^{1}

^{2}

^{1}

^{2}

Aging as the process in which the built-in entropy decreasing function worsens as
internal time passes. Thus comes our definition, “life is a one way
flow along the intrinsic time axis toward the ultimate heat
death, of denumerably many metabolic reactions, each at local
equilibrium in view of homeostasis”. However, our disposition is not of reductionismic as have been
most of approaches, but it is to the effect that such a complicated
dynamic system as lives are not feasible for modelling or reducing
to minor fragments, but rather belongs to the whole-ism.
Here mathematics can play some essential role because of its
freedom from practical and immediate phenomena under its own nose.
This paper is an outcome of hard trial of
mathematizing scientific disciplines which would allow description of life in terms
of traditional means of mathematica, physics. chemistry, biology etc.
In the paper, we shall give three basic math-phys-chem approaches
to life phenomena, entropy, molecular orbital method and formal language
theory, all at molecular levels. They correspond to three mathematical dsciplines—probability, linear algebra and free groups, respectively. We shall give some basics for the Rényi (

Life science seems to have been prevailing the modern science, which incorporates a great number of relevant subjects ranging from molecular biology to medicine, all of which seem to belong to “reductionism,” that is, “the whole is the totality of its parts.” Molecular biology presupposes, “genotype determines phenotype,” namely, that the gene codes (codons for amino acids) preserved in DNA determine all the phenomenal aspects of the living organisms which are designed by these codes. A traditional way that biology has been tracking is that of “classifying” creatures according to their “species” and molecular biology has been classifying the ingredients in the same spirit but at much smaller, ultramicroscopic level.

Classification is one of the most effective powers of mathematics. This is because one of the main objectives of mathematics is to make a classification of objects of study by sorting out some common features—structures, symmetries—from them to classify them, thereby use is made of neglecting irrelevant specificities and extracting the properties uplifted to absolute abstraction.

In extracting common features of a class of objects of study, mathematicians often appeal to analogy. This reminds us of a seemingly forgotten way of thinking in [

The description of roles of disciplines stated in [

In this paper, we will confine ourselves to a few selected constituents of living organisms. As one of main objects of study, one may take up cells and their functions. The reason can be given plenty. They are first of all still visible by microscopes and can be studied as manifestation of reductionism. There are 7 thousand billion cells in the human body and cell membranes play essential roles in maintaining life. The cells have internal and external membranes mainly made of lipids, polysaccharides, proteins, and so forth. Among these ingredients, we will be most interested in lipids and proteins, the first because the oxidation of lipids would lead to malfunction of the cells and the second because the proteins are polymers consisting of 20 basic amino acids joined by peptide bonds and it has been made clear that the production and properties of amino acids are dependent on the codons which are used (see e.g., [

An amino acid is a compound consisting of two parts—constant and variable, where the constant part comprises of ACH, an amino group, a carboxyl group, and a hydrogen atom, while the variable part consists of a side chain which appears in 20 flavors, thus yielding 20 basic amino acids.

A heteropolymer is an assemblage of several kinds of standard molecules—monomers—building a connected chemically homogeneous backbone with short branches attached to each monomer of the backbone.

We may summarize this in Table

Polymers | Alphabets | Backbone | Branches | Bond |

Polynucleotide (single-stranded DNA) | Sugar-phosphate | Bases | Phosphate covalent | |

Polypeptide (linear protein) | Codons (triplets) | ACH | Side chains | Peptide |

Motivated by the way in which the three important factors are treated, that is, circular and linear DNA strings [

In Section

In Sections

In Section

In Section

One of the objectives of this paper is to show freedom as well as power of mathematics for treating seemingly irrelevant disciplines. It is freed from realistic restrictions which always show their effect on researches in other akin science, physics, chemistry, and so forth. We hope we have shown that the more complicated the situation is as life, the more feasible for it is mathematics.

