Order, structure, physical organization and biological organization.
In order to understand the nature of life, we exemplify and define the concepts of order, structure, physical organization and biological organization. Pointing to the dominant role of possible microstates over the realized ones in the concept of entropy, we obtain further quantitative insight into the relation of biological organization with the realm of possibilities. We illustrate the entropic behavior of living organisms on the S(E) diagram. Our results tell that entropy increase generates further possibilities that are favorable for biological organization.
Self-organization as ordering
Until now, the concepts of self-ordering and self-organization have been somewhat confused. This confusion is devastating since the way to understand life apparently leads through these two concepts. Recently, there has been a trend toward clarification of the concept of entropy to free it from misconceptions identifying entropy with disorder. (1-6) "Of all the difficult concepts of classical physics ... the most difficult is entropy." (2) Biological organization and order should be added to the list of most difficult concepts. Self-organization is a term frequently used to cover ordering processes, as well. Schrodinger thought that "life seems to be orderly and lawful behavior of matter." (7) He identified entropy with disorder (though we now know that entropy and disorder are not identical), and so obtained that order is the negative of entropy. (7) Prigogine attempted to describe self-organization as 'order through fluctuations.' 8 Following Prigogine, Jantsch, in his book The Self-Organizing Universe, attempts to describe prebiotic evolution, the functioning of bioorganisms, neurophysiology, ecology, sociobiology, and cosmic evolution, in terms of dissipative structures and the development of order from fluctuations. (9) It is widely thought that order may come "free," that order may come out of chaos. (10-11) Recently, the view that biological organization also comes for free has become popular. (10) Haken stated that a system is self-organizing if it acquires a spatial, temporal or functional structure without specific interference from the outside. (12) At the same time, he regards as typical examples of self-organization fluids heated from below, or lasers. On the basis of synergetics, he approaches biological organisms as physical systems. Knyazeva notes that synergetics is the search for laws of the formation of structures, of the emergence of order out of chaos. (13)
Ordering is Different from Organization
Conversely, there are people who observe that the concept of order and organization is confused because biological organization is not identical with physical ordering. Denbigh already realized the fundamental difference between order and biological organization. (14) For example, a crystal is more ordered but less organized than a living cell. Elitzur realized that an unexposed film is in a very special state that can be thought of as arranged by a refined ordering of the molecules, while its exposure to light makes it less ordered (losing its special original order) but full of information. (15) Applying Schrodinger's measure of order, the unexposed film is more highly ordered since it has less entropy than the final state that develops due to exposure. At the same time, the potential chemical energies of molecules to be liberated upon exposure are evenly distributed. After exposure, the situation reverses. Part of the potential chemical energies of the molecules will be liberated, but, at the same time, the distribution of different molecules in the end state will be highly heterogeneous.
In contrast, living cells also have high chemical energies, which are liberated at decay, like the film molecules under exposure. At the same time, the cells have the capability to regenerate or reassemble themselves. In terms of Elitzur's film example, this would be like a film's ability to self-regenerate its original unexposed state, while simultaneously generating different, global patterns from time to time and from site to site, as living cells do permanently. Transforming Elitzur's film example to the case of living cells, we realize that the origin of structures seems to be only the physical half (thermodynamically downhill path) of life. The other half (thermodynamically uphill path) is the self-regenerative half. In this paper, we trace the path from order to structure, physical organization and biological organization, and point out the significance of entropy for our understanding of organization.
Ordering and Structure Generation as Partial Aspects of Biological Organization
The general belief is that complexity arises at a threshold somewhere between chaos and order. It seems that this notion does not exclude some ambiguities. For example, crystals show order; nevertheless, crystals arise from liquids or gases. Gases are chaotic ensembles of atoms or molecules. Now if complexity is between chaos and order, then complexity is similar to liquids. Liquids are either homogenous or amorphous. Neither of these properties are characteristic to complex systems and life. Moreover, beginning with Descartes, it has been widely thought that living organisms are nothing more than highly complex machines. Notwithstanding, Yockey pointed out that when complexity is considered as the non-compressible algebraic information content, high complexity means that a long algorithm is needed to describe the system, and therefore, highly organized systems have large entropy. (3) This means that highly organized systems and highly ordered ones occupy the opposite ends of the entropy scale; among the random sequences, the highly ordered are at the low end of the spectrum and the highly organized ones at the high end.
