# The theory of happiness. Accidents are nonrandom

3r3618. 3r3-31. 3r33584. I continue to acquaint Habr's readers with chapters from his book “The Theory of Happiness” with the subtitle “Mathematical foundations of the laws of meanness”. This is not a published popular science book, very informally telling how mathematics allows you to look at the world and life of people with a new degree of awareness. It is for those who are interested in science and for those who are interested in life. And since our life is complex and, by and large, unpredictable, the emphasis in the book is mainly on probability theory and mathematical statistics. Theorems are not proved here and the fundamentals of science are not given; this is by no means a textbook, but what is called recreational science. But it is this almost playful approach that allows us to develop intuition, brighten up with vivid examples lectures for students and, finally, explain to nemathematicians and our children what we found so interesting in our dry science. 3r33587.

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Published chapters: 3r33552. 3r311. • Introduction to Merrology

3r3618. • The law of watermelon peel and normality of abnormality 3r33333.

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The law of the zebra and foreign line

3r3618. • The curse of the director and damn printers 3r3333311.

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Thermodynamics of class inequality

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3r3618. In this chapter, we discuss the predestination of the flight of a coin, topographic maps, mathematical disasters, and the nature of randomness. And along the way we will look at such areas of mathematics as the theory of measures and the theory of dynamic chaos.

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3r3618. Talking about the laws of meanness, as the source of everyday turmoil, often begins with the famous 3r3551. sandwich law 3r???. . It is simply formulated, easily verified, and widely known:

3r3618. 3r33555. A sandwich always drops buttered down. 3r33558. It is clear that the word "always" here is an exaggeration. It is easy to imagine the conditions in which the sandwich will fall, leaving the buttered side intact. What do people understand by this law? Most likely, the sandwich drops the oil down often enough to make it noticeable. But does the unfavorable outcome of a fall occur more often than a favorable one? Sandwiches are different, they fall under different circumstances, from different heights There are so many parameters that it makes no sense to talk about patterns in such a task. Anything can happen. It happens that the oil falls down, then it becomes a shame, we remember the law and remember it. And if a sandwich falls uninteresting - butter up, or if it was left without butter at all, then there is nothing to talk about - it’s clear that the law is comic! In the end, a sandwich is like a coin that mathematicians use to produce random variables with two possible values: "eagle" and "tails". If the coin is “honest,” then it absolutely does not matter which side to fall, and we say that the probability of a tail and tails are the same and equal to 3r359595. 3r3-3597. . In theory, with sandwiches, things should be the same. We will return to them in the next chapter, but for now let's take a closer look at the most probable simple probabilistic system - the coin.

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3r3618. A coin in theoretical probabilistic experiments is thrown in some special magical way, so that the choice of the initial position, initial speed and speed of twisting when tossing does not affect the probability of a particular outcome. But obviously, this is impossible! A coin is a mechanical system and obeys the laws of mechanics, but they do not contain random variables. The future in the laws of motion of such a simple body as a coin is uniquely determined by the past state of this body. If the coin is tossed by a robot, or a Laplace demon, a mythical creature that has complete information about the coordinates and speeds of any mechanical system, then with constant initial data, identical results will be obtained. We, of course, are not robots and not demons, but do people really throw up coins so carelessly and unpredictably that the laws of mechanics can lead to accidents?

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3r3618. And from where, in general, is randomness taken in the world described by the laws of mechanics? Where does the accident come from? What is the difference between truly chaotic or stochastic systems, fundamentally unpredictable, and systems in which it is simply difficult to guess the behavior, but it can be calculated?

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3r3618. The coin problem was considered in 1986 by Joseph Keller. We will give a simple explanation for the occurrence of uncertainty in this process, based on the reasoning from Keller's article. Which side a coin falls on depends on its flight time 3r33595. 3r3-3597. and from the angular velocity

3r3-3597. . If you measure the angular velocity in revolutions per unit of time, then the number of revolutions performed by a coin expresses extremely simply 3r3-3595. 3r3-3597. . This dependence sets the lines of equal speed in coordinates 3r33595. 3r3-3597. , and they, in turn, limit the areas corresponding to even and odd number of turns.

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3r31414. 3r33584. 3r33585. Chart showing the parity of the number of turns made by a coin in flight. The rectangle shows the area in which the process of fortune-telling on a coin occurs most often. 3r33586. 3r33587.

