Localization of target vertebrae is an essential step in minimally invasive spine surgery, with conventional methods relying on “level counting” – i.e., manual counting of vertebrae under fluoroscopy starting from readily identifiable anatomy (e.g., the sacrum). The approach requires an undesirable level of radiation, time, and is prone to counting errors due to the similar appearance of verteb...

We show that the index of a lightlike geodesic in a conformally standard stationary spacetime (M0 × R, g) is equal to the index of its spatial projection as a geodesic of a Finsler metric F on M0 associated to (M0×R, g). Moreover we obtain the Morse relations of lightlike geodesics connecting a point p to a curve γ(s) = (q0, s) by using Morse theory on the Finsler manifold (M0, F ). To this end...

The purpose of the following is to try to make sense of the stereographic projection in a non-commutative setup. To this end, we consider the open unit ball of a ternary ring of operators, which naturally comes equipped with a non-commutative version of a hyperbolic metric and ask for a manifold onto which the open unit ball can be mapped so that one might think of this situation as providing a...

We prove sharp inequalities for the average number of affine diameters through the points of a convex body K in Rn. These inequalities hold if K is a polytope or of dimension two. An example shows that the proof given in the latter case does not extend to higher dimensions. The example also demonstrates that for n ≥ 3 there exist norms and convex bodies K ⊂ Rn such that the metric projection on...

Introduction. A general question which is of interest is the following. Suppose that ƒ is a mapping of a compact metric continuum X onto a metric space F. Under what conditions is there an embedding of X and Y in E (Euclidean n-space) or H (Hubert space) so that ƒ is topologically equivalent to a projection onto Y defined by some collection of parallel hyperplanes? Theorem 1 below provides an a...

The algorithm for finding the optimal consistent approximation of an inconsistent pairwise comaprisons matrix is based on a logarithmic transformation of a pairwise comparisons matrix into a vector space with the Euclidean metric. Orthogonal basis is introduced in the vector space. The orthogonal projection of the transformed matrix onto the space formed by the images of consistent matrices is ...

The polarization tomography problem consists of recovering a matrix function f from the fundamental matrix of the equation Dη/dt = πγ̇fη known for every geodesic γ of a given Riemannian metric. Here πγ̇ is the orthogonal projection onto the hyperplane γ̇⊥. The problem arises in optical tomography of slightly anisotropic media. The local uniqueness theorem is proved: a C1-small function f can be re...

Various results appear in the literature for deriving existence and uniqueness of fixed points for endofunctors on categories of complete metric spaces. All these results are proved for contracting functors which satisfy some further requirements, depending on the category in question. Following a new kind of approach, based on the notion of η-isometry, we show that the sole hypothesis of contr...

Recently, two retractions (projections) which are different from the metric projection and the sunny nonexpansive retraction in a Banach space were found. In this paper, using nonlinear analytic methods and new retractions, we prove a nonlinear ergodic theorem for positively homogeneous and nonexpansive mappings in a uniformly convex Banach space. The limit points are characterized by using new...

Data visualization is a core approach for understanding data specifics and extracting useful information in a simple and intuitive way. Visual data mining proceeds by projecting multidimensional data onto two-dimensional (2D) or three-dimensional (3D) data, e.g., through mathematical optimization and topology preserved in multidimensional scaling (MDS). However, this projection does not necessa...