Elitzur pointed out that randomness is related to the absence of relations between the set of elements representing the system, and one would expect that organisms, as their elements are significantly related by biological organization, are in a large distance from randomness. (16) The arising Yockey-Elitzur paradox has a crucial importance and may be resolved when realizing that organization acts between different molecules, all of which have relatively high entropies and randomness, yet simultaneously are related to each other. Certainly, the conformational states of proteins change dynamically in cells, for they are related to the government of cellular reactions occurring at a typical rate of more than [10.sup.5] reactions per second. (17) Individual proteins have a very large set of possible conformational states (~[10.sup.60]), and therefore relatively high entropies. At the same time, the conformational states of different proteins have to be coordinated; therefore their relation to each other is not random. More concretely, the conformational state represents a highly special state obtained from an extremely large set of possibilities, and so it has high entropy; thus while its selection from the set of possibilities can be regarded as random, the relation of a protein to other molecules (which also represent high entropy) is not completely random, but a very special type of relation, also highly entropic. Biological organization creates very special couplings from an extremely large set of possibilities of the interactions of proteins with other chemicals. High entropy is favorable for more possible states of molecules, as well as for more possible states of the interactions themselves. Therefore, even if the relations between molecules have an aspect of structure, if we regard these relations as exemplifying a kind of order, it would not be a type of order characterized by low entropy.
Order, Structure, Physical Organization and Biological Organization
'Order' is frequently described as showing structures (A). Structures are widely regarded as arising in self-organization (B). Self-organization is widely regarded as being identical with biological organization (C). Now if A=B=C, then A=C, therefore, biological organization and ordering can be identical. We disagree, and try to call attention to the basic differences between these concepts besides the similarities. To avoid misunderstandings, we exemplify the instances of order, structure, physical organization and biological organization.
Order is exemplified by crystals, magnets, etc. We define order as a constant relation between neighboring constituent elements of a system. Order usually arises by repeating the basic, simple constant relation or arrangement of neighboring elements, and therefore crystals, magnets, etc., show simple patterns, repeated basic arrangements of neighborhood elements. If we denote one unit (denoted by 1) the atoms or molecules of crystal, or the spins of a magnet, etc., the order can be characterized by the number sequence 111 ... Structures are exemplified by snowflake, molecular structure, stellar structure, or landscape, etc. We define structure as a more complex arrangement of elements. Therefore, we regard 'structure' as 'complex order.' Denoting the similar units by the same digits, a structure is described by a series of digits like 121212 ... or more complex sequences or fields of sequences.
We exemplify 'physical organization' by turbulence, Benard convection, or reaction-diffusion systems like the Belousov-Zhabotinsky reaction. Such phenomena are widely regarded as showing 'dynamic order,' 'dynamic structure' or 'dissipative structure,' but in these cases the concepts of order and structure are used as interchangeable concepts. We disagree because, e.g., a landscape has a structure but does not have order. We identify physical organization with units of changes that are (more or less) constant in time. Visibly, Benard cells of a given medium are very similar to each other, as well as the subsequent cycles of the Belousov-Zhabotinsky (BZ) reaction. Therefore, denoting a unit of dynamism, e.g., a (standing or moving) Benard cell, a cycle of BZ reactions, etc., by the digit 1, a convection zone or a pattern of BZ cycles are denoted by sequences like 111 ... The differences between order and physical organization are that in order, the units or elements are constant, and their lifetime is apparently not limited, while in physical organization, the units are dynamic, and their lifetime is short. In the case of Benard convection, the number of generated Benard cells in a convective zone is apparently not limited, since convective zones can have astronomical lifetimes. In contrast, in BZ reactions, the number of repetitions is very much limited, and is usually below [10.sup.5]. What is even more remarkable, is that for excitable BZ systems, and for so-called "autonomous" reaction-diffusion systems, the number of oscillations becomes even more limited; for oscillation periods around 50 s, the oscillation persists only for 1-2 hours, and so the number of oscillations is small, <150. (18) In contrast, for biological organisms--that are certainly highly excitable--the number of heart cycles is cca. [10.sup.9].
