Endüstri Mühendisliği Bölümü Koleksiyonu

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    An equivalence class decomposition of finite metric spaces via Gromov products
    (Elsevier Science Bv, 2017) Bilge, Ayşe Hümeyra; Çelik, Derya; Koçak, Şahin
    Let (X, d) be a finite metric space with elements P-i, i = 1,..., n and with the distance functions d(ij) The Gromov Product of the "triangle" (P-i, P-j, P-k) with vertices P-t, P-j and P-k at the vertex Pi is defined by Delta(ijk) = 1/2(d(ij) + d(ik) - d(jk)). We show that the collection of Gromov products determines the metric. We call a metric space Delta-generic, if the set of all Gromov products at a fixed vertex P-i has a unique smallest element (for i = 1,., n). We consider the function assigning to each vertex P-i the edge {P-i, P-k} of the triangle (P-i, P-j, P-k) realizing the minimal Gromov product at P-i and we call this function the Gromov product structure of the metric space (X, d). We say two Delta-generic metric spaces (X, d) and (X, d') to be Gromov product equivalent, if the corresponding Gromov product structures are the same up to a permutation of X. For n = 3, 4 there is one (Delta-generic) Gromov equivalence class and for n = 5 there are three (Delta-generic) Gromov equivalence classes. For n = 6 we show by computer that there are 26 distinct (Delta-generic) Gromov equivalence classes. (C) 2017 Elsevier B.V. All rights reserved.
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    Generalized Einstein tensor for a Weyl manifold and its applications
    (Springer Heidelberg, 2013) Özdeğer, Abdülkadir
    It is well known that the Einstein tensor G for a Riemannian manifold defined by R (alpha) (beta) = g (beta gamma) R (gamma I +/-) where R (gamma I +/-) and R are respectively the Ricci tensor and the scalar curvature of the manifold plays an important part in Einstein's theory of gravitation as well as in proving some theorems in Riemannian geometry. In this work we first obtain the generalized Einstein tensor for a Weyl manifold. Then after studying some properties of generalized Einstein tensor we prove that the conformal invariance of the generalized Einstein tensor implies the conformal invariance of the curvature tensor of the Weyl manifold and conversely. Moreover we show that such Weyl manifolds admit a one-parameter family of hypersurfaces the orthogonal trajectories of which are geodesics. Finally a necessary and sufficient condition in order that the generalized circles of a Weyl manifold be preserved by a conformal mapping is stated in terms of generalized Einstein tensors at corresponding points.