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3r3618. On such a chart, it is possible to show how the result of a coin flip, twisted by a known number of revolutions per second, and caught after a known flip time, will be. If we fall into a white strip, then the same side will fall out that was on top when tossing, if the orange side is the reverse. Lines of equal speed are hyperboles and it can be seen that as the number of turns increases, the alternation of areas becomes more and more frequent, and the areas themselves become thinner. The human hand is imperfect and a very small variation of the initial values overlaps many areas at once, making the outcome unpredictable. In the range of the hand (the rectangle in the diagram), an offset of 3r3-3595 is sufficient. 3r3-3597. to jump from white stripe to orange. The question remains: how does the “integrity” of a true mechanical coin follow from this construction? How to get the probability of falling heads and tails from the resulting diagram?

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3r3618. 3r33535. Take urgent action! 3r33535.

3r3618. Let us dive a bit into such mathematics that we don’t go through in school in order to better understand what we are talking about. We talked in 3r3106. introduction of 3r3107. that mathematicians do not study numbers or geometric shapes, as it may seem after studying a school course. They work with mathematical structures (abstract algebras, semirings, fields, monoids, topological spaces, and other abstract things), describe them, as it seems, completely unattached to practice, define them, study their properties, prove theorems. And then they hone their skills in the search for such structures in the most diverse areas of knowledge, making surprisingly useful breakthroughs, including in purely applied industries. We will now deal a little with such mathematics and consider how the basis of the theory of probability is built, based on a very abstract notion of measure.

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3r3618. We described the mechanics of the coin and obtained regions describing solution sets with certain properties. Areas are flat figures, how to move from them to probabilities correctly? We need to measure our areas and we naturally come to their area. The area is 3r33584. measure 3r3587. flat figure. This is an exact mathematical term for a function that associates a set of non-negative number values. Examples of measures are 3r3–3584. quantities in enumerable sets 3r33587. (the number of apples in the bag, for example), as well as 3r33584. length [/i] , 3r33584. square 3r33587. , 3r33584. volume figures [/i] .

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3r3618. In mathematics, there is a whole section called 3r33584. theories of measures 3r33587. . This theory was born at the turn of the 9th – 20th centuries (Emile Borel and Henri Lebesgue stood at its source) and opened up wide possibilities for mathematicians to analyze very complex objects: cantor and fractal sets. It formed the basis of functional analysis and modern probability theory, the foundation of which was laid by the remarkable Russian mathematician Andrei Kolmogorov. The definition of probability, as a measure, allows you to see all the basic properties of probability for both discrete and continuous sets.

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3r3618. Although our book is not a textbook, but it is worthwhile to dwell a bit on this to take a look at the concepts of the theory of probabilities, as it were, from “bird's-eye view” and get a taste of mathematics. To begin with, we list the basic properties of 3r3353584. any 3r33587. measures. In order to better imagine them, you can use instead of the word “measure” the words “quantity” or “length” or “square”.

3r3618. 3r33555. 1. The measure of the empty set is zero.

3r3618. 2. The measure of the entire measurable set is finite for finite measures.

3r3618. 3. The measure of the subset does not exceed the measure of the set

3r3618. 4. The measure of the union of two arbitrary sets is equal to the sum of the measures of these sets minus the measure of their intersection (additivity).

3r3618. 5. The measure of the complement of a subset is equal to the difference of the measures of the whole set and the measure of the subset. 3r33558. Can any non-negative numeric function be a measure? Not at all. For example, age puts a certain number in correspondence with a person. But the age of two people cannot be defined as the sum of their ages. And the speed of running is not a measure - two people do not run twice as fast. But the impulse (the amount of movement) or energy already has the properties of a measure. The weight, the amount of money, the volume of knowledge, the loudness of the cry, though not always easily measurable things, can also serve as a measure on many people.

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3r3618. At the intuitive level, the concept of probability is familiar now, practically, everything. It is assessed by political scientists and journalists on a talk show, it is discussed referring to global warming or tomorrow's rain, jokes about it: 3r-5384. 3r33560. What is the probability to meet on the Tver dinosaur? - One second: either a meeting or not. 3r33586. 3r33587. In modern mathematics, the concept of 3r33584. probability of [/i] is defined as a measure on a special set called 3r3-3584. probability space [/i] . It includes both elementary events and their combinations, which are obtained using join, intersection and exclusion operations. An example of an elementary event: “the loss of a troika when throwing a dice”. An example of an event that is not elementary: "the loss of any even number except two." So, we list the properties of the probability:

3r3618. 3r33555. 1. The probability of an impossible event is zero.

3r3618. 2. The probability for the entire probability space is equal to one.

3r3618. 3. If one event entails another, then the probability of the second does not exceed the probability of the first. The relation “entails” for events is equivalent to “is a subset” for sets.

