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GABRIEL KRON’S UNIVERSAL ENGINEERING

V. Popkov

International Alexander .Bogdanov Institute, Ekaterinburg, Russia

In March 1968, in a small American town Schenectady of New York state, there passed

away Gabriel Kron – a scientist and engineer, whose ideas still agitate explorers in

different countries of the world.

A memorable article, placed in this occasion in journal “Electrichestvo”, [1] noted merits

of Kron, as a pioneer of tensor analysis application in electrical machines and networks,

and originator of generalized theory of electric machinery and diakoptics – the method of

piecewise solution of large-scale systems.

However, such view about Kron’s works wasn’t formed at once. In 1950 in the same

journal “Electrichestvo” [2] there was published a letter by A.Berendeev “About Kron’s

works on tensor calculus application in electrical engineering”, where the author wrote:

“Kron’s articles have made an undeserved impression on those, not well enough

acquainted with tensor mathematics, and also predisposed to obsequiousness at American

sensations”, and editorial staff, sharing the authors opinion about methodological solidity

of Kron’s concepts, called up the readers for their critical analyses.

Kron’s works provoked acute discussions not only in our country. It was in particular

caused by the fact that his ideas were based on engineering intuition, not always backed

with strict proofs. Both this circumstance and the use of mixed mathematical and

electrotechnical nomenclature in Kron’s works scared “pure” mathematicians away from

his works. From the other hand, Kron supports his reasoning with fundamental concepts

of contemporary physics and mathematics with application of tensor analyses in an

unusual form which prevents an engineer, having ordinary qualification, from gaining an

understanding of Kron’s ideas and putting them into practice. Kron himself wrote about

his theory the following: [3] “When in the beginning of 30-s the author appeared with the

entire tensor and topological theory of rotating electrical machines, and in some years –

with the tensor and topological theory of fixed electrical networks, he came over a very

unpleasant surprise. In most technical journals new concepts invented by the author were

absolutely unexpectedly and vigorously announced useless and erroneous… From the

other hand the some workers of Institute of advanced research in Prinston (O.Veblen,

N.Veil, J. fon Neighman) and some ex-workers of the Institute (B.Hoffman, P.Langeven

and others) persistently advised the author to keep on further investigation. Even Einstein

told to the author, that he knew about his works from his collaborators (as the latter in his

practical problems used Einstein’s non-Riemannian dynamics of the general theory of

electric and gravitation fields). The opinions of the respected scientists had nothing in

common with absurd statements of that group of engineers.”

Although in his works Kron utilized the language of electrical engineering, he repeatedly

underlined that the nomenclature was not obligatory, and his method could be stated in the

language of the most modern mathematical theories, such as algebraic topology, geometry

of differentiated diversities, homologies and cohomologies groups, not to mention usual

tensor and matrix analysis.

Kron’s works, comprising 5 monographies and more then 100 articles published during 35

years, found a broad response in foreign literature. In numerous works of various authors

his methods were applied to most different problems. In a number of countries there exist

special scientific societies of scientists developing tensor methods: “Tensor club” in Great

Britain or Exploratory association of problem geometry in Japan.

In home literature Kron’s works became far less popular. Only in 1955 considerably

abridged there was translated into Russian his “Short course of tensor analysis for

engineers-electricians”, based on works of 1932-39 [4] and in Russian variant called

“Primeneniye tenzornogo analiza v elektrotekhnike” [5]. Numerous articles in various

foreign journals were still almost inaccessible, and without knowing general Kron’s

methodology it was pretty hard to understand them.

In 1972 there was made an attempt to fill in the gap by publishing Kron’s monography

“The piecewise solution of large-scale systems – diacoptics” [6] in Russian, that

generalizes the author’s research of many years. However, a very laconic style of the

narration, that presupposes awareness of previous works of the author, still left no hope to

become proficient in tensor methods to the addressees of the work – to engineers. Finally,

in 1978 there was published a translation of volumetrical Kron’s work – “Tensor Analysis

of Networks” [3], published in original as early as 1939. In 1985 there was also published

a book by A.Petrov “Tensor methodology in systems theory” (“Tensornaya metodologiya

v teorii sistem”) [7], where Kron’s ideas are given in a very clear form, and there also

have place examples of economic systems calculations with the help of tensor methods.

The bibliography given by him shows, that Kron’s methods are still not widely spread in

our country.

This review article aims to acquaint readers with Kron’s personality, to attract attention of

scientists to wealth of his ideas, to show that importance of his work never decreases,

more then that, it will constantly grow.

