Graph of Fuzzy Topographic Topological Mapping in relation to k-Fibonacci Sequence

A generated n-sequence of fuzzy topographic topological mapping, FTTM n , is a combination of n number of FTTM’s graphs. An assembly graph is a graph whereby its vertices have valency of one or four. A Hamiltonian path is a path that visits every vertex of the graph exactly once. In this paper, we prove that assembly graphs exist in FTTM n and establish their relations to the Hamiltonian polygonal paths. Finally, the relation between the Hamiltonian polygonal paths induced from FTTM n to the k-Fibonacci sequence is established and their upper and lower bounds’ number of paths is determined.

Processing of measures measured in underground polygonations

The information, which is the concrete basis for solving geodetic and topographic problems, comes from measurement observations made on quantities that are mainly angles and distances. The quality of the observations has an important role in achieving the objectives for which they are executed, in conditions of efficiency and safety. As topographically, underground works are conducted using polygonal paths, the methods used for processing measurements are of great interest.

Kloosterman paths and the shape of exponential sums

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums$\text{Kl}_{p}(a)$, as$a$varies over$\mathbf{F}_{p}^{\times }$and as$p$tends to infinity. Using independence of Kloosterman sheaves, we prove convergence in the sense of finite distributions to a specific random Fourier series. We also consider Birch sums, for which we can establish convergence in law in the space of continuous functions. We then derive some applications.

Closed paths whose steps are roots of unity

International audience We give explicit formulas for the number $U_n(N)$ of closed polygonal paths of length $N$ (starting from the origin) whose steps are $n^{\textrm{th}}$ roots of unity, as well as asymptotic expressions for these numbers when $N \rightarrow \infty$. We also prove that the sequences $(U_n(N))_{N \geq 0}$ are $P$-recursive for each fixed $n \geq 1$ and leave open the problem of determining the values of $N$ for which the $\textit{dual}$ sequences $(U_n(N))_{n \geq 1}$ are $P$-recursive. Nous donnons des formules explicites pour le nombre $U_n(N)$ de chemins polygonaux fermés de longueur $N$ (débutant à l'origine) dont les pas sont des racines $n$-ièmes de l'unité, ainsi que des expressions asymptotiques pour ces nombres lorsque $N \rightarrow \infty$. Nous démontrons aussi que les suites $(U_n(N))_{N \geq 0}$ sont $P$-récursives pour chaque $n \geq 1$ fixé et laissons ouvert le problème de déterminer les valeurs de $N$ pour lesquelles les suites $\textit{duales}$ $(U_n(N))_{n \geq 1}$ sont $P$-récursives.

FINDING POPULAR PLACES

Widespread availability of location aware devices (such as GPS receivers) promotes capture of detailed movement trajectories of people, animals, vehicles and other moving objects. We investigate spatio-temporal movement patterns in large tracking data sets, i.e. in large sets of polygonal paths. Specifically, we study so-called 'popular places', that is, regions that are visited by many entities. Given a set of polygonal paths with a total of [Formula: see text] vertices, we look at the problem of computing such popular places in two different settings. For the discrete model, where only the vertices of the polygonal paths are considered, we propose an [Formula: see text] algorithm; and for the continuous model, where also the straight line segments between the vertices of a polygonal path are considered, we develop an [Formula: see text] algorithm. We also present lower bounds and hardness results.