3r3618. 4. The probability of occurrence of at least one of two arbitrary events is equal to the sum of the probabilities of each of these events, minus the probability that events will happen simultaneously.

3r3618. 5. The probability of non-occurrence of an event is one minus the probability of an event occurring. 3r33558. Look closely at the properties of measures and probabilities and it will be seen that we are talking about the same properties.

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3r3618. Not all properties of probability follow from its definition, as measures: the concepts of independence of events and the method of calculating the probability of two or more independent events simultaneously as products from probabilities are introduced through 3r3584. conditional probability 3r33587. , but these concepts are consistent with the Kolmogorov definition.

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3r3618. Discrete random variables correspond to finite countable sets, in them a natural measure is an ordinary count of the number of elements. Accordingly, the probability in a discrete probabilistic space is the combinatorial counting of variants familiar to each student. For continuous random variables, probability, as a measure, is more like a length or an area, and here we are talking about r3r3584. probability densities of 3r3587. .

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3r3618. The analogy of probability with the measure does not end there. What is 3r33584. the average is 3r33587. ? This is an analogue of r3r3584. the position of the center of mass [/i] figures consisting of point masses or a solid body with a known density. And these values are calculated equally. And how is the variation of random variables around the average characterized: 3r3353584. dispersion 3r33587. ? Also as 3r33584. moment of inertia [/i] characterizes the distribution of mass around the center of mass. And again, the formulas for calculating the variance for a sample or distribution coincide with the formulas for the moment of inertia of a set of bodies or a solid body of a cunning form.

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3r3618. By the way, if we replace the sum in “definitions” and properties of probability with “maximum” and product with “minimum”, then we can construct an alternative theory, it is called 3r384. theory of opportunity 3-333587. . This is how math works. We start with abstract reasoning: numbers form an algebra with operations of addition and multiplication, but on a limited number interval one can construct a similar algebra with operations minimum and maximum. We build the concept of measure on a new algebra and find out that it opens up a new view of the world! In contrast to probability theory, in such a theory, it is possible to construct two coordinated measures - 3r-3584. opportunity 3r33587. and 3r?384. the need for 3r33587. and, unlike probability, they are in good agreement with both the operations of combining and intersection of events. This direction, created by the American Lotfi Zade, an Azerbaijani by origin, serves as a basis for 3-33584. Fuzzy Logic [/i] and is used in systems of automatic pattern recognition and decision making.

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3r3618. 3r33535. Unbelievable, but it is a fact! 3r33535.

3r3618. The first property of measures seems trivial, but it is interesting for its asymmetry. If the measure of a subset is zero, it does not mean that it is empty! For example, a line is a subset of points of a plane, but its area (measure) is zero. There are also more exotic examples - Cantor and fractal sets, having a complex structure, containing an infinite number of points, visibly "occupying" a certain area or volume, but nonetheless having a zero measure.

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3r3618. While preparing this illustration, I found a wonderful image of a disconnected Julia set on a transparent background with high resolution. Having inserted it into the vector editor, I faced an amusing difficulty - it was very difficult to get a mouse to this image in order to select it. It is so “loose” that the probability of hitting the filled pixel was noticeably less than hitting the transparent background. In probability space, subsets of zero measure may also exist, but this does not mean that events from these subsets are impossible. From the fourth to fifth attempts, I still could select the image, since the pixels have a finite size. But what would happen if I got at my disposal a real incoherent Julia set with infinite resolution?

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3r3618. Imagine that you are using a software random number generator, which produces an arbitrary real number from 3-335959. 3r3-3597. to 3r?595. 3r3-3597. . What is the probability of dropping the number 3r33595. 3r3-3597. ? and the numbers 3r359595. 3r3-3597. or 3r39595. 3r3-3597. ? In all these cases, the answer will be - zero! Rather, the smallest positive number available to the computer, the so-called machine epsilon, because the computerter operates with a finite number of decimal places. Wait, you say, in what sense - zero? These same numbers are not impossible. Let's conduct an experiment, as a result we will get some specific number and when we get it, then “by construction” the probability of its occurrence cannot be zero. That's right, but how long do you need to wait until exactly 0 drops out? Almost infinite! The fact is that a single number, like a point on an interval, has zero measure and an honest zero probability. Only the measure of a continuous segment, even a very small one, is nonzero. So we are not talking about probability, but about probability density, which, when multiplied by a finite measure of a subset in a probability space, will give a finite value — the probability to fall into this subset. By the way, if we had the perfect random number generator with infinite accuracy, the probability of obtaining any rational number with it will also be zero.