Gabriel Kron , the eighth and the last child in the family, was born on 23 of July in 1901

in a small town of Nagybanya, later renamed into Baya Mare, in a distant district of the

Carpathians in Hungary (Transylvania, which later became a part of Rumania). [8] Kron’s

thirst for knowledge and his purposefulness became apparent as early as at his school age.

In high-school he intensively studied physics and mathematics, found much time to study

astronomy, stenography, and languages: English and German. Being a high-schoolboy,

Gabby (as his friends called him) was invited as a house teacher to Felsobanie, seven

miles away from home. He kept on teaching until he finished school, every day

overcoming a fourteen-mile way. At seventeen Gabby was already able to work hard. He

was teaching till noon, practicing stenography from 1 till 2, solving algebraic problems

from 2 till 3, studying French from 3 till 4, doing physics from 4 till 5, translating from

German from 5 till 6, and so forth until 10 o’clock in the evening.

In June 1919 G.Kron finished the high-school and got a diploma. The question arose:

where to continue further education. Following the First world war Austro-Hungary fell to

pieces and Transylvania became a part Rumania. Kron didn’t know Rumanian language,

so he decided his future to be connected with other countries. But he needed money to

continue education abroad. Little money and hunger for studying were brought from war

by an elder brother Joseph. Gabriel talked his brother to go study to America by

promising that within one year he would make him prepared for examinations and getting

high-school diploma. In the end of October of 1919 Joseph got down to intensive

studying. Gabby picked for him not more then 10 per cent of pages from the books

necessary to read, and only them were studied by Joy. In the end of January Joy passed

exams for four grades, in April for the 5th and the 6th ones, in June for the 7th and the 8th,

and in August he passed exams for the whole high-school course. The way to America

was now open, and started on from Antverpern, in January of 1921 the brothers arrived at

New-York. In September they set to studying in the University of Michigan, at the same

time working for living and studies. Gabriel washed dishes in a restaurant, Joy worked in

a fur-shop.

At the last but one year in the University Gabriel wrote a small work “Fundamentals of

New Cosmology”, where he tried to consider the Universe from the point of view of an

engineer, disregarding such obstacles as laws of gravitation and relativity. At the same

time he started to dream about going round the world. After graduating, having earned

within four weeks a little sum of money for the most necessary things, having twenty

eight dollars in the pocket he leaves for California.

By the time he approached Los-Angeles he had twenty seven cents left, and he found a

job in American manufactory as an engineer of electric motors development. Soon he

moved to another company (“Robbins and Mayer”) in Springfield, Ohio, where he

worked under V.Branson. (In 1938 Kron devoted to him a book “Tensors application to

analysis of rotating electric machines”). In 1926 Kron came back to California and,

having three hundred dollars and a textbook on differential equations in his knapsack,

boarded a tanker bound for Tahiti.

After several weeks on Tahiti, Kron left for Fiji. While travelling he devoted his afternoon

hours to studying maths. After Fiji islands he headed for Sidney, where he stayed for a

while to earn money for the further trip through Australia, and then to Asia.

He wrote: “During the travelling I started to figure out basic ideas of multidimensional

vector analysis. Working every day with maps of unknown territories I got the similar

picture of engineering patterns, such as electric machine, or a bridge, or a plane, fixed in

my mind. I saw them as a network of spaces, connected into the whole, just like numerous

countries, islands, and continents, which are connected with interlacing of roads, laws,

and traditions”.

“If the connection between different members disappears, there also disappears

something, that turns forty eight independent states into united America, or thousands

separate details into a plane. Some years later I found out that mathematicians had already

made such kind of calculations, called “tensor analysis”.

The further Kron’s way went through Borneo, Manila, Hon-Kong and Saigon, where he

started for Burma, and then to Calcutta. Having crossed India on train he reached Karachi.

He crossed Persian Gulf onboard a vessel, visited Baghdad, Damascus, Cairo and from

Alexandria came to Constantsa, Bucharest, and then to his parents to Baya Mare.

Kron spent several months at home, studying and courting his wife to be Ann. After

returning to America in the end of 1928 he started to work in Lincoln electrical company

in Cleveland, and in 1930 he published the first of his more then hundred scientific

articles.

This work, called “Generalized theory of electric machinery” [9] started the series of his

works, representing more and more detailed analysis of machines and systems.

He suggested an idea that all types of electric machines should be special cases of a

generalized machine, and understanding of the general machine should cause invention of

new types of machines.

After he had moved to “Vestingaus” company to Springfield, he wrote his second work

[10], devoted to consideration of a coil conduct in an air gap under effect of a sine field.

