On Periodic Points of the Order of Appearance in the Fibonacci Sequence

Let ( F n ) n ≥ 0 be the Fibonacci sequence. The order of appearance z ( n ) of an integer n ≥ 1 is defined by z ( n ) = min { k ≥ 1 : n ∣ F k } . Marques, and Somer and Křížek proved that all fixed points of the function z ( n ) have the form n = 5 k or 12 · 5 k . In this paper, we shall prove that z ( n ) does not have any k-periodic points, for k ≥ 2

Use of Automatic Image Classification for Analysis of Landscape Evolution in the Village of Staré Jesenčany

Tato práce je zaměřena na metody digitálního zpracování obrazu z dálkového průzkumu Země, a to konkrétně na metody automatické klasifikace obrazu. Toto téma je nejprve přiblíženo v teoretické rovině a poté je na něj navázána problematika sledování změn v krajině. Významným dílem této práce je aplikace automatické klasifikace na letecké i družicové snímky a analýza změn krajiny v obci Staré Jesenčany pomocí programu ArcGIS Desktop.This work focuses on methods of digital image processing of remote sensing and namely on the methods of automatic image classification. This topic is at first indicated in the theory level and then tied to the problems of monitoring changes in the landscape follow. An important part of this work is the application of automated classification for air and satellite images and analysis of landscape changes in the village of Old Jesenčany using ArcGIS Desktop.Ústav systémového inženýrství a informatikyStudentka seznámila komisi s tématem své diplomové práce Využití automatické klasifikace obrazu pro analýzu vývoje krajiny v obci Staré Jesenčany. Jaká jiná metoda by byla vhodná pro klasifikaci leteckého snímku? Proč porovnáváte dvě naprosto odlišné metody? V čem se liší metody K-Keans a Fuzzy K-Means? Studentka na otázky odpovídala

On Some Properties of the Limit Points of (z(n)/n)n

Let (Fn)n≥0 be the sequence of Fibonacci numbers. The order of appearance of an integer n≥1 is defined as z(n):=min{k≥1:n∣Fk}. Let Z′ be the set of all limit points of {z(n)/n:n≥1}. By some theoretical results on the growth of the sequence (z(n)/n)n≥1, we gain a better understanding of the topological structure of the derived set Z′. For instance, {0,1,32,2}⊆Z′⊆[0,2] and Z′ does not have any interior points. A recent result of Trojovská implies the existence of a positive real number t<2 such that Z′∩(t,2) is the empty set. In this paper, we improve this result by proving that (127,2) is the largest subinterval of [0,2] which does not intersect Z′. In addition, we show a connection between the sequence (xn)n, for which z(xn)/xn tends to r>0 (as n→∞), and the number of preimages of r under the map m↦z(m)/m

On the Diophantine Equation z(n) = (2 − 1/k)n Involving the Order of Appearance in the Fibonacci Sequence

Let ( F n ) n ≥ 0 be the sequence of the Fibonacci numbers. The order (or rank) of appearance z ( n ) of a positive integer n is defined as the smallest positive integer m such that n divides F m . In 1975, Sallé proved that z ( n ) ≤ 2 n , for all positive integers n. In this paper, we shall solve the Diophantine equation z ( n ) = ( 2 − 1 / k ) n for positive integers n and k

The Proof of a Conjecture Related to Divisibility Properties of z(n)

The order of appearance of n (in the Fibonacci sequence) z(n) is defined as the smallest positive integer k for which n divides the k—the Fibonacci number Fk. Very recently, Trojovský proved that z(n) is an even number for almost all positive integers n (in the natural density sense). Moreover, he conjectured that the same is valid for the set of integers n≥1 for which the integer 4 divides z(n). In this paper, among other things, we prove that for any k≥1, the number z(n) is divisible by 2k for almost all positive integers n (in particular, we confirm Trojovský’s conjecture)

Repdigits as Product of Fibonacci and Tribonacci Numbers

In this paper, we study the problem of the explicit intersection of two sequences. More specifically, we find all repdigits (i.e., numbers with only one repeated digit in its decimal expansion) which can be written as the product of a Fibonacci by a Tribonacci number (both with the same indexes). To work on this problem, our approach is to combine lower bounds from the Baker’s theory with reduction methods (based on the theory of continued fractions) due to Dujella and Pethö

Algebraic Numbers of the form αT with α Algebraic and T Transcendental

Let α≠1 be a positive real number and let P(x) be a non-constant rational function with algebraic coefficients. In this paper, in particular, we prove that the set of algebraic numbers of the form αP(T), with T transcendental, is dense in some open interval of R

The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of z(n)

Let (Fn)n be the sequence of Fibonacci numbers. The order of appearance (in the Fibonacci sequence) of a positive integer n is defined as z(n)=min{k≥1:n∣Fk}. Very recently, Trojovská and Venkatachalam proved that, for any k≥1, the number z(n) is divisible by 2k, for almost all integers n≥1 (in the sense of natural density). Moreover, they posed a conjecture that implies that the same is true upon replacing 2k by any integer m≥1. In this paper, in particular, we prove this conjecture

On Homogeneous Combinations of Linear Recurrence Sequences

Let (Fn)n≥0 be the Fibonacci sequence given by Fn+2=Fn+1+Fn, for n≥0, where F0=0 and F1=1. There are several interesting identities involving this sequence such as Fn2+Fn+12=F2n+1, for all n≥0. In 2012, Chaves, Marques and Togbé proved that if (Gm)m is a linear recurrence sequence (under weak assumptions) and Gn+1s+⋯+Gn+ℓs∈(Gm)m, for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on ł and the parameters of (Gm)m. In this paper, we shall prove that if P(x1,…,xℓ) is an integer homogeneous s-degree polynomial (under weak hypotheses) and if P(Gn+1,…,Gn+ℓ)∈(Gm)m for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on ℓ, the parameters of (Gm)m and the coefficients of P

Využití automatické klasifikace snímků pro analýzy krajiny, případová studie Staré Jesenčany, Česká republika

The landscape is changing, mainly due to man. The changes in the landscape are increasingly important, so it is important to monitor these changes, mainly due to the use of resources in the country in the future. The article deals with the use of automatic classification of data from remote sensing to analyse the development of the landscape on the case study in the village Staré Jesenčany, the Czech Republic.Krajina se neustále mění. Tyto změny je nutné monitorovat, zejm. pro využití přírodních zdrojů v budoucnu. Článek je zaměřen na využití automatické klasifikace dat z dálkového průzkumu země k analýze rozvoje krajiny. Článek obsahuje případovou studii zaměřenou na Staré Jesenčany