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THE INSOLVENCY OF THE THEORY OF ELECTROMAGNETICISM AND THE EXIT FROM THE COMPLETED DROP.

"SPECIAL" THEORY OF RELATIVITY
(SRT * - new edition, SOTO and Quaternary Universe)

To the 100th anniversary of the theory of relativity

Physics. Discoveries in physics.

V.M. Myasnikov

Welcome to the forum

SPECIAL THEORY OF RELATIVITY
(SRT * - new edition)

So, let - Minkowski inertial physical reference system, i.e. In a homogeneous and isotropic space, the physical reference point O is chosen and the direction (ray) given by the unit vector . Space-time in the reference frame Define as a quaternary set of events

, (1)

Where c is the speed of light, the asterisk means multiplication by an imaginary unit, - the time measured from some origin at the reference point, and Is the radius vector from the reference point. Let in the reference frame A Cartesian rectangular coordinate system is chosen With the origin at the reference point O and so that the coordinate axis Was directed along the chosen direction. If you enter the coordinates of the coordinate axes , Then the quaters can be written in the form And event , If necessary, interpreted as a point with coordinates At the time .

Further, let another frame of reference - with reference point , Located on the axis , And the same chosen direction - moves, moving away or approaching the reference point O, with a constant speed Along the axis . Mobile system reference point Is determined by an event In an old (fixed) frame of reference.

Spin of rotation In the quaternion space ( 1 ) we define by normalizing the event , I.e.

Quater We call the rotation spinor (hyperbolic, since the angle of rotation - imaginary), Is determined from , , - the unit of the selected ray, and - the projection of the vector-velocity on the beam, i.e. In the case of removal of a moving reference point from a fixed point , In the case of approximation - .

Space-time as a quaternary set of events in a moving reference frame

(2)

Or, what is the same, dashed coordinates and time at We determine, using the Lorentz transformations, which in a quaternary space are defined as the right and left half-rotations ( see Chapter III ),

. (3)

Having done all the calculations and writing the result for the difference of events in the Cartesian coordinate system, and - respectively, we obtain

. (4)

The dashed values ​​in ( 4 ) are determined (calculated) from the unprimed reference frame With the help of "unprimed" standards, and yet do not have the sense of "proper" values ​​of the primed reference system , I.e. They can not be given a physical meaning. This problem of Lorentz transformation, by themselves, can not be solved, some additional condition is required. One of these conditions is the condition of simultaneity of spatially separated events and at , More precisely, the condition for the preservation of this simultaneity in the transition from a fixed frame of reference to a moving one (more precisely, the condition of the reality of events and the preservation of reality when passing from one reference frame to another.) The condition of simultaneity is only a necessary condition for reality.

Definition of simultaneity : If two events in the reference frame are defined by radius vectors and At moments of time, respectively, and , These events are called simultaneous with respect to the reference point, if

, (5)

Where - speed of light, and - the length of the radial (relative to the reference point) component of the vector (See the Quaternary Universe below for more details ).

In our notation, events and Are simultaneous with respect to the reference point of the system , if

. (6)

Substituting From (6) into the first formula (4), we have

Or, finally

, (7)

Where

or . (8)

We note that in practically important cases , Relations ( 8 ) can be written down to small First order of smallness, in the form

. (9)

We draw attention to the fact that here the speed - algebraic (the projection of the vector-velocity on the chosen ray (on the axis )), I.e. If the mobile reference point moves away from the fixed point, And from ( 7 ) it follows that , I.e. Time accelerates (second becomes shorter). If the mobile reference point approaches a fixed reference point, then vice versa And from ( 7 ) it follows that , I.e. Time slows down.

Similarly, substituting From ( 6 ) to the second formula ( 4 ), we have

, (10)

Ie the length in the direction of motion decreases in the case when the mobile reference point is removed from the fixed point and increases in the case of approach.

Finally, note that the last two formulas ( 4 ) show that the lengths of segments perpendicular to the chosen direction are independent of the motion of the reference frame.

Some consequences:

The Doppler effect. Aberration of light. If in ( 10 ) - the wavelength of emission of a moving light source, and - the observed wavelength in the stationary reference frame, then ( 10 ) with regard to ( 9 ) is written in the case of removal of the light source in the form

, (eleven)

And in the case of a light source approaching -

. (12)

Formulas ( 11 ) and ( 12 ) describe the so-called. The Doppler effect of the shift of the spectral lines, respectively, to the red end of the spectrum in the case of a receding light source (formula ( 11 )) and to the violet one in the case of an approaching source (formula ( 12 )).

