Math-Phys-Chem Approaches to Life

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 manymetabolic 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 α -entropy, Chebyshev polynomials and the notion of free groups inrespective places. Toward the end of the paper, we give some of our speculations on life and entropy increase principle therein. Molecular level would be a good starting point for constructing plausible math-phys-chem models.


Introduction
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 Motivated by the way in which the three important factors are treated, that is, circular and linear DNA strings 4, page 19 and in 5, page 741 , entropy 4, page 55 , coupled with a rather speculative definition of life in 6, pages 124-128 as information preserved by natural selection, we will dwell on the following mathematical stuff which correspond to the respective notions.
In Section 2, we adopt Renyi's theory of incomplete probability distribution to be compatible with and match the real status of life, expounding the notion of entropy in, and evolution-theoretic aspects of, life.
In Sections 3 and 4.2, we will outline the theory of energy levels of carbon hydrides based on the theory of Chebyshëv polynomials as developed in 7, Chapter 1 comparing the levels of polygonal and circular carbon hydrides. It is hoped this analysis will shed some light on the corresponding problem of linear and circular DNA. In Section 5, we provide some unique exposition of the Chebyshëv polynomials to such an extent that will be sufficient for applications.
In Section 6, we state mere basics of free groups as opposed to direct i.e., Cartesian products 4, page 44 of many copies of an attractor.
In Section 7, we assemble some meaningful definitions of life from varied disciplines. 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.

Shannon's Entropy
In 8 , Shannon developed mathematical theory of communication. Suppose, we have a set of possible events whose occurring probabilities are p 1 , . . . , p n , 0 < p k < 1 with We note that simultaneously with and independently of, Shannon, the same result was obtained by N. Wiener. It was Fadeev 9 who formulated Shannon's theorem in the axiomatic way as above. The base 2 is preferred because they were interested in the switching circuit, on and off. For postulate iii , c.f. 2.36 below and Remark 2.7, i .
The proof of a more general theorem of Rényi Theorem 2.5 below as well as this theorem is easy except for one intriguing number-theoretic result originally due to Erdös 10 . We give a proof slightly modified yet in the spirit of Rényi's well-known proof in the case of additive functions. for all relatively prime pairs m, n, that is, the gcd of m and n, denoted by gcd m, n is 1. If f satisfies 2.5 for all m, n, it is called a completely additive function.
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 n p α 1 1 · · · p α k k is the canonical decomposition into prime powers of n, then we have  Proof. It suffices to prove 2.8 for n a prime power, that is, for all prime powers p k . We fix p k and prove that as n → ∞, where we set g n f n − f p k log n log p k .

2.11
Since Δg n Δf n − f p k / log p k log 1 1/n , 2.8 for g also holds true by 2.8 for f. Further, g vanishes at n p k :g p k 0. We construct the strictly decreasing sequence {q j } of successive quotients of n divided by p kj . By the Euclidean division, q j p k q j 1 r j , 0 ≤ r j < p k , 2.12 starting from j 0 with n q 0 , where q j 1 q j /p k . Let r denote the greatest integer such that p k r−1 ≤ n. Then solving this inequality, we get r ≤ log n/ log p k 1, with y indicating the integral part of y, that is, the greatest integer not exceeding y. Then q r < p k . From this sequence q n/p k r j 0 we construct a sequence all of whose terms are relatively prime to p k or p by subtracting a fixed positive integer a < p from the quotient; n j q j − a n j−1 /p k − a if p | q j . Then by the way of construction we have n j p k n j 1 r j , 0 ≤ r j < a 1 p k , j 1, . . . , r − 1.
2.13 6 International Journal of Mathematics and Mathematical Sciences By the additivity of g and gcd n j , p k 1, we obtain g p k n j 1 g p k g n j 1 g n j 1 , 2.14 by the vanishingness condition. Hence, noting that g n j g n j g n j 1 − g p k n j 1 g n j 1 g n j − g p k n j , 2.15 and that g n j − g p k n j 1 g n j − g n j − 1 g n j − 1 − g p k n j 1 we may express g n j − g n j 1 as a telescoping series g n j − g n j 1 Δg i .

