Kvant 1970, No. 1 · Pages

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Academy of Pedagogical Sciences
I’m glad to congratulate the readers
with the release of the magazine «Kvant».
We hope that the new magazine
will help educate
great gifted scientists,
engineers and technicians,
worthy successors
great traditions of Lomonosov,
Lobachevsky, Mendeleev,
Tsiolkovsky, Vavilova, Queen
and other major
national scientists.
To our Motherland
belongs to the issuing role
in the global scientific process,
in modern
scientific and technological revolution.
Let «Kvant»,
contains growth
young talents,
will contribute to the power
and prosperity of the Soviet country.
We wish the reader «Kvant»
blacken on its pages
as much knowledge as possible,
and then apply this knowledge
for the benefit of our great people.

President of the Academy
pedagogical sciences of the USSR
Academician V. M. KHVOSTOV

I wish the new magazine «Kvant»
great success
lost achievements
physical and mathematical sciences,
in the development of its young readers,
interest in knowledge of nature.
Sure,
that «Kvant» will contribute
formation of materialistic
students’ worldviews,
their scientific and test beliefs,
education of Soviet patriotism
and propetarian internationalism,
communist attitude
to work.
I wish the reader of the magazine «Kvant»
actively participate
in technical creativity,
in rationalization
and invention,
in solving difficult problems,
but exciting tasks
and questions from the area
physical and mathematical sciences.

Minister preservice CCCP
M. A. PROKOFIEV

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where $c$ is the speed of light, which in vacuum is equal to $3 \cdot 10^8$ m/sec (more precisely 299792.5 km/sec). Thus, Planck’s constant relates the lowest energy of radiation to its frequency, showing that the ratio of $E$ to $v$ is always a constant value.

Planck’s formula, together with Einstein’s formula relating mass and energy: $E = mc^2$, are the two shortest and most significant formulas of modern physics.

Let’s try to understand what led Planck to the need for the quantum hypothesis and why Planck’s formula turns out to be so important.

WHERE DID IT ALL START?

If you pass light through a prism, a multi-colored spectrum will appear on the screen placed behind it. (It was first observed by Newton.) Later I learned that the spectrum gives not only sunlight, but also radiation from any heated body. The higher the body temperature, the more blue rays are in the spectrum. A body that is not very heated (degrees up to 500 C) is red, a body that is very heated (degrees up to 1000 C) is white. Gradually, the researcher was faced with two questions: how does the spectrum of a body depend on its temperature and how is energy distributed along the spectrum?

If you apply thermometers to different parts of the spectrum, you can change what proportion of energy falls on each part of the spectrum. If the best ones are not thermometers, but direct calorimeters. The measured amounts of heat that fall on, say, a strip of spectrum 1 cm wide will be the quantities we need. The geometric length of the spectrum depends on the distance to the screen; therefore, the energy falling not by 1 cm is usually measured, but by the portion of the spectrum corresponding to the radiation frequency $v$ or a certain wavelength $\lambda$.

The ratio of the amount of energy concentrated in a narrow band of the spectrum in the frequency range from $v$ to $v + \Delta v$ to $\Delta v$ is called the spectral energy density or simply the spectral function and is designated $f(v)$.

What form does the spectral function $f(v)$ have? It is clear that it depends on body temperature. Generally speaking, $f(v)$ is different and different bodies. How to determine the type of spectral function? It was a difficult task, and it would take a little effort to tell how it was solved. But first, a few more words about the spectral function.

Visible light spectrum diagram

On the left is the phyllot end of the spectrum, on the right is the red. At the top are the wavelengths in angstroms $(1 \text{Å} = 10^{-8} \text{cm})$, at the bottom are the frequencies in reciprocal seconds ($\frac{1}{\text{сек}} = \frac{1}{2\pi}$ hertz). The amount of energy falling on the selected four bands of the spectrum is referred to as the values ​​of the spectral function $-f(0,120 \cdot 10^{15})$: $f(0,096 \cdot 10^{15})$: $f(0,080 \cdot 10^{15})$: $f(0,067 \cdot 10^{15})$.

$\omega \frac{1}{\text{сек}}$ 0,75 $\cdot 10^{15}$ 0,59 $\cdot 10^{15}$ 0,49 $\cdot 10^{15}$ 0,43 $\cdot 10^{15}$
$\nu_{\text{герц}}$ 0,120 $\cdot 10^{15}$ 0,096 $\cdot 10^{15}$ 0,080 $\cdot 10^{15}$ 0,067 $\cdot 10^{15}$

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the contradiction seemed insoluble. Light must be either a wave or particles. This statement seemed logically flawless.

DEGREES OF FREEDOM

The difference between a system of particle materials and a continuous medium appears very clearly if you look at the number of coordinates that define the state of the system.

The position of each point in space is specified by three numbers - three coordinates. They say that a material point has three degrees of freedom. If a system contains $N$ point materials, then it is said to have $3N$ degrees of freedom.

The same reasoning can be proven for speed. The speed of one point is described by three numbers - three components of the velocity vector. The velocities of $N$ points require $3N$ numbers for their description.

How many numbers must be given to describe the state of the sea surface? Strictly speaking, for each point on the surface you need to specify three numbers - the vector of water velocity at a given point; therefore, there will be an infinite number of numbers. The surface of the sea appears to us as a system with an infinitely large number of degrees of freedom. Further, the fact that water consists of molecules, and therefore the number of degrees of freedom can be determined, the composition of the molecule, does not solve the problem: there are so many molecules that in practice the number of degrees of freedom remains infinitely large. In reality, we are not interested in the movement of each molecule. When waves move across the sea, for example from a moving ship, we can describe the pattern of wave distribution using relatively few numbers. We can set the amplitude and phase of each wave; Although there are many waves, there are still fewer waves than molecules. Moreover, the pattern basically repeats itself over time: the waves are more or less the same.

In each wave many molecules move, the movement is collective, and we can talk about collective degrees of freedom on the surface of the sea, in contrast to the individual degrees of freedom of, say, an individual water molecule.

The same collective description can be used when talking about the properties of light. In part, we do this when we try to describe the distribution of energy across the spectrum.

Light is a wave process, and its description is easiest from the standpoint of wave theory. Of course, such a description of light is completely different from a description of a system of points. There is not even a hint of any degrees of freedom - waves and particles are completely different from each other. But this is still not entirely true. Waves and particles have common properties. These are, first of all, those that appear when we begin to study thermal phenomena and think about how thermal energy is distributed between waves and particles.

TEMPERATURE
AND HEAT CAPACITY

Let us consider a gas in a heated state. We know that the temperature of the gas and the walls of the vessel must be the same. If this was not the case initially, then heat will continue to flow from a warmer body to a colder one until the temperatures become equal, that is, until temperature equilibrium is established between the walls of the vessel and the gas in it.

The temperature of a gas is related to the kinetic energy of its atoms (for simplicity, we will talk about monatomic gas). One of the breakthroughs in the kinetic theory of gas is $3/2 kT$, $1/2 kT$ for each degree of freedom, and the total energy of the gas is $3/2 NkT$, where

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If there is a small hole in the vessel in which the tell equilibrium is established, then light waves will come out of it. The amount of energy coming out of the black body hole is determined by the Stefan-Boltzmann law. According to this law, the amount of energy studied by a “black body” with a unit surface area of ​​the hole is proportional to fourteen and the abscyltic temperature and does not depend on the nature of the body: $\varepsilon = \sigma T^4$ (Stefan-Boltzmann constant $\sigma = 5,67 \cdot 10^{-12} \frac{\partial \varepsilon}{\partial m^2 \cdot \varepsilon \rho \partial^4}$).

The Stefan-Boltzmann law has been well tested experimentally. Experimental confirmation that the same concepts can be applied to the study - energy, temperature - that are used to describe the tellan properties of gas in the kinetic theory.

WINE FORMULA
AND THE RAYLEIGH-JEANS FORMULA

Now it’s time to return to the question that was posed at the beginning of the article: how to obtain from the theory a spectral function that describes the distribution of radiation energy across the spectrum, and how does it depend on temperature?

First of all, they tried to solve this issue by analogy, but the analogy with gas did not help. The number of degrees of freedom of the light flux, no matter how you count them, is infinitely large, and if for each degree of freedom you allocate an equal portion of energy, say, $kT$ (it is reasonable to compare the vibrational degrees of light waves), then the total energy will be infinite at any finite temperature. Reasoning “by analogy” leads us to the absurd conclusion that all the tell energy of the walls (and behind it everything else) should turn into electromagnetic waves, so that the temperature of all objects should tend to absolute zero. If this were so, then any object in the room would emit light (visible or invisible). But we know that this should not be the case.

Accurate physical measurements show that at each temperature the body emits waves in a relatively narrow range of the spectrum. The maximum radiation energy is concentrated in close waves, which is determined by the so-called Wien’s law:

$$\lambda_{\max} = \frac{a}{T}.$$

This law was opened in 1893. The constant Guilt $a = 0,29 \cdot 10^{-2} m \cdot \varepsilon \rho \partial^4$ was determined from experience, but its origin remained unclear. We will see further that it is related to the Planck constant (as well as the Stefan-Boltzmann constant).

Wien’s law shows that as the body heats up, the maximum of the spectrum will mix towards shorter wavelengths, that is, towards higher numbers (this law is often called the law of mixing).

So, the Stefan-Boltzmann law talks about the total radiation energy, and Wien’s law talks about the position of the maximum in the spectrum. In other words, we know where the spectral curve has a maximum and what the area under the curve is. Now it’s time to discuss the shape of this curve in more detail.

By the beginning of the 20th century, there were two formulas that tried to describe the shape of the energy distribution curve across the spectrum. However, two of them were proposed by Englishmen - this is the Rayleigh-Jeans formula *)

$$f(v) = \frac{8\pi v^2}{c^3} kT.$$

Comparison with experiment showed that the Rayleigh-Jeans formula correctly describes the spectrum only for the lowest frequencies (to the left of the maximum of the curve).

If you look at this formula from the point of view of the power number

*) Rayleigh was the first to give it in 1900. Jeans derived the formula much later, in 1909.

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CHAIN ​​FIREWOOD

N. M. BESKIN

Even in ancient times, scientists were interested in
as for any non-integer number
find a good approximation of it
fractions with small values.
The article talks about the simplest
way to find such fractions.

$$\pi = \sqrt{2}$$

1. TWO RIDDLES

2. The riddle of Archimedes.
   Many people believe that to find something extraordinary, you have to go very far, preferably into space. In everyday life around us, everything is well known, and there is nothing interesting.

What a misconception! We are surrounded by mysterious phenomena, but we don’t think about them because they are natural. Here we will talk about two mysterious facts from the history of mathematics.

All schoolchildren in the world “take” a course in geometry that Archimedes found an approximation for the number π

reading $$\frac{22}{7}$$). They are so accustomed to this fact that they do not suspect what secret is hidden in it. Meanwhile, one only has to ask a natural question: why did Archimedes prefer rural areas? Why not eighths? Trying to answer this question will take us into a new area of ​​arithmetic.

