The Retention of Information in the Presence of Increasing Entropy Using Lie Algebras Defines Fibonacci-Type Sequences

Abstract

In the general linear Lie algebra of continuous linear transformations in n dimensions, we show that unequal Abelian scaling transformations on the components of a vector can stabilize the system information in the presence of Markov component transformations on the vector, which, alone, would lead to increasing entropy. The more interesting results follow from seeking Diophantine (integer) solutions, with the result that the system can be stabilized with constant information for each of a set of entropy rates ($k=\mathrm{1,2},3,\dots$ ). The first of these—the simplest—where $k=1$, results in the Fibonacci sequence, with information determined by the olden mean, and Fibonacci interpolating functions. Other interesting results include the fact that a new set of higher order generalized Fibonacci sequences, functions, golden means, and geometric patterns emerges for $k=2, 3, \dots$ Specifically, we define the $k$^th order golden mean as ${\Phi }_k=k/2+\sqrt{{(k/2)}^2+1}$ for $k =1, 2, 3, \dots .$. One can easily observe that one can form a right triangle with sides of 1 and $k/2$ and that this will give a hypotenuse of $\sqrt{{(k/2)}^2+1}$. Thus, the sum of the $k/2$ side plus the hypotenuse of these triangles so proportioned will give geometrically the exact value of the golden means for any value of k relative to the third side with a value of unity. The sequential powers of the matrix $(k^2+1,k,k,1)$ for any integer value of $k$ provide a generalized Fibonacci sequence. Also, using the general equation expressed as ${\Phi }_k={\displaystyle \frac{k}{2}}+\sqrt{{(k/2)}^2+1}$ for $k =\mathrm{1,2},3, \dots ,$ one can easily prove that ${\Phi }_k=k+1/{\Phi }_k$ which is a generalization of the familiar equation expressed as $\Phi =1+1/\Phi$. We suggest that one could look for these new ratios and patterns in nature, with the possibility that all of these systems are connected with the retention of information in the presence of increasing entropy. Thus, we show that two components of the general linear Lie algebra ($GL(n,\mathbb{R})$), acting simultaneously with certain parameters, can stabilize the information content of a vector over time.

1. Introduction

The identification of symmetries in nature, as well as their violation, is of the highest importance not only in the sciences but also in art; architecture; music; and, certainly, in mathematics, physics, and engineering. Perhaps one of the most famous and early examples of a symmetry concerns the golden ratio and the associated Fibonacci numerical sequences that begin with 0, 1, with all succeeding values as the sum of the two preceding values and continuing as such: 1, 2, 3, 5, 8, 13, … The Fibonacci sequence was first described by Pingala in 200 BC relating to pottery patterns. But the sequence is named for Italian mathematician Leonardo of Pisa, known also as Fibonacci, who discussed the sequence in his Liber Abaci book in 1202 as he related the sequence to the growth of hypothetical rabbit populations. This sequence is so universal that a journal, Fibonacci Quarterly [1], is dedicated to its study. The sequence and the golden ratio appear in many living systems, such as trees, leaves, pineapples, pinecones, sunflowers, and other living systems. An extensive review can be found on Wikipedia, available with a search for Fibonacci Sequences [2]. Although there are extensive examples in pure mathematics, the fact that the sequence and its limiting ratio of successive values (the golden ratio of $\Phi =1.618\dots )$ can be found in many living systems remains puzzling. Such extensive examples have led to speculations on possible origins, such as optimal packing of seeds and other reasons leading one to question what basis in life forms would lead to their presence.

Symmetries are fundamental to the foundations of physics, where the theory of Lie algebras and groups is used to study them. The theory of quantum mechanics rests upon the Heisenberg Lie algebra for the four momentum and space time and in special relativity on the Lorentz Lie algebra, which contains both the rotation and Lorentz transformation symmetries, jointly called the Poincare Lie algebra, whose representations define the fundamental particles, along with the inversions of space, time, and fundamental particle groups. These were combined into an extended Poincare Lie algebra by the author [3]. The vast array of fundamental particles and their interactions is now based upon the $SU\left(3\right)\times SU\left(2\right)\times U\left(1\right)$ “Standard Model”, which has had phenomenal success [4], leaving only the Theory of General Relativity otherwise framed in nonlinear differential equations. The author recently proposed an extended form of Lie algebra to include Einstein’s general relativity equations for gravitation [5]. The author has also separately studied the general linear Lie algebra [6] ($GL(n,\mathbb{R}$)), which describes the Lie algebra that generates all continuous nonsingular matrix transformations in n dimensions over the real numbers. He proved that they are composed of exactly two subalgebras: (1) an Abelian Lie algebra ($A\left(n\right)$), which consists of matrices with a “1” in the $n$ different positions on the diagonal, which generate continuous exponential expansions and contractions of the associated n axes, and (2) a second “Markov-type” ($MT(n^2-n)$) Lie algebra that consists of a “1” in the (i, j) position, along with a “−1” in the (j, j) position that generates transformations that move one over the plane perpendicular to the (1, 1, … 1) vector in an $n$-dimensional space. The actions of this associated group transfer a fraction of the vector component at one position and adds it to another component, thereby conserving the sum of the components of the vector. If one only takes the positive linear combinations of these Lie generators, one obtains a Lie monoid (a group without an inverse) whose transformations exactly give the Markov Monoid ($MM$) transformations on the vector. These Markov transformations describe increasing entropy, thereby lowering the “information content” in the vector, while the diagonal transformations can increase the information content in a vector with their expansions and contractions of components. The $MM$ transformations describe the diffusion of one liquid in another and the dispersal of order in a system of increasing entropy. The author noted that the Fibonacci sequence could be described and generated in his decomposition of the general linear group as a special combination of exponential growth and increases in entropy (disorder). Living systems constantly fight the increase in entropy in order to live and must develop system order in their body and, for animals, their brain [7].