If the light source is not on the selected beam and the direction to the source forms an angle With the selected beam, then passing to the new Minkowski reference frame with the same reference point (the light receiver) and the new selected direction to the source, we have the only difference from the "old" Minkowski reference system in that now the projection of the velocity vector on the selected direction is equal to . In this case ( 11 ) and ( 12 ) are generalized by formula

.

Lateral Doppler effect ( ) In our theory there. We have reason to assert that it does not exist in Nature.

The transverse component (relative to the direction to the source) of the vector-velocity Gives an effect called aberration of light, in particular, takes place

(13)

- A formula for the observed deviation from the Earth of the position of the stars on the celestial sphere forward along the run Movement. Here - vector-velocity of the Earth and - the angle of the direction of the star relative to the vector-velocity. (For more details, see [1] , Chapter XIII ).

Thus, the Doppler effect and light aberration are direct and direct experimental confirmation (or, if you like, a consequence) of the special theory of relativity.

"Paradox" of the twins. We write in quotes, because within our theory there is simply no such paradox. Movable twin the first half of the path is removed from the Earth and its time, from the point of view of the motionless, is accelerated. The second half of the way the moving twin approaches the Earth, and its time slows down and by the time of return completely compensates the acceleration of the time of the first half of the path, i.e. From the point of view of the immobile twin their age is the same. And this situation with twins is absolutely symmetrical, i.e. And from the point of view of a traveling twin their age is the same. Travelers of the future can not be afraid of returning to Earth to get into the distant future of the Earth. .

The rotation of the starry sky. As is known, the rotation is kinematically relative, i.e. The rotation of the Earth around its axis and the rotation of the starry sky from the Earth in the opposite direction are kinematically equivalent. In other words, the observed rotation of the starry sky is just as real (kinematically, that is, without the dynamic parameters of the motion of stars-masses, inertia forces, etc.), as well as the rotation of the Earth around its axis. And then a lot of questions arise concerning the kinematics of such motions : What is the linear (tangential) velocity of the stars? Is it more or less than the speed of light? What kinematic effects of the special theory of relativity (the lateral Doppler effect?) Take place under such motions? Etc. Traditional SRT, as far as we know, ignores these questions.

We solve these questions radically : the observed motions of stars on the celestial sphere during the Earth's rotation are not physical , but quite real . And if these movements are not physical, then the answers to the questions posed can be formulated quite arbitrarily, relying, for example, on "classical common sense." Thus, we can assume that the tangential velocity of stars is determined by the classical law of rotation, without limiting the speed of light, , Where Is the radius vector of the star, and - the angular velocity of the Earth (the problem here is that the tangential velocities determined by this formula are greater than the speed of light, even for the nearest stars). Note that the special theory of relativity does not give a single reason against our proposal, because Its effects take place only in the radial direction (this does not apply to the traditional SRT, in which the lateral Doppler effect takes place). Of course, this should not be considered as a proof of our proposal, but the fact that SRT * does not reject our proposal, gives us additional confidence in its fairness

Addition of STR effects. Addition of velocities. In Chapter XI we posed another question about the "addition" of SRT effects, which is absent in the traditional theory. We are talking about the following.

Consider three reference frames , and , In which coordinate systems are chosen so that their abscissa axes lie on a common straight line and the origin is chosen at the reference points. Let the system Motionless, Moves along a straight line with a speed On the system , And the system - along the same straight line with speed On the system And with the resultant speed On the system . Consider, for example, the effect of shortening lengths. Let - the length of the segment (in the direction of travel) in the system , - the length of the same segment in the system and - in system , Then ( see (10) and (8) )

, (14)

, (15)

, . (16)

Comparing From (16) and from (15) with allowance for (14) , we conclude

,

Whence

(17)

- formulas for "addition" of SRT effects. Calculating the tangent of the sum and substituting the values ​​of the tangents from (14) - (16) , we obtain

(18)

- the formula for the addition of velocities.

The formula for the addition of velocities is also derived in the traditional SRT , whereas the formulas for "addition" of effects in the traditional SRT are not. This is due to the fact that in traditional SRT relativistic effects depend on the square of the velocity and, therefore, do not depend on the sign of the velocity, which sometimes leads to contradictions in their interpretation.