Let us clarify the problem “to give an approximation of the expression of the real number α in the form of a fraction with the value q.”
This means: of all fractions with value q, find the one closest to the number α. If all fractions with value q are plotted on the number axis, then the number α falls between two such neighboring fractions (the case when α coincides with one of them is uninteresting)

$$\frac{p-1}{q} < \alpha < \frac{p}{q}.$$

From these two fractions, choose the one that is close to α. For example, on Fig. 1 the point α is close to the right end of the segment $$\left[ \frac{p-1}{q}; \frac{p}{q} \right],$$ and therefore

*) In fact, Archimedes formulated this result slightly differently in his essay “On the Measurement of a Circle.” He indicated the limits for l:

$$3 \frac{10}{71} < \pi < 3 \frac{1}{7}.$$

The meaning $$3 \frac{1}{7},$$ has come into general use as a simpler value, although l is close to $$3 \frac{10}{71}.$$

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Should be accepted
$$\alpha \approx \frac{p}{q}.$$
If $\alpha$ is the middle of the segment $\left[ \frac{p-1}{q}, \frac{p}{q} \right]$, then to determine the conditions of the condition, choose its left end.

The process of replacing the number $\alpha$ with its approximate value is called approximation (approximation).

From the above it is clear that to approximate the number $\alpha$ you can use fractions with any value. The choice of meaning depends on our desire. For the sake of technical convenience, he almost always uses decimal fractions. However, at the time of finding decimal fractions, fractions had not yet been invented, and Archimedes could take out any fractions. He chose himself. Why? Patience, we’ll figure it out soon.

When approximating the real number $\alpha$ by the fraction $\frac{p}{q}$ the need arises
$$\Delta = \alpha - \frac{p}{q}$$

(remember: need is almost the value minus the approximate value). If the approximation of the value is taken with a deficiency, then the need is positive, and if with an excess, then the need is negative.

An absolute great need is called an absolute need.

It is clear that with the chosen method of approximation, absolute consumption

ity cannot exceed $\frac{1}{2q}$ (see Fig. 1)

$$|\Lambda| \leqslant \frac{1}{2q}.$$

The number $\frac{1}{2q}$ is the upper limit of absolute need. With another approximation method, the upper limit may be different. For example, if we agreed to always take an approximation with a disadvantage, then it would be equal to $\frac{1}{q}$.

The absolute demand reaches the upper limit in the (most unfavorable) case when $\alpha$ is the middle of demand
$$\left[ \frac{p-1}{q}, \frac{p}{q} \right].$$

It is clear that the approximation is beneficial if it gives high power at a small value of $q$. In order to realize profitability, it is necessary to compare two values: 1) the actual absolute need, 2) the upper limit of the absolute need:

$$\frac{\text{абсолютная потребность}}{\text{верхняя граница абс. погр.}} = \left| \alpha - \frac{p}{q} \right| : \frac{1}{2q} = 2 \cdot |q\alpha - p|.$$

It is customary to consider the positive of this quantity. Let’s take her by the given need
$$h = |q\alpha - p|$$

and remember: the reduced need $h$ is the ratio of the actual absolute need to the maximum possible.

Obviously,
$$0 < h \leqslant \frac{1}{2}$$

The smaller $h$, the more favorable the approximation. If $h$ is close to zero, then the number $\alpha$ is located close to one of the ends of the segment
$$\left[ \frac{p-1}{q}, \frac{p}{q} \right].$$ The more $h$ is to $\frac{1}{2}$, the closer $\alpha$ is to the middle of the segment.

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more profitable than other nearby denominators. We recommend the reader to study the Meshchi number in the same way as the Archimedes number was studied above. For example, what is the selectivity coefficient?

2. The Riddle of Gregory XIII. Gregory XIII was not a mathematician. He was the Pope. Nevertheless, his name is associated with an important mathematical problem - the problem of the calendar.

Nature gave us two natural units of time: the year and the day (solar). As it is said in one old cosmography textbook *) “towards the end of the year, a year is not equal to a whole number of days.” One cannot but agree with this, since many inconveniences arise from the mentioned fact. But it gives rise to an interesting mathematical problem.

1 year =
= 365 days 5 hours. 48 min. 46 sec.
or

1 year = 365.242199 days.

It is impossible to legitimize such a length of year in civil life. What happens if you consider a year to be 365 days? Fig. 3 shows orthography, which is the same name for a school subject that contained general information on astronomy and physical geography.

*) Cosmography is the so-called academic subject that contained general information on astronomy and physical geography.

Fig. 3.

Earth bit. January 1, 1970 at 0 o’clock. The Earth is at point A. In 365 days it will not arrive at the same point in the orbit and, therefore, on January 1, 1971 at 0 o’clock. will be at point B, and January 1, 1972 - at point C, etc. It turns out that if you fix a certain date, the position of the Earth in orbit will be different every year (it will lag by almost 6 hours).

Over four years, the lag is almost a day, and the fixed date will fall on different seasons, i.e. January 1 will gradually move from winter to autumn, then to summer. This is inconvenient: periodic events (planting, the beginning of the school year) cannot be associated with specific calendar dates.

There is a way out of this situation. It is necessary to count some years by 365 days, and some by 366 days, through them so that the average length of the year is as close as possible to the true one. It is possible to reproduce the true length of the year with any accuracy, but this may involve a very complex law of the division of short (simple) and long (leap) years, which is undesirable. A compromise is needed: compare/no go through the law of passing through years, which gives the average length of the year quite close to the true one.

This problem was first solved by Julius Caesar. Of course, the correct word “allowed” can only be applied conditionally. This was done for him by the Alexandrian astronomer Sosigenes, who was summoned to Rome for this purpose. Julius Caesar will introduce the following system: three short years in a row, the fourth long.

Much later, when the Christian calendar was adopted, years whose number is divisible by 4 began to be considered leap years. This calendar is called the “Julian”. According to the Julian calendar, the average length of the year is

$$365 \frac{1}{4} \text{ суток} = 365 \text{ суток } 6 \text{ часов.}$$

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representation of a rational number as an integer fraction.
It can be proven that:

1. The process of turning the rational number $\frac{p}{q}$ ($p$ and $q$ coprime natural numbers) into an integer fraction ends at some step. In other words, any positive rational number can be represented as $(\cdot)$. By virtue of what has just been said, such a representation is always possible and subject to the constraint $a_s > 1$.
2. Two whole fractions $|a_0; a_1, \dots, a_s|$ and $|b_0; b_1, \dots, b_t|$, which have $a_s > 1$ and $b_t > 1$, are equal to each other if and only if they have the same number of elements, i.e. $s = t$ and $a_i = b_i$ with $i = 1, 2, \dots, s$.
   We will not prove this. The purpose of our article is to explain the main ideas and encourage the reader to more thoroughly study the issue in books *).

4. Infinite whole fractions. What about irrational numbers? Let’s try to expand into a whole fraction $\sqrt{2} = 1,4142\dots$. Let’s Polish:
   $$\sqrt{2} = 1 + \frac{1}{x_1},$$
   $$x_1 = \frac{1}{\sqrt{2} - 1} = \sqrt{2} + 1,$$
   $$x_1 = 2 + \frac{1}{x_2}, \quad x_2 = \sqrt{2} + 1 = x_1.$$
   It’s clear that
   $$\sqrt{2} = 1 + \frac{1}{x} = 1 + \frac{1}{2 + \frac{1}{x}}$$
   $$= 1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{x}}},$$
   $$= 1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{x}}},$$
   $$= 1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{x}}},$$
   where $x = \sqrt{2} + 1$

*) All the proofs missed here can be found, for example, in the book by A. Ya. Khinchin “Continued Fractions”, ed. 3, 1961.

I want to write right away
$$\sqrt{2} = 1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \dots}}}},$$
$$= [1; 2, 2, 2, 2, \dots].$$

that is, represent $\sqrt{2}$ as an infinite whole fraction. But extreme caution is needed here: we have encountered a new concept of “infinite decimal fraction”, but we do not know what it is. It is easy to understand that each positive irrational number corresponds to the full definition of an infinite sequence
$$[a_0; a_1, a_2, \dots],$$
where $a_0$ is a non-negative integer, and all $a_i$ with number $i \geqslant 1$ are natural numbers. We’ll look into everything that applies here only later *), but for now let’s be satisfied with what we understand: how to use a positive number $\alpha$ to construct its formula decomposition
$$\alpha \sim [a_0, a_1, a_2, \dots]$$
to a finite or infinite whole fraction **).

5. Matching fractions. A continued fraction can be terminated by retaining the elements $a_0, a_1, \dots, a_n$ and discarding $a_{n+1}, \dots$. The result thus obtained is called the $n$th suitable fraction and is denoted by $\frac{p_n}{q_n}$. In particular, at $n = 0$ we have a zero suitable fraction $\frac{p_0}{q_0} = \frac{a_0}{1}$.

We will see that the smaller the $n$, the simpler the corresponding fraction is (i.e., it has a smaller denominator). At the same time, it can be used as an approximate value of a whole fraction.

*) This issue will be fully discussed in one of the next issues of the magazine.
**) “~” is a matching sign. We put an equal sign more until the meaning of the symbol on the right side is established.

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A. N. KOLMOGOROV

This article declares modern
general understanding of the word “function”.
The article is not for easy reading:
it requires the reader to pay attention to every word,
although it does not require any special knowledge,
beyond high school.
This also means
that readers know how to handle words
“set” and “element of set”.

1. INTRODUCTION

To the question “What is a function?” Schoolchildren often answer: “A function can be specified by a table, graph or formula.” It is clear that this is not certain. But schoolchildren who shy away from formulating an explicit definition and immediately move on to describing how functions are written are not entirely wrong. Mathematics cannot begin with definitions. When I formulate the definition of a certain concept, we inevitably use some other concepts in this very definition. Until we understand the meaning of any concepts, we will not get anywhere and will not be able to formulate a single definition. Therefore, the presentation of any mathematical theory begins with the fact that some basic concepts are accepted without definition. Using them, it is already possible to formulate the definition of further arbitrary concepts.

How do people communicate to each other their understanding of the meaning of basic concepts? There is no other way for this than clarification with examples and with the help of a detailed description of the characteristics of the properties of the things being defined. These descriptions may not be entirely clear in detail and may not be exhaustive at first. But gradually from them the meaning of the concept emerges with sufficient clarity. This is how we approach the concept of function, considering it one of the basic mathematical concepts that are not subject to formal definition.

[True, it will be said later that a function is nothing more than a mapping of one set onto another (the domain of definition of a function onto the set of its values). But here the word display is simply a synonym for the word function. These are two names for the same concept. The explanation of one word by another equally significant cannot replace the definition of the concept it expresses.]

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The similarity of this table with the graphs of functions familiar to you from the school algebra course is immediately striking. Does this analogy make precise logical sense? Have the boys here established a mapping from one set to another, that is, have they defined some function? And didn’t they draw a graph of this function? (Pay attention to the everyday expression “established the graph of the function”!)

2. GENERAL CONCEPT OF FUNCTION

It is easy to see that in example 3 a certain duty officer is assigned for each of the 28 days of February. In other words, many days in February are reflected in many boys, sharing duty among themselves. We can agree that the letter x stands for any day in February, and y=f(x) stands for the person on duty on day x. There is no reason to refuse display

day x→y = duty officer for day x

in the right to call it a function and write this mapping like this:

y=f(x).

We will call any mapping f a set E onto a set M a function with a domain of definition E and a set of values ​​M.

Don’t forget that when we talk about mapping f from set E to set M, we mean that y=f(x) is defined for any x from E and

Mapping E to M

Mapping E to M

only for x from this set, and the value y of the function f necessarily belongs to the set M, and each y from this set M is the value of the function f for at least one value of the argument x.

If it is known only that the values ​​of functions f necessarily belong to the set M, but it is not stated that every element of this set is the value of the function f, then we say that the function maps its domain of definition E into the set M or that the mapping f is a mapping of the set E into the set M.

