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Renovation of coastal industrial zones with possibility of using engineering geodesic dome structures made of wood and polymer materials
Published in Sergey Sementsov, Alexander Leontyev, Santiago Huerta, Ignacio Menéndez Pidal de Navascués, Reconstruction and Restoration of Architectural Heritage, 2020
Geodesic dome-an architectural structure in the form of a sphere formed by connecting rods in triangles on the cellular principle. The principle of building a dome-shaped frame was developed by the American architect Richard Fuller based on the geometric shape of the Earth in the 1950s. The surface of such a dome consists of steel ribs of different lengths, which when combined form the shape of a dome. The practical application of the building contour geometry proposed by Fuller is based on dividing space by vectors. The basic unit of this division is the tetrahedron. The above separation allows you to achieve optimal space filling and the most complete use of the structural strength of materials.
Karush-Kuhn-Tucker optimality conditions for non-smooth geodesic quasi-convex optimization on Riemannian manifolds
Published in Optimization, 2023
Feeroz Babu, Akram Ali, Ali H. Alkhaldi
Assume that , which is known as hyperbolic 1-space. Since is a complete Riemannian manifold under the Riemannian metric given by Riemannian distance is defined as The normalized geodesic starting from is defined by where is a unit vector. The expression of the inverse exponential map is defined as follows It is also known as . Consider a geodesic convex set in the Riemannian manifold . Define a real-valued function as Note that f is geodesic quasi-convex on K. Since is a minimizer of f in K, then f is essential geodesic quasi-convex. Whereas f is not upper-semicontinuous.
The KKT optimality conditions for optimization problem with interval-valued objective function on Hadamard manifolds
Published in Optimization, 2022
On the other hand, recent interests of a number of researchers are focused on extending some concepts and techniques of nonlinear analysis in Euclidean spaces to Riemannian manifolds (see [15–26]). In general, a manifold is not a linear space. In this setting, the linear space is replaced by a Riemannian manifold and the line segment by a geodesic (see [22,23,25]). As we know, the generalization of optimization problems from Euclidean spaces to Riemannian manifolds has some important advantages, for example, nonconvex minimization problems can be reduced to convex problems and nonmonotone vector fields can be transformed into monotone by choosing an appropriate Riemannian metric (see [17,22,23]). Udriste [25] and Rapsck [23] considered a generalization of convexity called geodesic convexity. Wang et al. [26] established the equivalence between strong (geodesic) convex functions and strong monotonicity of its subdifferential on Riemannian manifolds. A necessary optimality condition for global minimization of a locally Lipschitz function on Riemannian manifold was presented in [19]. Recently, Ruiz-Garzn et al. [24] present the necessary and sufficient optimality conditions in the Riemannian manifolds context for both the scalar and vector optimization problems. However, to the best of our knowledge, there is no paper to study the interval-valued optimization problem on Hadamard manifolds. Therefore, it is an interesting problem to consider the optimization problem with interval-valued function on Hadamard manifolds.
Multifractal analysis of geodesic flows on surfaces without focal points
Published in Dynamical Systems, 2021
We prove the first statement; the second statement can be proved analogously. As above, there exist small such that and that . In particular, the line segment connecting v and along is mapped under the time map of the geodesic flow to the line segment connecting and along . Since the entire process lies completely within from the construction, Lemma 4.1 applies.