In this paper we describe how the matrix generators [6] of Fibonacci sequences are a combination of both of these transformations by increasing information with diagonal transformations ($A\left(n\right)$) while the $MM(n^2-n)$ transformations attempt to lower the system information. Thus, the Fibonacci sequence can maintain constant information (for a period of time) using $A\left(n\right)$ to counter increasing entropy from $MM$. It is well known that closed systems never increase their order, and, except for cyclic systems, they continuously increase their entropy, which is a metric of the total system disorder. This is true both in thermodynamics and in information systems, as information is stored as special physical states of matter and energy. Thus, on the surface, it is perplexing that living things, as subsystems of larger closed systems, can become increasingly organized and that groups of living things such as humans and their societies can dramatically increase their order as subsystems of a larger closed system. However, this does not violate the increase in entropy of the system as a whole. We understand that this is accomplished by utilizing two sources of relative order for such organizing systems, where, for example, energy flows from a more ordered domain to states of lower order, such as from a hot reservoir (e.g., the Sun) to a colder one (e.g., the Earth or outer space). In some sense, we could visualize the self-organizing subsystem to be “feeding” on the available source of higher order so as to maintain and increase its own order. It is natural and customary to imagine this entire process of retaining and even increasing order in the presence of increasing entropy to be extremely complex (in the technical sense) or certainly highly nonlinear. But we will show that, within the framework of an extremely simple and fully linear system, while entropy is increasing, it is possible to utilize hot- and cold-type reservoirs (more precisely, an information source and sink) for the subsystem to retain order (or information) as entropy for the combined system increases. To accomplish this, we need to carefully define order, information, entropy, and the mathematical structure of the system to be studied. Our work rests upon previous work by the author concerning entropy and diffusion using continuous (Lie) groups with a decomposition of the general linear group in n dimensions. Beginning with the works of Einstein on random motion of molecules and Markov on diffusion using Markov transformations over a century ago, diffusion with the associated increases in entropy is described by Markov transformations and the associated diffusion equations in the continuous limit. It is the very essence of a diffusion process that these transformations are irreversible and, thus, have no inverse transformation. Thus, our approach using continuous group theory seems, at first, paradoxical, as all groups have an inverse for each transformation but becomes apparent upon closer analysis in the following when the inverse must be removed.

2. The General Linear Group in n Dimensions

We will explore the general linear group in n dimensions ($GL(n,\mathbb{R})$) over the real numbers, which we have previously shown [6] can be decomposed into a Markov-type Lie group ($MT(n^2-n)$), which models increasing entropy (disorder), and an Abelian scaling group ($A\left(n\right)$), which can model increased order. First, consider the Abelian scaling group, which is generated by the Lie algebra matrix representation that consists of the $n$ elements ($L^{ii}$) that have a value of ‘1’ on a single diagonal position ($ii$) and ‘0’ in every other position. Thus, the group consists of elements like

$$\mathrm{A}\left(\mathrm{a}\right)=\exp \left(a_{ii}{L}^{ii}\right),$$

(1)

which is a matrix with all off-diagonal elements with values of ‘0’ and diagonal values of $e^{aii}$. Thus, these transformations simply scale each axis ($i$) by these multipliers to increase system order.

One recalls that a Markov transformation is a linear transformation on a vector space that preserves the sum of the elements of the vector and that, when acting upon a vector with non-negative components, it transforms it into a new vector with all non-negative components. The $MT$ (Lie) group (the non-Abelian part of the general linear group) is the group of transformations in n dimensions that preserves the sum of the elements of the vector upon which they act without regard to whether these elements are positive or negative. The $MT$ Lie group was found to contain all valid Markov transformations when a particular basis is used for the generating Lie Algebra. The $MT$ Lie algebra basis consists of all linear combinations of elements that have a value of ‘1’ in the $ij$ off-diagonal position and ‘−1’ in the $jj$ position, with all other elements being ‘0’:

$$MM\left(a_{ij}\right)=\exp \left(a_{ij}{L}^{ij}\right).$$

(2)

Thus, each of these generating elements has columns that each sums to ‘0’ and that closes as a Lie algebra with $n(n-1)$ basis elements. By exponentiation, they generate all linear transformations that preserve the sum of components of a vector upon which it acts. As the sum of components is invariant (as opposed to the sum of squares for the motion on a sphere), we can say that this sum represents a conserved entity (such as probability, money, or substance) being redistributed by the transformation. But as many of these transformations generate new vectors that have some negative components, these are unacceptable as Markov transformations for probabilities.

3. Continuous Markov Transformations—The Markov Monoid (MM)

It can then be shown that, with the basis elements just defined, if one takes only non-negative linear combinations, i.e.,

$$L\left(a\right)=\exp \left(a_{ij}{L}^{ij}\right) \mathrm{where} a_{ij}\ge 0,$$

(3)

then one gets all continuous Markov transformations in $n$ dimensions that are continuously connected to the identity. This process removes the (unacceptable) inverse transformations and, thus, makes this Lie group into a Lie monoid (which we call the Markov Monoid ($MM$)), which is a group without inverses. The values of $a_{ij}$ give the exponential rate of transfer, redistribution, and diffusion of the conserved substance from component $j$ to component $i$. It is easy to show that the $MT$ transformations are linear transformations that move one over the hyperplane perpendicular to the vector (1,1,1, …1) in $n$ dimensions and that the $MM$ transformations are those which constrain the transformations to the positive hyperquadrant of the $n$-dimensional space. These transformations are very intuitive, and if one chooses only elements that are just below or just above the diagonal for $a_{ij}$, then one gets a random walk in one dimension over a lattice of positions represented by the components ($x_i$) as probabilities or amounts of substances. Reframed as a Hilbert space with continuous positions replacing the discrete values of ‘$i$’, one obtains continuous diffusion in one dimension.