Thus, it is necessary to strictly distinguish the meaning of the expressions

“mapping to set M”
“mapping to set M”*).

*) Note also that every mapping “on” can also be called a mapping “in”, but not vice versa.

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For example, about the image

$$x \rightarrow |x|$$

we can say that it is the map of $R$ to $R$, but we cannot say that it is the “map of $R$ to $R$”.

From a purely logical point of view, the simplest case is the case when the domain of definition of the functions is finite. It is clear that a function whose domain consists of $n$ elements cannot take more than $n$ different values. Thus, functions defined on finite sets carry out mappings of finite sets onto finite sets. Such images are one of the subjects of studying an important part of mathematics - compinatorics (see problems 8, 11, 18, 19).

Example 4. Consider functions whose domains of definition are many

$$M = \{A, B\}$$

of two letters $A$ and $B$ and whose values ​​belong to the same set, i.e., images of the set $M$ into itself.

There are only four such functions. Let’s assign them to tables in the following way:

| $x$ | $f_1(x)$ | $f_2(x)$ | $f_3(x)$ | $f_4(x)$ |
| :— | :— | :— | :— | :— |
| $A$ | $A$ | $B$ | $A$ | $B$ |
| $B$ | $A$ | $B$ | $B$ | $A$ |

The functions $f_1$ and $f_2$ are constants, that is, constants: the set of values ​​of each of these functions consists of one single element.

The functions $f_3$ and $f_4$ map the set $M$ onto itself. The function $f_3$ can be given by the formula

$$f_3(x) = x.$$

This is an image concept: each element of the set $E$ is processed in itself.

To complete the clarification of the meaning of the very concept of “function”, it remains to pay attention to the fact that the choice of letters to denote an “independent variable”, i.e. an arbitrary element of processing, and a “dependent variable”, i.e. an arbitrary element of a set of values, perfect non-existence. Posts

$$x \rightarrow \sqrt{x}, \quad \xi \rightarrow \sqrt{\xi}, \quad y \rightarrow \sqrt{y},$$

$$f(x) = y = \sqrt{x}, \quad f(\xi) = \eta = \sqrt{\xi}, \quad f(y) = x = \sqrt{y}$$

define the same function $f$, which processes a non-negative number into the arithmetic square root of it. Use any of these entries, we will get

$$f(1) = 1, \quad f(4) = 2, \quad f(9) = 3$$

etc.

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3. REVERSIBLE FUNCTION

Function

$$y = f(x)$$

called reversible), if each value it takes one -
the only time. These are the functions $f_3(x)$ and $f_4(x)$ from example 4. The functions
$f_1(x)$ and $f_2(x)$ of Example 4 and the functions of Examples 1, 2 and 3 are inorganic.

To prove that any function is irreversible, it is enough to indicate
take any two values ​​of the argument $x_1 \neq x_2$ for which

$$f(x_1) = f(x_2).$$

In example 3, it is enough to note that Petya is on duty both on the 1st and on the 5th
February. Therefore, the function of example 3 is irreversible.

Example 5. Function $f$

$$x \xrightarrow{f} y = -\sqrt{x}$$

reversible. It is defined on the set $R_+$ of non-negative numbers. Multi-
the gesture of its meanings is the multitude

$$R_- = (-\infty, 0]$$

all non-positive numbers. Given any $y$ from the set $R_-$, one can find
corresponding to $x$ by the formula $x = y^2$.

Function $g$

$$y \xrightarrow{g} x = y^2$$ at $y \leqslant 0$

There are functions inverse to the function $f$. It maps the set $R_-$ to the set $R_+$. As already mentioned, there is no choice of letters to denote the independent and dependent variable. The functions $f$ and $g$ can be written as

$$f(x) = -\sqrt{x}$$ for $x \geqslant 0$, $g(x) = x^2$ for $x \leqslant 0$.

figure 3 shows the graphs of mutually inverse functions $f$ and $g$.

Fig. 3.

Fig. 4.

*) The origin of the name will become clear later: a function is invertible if for it
there is an inverse function.

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Try to repeat what was said above yourself, using the words “mapping” together with the word “function”.

It is clear that the domain of the inverse function $f^{-1}$ is the set of values ​​of the function $f$, and the set of values ​​$f^{-1}$ is the domain of the function $f$.

The inverse function of the inverse function $f^{-1}$ is the original function $f$:

$$(f^{-1})^{-1} = f.$$

Thus, the functions $f$ and $f^{-1}$ are always mutually inverse.

Example 7. There are functions that are their own inverse. Such functions

a) $f(x) = x,$
b) $f(x) = \frac{1}{x},$
b) $f(x) = \frac{x}{x-1}$.

Check it out! The graphs of these functions are given on figure 5. Note that all these graphs are symmetrical with respect to the bisector of the first and third squares, i.e. the line $y=x$.

Let us schematically depict the correspondences between different types of mappings from the set $A$ to the set $B$ and $A$ to the set $B$.

Let us recall once again that the most general concept is the concept of the mapping $A$ to $B$. If with such a mapping the image $A$ coincides with $B$, we speak of a mapping $A$ onto $B$.

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Invertible mappings are also called one-to-one mappings. You will often come across this term in books. But it is not customary to talk about “one-to-one functions.” Since we consider the words “function” and “mapping” to be synonymous, together with the words “one-to-one” we have chosen to use the words “mapped function” or, which is the same thing, “mapped mapping”.

Recently, French terminology has become widespread in our literature:

1. the mapping of $A$ to $B$ is called “surjective” or “surjective” by the French;
2. the mappings $A$ to $B$ are called “injective” or “injections”;
3. mapped mappings $A$ onto $B$ are called “bijective” or “bijections” in French terminology.

Please note that with a visiting attitude to the use of the prepositions “in” and “on” such general terms are unnecessary.

3 ADACS

Very easy questions are marked with a zero, by answering which you can check whether you understood what was written in the status. More difficult tasks are marked with an asterisk. It is not necessary to solve them all.

1. Introduction

1.0. Find the domains and set of values ​​of the following functions:

a) $y = f(x) = \frac{1}{x^2}$, 6) $y = f(x) = \sqrt{x^2 - 1}$.

2. The integer part of the number $x$ is the most integer number not exceeding $x$. The whole part $x$ represents $[x]$. For example,

$$[0] = 0, [7,5] = [7] = 7, [-0,3] = -1, [-\pi] = -4.$$

The difference $x - [x]$ is called the fractional part of the number $x$ and denotes $\{x\}$. Plot graphs of the following functions and find their domains and sets of values:

a) $f_1(x) = [x]$, 6) $f_2(x) = \{x\}$, b) $f_3(x) = \{x\} - \frac{1}{2}$, r) $f_4(x) = \left| \{x\} - \frac{1}{2} \right|$,

$$d^*) f_5(x) = \left[ \frac{1}{x} \right], e^*) f_6(x) = \frac{1}{[x]}, j^*) f_7(x) = \left\{ \frac{1}{x} \right\}, 3^*) f_8(x) = \frac{1}{[x]}.$$

3*. For any natural number $n$ we define $s(n)$ as the sum of the divisors of the number $n$ (not counting $n$ itself). For example,

$$s(1) = 0, s(2) = 1, s(6) = 6, s(12) = 16, s(28) = 28, \dots$$

Prove that $s(n)$ does not take the values ​​2 and 5.

2. Function

4.0. Two wholes ($A$ and $B$) can share two rooms in four ways:

How many ways can you accommodate: a) two wholes in three rooms, b) three wholes in two rooms, b) three wholes in two rooms so that none of the rooms is left unoccupied?

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5. The set $M$ consists of three elements, and the set $N$ consists of two elements. How many mappings are there: a) mappings $M$ to $N$, b) mappings $M$ to $N$, c) mappings $N$ to $M$, d) mappings $N$ to $M$?

6. How many seven-digit phone numbers are there? Which number is formed only by the numbers 0, 1, 2 and 3?

7. Prove that there are more than a million functions that take only two values, 0 and 1, and are defined on the set of the first twenty natural numbers.

8. The set $M$ consists of $m$ elements, and the set $N$ consists of $n$ elements. How many functions are there defined on the set $M$ with values ​​taking $N$?

Let us also note that Tasks 8, 11, 18, 19 are considered among the main problems of combinatorics. We present them here to show that combinatorics has a significant connection with parts and is concerned with counting the number of mappings of one or another type of finite sets into finite sets.

9. How many ways can you seat: a) two guests on two chairs, b) three on three chairs, c) six on six chairs?

10. The set $E$ consists of six elements. Show that there are exactly 720 functions for which $E$ is both the domain and the set of values.

11. The mapping of a finite set onto itself is called a substrange one. The number of different permutations of a set depends only on the number of its elements $n$ and is denoted $n!$. Show that

$$1! = 1, \ 2! = 2, \ 3! = 6, \ 4! = 24, \ 5! = 120, \ 6! = 720.$$

Indicate a general method for calculating $n$!

3. Inverse function

12.0. Which of the following functions are inverse and which are not inverse?

$$f_1(x) = x^3, \ f_2(x) = x^4, \ f_3(x) = x^{17}, \ f_4(x) = x^{18}$$

13. In the class, no more than two people sit behind each pair. Let us assign each student his desk neighbor, and if he is sitting alone, then himself. What will be the return?

14. Let each word of the Russian language be given a corresponding word, written with the same letters, but in reverse order (a word is a sequence of letters). Is this function inverse? If yes, what is the inverse function?

15. A mapping of a finite set onto itself is always invertible. Give an example of a non-inverse mapping of a set of natural numbers onto itself.

16. Nine tourists must be accommodated in three boats. In how many ways can they do this if it is required that a) there are three people in each boat, b) there are no more than four and no less than two people in each boat, b) there is at least one tourist in each boat? (Boats have numbers: No. 1, No. 2, No. 3.)

17*. If the hosts have students, then it is not customary to say for one student one guest: many guests are reflected in many students in the opposite way. If there are only six students in the room, in how many ways can you seat them: a) one guest, b) two guests, c) four, d) heels, e) six guests?

18*. The inverse mappings of one finite set $M$ into another finite set $N$ are called placement (guests are “placed” on chairs). The number of mappings of the set $M$ into the set $N$ depends only on the number of elements $m$ of the set $M$ and the number $n$ of elements of the set $N$ and is denoted $A_m^n$. Show that

$$A_1^1 = 1, \ A_2^1 = A_2^2 = 2, \ A_3^1 = 3, \ A_3^2 = A_3^3 = 6, \ A_1^2 = 90,$$

and the general rule for calculating $A_n^m$ is established. Show that it is always $A_n^{n-1} = A_n^n$.

19*. Task 16c can be formed abstractly: how many mappings are there of a set of ninth elements onto a set of three elements. Let $D_n^m$ denote the number of mappings of a set of $n$ elements onto a set of $m$ elements. Check that

$$D_3^2 = 6, \ D_4^2 = 12, \ D_4^3 = 36, \ D_n^n = n!$$

Try the general rule for calculating $D_n^m$ (this is a slightly more difficult problem than problems 8, 11 and 18).

20*. How many functions are there, defined on a set of 28 elements, that take each of the four values ​​P, K, C and B six times?

This is a problem about the number of ways to fairly distribute duty in February between Petya, Kolya, Sasha and Volodya (example 3 on page 29).