2.17
By the same telescoping technique, we obtain g n g n 0 r−1 j 0 g n j − g n j 1 g n r , 2.18 whence substituting 2.17 , we deduce that g n g n r Δg i .

2.19
Now the double sum on the right of 2.19 may be written as N r k 1 Δg m k with the increasing labels {m k }, m 1 n r p k , m N r n − 1.
In view of 2.8 and regularity of the C, 1 -mean, it follows that Δg m k 0.

2.20
International Journal of Mathematics and Mathematical Sciences 7 Also the number N r of terms is estimated by with a constant c > 0, by 2.13 and the estimate on r. It remains to estimate 2.19 divided by log n, thereby we note that since n r < p k , |g n r | ≤ max 1≤j≤p k |g i | : C, say. Hence it follows that as n → ∞, thereby proving 2.10 . Hence it follows that lim n → ∞ f n / log n c, say, must exist and be equal to f p k / log p k , that is, 2.9 follows, completing the proof.

2.24
Let Δ denote the set of all finite discrete generalized probability distributions P. For P, Q in Δ, define their Cartesian product and union by P × Q p j q k , P ∪ Q p j , q k , 2.25 the latter defined for W P W Q < 1 only.
We will characterize the entropy of order 1 S S P by the following 4 postulates: i S P is a symmetric function of the elements of P, ii if {p} indicates the singleton, that is, the generalized probability distribution with the single probability p, then S {p} is a continuous function in p in the interval 0 < p ≤ 1, iii for P, Q ∈ Δ, we have it follows from Postulate iv that Indeed, writing p 1 tp 1 , p 2 1 − t p 1 , whence p 1 p 2 p 1 , we may rewrite 2.2 as which is 2.33 .
ii As stated in 12, page 503 , one of the advantages of the notion of entropy of incomplete probability distribution is that as indicated by 2.30 , the factor log p k in 2.4 may be regarded as the entropy of the singleton {p k }, and so 2.4 or for that matter, 2.37 with W P 1 is the mean entropy average . iii Definition 2.4 is to be stated in a mathematical way as follows. Let Ω denote the set of elementary events, B the set of events, that is, a σ-algebra of subsets of Ω containing Ω, and P a probability measure, that is, a nonnegative, additive set function with P Ω 1. The triplet Ω, B, P then is called a probability space and a function ξ ξ ω defined on Ω and measurable with respect to B is called a random variable. What Rényi introduced is an incomplete random variable, that is, taking a subset Ω 1 of Ω, he introduced ξ ξ ω defined on Ω 1 such that 0 < P Ω 1 < 1. An incomplete random variable may be interpreted as a quantity describing the results of an experiment depending on a chance, all of which are not observable. We use the notion of incomplete random variable to describe the results of evolution, the capricious experiment by the Goddess of Nature, in which not all species are observable since the species which we now see are those which have been chosen by natural selection.

Rényi's α-Entropy
It would look natural to extend the arithmetic mean in 2.37 by other more general mean values. Let g be an arbitrary strictly monotone and continuous function with its inverse function g −1 . General mean values of {x l , . . . , x n } are described as in which case g is called the Kolmogorov-Nagumo functions associated with 2.35 .
We may replace Postulate iv above by iv' If W P 1 · · · W P n < 1, then we have

2.36
Theorem 2.8 Rényi . The only S P defined for all P {p 1 , . . . , p n } ∈ Δ and satisfying the is the order α entropy of Rényi.
Since lim α → 1 S α P S α P S 1 P , order α entropy of Rényi would suit a measure for the incomplete random variables and would be in conformity with Carbone-Gromov notion of dynamical time of variable fractal dimension in Section 8.
A complete characterization of S in Theorem 2.8 with general g was made by Daróczy in 1963 to the effect that the only admissible g are linear functions and linear functions of the exponential function see e.g., 13, page 313 .
As is stated in 14, page 552 11 , the most significant order α information of Rényi is the "gain of information," which would also work in comparing the microstates of the body. We hope to return to this in the near future.