Weight in the SRT *. The kinematic mass. Dynamic mass. There are problems in which the mass plays the role of a passive parameter and does not affect actively the physical conditions, for example, the mass of the test body, which (by the test body definition) reacts to physical conditions, but does not affect these conditions. Such a mass is called kinematic . In our theory, the kinematic mass is transformed in the same way as time and length ( see (7) and (10) ), decreases with removal and increases with approach

, (19)

Where Is defined in ( 8 ). (For the conclusion (19), see [1] ).

If the mass is estimated as a measure of the interaction of the body, for example, with the field, then such a mass should be treated differently. As an example, we consider the mass of a moving electron in a known experiment by W. Kaufman to verify the dependence of the electron mass on velocity. We propose the interaction of an electron with each point of the field as a transient process in which the electron first approaches the point of the field, then merges with it and then is removed. The transition characteristic of such an interaction is unknown to us, but we can consider an idealized transition process with an ideal transition characteristic in the form of a "single step". And then our theory gives ( Is the rest mass of the electron)

(20)

- a result that coincides with the conclusions of A. Einstein, confirmed by V. Kaufman and on modern accelerators (the dependence of mass on velocity ( 20 ) was found by us in the Newtonian model of the Universe in the proof of the Mach principle, Chapter V ). We propose to call such a mass dynamic .

The question of whether the mass of the experimental body is considered kinematic or dynamic, under conditions of real experience or in theory , remains at the investigator's discretion.

Measurements. Standards. Theory of dimensions in SRT * . Any measurement in physics, in the final analysis, amounts to a comparison with the standard. The principle scheme of measuring a physical quantity is reduced to finding a number that indicates how many times the standard fits into the measured quantity. Let - length (length) and - the standard of length (meter, 1 meter), then the length Define as follows:

. (21)

Here - a number that indicates how many times the standard Fit in the size . We shall call - the dimensionless value of the quantity And denoted by the same letter in square brackets. Similarly, we define time and mass, denoting and - respectively, the standards of time and mass,

. (22)

Recall that if the reference frame (Dashed) moves with a constant speed Along the selected direction of the fixed system , Then ( see (7), (10) and (19), a and (8) )

.

We rewrite the latter with allowance for ( 21 ) and ( 22 )

,

But dimensionless quantities, as "the number of times ... ", Do not depend on physical conditions, on movement, etc., ie. Under any transformations. And then we get

, , (23)

- conversion of standards of time, length and mass.

Further, we assume that the standards of all physical quantities, compiled from the fundamental standards of time, length, and mass, are transformed as a whole so that the constituent standards are transformed according to the law ( 23 ). The meaning of this statement will be clear from the examples.

For example, the gravitational constant

Does not change during the transition from the fixed to the moving reference frame, while the Planck constant

(24)

It decreases with the transition to the moving reference frame and increases with the transition to the approaching system.

The latter must inevitably lead to new ideas and possibilities in quantum physics, especially considering that collision of particles is one of the most important "tools" in the study of elementary particles ( see and [2] ).

The energy. Why do bodies move (rotate) in energy? Let us return to the Kaufmann experiment and ask the following question: does the role of the electron, that is, the "experimental" particle, play any role for the proposed derivation of formula ( 20 )? A particle that has an electric charge interacting with the electromagnetic field of the device?

The charge of the electron, electric and magnetic fields in the Kaufmann device can be considered only as the "technical part" of the device providing the relativistic controlled velocities of material particles (in this particular experiment, electrons). Therefore, preserving the basic idea of ​​the concept of the dynamic mass as a measure of the interaction of a material particle with a field, we are distracted from the specific nature of the field and from the method of making the particle of constant velocity.

So, we believe that the inertial frame of reference is a kind of constant field . And let the material particle of mass (Rest mass) moves at a constant speed Along a straight line passing through a fixed point of the field (the inertial system). Further, arguing in exactly the same way as in the case of the interaction of an electron with a field in the Kaufmann experiment (as before, with an ideal transition characteristic in the form of a single step, that is, the field at a fixed point "turns on" when the center of the particle has reached this point, while Half of the particle has not yet reached this point and is approaching it, whereas the second half has already passed this point and is moving away from it), we find

, (25)

those. The inertial system "acts" on a particle of mass m moving relative to the inertial system at a constant speed , As if its mass were determined by the expression ( 25 ). The conclusion about the dynamic mass ( 25 ) is the final one upon completion of the transient process of interaction at one point of the inertial system, then to the other, and so on.