During the American depression Kron together with his wife comes back to Baya Mare,

where he continues studying maths, and in particular, for the first time deals with non-

Riemannian geometry. He found analogy between abstract notions and complex

interaction of electric, magnetic, and mechanical forces in machines, and in 1934 he wrote

his classical work “Non-Riemannian dynamics of rotating electrical machinary” [11]

which in 1935 was honored with the Montefiore prize of the University of Liege.

This work caused at once broad discussions and contradictions. Using mathematical

notions in such a way nobody done before, Kron proposed a new meaning for equations

and settled rules, and that was the reason for many specialists to criticize his work,

considering them unreasonably complicated and unsuitable for practical use. When Kron

first proposed his ideas there were no big computers and engineers were not much

interested in big systems. It took time for their significance to be appreciated.

Kron always aimed at his theories being applicable to the largest possible range of

phenomena, strove for maximum generalization. That’s why his methods were more

complicated then a separate problem required. This, in particular, is one of the reasons

why Kron’s methods are not widely spread. Engineers, dealing with a single particular

machine, prefer to use simplest methods and feel little interest to elegant generalized

Kron’s theories.

Since 1934 till his last days Kron was working in General Electric Co., where he dealt

with electric machinery, power generators, and computers.

Within the period from 1936 to 1942 Kron, working hard, at the same time publishes in

“General Electric Revue” numerous articles, devoted to construction and utilization of

equivalent electrical circuits for different kinds of machines and systems. These articles

were published as his third book “Equivalent circuits of electrical machinery” [12] in

1951. His previous books “Tensor analysis of networks” and “Short course of tensor

analysis for engineers-electricians” appeared correspondingly in 1939 and 1942.

In 1942 Kron was transferred into steam turbines department, where he worked at

studying tension, appearing in steelworks. In 1945 he began working in the research

laboratory, where he solved problems of temperature distribution, nuclear reactors control

and so on.

In the lab Kron had worked up to 1963 (with a small pause from 1950 to 1953). From

1963 he was connected with engineering analytical branch, where he retired at the age of

65. He died after a short disease on the 25 of March, 1968.

Kron was a bright personality, a pioneer, whose achievements were appreciated by a few

in the time his works were first published, but whose methods of large-scale systems

analysis are widely used now.

His awards include the already mentioned Montefiore prize, he is an honorary M.Eng. of

University of Michigan (1936), an honorary doctor of Nottingham University (1961), a

Patron and Honorary Member of the Tensor club of Great Britain and Research

association of practical geometry in Japan.

Kron’s eccentricity is the result of his tensor point of view. Tensor methodology brought

him to working out of many powerful methods of analysis. Everything done by him could

be conditionally divided into three parts: 1) generalized theory of electrical machinery,

analysis of power transition systems and diacoptics (the piecewise solution of large-scale

systems), which usually applied in a simplified matrix form, 2) tensor philosophy and

mathematics that came into collision with indifference and misunderstanding, 3) works on

adaptive wave network, based on (1) and (2). There are few publications on the last part of

the investigation, although Kron in his last-years works discussed in general the principles

and the results received.

The first two interdependent parts represent a rather full theory. Let’s mark first of all, that

engineers are ready to accept results which join easily existing technologies, and have no

desire to deal with some abstract theoretical research, that tensor methods may seem.

But tensors have very special properties, which allow to consider them being certain

essences.

Let’s consider an example of a vector, which, as is known, can be shown graphically as an

arrow. If we introduce a coordinate system, we can define the vector components and

write them down in matrix-column or matrix-line. But we cannot identify matrix and

vector itself. When we change the coordinate system, the vector components change as

well, so we would deal with different matrices. But the vector remains the same,

irrespective of coordinate system changes. Thus, a vector is not just a matrix. Actually, it

is not a matrix at all. This is objective essence, which can be represented by an infinitely

big number of matrices, each corresponding to a certain coordinate system, so when we

move from one coordinate system to another, vector components transform in accord with

a special rule which reflects objectivity of this invariable essence.

Vector is an example of the simplest tensor of the first rank. In three-dimensional space it

has three components, corresponding to three axes of coordinate system. In n-dimensional

space it has “n” component. Tensor of “r” rank has actually the same properties as vector

does, excluding the fact that its number component equals nr and for “r” larger then 2 it

cannot be visualized.

The decisive statements are: 1) tensor reflects objective reality; 2) its components

transform in accord with a special rule, when the coordinate system changes. The tensor

property (2) shows that in some technical structure behavior equations are most

convenient when they are written down in tensor form, as tensor properties, to which

physical essences correspond, neither appear nor disappear while transforming.