Now we have arrived to the most complex of all these concepts, namely, to biological organization. We exemplify biological organization with the more than 10 5 chemical reactions occurring in a cell, and with the multicellular organisms having [10.sup.14] cells (humans). Therefore, in this first approximation, biological organization is the process that governs our >[10.sup.19] cellular reactions occurring in a second. Denoting the units of activity (chemical reactions) with digits 1, 2, 3, etc., biological organization is described by dynamic fields of number sequences. Visibly, genuine biological organization generates new reactions in every cells in every time steps, therefore, biological organization is different from order, structure or physical organization, which are all (more or less) constant in time. Naturally, our bones are constant in time, and there are repeated reactions and redundant information in the life of cells. Therefore, order, structures and physical organization accompany biological organization. Nevertheless, this does not mean that genuine biological organization always repeats the same chemical reactions in every cell, without ever replacing the previous chemical reaction at a certain part of a cell by another one. Biological organization is a process using the free energy content of the organism in order to maintain its entropic distance from the equilibrium. (19) In a second approach, biological organization is a process that couples endergonic processes to the exergonic ones, therefore thermodynamically uphill processes to downhill ones. In contrast, in maintained systems like turbulence, Benard convection or reaction-diffusion systems, the thermodynamically downhill processes dominate and the structures are quickly dissipated. In a third formulation, biological organization is the process generating a system of collective degrees of freedom, involving global degrees of freedom having significant free energy for governing the system to be able to actively maintain the vitality of the organism, where vitality V=G/T (G is the Gibbs free energy and T is the temperature) is its entropic distance from equilibrium (see Figure 1). In contrast, Prigogine realized that physical self-organization is a process in which "we can never determine when the next bifurcation will arrive." (11) Now if biological organization would apply bifurcation points when selecting the reactions to be realized in the next timestep, it would be an enormous number of bifurcation points available necessary at every timestep--which seems not to be the case, as it is not possible to determine when the next bifurcation point will arrive.
[FIGURE 1 OMITTED]
Entropy as a Measure of Possibilities
To clarify the situation further, let us consider the concept of entropy. First of all, entropy is not a measure of disorder, because entropy can decrease when order decreases as well. (4, 5) Now if entropy is not a measure of order, what then is its meaning? We propose to consider that the meaning of entropy is intimately connected with something that does not exist in the material sense of the word: with possibility. Entropy is given by the formula S=ln [DELTA][GAMMA]. (20) Here, [DELTA][GAMMA] is the number of quantum states having energies in the [DELTA]E range, and it characterizes the rate at which the energy is smeared out around the average energy E. Lambert formulates the problem as "the number of microstates that are possible in a system indicates all the different ways that energy can be arranged in that system; the more the number of microstates possible, the greater is the system's entropy at a given temperature." 6 Formulating the concept of entropy as measuring the number of possible microstates corresponding to the given macrostate, the second law of thermodynamics will tell us that in spontaneous natural processes, the macrostates are shifted towards new macrostates in which the number of possible microstates is higher. In a simplistic, but compact and preliminary form: spontaneous natural processes are directed towards increasing possibilities.
Entropy Decreasing Order and Organization with High Entropy
If we were to conceive the concept of order as a state in which a special relation is fixed between the neighboring constituent elements of the system, it would become clear that order acts as delimiting the possibilities of a given system, effectively excluding those possibilities that would arise in the absence of order. At the same time, in most ordered systems, not all of the possibilities are fixed, and the not-fixed possibilities of the system may increase, and possibly may overbalance the number of newly limited possibilities that resulted from the antecedent generation of order. This would mean that decrease of order and decrease of entropy are not necessarily parallel processes.
Regarding the origin of order, we can distinguish between order arising in near-equilibrium thermodynamics, such as freezing, crystallization, snow-flake formation, spontaneous magnetization, superconductivity, laser, or Bose-Einstein condensation; and structure formation in thermodynamic systems maintained in a state far from equilibrium, like turbulence, Benard cells, or reaction-diffusion systems. In the first case, the origin of order may be explained as a phenomenon accompanying entropy decrease. For example, we can understand the latent heat formation of freezing as a trick to increase entropy optimally, thereby producing a greater number of possible microstates.
In systems that are far from equilibrium, it is a widespread belief that physical self-organization is a step towards biological organization. This view has its basis in the fact that while in cases of near-equilibrium systems, the order develops due to a decrease of energy; in systems far from equilibrium, as well as in living organisms, the order is due to an opposite process: to energy inflow. Therefore, it appears we need to investigate these respective cases in greater detail.