Either of these models takes one from a highly organized state of perfect information and minimum entropy into one of maximum entropy at the final equilibrium state if $a_{ij}$ is multiplied by a continuous parameter ($t$). In our work here, we will assume that all states are equally likely and, thus, that the equilibrium is the uniform distribution over all states. One can use the Shannon definition of entropy, i.e.,

$$S={\sum }_i{x}_i{log}_2{x}_i,$$

(4)

or the Renyi second-order entropy, which is defined as follows:

$$R_n={log}_2\left\{n{\sum }_i{x}_n^i\right\}$$

(5)

We will use the Renyi second-order entropy ($n=2$), which, in fact, differs only slightly in value from the Shannon formula. For a two-component system this becomes

$$R_2={log}_2\left\{2\left(x_1^2+x_2^2\right)\right\}.$$

Thus, when the substance or probability is in one or the other state (i.e., x₁ = 1 or, conversely, x₂ = 0), then one obtains

$$R_2={log}_2\left\{2(1+0)\right\}=1 (\mathrm{i}.\mathrm{e}., \mathrm{one} \mathrm{bit} \mathrm{of} \mathrm{information})$$

(6)

Then, after an infinite time with maximum diffusion equally in both directions, $M\left(t\right)=\exp \left(t(L^{12}+L^{21})\right)=\left({\displaystyle \frac{1}{2}}\right)\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$ as $t\to \infty$ and, thus,

$$R_2={log}_2\left\{2{(1/2)}^2+{(1/2)}^2\right\}=0 (\mathrm{i}.\mathrm{e}., \mathrm{zero} \mathrm{bits} \mathrm{of} \mathrm{information}),$$

(7)

Thus, we can conclude that the Markov transformations will distribute the conserved entity (such as probability) equally to all available states in an irreversible way that perfectly mimics the diffusion of dye in a liquid or dust in a room, with our general view of entropy being maximized irreversibly. The inverse transformations in $MT$ are not available, as they can lead to unphysical states of negative probability.

4. The Abelian Group Combined with the Markov Monoid

However, although the Abelian group does not conserve the total substance or probability as the $MM$ transformations do, it still does not lead to negative states; rather, these transformations can expand or contract any axis independently with an exponential multiplier ($A\left(a\right) = \exp \left(a\right)$, where $a$ can be a positive or negative real number, making the quantity larger or smaller in that direction). It is easier to think of the conserved entity as money rather than probability; then, the Abelian group would be like having a bank infuse or remove the money supply for any component at a rate proportional to its values. One visual model in two dimensions would be to have a tank with water divided into two halves separated by a membrane—one half clear and the other half with a red dye. If the dye is on the left, this could represent a bit value of ‘1’, and if the dye is on the right, it would represent a ‘0’. Larger numbers of tanks would represent the physical storage of any number of bits of information. If a small hole is made in the membrane, then diffusion will occur so that, no matter what original state the red dye is in, eventually, the two sides will be evenly colored, and all information will be lost, as represented by the $MM$ transformation above for a two-component system representing the amount of dye in each side. The information of the system can be measured by the Renyi entropy. Now, the action of the Abelian group can allow the amount of water to grow exponentially on one side while decreasing exponentially on the other side, simultaneously with the action of the $MM$ transformation. This can be done at a rate that will exactly cancel the increase in entropy and, thus, keep the information of the system constant, as determined by the ratio of color in the two sides (effectively renormalizing the length of the vector continuously to make it have a value of unity in the computation of information).

Thus, the linear transformations that we wish to study are combinations of unequal scaling transformations acting simultaneously with the $MM$ transformations. Unequal scaling is expressed as follows:

$$A\left(t\right)= \exp \left(t\rho \left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\right)=\left(\begin{array}{cc}{e}^{+\rho }& 0\\ 0& e^{-\rho }\end{array}\right)$$

(8)

Likewise, if we assume that each component is equally likely to have the red dye (the tank is divided into two equal parts with no preference), then if follows that the $MM$ transformation is given simultaneously by

$$M\left(t\right)=\exp \left(t\sigma \left(\begin{array}{cc}-1& +1\\ +1& -1\end{array}\right)\right).$$

(9)

Acting simultaneously, we wish to study the following two-parameter transformation:

$$F\left(t\right)= \exp \left(t\left(\begin{array}{cc}\rho -\sigma & \sigma \\ \sigma & -\rho -\sigma \end{array}\right)\right)$$

(10)

5. Retention of Information in the Presence of Entropy

We have simplified the four-parameter general linear transformation by making the equilibrium state one with equal values of $x$ and $y$ and, thus, removing any asymmetry of the Markov transformation. Secondly, since information is only dependent upon the ratio of $x$ to $y$, then we can remove the overall growth transformation represented by the same exponential factor for both components, leaving only the asymmetric Abelian transformation, which is useful in increasing the information. We now explicitly understand the two components of this transformation, where the Markov component proceeds at a relative rate of ‘t’ to transform the two components of a vector into a new vector with ever-increasing nearness to the state of maximum entropy, where both components are equal and, thus, where we could not identify whether the ‘bit’ of information so represented was a ‘1’ or a ‘0’. But simultaneously with the increase in entropy, the Abelian transformation unequally acts upon the two components, reducing one by a proportional factor and increasing the other. The ($A + MM$) infinitesimal transformation is expressed as follows:

$$F\left(dt\right)=\left[\begin{array}{cc}1+dt(\rho -\sigma )& dt\left(\sigma \right)\\ dt\left(\sigma \right)& 1+dt(-\rho -\sigma )\end{array}\right]$$

(11)

where σ is the multiplier of the $MM$ transformation and $\rho$ represent the unequal growth for the Abelian group. When acting upon a vector $(x, y)$ and treating $t$ as infinitesimal, we get the following two coupled first-order differential equations:

$${\displaystyle \frac{dx}{dt}}=\left(\rho -\sigma \right)x+\sigma y$$

(12)

and

$${\displaystyle \frac{dy}{dt}}=\sigma x+\left(-\rho -\sigma \right)y,$$

(13)

which can be combined into the following single second-order equation:

$${\displaystyle \frac{d^{2}x}{{dt}^2}}+2\sigma {\displaystyle \frac{dx}{dt}}-{\rho }^{2}x=0.$$