Let us consider in more detail the transient process (as before with an ideal transient response) of the action of the point of the inertial system (field) on a particle of mass m moving with constant velocity. The point of the inertial system first "meets" the moving particle (half the particle at the moment of "turning on" the field) and then, in accordance with ( 19 ), , Then - "escorts" with a smaller mass . Where does the mass disappear

? (26)

Since the inertial system, by definition, does not change after passing through it a material particle with a constant velocity, there remains the only possibility that this mass is "carried away" by the particle. Below is shown what meaning it is possible to enclose in this statement.

In Chapters IV and V ( see and [2] ), we considered the model of the universe as the inner space of the "material point of the mass of the universe" under the assumption that the entire substance of the universe is localized on its gravitational sphere. In this model, all points of space are equal and any of them can be chosen as the geometric center of the universe, and the space with respect to the geometric center (or any point) is homogeneous and isotropic. The reference system, with respect to any point in a homogeneous and isotropic space, was called an inertial frame of reference. Movement with a constant speed does not violate the homogeneity and isotropy of space and, therefore, the inertiality of reference frames. In real physical conditions, any frame of reference can be considered inertial insofar as in these physical conditions it can be assumed that the space is homogeneous and isotropic.

Modern ideas about the universe as a whole come from the ideal model of a homogeneous and isotropic universe with a constant average density of matter (and radiation), which is determined by a paper-pencil operation (the term of PW Bridgman ) dividing the mass of matter in some area of ​​the universe by the volume of this Region, as a result of which the average density of different regions is leveled, and with the further increase in the sizes of the regions, up to the largest single region - the Metagalaxy, gives the average density of matter (and radiation) in the universe. The "physical realization" of such a model is the Universe, in which all matter is uniformly distributed over its volume. Are there any regions in the universe that could be considered, at least approximately, as an "example of realization" of the space of an ideal universe? Today, science does not know this, at least in the Galaxy and its environs there are apparently no such regions.

We propose another ideal model of a homogeneous and isotropic universe (for more details, see Chapters V, VIII and XV ), in which the concept of the gravitational sphere of the Universe is introduced, and all matter is "pushed" to the horizon and "localized" on the gravitational sphere , But there is no substance in the space inside the sphere. In the real universe, observed from the Earth, even if we take Large (say, the limiting distance available to an average telescope), then still , And - mass of substance inside the sphere of radius Still much smaller than the mass of the universe, i.e. And the bulk of the substance of the universe, which determines the dynamics of the universe as a whole, is still beyond our observational capabilities. From here one step to the representation of the ideal Universe - neglected And push To the horizon. As a "physical realization" of the space of such a model, one can consider any region of outer space that is far removed from massive bodies and whose space can be considered homogeneous and isotropic, for example, intergalactic space, the space of the solar system far from the Sun and planets or even on the Earth's surface at Very rude experiments (for example, in everyday life).

On the other hand, calculating the Newtonian gravitational potential created by the gravitational sphere (the entire substance of the universe) at an arbitrary point in the universe ( ), We find ( see Chapter V )

(27)

( - the mass of the universe and - its gravitational radius), i.e. The gravitational potential of the universe is equal to the constant (and not to infinity, as in the Newtonian universe) and, consequently, the universe at the observation point ( ) Does not create a gravitational force field (we recall that in the Newtonian gravitation the physical potential has a potential increment, but not the potential itself, more precisely, the gravitational forces, defined as the potential gradient, appear only in the alternating potential field). We call the gravitational field ( 27 ) equipotential . Thus, at every point in space in the universe, the aggregate substance of the universe determines, along with the inertial space, the gravitational ( not force, inertial ) gravitational field ( 27 ). Moreover, we believe that it is the aggregate of the universe that determines , through the inertial field ( 27 ), the inertial space and thereby the inertial frames of reference . The basis for such an assertion is that a motion with a constant velocity in an equipotential field does not violate its equipotentiality i.e. And in the moving frame the field remains equipotential. The latter can be regarded as another definition of the inertial frame of reference. In accordance with this definition, a reference frame can be referred to inertial systems (in reasonable approximation), inside a space station moving along the Earth's orbit, i.e. On the equipotential trajectory, which determines the equipotential field inside the station. We observe, for example, the phenomenon of weightlessness as a manifestation of inertiality in the reference frame of the station. For the same reason, it is possible in terrestrial physics not to take into account the gravitational influence of the Sun, etc.