In [

If a choice is broken down into two successive choices, then the original

The only

We note that simultaneously with and independently of, Shannon, the same result was obtained by N. Wiener. It was Fadeev [

The proof of a more general theorem of Rényi (Theorem

An arithmetic function, that is, a function defined on the set of natural numbers with complex values, is called an

By the fundamental theorem in arithmetic it is clear that an additive function is completely determined by its values at prime power arguments, and a completely additive function by its values at prime arguments. Indeed, if

If an additive arithmetic functions

It suffices to prove (

We construct the strictly decreasing sequence

By the additivity of

Now the double sum on the right of (

In view of (

Also the number

It remains to estimate (

A finite discrete generalized probability distribution

We will characterize the entropy (of order 1)

if

for

if

The only

Let

To prove (

Thus by Theorem

For an ordinary distribution, Theorem

It suffices to deduce (iii) in Theorem

(i) We note that (

Indeed, writing

(ii) As stated in [

(iii) Definition

It would look natural to extend the arithmetic mean in (

We may replace Postulate (iv) above by (iv’) If

The only

Since

A complete characterization of

As is stated in [

Quantities of the form

We now give a brief description of elements of thermodynamics from Boltzmann’s standpoint (see e.g., [

All the natural phenomena have the propensity of transforming into the state with higher probability, that is, to the state with higher entropy. This is often recognized as the entropy increase principle.

Let

The

He proved.

We state a heuristic argument [

In statistical mechanics, macrostates (properties of large number of particles such as temperature

Suppose that the

Since

Boltzmann proved.

We have the relation:

Theorems

The maximum of the entropy (

Since we have the constraint

Equation (

This section is devoted to a clear-cut exposition of energy levels of molecular orbitals of hydrocarbons (carbon-hydrides) and is an expansion of [

We will consider the difference between energy levels of molecular orbitals (MOs) of a chain-shaped polyene (e.g., 1,3-butadiene) and a ring-shaped polyene (e.g., cyclopentadienylanion) in Section

In quantum mechanics, one assumes that the totality of all states of a system form a normed

We deduce the secular determinant for the molecular orbital

Hereby we also incorporate the simple

With all above simplifications incorporated, the secular determinant reads

In Section

This section is an extract from [

There is enormous amount of literature on the golden ratio and the Fibonacci sequence most of which are speculative. We mention a somewhat more plausible and persuasive statement in [

Living organisms, and a fortiori, their descriptions in various media such as paintings, sculptures, and so forth are to be inscribed into pentagons, which are the governing frame of living organisms and which control their structure as a hierarchical overstructure and, as a result, the golden ratio appears as the intrinsic lower structure wherever there are pentagons.

By Theorem

Putting

By Theorem

As we will see in Section

Hence numerical values of energy levels are

On the other hand, to find energy levels of molecular orbitals of the cyclopentadienylanion, direct computation is possible, but we prefer to apply the theory of circular matrices as in Section

By Theorem

As in Section

Regarding

Thus we appeal to the following theorem making use of Chebysëv polynomials.

Let

By standard technique,we may deduce the recurrence

For

In the case of

In the case

Hence it follows that

On the other hand, to find molecular orbitals of the benzene, we may apply the theory of circulant matrices.

For

Note that

Letting

Any circulant matrix

Letting

For

In deducing Theorem

In this section we assemble some basics on the Chebyshëv polynomials to an extent for enabling to understand the computations in Section

If

The notation is after Tchebyshef (or Tschebyscheff) who first introduced them, proper transcription being

We point out that most of the identities for the Chebyshev polynomials are rephrases of the well-known trigonometric identities. For example, the second recurrence in (

As an important case, we rephrase the identity (which follows from addition theorem)

Thus, all the results on

Since it turns out that it is usually easier to work with

We note that although (

We have the following concrete expressions:

If

We find the values of

Solving the equation

We have a companion formula to (

Since the coefficients in (

In [

As opposed to the familiar Cartesian product, the free product is the most general construction from a given family of sets. It is indeed a dual concept of the direct product in case of groups.

Let

In the case of codons, we have

Now we go on to the notion of free groups. Given a family of groups

Thus, as stated in [

On [

A penetrating definition is essential to describing the whole realm of a discipline. We may recall the first passage from Pauling [

The universe is composed of substances (forms of matter) and radiant energy.

As in [

It may be true, however, that the passage is to be modified according to the modern 20th century physics that matter and energy are verbatim—fermions and add information—bosons to rephrase it:

The universe is composed of energy and information,

Still the first passage helps to have a grasp of the whole picture.

The ultimate objective of all sciences would be attaining “immortality” or at the very least “longevity in good health.” To achieve this, it is necessary to know what life (process) is. In this section we will try to formulate a proper enlightening definition of life by incorporating several ones claimed before.

