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In this paper we study geodesic Ptolemy metric spaces X which allow proper and cocompact isometric actions of crystallographic or, more generally, virtual polycyclic groups. We show that X is equivariantly roughly isometric to a Euclidean space.
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We present an approximation method for geodesic circles on a spheroid. Our approximation curve is the intersection of two spheroids whose axes are parallel, and it interpolates four points of the geodesic circle. Our approximation...
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We present an approximation method for geodesic circles on a spheroid. Our approximation curve is the intersection of two spheroids whose axes are parallel, and it interpolates four points of the geodesic circle. Our approximation method has two merits. One is that the approximation curve can be obtained algebraically, and the other is that the approximation error is very small. For example, our approximation of a circle of radius 1000 km on the Earth has error 1-13 cm or less. We analyze the error of our approximation using the Hausdorff distance and confirm it by a geodesic distance computation.
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We look at geodesic foliations on the Lorentzian 2-tori which are lightlike on a proper subset. We prove that they do not exist if the torus is geodesically complete. We describe some properties of their orthogonal foliations and ...
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We look at geodesic foliations on the Lorentzian 2-tori which are lightlike on a proper subset. We prove that they do not exist if the torus is geodesically complete. We describe some properties of their orthogonal foliations and we give several new examples.
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In this paper, the edge version of the geodesic number of a fuzzy graph is introduced and the properties satisfied are identified. A comparison between the vertex and edge version of the geodesic number of fuzzy graphs is obtained...
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In this paper, the edge version of the geodesic number of a fuzzy graph is introduced and the properties satisfied are identified. A comparison between the vertex and edge version of the geodesic number of fuzzy graphs is obtained. The edge geodesic number of fuzzy trees, complete fuzzy graphs, complete bipartite fuzzy graphs and of fuzzy cycles are identified. A necessary and sufficient condition for the existence of an edge geodesic cover in a fuzzy graph is obtained. An application of edge geodesic sets in transportation systems in optimizing the number of traffic inspectors patrolling an urban road network is demonstrated. The fuzziness in the problem helps to identify routes receiving less priority among passengers, elimination of which minimizes the loss suffered by various transport corporations due to lack of collection.
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The paper surveys open problems and questions related to geodesics defined by Riemannian, Finsler, semi-Riemannian and magnetic structures on manifolds. It is an extended report on problem sessions held during the International Wo...
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The paper surveys open problems and questions related to geodesics defined by Riemannian, Finsler, semi-Riemannian and magnetic structures on manifolds. It is an extended report on problem sessions held during the International Workshop on Geodesics in August 2010 at the Chern Institute of Mathematics in Tianjin.
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If Λ is the Cayley graph of a Gromov hyperbolic group, then it is a fundamental fact that quasi-geodesics in Λ are tracked by geodesics. Let (W, S) be a finitely generated Coxeter system and Λ be the Cayley graph of (W, S). For...
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If Λ is the Cayley graph of a Gromov hyperbolic group, then it is a fundamental fact that quasi-geodesics in Λ are tracked by geodesics. Let (W, S) be a finitely generated Coxeter system and Λ be the Cayley graph of (W, S). For general Coxeter groups, not all quasi-geodesic rays in Λ are tracked by geodesics. In this paper, we classify the Λ-quasi-geodesic rays that are tracked by geodesics. As corollaries we show that if W acts geometrically on a CAT(0) space X, then CAT(0) geodesics in X are tracked by Cayley graph geodesics (taking the Cayley graph as equivariantly placed in X) and for any A ? S, the special subgroup is quasi-convex in X. We also show that if g is an element of infinite order for (W, S), then the subgroup is tracked by a Cayley geodesic in Λ (in analogy with the corresponding result for word hyperbolic groups).
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We determine in R-n the form of curves C for which also any image under an (n - 1)-dimensional algebraic torus is a geodesic or an almost geodesic with respect to an affine connections del with constant coefficients and calculate ...
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We determine in R-n the form of curves C for which also any image under an (n - 1)-dimensional algebraic torus is a geodesic or an almost geodesic with respect to an affine connections del with constant coefficients and calculate explicitly the components of del. In this paper we consider the special case for the connection del when any curve from a set of images of C is almost geodesic with respect to del.
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We construct a geodesic net in the plane with four unbalanced (boundary) vertices that has sixteen balanced vertices and does not contain nontrivial subnets. This is the first example of an irreducible geodesic net in the Euclidea...
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We construct a geodesic net in the plane with four unbalanced (boundary) vertices that has sixteen balanced vertices and does not contain nontrivial subnets. This is the first example of an irreducible geodesic net in the Euclidean plane with four boundary vertices that contain cycles of balanced vertices.
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In the present paper we continue to study almost geodesic curves and determine in R-n the form of curves C for which every image under an (n - 1)-dimensional algebraic torus is also an almost geodesic with respect to an affine con...
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In the present paper we continue to study almost geodesic curves and determine in R-n the form of curves C for which every image under an (n - 1)-dimensional algebraic torus is also an almost geodesic with respect to an affine connection del with constant coefficients. We also calculate explicitly the components of del. For the explicit calculation of the form of curves C in the n-dimensional real space Rn that are almost geodesics with respect to an affine connection del, we can suppose that with C all images of C under a real (n - 1)-dimensional algebraic torus are also almost geodesics. This implies that the determination of C becomes an algebraic problem. Here we use E. Beltrami's result that a differentiable curve is a local geodesic with respect to an affine connection del precisely if it is a solution of an abelian differential equation with coefficients that are functions of the components of del. Now we consider the special case for the connection del in which every curve is almost geodesic with respect to del.
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We look at geodesic foliations on the Lorentzian 2-tori which are lightlike on a proper subset. We prove that they do not exist if the torus is geodesically complete. We describe some properties of their orthogonal foliations and ...
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We look at geodesic foliations on the Lorentzian 2-tori which are lightlike on a proper subset. We prove that they do not exist if the torus is geodesically complete. We describe some properties of their orthogonal foliations and we give several new examples.
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