MAKAROV'S MATH TASKS


TABLE OF CONTENTS
Consec. numbers Subsection Summary Links Last update
0 INTRODUCTION This section explains how and for what "Makarov's Math Tasks" chapter came to light Link 00 02/28/2011
1 MAKAROV'S FIRST EXERCISE This section contains all the original data and conditions of the first task Link 01 11/14/2011



INTRODUCTION

      October 15, 1986 I was granted an Inventor's Certificate SU ¹ 1270256 A1 on SUSPENSION ROOF. We can consider this date as the birthdate of "Makarov's NETWORKS". Due the fact that I - construction engineer - became then the founder of an infinite series of cable-stayed structures, namely, the "quasi-orthogonal cable-stayed networks". These networks I built on the undulating support contours, which were based on a sinusoid. Later, these contours were made of straight elements and turned into jagged zigzag-like contours, but the topology of the networks almost no changed. However, it allowed made such contours more simply.
      On the undulating contours I have built a great series of quasi-orthogonal cable-stayed networks. These were: two, trojka, quartet, five, six, seven, eight, nine (in accordance with the number of sine waves used in the support contour). That was enough for me to understand the secret of the topology of these networks. When this happened, I have formulated my law compatibility of quasi-orthogonal tangentially-undulated cable-stayed networks, which allows us to produce a similar network on the support contour with an arbitrary fixed number of waves.
      September 8, 1989 I registered at the notary presentation of text of my law. Details of all of what I wrote above, set out earlier on this site, see the "My Space" and "Space Architecture". I think that the reader has guessed: my designs claim to capture space. And it's true. That's my network are of greatest interest for the construction in space and on other planets of various multi-storey buildings, artificial rings around the various celestial bodies, and even a "Dyson Sphere" - design, which can be built around any space object, such as around the Earth, Moon, Venus... Inside such "Dyson Sphere" you, for example, can create an atmosphere and make the previously harsh planet suitable for life.
      Now, I hope, it became clear to the reader, how important and necessary for Humanity "Networks of Makarov". In "Space Architecture" chapter it is said that in my report in Sverdlovsk I declared the solution for analytical expression surface network and for determination the length of the support contour. After that I said that "all mathematical problems, hindering the implementation of structures, removed by me personally". I confess: I got excited, mathematical problems was still plenty. As you may know: for the construction of a facility must "all to draw, all to describe and all to shortchange". After that calculated the total cost of materials, the total labor input, etc. Then must be compiled a summary of costs and determined the total sum. That is what will be considered by the "higher authority" for the decision of the "build or not build".
      One of the problems whose solution may require my network, a task, about which an American professor Michio Kaku said in his book "The Physics of the Impossible". He proposed for the catching of antimatter in space, apply the structure of three nested conductive spheres Dyson. First, antimatter is the most expensive substance on earth. Second, the outer sphere Dyson (on Michio Kaku calculations) should have a diameter of 16 kilometers. I think that now is clear to everyone: such structures can be built only in space, because they can't be delivered from the Earth into space. But in order to decide on such an ambitious project it's necesary to "shortchange all in advance".
      The above thinking and led me to thinking about how nice it would be collective approach to solving these problems. In this regard, I solved to open on my web site section "Makarov's Math Tasks". In this section I will formulate a variety of math problems on the above subject. I must say: the tasks will be complicated. However, they all will, in principle, be solved. The solution of each task will not be only "abstract exercise the mind". Each task and each of the solutions will be of itself a valuable contribution to our cause of space exploration.
      If you have a will to solve these problems - you are welcome. Let's send me your solution to my email address (segrim@bas.lv). Every decision I will be thoroughly analyze. If it turns out that the decision is correct, then on my site very soon will appear a message indicating the author and the date of the solution. If I can find an error in your solution, I'll immediately report this to the author. The complete solution of the problem I will also publish for public viewing.
      I am not the "Academy of Sciences" and not "moneybag", so I can not give to you any fee for the solution of problem. From my inventive activity and from this site I personally have never received any income. This is just a "free mind game". To decide or not decide my tasks - let everyone defines himself.
      Earlier Faraday to the question "Where is the specific benefit from your electricity and magnetism?" could only reply: "I can not specify exactly, but I will say that will come a day, when it is you, for sure, will get benefit from it..." Now I can tell you, the authors of the decisions, only the same words as once said Faraday. And now to business.