Thermodynamic Intermission a là Boltzmann
Quantities of the form or any analogue thereof, played a central role in Boltzmann's statistical mechanics much earlier than the information entropy. In Boltzmann's formulation of thermodynamics, p k is the probability of the system to be in the cell k of its phase space. See also the heuristic argument of 15, page 18 below. We now give a brief description of elements of thermodynamics from Boltzmann's standpoint see e.g., 16 .

Entropy Increase Principle
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 v v x , v y , v z denote the velocity of molecules of the same kind and let f v, t f v x , v y , v z , t denote the velocity distribution function. Then the total number N of molecules is given by Boltzmann introduced the Boltzmann H-function The S in 2.38 may be regarded as −1 times the Boltzmann H-function: S −H. For we may view S as a Stieltjes integral which in turn may be thought of as −H. See Theorem 2.10 below.
He proved.
that is, H decreases as time elapses.
We state a heuristic argument 15, page 18 toward the natural introduction of the H-function.
In statistical mechanics, macrostates properties of large number of particles such as temperature T , volume V , pressure P are contrasted with microstates properties of each particle such as position x, momentum M, velocity v . Given a macrostate Σ, there are N microstates σ r corresponding to Σ: Σ ↔ ∪ N r 1 σ r . Then the entropy S of Σ is defined as Suppose that the rth microstate σ r occurs with probability p r . Consider the system Σ v consisting of a very large number v of copies v-dimensional Cartesian product of Σ. Then on average there will be v r vp r copies v r -dimensional Cartesian product of σ r in Σ v , where the norm symbol · indicates the nearest integer to "·". Hence for the total number Applying the Stirling formula 7, 2.1 , page 24 : International Journal of Mathematics and Mathematical Sciences we find that Under the normality condition N r 1 p r 1, 2.43 simplifies to Since S may be regarded as the arithmetic mean of S v 's, it follows from 2.47 that The first law of thermodynamics or the law of conservation of energy is one of the most universal laws that governs our space. We consider an isolated thermodynamical system, where isolated means that the system does not give or receive heat from outside sources: i Q means the heat, ii T means the absolute temperature, iii S Q/T means the entropy, Boltzmann proved. Theorem 2.10. We have the relation: Theorems 2.9 and 2.10 together imply that entropy increases, which is the second law of thermodynamics.
Proposition 2.11. The maximum of the entropy 2.38 for a probability distribution of (an information system) {p 1 , . . . , p n }, 0 < p k < 1 is attained for with maximum log n.
where λ is a parameter. We may find the extremal points of L among stationary points which are the solutions to the equation ∇L o: that is, they are the solutions of the system of equations From 2.52 , we have p k e k−1 . Substituting these in 2.23 , we conclude that the stationary point is 1/n, . . . , 1/n . Since the entropy always increases, we conclude that it is attained for 2.50 . Equation 2.50 is in conformity with our intuition that the entropy becomes the maximum when all the variables have the same value. Consider, for example, the case "The dice is cast."

Molecular Orbitals
This section is devoted to a clear-cut exposition of energy levels of molecular orbitals of hydrocarbons carbon-hydrides and is an expansion of 7, Section 1.4 .
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 4.1 in contrast to the chain-shaped 1,3,5-hexatriene and the ring-shaped benzene treated in Section 4.2.
In quantum mechanics, one assumes that the totality of all states of a system form a normed C-vector space V and that all quantum mechanical quantities are expressed as hermitian operators A : V → V . For a Hermitian operator A, the eigenvectors v belonging to its eigenvalue λ ∈ R are viewed as the quantum state whose mechanical quantity is equal to 14 International Journal of Mathematics and Mathematical Sciences λ. The Hermitian operator H expressing the energy of a system is called the Hamiltonian and its quantum state v v t varies with time variable t according to the Schrödinger equation where ħ h/2π and h > 0 is called the Planck constant. If Hv t Ev t , E being real and called the energy levels of the system, the solution is given by v t e −iEt/ħ v 0 and is called the stationary state on the ground that its expectation does not change with time. The energy level means the values of the energy which the stationary state can assume.
Example 3.1. We deduce the secular determinant for the molecular orbital Ψ consisting of n atomic orbitals: where φ k are atomic orbitals and c k are complex coefficients. Let H denote the Hamiltonian of the molecule and let where, in general, Ψ is to be treated as a complex vector, in which case ΨHΨ respectively Ψ 2 are to be regarded as ΨHΨ respectively |Ψ| 2 and the integrals are over C n . We write and refer to H ij and S ij as the overlapping integral and the resonance integral between φ i and φ j , respectively. Then For 3.9 to have a nontrivial solution c i , the coefficient matrix must be singular, so that We apply the simple LCAO linear combination of atomic orbitals method with the overlapping integrals S ij δ ij , where δ ij is the Kronecker delta, that is, S ij 0 for i / i and S ii 1, so that 3.10 reduces to which is the secular determinant for Ψ.
Hereby we also incorporate the simple Hückel method with the Coulomb integral of the carbon atom in the 2p orbit be α, and the resonance integral H ij between neighboring C-C atoms in the 2p orbit be β, and others are 0.