We also recall ( see Chapter IX ) that we call the "intrinsic" potential energy of a particle of mass m (negative) energy

,

Where Is the gravitational radius of mass m . The potential energy of a particle can also be interpreted as the potential energy of interaction of a particle with the entire substance of the universe ( see (27) )

.

And then the mass of the material particle can be defined as

.

Here - the kinetic (relativistic) energy of the particle m , determined by the virial theorem, see Ch. IX . It is in this sense that we affirm that the mass of a material particle can be interpreted as a measure of the interaction of a particle with the entire substance of the universe, and ( 25 ) as a "mechanism" of such interaction, while we believe that there is no need to divide such an interpretation of the mass into "subspecies" (Inert, gravitational active and passive, electromagnetic, etc.).

Let us say a few more words about the famous Einstein formula . We believe that the significance of this formula, as a formula establishing the connection between mass and energy , is greatly exaggerated. The fact is that it is in principle impossible to measure independently the mass And energy For the same object in one experiment. The connection between mass and energy in this formula is purely terminological . If - mass in the "language of the masses", then - the same mass in the "energy language", meaning that - a fundamental physical constant ( in this sense it is necessary to understand the statement about the equivalence of mass and energy .). You can also give a lot of other names of the mass, for example, - in the "language of momentum" - in the "language of lengths" - in the "language of time", etc. Any of the listed mass names can be used provided that the physical dimensions are reconciled and the terminology used is corrected or even another physical interpretation of the phenomenon.

We return to ( 26 ). If the mass of a moving particle is given by the scalar momentum , Then ( 26 ) can be rewritten in the form

, (28)

those. If the particle of mass Moves with a constant speed Relative to some inertial system, then every point of the inertial system (or, which is the same, every point of the inertial gravitational field of the Universe at the corresponding point of the inertial system) informs the particle of the amount of motion ( 28 ). Taking into account that the mass and velocity are constant (the velocity is constant and in the direction, that is, ( 28 ) is easily generalized to the vector velocity), the inertial system at each point informs the particle of the same amount of motion (momentum - for vector velocity) , I.e. The particle moves with an unchanged amount of motion. Such a move since Newton's time is called a movement of inertia ( Newton's 1st law ). The proposed interpretation ( 28 ) can be considered as the "cause" of motion by inertia.

Thus, the inertial gravitational field ( 27 ), generated by the aggregate substance of the Universe, is the cause of the motion of material bodies by inertia. The latter is naturally connected with the Mach principle, i.e. Motion by inertia is one component of Mach's principle . Another component of Mach's principle refers to the emergence of inertia forces. In the case of violation of the conditions of homogeneity and / or isotropy, the frame of reference is not more inertial, which, on the one hand, leads to a change in the amount of motion during the motion of the material body, on the other hand, to a violation of the constancy of the potential in the reference frame of the body; The potential difference at the points of space with the motion of the body becomes different from zero, which leads to the appearance of gravitational forces in the frame of reference associated with the body. The latter are interpreted as inertia forces (for more details see Chapters V and IX ).

Note that the listed physical phenomena occur simultaneously, and it is not always possible to distinguish between causes and effects among them. It is us, people, creating a physical theory, we build physical phenomena in a cause-and-effect chain - another physical (and sometimes non-physical, for example, God) phenomenon is called to explain (understand) one physical phenomenon called the cause. For example, we can suggest the following causal chain of physical phenomena explaining the emergence of inertia forces : Newton introduced the concept of force as the cause of the change in the amount of motion, and then -

1. External, in relation to the body, force leads to a change in the amount of motion of the body ;

2. The change in the amount of motion with a constant mass leads to acceleration of the body relative to the inertial system proportional to the force (Newton's 2nd law) ;

3. Acceleration violates the isotropy of space in the reference frame of the accelerated body ;

4. anisotropy of space leads to a violation of the constancy of the gravitational potential of the universe at various points in the body's reference frame ;

5. the inconstancy of the potential leads to a non-zero potential difference in the reference frame of the body when the body moves ;

6. Nonzero potential difference leads to the appearance of gravitational forces in the body reference system ;

7. The latter are interpreted as inertia forces.

The last 4 points ( 4 - 7 ) we call the second component of the Mach principle . The first component explains how the aggregate of the universe supports the free (by inertia) motion of material bodies (see above), the second explains the response of the universe to the action of external forces relative to the material body ( see Chapter V ).

