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  • International Alexander Bogdanov Institute, Ekateriburg, Russia
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
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
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
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 completeequations 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”.
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
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
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, 6its first differentials, and 3second
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.
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,
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.
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