Theorem 3.2. With all above simplifications incorporated, the secular determinant reads
where H 1n H n1 0 or β according as the molecule is chain-shaped or ring-shaped.

Concrete Examples of Energy Levels of MOs
In Section 4.1, we dwell on 1,3-butadiene and cyclopentadienylanion in 21, Section 3 while in Section 4.2, we mention 1,3,5-hexatriene and a ring-shaped benzene treated in 7, Chapter 1 .

Golden Ratio in Molecular Orbitals
This section is an extract from 17, Section 3 , referring to the golden ratio in the context of molecular orbitals. We will use the notation therein. Let τ 1 √ 5 /2 1.618 · · · be the golden ratio. In 17, Section 3 , we considered the relation between Fibonacci sequence {F n } and the golden ratio, known as Binet's formula: 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 18 referred to as an aesthetic theorem in 17 , where it is divided into two descriptive statements.

Theorem 4.1
The hierarchical over-structure theorem . 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.

4.11
Thus the golden ratio appears in this context. It would be just natural that it appears for the pentagonal molecule but it is remarkable that the golden ratio appears for 4 carbon atoms case for a chain-shaped hydrocarbons. For the ring-shaped 1,3-cyclobutadiene see the end of Section 4.2.

Linear and Hexagonal MOs
By Theorem 3.2, the secular determinant of the 1,3,5-hexatriene is On the other hand, the secular determinant of benzene is

International Journal of Mathematics and Mathematical Sciences
Substituting in −λ α − ε /β, one sees that the energy levels of a chain-shaped toluene are ε α 2β cos k n 1 π, 1 ≤ k ≤ n.

4.19
Proof. By standard technique,we may deduce the recurrence On the other hand, to find molecular orbitals of the benzene, we may apply the theory of circulant matrices.  one calls it the shift forward matrix which plays a fundamental role in the theory of circulant matrices , where e k δ k,1 , . . . , δ k,n with δ k, denoting the Kronekcer symbol, are fundamental unit vectors π is for push . Using this, we conclude that C c 1 c 2 π · · · c N π N−1 . Viewing this as a polynomial, one calls a representor of C.
Note that n × n circulant matrices are matrix representations of the group ring over C or GF q as the case may be, of the underlying cyclic group 19, 20 . For example, {π, π 2 , I} is the matrix representation of the group ring C r , where be the piervot'ny primitive nth root of 1, we define a Fourier matrix F by means of its conjugate transpose F * :

4.30
Remark 4.8. In deducing Theorem 4.7, full force of Theorem 4.6 is not used. It may also be used in another setting to give a few-lines-proof of the celebrated Blahut theorem in coding theory to the effect that the Hamming weight of a code is the rank of its Fourier matrix cf. 22 .

Chebyshëv Polynomials
In this section we assemble some basics on the Chebyshëv polynomials to an extent for enabling to understand the computations in Section 3 Chebyshëv polynomials may most easily be introduced by the de Moivre formula cos nθ i sin nθ e inθ cos θ i sin θ n .

