A tropical ball is a defined by the metric over projective torus. In this paper we show several properties of balls torus and also space phylogenetic trees with given set leaf labels. Then discuss its application to K nearest neighbors (KNN) algorithm, supervised learning method used classify high-dimensional vector into categories looking at centered vector, which contains vectors in space.

Definition 1.2 A real vector space X is called a real normed space if there exists a map || · || : X → IR, such that, for all λ ∈ IR and x, y ∈ X, (i) ||x|| = 0⇔ x = 0; (ii) ||λx|| = |λ| ||x||; (iii) ||x + y|| ≤ ||x|| + ||y|| (triangle inequality). The map || · || is called the norm. If || · || only satisfies (ii) and (iii) then it is called a seminorm. Note that X is automatically a metric spa...

According to the formula of translational motion of vector along an infinitesimal closed curve in gravitational space, this article shows that the space and time both are quantized; the called center singularity of Schwarzschild metric does not exist physically, and Einstein’s theory of gravity is compatible with the traditional quantum theory in essence; the quantized gravitational space is ju...

We wish to describe how the hyperbolic geometry of a Riemann surface of genus g y g > 2, leads to a symplectic geometry on Tg, the genus g Teichmüller space, and ~Mg, the moduli space of genus g stable curves. The symplectic structure has three elements: the Weil-Petersson Kahler form, the FenchelNielsen vector fields t+, and the geodesic length functions I*. Weil introduced a Kahler metric for...

We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let X and Y be two smooth vector fields on a two-dimensional manifold M . If X and Y are everywhere linearly independent, then they define a classical Riemannian metric on M (the metric for which they are orthonormal) and they give to M the structure of metric space. If X and Y become l...

We introduce the notion of λ-spaces which is much weaker than cone metric spaces defined by Huang and X. Zhang 2007 . We establish some critical point theorems in the setting of λ-spaces and, in particular, in the setting of complete cone metric spaces. Our results generalize the critical point theorem proposed by Dancs et al. 1983 and the results given by Khanh and Quy 2010 to λ-spaces and con...

We show that the half-line m functions associated with vector-valued discrete Schrödinger operators are elements in Siegel upper half space. introduce a metric on space of to these operators. Then, we action transfer matrices is distance decreasing.

We prove a related fixed point theorem for n mappings which arenot necessarily continuous in n fuzzy metric spaces using an implicit relationone of them is a sequentially compact fuzzy metric space which generalizeresults of Aliouche, et al. [2], Rao et al. [14] and [15].

The paper presents an extension of the geometric quantization procedure to integrable, big-isotropic structures. We obtain a generalization of the cohomology integrality condition, we discuss geometric structures on the total space of the corresponding principal circle bundle and we extend the notion of a polarization. 1 Big-isotropic structures Weak-Hamiltonian functions belong to the framewor...