scholarly journals On the Plane Geometry with Generalized Absolute Value Metric

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
A. Bayar ◽  
S. Ekmekçi ◽  
Z. Akça
Keyword(s):  
Metric Spaces ◽  
Plane Geometry ◽  
Trigonometric Functions ◽  
Absolute Value ◽  
Euclidean Metric

Metric spaces are among the most important widely studied topics in mathematics. In recent years, Mathematicians began to investigate using other metrics different from Euclidean metric. These metrics also find their place computer age in addition to their importance in geometry. In this paper, we consider the plane geometry with the generalized absolute value metric and define trigonometric functions and norm and then give a plane tiling example for engineers underlying Schwarz's inequality in this plane.

scholarly journals Improved Performance of Asymptotically Optimal Rapidly Exploring Random Trees

2018 ◽  
Vol 141 (1) ◽  
Author(s):  
Beth Boardman ◽  
Troy Harden ◽  
Sonia Martínez
Keyword(s):  
Degrees Of Freedom ◽  
Metric Spaces ◽  
Dynamic Environments ◽  
Random Trees ◽  
Asymptotically Optimal ◽  
Euclidean Metric ◽  
The Third ◽  
Dubins Vehicle ◽  
Improved Performance ◽  
Dubins Vehicles

Three algorithms that improve the performance of the asymptotically optimal Rapidly exploring Random Tree (RRT*) are presented in this paper. First, we introduce the Goal Tree (GT) algorithm for motion planning in dynamic environments where unexpected obstacles appear sporadically. The GT reuses the previous RRT* by pruning the affected area and then extending the tree by drawing samples from a shadow set. The shadow is the subset of the free configuration space containing all configurations that have geodesics ending at the goal and are in conflict with the new obstacle. Smaller, well defined, sampling regions are considered for Euclidean metric spaces and Dubins' vehicles. Next, the Focused-Refinement (FR) algorithm, which samples with some probability around the first path found by an RRT*, is defined. The third improvement is the Grandparent-Connection (GP) algorithm, which attempts to connect an added vertex directly to its grandparent vertex instead of parent. The GT and GP algorithms are both proven to be asymptotically optimal. Finally, the three algorithms are simulated and compared for a Euclidean metric robot, a Dubins' vehicle, and a seven degrees-of-freedom manipulator.

scholarly journals On lp-Complex Numbers

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 877
Author(s):  
Wolf-Dieter Richter
Keyword(s):  
Symmetry Group ◽  
Complex Number ◽  
Common Property ◽  
Complex Numbers ◽  
Trigonometric Functions ◽  
Absolute Value ◽  
Basic Properties ◽  
The Common

Dispensing with the common property of distributivity and replacing classical trigonometric functions with their l p -counterparts in Euler’s trigonometric representation of complex numbers, classes of l p -complex numbers are introduced and some of their basic properties are proved. The collection of all points that leave the l p -absolute value of each l p -complex number invariant under l p -complex numbers multiplication is shown to be a group of elements that have l p -absolute value one but not the symmetry group.

scholarly journals Union of Euclidean Metric Spaces is Euclidean

2016 ◽  
pp. 1-14 ◽  
Author(s):  
Konstantin Makarychev ◽  
Yury Makarychev
Keyword(s):  
Metric Spaces ◽  
Euclidean Metric

Pointless metric spaces

1990 ◽  
Vol 55 (1) ◽  
pp. 207-219 ◽  
Author(s):  
Giangiacomo Gerla
Keyword(s):  
Lower Bound ◽  
Metric Space ◽  
Upper Bound ◽  
Metric Spaces ◽  
Space Theory ◽  
Research Projects ◽  
Natural Numbers ◽  
Real Numbers ◽  
Euclidean Metric

In the last years several research projects have been motivated by the problem of constructing the usual geometrical spaces by admitting “regions” and “inclusion” between regions as primitives and by defining the points as suitable sequences or classes of regions (for references see [2]).In this paper we propose and examine a system of axioms for the pointless space theory in which “regions”, “inclusion”, “distance” and “diameter” are assumed as primitives and the concept of point is derived. Such a system extends a system proposed by K. Weihrauch and U. Schreiber in [5].In the sequel R and N denote the set of real numbers and the set of natural numbers, and E is a Euclidean metric space. Moreover, if X is a subset of R, then ⋁X is the least upper bound and ⋀X the greatest lower bound of X.

scholarly journals P-Adic metric preserving functions and their analogues

2021 ◽  
Vol 71 (2) ◽  
pp. 391-408
Author(s):  
Robert W. Vallin ◽  
Oleksiy A. Dovgoshey
Keyword(s):  
Rational Numbers ◽  
Real Numbers ◽  
Absolute Value ◽  
The Real ◽  
Euclidean Metric ◽  
Adic Completion ◽  
Compare And Contrast

Abstract The p-adic completion ℚ p of the rational numbers induces a different absolute value |⋅| p than the typical | ⋅| we have on the real numbers. In this paper we compare and contrast functions f : ℝ+ → ℝ+, for which the composition with the p-adic metric dp generated by |⋅| p is still a metric on ℚ p , with the usual metric preserving functions and the functions that preserve the Euclidean metric on ℝ. In particular, it is shown that f ∘ d p is still an ultrametric on ℚ p if and only if there is a function g such that f ∘ d p = g ∘ d p and g ∘ d is still an ultrametric for every ultrametric d. Some general variants of the last statement are also proved.

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