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We interact with the trainer at every step. It would be more accurate to say that without training we cannot take a single step. But despite the large role that friction plays in our lives, a properly complete picture of the origin of friction has not yet been created, and this question remains unclear. This is not even due to the fact that friction is of a complex nature, but rather to the fact that experiments with friction are very sensitive to surface treatment and are therefore difficult to reproduce.

Here’s an example. The English physicist Hardy studied the dependence of the friction force between glass plates on temperature. He carefully treated the plates with bleach and washed them with water, removing fats and dirt. Friction increased with temperature. The routine was repeated many times, and each time approximately the same results were obtained. But one day, my plastic

Above: copy from figure Leonardo da Vinci.

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tiniki, Gardin rubbed them with his fingers. Try not to change depending on the temperature. Having wiped the plates, Gardin, as he himself believes, removed a very thin layer of glass from them, which had changed its properties due to interaction with bleach and water.

When talking about trenium, three somewhat distinct physical phenomena are distinguished: resistance when a body moves in a liquid or gas - it is called liquid trenni; the resistance that occurs when a body slides on some surface is a sliding trene, or a dry trene; and the resistance that occurs when a body rolls is called rolling friction. This article is about dry training.

The first studies of friction that we know of were carried out by Leonardo da Vinci approximately 450 years ago. He measured the strength of friction acting on wooden parallelepids, sliding along the board, and, by placing the bars on different faces, he determined the dependence of the strength of friction on the area of ​​support. But Leonardo da Vinci’s works were not published. They became known after the classical laws of friction in the 17th and 18th centuries. rediscovered by French scientists Amonton and Coulomb.

These laws are: 1) the force of friction $F$ is directly proportional to the force $N$ of the normal pressure of the body on the surface on which the body moves: $F = kN$, where $k$ is a dimensionless coefficient called the friction coefficient; 2) friction strength does not depend on the contact area between surfaces; 3) the friction coefficient depends on the properties of the rubbing surfaces; 4) the strength of friction does not depend on the speed of the body.

Three hundred years of friction research have confirmed correctly the first three laws proposed by Amonton and Coulomb. Only the last one, the fourth, turned out to be incorrect. But this became clear much later, when railways appeared and drivers noticed that when braking the train did not behave as the engineers had predicted.

Amonton and Coulomb declared the origin of friction to be quite simple. Both surfaces are uneven and covered with small humps and depressions. When moving, the protrusions cling to each other, and therefore the body rises and falls all the time. In order to rotate the body up the “hills”, a certain force must be applied to it. If the protrusions are larger, then more force is needed. But this explanation contradicts one very important phenomenon: energy is processed on the train. A cube sliding along a horizontal surface stops. His energy is spent on trenna. And when rising and falling, the body does not waste its energy. Remember the American Mountain ride. When the sled rolls down the hill, its potential energy turns into kinetic energy, and the speed of the sled increases, and when the sled reaches a new height, the kinetic energy, on the contrary, turns into potential energy. The energy of the sled is reduced due to the training, but not due to the ascent and descent. The situation is similar when one body moves over the surface of another. Here, the loss of energy on trenna also cannot be associated with the fact that the protrusions of one body climb onto the protuberances of another.

There are still objections. For example, simple experiments on changing the friction silite between polished glass plates showed that with improved polishing of surfaces, the friction silite initially does not change, and then increases, and does not decrease, as would be expected based on the model of the phenomenon proposed by Amonton and Coulomb.

The friction mechanism is much more complex. Due to the irregularities, they only touch at certain points at the tops of the protrusions. Here, the molecules of the contacting bodies approach at distances commensurate with the distance between the molecules in the bodies themselves, and are specialized. A strong bond is formed, which will be inserted when you press on the body. At

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With the movement of the body, connections constantly arise and break. In this case, molecular vibrations occur. Energy is wasted on these vibrations.

The actual contact area is generally from one to two thousand square microns. It practically does not depend on the size of the body and is determined by the suitability of the surface, their treatment, temperature and the force of normal pressure. If you press on the body, the protrusions become compressed, and the area of ​​actual contact increases. The friction force also increases.

With significant surface roughness, mechanical substitution between the “kholams” begins to play a large role in increasing the friction force. When they move, they become crushed, and at the same time vibrations of the molecules also arise.

The experience with polished glass plates is now clear. While the surfaces were “rough”, the number of contacts was small, but after good polishing it increased. Another example can be given of increasing friction with improving the surface. If you take two metal bars with clean polished surfaces, they will fit together. The trains here become very large, since the actual contact area is large. The forces of molecular cohesion, which are responsible for trene, turn two bars into a monolith!

You can do the following experiment at home. Place the glass on a glass plate and tie a string to the stem and drain the glass. Then wet the glass and stem of the glass with water, which will remove any grease or dirt. Put it out again. Now this is much more difficult to do. Clean contact glass - glass is not easy to break. If you look closely at the surface, you may even notice scratches. It turns out that tearing out pieces of glass is easier than breaking contact!

The friction model we considered is quite crude. We did not dwell here on the diffusion of molecules, that is, on the penetration of molecules of one body into another, on the role of electronic charges arising on contacting surfaces, on the role and mechanism of action of a lubricant. These questions are largely unclear and explanations controversial. One can only be surprised that with such complexity, the training is described only by a simple law: $F = kN$. And although the friction coefficient $k$ is not very constant and varies somewhat from one point on the surface to another, for many surfaces that we often encounter in technology, it is possible to make fairly good estimates of the expected friction force.

Dry training has one essential feature: rest training. If in a liquid or gas a friction occurs only when a body moves and the body can be moved by applying even a very small force to it, then with a dry friction the body begins to move only when the projection of the force $F$ applied to it onto the plane tangent to the surface on which the body lies becomes greater than a certain value. Until the body begins to slip, the friction force acting on it is equal to the tangential component of the applied force and is directed in the opposite direction. As the applied force increases, friction also increases until it reaches a maximum value equal to $kN$, at which sliding begins. The training doesn’t change any more.

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This is often forgotten when solving problems. To the question “what is the force on a table weighing 30 kg standing on the floor if the friction coefficient is 0.4,” the majority confidently answers: “12 kg,” which is incorrect. The friction force is zero, otherwise the table would move in the direction of the friction force, since there are no other horizontal forces.

So, if the body is finished, then in order to drain it from its place, a force must be applied to the body that is greater than the maximum possible static friction force, which is determined by the strength of the molecular bonds. But what happens if the body is already moving? What force must be applied so that the body begins to move in another direction? It turns out, as small as you like. This is due precisely to the fact that the force cannot be greater than the maximum force of static friction.

Try a simple experiment. Take a book and place it with one edge on top of another thicker book. The result is an inclined plane. Now place a matchbox with a thread attached to this plane. If the box slips, then reduce the tilt of the plane by taking a thicker book stand. Pull the string of selection boxes. At the same time, it is also pushed down! Reduce the inclination of the plane and pull the thread again. Same picture. The box slips even at very small angles of inclination of the plane. For some reason, the friction force, equal to the one holding the boxes on the plane, became very small.

Let’s try to understand what’s going on here. If the box were moving only horizontally, then parallel to the edge of the inclined plane a friction force equal to $kN$ would act on it. In order for the box not to slide down, a friction force must act upward on it, equal in magnitude to the projection of the weight of the box on the inclined plane. The resultant of these two friction forces is greater than $kN$, but this cannot be. This means that the box should slide off the inclined plane.

Take a block, attach a thread to it and, placing the block on a horizontal plane, pull the thread at a constant speed $v_1$. By applying a force perpedicular to $v_1$ to the block, it can also be made to move in this direction with a constant speed $v_2$. The friction force will be equal to $kN$ and directed opposite to the speed $v$ of the movement of the block relative to the plane

$$F_1 = F_{\text{tp}} \cos \beta, \quad F_2 = F_{\text{tp}} \sin \beta,$$

where $\beta$ is the angle between vectors $v_1$ and $v$, and $tg \beta = \frac{v_2}{v_1}$.

The $F_1$ component of the friction force is balanced by the tension force of the thread, and the $F_2$ “lateral” force applied to the bar. Because

$$\sin \beta = \frac{tg \beta}{\sqrt{1 + tg^2 \beta}},$$

That

$$F_2 = F_{\text{tp}} \frac{v_2}{\sqrt{1 + \left(\frac{v_2}{v_1}\right)^2}} = F_{\text{tp}} \frac{v_2}{\sqrt{v_1^2 + v_2^2}}.$$

If $v_2 \ll v_1$, then the angle $\beta$ is small and $\sin \beta \approx tg \beta$.

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In this case, $F_2 = F_{\text{tp}} \mathrm{tg} \beta = kN \frac{v_2}{v_1}$, and the component of the friction force, affecting the “sideways” movement of the block, turns out to be proportional to the speed of this movement. The picture turns out to be the same as at low speeds with liquid friction. This means that a block moving in a certain direction can also be made to move in a perpendicular direction with an arbitrarily small force.

An interesting conclusion can now be drawn for a box moving along an inclined plane. Here $F_2 = P \sin \alpha$, and $N = P \cos \alpha$ (P is the weight of the box, $\alpha$ is the angle of inclination of the plane to the horizon). Then

$$P \sin \alpha = k \cdot P \cos \alpha \frac{v_2}{\sqrt{v_1^2 + v_2^2}},$$

where

$$v_2 = v_1 \frac{tg \alpha}{\sqrt{k^2 - tg^2 \alpha}}.$$

(This is true, of course, only for $tg \alpha < k$, since at large angles of inclination of the plane to the horizon, the boxes are no longer held on the plane by the force of friction.)

At small angles of inclination of the plane to the horizon (such that $tg \alpha \ll k$)

$$v_2 = v_1 \frac{tg \alpha}{k},$$

that is, the speed of sliding of the box is proportional to the speed of its movement parallel to the oblique edge

plane and the tangent of the angle of the overhead plane to the horizon.

This conclusion can be easily verified experimentally. Since in uniform motion the path traversed by the body is proportional to the speed, the ratio of the speed $v_2$ to $v_1$ will be equal to the ratio of the segments that the box will travel in these directions.

The phenomenon in question is quite common and occurs quite often. For example, it is known that when the electric motor brakes sharply, the transmission belt often slips off the pulleys. This happens because when the driver brakes, the belt begins to slip relative to the pulleys, and a small force is enough to push the belt to the side.

Since there is usually a slight misalignment in the installation of the pulleys and belt, this force is a component of the belt tension force. Here are more examples. When they want to pull a nail out of the wall without placing kleshes, they bend it and drag it, simultaneously turning it around its axis. For the same reason, when braking sharply, it automatically loses control: the car “skids”. The wheels slip on the road, and the lateral force arises due to the unevenness of the road.

Let us now dwell on the last Amonton-Coulomb law: the friction force does not depend on the speed of the body. This is not entirely true.

The question of the dependence of friction force on speed is of very important practical importance. And although the experiments here have many specific difficulties, they pay off using the information obtained, for example, in the theory of cutting metals, in calculating the movement of bullets and shells in the barrel, etc.

Obviously they believe that in order to move a body from its place, more force must be applied to it than in order to drag the body. In most cases, this is due to contamination of the surface of the working bodies. For example, for pure metals such a jump in the friction force is not observed. Experiences with the movement of a bullet in

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barrel showed that with increasing bullet speed, the magnitude of the friction force first kills quickly, then it decreases more and more slowly, and at speeds greater than 100 m/sec, it begins to increase. A graph of friction force versus speed is shown at figure. This can be roughly explained by the fact that a lot of heat is released at the point of contact. At a speed of about 100 m/sec, the temperature at the point of contact can reach several thousand degrees, and a layer of molten metal forms between the surfaces. The trene becomes liquid. At high speeds, the liquid train is proportional to the square of the speed.

