Some Speculations about Inverse Functions

Milan Kunz

JurkoviRova 13, 63800 Brno, The Czech Republic

On page from February 6, 2001

Abstract

Topological distances (the number of bonds) between atoms i and j in a molecule are inverse functions of the structure of the given compound, similarly as conductivities are inverse functions of the structure of an electrical circuit. The practicable applicability of results leads to the question about topological dimensionality of the universe which should be infinite but it is projected on the three dimensional geometry. Further, there are two possible matrix analogies for the relation between the brain and knowledge. Knowledge can be some kind of an inverse function of the physical state of the brain or eigenvalues of this matrix.

Key words: Matrix inverses, Wiener index, topological dimensionality

Introduction

Mathematics knows two types of inverse functions: additive:

and multiplicative:

what corresponds to

A more complicated function is the matrix inverse. Essentially, there are two types of inverse matrices, too. Moreover, there exists the third type of this function.

It is not possible to speculate about mere mathematical functions, but our thoughts and philosophy must explain the world we live in. Therefore, we start with chemistry.

The Wiener number

Somewhat more than fifty years ago, an otherwise unknown chemist Harry Wiener (1948) noticed a relation between boiling points of alkanes, then determined carefully with a greater precision than before, and the sum of products of the number of carbon atoms on both sides i and j of all bonds between carbon atoms in an alkane

Boiling points are a rather complicated physicochemical property characterizing chemical compounds. Distillation was one from the first chemical operations, it was used for purification of chemicals, and in alchemy, it got a symbolic meaning of purification. The Wiener number connected this property with rather abstract feature of the structure of the given compound. The importance of the Wiener number was recognized only slowly [1]. It was found that the Wiener number correlated well with other properties of molecules and it became the first of the topological indices.

At first, the Wiener number was connected with the half of the sum of the elements of the quadratic distance matrix D, which elements dij are topological distances (the number of bonds) between carbon atoms i and j (Mihalic et all, 1992).

These relations are elementary. One can only wonder why such simple counts of the length of all paths in a molecule correlate well with its physicochemical properties.

Nevertheless, this simplicity can be removed easily. We start with the most recent findings.

Searching more discriminating topological indices, Diudea (1995) introduced asymmetrically weighted distance matrices, the Cluj matrices, by the Wiener weights n_j (the number of vertices on the end j of the path p_ij from the diagonal vertex i to the off-diagonal vertex j. From these Cluj weighted distance matrices D_C can be derived the Cluj weighted adjacency matrices A_C.

Path and walk matrices

It is well known (Cvetkovic et all) that the Laplace-Kirchhoff matrices have (n-1) nonzero eigenvalues. The Laplace-Kirchhoff matrices are quadratic forms S^TS, where S^T is transposed incidence matrix S (s_ij = -1, if the arc i goes from the vertex j, s_ij = -1, if the arc i goes to the vertex j, s_ij = otherwise.

Both quadratic forms S^TS and SS^T have the same set of eigenvalues, except zero ones. In trees (acyclic connected graphs, an example: alkanes) with (n - 1) arcs, SS^T is (n - 1) dimensional square matrix, therefore it must be nonsingular. Inverting these matrices (Kunz, 1989), inverse matrices of trees were found. Their trace equals to the Wiener number.

The elements of (SS^T)^-1 are distances. More precisely, the Wiener number on the diagonal is obtained by counting how many times the arc (edge) j was used in all paths or walks between all vertices of the tree. This number is equal to the product of the number of vertices in subgraphs on its both ends. The off-diagonal elements are the counts of how many times the pairs of arcs ij were used in all paths together. The inverse was defined in the form W^TW, where W is a matrix which rows represent path (walk) i between all pairs of vertices. The columns represent the arcs. The elements w_ij are 1 if arc j is incident with the path i, zero otherwise. Walk matrices, defined for unoriented graphs, must have some elements negative and the sign alternates in the walk.

This definition does not give the inverse directly but as its n multiple

This form is preferred to keep the Wiener number intact and to avoid fractions. The distance matrix D of trees, is a part of the left hand inverse of its incidence matrix S:

(Rutherford 1990, Kunz, 1992). This relation is given by orthogonality of arcs in trees. The distance matrix D of a tree is the inner inverse the Laplace-Kirchhoff matrix of the tree. Or otherwise, the incidence matrix S of the tree divided by its eigenvalues is the eigenvector of its distance matrix D. The adjacency matrices A can be related to the distance matrices D as their direct infinite negative moments (Kunz, 1997).

