Open access peer-reviewed chapter

Nature of Phyllotaxy and Topology of H-matrix

By Ab. Hamid Ganie

Submitted: October 19th 2017Reviewed: January 30th 2018Published: August 29th 2018

DOI: 10.5772/intechopen.74676

Downloaded: 308

Abstract

The main purpose of this chapter is to introduce a new type of regular matrix generated by Fibonacci numbers and we shall investigate its various topological properties. The concept of mathematical regularity in terms of Fibonacci numbers and phyllotaxy have been discussed.

Keywords

  • sequence spaces
  • infinite matrices
  • Fibonacci numbers
  • phyllotaxy

1. Preliminaries, background and notation

In several branches of analysis, for instance, the structural theory of topological vector spaces, Schauder basis theory, summability theory, and the theory of functions, the study of sequence spaces occupies a very prominent position. There is an ever-increasing interest in the theory of sequence spaces that has made remarkable advances in enveloping summability theory via unified techniques effecting matrix transformations from one sequence space into another.

Thus, we have several important applications of the theory of sequence spaces, and therefore, we attempt to present a survey on recent developments in sequence spaces and their different kinds of duals.

In many branches of science and engineering, we deal with different kinds of sequences and series, and when we deal with these, it is important to check their convergence. The use of infinite matrices is of great importance, we can bring even the bounded or divergent sequences and series in the domain of convergence. So we can say that the theory of sequence spaces and their matrix maps is the bigger scale to measure the convergence property. Summability can be roughly considered as the study of linear transformations on sequence spaces. The theory originated from the attempts of mathematicians to assign limits to divergent sequences. The classical summability theory deals with the generalization of the convergence of sequences or series of real or complex numbers. The idea is to assign a limit of some sort to divergent sequences or series by considering a transform of a sequence or series rather than the original sequence or series.

The earliest idea of summability theory was perhaps contained in a letter written by Leibnitz to C. Wolf (1713) in which he attributed the sum 1/2 to the oscillatory series −1 + 1−1 + …. Frobenius in (1880) introduced the method of summability by arithmetic means, which was generalized by Cesàro in (1890) as the (C,K) method of summability. Toward the end of the nineteenth century, study of the general theory of sequences and transformations on them attracted mathematicians, who were chiefly motivated by problems such as those in summability theory, Fourier series, power series and system of equations with infinitely many variables.

Presenting some basic definitions and notations that are involved in the present work, the author proposes to give a brief resume of the hitherto obtained results against the background of which the main results studied in the present chapter suggest themselves.

2. Notations and symbols

Here, we state a few conventions which will be used throughout the chapter.

2.1. Symbols N, C, RandA

The symbols are denoted as follows:

N: Set of non-negative integers.

C: Set of complex numbers.

R: Set of real numbers.

A: The infinite matrix ank,nk=12.

2.2. Summation convention

By αβfn, we mean the sum of all values of fnfor which αnβ. In the case β<α, then we take this to be zero.

Summations are over 0,1,2,, when there is no indication to the contrary. If xk=x1x2is a sequence of terms, then, by kxkwe mean k=1xkand we shall sometimes write as xkincase where no possible confusion arises.

2.3. The spaces ω, l, c, c0, lp

A sequence space is a set of scalar sequences (real or complex) which is closed under coordinate-wise addition and scalar multiplication. In other words, a sequence space is a linear subspace of the space ωof all complex sequences, that is,

ω=x=xk:xkRorC.

The space l: The space lof bounded sequences is defined by

x=xk:supkxk<

The spaces c: The spaces cand c0of convergent and null sequences are given by

x=xk:limkxk=llC

The space c0: The space c0of all sequences converging to 0 is given by

x=xk:limkxk=0

The space lp: The space lpof absolutely p-summable sequences is defined by

x=xk:kxkp<,0<p<

The spaces l,c,and c0are Banach spaces with the norm,

x=supkxk

The space lpis a Banach space with the norm,

xp=kxkp1p,1p<

2.4. Cauchy sequence

A sequence x=xkis called a Cauchy sequence if and only if xnxm0mnthat is for any ϵ>0, there exists N=Nϵsuch that xnxm<ϵfor all n,mN. By C, we denote the space of all Cauchy sequences, that is,

C:x=xk:xnxm0asnm

2.5. FK-space

A sequence space Xis called an FK-space if it is a complete linear metric space with continuous coordinates pn:XCdefined by pnx=xnfor all xXand every nN[1, 2].

2.6. BK-space

A BK-space is a normed FK-space, that is, a BK-space is a Banach space with continuous coordinates [3, 4, 5, 6].

2.7. Fibonacci numbers

In the 1202 AD, Leonardo Fibonacci wrote in his book Liber Abaci of a simple numerical sequence that is the foundation for an incredible mathematical relationship behind phi. This sequence was known as early as the sixth century AD by Indian mathematicians, but it was Fibonacci who introduced it to the west after his travels throughout the Mediterranean world and North Africa. He is also known as Leonardo Bonacci, as his name is derived in Italian from words meaning son of (the) Bonacci.