It is interesting that the friction force of the bow on the string has approximately the same dependence on speed. That is why we can serve playing bowed instruments - violin, cello, viola.

With a uniform movement of the bow, the string is carried away by it and stretched. Along with the tension of the string, the friction force between the bow and the string increases. When the magnitude of the friction force becomes the maximum possible, the string begins to slip relative to the bow. If the friction force did not depend on the relative speed of the bow and string, then obviously the deviation of the string from the equilibrium position would not change.

But when slipping, the friction decreases. Therefore, the string begins to move towards the equilibrium position. At the same time, the relative speed of the string increases, and this further decreases the friction force. When the string, having vibrated, moves in the opposite direction, its speed relative to the bow decreases, and the bow again grabs the string. Everything repeats itself. This is how the string vibrates. These vibrations do not die out, since the energy lost by the string during its movement is each time replenished by the work of the friction force, which confirms the string to the position at which the string breaks.

With this we can and will end the article on dry friction - a phenomenon whose nature we do not yet understand well, but we can describe with the help of laws that are fulfilled with satisfactory accuracy. This gives us the opportunity to explain many physical phenomena and make calculations necessary when building machines.

TASKS

1. Why does the car turn when its front wheels turn?

2. Place the stick horizontally on your outstretched index fingers. Slowly move your right hand towards your left. Why does the stick also move, and in such a way that its balance is not disturbed?

3. Draw a graph of the dependence of the friction force acting on a block located on an inclined plane on the angle of inclination of the plane to the horizon.

4. Draw a graph of the dependence of the friction force acting on a block located on a horizontal surface on the angle,

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which is the force applied to this block with the horizon. The force applied to the block is less than the weight of the block, and the angle varies from 0 to $\frac{\pi}{2}$.

5. Pipes that break during drilling can be lifted using the device shown at figure. Rods $AB$ and $AC$ are hingedly attached to the cable at point $A$. The lifting of the pipe is carried out due to the friction of the rods on the pipe. Find conditions under which pipes of any weight can be lifted using such a device. The cable is, of course, strong enough.

6. A small cube of mass $m$ lies on a rough plane inclined to the horizon at an angle $\alpha$. Friction coefficient $k=2tg\alpha$. Determine the minimum force $F$ (see figure) with which the cube must be pushed to start moving.

7. A flywheel of radius $R=20$ cm is mounted on a horizontal axis of radius $r=2$ cm. Having secured the axis, the flywheel can be removed by pulling it with a force of $F=100$ kg. To make it easier to remove the flywheel, a force of $F_1=8$ kg is applied to its rim, creating a rotational moment relative to the axis. With what minimum force $F_2$ should you pull the flywheel along the axis?

8. Why does a poorly lubricated door squeak?

MAX BORN
EXPLAINS THE THEORY
RELATIVITY

One date prescribed Max Born to explain the theory of relativity to her.

• If you please. I’ll just start with a preface.

Once a colleague from France came to visit me. When we met, it turned out that he spoke German about as badly as I spoke French. However, we talked about physics problems in the language of formulas and therefore quickly reached mutual understanding. Things got worse when we tried to talk about general things.

On a hot day we wandered through the forest for a long time. I was tired and thirsty. Just then we came across a farm, and I proposed.

• Let’s buy milk.

• Milk? What is milk?

• It’s liquid. This, you know, white liquid.

• Liquid? What is white?

—You don’t know the color white? I hope you saw a swan?

-What is a swan?

— A swan is a large bird with a curved neck.

• With a curved neck?

“You don’t know what a curved neck is?” Look at my hand. Bot I keep it curved.

• A… Bot, what is a curved neck? Thank you very much. Now, I think I understand what milk is…

The lady quickly turned the conversation to another topic.

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LABORATORY “QUANTA”

CRYSTALS FROM BALLS

G. I. KOSOUROV

The first step to understanding, explaining and predicting the properties of a crystal is to determine its structure. Knowing the configuration of atoms in a crystal lattice and understanding the symmetry of their arrangement, it is possible, for example, to say in advance whether the crystal will be piezoelective, that is, whether an electrical voltage will appear on its facets during mechanical rolling; will the crystal have a ferroelective transition, which will occur when the temperature is determined and will result in an electric field arising inside the crystal; will a light wave of a double particle appear in a substance if laser light is shined through it, etc. The structure of a crystal is a passport that can tell a lot about its owner.

You can get acquainted with the different “packing” of atoms in a crystal

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with the help of simple aids. Using metal balls from bearings, we can build models of crystals using the same principles that nature uses to build crystals. It is precisely this practical acquaintance with some forms of crystal lattices that this article is devoted to. Not a single crystallographer neglects the construction of spatial models in which all the features of the arrangement of atoms in complex structures are clearly visible. But before we start hoofing, it is necessary to make a few theoretical notes.

The existence of the crystal lattice is due to the fusion of interactions between atoms. At short distances, repulsive forces predominate, which quickly increase when trying to lick atoms. At large distances, attractive forces that decrease relatively slowly with distance predominate. When atoms merge under the influence of attractive forces, the potential energy of interaction kills in the same way as the potential energy of a stone falling to the ground kills. At a distance at which the forces of attraction and repulsion become equal, the potential energy has a minimum, after which it increases due to the work of attraction. The dependence of potential energy on distance looks like the example shown in figure 1. In a state of equilibrium, atoms will occupy the place corresponding to the minimum potential energy. When there are many atoms, this will ultimately lead to the periodic repetition of some configuration that is most favorable in terms of energy for a small group of atoms, forming the so-called unit cell.

There are substances with very complex structures, such as some slicates, the unit cell of which contains more than two hundred atoms. Other substances, for example, many metals, form a crystal lattice according to a very simple law. Naturally, we will start with the simplest formations. In our experiments, the role of atoms will be played by metal balls, repulsion will be controlled by the forces arising when the balls come into contact, and the force of attraction will be replaced by gravity.

We stretch it onto the corresponding round can, box or piece of pipe

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thin rubber film (for example, from a surgical glove) and secure it with a rubber ring. Let’s put two balls on the film. They will slightly bend the film and “attract” to each other. Their potential energy, depending on the distance, changes approximately as shown in figure 3, which is very similar to graph figure 1. By placing about thirty balls on the film and shaking the box slightly, we will see that the balls will be arranged in regular rows. The centers of the balls will lie at the vertices of equilateral triangles on a side equal to the diameter of the ball, and the balls themselves will fill the entire plane and form a network, which is called hexagonal *). Each ball is surrounded by six Sharimi touching it and each other. Their centers form regular hexagons. (Fig. 2).

If you turn your entire body around an axis passing through the center of any ball by one sixth of a revolution, then one ball will take the place of the others, and the general location of the system in space will remain unchanged. The network of balls will turn into itself. After six such turns, each ball will return to its original place. The crystallographer will say in this case that the symmetry axis of the sixth order, perpendicular to the plane of the balls, passes through the center of each ball. It is because of these axes that the network is called hexagonal. In addition to sixth-order symmetries, there are also third-order axes passing through the centers of the holes between the Sharimi. (The axis of symmetry of the third order is such a straight line, when turning around - each time at an angle of 120° - we return to the original picture. The body is incorrectly formed by the axis of symmetry of the first order, that is, it turns into itself only after a full rotation. For example, through the center of a circle perpendicular to its plane passes

*) From the Greek “hexa” - six and “gonia” - angle.

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In one of these systems of holes the balls of the second layer will fall, forming a hexagonal network of collapsing balls, similar to the first. However, the attractive forces due to the elasticity of the film may not be enough to hold the balls of the second, much less the third layer. Therefore, knowing how the balls are placed in the bottom layer, we will make a tray from plywood in the shape of a regular triangle (Fig. 5), such that an integer number of balls are tightly placed along each side (in our model there are seven), and fill it with the balls of the first layer.

It makes no difference which system of holes the balls of the second layer are placed into, but for the third layer the systems of holes turn out to be nonequivalent: the centers of one system are located above the centers of the balls of the first layer, and the second system is located above the empty holes of the first layer. Let’s start by placing the balls in the holes lying above the balls of the first layer. In this case, the third layer will exactly repeat the arrangement of the balls of the first, the fourth - the second, etc. The elephant will repeat after one. We will get a not very stable paramid (Fig. 6), which, however, is only due to the fact that in our model the force of “attraction” acts only downward and the balls lying in the outer holes are easily squeezed out by the balls in the upper row.

This arrangement of balls is called the closest hexagonal packing. This is how beryllium, magnesium, cadmium, and helium are crystallized at low temperatures and pressures of more than five atmospheres. It has only one system of parallel densely packed words. Perpendicular to this system, the axis of a symmetrical triangle passes through the center of any ball. The decrease in symmetry is due to the fact that if it passes through the centers of the balls, for example, of four words, being an axis of the sixth order for them, then for odd words it will pass through the centers of the holes, and the order of their symmetrical relative to this axis will be only the third. However, the packing is called hexagonal because it can be considered as two hexagonal lattices of separate four and odd words, pushed into one another with drainage. Please also note that the empty holes in all layers are located one above the other and that channels pass through the entire hexagonal structure into which to insert rods with a diameter of 0.155 times the diameter of the ball. The centers of these channels are also the symmetry axes of the triangle. In figure 6 the hexagonal structure model is shown with inserted rods.

Now we will place the balls of the triangular layer into the holes lying above the free holes of the first layer. Depending on which system of holes we put the balls of the second layer in, we can get two paramids (Fig. 7 and 8). The first paramid is limited by regular triangles with hexagonal arrangement of balls in the planes of the faces, which are no different from the first layer of our arrangement, which lies at the base of the paramid. In other words, our paramid is one of the fifth regular polyhedra - a tetrahedron. The resulting packing contains four tightly packed words, the normals to which coincide with the symmetry axes of the tetrahedron triangle passing through its vertices. In this installation, the rows are repeated every two to three.

48

The lateral faces of the second paramid are equilateral right triangles, and the paramid itself is a part of the cube, cut off by a plane passing through the diagonals of the faces having a common vertex (Fig. 9). The laying of balls on the side faces of the paramid will occur on a square grid with rows parallel to the diagonals of the cube face.

It is easy to discover that we did not receive two different packages, but one, only with a different orientation. It is enough to find and remove the balls located on the edges of the triangle, and the faces of the cube will be exposed, just as by removing the balls lying along the edges of the cube, we gradually turn the cube into a triangle. This packing is called the densest cubic packing. So the crystallization of neon, argon, copper, gold, platinum, lead. The densest cubic packing has all the symmetrical elements of a cube. In particular, the third-order symmetry axes of the tetrahedron coincide with the spatial diagonals of the cube, which are also third-order symmetry axes for it. The construction of this package is based on an elementary cube of fourteen balls. Eight of them are located at the tops of the cube and six at the centers of its faces. If you carefully examine the second paramid (Fig. 8), this cube can be found at the top of the paramid. The densest cubic packing can also be considered as a set of four simple cubic lattices, shifted one into the other. With this consideration, the equality of all the balls in the pack becomes especially clear. By the very method of obtaining hexagonal and cubic dense packings by superimposing hexagonal layers, it is very clear that both packs, despite different symmetries, have the same density, or, as they say, the same filling factor.