Paths from the root

Arcs (edges) are coded in path (walk) matrices in their own space by only one element. In the space of vertices, an arc must be coded on its both ends.

To code trees, it is necessary to choose one vertex as the root and mark it (Aissen et all, 1989). The code matrix C has n columns and (n -1) rows, since there are (n - 1) paths from the root to other vertices. Thus, code matrices of trees are singular.

To remove this singularity, a row with one unit element e₁ is added. The code matrix C has lower triangular form (that is at least after suitable permutations) (Kunz, 1996), and the unit diagonal. Therefore, it has the inverse. This is the incidence matrix S^T of the given oriented tree which singularity was removed same way by adding a row with one unit element e₁.

The off-diagonal elements of the quadratic form of (S^TS)^-1 are resistance distances (Klein et all, 1993, Kunz, 1995a). The resistance distances d_ij can be calculated directly, for example in simple cycles as dij = (1/d_l + 1/d_r)-1, where d_l and d_r are the left and right distances from the root i to the vertex j, respectively. The fractions 1/dl are known as conductivities. The rooting technique is deforming the original form of the graph and disturbs the eigenvalues.

Eichinger matrices

In the theory of mechanics of polymer chains (Eichinger, 1980, 1985), two matrices of linear chains Ln have been used. They are known as Rouse R and Forsman F matrices. The Rouse matrix is identical with the Laplace-Kirchhoff matrix of the linear chain L_n. The Forsman matrix F is the generalized inverse (again its n multiple) of the Rouse matrix R

The trace of a Forsman matrix gives the Wiener number, more precisely, its double, since its diagonal elements are sums of distances of the vertex i to other vertices j. The off-diagonal elements are distances unpassed in the paths between pairs of vertices i and j. It was then rather easy to generalize the relation on all trees and to give its proof (Kunz, 1992). In graphs with cycles, the existence of the generalized inverse is based on reconstruction of the eigenvalue polynomial from the Ulam subgraphs polynomials. Indeed there are infinitely many generalized inverses (Kunz, 1995b) but generalized inverses of trees defined by distances are unique since they are derived as sums of partial inverses. They can be found for all Laplace-Kirchhoff matrices as Eichinger matrices E

There are n partial inverses with (n - 1) unit elements in the sum therefore we get (n - 1) multiple of I on the diagonal of the generalized inverse. The eigenvalues of the generalized inverse are just the inverse eigenvalues of the Laplace-Kirchhoff matrix, except the eigenvalue corresponding to the zero eigenvalue which is equal to the sum of other (n -1) eigenvalues.

Distances in information strings

Tossing a coin is a model of the binomial distribution, interpreting heads and tails as zeroes and ones, respectively. The distances between consecutive events form the inverse of the binomial distribution, the negative binomial distribution. This distribution was until recently a mere mathematical curiosity due to difficulties with computations. Now it is implemented in many statistical programs.

For example, the decadic numerals in the number e are well described by the negative binomial distribution [3]. The negative binomial distribution appeared in the distributions of distances between consecutive symbols and nucleic or aminoacids in texts and DNA (Kunz et all, 1998a, this page) at some information units in tails of distributions together with tree other extremely skewed distributions, the lognormal distribution, and the Weibull distribution and the exponential distribution.

DNA determines the structure of our body and our brain. Words and their strings which are signals triggering production of different chemicals or starting of different physicochemical processes have similar statistical properties. In both cases, the distributions show disturbances, the values in some ranges are lower or higher than could be expected.

The distances in information strings can be viewed as time needed to read them or to synthesize them. The time intervals can be measured directly, for example as times when publications are received by journals. At prolific authors, where the statistical tests can be applied, these time intervals are again described the Weibull distribution (Kunz et all, 1998b). Fourier analysis gave frequencies corresponding to the range of mean intervals between longer intervals. It could be assumed that short intervals were induced by the author whereas the longer ones by the academic surrounding.

Some speculations

When somebody deals longer with something there always appears some vague ideas and speculations, in my case about inverse functions and distances, and one tries to find some generalizations which go behind the scope of studied matter.