(14)

Assuming a solution ($x\left(t\right)=A{e}^{\alpha t}$), then one finds

$$\alpha =-\sigma \pm \beta \mathrm{where} {\beta }^2=({\sigma }^2+{\rho }^2).$$

(15)

One can also expand the general linear transformation:

$$G\left(t\right)= \exp \left(t\left(\begin{array}{cc}-\sigma +\rho & \sigma \\ \sigma & -\sigma -\rho \end{array}\right)\right)$$

(16)

And by collecting terms one gets

$$G\left(t\right)=e^{\sigma t\left(\begin{array}{cc}cosh\left(\beta t\right)+{\displaystyle \frac{\alpha }{\beta }}sinh(\beta t)& {\displaystyle \frac{\left(\sigma +\nu \right)}{\beta }}sinh(\beta t)\\ {\displaystyle \frac{\left(\sigma -\upsilon \right)}{\beta }}sinh(\beta t)& cosh\left(\beta t\right)-{\displaystyle \frac{\alpha }{\beta }}sinh(\beta t)\end{array}\right)}.$$

(17)

One can prove that the removal of the overall exponential factor ($\sigma$) yields $G^{\prime }\left(t\right),$which always has unit a determinant of $\left|G^{\prime }\right(t\left)\right| = 1.$ We now wish to make two simplifications: (1) We, again, restrict the system to have a symmetric equilibrium and, thus, ${\beta }^2={\rho }^2+{\sigma }^2$. (2) Also, since the overall exponential factor does not affect the information content, which is determined by the ratio of the two components, we can remove $e^{\sigma t}$, leaving the solution as

$$G\left(t\right)=\left(\begin{array}{cc}cosh\left(\beta t\right)+{\displaystyle \frac{\sigma }{\beta }}sinh(\beta t)& {\displaystyle \frac{\sigma }{\beta }}sinh(\beta t)\\ {\displaystyle \frac{\sigma }{\beta }}sinh(\beta t)& cosh\left(\beta t\right)-{\displaystyle \frac{\sigma }{\beta }}sinh(\beta t)\end{array}\right).$$

(18)

Also, in the limit of a very large $t$, the cosh and sinh terms become equal and can be factored out with the normalization of the sum as $x^{\prime }+y^{\prime }=1$, leaving the information as

$$I={log}_2\left\{2({x^{\prime }}^2+{y^{\prime }}^2)\right\}, \mathrm{where}$$

(19)

$$x^{\prime }={\displaystyle \frac{\left\{1+\frac{\alpha }{\beta }\right\}x+\frac{\sigma }{\beta }y}{N}}\mathrm{and} y^{\prime }={\displaystyle \frac{\frac{\sigma }{\beta }x+\left(1-\frac{\alpha }{\beta }\right)y}{N}},$$

(20)

where $N =x^{\prime }+ y^{\prime }$ renormalizes the sum to unity.

If one begins with a state of perfect information ($x=1, y=0$), then using the previous equations, as $t\to \infty$ the information becomes

$$I={log}_2\left\{2{\displaystyle \frac{\left[{\left(1+\frac{\alpha }{\beta }\right)}^2+{\left(\frac{\sigma }{\beta }\right)}^2\right]}{{\left[1+\frac{\alpha }{\beta }+\frac{\sigma }{\beta }\right]}^2}}\right\}={log}_2\left\{4\left({\displaystyle \frac{(\alpha +\beta )\beta }{{(\alpha +\beta +\sigma )}^2}}\right)\right\}.$$

(21)

Thus, if we only have Markov diffusion via σ, with no Abelian unequal growth, one gets

$$I={log}_2\left(1\right)=0$$

(22)

Thus, no information and maximum entropy occur at an infinite time in the future. Thus, in the case of a pure Markov transformation with a nonzero σ but with no Abelian transformation (thus$, \alpha =0$ and $\beta =\sigma$), then $x^{\prime } = y^{\prime }$, and the limiting case has no information taken, for example, from an initial state of perfect information (1,0).

But when α is nonzero, there is always some information, as $x^{\prime }$ and $y^{\prime }$ are not zero. By adjusting the value of α, we can maintain the information level as we desire; however, the price that the system is taking a substance (e.g., energy, money, etc.) from some source at an exponential rate that cannot be maintained indefinitely.

6. Restriction to Integer Solutions (Diophantine Type Equation)

Quantization to discrete values is not just a property of quantum mechanics and the atomic and nuclear domains. Quantization occurs in almost all branches of science and for diverse reasons. Quantization specifically occurs in life forms [5]. A tree or bush does not grow one huge trunk with a single leave of ever-increasing size growing in a continuous fashion. In fact, discreteness is at the core of social and biological systems that thrive on diversity and numbers rather than a single massive entity that grows continuously. This requires a quantization into integer units of leaves, stems, seeds, pods, and component organs.

For those reasons, we now leave the continuous case and ask what integer solutions there are to a system where entropy is countered by uneven growth. Thus, we set

$$G\left(t\right)=\left(\begin{array}{cc}n& k\\ k& m\end{array}\right),$$

(23)

where $n, k, m$ are all integers for some value of t. Since we have removed the overall exponential growth, it is easy to prove that the remaining $G\left(t\right)$ has a unit determinant for all values of t as the system evolves, and this restricts us to

$$nm-k^2=1.$$

(24)

Beginning with the simplest possible solutions, we must have some minimal value of k, we can take as $k = 1, 2, 3, \dots$. One series of integer solutions for these values of $k$ is $n=k^2+1$ and $m = 1$, as this gives a unit determinant for every value of $k$ in this series, i.e.,

$$\mathrm{Det} G\left(t\right)=\left(k^2+1\right)\left(1\right)-k^2=1.$$

(25)