We first state rather virtual and speculative definition in [

A “living being” is any entity which codes information (in the physics sense of this word) with the information coded being preserved by natural selection. Thus life is a form of information processing, and the human mind—and the human soul—is a very complex computer program. Specifically, a “person” is defined to be a computer program which can pass the Turing test.

This is rather against the classical definition of life as a complex process based on the chemistry of carbon atoms. In [

Life is information preserved by natural selection.

As to the classical definition in terms of carbon atoms, it would be quite natural to go on to the booklet of Carbone and Gromov [

“The dynamics of the cell is a continuous flow of small molecules channeled by the interaction with macromolecules: DNA, RNA and proteins. The behavior of small molecules obeys the statistical rules of chemical kinetics,

As mentioned in Abstract, we adopt the notion of entropy to view it, incorporating the ideas of Schoenheimer of “dynamic state of body constituents” [

On [

Life is a flow in dynamic equilibrium.

This definition resembles the Carbone-Gromov definition of cell dynamics in that both refer to “flow.” It gives, however, an impression that equilibrium is already attained and it should mean local equilibrium. We need to incorporate the ultimate equilibrium, death, which could be compared to heat death [

However, we have a much better and penetrating metaphor in beautiful prose by a Japanese hermit-essayist in the 16th century. It reads:

The river never ceases to flow, its elements never remaining the same.

The foams that it forms appear and disappear constantly and never be stable.

As such are the life and its vessel.

The river is a human adult body with water supply corresponding to food supply. The foams correspond to various chemical reactions that take place in the body: regeneration and degradation. Only oxidation part is missing which is replaced by intensity of flow generated by the mass of water. Although this prose originally was to express the frailty of life, it literally describes the life process as seen by Schoenheimer.

Thus comes our definition of life:

Life is a constant irreversible flow, along the axis of internal time, of resistance against the entropy increase leading to the ultimate heat death, in terms of homeostasis to keep the local equilibrium which works to balance the regeneration and degradation of molecules using the energy produced by oxidating the intake material, where the synthesis is conducted according to the

Life is a dynamic system with which the negentropy is supplied by degrading and regenerating its components and excreting the waste before they could be damaged by disturbances from outside, making the inner entropy increases.

We will explain why we have come to this definition which incorporates many ingredients scattered around in the literature.

Internal time clock idea came from [

In [

Life is an irreversible flow of dynamically integrated aggregates of local equilibria maintained by homeostasis.

Aging is a malfunction of homeostasis caused by the elapses of internal time.

We do hope by elucidating life activities to get the process of aging back, that is, our wishful definition of life is the following.

Life is a one-way flow of dynamically integrated aggregates of local equilibria maintained by homeostasis, the flow being slowed down by due care of body and mental health.

To formulate “replicative stability of dynamical systems’’ a slightly modified Carbone-Gromov suggestion [

There is criticism about the evolution theory that it is a tautology saying that those which are likely to survive, or those which survived are judged to be the most fitting. However, it seems that those which are likely to occur, that is, with higher occurrence probability occur more frequently than those which are less likely to occur (with lower probability). When there are several events which are equally likely to occur, then it will be the most natural that all events occur in the long run. The more the events, the more the choices, or uncertainty, whence if there is means of measuring the tendency of occurrence of events, then it is to be an increasing functions of the number of events. Shannon [

On [

We adopt the standpoint of [

We take in food—material of smaller entropy—in our bodies to burn (oxidize, oxidate) it to produce energy. Here a remark is due on the entropy description. Food is material of smaller entropy for the sources it come from, but for our body it may be a big noise and therefore, our oxidation system oxidizes it to produce material of bigger entropy which is to be excreted from the body. For example, glucose (of lower entropy) is absorbed through cell membranes and will get oxidized to become carbon dioxide

In [

A biological system represents one big cycle of closely linked chemical reactions.

After death, when the oxidative systems disappear, the synthetic systems also cease, and the unbalanced degenerative reactions lead to the collapse of the “thermodynamically unstable structure elements.”

Thus we may duly call the ultimate death “heat death” and understand the life process as a flow of many chemical reactions in local equilibrium.

Thus aging is to mean the

There may be many causes that give rise to the malfunction of the homeostasis. One typical example is the attacks of