The Fibonacci numbers have been introduced [7, 8, 9, 10, 11, 12, 13, 14]. The Fibonacci numbers are the sequence of numbers fn,nNdefined by recurrence relations

f0=0,f1=1andfn=fn1+fn2;n2

First derived from the famous rabbit problem of 1228, the Fibonacci numbers were originally used to represent the number of pairs of rabbits born of one pair in a certain population. Let us assume that a pair of rabbits is introduced into a certain place in the first month of the year. This pair of rabbits will produce one pair of offspring every month, and every pair of rabbits will begin to reproduce exactly 2 months after being born. No rabbit ever dies, and every pair of rabbits will reproduce perfectly on schedule.

So, in the first month, we have only the first pair of rabbits. Likewise, in the second month, we again have only our initial pair of rabbits. However, by the third month, the pair will give birth to another pair of rabbits, and there will now be two pairs. Continuing on, we find that in month 4, we will have 3 pairs, then 5 pairs in month 5, then 8, 13, 21, 34, …, etc., continuing in this manner. It is quite apparent that this sequence directly corresponds with the Fibonacci sequence introduced above, and indeed, this is the first problem ever associated with the now-famous numbers.

Fibonacci numbers have many interesting properties and applications in arts, sciences and architecture. Also, following [7], some basic properties are as follows

k=0nfk=fn+21;nN,

and

k=0nfk2=fnfn+1;nN

Everything in Nature is subordinated to stringent mathematical laws. Prove to be that leaf’s disposition on plant’s stems also has stringent mathematical regularity and this phenomenon is called phyllotaxis in botany. An essence of phyllotaxis consists in a spiral disposition of leaves on plant’s stems of trees, petals in flower baskets, seeds in pine cone and sunflower head, etc.

This phenomenon, known already to Kepler, was a subject of discussion of many scientists, including Leonardo da Vinci, Turing, Veil, and so on. In phyllotaxis phenomenon, more complex concepts of symmetry, in particular, a concept of helical symmetry, are used. The phyllotaxis phenomenon reveals itself especially brightly in inflorescences and densely packed botanical structures such as pine cones, pineapples, cacti, heads of sunflower and cauliflower, and many other objects [11].

On the surfaces of such objects, their bio-organs (seeds on the disks of sunflower heads and pine cones, etc.) are placed in the form of the left-twisted and right-twisted spirals. For such phyllotaxis objects, it is used usually the number ratios of the left-hand and right-hand spirals observed on the surface of the phyllotaxis objects. Botanists proved that these ratios are equal to the ratios of the adjacent Fibonacci numbers, that is,

fi+1fi:21,32,53,85,138,=1+52

By using hyperbolic Fibonacci functions, he had developed an original geometric theory of phyllotaxis and explained why Fibonacci spirals arise on the surface of the phyllotaxis objects namely, pine cones, cacti, pine apple, heads of sunflower, and so on, in process of their growths. Bodnar’s geometry [15] confirms that these functions are ‘natural’ functions of the nature, which show their value in the botanic phenomenon of phyllotaxis. This fact allows us to assert that these functions can be attributed to the class of fundamental mathematical discoveries of contemporary science because they reflect natural phenomena, in particular, phyllotaxis phenomenon.

From above discussion, it gave us motivation to see the behavior of the infinite matrices generated by Fibonacci numbers.

In the present chapter, we have introduced a new type of matrix H=hnkun,kNby using Fibonacci numbers fnand we call it as H-matrix generated by Fibonacci numbers fnand introduce some new sequence spaces related to matrix domain of Hin the sequence spaces lp,l,cand c0, where 1p<.

2.8. The space rqup

Sheikh and Ganie [16] introduced the Riesz sequence space rqupand studied its various topological properties where u=ukis a sequence such that uk0for all kNand qkthe sequence of positive numbers and

Qn=k=0nqk,nN

Then, the matrix Ruq=rnkqof the Riesz mean Ruqnis given by

rnkq=ukqkQnif0kn,0,ifk>n.

The Riesz mean Ruqnis regular if and only if Qnas n.

3. H-matrix generated by Fibonacci numbers

Let Xand Ybe two subsets of ω. Let A=ankbe an infinite matrix of real or complex numbers ank, where n,kN. Then, the matrix Adefines the A-transformation from XintoY, if for every sequence x=xkXthe sequence Ax=Axn, the A-transform of xexists and is in Ywhere

Axn=kankxk.

For simplicity in notation, here and in what follows, the summation without limits runs from 0 to . By XY,we denote the class of all such matrices. A sequence xis said to be A-summable to lif Axconverges to lwhich is called as the A-limit of x.

For a sequence space X, the matrix domain XAof an infinite matrix Ais defined as

XA=x=xkω:AxX,E1

which is a sequence space.