One notes that with each increasing value of k, one is introducing larger and larger rates of entropy increase over time; thus, the required values must increase accordingly. Solutions for $n=k^2+1$ and $m = 1$ and $k = \mathrm{1,2},3, \dots$ yield

$$G\left(t\right)=\left(\begin{array}{cc}cosh\left(\beta t\right)+{\displaystyle \frac{\alpha }{\beta }}sinh(\beta t)& {\displaystyle \frac{\sigma }{\beta }}sinh(\beta t)\\ {\displaystyle \frac{\sigma }{\beta }}sinh(\beta t)& cosh\left(\beta t\right)-{\displaystyle \frac{\alpha }{\beta }}sinh(\beta t)\end{array}\right)=\left(\begin{array}{cc}n& k\\ k& m\end{array}\right),$$

(26)

one obtains the following:

$$2\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(\beta t\right)=n+m=k^2+2 \mathrm{or} \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(\beta t\right)={\displaystyle \frac{\left(k^2+2\right)}{2}}$$

(27)

$$\left({\displaystyle \frac{2\alpha }{\beta }}\right)\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\beta t\right)=k^2 \mathrm{or} \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\beta t\right)={\displaystyle \frac{\beta k^2}{2\alpha }},$$

(28)

$$\left({\displaystyle \frac{\sigma }{\beta }}\right)\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\beta t\right)=k \mathrm{or} \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\beta t\right)={\displaystyle \frac{\beta k}{\sigma }}, \mathrm{and} \mathrm{thus} 2\alpha =\sigma k, \mathrm{where}$$

(29)

$${\beta }^2={\alpha }^2+{\sigma }^2={\alpha }^2+{\left({\displaystyle \frac{2\alpha }{k}}\right)}^2.$$

(30)

The $t$ parameter is not determined explicitly, as it is adjustable with time. One can select units of time that correspond to a $\sigma =1$ rate of diffusion and let that rate scale the time parameter. Our results do not depend upon an explicit value of $t$. Thus, the equilibrium ratio at $t=\infty$ is determined beginning with a vector of compete information (1,0) and transformed to $(x^{\prime }, y^{\prime })$ at $t=\infty$ to give

$$x^{\prime }=\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(\beta t\right)+\left({\displaystyle \frac{\alpha }{\beta }}\right)\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\beta t\right)\mathrm{and} y^{\prime }=\left({\displaystyle \frac{\sigma }{\beta }}\right)\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\beta t\right);\mathrm{t}\mathrm{h}\mathrm{u}\mathrm{s},$$

(31)

$${\displaystyle \frac{x^{\prime }}{y^{\prime }}}={\displaystyle \frac{\beta }{\sigma }}+{\displaystyle \frac{\alpha }{\sigma }}={\displaystyle \frac{k}{2}}+\sqrt{{\left({\displaystyle \frac{k}{2}}\right)}^2+1} \mathrm{for} k=\mathrm{1,2},3,\dots$$

(32)

$$\mathrm{With} \mathrm{information} I={log}_2\left\{{\displaystyle \frac{2\left({x^{\prime }}^2+{y^{\prime }}^2\right)}{{\left(x^{\prime }+y^{\prime }\right)}^2}}\right\}.$$

(33)

7. Fibonacci and Related Sequences as Solutions ($\mathit{k}\mathbf{=}\mathbf{1}$)

We will now explore the limiting value of the ratio of

$${\displaystyle \frac{x}{y}}={\displaystyle \frac{k}{2}}+\sqrt{{\left({\displaystyle \frac{k}{2}}\right)}^2+1} \mathrm{for} k=\mathrm{1,2},3,\dots ,$$

(34)

(which we define as ${\Phi }_k$), which shows that the information level is stabilized at these fixed values as $t\to \infty$, beginning with a state of perfect information (1,0). Thus, the simplest result of combating diffusion with uneven growth is with

$$k=1, \mathrm{where} \mathrm{we} \mathrm{get} {\displaystyle \frac{x^{\prime }}{y^{\prime }}}$$

(35)

Setting the ratio of the limited values to be

$${\Phi }_1={\displaystyle \frac{1}{2}}+\sqrt{{\left({\displaystyle \frac{1}{2}}\right)}^2+1}={\displaystyle \frac{1+\sqrt{5}}{2}}=1.618034\dots$$

(36)

which is often called the golden ratio or golden mean, which is the limiting ratio of adjacent Fibonacci numbers.

By taking the matrix expressed as

$$G=\left(\begin{array}{cc}{k}^2+1& k\\ k& 1\end{array}\right) \mathrm{for} k=1,$$

(37)

one obtains the Fibonacci sequence in higher powers of this matrix as follows:

$$\left(\begin{array}{cc}2& 1\\ 1& 1\end{array}\right),\left(\begin{array}{cc}5& 3\\ 3& 2\end{array}\right),\left(\begin{array}{cc}13& 8\\ 8& 5\end{array}\right), \mathrm{etc}., \mathrm{or}$$

(38)

$$\mathrm{1,1},\mathrm{2,3},5, 8, 13, 21\dots , \mathrm{where} n=n_{-1}+n_{-2}(\mathrm{i}.\mathrm{e}., \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{s}\mathrm{u}\mathrm{m} \mathrm{o}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t} \mathrm{t}\mathrm{w}\mathrm{o} \mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}\mathrm{s}).$$

(39)

Let us now explore higher order Fibonacci-type sequences with $k= 2$ to obtain the next-order golden ratio:

$${\displaystyle \frac{k}{2}}+\sqrt{{\left({\displaystyle \frac{k}{2}}\right)}^2+1} \mathrm{with} k=2 \mathrm{or}$$

(40)

Φ₂ = 1 + √2 = 2.4142.

(41)

The $k=2$ associated sequence is given by the powers of $\left(\begin{array}{cc}5& 2\\ 2& 1\end{array}\right)$, which yield the following sequence:

$$\mathrm{0,1},\mathrm{2,5},\mathrm{12,29,70,169},\dots \mathrm{where} n-2n_{-1}+n_{-2},$$

(42)

i.e., twice the last value plus the value before that.