Let us take the example of Benard-cells. The general view holds that the order manifested in the cell's structure comes for free. Prigogine and Stengers stated that the Benard instability is due to a vertical temperature gradient set up in a horizontal liquid layer. (11) Morowitz disagrees, and states that the structural formation of Benard cells arises from the kinetic equations of hydrodynamics and chemical kinetics rather than from thermodynamic considerations. 21 Actually, the calculations of Chandrasekhar have already shown that changing the boundary conditions will drastically change the structure of the cells. (22) If the fluid is confined between a lower and an upper surface, the boundaries may be rigid surfaces, on which no slip occurs, or free surfaces, on which no tangential stresses act. The sizes and flow patterns of cells will be different, depending on the fact that the surfaces are rigid or free. The pattern, which the convection cells will exhibit, remains still unspecified and will be determined only when symmetries are considered. In cases when there are no points or directions in the horizontal plane, which are preferred, the entire layer in the marginal state must be tessellated into regular polygons. Therefore, the boundary conditions and the symmetries of the problem also contribute as factors introducing macroscopic order into the system. Not only boundary conditions, but gravitation, thermal gradients, etc., may also introduce symmetries into the system. Certainly, it is the energy inflow that drives the internal flows present in the convective cells. Therefore, we can regard any system having Benard cells as a maintained system; the energy inflow maintains the distance from the thermodynamic equilibrium. As the energy inflow changes, accordingly the Benard cells will change. The dependence of Benard cells on their maintaining factor--energy inflow--is immediate and direct. This circumstance is crucially different from living organisms, since the latter can maintain autonomous behavior independently of their immediate environmental energy inflow.
Now let us characterize the order characteristic to systems far from equilibrium maintained by energy inflows. Turbulence, Benard cells and reaction-diffusion systems like the Belousov-Zhabotinsky reaction represent characteristic patterns, the character of which does not change in space and time, if the environment is constant, including the boundary conditions. They represent simple, repetitive patterns of a static character. We can understand the nature of this physical organization when comparing it with biological organization. Biological organization couples the individual degrees of freedom into collective ones, and since the environment and the organism changes from instant to instant, the couplings, therefore, have to change from instant to instant, as well as in terms of spatial relations. One can believe that organization is a dynamic order, as Haken proposed, defining self-organization as generating not only spatial, but temporal and functional structures, as well. (12) But this view would not clarify, but would confuse the relation of order and organization. The reason is that not only the similarities, but also the differences that principally obtain between order and organization are significant and complex. The complexity of the relation of order to organization becomes clear if we take into account that they can be processes engaged in mutually opposite directions. Structure generation can be an inclusive part of self-organization; thus their relation can be inclusive and opposite at the same time. Let us here consider some examples.
Take first the process of freezing of a living organism and consider the rate of order. In this case, the higher degree is the order of the crystallized materials of the body, the lower degree will be the organization, as judged from the decreasing ability to act, exert work, and manifest motion and change with a body frozen in a higher rate. Certainly, below a critical value of order, the organism will cease to be living. In the opposite case, the hibernated organism may recover, and in the process of slow recovery, the degree of order will decrease. At the same time, if the temperature were to grow above the optimal level, the vitality of the organism will decrease again, in parallel with the degree of order. Moreover, even in the parameter range optimal for biological functions, order is present, as a result of certain organizational processes. Since biological order is the product of biological organization, their relation can be inclusive, in a unidirectional way.
Most importantly, we can understand the real nature of biological organization only in the light of possibilities. Biological organization acts not only to create momentary relations between microstates, but it creates wholly new possibilities with the help of energy investment. Living organisms permanently mobilize their Gibbs free energy in order to create new possibilities that have direct biological significance. Therefore, from timestep to timestep, biological organization renews the set of possibilities, and in the next instant, it generates a new set of different possibilities to the then-prevailing set. In this way, biological organization governs the possible states, and its primary action is generating new possibilities.
Order and Physical Organization--Reformulated
Based on these insights, we may re-formulate the definition of physical organization and biological organization. We distinguish between physical ordering occurring near equilibrium and physical organization occurring far from equilibrium. Physical organization occurs in maintained systems due to energy inflow and is manifested in structures generated in spontaneous physical processes. The dynamic units, e.g., Benard cells, contain in themselves energies at the collective degrees of freedom, in the form of kinetic energies of circulation currents, or the global kinetic energies of moving Benard cells. Certainly, the energies of these collective degrees of freedom are dispersed with the decay of Benard cells, and so they give back their collective energies to the environment. Nevertheless, the formation of Benard cells in the convective zone is a process supplied with energy by the energy inflow. During a time much longer than the lifetime of Benard cells, the amount of these collective energies is closely constant in the convective zone. In physical organization, the energies of collective modes are used to maintain fixed flow patterns.