5.1
Definition 5.1. If cos θ x, then cos nθ is a polynomial in x of degree n and is known as the Chebyshëv polynomial of the first kind and denoted by T n x . Similarly, sin n 1 θ/ sin θ is a polynomial U n x in x of degree n known as the Chebyshëv polynomial of the second kind: T n x cos n arccos x , U n x sin n 1 arccos x sin arccos x . 5.2 The notation is after Tchebyshef or Tschebyscheff who first introduced them, proper transcription beingCebyšëv. T n x and U n x satisfy the recurrences by which they may be also so defined.
respectively, with initial values 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 5.3 is a consequence of the trigonometric identity sin n 2 x 2 sin n 1 x cos x − sin nx.

5.5
International Journal of Mathematics and Mathematical Sciences

23
As an important case, we rephrase the identity which follows from addition theorem sin n 1 θ sin n − 1 θ cos nθ sin θ.

5.6
Dividing this by sin θ, we obtain Thus, all the results on U n may be transferred to T n through 5.7 , which fact will show its effect in elucidating the coefficients in 5.8 .
Since it turns out that it is usually easier to work with U n x , we will mainly treat the second kind. The reason, which is not made clear in the preceding literature, is that the sine function corresponding to U n is set on basis as the fundamental wave which vanishes at the origin, and "cosine" is its counterpart cosine, corresponding to T n cf. 5.12 below .
We note that although 5.8 are initially obtained for x ∈ −1, 1 , they are valid for all values of x ∈ C by analytic continuation. If in the substitution cos θ x, we regard cos θ as a complex analytic function, there is no range restriction, but then we need to take into account the multivaluedness of the inverse cosine. It is instructive to consider the situation as a limiting case of the mapping w 1/2 z 1/z .

5.8
ii If n is odd, then sin nz is a polynomial P · in sin z and if n is even, then sin nz/ cos z is a polynomial in sin z.
In the case n 2m 1, iii We find the values of sin π/5 and cos π/5 . We apply the pentatonic formula 5.9 for sin θ:

5.13
We have a companion formula to 5.10 : Here is a point that distinguishes U n from T n : x cos θ cos k n 1 π, 1 ≤ k ≤ n.

5.17
Since the coefficients in 5.8 are rather involved, it is natural to seek for more concise form for them. The easiest method is to use the DE satisfied by U n and T n , which is widely known. But, since the Chebyshev polynomials are special cases of Gegenbauer polynomials, which in turn are special cases of hypergeometric functions, we are to work with the last to apply the method of undetermined coefficients.
In 17 we appealed to the generatingfunctionology as stated in Comtet 23, page 87 . proving that if we assume the second recurrence formula in 5.3 with the second initial condition 5.4 , then we may deduce a universal expression for U n x .

Free Groups versus Formal Language Theory
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 A be a given nonempty set, called alphabets. We call any finite sequence a 1 , a 2 , . . . , a n a word w or a string : written w a 1 a 2 · · · a n , where we also call the void sequence a void word, written ∅. Let W denote the set of all words on A. On W there is a concatenation operation, that is, given two words w a 1 · · · a n , w a 1 · · · a m we catenate them to get a new word ww a 1 · · · a n a 1 · · · a m . Since the associative law holds true, W forms a monoid with ∅ the identity.
In the case of codons, we have A {A, T, G, C} and W is the set of all single-stranded DNAs. We refer for example, to 5 , where the difference between circular and linear DNAs is remarked and also that the present language theory deal with linear strings. Therefore, the codons are treated in pairs. Now we go on to the notion of free groups. Given a family of groups {G λ } λ∈Λ , A is the disjoint union of G λ 's and W is the set of all words on A. W is a monoid as above.
To introduce the group structure, we define the relation w → w if either i the word has successive members a, b in the same group G λ and w is obtained from w by replacing a, b by their product, or ii some members of w is an identity and w is obtained by deleting them. For two words w, w we write w ≡ w if there is a finite sequence w w 0 , . . . , w n w such that for each j, 1 ≤ j ≤ n, either w j → w j−1 or w j−1 → w j holds. Then we may prove that this relation is an equivalence relation and so we may construct the quotient set G W/ ≡ on which we may define the multiplication and G becomes a group, the free product of G λ 's.
Thus, as stated in 24, page 13 , in order to multiply the word w by another word w , we write them down in juxtaposition and carry out the necessary cancellations multiplications in a group and contractions deleting identities .
On 4, page 20, page 56, etc. , one finds some interesting arguments on the singlestranded DNAs as words in the free group F 2 generated by two alphabets A and G with T A −1 , C G −1 . The ablianized group F 2 / F 2 , F 2 , where the modulus is the commutator group, is isomorphic to Z 2 , an infinite cyclic group and would result in excessive cancellation hybridization . In addition to these 4 natural alphabets, there are synthesized ones including X, Y . It would be an interesting problem to find the reason why creatures use only 4 alphabets. We may need to use formal language theory developed so that it can treat both circular and linear strings to consider such a problem and we hope to return to this at another occasion.

