Fibonacci Length Research Articles

Electromagnetic wave transmission in hybrid fractal plasma photonic crystals through developing new kinds of productive quasi-periodic multilayers

ABSTRACT In this study, for the first time, we present a new way of producing novel quasi-periodic multilayers based on the multiplication of some primitive available sequences such as Thue–Morse, Fibonacci (of two types N and P), Cantor, and constant length square sets. We also present some pseudo-codes to create these multilayers. Using a one-dimensional transfer matrix method, we obtain the transmission coefficient for some plasma-dielectric photonic crystals. Although, employing the above-mentioned approach many new sequences may be obtained, but for conciseness purpose, we analyze the photonic band gaps of a limited number of multilayers obtained in this way. We consider the effect of the number of layers, the refractive index of the dielectric later, multiplication of different fractal sets, etc., on the electromagnetic wave transport properties.

Fibonacci length and the generalized order k-Pell sequences of the 2-generator p-groups of nilpotency class 2

In this paper, we consider the 2-generator p-groups of nilpotency class 2. We find the Fibonacci length and the period of the generalized order k-Pell sequences of these groups.

Factorization of the $t$-extension of the $p$-Fibonacci and the Pascal matrices

In this paper, we consider the 2-generator $p$-groups of nilpotency class 2. We find the Fibonacci length and the period of the generalized order $k$--Pell sequences of these groups.

Classes of non-commutative semigroups with finite Fibonacci invariants

For a $2$-generated semigroup $S$ we define the invariant $\lambda (S)$ as the minimum of the Fibonacci lengths over all generating pairs of the semigroup $S$, where the Fibonacci length with respect to a generating pair $(x, y)$ is the fundamental period (if exist) of the truncated periodic sequence $z_0=x, z_1=y, z_k=z_{k-2}z_{k-1}, (k\geq 2)$ of the elements of $S$. We name this invariant as the Fibonacci invariant of $S$. Our used notation is the same as of the celebrated work of D.L. Johnson in $2005$ on infinite groups. In this paper we examine two classes of semigroups for existence of this invariant. The considered semigroups are the finite semigroup $S=\langle a, b\mid a^{p^\alpha}=a, b^{q^\beta}=b,ab=a\rangle$ of order $p^{\alpha}q^{\beta}-1$ and the infinite semigroup $T=\langle a, b\mid a^{p^\alpha}=a, ab=a\rangle$, for all integers $\alpha, \beta \geq 2$ and all distinct primes $p$ and $q$. We prove the existence of the Fibonacci invariants of these semigroups, for all parameters. As a numerical result we show that $\lambda (T)\leq \lambda (S)=p^{\alpha +1}$, if $p$ is even.

Fractality in nonequilibrium steady states of quasiperiodic systems.

We investigate the nonequilibrium response of quasiperiodic systems to boundary driving. In particular, we focus on the Aubry-André-Harper model at its metal-insulator transition and the diagonal Fibonacci model. We find that opening the system at the boundaries provides a viable experimental technique to probe its underlying fractality, which is reflected in the fractal spatial dependence of simple observables (such as magnetization) in the nonequilibrium steady state. We also find that the dynamics in the nonequilibrium steady state depends on the length of the chain chosen: generic length chains harbour qualitatively slower transport (different scaling exponent) than Fibonacci length chains, which is in turn slower than in the closed system. We conjecture that such fractal nonequilibrium steady states should arise in generic driven critical systems that have fractal properties.

Fibonacci lengths of all finite p-groups of exponent p2

The Fibonacci lengths of finite p-groups were studied by Dikici and coauthors since 1992. All considered groups are of exponent p and the lengths depend on the Wall number k(p). The p-groups of nilpotency class 3 and exponent p were studied in 2004 also by Dikici. In the paper, we study all p-groups of nilpotency class 3 and exponent p 2. Thus, we complete the study of Fibonacci lengths of all p-groups of order p 4 by proving that the Fibonacci length is k(p 2).

Harmonic Series Meets Fibonacci Sequence

SummaryThe terms of a conditionally convergent series may be rearranged to converge to any prescribed real value. What if the harmonic series is grouped into Fibonacci length blocks? Or the harmonic series is arranged in alternating Fibonacci length blocks? Or rearranged and alternated into separate blocks of even and odd terms of Fibonacci length?