First of all, it is the question about topological dimensionality of the universe.

Mathematicians do not connect their speculations with the physical reality. They suppose something to be n-dimensional and make no consequences from it.

The Czech mathematician Vopýnka (1993) proposed as an explanation, why Gauss did not publish his results about non-Euclidean geometry: It was a freemason secret and Gauss as a freemason was not allowed to explain it to nonmembers. The same explanation holds why he did not explain his normal distribution by geometry of the multidimensional space. Mathematicians form a secret society, which does not reveal its secrets.

The graph theory considers graphs to be dimensionless objects or one-dimensional objects, even if they are connected with n dimensional matrices.

The equation (2) gives at graphs with cycles the matrix of angles between arcs (Kunz, 1994). The result should be interpreted in such a way, that trees with n vertices exist in (n-1)-dimensional space, the distances in the distance matrix D being the squared Euclidean distances.

To appear in a 3-dimensional space, the molecules are deformed from their ideal shape. The complete graphs until K₅ form regular bodies (triangle and tetrahedron). The higher complete graphs cannot be arranged in such a way [4]. If elementary particles are more-dimensional objects, then they may try to avoid such deformations by changing their projections constantly and appearing as waves. The enigma of transformation of a wave function into a geometrical shape was already discussed long ago (Wooley, 1978) and problems of transition from topology to geometry connected with chirality studied Balaban (1997).

I do not include any references to physics and its string theory since it were a long list. Specialists already know them and for a layman, it were a difficult reading.

I proposed a rather primitive explanation of very complicated processes and relations. The idea is not mine; it is already very old. Plato's shadows on the wall of the cave we live in [5] were a better formulation, if they were 3-dimensional.

However, there is some other interesting generalization, less provocative.

Both inverse functions, additive and multiplicative, have different properties. The additive inverse can be complementary, for example when an electron and positron annihilate; their charges are annulled. The additive inverses are separate objects.

The multiplicative inverses are separate entities, too, but in our examples they represented different aspects of the same objects as if the object were seen one time from outside, the other time from inside.

A molecule or an electrical circuit are objects which inverses are rather easily interpreted.

But I must admit that I have no notion what is, for example, the multiplicative inverse of an electron having no practical experience with them as with chemicals.

A molecule seems to be, when we look at its figure or its model a rather simple structure, and its matrices rudimentary, compared with the structure and complexity of the brain.

The brain (Crick, 1994) is a complicated net of axions analogical to an electrical net. Chemical states of individual axions, molecules forming their membranes, determine properties of this net. The inverse functions of this state are conductivities, which determine processes going in this net. These processes can be chemical reactions, transport of molecules or electrons, or mere changes of configuration of the molecules. These processes form base for thoughts, for the knowledge or the soul, using the Crick's term.

The body and the soul are not identical, and they are not complementary. Their relation can be characterized as an inverse function, of course not strictly as a mathematical definition of this relation but as its analogy.

Similarly as the behavior of the molecules when exposed to some temperatures and pressures is determined by the inverse of their structure, so our behavior in all situations depends on the processes going in our brain. These processes depend on the structure of the brain.

The proposed analogy has many faults but it has at least one advantage. To characterize the relation between the knowledge and the mass is an old philosophical problem. Defining them as inverses removes the question, what is primary. Inverses do not exist separately.

The other possible relation between the brain and knowledge is the relation between a matrix and its eigenvalues: Our thoughts are some kind of eigenvalues corresponding to states of our brain.

It is possible to formulate still other inverse relation, concerning the knowledge, defining the truth. Searching the truth, our thoughts are boiling. Using this analogy, the world W and our knowledge K, when formally multiplied, must give WK = I, I being the identity.

Notes

[2] The term x_n-2 of the characteristic polynomial of the Cluj weighted adjacency matrices A_C is calculated as by the equation (1).

[3] The significance of CHI square test goes from 0.04058 for the numeral 6 till to 0,99335 for the numeral 8. Disturbances increasing the value of CHI square are local lows of some intervals. I used the number e calculated to 10 000 places by Ventluka (1998).

[4] I published this problem as an April's fool joke (Kunz, 1994), but I meant it seriously.

[5] The cave we live in is essentially the scull cave where our brain is located.

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