Definition of Life
A penetrating definition is essential to describing the whole realm of a discipline. We may recall the first passage from Pauling 25 .
The universe is composed of substances forms of matter and radiant energy. As in 6, page 71 , since the beginning of time at the Big Bang singularity to the present, there has been only finite amount of entropy generated, most of which is in the form of cosmic background radiation. Thus in the sense of classical physics, this is a comprehensive definition.
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 informationbosons 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 International Journal of Mathematics and Mathematical Sciences 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 6, pages 124-128 , though we intend to pursue longevity in vivo.
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 26 it is suggested that the first living beings-our ultimate ancestors-were self-replicating patterns of defects in the metallic crystals, not carbon. Over time, the pattern persisted and transferred to carbon molecules. Thus, one key feature of life is a dynamic pattern that persists over time, the persistence being due to a feedback with their environment: the information coded in the pattern continually varies, but the variation is constrained to a narrow range by this feedback. Thus: 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 4 as carbon is one of the main constituents of the living organisms and the first author's name is Carbone, meaning carbon. We are particularly interested in 4, pages 12-14 . On 4, page 12 "Crick's dogma" is stated to which we will return later. As part of definition of life, 4, ll. 1-3, page 13 may be taken into account, which reads: "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" 27 , where a simile is given of a military regime and an adult body.
On 28, page 107 the author elaborates on Schoenheimer's definition of life and states 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 6, pages 66∼73 .
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 complementarity principle. Or more physically speaking, 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 29 and this explains the difference between biological and chronological ages.
In 30 , although the notion of entropy is introduced to interpret aging, the mechanism is not elucidated as to how life in vivo can continue much longer than the experiments in vitro, which is the notion of dynamic state of constituents first invented by Schoenheimer as alluded to above.
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 4, page 44 would be suitable. Different internal time-clocks might use dynamical time of variable fractal dimension taking into account the number of population in the species. See Section 8.
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 8 proved a uniqueness theorem for such a measure to the effect that those measures which satisfy some more conditions must be of the form of an entropy times a constant, cf. Theorem 2.1 .
On 31, page 199 some more important notion is mentioned, that is, assimilation and dissimilation.

Entropy Increase Principle in Life Activities
We adopt the standpoint of 30, pages 105-116, 213-215 to interpret aging as the increase of entropy in the body. As is stated in earlier sections, in all autonomous systems, the reaction proceeds in the direction of entropy increase. In living organisms-human bodies in particular, there may be internal time which is governed by the amount of entropy as opposed to outer time. With lower entropy the body can remain young irrespectively of the outer time that elapses. This may explain the big difference between biological and chronological age. There may be the difference up to one generation-25 years among individuals.

International Journal of Mathematics and Mathematical Sciences
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 CO 2 which is to be excreted as substance of bigger entropy.
In 30 the understanding is that when entropy attains its maximum, the reaction stops and the system comes to equilibrium; in a living organism, it means the death of that individual. Thus there must be a function which makes the inner entropy lower, called "Homeostasis" which controls the amount of entropy to be lower. When one ages, the functions stops working well and then entropy starts increasing to come to the end of the living reaction. With insight of Schoenheimer, the process may be refined as follows.
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 malfunction of homeostasis.
There may be many causes that give rise to the malfunction of the homeostasis. One typical example is the attacks of free radicals, to which we hope to return at another occasion cf. 32 .