On the k-Nacci Sequences in Finite Binary Polyhedral Groups

A k-nacci sequence in a finite group is a sequence of group elements x0,x1,…,xn,… for which, given an initial (seed) set x0,x1,…,xj-1, each element is defined by [Formula: see text] It is important to note that the Fibonacci length depends on the chosen generating n-tuple for a group. The binary polyhedral groups have been studied recently by C.M. Campbell and P.P. Campbell for their Fibonacci lengths. In this paper, we obtain the period of k-nacci sequences in the binary polyhedral groups 〈2,2,2〉, 〈n,2,2〉, 〈2,n,2〉 and 〈2,2,n〉 for any n > 2.

Formula omitted]-nacci sequences in some special groups of finite order

In this paper, we obtain the period of k -nacci sequences in the semidihedral groups S D 2 m and in the ordinary nonabelian group having the presentations W 2 m = 〈 a , b : a 2 m − 1 = b 2 = 1 , a b = a 1 + 2 m − 2 〉 of order 2 m , m ≥ 4 . It is important to note that the Fibonacci length depends on the chosen generating n -tuple for a group.

A subgroup involvement of the Fibonacci length

For a non-Abelian 2-generated finite group G=〈a,b〉, the Fibonacci length of G with respect to A={a,b}, denoted by LENA(G), is defined to be the period of the sequence x1=a,x2=b,x3=x1x2,…,xn+1=xn−1xn,… of the elements of G. For a finite cyclic group Cn=〈a〉, LENA(Cn) is defined in a similar way where A={1,a} and it is known that LENA(Cn)=k(n), the well-known Wall number of n. Over all of the interesting numerical results on the Fibonacci length of finite groups which have been obtained by many authors since 1990, an intrinsic property has been studied in this paper. Indeed, by studying the family of minimal non-Abelian p-groups it will be shown that for every group G of this family, there exists a suitable generating set A′ for the derived subgroup G′ such that LENA′(G′)|LENA(G) where, A is the original generating set of G.

An application of the Fibonacci length

For every integer n ≥ 2 we define the Fibonacci class of degree 2 of 2-generated groups, and study certain properties of this class. Our method of study is applicable for the groups with more than two generators. This investigation is indeed an application of FLA(G), the well-known Fibonacci length of the finite group G = � A� , where A = {a1,a2,...,ak}.

Fibonacci Lengths of Certain Nilpotent 2–Groups

In this paper, we study two classes of 2–generated 2–groups of nilpotency class 2 classified by Kluempen in 2002 and also a class of finite 2–groups of high nilpotency class for their Fibonacci lengths. Their involvement in certain interesting sequences of Tribonacci numbers gives us some explicit formulas for the Fibonacci lengths and this adds to the small class of finite groups for which the Fibonacci length are known.

Fibonacci lengths involving the wall number k(n)

Two infinite classes of special finite groups considered (The group G is special, if G 0 and Z(G) coincide). Using certain sequences of numbers we give explicit formulas for the Fibonacci lenghts of these classes which involve the well-known Wall numbers k(n).

The Fibonacci length of certain centro-polyhedral groups

We examine the Fibonacci length of certain centro-polyhedral groups and show that in some cases the lengths depend on tribonacci sequences. Further we obtain specific examples of infinite families of threegenerator groups with constant, linear and (3-step) Wall number dependent Fibonacci lengths.

Fibonacci Sequences in Groups

When this sequence is periodic, its fundamental period is called the Fibonacci length of (x1, x2) in G. When G is a finite 2-generator group, the minimum of these lengths over all generating pairs defines an invariant λ(G) of G. After briefly listing some known results, we launch the quest for infinite groups of finite Fibonacci length by giving three modest examples and conclude with a selection of open problems.

Search Techniques and Epimorphisms Between Certain Groups and Fibonacci Groups

We examine the Fibonacci lengths of all generat- ing pairs for certain centro-polyhedral groups. The problem requires a variety of approaches both exhausive and random search.