In his generalized theory of electrical networks and machines Kron demonstrated

applicability of this approach of considering tensors’ physical nature.

The basic Kron’s idea on tensor analysis application is that two different networks, which

have the same branches, are considered to be two different coordinate systems of one and

the same physical essence. The idea that network is the set of branches, bound up into a

single whole through tensor of joint, brings us to the following fundamental conclusion:

tensor of joint within transformations, can be interpreted as tensor of transformations

joining different sub-nets into one big network.

One of the most significant achievements of Kron is his theory of non-Riemannian

dynamics of electrical machinery. The outstanding importance of this theory lies not just

in inventing tensor of joint, which shows possibilities of the entire approach in creation of

general machine theory, but also in receiving such solution procedure that would

transform dynamic systems equations for a statistical case. And through tensor

transformations, from a machine with fixed axes, we can get equations of any rotating

machine.

The starting point for equations describing behavior of electrical machines of any type, is

Lagrange dynamic equations, which, as is known, set up proportion between generalized

moments and generalized forces.

Lagrange equations can take tensor form only when usual differentiation is replaced by so

called covariant differentiation, which takes into account change of component tensors at

parallel transfer within curvilinear Riemannian space. However, usual covariant

differentiation formula can be applied only in case of holonomic coordinate system

(systems with geometrical connections, i.e. connections, depending only on mutual

disposition, not on speed). In non-holonomic systems there appear additional members,

but Kron successfully got over this obstacle, having shown that in case of electrical

machine additional members behave just like normal tensors. But their presence in

covariant differentiation changes geometry of space from Riemannian to non-Riemannian.

Thus, Kron managed to get from Maxwell-Lagrange equations engineering formula for

calculation of any electric network, having overcome non-holonomity nuisances

appearing when electrical axes change, by a simple transition from Riemannian to non-

Riemannian geometry.

In his further publications Kron kept on improving the theory of transformations, applying

it to various types of machines, obtaining the theory generalization by integrating results

into the theory of piecewise solution of electric network and systems (diacoptics).

Diacoptics was proposed by G.Kron in the beginning of 1950. The basic idea of

diacoptics consists in a system solution by tearing it into isolated parts. The total solution

of the whole system is received from the earlier solved individual torn parts; the second

step is aggregating the parts by transforming solutions for the parts received earlier.

The method of tearing was applied by Kron for determining equations of a completely

torn model, which he called “primitive”; it is usually the simplest, disconnected system.

Solutions of separate sub-systems, forming the primitive system, and consequently, the

resultant solution, can be exact or approximate, can represent linear or, with certain

precautions, nonlinear systems. They can be expressed in the numerical form or in

language of matrices, their elements being real and complex numbers, functions of time,

differential or other operators, and so on. Kron, using example of numerous problems,

convincingly proved that the method of tearing can be applied to solve algebraic

equations, equations in usual and partial derivatives with different boundary conditions,

problems on finding out eigenvalues. The method assumes continuous expansion and

generalization. This process, according to Kron, is similar to sky-scraper construction by

erecting a steel framework first, and only after that filling room between beams the way it

is required. [3]

The great advantage of the method is saving calculation time. For big, intricate systems

typical is the problem of matrices manipulations with big number of lines and columns

(form tens to thousands), which takes many hours of work even from modern computers.

If we use diacoptics, a system tearing on n parts saves time at matrices manipulations to

the value

where T is the time of usual (without tearing) manipulation.

Kron’s diacoptics is based on several key principles. [15] The first one requires a

complete record of all equations for generalized forces and responses for them, knowledge

of additional equations for responses and interrelations, that exist between forces and

responses. In other words, the definition “the equation of state” of some physical system

presupposes existence of set of equations, whose number equals to the full number of

forces and responses in the system. In a similar way “the equations of solutions”

presuppose existence of the same number of equations. Such kind of calculations is

carried out “automatically” by writing down equations in tensor form. The excess of

equations is more then compensated by the advantages, given by full mathematical model

and first and foremost the possibility of tensor apparatus application.

The second principle lies in utilization of electric network model for recording equations

of forces and responses. Electric networks exist on paper, so there is no need in their

physical realization. They are just easily observed models of more general topological

notions forming Kron’s method of reasoning.

The third key moment is the discovery of an “orthogonal nature” of electric (and other

types of) networks, having always non-singular (square) matrices of transformation. Due

to the use of configuration space of generalized forces variables, side by side with space

of responses (orthogonal to the space of force variables), there is an opportunity provided

for the change from complete “equations of state” to complete “equations of solution”,

and visa versa, at any stage of investigation.