Now, for the next level with k = 3, we obtain the next-order golden ratio as follows:

$${\Phi }_3={\displaystyle \frac{k}{2}}+\sqrt{{\left({\displaystyle \frac{k}{2}}\right)}^2+1}={\displaystyle \frac{3+\sqrt{13}}{2}}=3.3027756.$$

(43)

The associated $k=3$ sequence is given by the powers of $\left(\begin{array}{cc}10& 3\\ 3& 1\end{array}\right)$, which yield the following sequence:

$$\mathrm{0,1},\mathrm{3,10,33,109},\dots \mathrm{where} n=3n_{-1}+n_{-2}.$$

(44)

Recalling that we defined the $k$th-order golden mean as

$${\Phi }_k={\displaystyle \frac{k}{2}}+\sqrt{{\left({\displaystyle \frac{k}{2}}\right)}^2+1} \mathrm{for} k=\mathrm{1,2},3,\dots ,$$

(45)

one can easily observe that one can form a right triangle with sides of 1 and $k/2$and that this will give a hypotenuse of √((k/2)² + 1) $\sqrt{{\left(k/2\right)}^2+1}$. Thus, the sum of the k/2 side plus the hypotenuse of these triangles so proportioned will give geometrically the exact value of the golden mean for any value of k relative to the third side with a value of unity. That type of construction for the Fibonacci golden mean is well known and is the foundation of multiple geometrical constructions, which can now be easily extended to these other Fibonacci-like sequences and ratios.

8. General Discussion of the Associated Generating Functions and Differential Equations for the Fibonacci Sequences of Different Orders

The general form of interpolating functions for Fibonacci sequences of different order and the associated differential equations have already been shown to be

$$G\left(\epsilon \right)=\exp \left(\begin{array}{cc}1+\epsilon (\alpha -\sigma )& \epsilon (\sigma +\upsilon )\\ \epsilon (\sigma -\upsilon )& 1+\epsilon (\alpha -\sigma )\end{array}\right),$$

(46)

and when acting upon a vector $(x, y)$ and treating ε as an infinitesimal value of $t$, one can get two coupled first-order differential equations:

$${\displaystyle \frac{dx}{dt}}=\left(\alpha -\sigma \right)x+\left(\sigma +\upsilon \right)y \mathrm{and} {\displaystyle \frac{dy}{dt}}=\left(\sigma -\upsilon \right)x+\left(\alpha -\sigma \right)y.$$

(47)

These can be combined into the single second-order equation:

$${\displaystyle \frac{d^{2}x}{{dt}^2}}+2\sigma {\displaystyle \frac{dx}{dt}}+\left(-2\gamma \sigma -{\alpha }^2+{\upsilon }^2\right)x=0.$$

(48)

Assuming a solution of

$$x\left(t\right)=A{e}^{\rho t}, \mathrm{then} \mathrm{one} \mathrm{finds}$$

(49)

$$\rho =\left(\sigma -\gamma \right)\pm \beta , \mathrm{where} {\beta }^2={\alpha }^2+{\sigma }^2-{\upsilon }^2.$$

(50)

The general solution to this expansion is given by

$$G\left(t\right)=\exp \left(t\left(\begin{array}{cc}\gamma +\alpha & \sigma +\upsilon \\ \sigma -\upsilon & \gamma -\alpha \end{array}\right)\right) \mathrm{or}$$

(51)

$${G\left(t\right)=e}^{\left(\sigma -\gamma \right)t\left(\begin{array}{cc}cosh\left(\beta t\right)+{\displaystyle \frac{\alpha }{\beta }}sinh(\beta t)& {\displaystyle \frac{\left(\sigma +\upsilon \right)}{\beta }}sinh(\beta t)\\ {\displaystyle \frac{\left(\sigma -\upsilon \right)}{\beta }}sinh(\beta t)& cosh\left(\beta t\right)-{\displaystyle \frac{\alpha }{\beta }}sinh(\beta t)\end{array}\right)}.$$

(52)

The parameters are set to the discrete values as

$$\left(\begin{array}{cc}n& k\\ k& m\end{array}\right)=\left(\begin{array}{cc}{k}^2+1& k\\ k& 1\end{array}\right) \mathrm{for} \mathrm{any} \mathrm{integer} \mathrm{value} \mathrm{of} k.$$

(53)

It follows that as time evolves, the continuous functions pass through each of the new discrete quantized sets of integer values when time itself is an integer multiple of the first value. This follows because the first value of the time gives the first matrix, so multiples of that time value give products of the original matrix. It also follows that the functions that are elements of $G\left(t\right)$ are interpolating functions of all of the multiple Fibonacci sequences, although it should be noted that these functions pass through alternate values of each of the Fibonacci sequences.

9. Prediction of a Possible Occurrence of ${\mathit{\Phi }}_{\mathit{k}}\mathbf{=}\mathit{k}\mathbf{/}\mathbf{2}\mathbf{+}\sqrt{{\mathbf{(}\mathit{k}\mathbf{/}\mathbf{2}\mathbf{)}}^{\mathbf{2}}\mathbf{+}\mathbf{1}} \mathbf{f}\mathbf{o}\mathbf{r} \mathit{k} \mathbf{=} \mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}$ in Nature