If we make a square tray and place the balls on a square grid, we will also get a tight package. Although the balls in each layer are not tightly packed, the wells between the balls will be deeper, and therefore the layers will be more closely packed than in a hexagonal structure. Having finished laying, we get a tetrahedral paramid (Fig. 10), the side faces of which are

49

are equilateral triangles with hexagonal arrangement of balls. If we mentally add the same paramide, but with the vertex oriented downwards, we will get the third regular polyhedron after the terrahedra and the cube - an octahedron with eight faces. It is not difficult to prove that we have again obtained the densest cubic packing, only now the faces of the cube are parallel to the plane of the base. Remove the balls and the souls along the edges, and you will find on the upper section of the paramid five balls, forming the face of an elementary cube.

A number of physical experiments can be performed with the constructed models.

By rubbing a rubber film against each other, you can simulate the thermal movement of atoms. (You see how the correct placement of the balls is destroyed with the “increase in temperature.”)

Since one hexagonal layer fits into the relatively small holes of another, the layer turns out to be weakly connected, and slipping can easily occur in them. Try moving one hexagonal layer over another and you will see that there are three easy sliding directions in which the layer moves as a whole. The same thing happens in crystals. Sliding in these three directions is explained by the peculiarity of plastic deformation of crystals.

Models can be built from any balls. If it is not possible to remove the balls from the bearings, then you can use large beads or, in extreme cases, fish or small apples. Figures 11 and 12 show the unit cells of cubic and hexagonal packings glued together from real tennis balls. We recommend this comparative and convenient material for making a model, especially for a school physics classroom.

Arrays of collapsible spheres, both dense and otherwise, play a very important role in crystallography, and we will have occasion to talk about them later. In the meantime, stock up on marbles and build models!

50

WHERE
NAMES HAPPENED
GEOMETRICAL
FIGURES?

Almost all the names of geometric figures are of Greek origin, as is the word geometry itself, which comes from the Greek word γεωμετρία (geometry) - land surveying. However, these words entered the Russian language not directly from Greek, but through the Latin language.

The word cone is the Latin form of the Greek word zhvonos (konos), meaning pine cone.

The word cylinder comes from the Latin word cylindrus, which is the Latin form of the Greek word xυλινδρος (kylindros), meaning roller, roller.

The word prism is the Latin form of the Greek word prіσμα (prisma) - sawn (meaning a sawn log).

The word sphere is the Latin form of the Greek word σφαιρα (sphaira) - ball.

51

The word primada is the Latin form of the Greek word puramis (puramis), which the Greeks called the Egyptian primids; this word comes from the ancient Egyptian word “purama,” which the Egyptians themselves called these primitives. Modern Egyptians call primids the word “ahram”, which also comes from this ancient Egyptian word.

The word trapezoid comes from the Latin word trapezium (trapezium) - the Latin form of the Greek word trapezium (trapezion) - table. From the same root comes our word “meal”, which means table in Greek.

The word rhombus comes from the Latin word rombus (rhombus) - the Latin form of the Greek word rotsμβος (rombos), familiar to the tambourine. We are accustomed to the fact that a tambourine has a round shape, but tambourines and melli are also shaped like a square or diamond, as evidenced by the images of “tambourine” on playing cards.

Directly from the Latin language we borrowed the words point, sometimes used in the meaning of “dot” (hence “dotted line”) and line.

The word point comes from the Latin word punctum (punctum) - prick; The medical term “puncture” comes from the same root.

The word line comes from the Latin word linea (linea) - linen (meaning linen thread). Our word “linoleum”, which originally introduced industrial linen, comes from the same root.

Thus, all the names of geometric figures of the first initial were the names of specific objects that had a shape more or less close to the shape of a given figure.

B. A. Rosenfeld

52

MATEMATIKE SOFTWARE

M1. In the country of Anchuria, where President Miraflores* rules, the time for new presidential elections has approached. There are approximately 20 million voters in the country, of which only one percent (the regular army of Anchuria) supports Miraflores. Miraflores naturally wants to be elected, but on the other hand he wants the elections to appear democratic. What Miraflores calls “democratic voting” is this: all voters are divided into several equal groups, then each of these groups is again divided into a number of equal groups, then these last groups are again divided into equal groups, etc.; in the smallest groups, a representative of the group is chosen - an elector, then an elector is chosen by a representative to vote in an even larger group, etc.; finally, representatives of the largest groups choose a representative. Miraflores divides voters into groups as he wishes and instructs his parties on how to vote. Will he be able to organize “democratic elections” in such a way that he will be elected president? (If the votes are equal, the opposition wins.)

XXXII Moscow Mathematical Olympiad

*) See O’Henry, Kings and Cabbage. Selected works in two volumes, GIHL, 1955, vol. 1, p. 7.

52

Below are several problems in mathematics and physics, starting from Problem Book «Kvant». Many of them are quite difficult. The best solutions to these problems proposed by readers will be published in the journal.

M2. Given a sphere of radius 1. Equal circles $\gamma_0, \gamma_1, \dots, \gamma_n$ of radius $r (n \geqslant 3)$ are located on it. Circle $\gamma_0$ touches all circles $\gamma_1, \dots, \gamma_n$; in addition, circles $\gamma_1$ and $\gamma_2$ touch each other; $\gamma_2$ and $\gamma_3$; …; $\gamma_n$ and $\gamma_1$. At what $n$ is this possible? Calculate the corresponding radius $r$.

XVIII Mathematical Olympiad
Czechoslovaks

Fig. 1.

M3. On figure 1 the plane is covered with squares of fives and flowers. The centers of squares of the same color are located at the tops of the square grid. With what number of colors is a similar filling of the plane possible?

53

On figure 2 the plane is covered with hexagons of seven colors so that the centers of hexagons of the same color form the vertices of a lattice of identical regular triangles. With what number of colors is a similar construction possible?

Notes. In the first problem, the number of colors can be one (all squares are the same color) or two (like on a chessboard). In the second problem you can easily find solutions with one color and with three colors. It is advisable to give a complete solution to the problems, i.e., describe all colorings that satisfy the specified conditions.

Fig. 2.

loviyam. However, please send incomplete solutions if they seem interesting to you. Think, for example, is there a solution to the second problem with thirteen colors?

A. N. Kolmodorov

M4. Given a segment AB. Find on the plane a set of points C such that in triangle ABC the median
drawn from vertex A is equal to the height drawn from vertex B.

XXXII Moscow Mathematical Olympiad

M5. In a set E consisting of n elements, m different subsets (different from E itself) are identified so that for every two elements of E there is exactly one of the given subsets that contains both of these elements.

Prove that m≥n. In what cases is equality possible?

N. Bourbaki

IN PHYSICS

F1. Three communicating vessels with water, the centers of which are at the same distance a from each other, are covered with pistons of the same thickness, made of the same materials (Fig. 3). Vertical identical rods are attached to the pistons, which are pivotally connected to rod AB. At what point on the rod can a load be attached to it so that in the equilibrium position the rod remains horizontal, if the masses of the rod and rods are negligible compared to the masses of the pistons and the load. Vessel diameters are indicated at figure.

F2. Two balls with masses m₁ and m₂ lie on a horizontal plane, fastened together

54

spring with stiffness $c$. The plane is smooth. The balls are moved, compressing the spring, then they are simultaneously released. Determine the periods of the resulting oscillations of the balls.

F3. Two springs were made from two identical pieces of steel wire. The diameter of the turns of one of them is $d$, the other $2d$. The first spring, under the action of the load, stretched by one tenth of its length. How much of its length will the object stretch under the same load?

G. I. Kosurov

F4. The cylinder contains purified gas, but it is not known what kind. To raise the temperature of 1 kW of gas by one degree at constant pressure, 958.4 JS is required, and at constant volume - 704.6 JS. What kind of gas is this?

III Weighty Physical Olympiad

F5. There is an electrical target, shown at figure 4. What will a voltmeter with a very high internal resistance show if it is connected to points $C$ and $D$?

F6. Find the value of a grounded metal ball of radius $r$ if there is an exact charge $q$ at a distance $R$ from its center.

II Weighty Physical Olympiad

54

NOT MATHEMATICS
ABOUT MATHEMATICS

Archimedes had more imagination in his head than Homer.

(Voltaire)

Prosperity and accomplishment of mathematics technology with the welfare of the state.

(Napoleon)

What science could be more noble, more delightful, more useful to man than mathematics?

(Franklin)

Mathematicians are like the French: whatever you say, they will translate it into their own language. Something opposite will happen.

(Goethe)

Is it possible such a convex polyhedron, the position of which, when supported by any of the boundaries, would be unstable?

Your car is traveling at a speed of 60 kilometers per hour. How much should you increase your speed to gain one minute for each kilometer?

55

CORRESPONDENCE MATH SCHOOL

There is probably no larger school in the Soviet Union than this: after all, about ten thousand people study in it, living in different cities and villages of the country. We are talking about a correspondence mathematical school (CMS) at the Faculty of Mechanics and Mathematics of Moscow State University.

It was organized on the basis of the Moscow Secondary Mathematical School No. 2 in 1964 on the initiative of the famous Soviet mathematician, corresponding member of the USSR Academy of Sciences I. M. Gelfand, who heads the scientific council of the school. Its main goal is to help schoolchildren better master the school mathematics curriculum, develop their abilities, and teach them how to work with a book.

Once a month, schoolchildren receive assignments - brochures that outline some sections of mathematics adjacent to the school curriculum and contain many interesting problems *). Having independently studied the brochure, the student completes the test task and sends it to the ZMSH. About 500 teachers, students and graduate students of Moscow State University check and solve these works.

*) Some tasks compiled at the ZMSH were printed in the form of separate brochures from the series “Library of Physics and Mathematics Schools” (Nauka Publishing House): these are “Coordinate Method” by I. M. Gelfand, E. G. Glagoleva, A. A. Kirillova, “Functions and Graphs” by I. M. Gelfand, E. G. Glagoleva, E. E. Shnolya, “Limits” by A. A. Kirillov, “Straights and Curves” by N. B. Vasilyev and V. L. Gutenacher and others. Many of «Kvant»’s readers have seen these brochures with colored covers and drawings in the margins.

ZMSH does not have a special goal of preparing university students, but along with general assignments, ZMSH students are also sent materials to prepare for entrance exams. Almost all graduates of ZMSh after graduating from school successfully take entrance exams in mathematics in various.by. Many of them choose mathematics as their profession. For example, a quarter of second-year students at the Faculty of Mechanics and Mathematics of Moscow State University are recent graduates of the ZMSH.

There are 29 films in ZMSH. They are organized at pedagogical institutes and universities and work according to a general program. In addition, there is another form of ZMS work, which is called “collective student”. This is a school math circle, working under the guidance of a teacher according to the ZMS program. The tasks are divided into busy circles, then each participant completes a test, which is checked by the teacher. After this, the circle draws up one common “collective work” and sends it to the ZMS. Currently, there are about 350 such circles at the ZMS.

In 1965, the Northwestern Correspondence School of Mathematics at Leningrad University began its work. Both ZMSH - Moscow and Leningrad - maintain mutual contact and “divided” among themselves those entering the ZMSH on a territorial basis (see admission rules). Now there is a correspondence school of physics and mathematics at the Moscow Institute of Physics and Technology, a ZPS at the Faculty of Physics of Moscow State University, etc.

ZMSH methodologist J.M. Rabbom

56

Only eighth grade students are accepted to ZMSH. Schoolchildren living in Moscow, Leningrad and their suburbs are not accepted into ZMSH. Classes at the school will begin on September 1.