The most important key notions are the notions of “tearing” into small parts of complex

systems on arbitrary number of subsystems, and a reversed notion “unification” of them

into initial or any other possible system. It should be noted, that topology – the science,

that deals with properties of interconnected spaces – originates in Kirkhgoff research

which concerned electric currants transit through a network. Here we could have

concluded, that Kron’s approach is just a practical application of a certain branch of

topology. However, it’s not like that. Actually, topology studies only those network

properties which stay invariant during wriggle, protraction or torsion (“rubber geometry”).

The properties that stay invariant during networks tearing into isolated parts, lie outside

topology interests.

As opposed to topology, the theory of tensor analysis and piecewise solution of complex

physical systems is primarily based on the use of notions which stay invariant, when the

network of spaces brakes up into initial space components, and after that join into all kind

of configurations, including initial network. This radically new point of view declares,

that we can move from equations of a certain possible configuration to equations of any

other configuration composed of the same components, using the system of non-singular

matrices of transformation, forming a so called group of connections. It is the

representation of the group, that Kron called “tensor of connection”.

2

2

n

Τ

The key notions “transformation”, “invariance”, “group” form a base for application of

tensor analysis of complex systems, which before Kron was applied in classical physics

for problems solution in the theory of fields of various nature and dynamic problems.

Depending on the problem type, invariant could be either full conductive power in the

circuit, or a part of intake, or a set of voltages applied, and so on.

The tensor approach breaks down barriers between the notions “equations of state” and

“equations of solution”, which first seem isolated, but actually are dual, by connecting

them with matrices of connection tensor, convenient to be calculated on electronic

computers.

Physical systems, used for connection, may be of absolutely different nature. They may

consist of devices, where electric, thermal, chemical, mechanical phenomena take place in

their poured speed key. Non-physical problems, which can be represented by tensor

equations, can also be solved by the method of diacoptics. In works of Kron and his

followers there are given numerous examples of calculations in various fields of

engineering, including force constructions calculations, aerodynamics, systems of control,

modern electronics, and among non-physical systems – economic problems, operation

analysis, and so on.

Tensor method has a number of advantages. One of them is what Kron called “mass

production” of solutions. Indeed, is a piecewise torn system consists of similar, or a few

repeated forms, one solution for the subsystem is applicable for most parts. The process of

connection is often the same for most analogues devices.

The system solutions can be “stored up” either in numerical form, or in the form of

structural tensors, so that they could be used again every time there appears a need in

solving the bigger system. Standard solutions can be stored and then applied to problems

of various types – a procedure, not arising from other methods of solution.

When a large-scale system is already calculated and its certain parts change after that, the

same change can be carried out on the corresponding parts of the solution.

There is no need in analyzing and solving the changed system every time anew, we should

just make calculations in the changed part. Thus, tensor solutions can correspond the same

type of growth and evolution the analyzed system overcomes.

As for applying models as electric circuits, Kron wrote: [15] Evidently, from a strictly

scientific point of view, the model of physical phenomena should be either algebraic or

geometric (topological). The author, being an engineer-electrician, neither mathematician

nor topologist, should express his ideas in terms of the science most familiar to him. Of

course, the use of electric circuit model is not absolutely necessary. It’s just a ruse to

substitute records of extremely big number of equations and manipulate with them”. It is

really a ruse, that verges on art. Kron constructed a lot of models of electric networks for

the most various types of problems: distribution of a thermal current, wave equation of

Schroedinger, current of a liquid, neutron diffusion in a nuclear reactor, stresses in elastic

girders, forces of basicity in polyatomic molecules, lines of electric power transfers,

systems of linear and nonlinear equations in usual and partial derivatives and others.

However, after his death, there appeared no new models of electric circuits, proposed by

other authors.

Along with electric circuits Kron used model representations in the form of algebraic

diagrams of Ross, which allow to visualize the structure of problems, requiring for their

solution the use of interconnected multidimensional spaces. The possibility of

constructing models, reflecting the problem structure was also found by Kron in using

theory of groups and symbolical logic.