It occurs to one that these higher order Fibonacci type sequences of numbers, as well as generalized golden mean values, might also be observed in nature, just as we find many instances of Fibonacci numbers and the first golden mean of ${\Phi }_1=1.618$. We can conjecture that this might occur for reasons like those that initiated this investigation—namely, the fact that some systems that experience increasing entropy in nature could stabilize their ‘information content’ or ‘internal order’ by utilizing unequal growth from sources and sinks utilizing the derived Abelian scaling transformations. It is probably more unusual to find such higher order generalized Fibonacci-type sequences and the next few golden means, but it could be an important prediction. That is because this prediction is of something that could be observed in nature that is based almost exclusively on mathematical reasoning and devoid of experimental data. It simply rests on the fact that entropy increases in nature and that living things must be able to sustain their information content and internal structure over time in order to ‘survive’, not unlike how standing waves on a string give f_n = nf₁. In fact, this is the most fundamental aspect of life: that it can maintain order. One of the very interesting properties of the golden mean (${\Phi }_1$) is that ${\Phi }_1=1+{\displaystyle \frac{1}{{\Phi }_1}}$. Now, using the general equation expressed as

$${\Phi }_k={\displaystyle \frac{k}{2}}+\sqrt{{\left({\displaystyle \frac{k}{2}}\right)}^2+1} \mathrm{for} k=\mathrm{1,2},3, \mathrm{one} \mathrm{can} \mathrm{easily} \mathrm{prove} \mathrm{that} {\Phi }_k=k+{\displaystyle \frac{1}{{\Phi }_k}}.$$

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10. Another Generator and Fibonacci Interpolating Functions

The generator that we have been using,

$$L=\left(\begin{array}{cc}1/2& 1\\ 1& -1/2\end{array}\right), \mathrm{has} \upsilon =0 (\mathrm{no} \mathrm{asymmetry} \mathrm{in} \mathrm{the} \mathrm{entropy} \mathrm{equilibrium}), \mathrm{and}$$

(55)

$$\gamma =1 (\mathrm{thus}, \gamma +\sigma =0 \mathrm{and} \mathrm{no} \mathrm{overall} \mathrm{exponential} \mathrm{growth} \mathrm{or} \mathrm{decay} \mathrm{of} \mathrm{the} \mathrm{system}).$$

(56)

We now wish to look at this same matrix but with an additional overall growth factor of $\gamma =2$, giving $L=\left(\begin{array}{cc}1& 1\\ 1& 0\end{array}\right)$. This new $L$ is an interesting Lie algebra generator because when acting as a discrete transformation on (# of old rabbit pairs, # new rabbit pairs), it gives the original Fibonacci numerical sequence in both components when beginning with $\left(\mathrm{0,1}\right)$, i.e., one new pair of rabbit pairs. The sequential discrete action is ($0.1), (\mathrm{1,0}), (\mathrm{1,1}), (\mathrm{2,1}), (\mathrm{3,2}), (\mathrm{5,3}), \dots .$etc., obeying the original rule that a new pair must age into an older pair for one cycle and that every old pair creates a new pair every cycle. When taken as the generator of a continuous Lie group transformation, it gives an interesting set of Fibonacci interpolating functions that are exponential expansion terms multiplied by Fibonacci numbers:

$$G\left(t\right)= e^{tL}=\left(\begin{array}{cc}{f}^{″}\left(t\right)& f^{\prime }\left(t\right)\\ f^{\prime }\left(t\right)& f\left(t\right)\end{array}\right) \mathrm{where}$$

(57)

$$f\left(t\right)=1+0t+1{\displaystyle \frac{t^2}{2!}}+1{\displaystyle \frac{t^3}{3!}}+2{\displaystyle \frac{t^4}{4!}}+3{\displaystyle \frac{t^5}{5!}}+5{\displaystyle \frac{t^6}{6!}}+\cdots ,$$

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where $f^{\prime }$ and $f″$ are the first and second derivatives of this function, which interpolates the Fibonacci numbers.

By returning to the original form of $G\left(t\right)$ in terms of the cosh and sinh functions, one can write this in that previous form. One notes that this Lie generator, i.e.,

$$L\left(1\right)=\left(\begin{array}{cc}1& 1\\ 1& 0\end{array}\right), \mathrm{has} \mathrm{the} \mathrm{property} \mathrm{that} {L\left(1\right)}^2=\left(\begin{array}{cc}2& 1\\ 1& 1\end{array}\right)$$

(59)

and, thus, is the ‘square root’ of the defining Diophantine equation.

11. Generators of the Generalized Fibonacci Numbers

One recalls that we found the generalized Diophantine sequential matrices to be of the form of $R=\left(\begin{array}{cc}{k}^2+1& k\\ k& 1\end{array}\right)$, and one can easily show that $L\left(k\right)=\left(\begin{array}{cc}k& 1\\ 1& 0\end{array}\right)$ is such that ${L\left(k\right)}^2=R$. This suggests that this Lie generator will provide the expanded forms of the Fibonacci generalized functions. One notes that this $L\left(k\right)$ for $k=1$ gives the previous traditional Fibonacci sequence where each number is the sum of the two previous values. This also follows from the fact that this $L\left(k\right)$ gives a sequence where each generalized Fibonacci number is equal to $k \ast$ the previous value plus the second previous value (i.e., $F\left(n\right)=k\ast F\left(n-1\right)+F(n-2)).$ Thus, we can generalize our previous result as

$$G\left(t\right)=e^{tL\left(k\right)}=\left(\begin{array}{cc}{f_k}^{″}\left(t\right)& {f_k}^{\prime }t)\\ {f_k}^{\prime }\left(t\right)& f_k\left(t\right)\end{array}\right), \mathrm{where} \mathrm{now},$$

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$$f_k\left(t\right)=1+0t+1{\displaystyle \frac{t^2}{2!}}+k{\displaystyle \frac{t^3}{3!}}+\left(k^2+1\right){\displaystyle \frac{t^4}{4!}}+\left(k^3+2k\right){\displaystyle \frac{t^5}{5!}}+\left(k^4+3k^2+1\right){\displaystyle \frac{t^6}{6!}}+\cdots ,$$

(61)

where, again$, f″$ and $f^{\prime }$ represent the second and first derivatives of $f$.