In general, the "action" of Mach's principle can be schematically described as follows : the first component "acts" always ( Newton's 1st law ) ; With the advent of external forces ( Newton's second law ), the second component is superimposed on the first and they "act" together (inertia forces) ; To stop the action of external forces, the second component is turned off and again only the first component, already with new constant parameters, "acts".

The principle of Mach naturally explains the rotation by inertia. Consider a homogeneous solid body with axial symmetry, and let this body rotate relative to some inertial system with a constant angular velocity about the axis of symmetry. We assume for simplicity that the linear velocity of the body, as a whole, relative to the inertial system is zero. Each point of the body (more precisely, a small element of the volume of the body) moves relative to the fixed point of the inertial system (and in a small neighborhood of this point) with a constant tangential velocity and constant centripetal acceleration, i.e. The Mach principle "acts" both the first and second components. The second component of the Mach principle leads to the appearance of centrifugal forces of inertia, but the "hardness" of the body and, mainly, the symmetry about the axis of rotation completely counterbalance the centrifugal forces of inertia. Thus, we can assume that the second component of the Mach principle is "turned off" ("neutralized"), and only the first component remains, which ensures the rotation of the body at a constant angular velocity. It is possible to calculate the moment of the impulse (the rotational moment) of the body of revolution, constant at a constant angular velocity, and then explain the rotation by inertia as follows : the aggregate substance of the universe informs the rotating body of the constant angular momentum, i.e. Support rotation with constant angular velocity (Mach principle for rotation).

"Journey" on the timeline. Strictly speaking, the possibility of such a "journey" is justified within the framework of the Quaternary Universe, but, as regards the special theory of relativity, it makes sense to say about it here. At the same time, we draw attention to the fact that we are not talking about the "time travel" of the observer (us and you), but of the observed objects , i.e. Real physical processes (objects) that actually occur (not "happened in the past" or "will happen in the future," but occur !) In the past or the future.

In fact, there is nothing paradoxical about what has been said. In the theory that we called the Quaternary Universe, one of the consequences for local (terrestrial) physics is ( 1), Chapter XV :

In the special theory of relativity, the transition from a fixed frame of reference (the epoch ) Into a retreating or approaching with a constant speed The frame of reference should be interpreted as a transition (in time) to the past - in the era Or, accordingly, in the future - in the era . ( - speed of light, - Hubble's constant in the modern era).

Thus, observing from a fixed frame of reference a certain physical process in a moving frame of reference, we observe this process as occurring in a different time period, in the past or in the future, depending on the speed of the mobile system. On the other hand, under certain conditions, we can regard this process as one-moment ( see section Quaternary Universe ) with a fixed reference point (where "we are with you"), i.e. What is happening "now", in our era.

As for the "journey through time", i.e. The physical movement of the observer in a different time period, then we believe this is impossible in principle.

USED ​​BOOKS

V.M. Butchers. Natural philosophy. (book, 400 pages, unpublished).

V.Myasnikov. Expansion of the universe => local physics. Proceedings of Congress-98 "Fundamental Problems of Natural Science". Volume II. Series "Problems of the Universe" vol. 22. St. Petersburg, 2000, p. 353-370

V.Myasnikov. Mathematical principles of modern natural philosophy. Proceedings of the Congress-2002 "Fundamental Problems of Natural Science and Technology." Part II. Series "Problems of the Universe" vol. 25. St.Petersburg, 2002, p. 135-167.

V.Myasnikov. Mathematical principles of modern natural philosophy. Abstracts of the report. Fundamental problems of science and technology. Program and theses of the reports of the Congress-2002. St. Petersburg. 2002, p.74

The article uses only the author's original ideas that do not require third-party information, so the list includes only the author's work.

See also the site of the author http://Quater1.narod.ru

print version
Author: V. M. Myasnikov
PS The material is protected.
Date of publication 13.01.2005гг