During the last decade Kron was intensively working on the theory of self-organized

polyhedral networks or wave automatic machines. Here, the starting point in Kron’s

reasoning is still his belief that even linear (one-dimensional) network contains much

more information about modelled problem, then just a set of points in space (a

description, typical for usual mathematics), as the network branches penetrate

surroundings of a given point, and thus, it turns out to be connected with the nearest and

more distant environment. Further generalization consists in considering a network,

consisting of planes, where each plane connects three (or more) neighbor lines along with

linear network ( whose branches are 1-symplexes, connecting two neighbor points). Now,

in n-dimensional space there appear an additional network, consisting of such triangle

planes (2-symplexes), which, as well as in case of linear network, can form open and

closed paths. Since four triangles with common edges form tetrahedron (3-symplex), it

becomes evident, that even more complete information can be received from the neighbor

surroundings of the given point, if we fulfil utterly n-dimensional space with tetrahedral

network, forming open and closed paths. This process can be continued provided the

dimensionality of elements (p-simplexes), composing resultant additional network, will

grow until the entire n-dimensional space is fulfilled with n+1 various networks (the

initial system of points can be considered to be 0-symplex). In combinatorial topology

such universal structure, composed of 0-,1-,2-,…n-dimensional interconnected networks

is called polyhedron. However, there are no such notions as either “impendance”, or

“current” and “stress” in combinatorial topology, but Kron accepted the point of view that

“area” of each p-simplex is represented by its impendance.

Then, to complete description of n-dimensional space, Kron also invented a notion of

mutually-orthogonal to the initial, “dual” to it polyhedron. It turned out that with each p-

simplex of the initial polyhedron there is connected n-p simplex of dual polyhedron, and

these two simplexes represent a certain part of n-dimensional space, and now

surroundings of a separate point are completely described by n+1 different doubled

simplexes of various dimensions that surround the point.

Thus, it is accepted, that two “non-brisk” polyhedrons: initial and dual, represent the set of

independent variables more complete, even in the case, when there is given a

comparatively small number of points. However, neighbor surroundings of each point

deliver now more information for analysis.

The task was “to revive” the received polyhedral network in accord with accepted

methodology. The fact is that electric currant cannot flow in polyhedron, as currant

vectors do not satisfy Stokes’ Theorem, while passing boundaries of networks with

different dimensions (vector linear integral should on closed trajectory be equal to surface

integral of vector curl). However, this theorem is satisfied by a full set of Maxwell

equations. Thus, a complete electromagnetic wave, characterized with four types of

adaptive parameters e, h, b, d, can spread through a polyhedron and the one, dual to it.

Trying to satisfy Stokes’ Theorem in wave passage through networks of different

dimensions Kron determined the fact (well-known in geometry), that even-dimensional

spaces behave differently from odd-dimensional spaces, and therefore, in polyhedron

there should be invented two complete networks of various physical nature for generating

one electromagnetic wave. In this connection Kron invented the generalization, that all

even-dimensional spaces are constructed of magnetic material, while all odd-dimensional

networks – of dielectric material. In dual polyhedron the physical role of spaces of even

and odd dimensions is mutually reversed.

Thus, for one electromagnetic wave diffusion there should be two networks of p and p-1

dimensions (even and odd), therefore n-dimensional polyhedron and dual polyhedron

comprise n/2 sequence of space waves, each spreading through spaces of growing

dimensions.

The constructed polyhedral structure represents a multidimensional space-filter with a

fixed set of natural frequencies. If we make up our minds to model (to adjust) some

function, the problem would be solved with the least mistake for the functions close to

proper solutions, corresponding to this certain set of frequencies.

Kron asked himself a question, if it was possible to construct such multidimensional

space-filter which would help to adjust not one, but a big number of functions

simultaneously, or, in other words, if it was possible to construct such a structure which

could oscillate not only with one certain set of frequencies, but with all possible

frequencies. Such structure should differ from a usual filter with discrete spectrum by

having a continuous spectrum.

It was found out that this problem could be solved, if we placed straight and dual

polyhedron into stationary magnetohydrodynamic plasma, accompanied by four

additional adaptive parameters, which characterize densities of electric charge, magnetic

fluxes, and corresponding currents (ρe ,ρm, Je, Jm). The presence of already eight types of

adaptive parameters in each p-network gives no promise that such structure will be able to

oscillate. It is also necessary for various straight and dual networks to be connected by

ideal transformers. Such non-mechanical oscillating structure (having neither velocities

nor moments) was called by Kron “oscillating polyhedron”, which represents the simplest

possible form of “self-organizing automate of dynamo type”, i.e. whose operation is based

on complex interaction of generalized rotating electric machines (dynamos). Oscillating

automate can be a model for solving numerous statistical, economical, and physical

problems of large dimensions. [16]

Kron demonstrated possibilities of self-organizing automate by an example of simulation

(adjustment) of six arbitrary functions on four, given on a plane, points. The task required

was that polyhedron could simultaneously reproduce all six proposed functions plus their

first and second derivatives as exactly, as it was possible without the model change every

time the adjusted function was changed. The model consisted of 13 structures,

representing generalized rotating electric machines with 13 usual and 13 quadratic axes.