In contrast, the spontaneous processes of biological organization are directed towards permanently generated new possibilities on the collective degrees of freedom. Living organisms continuously mobilize their vitality in order to generate new possibilities for the required new reactions. The collective energies are not used to generate fixed patterns, but to generate permanently novel possibilities. In physical organization, collective energies are passive, while in biological organization, collective energies are active, continuously generating new possibilities, and selecting from these biologically generated possibilities the ones that allow the compensation of thermodynamically downhill processes by thermodynamically uphill ones.
Biological Organization as Possibility Generating and Selecting Process
We suggest that biological organization acts in three 'elementary steps.' The first step (1) is termed as 'generation and selection.' We know that the number of collective degrees of freedom is equal to the number of all possible subsystems of the given system. From set theory, we know that if a system has a number of individual degrees of freedom N, the number of possible collective degrees of freedom is 2N. In a physical system near equilibrium, the number of possible entropic microstates is [e.sup.S/k]. In a living human organism with vitality V=G/T (measuring the entropic distance from the equilibrium), V>>S. (17) With the help of vitality, the living organism can generate a number of collective (coupled) states [W.sub.biol1]~2^[e.sup.V/k] (a hyperastronomical number!) and select the set of possible uphill processes that are useful for the regeneration of the organism.
This first step of biological organization is also selective because vitality can generate different cones of directions for the future behavior of the organism; when it actually generates certain directivity, it selects from a larger set of possibilities, the realization of any one of which would lead to a certain actual pathway of the organism. Step (2) is termed as 'activation.' In this step, an input free energy [DELTA]A is used to activate some selected endergenic processes, selecting from [W.sub.biol1] a much smaller set of possible states [W.sub.biol2]~[e.sup.[DELTA]A/kT], activating them, then adding up their energy quanta to the biologically optimal excitation levels. Step (3) is termed as 'realization,' and corresponds to a further step of selection, realizing the optimal reactions and the optimal coupling of endergenic and exergenic processes from the set of possibilities activated in step (2). Step (3) realizes the government of collective modes, selecting the to-be-manifested state ([W.sub.biol3]=1, if we accept the view that in a certain instant any macroscopic system has only one microstate realized) from [W.sub.biol2]<<[W.sub.biol1], organizing the collective states into global modes, governing all the >[10.sup.19] reactions [s.sup.-1] occurring in the cells of the human organism, and manifesting them. The following step (4) does not itself represent biological organization, but its physical counterpart: the downhill processes. Step (4) already corresponds to spontaneous physicochemical reactions, to separation of collective states into individual ones, and to the decrease of Gibbs free energy and vitality. These 3+1 elementary steps, occurring continuously and simultaneously, together form an "elementary timestep," which altogether represents the biological organization of microstates into macrostates that correspond to biological behavior.
Let us exemplify the changes in the entropy vs. internal energy by means of the following S(E) diagram. We learned about the S(E) diagram from Martinas. (23)
How will the living organism be influenced by the photons present in the organism in the activated electronic levels? In our previous paper, we pointed out that the Gibbs-energy of photons is high, around 3.2 MJ/mol. (17) Therefore, if we would think for a moment taking all the photons out from the living organism, the huge energy 3.2 MJ/mol would leave the body in an instant, and the position of the organism would be shifted from point 2 to point 1. If the organism would regain all of its photons in an instant, it would return from point 1 to point 2. Now if the organism will gain internal energy from food and breathing, it will be shifted from point 2 to point 2' (the distance 22' is exaggerated for illustrative purposes only). When the organism utilizes its new internal energy income, usually for biological aims, for internal and outer work, at the end it will return to point 2. If it will not take energy income, and its internal energy would remain constant, its internal structure and constituent elements like proteins, etc., would slowly decay, and the organism would move from point 2 towards point 3, increasing its entropy content and decreasing its vitality. Due to the utilization of the photons' energy, the organism regains its vitality. Therefore, vitality turns out to be constant on the timescales of days, months, years, and decades for adult and healthy humans.