An infinite matrix A=ankis said to be regular if and only if the following conditions (or Toplitz conditions) hold [17, 18, 19]:

  1. limnk=0ank=1,

  2. limnank=0,k=0,1,2,

  3. k=0ank<M,M>0j=0,1,2.

In the present paper, we introduce H-matrix with H=hnkun,kNas follows:

hnku=ukfk2fnfn+1if0kn,0,ifk>n.

Thus, for uk=1and for all kN, we have

H=100001/21/20001/61/64/6001/151/154/159/150.

It is obvious that the matrix His a triangle, that is, hnnu0and hnku=0for k>nand for all nN. Also, since it satisfies the conditions of Toeplitz matrix and hence it is regular matrix.

Note that if we take qk=fk2, then the matrix His special case of the matrix Ruq, where

Qn=k=0nfk2=fnfn+1,

introduced by Sheikh and Ganie [16].

The approach of constructing a new sequence space by means of matrix domain of a particular limitation method has been studied by several authors [17, 18, 19, 20, 21, 22, 23, 24, 25, 26].

Throughout the text of the chapter, Xdenotes any of the spaces l,c, c0and lp1p<. Then, the Fibonacci sequence space XHis defined by

XH=x=xkω:y=ykX,

where the sequence y=ykis the H-transform of the sequence x=xkand is given by

yk=Hkx=1fkfk+1i=0kfi2uixiforallkN.E2

With the definition of matrix domain given by Eq. (1), we can redefine the space XHas the matrix domain of the triangle Hin the space X, that is,

XH=XH.

Theorem 1: The space XHis a BK-space with the norm given by

x=HxX=yX=k=0ykp1pforforXlp.supkykforXlcc0.E3

Proof: Since the matrix H=hnkuis a triangle, that is, hnnu0and hnku=0for k>nfor all n. We have the result by Eq. (3) and Theorem 4.3.2 of Wilansky [6] gives the fact that XHis a BK-space.◊

Theorem 2: The space XHis isometrically isomorphic to the space X.

Proof: To prove the result, we should show the linear bijection between the spaces XHand X. For that, consider the transformation Tfrom XHto Xby xy=Tx. Then, the linearity of Tfollows from Eq. (2). Further, we see that x=0whenever Tx=0and consequently Tis injective.

Moreover, let y=ykXbe given and define the sequence x=xkby

xk=fk+1ukfkykfk1ukfkyk1;kN.E4

Then, by using (2) and (4), we have for every kNthat

H(x)=1fkfk+1i=0kfi2uixi=1fkfk+1i=0kfi(fi+1yifi1yi1)=yk.

This shows that Hx=yand since yX, we conclude that HxX. Thus, we deduce that xXHand Tx=y. Hence, Tis surjective.

Furthermore, for any xXH, we have by (3) that

Tx=y=HxX=xX

which shows that Tis norm preserving. Hence, Tis isometry. Consequently, the spaces XHand Xare isometrically isomorphic. Hence, the proof of the Theorem is complete.◊

Theorem 3: Let fjbe Fibonacci number sequences. Then, we have

supifi2j=i1fjfj+1<.

Proof: We have,

k=n1fk1fk+1=1fn

This gives,

1=fnk=n1fk1fk+1=fn21fnk=nfk+1fkfkfk+1=fn21fnk=nfk1fkfk+1fn2fn1fnk=n1fkfk+1

and the conclusion follows because fnfn1is bounded since it converges to 5+12.◊

Theorem 4: XXHholds.

Proof: It is obvious that c0c0Hand ccH, since the matrix His regular matrix. Now, let xl. Then, there is a constant K>0such that xj<Kujfor all jN. Thus, we have for every iNthat

Hix1fifi+1j=0ifj2ujxjKfifi+1j=0ifj2=K

which shows that Hxl. Therefore, we deduce that xlimplies xlH.

We now consider the case 1p<. We only consider the case 1<p<and by similar argument will follow for p=1. So, let xlp. Then, for every iNand by Holder’s inequality, we have

Hixpj=0ifj2fifi+1ujxjpj=0ifj2fifi+1ujxjpj=0ifj2fifi+1p1=1fifi+1j=0ifj2ujxjp.

Hence, we have

i=0Hixpi=01fifi+1j=0ifj2ujxjp=i=0xjpujpfj2i=j1fifi+1.

Hence, the right-hand side of above inequality can be made arbitrary small, since, supjfj2i=j1fifi+1<by Theorem 3 (above) and xlp. This shows that xlpH. This completes the proof of the theorem.◊

Acknowledgments

The author would like to express his sincere thanks for the refree(s) for the kind remarks that improved the presentation of the chapter.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Ab. Hamid Ganie (August 29th 2018). Nature of Phyllotaxy and Topology of H-matrix, Matrix Theory - Applications and Theorems, Hassan A. Yasser, IntechOpen, DOI: 10.5772/intechopen.74676. Available from:

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