Abstract:
Inscribed angles are investigated in taxicab geometry with application to the existence and uniqueness of inscribed and circumscribed taxicab circles of triangles.

Abstract:
The set of trigonometric functions in taxicab geometry is completed and derivatives of all of the taxicab trigonometric functions are explored.

Abstract:
A natural analogue to angles and trigonometry is developed in taxicab geometry. This structure is then analyzed to see which, if any, congruent triangle relations hold. A nice application involving the use of parallax to determine the exact (taxicab) distance to an object is also discussed.

Abstract:
We present and study the Static-Routing-Resiliency problem, motivated by routing on the Internet: Given a graph $G$, a unique destination vertex $d$, and an integer constant $c>0$, does there exist a static and destination-based routing scheme such that the correct delivery of packets from any source $s$ to the destination $d$ is guaranteed so long as (1) no more than $c$ edges fail and (2) there exists a physical path from $s$ to $d$? We embark upon a systematic exploration of this fundamental question in a variety of models (deterministic routing, randomized routing, with packet-duplication, with packet-header-rewriting) and present both positive and negative results that relate the edge-connectivity of a graph, i.e., the minimum number of edges whose deletion partitions $G$, to its resiliency.

Abstract:
In this paper we present geometry of some curves in Taxicab metric. All curves of second order and trifocal ellipse in this metric are presented. Area and perimeter of some curves are also defined.

Abstract:
European singles charts are important part of the music industry responsible for creating popularity of songs. After modeling and exploring dynamics of global album sales in previous papers, we investigate patterns of hit singles popularity according to all data (1966-2015) from weekly charts (polls) in 12 Western European countries. The dynamics of building popularity in various national charts is more than the economy because it depends on spread of information. In our research we have shown how countries may be affected by their neighbourhood and influenced by technological era. We have also computed correlations with geographical and cultural distances between countries in analog, digital and Internet era. We have shown that time delay between the single premiere and the peak of popularity has become shorter under the influence of technology and the popularity of songs depends on geographical distances in analog (1966-1987) and Internet (2004-2015) era. On the other hand, cultural distances between nations have influenced the peaks of popularity, but in the Compact Disc era only (1988-2003). We have also indicated the European countries in line with global trends e.g. The Netherlands, the United Kingdom and outsiders like Italy and Spain.

Abstract:
In this article, we make use of some known method to investigate some properties of the numbers represented as sums of two equal odd powers, i.e., the equation $x^n+y^n=N$ for $n\ge3$. It was originated in developing algorithms to search new taxicab numbers (i.e., naturals that can be represented as a sum of positive cubes in many different ways) and to verify their minimality. We discuss properties of diophantine equations that can be used for our investigations. This techniques is applied to develop an algorithm allowing us to compute new taxicab numbers (i.e., numbers represented as sums of two positive cubes in $k$ different ways), for $k=7...14$.

Abstract:
Visualization and interpretation of contingency tables by correspondence analysis (CA), as developed by Benzecri, has a rich structure based on Euclidean geometry. However, it is a well established fact that, often CA is very sensitive to sparse contingency tables, where we caracterize sparsity as the existence of relatively high-valued counts, rare observations discussed by Rao (1995), and zero-block structure emphasized by Novak and Bar-Hen (2005) and Greenacre (2013). In this paper, we aim to emphasize the important roles played by L1 and L2 geometries. This will be done by comparing the maps obtained by CA with the maps obtained by taxicab correspondence analysis (TCA), where TCA is a robust L1 variant of correspondence analysis. If the projections of view of both maps are quite different, we refer to this phenomenon as parallax. In astronomy, parallax means the apparent change in the position of an object as seen from two different points. In our case the two different points correspond to the two different geometries, Euclidean and Taxicab. The existence of a parallax highlights the important, but hidden, role of the underlying geometry in the interpretation of the maps obtained in multivariate data analysis. We emphasize the following fact: Only by comparing CA and TCA graphical displays, we are able to reveal the phenomenon of parallax. Examples are provided.