In particular, 4 machines represented a function, 6 – its first differentials, and 3 – second

differentials. [17]

Calculations showed very high accuracy of adjustment, especially in case of oscillating

polyhedron. It is important to underline, that all six functions gave the same results with

equal accuracy without changing model, when the functions changed, for which the

evaluations were done. A big number of adaptive parameters (waves) adjusted themselves

to the change of boundary conditions. Thus, self-organizing polyhedral automate is,

actually, a universal model, able to simultaneously evaluate any number of functions,

when there is a fixed set of data (matrix data or independent variables).

Euclidean polyhedral automate with “straight” p-simplexes can first of all be applied for

receiving numerical solutions in multidimensional problems, such as multidimensional

adjustment of curves, interpolation, data smoothing, generalized harmonic analysis and

others. With the help of oscillating polyhedron there can also be studied the processes of

non-linear programming, problems of reliability, recognition of forms and others. The

further way of generalization proposed by Kron consists in insertion of polyhedrons into

moving magnetohydrodynamic plasma. In this case “straight” lines, planes, cubes and so

on in polyhedron become curve lines, planes, cubes. I.e. usual Euclidean spaces (and

simplexes) with metrics, set by the network elements, are changed to non-Euclidean,

Riemannian or non-Riemannian simplexes with the help of “affine connections” of

various types. These bent elements of the network cause appearance of new parameters,

characterizing their curvature and torsion.

Such a complex automate is useful first of all for studying the very magnetohydrodynamic

plasma. The possibility appears to analyze many phenomena, which occur in the plasma,

proceeding not only from usual field description, but from the discrete one as well.

Polyhedral structure, immersed into multidimensional fluid, is considered to be a

stationary coordinate system, where there are projected moving fragments and waves (to

be more exact, their linear , surface, volume and so on integrals). Although polyhedron

and plasma, taken separately, are described by a comparatively small number of variables,

additional multidimensional p-networks, nevertheless, provide detailed information about

all derivatives in various directions of all operating surfaces. Each p-simplex can be

considered in details on the analysis basis of p-dimensional field distribution with the help

of tensors of “p” rank.

Full description of plasma can be replaced by consideration of multidimensional networks

consequence of growing dimensions transmittance. Networks unite into one operating

system a big number of multidimensional rotating electric machines, excited by

electrostatic and magnetic flows and possessing liquid and gaseous rotors, each having

immediate speed, moments, and other attributes.

Tensor notions can be used for studying stability of plasma, considered as networks of

generation, transmittance and distribution of energy types. The already worked out tensor

notions point at existence of new types of stabilizing forces, which are not taken into

account by usual non-tensor methods and which can be unknown to practicing engineers.

The Kron’s idea about polyhedron, which in problems of cognitive type (such as

recognition of images) can play the role of “artificial brain”, where each “neuron” is

represented by magnetohydrodynamic generator (generalized rotating electric machine), is

almost the most prospective direction for development of Kron’s polyhedral wave

automate. Such type of artificial brain (dynamo type or a type of “energetic network”) is

based on a foundation, different from the models of artificial brain, presently being

developed on the basis of commutation networks (switching networks).

There are very few publications on this research, although Kron discussed in general the

principles and results received in his various works of last years. Kron himself considered

this part of work to be the crown of his research, especially the idea of “crystalline

computer”, using analogy between crystal optical properties under effect of light source

and mathematical properties of multidimensional polyhedron, submerged into plasma.

Very important is the fact that polyhedral system assumes statistical interpretation, i.e.

“non-brisk network” represents moments distributive functions of various types, and

distributed waves can be treated as presentation of means, variations, correlative

functions, spectral density, and many other notions.

For description of a big number of rotating electric machines Kron widely used the

notions, in their basis having statistical nature, such as boundary and free energy, entropy,

thermodynamic density, and others.

Here could be seen an extremely interesting connection between Kron’s self-organizing

polyhedral structures and those fruitful ideas, which are being worked out by the winner

of the Nobel Prize I.Prigozhin, lately having become widely spread among representatives

of various sciences and trends. [18]

The idea of developed by Prigozhin (a specialist on nonequilibrium thermodynamic

systems) approach is expressed in the title of one of his books – “Order from chaos”. He

convincingly showed that under certain conditions open nonequilibrium systems,

exchanging energy flows with environment, can arbitrary turn from chaotic, disorganized

state, to organized system, and can stay in such state till conditions change. Thus, Kron’s

adaptive polyhedrons, realizing nonlinear, multidimensional statistical phenomena can

serve as ideal models for analyses of such complex systems, changing from disorganized

to organized state.