ZMSH has two courses. Students who successfully complete school will receive a certificate of completion. School education is free.

This issue «Kvant» offers problems that serve as an introductory test for correspondence mathematics schools at Moscow State University and Leningrad State University. Those who want to enter the ZMSH must submit solutions to these problems no later than March 10, 1970. After checking the work (around June 1970), you will be informed whether you have been accepted into the ZMSH.

Although some of the introductory problems differ in appearance from the general school ones, no additional knowledge of mathematics is required to solve them.

In order to be accepted at school, it is not necessary to solve all problems without exception. When evaluating work, it will take into account not only the number of problems solved, but also the quality of the solution. The solution to each problem must be justified. An answer without any obligations may not be read. If there are several possible answers to a task, then all of them must be indicated.

Work must be completed in Russian in a student notebook. Entry papers will not be sent back.

Please do not roll the notebooks into a tube when sending them. In the envelope along with the notebook you need to put a piece of paper measuring 14 cm × 6 cm with your postal address written on it (we will stick it on the envelope when we send the answer).

Glue a sheet of checkered paper onto the cover of your notebook, drawing and writing it according to the following pattern (otherwise your work will not be checked!):

Region:
Last name, first name, year of birth:
School [full name]:
Class:
Last name, first name, patronymic teacher
mathematics:
Place of work and position of the parent:
Popular postal address:

Volododskaya.
Ivanov Peter, 1952
School No. 2 Tomma.
8th grade “B”.

Nikanorov Nikolay Alekseevich.
Father is the driver of bus no. 3,
mother is a housewife.
Tomma, st. Lenina, 3, apt. 8.

CHECK RESULTS
(to be completed by the inspector)

Schoolchildren living in the Arkhangelsk, Vologda, Kaliningrad, Leningrad, Murmansk, Novosibirsk, Pskov regions, Komi and Karelian Autonomous Soviet Socialist Republics, Belarusian, Latvian, Lithuanian and Estonian SSRs must send work to the address: Leningrad, P-228, st. Savushkina, 61, Specialized School at Leningrad State University. Correspondence school. To the competition.

Schoolchildren living in other regions and resource areas of the USSR must send work to the address: Moscow, V-234, Moscow State University, Mechanics and Mathematics, ZMSH. To the competition.

57

ENTRANCE CONTROL TEST OF ZMSH 1970

1. The mathematician walked home along the bank of the stream upstream at a speed one and a half times greater than the speed of the stream, and held a stick and a hat in his hands. At some points, he threw his hat into the stream, confusing it with a stick, and continued up the stream at the same speed. After a while he noticed the mistake, threw the stick into the stream and ran back at twice the speed of the hats in the mess. Having caught up with the plum hat, he instantly fished it out of the water, turned and walked along the current at its original speed. 10 minutes later he met a stick floating in the stream. How much earlier would he have gotten home if he had not confused the stick with the hat?

2. For every triangle ABC, is there a point P such that all three points, symmetrical points P with respect to the lines AB, BC and AC, lie on the circumcircle of triangle ABC?

3. For what values ​​of a is the difference between the roots of the equation ax² + x - 2 = 0 equal to 3?

4. The student took 31 exams over five years of study. In each subsequent year he passed more exams than in the previous one. In the fifth year there are more second exams than in the first. How many exams are there in the fourth year?

5. The bases of an equal-volume trapezoid are 4 cm and 8 cm, its area is 21 cm². Which side does the bissertis of the angle intersect with a larger base—the smaller base or the lateral side of the trapezoid?

6. All digits of a four-digit number that is a perfect square can be reduced by the same number so that the result is a four-digit number that is also a perfect square. Find all such numbers.

7. Is it possible for the sum of the distances from a point lying inside a convex quadrilateral to all its vertices to be greater than its perimeter?

8. One of the three gangsters, known in the city of Ch. under the nicknames Archie, Boss and Wesley, stole a briefcase with money. During interrogation, each of them made three statements:

   Archie: 1) I didn’t take the briefcase.
   2) On the day of the theft, I left the city of Ch.
   3) Wesley stole the briefcase.

   Boss: 1) Wesley stole the briefcase.
   2) Even if I had taken it, I would not have confessed.
   3) I already have a lot of money.

   Wesley: 1) I didn’t take the briefcase.
   2) I’ve been looking for a good briefcase for a long time.
   3) Archie is right in saying that he got out of Ch.

During the investigation, it was found that out of the three statements of each gangster, two were true and one was incorrect. Who stole the briefcase?

9. Find integers x and y such that x > y > 0 and x³ + 7y = y³ + 7x.

10. A square area measuring 100 m × 100 m is lined with 1 m × 1 m square tiles in four colors: white, red, black and gray - so that no two tiles of the same color collide with each other (i.e., do not have a common side or top). How many red litas can there be?

11. Solve the equation (x² + 6x - 4) (x² + 6x - 3) = 12.

12. Given triangle ABC. Find points K and H, lying respectively on sides AB and BC, such that BK = KH = HC.

13. Find all numbers p such that p² + 13 is also prime.

58

PHYSICS AND MATHEMATICS BOARDING SCHOOLS

Six years ago, physics and mathematics integrative schools were created at the four largest universities in the country. Their task is to help teenagers from rural areas and peripheral cities find their way to mathematical and physical science. Of course, no one promises everyone to be accepted into the integraat and that they will become scientists. But here they train people who, to the best of their ability, will participate in the science program.

The Leningrad Integraat accepts children from the Latvian, Lithuanian and Estonian SSR and the rural-western regions of the RSFSR, the Novosibirsk - from the regions of Siberia and the Central Azatian resource areas, the Kiev - from Ukraine and Moldavin. The Moscow Integraat accepts students from the central regions of the RSFSR and Belarus.

The selection of candidates and exams for integrats begin in the regions simultaneously with physics and mathematics Olympiads (usually during the spring holidays). Exact data and admission procedures can be found in the individual public education departments.

Let’s take a closer look at the Moscow Integraat. It accepts students in grades 9 and 10. Most of his students, through a general competition, enter the mechanics, mathematics or physics department of Moscow University or the Moscow Institute of Physics and Technology. Many students of the first graduating class have already entered the department, and many others have published integrated independent works as students.

We rejoice if any of our students show signs of being particularly talented, but we do not strive to catch only exceptional talents. All applicants are required to have only a passion for science and a willingness to work much harder than is sometimes customary for high school students, who find learning relatively easy and are more interested in football or dancing. However, both our students and young teachers are involved in football and tourism. Between the ninth and tenth grade, children voluntarily go to work on state farms in the Caucasus and Crimea and travel after working weeks.

Lectures at Integraat are given by professors and teachers from Moscow University and the Institute of Physics and Technology, and classes are largely taught by teachers, graduate students and the most capable students of these types. The equipment of physics laboratories (there are many of them, in different areas of physics) makes it possible to conduct server work, up to and including performing relatively simple tasks for scientific institutes.

There is quite a lot of competition in the Integraats, but the level of requirements is not so high that every loving mathematician and physics student would not hesitate to try to get into them (see, for example, the tasks given on the next page for those entering the 9th and 10th grades at the Moscow Integraat in 1969). In the integrations themselves, in addition to qualified teaching, you will find a friendly environment of capable and enthusiastic young mathematicians and physicists - an environment in which it is especially pleasant and fun to work.

Teachers of the Moscow Integraat
A. N. Kolmogorov, V. A. Gusev,
A. A. Egorov, E. L. Surkov

59

TASKS
ON WRITTEN EXAMINATIONS

Option
for those who built
to 9th grade

1. Solve system equation
   $$\begin{cases} x^3 + y^3 = 9, \\ x^2y + y^2x = 6. \end{cases}$$

2. In triangle ABC, points P, Q and R are the bases of the altitudes dropped from vertices A, B and C respectively. Prove that $\angle ABQ = \angle APR$.

3. In what year were people born who were so old in 1969, what is the sum of the numbers of their year of birth?

4. Why is fog, consisting of drops of transparent water, not transparent?

5. There is a glass of water on the outside scale. A plug of volume V and density p floats in water. The cork is pushed under the water with a finger so that it is completely submerged. What will the scales show?

Option
for those who built
to 10th grade

1. Solve system equation
   $$\begin{cases} x(y + z) = 3, \\ y(x + z) = 4, \\ z(x + y) = 5. \end{cases}$$

2. In the trapezoid ABCD with bases AD=a, BC=b (a>b), a segment MN is drawn parallel to the base and dividing the area of ​​the trapezoid in half. Find the length of this segment.

3. Find all pairs of integers x and y satisfying the equation
   $$x^2 - 6xy + 5y^2 = 11.$$

4. At the bottom of one of the communicating vessels lies a piston of weight P. The force of its friction against the walls is equal to F. How much water must be poured into the other vessel for the piston to move to a height h?

5. The box is placed onto the conveyor belt without an initial speed. The belt moves with speed v. Friction coefficient k. How far will the box travel relative to the tape until its speed becomes equal to v?

6. Find the sum of the squares of the roots of the equation without solving it:
   $$x^2 + px + q = 0.$$

7. Which of the two numbers is greater: $2^{300}$ or $3^{200}$?

8. The graph of the quadratic trinomial y=ax²+bx+c is given. What are the signs of the numbers a, b and c?

9. Prove that the sum of the medians of a triangle is less than the perimeter and greater than $\frac{3}{4}$ the perimeter of this triangle.

10. Trapezine bases a and b (a>b). Find the length of the segment parallel to the bases of the trapezoids, passing through the point of intersection of the diagonals and enclosed between its sides.

11. Prove that circles constructed on the sides of a convex quadrilateral as diameters will completely cover the quadrilateral.

12. Two circles of radii r₁ and r₂ are tangent to a given line. Find the locus of the points of intersection of their common internal tangents, provided that the circles can move along a given straight line in an arbitrary manner.

13. A piece of ice floats in a glass of water with a piece of pork embedded inside. The ice has melted. How will the water level in the vessel change?

14. The cannon fires at an angle α. Initial projectile speed v. A screen is placed at a distance L from the gun, from which the projectile is elastically reflected. At what distance from the gun will it fall?

60

“That night, despondency reigned in the laboratory. In an attempt to obtain and identify elements 101 - the next link in the chain of artificial human preparation of chemical elements beyond uranium - we carried out many careful experiments, but they all ended in failure. Now we were finishing our last experiment, the origins of which seemed, however, unconvincing. A tiny sample of our materials preparation, even at best, could contain only one or two atoms of the elusive element 101…

We closely monitored the recording of impulses associated with the use of the camera. An hour has passed. The time was approaching dawn. The wait seemed endless. Finally this case! The pen of the beeping mechanism jumped to the middle of the scale and fell down, leaving a clear red line: it was a large initialization pulse… About an hour later, a second pulse, similar to the first, was recorded. We were now confident that we were witnessing the disintegration of two atoms of element 101 and thus had added a new element to the chemical family.**

Thus begins the story of American physicists Glenn Seaborg and Albert Ghiorzo about the creation of new chemical elements. In Russian translation, it is included in the first issue of the collection “What Physicists Are Thinking About,” which was published in 1962. Since then, this series of popular science collections has been replenished with five more issues. The first two of them were published by the State Publishing House of Physics and Mathematics Literature, the rest by the Nauka Publishing House. Here are the titles of these releases:

Issue 1. Physics of the atomic nucleus (1962, price 23 kopecks).
Issue 2. Elementary particles (1963, price 25 kopecks).
Issue 3. Elementary particles (1965, price 43 kopecks).
Issue 4. Physics of the atomic nucleus (1965, price 29 kopecks).
Issue 5. Quantum macrophysics (1967, price 62 kopecks).
Issue 6. Astrophysics (1967, price 1 rub. 06 kopecks).
All these books are from the very cutting edge of science. That’s why they are called “What Physicists Think About.” Indeed, they talk about what scientists are doing these days, and sometimes about what they are still planning to do. It’s not for nothing that the article will end in much the same way as Robert Marshak’s “Nuclear Forces”, published in four issues:

“What can be said about the cherished dream of physicists - a true theoretical understanding of nuclear forces? All we can do at the moment is to express the hope that we are on the right track… However, we are confident that someday we will be able to write the right issue."