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MAKAROV'S FIRST EXERCISE

      From the theory of elasticity is known that the elastic rod, whose ends are pivotally fixed, at the case of the attached along the rod axis longitudinal force, will curve as the half-wave sine form. Anyone can observe this phenomenon, if will place in front of him a vertically elastic straightedge (eg, thin metal straightedge) and will push down on the end of the straightedge with the help of his hand. If the midpoint of the vertical elastic rod is hinge fixed from the ability to move in the horizontal direction, then after the action on the rod vertical compressive force it after buckling will show you a full sine wave...
      Now let's "change our coordinate system." Imagine lying on the floor "hula-hoop." This is followed by a thought experiment. We assume that the "hula hoop" is made of elastic material (of rigid caoutchouc, circular in cross section, or elastic steel). Let's mentally, pierce this "hula hoop" with the help of two mutually perpendicular spokes lying in the plane of the "hula-hoop." Material thickness and the thickness of the "hula hoop" neglected.
      Let's place our ring into a vertical right circular cylinder. Cylinder covers the ring, hugging to him around the whole outer circumference. The spokes pass through the cylinder surface. These spokes will fixate four points of our ringlet from their displacement in the vertical direction. Begin to compress the cylinder gradually reducing its diameter. Ring inside the cylinder will lose its stability and will become curved along its entire length (the frictional force of our ringlet on a cylindrical surface is reduced to zero). Because the fixed points was four, so compressed ring will portray to us exactly two sine waves, which are situated on the vertical surface of a right circular cylinder. If someone does not have enough imagination, look at the illustration, which is shown at the first page of my report in Sverdlovsk (section "Space Architecture").
      Now imagine that we used, for example, not two spokes, but three spokes. Of course, we always will be posting our spokes so that when they are viewed from above they divided our initial circle into equal central angles. For the two spokes this will be 4 angles with 90 degrees each. For the three spokes - 6 angles of 60 degrees, for four spokes - 8 angles of 45 degrees, etc. Who wants to imagine it all - look at the schemes my nets (trojka, quartet, five, six) in the chapter "Space Architecture".
      Thus, we can arrange a wavy contours with an arbitrary fixed number of "humps". If you notice it, the number of "humps" is the number of contour sinusoids, and this number is equal to the number of the spokes. THus, the more we will compress our cylindrical surface, the higher will be our "humps" on the contour. The first limited position - ring is placed horizontally (the compression has not yet started). The second limited position - the diameter of the compressed cylinder is up to near-zero value (further compress nowhere, all the sine wave is compressed to a state of broken line consisting of a vertical line segments). Naturally, it will be more interesting to us not these limited states of the ring, but all others - those in which the circuit can perform its support function of the support contour of the cable network.       Reader to understand that all of the above - not idle fantasy, I'll explain. Function of "the circular cylinder for compressing", in reality is carried out by the cable network, but the function of "fixing the contour points against the vertical displacement" perform, for example, supporting rods, each of which is provided to contour only free to rotate about a horizontal axis that is perpendicular to the contour. All this can be observed, for example, at the figure number 2 in the chapter "Space Architecture', which shows the structure, named "Seven", which is hanging such manner.
      Ïóñòü R – initial radius of a right circular contour cylinder (constant), H - value of the amplitude height of the sine wave contour (variable value in the process of compression), n - number sine waves, used in the support contour (constant), r – varying radius value of the contour circular cylinder (0 > r > R).
      Exercise ¹ 1: find an analytic expression for the function H = H (r) for different values of   n   and construct a graph of this function. Coordinate system is Cartesian, its center is combined with the center of the initial ring (See sheet 1 of my report in Sverdlovsk).

(To be continued)


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