12. Linear Transformations That Include the ‘Source’ and ‘Sink’ of Order

So far, we have only looked at a two-component system with internal diffusion $(x_1, x_2)$, and we have treated the ‘source’ and ‘sink’ of energy, money, or some entity or order as external to this system. Here, we close the system by adding a source component; $y_1$—the Sun, a bank, or some source of ‘order’) and a ‘sink’ component ($y_2$) for that same entity. Thus, now, we have a closed system that conserves the entity with transformations on the four-dimensional vector $(y_1, y_2, x_1, x_2).$The four-by-four infinitesimal transformation that does this is

$$L=\left(\begin{array}{cccc}1& 0& -\alpha & 0\\ 0& 1& 0& +\alpha \\ 0& 0& 1+\alpha -\sigma & +\sigma \\ 0& 0& +\sigma & 1-\alpha -\sigma \end{array}\right)$$

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One can verify that the sum of each column is equal to one and, thus, this is a Markov-type transformation. The transformation on $(x_1, x_2)$ is the same as before, but recall that we ‘infused’ a fraction of $x_1$, $\alpha$, into $x_1$ and removed a fraction $\alpha$ of $x_2$ from $x_2$. While the fractions were the same, they were fractions of two different numbers; thus, the quantities were not equal. In the new matrix transformation, fraction $\alpha$ of $x_2$ that was removed from $x_2$ is now transferred into (the sink of the order) $y_2$, thereby conserving that quantity. That action is still in keeping with a true Markov transformation because a positive fraction of one component ($x_2)$ is transferred to another component $\left(y_2\right)$. However, the transformation infuses $x_1$ at a positive rate ($+\alpha$) proportional to $x_1$itself, which, when exponentiated, leads to exponential growth of $x_1$. That infused substance comes from $y_1$; thus$, y_1$ loses substance at a rate proportional to $x_1$. If $x_1$ is larger than $y_1$, then $y_1$ will become negative; thus, this transformation is outside of the Markov monoid and is among the Markov-type transformations that preserve the sum of components but not their positive definiteness. So, this part of the transformation must rely on the fact that the ‘source of entity’ ($y_1$) must be so large that during the time of action of the transformation, $y_1$ will not be used up and become negative due to the exponential growth of $x_1$. This is not a problem if the resource ($y_1$) is many orders of magnitude larger than the subsystem $(x_1,x_2)$. An analogy would be the Sun as a source of energy ($y_1$) and the Earth (or empty space) as the sink ($y_2$). Thus, at some point, the subsystem that is maintaining the information level $(x_1,x_2)$ must transfer its information content to a much smaller system $(z_1,z_2)$ while maintaining the ratio of the two components with ${\displaystyle \frac{z_1}{z_2}} = {\displaystyle \frac{x_1}{x_2}}.$ Thus, the original system must cease to exist at some point and must reproduce, i.e., pass its ‘information content’ to a new small system that can then exponentially grow.

13. Conclusions

Within the framework of the continuous general linear (Lie) group acting just on a two-dimensional space, we have studied how the Abelian group of diagonal scaling transformations, acting unequally upon the components, can counter the loss of information and, thus, the natural increase in entropy that obtains from different levels (k) of continuous Markov transformations:

$$G\left(t\right)=e^{t\left(\begin{array}{cc}\alpha & \sigma +\upsilon \\ \sigma -\upsilon & -\alpha \end{array}\right)}.$$

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We concentrated on cases where the entropy equilibrium was equal in both components and, thus, $\upsilon =0$. We sought to what extent information would be retained if by different actions of the scaling transformation via the $\alpha$ parameter, and we found that any non-zero value altered the equilibrium and retained some information. An interesting result followed when we asked what integer (Diophantine)-type solutions might exist, and we found an infinite sequence based upon an integer expressed as $k = \mathrm{1,2},3, \dots ,$ where $k=1$ reproduced the Fibonacci sequence and other results such as the associated interpolating functions, the golden ratio, and geometric constructions. The natural conjecture is that perhaps the presence of the observation of the Fibonacci sequence in nature arises from an attempt to counter increasing disorder and loss of information by utilizing an unequal source and sink entity such as energy. While we know that this happens with all living things, this a possible source of the Fibonacci sequence in nature. Based upon this Lie algebra decomposition of the general linear transformations, we were able to consider higher order levels of entropy indicated by $k = 1, 2, 3, \dots$, and ask what Diophantine (integer) solutions issue. We found solutions for each value of k and the associated generalized Fibonacci sequence, generating and interpolating functions, generalized golden means, and geometric constructions. These generalizations provide much deeper insight into the Fibonacci numbers and the related mathematics.

The most interesting conjecture from this work is that if it is true that the Fibonacci sequence is a methodology of retaining order in living systems, then one might be able to observe higher order generalized sequences and their geometric constructs in terms of elemental triangular ratios and the generalized values of the golden means. If these could be observed, it would be very remarkable that this rather pure mathematical model based upon the concept of entropy and information retention and without any data input from nature could predict something observable in the physical world.

It is obvious that although this investigation has only looked at a two-component system, similar to a single bit of information stored in a binary form, such elementary systems could be collected to represent complexity at any level, as with molecular structures and vast sequences of coded magnetic bits of information. This two-node network is easily generalized, and very complex systems can be constructed from such elemental systems, allowing them to maintain any possible level of information and, thus, order and structure. The author has been able to show that every possible network [8] among nodes (as represented by non-negative, non-diagonal values of a square connection matrix ($C_{ij}$)) is isomorphic to a Markov monoid. Even more general networks allow for Abelian diagonal transformations, as discussed here. Thus, this simple exchange between the $x$ and $y$ coordinates could be thought of as an exchange between any two given nodes of a more complex network; thus, order could be maintained in a network of any complexity by appropriately ‘feeding substance’ to nodes at specific rates to counter entropy of the network diffusion. The organizational structure of such networks could represent interesting problems in complex systems that are more solvable due to the underlying coupled linear systems represented here. Thus, this investigation is relevant to other investigations in network theory. This is even more valid, since social networks form and expand in order to increase the survival of the system. We are currently studying information retention in the presence of entropy in larger networks.