Adaptive polyhedrons, transforming any (including chaotic) spaces of states into

“automatically” organized spaces of solutions can serve as models for quickly developing

lately theories, using such notions as “fractals” or “regulated chaos”.[19]

Actually, tensor methodology of Kron gives unlimited opportunities for constructing

various kinds of models, and their value consists not just in appearance of common

algorithm for their consideration, but also in possibility of receiving new knowledge.

Tensor approach can also be applied for solution of such important problem as forming

data bases and knowledge (tensor data bases). Various signs of some information massif

can be considered as tensor essence component (projections) into this or that coordinate

system (system of knowledge).Transition from one massif to another when the required

sign changes, can be carried out on the basis of tensor transformation rules, thus excluding

the need to search information every time anew. A possibility to keep ready solutions for

different subsystems and, if necessary, carry out synthesis of solutions form a basis for

constructing and controlling systems of automation of new technical means and

technologies projecting, including automated projecting of robototechnic systems.

Although Kron himself wasn’t engaged in the sphere of management, his ideas formed an

important part of conceptual notions.

Principle system notions, such as idea of change, existence of alternative methods for

receiving results, strong influence of environment and others, find adequate reflection in

the Kron’s theory.

The list of examples of tensor methodology application could be ever continued, but the

already given review shows fully enough the idea about strength, flexibility and breadth

of tensor approach to solution of complex systems. The author is convinced, that we

should keep on studying the scientific heritage of G.Kron, especially it concerns his works

on self-organizing polyhedrons, develop his methods of tensor analysis and networks

syntheses as a common model basis, make them available for wide range of specialists.

Literature

1. Electrichestvo, 1969 g. № 1, p.92.

2. Electrichestvo, 1950 g. № 12, p.78.

3. Kron G. Tensorny analiz setey, translation from English (pod red. L.T.Kusina,

P.G.Kyznetsova), M, Sov. Radio, 1978.

4. Kron G. A short course in tensor analysis for electrical engineers, N.Y. Wiley, London, 1942.

5. Kron G. Primeneniye tensornogo analiza v elektrotekhnike, trans. from English (pod red.

Meerovicha P.V.), M, Gostekhizdat, 1955.

6. Kron G. Issledovaniye slozhnykh sistem po chastyam – diakoptika, trans. from English (pod

red. A.V.Baranova), M, Nauka, 1972.

7. Petrov A.E. Tensornaya metodologiya v teorii sistem, M, Radio i svyaz, 1985.

8. Narr.N.N. Gabriel Kron and System Theory. – N.Y. Schenectady, Union College Press,

1973.

9. Kron G. Generalized theory of electric machinery, - AIEE (American Institute of Electrical

Engineers) Trans. v. 49 (1930), p.666.

10. Induction motor slot combinations; rules to predetermine crawling vibration, noise and hooks

in the speed – torque curve. AIEE Trans. vol. 50 (1931), p.757.

11. Kron G. Non-Riemannian dynamics of rotating electrical machinery. – J. Math. Phys. 1934,

v.13, № 2, p.103.

12. Kron. G. Equivalent circuits of electrical machinery, - N.Y. Wiley 1951, London, Dover,

1967, p.278.

13. Journal of the Franklin Institute, v.286, № 6, 1968.

14. Hoffman B. Kron’s Non-Riemannian Electrodynamics – Rev. Mod. Phys. (Einstein’s Birth

day commemorative issue), 1949, v.21 p.535.

15. Kron G. A set principles to interconnect the solutions of physical systems. – J. Appl. Phys.

1953, v.24, p.965.

16. Kron G. Self-organizing, dynamo-type automata Matrix and Tensor Quarterey, 1960, v. 11,

№ 2, p.42.

17. Kron G. Multi-dimensional curve-fitting with self organizing automata, J. Math. Analysis

and Appl. Phys. 1962, v.5, № 1, p.46-69.

18. I.Progozhin, E.Stengers, Poryadok is khaosa, Mir, 1968.

19. R.M.Kronover, Fraktaly i khaos v dinamicheskih systemah. M, Postmarket, 2000.

20. Kusin L.T., Armensky A.E., Petrov A.E., Abramov D.Yu., Ermakov A.N. Tensornye banki

dannykh. Izv. Vuzov SSSR. Priborostroyeniye. 1984. № 6, p.38.