function in the form of z = f (x,y). To find all the points with a given height c, we need to solve the implicit equation g(x, y) = f (x, y) – c = 0. As we all know, sometimes we may have the analytical solution of an implicit function, but there is no general solution to convert it to the explicit representation or represent one variable using the others. For a given c, the function g(x, y) describes a curve corresponding to a constant z = c in the equation of f(x,y), which is the contour curve or isocontour.
Generalized desirability functions: a structural and topological analysis of desirability functions
Published in Optimization, 2020
Başak Akteke-Öztürk, Gerhard-Wilhelm Weber, Gülser Köksal
In , the equivalent optimization problem of (5) can be written as follows: where is our vector in with , and is the corresponding height function minimized on the epigraph of : The constraint function of (17) is , where . Now, the corresponding feasible set can be written as: We note that is a linear function and our max-type function f appears now as a set of additional inequalities in , i.e., . Here, instead of being concerned with the optimization problem , we consider the minimization of the height function on the epigraph of .
Numerical investigations of two-dimensional irrotational water waves over finite depth with uniform current
Published in Applicable Analysis, 2021
Now we apply the hodograph transformation of Dubreil-Jacotin [9], i.e. and define the wave height function h as The governing equations then become with h of period in the q-variable satisfying It is noted that when u>c we have The transformation for the case of is illustrated in Figure 1.
Estimates of fundamental solution for Kohn Laplacian in Besov and Triebel-Lizorkin spaces
Published in Applicable Analysis, 2023
Tongtong Qin, Der-Chen Chang, Yongsheng Han, Xinfeng Wu
Denote be a pseudo-convex domain in with the boundary In this case, we may define the ‘height function’ . The domain can be considered as a ‘model’ domain of finite type in . In particular, when k = 1, is a very good approximation of strongly pseudoconvex domains in . (See e.g. Chern-Moser [1], Phong-Stein [2,3].)