61

equations and find a way to solve them. Until then, we seem to have no choice but to press ahead using the empirical approximations described in this paper.”
Nowadays, science is developing incredibly quickly, and textbooks, naturally, cannot keep up with this development. If you get acquainted with some modern science only from a textbook, then it may seem that everything has already been done in it, all the problems have been solved, the mine, so to speak, has already been worked out, the ore has been finished off and only waste rock is left to your share. It even becomes a bit of a shame that I was born late. All countries have already been discovered and there are no blank spots on the map of the Earth, all the laws have been found - Newton, Coulomb, Ampere, Joule - Lenz, etc., the theory of relative mechanics and quantum mechanics have been created. It would seem, what else? But in reality this is not the case, and physics today faces many unsolved problems.

What substance does elementary particles consist of (electrons, neutrons, protons, etc.) and what is their structure? Perhaps they are homogeneous components of unknown matter, or perhaps, as the famous Russian poet Valery Bryusov once wrote,

“these electrons are Worlds where there are five continents, Arts, knowledge, wars, thrones and the memory of forty centuries!”

What is the nature of gravitational forces, and how do they arise? What are nuclear forces? Where does the periodic system of elements end and what other atoms can be added to it? Is there somewhere in the bottomless depths of space at least one anti-world, built from anti-morality? How many elementary particles does nature have and what are their properties?

Such problems are discussed in the collection “What Physicists Are Thinking About.” Moreover, as a rule, it is told first-hand by major scientists who themselves conduct these studies. Therefore, reading the article, as if entering their laboratory, you learn many interesting details and begin to better understand how science is done.

Despite the fact that in these collections the conversation is always about the latest problems in physics, which are generally very difficult to understand, they invariably remain completely accessible to high school students. The authors with great skill use the entire arsenal of truly scientific means: they provide simple examples and analyses, use clear models of various physical processes, and practically do without complex mathematical apparatus. The stats are written in a very interesting, lively, figurative language and sometimes read like works of art, although they have a global physical meaning.

Here is a brief description of the main content of these collections.

Two issues (the first and fourth) talk about the research of atomic nuclei. First of all, it will decide about research tools - accelerating charged particles, sophisticated cameras, special highly sensitive photographic plastics, spectrometers. Other articles examine the results of studying the structure of atomic nuclei and the nature of nuclear forces. Work on the creation of transwound elements is also described in detail.

The second and third issues contain articles on elementary particles. In addition, it talks about many individual types of particles - pions, neutrinos, antiprotons, antineutrons,

K-mesons, hyperons, resonances, their discovery, properties and the place they occupy in the microamp.

The fifth issue includes an article on macrosoptical effects and phenomena whose nature is fundamentally quantum in nature. This is primarily quantum electronics, which created molecular generators and successfully produced various radiations, ultra-precise atomic clocks. This also includes the recently discovered quantum phenomena associated with superconductivity and superfluidity, as well as the famous Mössbauer effect.

The sixth issue is entirely devoted to the problems of modern astronomy. It talks about new branches of astronomy - infrared, reptgen, neutrino, gamma and radio astronomy - and adding amazing information to them. What, for example, makes hail worlds—Galaxies—explode? Why do stars pulsate? How old is the Orin Nebula? What physical processes occur in the Large Magellanic Region? These are some of the problems discussed in this book.

Unfortunately, among the authors of the collections there were very many Soviet scientists, many of whom made a worthy contribution to the development of modern physics. In some articles, this contribution is not even acknowledged not only in the main text, but also in the applying editors.

Most of these collections are available. They were published in relatively small editions, and now they can only be found in libraries. But everyone who takes the trouble to read at least one of these books will feel the powerful breath of modern science with its intersecting solved and unresolved problems.

V. A. Leshkovce

62

To the article “The Story of Quantum”
(pp. 6-15)

1. 9,8·10⁻⁷ ∂x/cm³.
2. 2.3·10⁻¹² ∂p ≈ 1.5 electron-volts.
   Maximum number of quanta: 1.7·10⁻¹² ∂p ≈ 1 electron-volt.
3. 4.4·10¹² quanta in one cubic centimeter.
4. Quantum energy: 9.6 kiloelectron-volts, electron energy: 0.1 kiloelectron-volt.

To the article “Continued Fractions”
(pp. 16-26)

1. λ = 5216,5.
2. [0; 1, 2, 3, 4, 5].
3. $\frac{5777}{1875}$.
4. a) [1; 1212⋯],

6. [2; 444⋯],
   b) [2; 2424⋯].

5. 6. $\frac{\sqrt{ab(ab-4)}-ab}{2a}$*.

To the article “What is a function”
(pp. 27-36)

1. Natural domain of definition:
   a) x ≠ 0; 6) x ≤ -1; x ≥ 1.
2. a) 9; 6) 8; b) 6.
3. a) 8; 6) 6; b) 9; d) 0.
4. 10⁷; 4⁷.
5. nⁿ.
6. f₁ and f₃ are reversible.
   13 and 14. The mapping coincides with the inverse one.
7. a) 1680; 6) 9240; b) 18 150=3⁹-3·2⁹+3
8. Aₙⁿ=n (n−1) … (n−m+1), if m≤n; Aₙⁿ=0 if m>n.
9. $\frac{28!}{(7!)^4}$.

*) This is the positive root of the equation
$$x = \frac{1}{a + \frac{1}{b+x}}.$$

To the article “Dry friction”
(pp. 37-43)

1. Let us decompose the friction forces acting on the front wheels automatically into two components: F₁, lying in the plane of the wheels, and F₂, perpendicular to the wheels (see figure). F₁ forces make the wheels turn, and F₂ forces turn automatically.

2. If the center of gravity of the stick is not in the middle between the fingers, then the pressure of the stick on the fingers is different. The friction forces acting on the stick from the fingers are also different. The stick moves to the side where there is less friction.

3. Until the block slides along the plane, the friction is equal in magnitude to the projection of the weight of the block onto the inclined plane F_TP = P sin α. The block begins to slip when the friction reaches its maximum rest value F_TP = kN = kP cos α. In this case, the condition kP cos α = P sin α is satisfied. Therefore, the sliding of the block begins at the angle of inclination of the plane to the horizon α = arctan k. After this, the friction will be equal to F_TP = kN = kP cos α.

The angle α = arctan k at which the block begins to slip is called the friction angle. It also has another geometric meaning: if a force is applied to a block lying on a horizontal plane that is less than the friction angle with the vertical, then the block cannot be moved from its place, no matter how much the applied force is applied. You can prove it like this. Let us place the observer on an inclined plane on which a block of wood lies.

63

juice, and we will increase the angle of inclination of the plane to the horizon. The observable will say that in his coordinate system, a body lying on a plane, which he considers horizontal, is acted upon by a layer that makes an angle α with the redistribution to the plane. If tgα < k (i.e. α < arctg k), then the block does not slide along the plane, no matter how great the applied force.
4. The answer is shown at figure.

The drawing inserted into the slot will be pressed against the board by the ball. If the angle α formed by the inclined edge is passed vertically, such that tan2α < k (where k is the coefficient of friction between the ball and the drawing), then the drawing will be held by friction against the ball, no matter how large the weight of the drawing and no matter how small the friction

5. The cable acts on the rods with forces F1 and F2, the magnitude of which depends on the weight of the pipe. For example, if the rods make an angle with the overload to the surface of the pipe less than α = arctan k (k is the friction coefficient of the rods on the pipe), then the rods will not slide along the pipe, no matter how large the forces F1 and F2 are, and therefore no matter how large the weight of the pipe is (see solution to problem 3). In such cases, engineers talk about jamming.

Using wedging, you can make a simple and convenient lock for hanging maps and drawings. You need to make two metal circles, as shown in figure (p. 64, above), and, having nailed the circles to the board, put a ball into them.

drawing on the board. To remove the drawing from the lock, just lift the ball with a pointer.

6. The cube begins to slide when the equal force F and the component of the weight of the cube, parallel to the inclined plane,

64

HOW TO DRAW
CIRCLE
WITHOUT COMPASSES!

There’s a geometry lesson going on.
You need to draw a circle in your notebook, but there is no compass.
You can get out of this situation by drawing a circle by hand and using only a checkered notebook. You just need to remember the numbers:
three is one, one is one, one is three.

Start drawing the circle at one of the points (A) of the intersection of the horizontal and vertical rulers of the notebook. Draw a smooth curved line from it “by eye” to the right and down, saying (aloud or mentally): three - one.

This means you move from point A to point B three cells to the right and one cell down (see figure; start moving so that your line touches the horizontal line at point A). Then from B go to C saying one to one (one square to the right and one to the bottom) and finally from C to D saying one to three. Line ABCD is a quarter circle!

From D go to E, again starting to say: three - one, this time moving three cells down and one to the left, then: one - one and one - three, moving in the same directions. The result will be similar to the DEFG circle. In the same way you will get the third quarter GHIJ (move left and up) and the quarter JKLA (up and right). If you followed this rule and never lost your way, then after point L you will end up at the starting point A. (On our figure the second half of the circle is not marked.)

After two or three attempts, you will quickly and confidently be able to draw an excellent circle without a compass.

What is this method based on?

CROSSWORD “PRIME NUMBERS AND SQUARE” *)

HORIZONTAL:

a. The square of the smallest two-digit prime number.
c. A square whose last digit is equal to the sum of the digits of the base.
d. A square whose sum of digits is equal to the sum of the digits of the base.
f. The largest three-digit number that is a square.
h. A prime number remains prime after rearranging its first two digits.
i. A prime number obtained from d horizontally by rearranging the digits.
l. Square the number q vertically.
m. A prime number whose second digit is 8.
o. A prime number with the same first and last digits, 70 greater than k vertically.
q. Prime number; its first digit is equal to the sum of the rest.
s. A square whose last two digits are equal.
t. A prime number 100 greater than i vertically.
u. A prime number whose first two digits add up to the third.

VERTICAL:

a. A square written in the same numbers as f horizontally.
b. A prime number whose first and last digits are equal.
c. A square whose first digit is equal to the sum of the digits of the base.
e. A square written in the same numbers as a vertically.
g. Square the number t horizontally.
h. A square whose first two digits are the same.
i. A prime number starting with WISEe and ending with WISEe by 1.
j. A prime number obtained from b vertically by rearranging the second and third digits.
k. A prime number whose second digit is 1.
n. A prime number whose first digit is twice as large as the second.
p. A prime number whose first two digits are equal.
q. A prime number is 10 less than t horizontally.
r. A prime number is 6 greater than a horizontally.

*) Each (uncrossed) cell must contain one number.

PRICE 30 kopecks.
INDEX 70465

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