Considering the nature of biological organization, Elitzur noticed that the joint order of two systems combined can be greater than the sum of their separate orders. (15) His example is mathematical: let us regard the number 5177274640 as random, having no order. Let us take it twice, and couple the two series of digits into one after the other: 51772746405177274640. The coupled series of digits already has order, although neither of its constituents displays any particular order itself. Now let us transform Elitzur's coupling example into physics. The two systems to be joined are now two liters of ideal gases. Neither has order in their initial states. Now, if we want to couple them in the same way as the series of digits were coupled in Elitzur's mathematical example, this would mean that all the molecules of the second liter of gas move strictly in the same way as the corresponding assemblage of molecules moved in the first liter of gas. Certainly, the entropy of the second gas will be zero in this case, since no molecule in the second gas would have any possible realization other than the one instructed by its corresponding molecule in the first liter of gas. The amusing thing is that the two liters of gas seemed to have the same entropy content, in which case their joined system would be [S.sub.1+2]= [S.sub.1]+ [S.sub.2]= 2[S.sub.1]=2[S.sub.2].
Yes, their initial entropies were indeed the same. But in the course of the unrealistic coupling (this is, after all, a thought experiment), the molecules of the second liter of gas are completely repeating the motion of the first one, and therefore, their entropies are already effectively zero. Due to the particular requirements of the precise order specifying the correlation of the two gases, the entropy of the second liter of gas had to decrease. This fact exemplifies that order production is indeed frequently (but not always!) accompanied by entropy decrease.
Let us consider how biological organization relates to order of this kind. Biological organization acts not on the individual degrees of freedom, but on the collective ones. Now if biological organization were to use the energies of the individual degrees of freedom, there would be less energy to distribute on the individual degrees of freedom, and so their entropy content would decrease. In order-generating processes utilizing the energies of the individual degrees of freedom, the produced order is parallel with a corresponding entropy decrease. But biological organization is not such a process. As our S(E) diagram shows, vitality V=G/T corresponds to the Gibbs free energy, above the thermal energy TS. Biological organization uses a completely different energy source, G, instead of TS. Therefore, biological organization does not require the diminution of the entropy of the individual degrees of freedom. Biological organization produces many more possibilities than entropy alone can provide, since in human organisms V>>S. It was shown that in the human body, H=G+TS=2.52 MJ/K, S=202.4 kJ/K, therefore, G=2.32 MJ/K>>S. (17) Therefore, it is misleading to think that since entropy increases during the decay of living proteins, the organizing process necessarily decreases the entropy.
No, the case is different. Since living organisms utilize not thermal energies, but the Gibbs free energy to govern their chemical reactions, and since they utilize their vitality to activate collective degrees of freedom in a level that is autonomous, which does not depend directly on thermal energies distributed on a different level, namely, on the level of individual degrees of freedom, therefore, biological organization does not use energy that could be distributed in entropic processes.
The results of our considerations show that entropy is intimately connected with a reality much more comprehensive than the realm of material, reified processes: namely, with the reality of the not-yet-reified individual possibilities. In contrast, living organisms operate with something quantitatively and qualitatively different than ordered systems in nature, namely, with the help of their vitality, they themselves permanently generate collective possibilities. Biological organization then acts to fill these collective modes with significant energies. And when global degrees of freedom are activated, the organism can behave globally, and can be able to initiate its own motion. Although biological organization is largely independent from entropy, having its primal power in its combinatorial nature ([W.sub.biol1][approximately equal to]2^[e.sup.V/T]!), when it acts on molecules with higher entropies, it can generate a larger number of entirely new possibilities. Therefore, biological organization and entropy increase are both possibility generators; the growth of entropy can be favorable for enhancing biological organization.
The differences between order, structure, physical organization, and biological organization are so fundamental that their understanding is crucial to our understanding of the nature of life. We point out here that physical ordering is frequently a process that is related to generation of actual structures, while biological organization corresponds primarily to a deeper realm: the realm of collective possibilities.
The author wishes to express his heartfelt thanks for Jean Drew for permanent inspirations and for the works preparing Figure 1 and lecturing the English of the text.
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Dr. Attila Grandpierre
Konkoly Observatory, H-1525 Budapest, P. O. Box 67, Hungary.
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