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3r3618. If someone patiently conducts a thousand experiments with a coin and happily tells you that he got as many “eagles” as he did “solved”, feel free to doubt or congratulate him with rare luck. Although throwing a coin and a discrete random process, but as statistics accumulate, the power of the probabilistic space will increase and the measure of the event: “the number of eagles coincides with the number of grids” will decrease. It can be shown, using the Stirling formula, that the probability of this “most probable” event tends to zero as the number of trials rises to 3-33595. 3r3-3597. . For a hundred throws, this is a little more than five percent, for ten thousand - just half a percent. In such cases, the mathematicians say: 3r?384. almost certainly the number of "eagles" will not be equal to the number of "tails" 3r-3587. . Strange as it may sound, “almost certainly” is an exact mathematical term meaning that an event is an addition to a subset of a zero-measure probability space. We will return to these arguments, in one of the following chapters, when we ask ourselves: how much each of us can consider ourselves to be normal.

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3r3618. 3r33535. Check the honesty of the real coin

3r3618. Let's return to the coin and to her honesty. The Kolmogorov definition of probability reconciled its frequency definition (as the relative frequency of the events) and the geometric (as a fraction of the "volume" of the event in the total "volume" of possibilities). Thus, the fraction of the area of white stripes on the chart calculated for a rotating coin reflects the probability of falling out of the same side with which we planted it.

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3r3618. But the trouble! The area of each strip in our diagram is infinite (if we look at the whole quarter of the coordinate plane). However, the additivity of the measure will allow us to accurately show that this does not prevent the areas of shaded and white areas to be the same. In explicit form, the equations for our curves are of the form 3r359595. 3r3-3597. . If the area under the curve is 3r39595. 3r3-3597. equal to r3r3595. 3r3-3597. then due to the additivity property, the area under the curve is 3r3359595. 3r3-3597. will be equal to 3r39595. 3r3-3597. . In turn, for individual strips we get: 3r33595. 3r3-3597. . It turns out that the area difference does not depend on the “number” of the hyperbola. This is not something special relating to hyperbolas; the same conclusion can be made for any curve like 3r33595. 3r3-3597. , if only function 3r33595. 3r3-3597. was measurable. And if this is the case, then for the whole domain of definition, hitting the white part of the diagram or the shaded one is equally likely, as is expected for an "honest" coin.

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3r3618. The arguments we have just led seem simple enough, but they give a very general result, applicable to any additive quantities. The abstract concept of measure allowed us to compare among themselves infinite values, remaining within the framework of logic and common sense.

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3r3618. Abstractions are good, but it can be argued that in reality we do not throw up coins with all the possible parameters. As shown by experiments with a high-speed camera, the angular velocities fall in the range of 3r-3595. 3r3-3597. to 3r?595. 3r33333. 3r3-3597. revolutions per second, and the duration of the flight - from half to one second. This area is highlighted by a rectangle in the diagram. In it, the total area of the white stripes is slightly larger than the orange ones, and it can be concluded that the probability of hitting the same side as when throwing up will be 3r33595. 3r3304. 3r3-3597. .

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3r3618. In 200? a group of Percy Diaconis and co-authors from Stanford published 3-333310. article

In which the detailed analysis of the process of coin toss is given. A detailed description of the mechanics of a flying and rotating disk, which not only rotates, but also precesses - the axis of rotation rotates itself in flight, shows that with manual throwing from the eagle position from above, the probability of an eagle being dropped is one-hundredth more than half.

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3r3618. Is it a lot or a little? How many experiments are needed to notice such a difference? With the accumulation of experimental data, the standard error of the mean, reflecting the error with which the average value can be calculated, decreases in proportion to the square root of the number of tests: 3r359595. 3r3333317. 3r3-3597. , here is 3r39595. 3r3-3597. - standard deviation for the studied distribution. In our case, for the Bernoulli distribution with probability 3r359595. 3r3-3597. which is equal to 3r39595. 3r3-3597. . In order to confidently single out one hundredth average deviation, this deviation must exceed 3r-3595. 3r3-3597. standard deviations. Thus, we can estimate the number of trials: 3r3 -9494. 3r3-3598. 3r? 3594. 3r33595. 3r33333. 3r3-3597. 3r3-3598. So many times you need to flip a coin to notice the mechanistic predestination of the result. To make it clearer what is meant, I will give an example of two hundred trials of an ideal and slightly non-ideal “coins” conducted in order to calculate the probability of a loss of, say, an eagle. Each test consists of 3r33595. 3r3-3597. "Toss". The words "coin" and "toss" are quoted, because in fact it was not a physical coin that was used, but a random number generator obeying the Bernoulli distribution.

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3r31414. 3r33584. 3r33585. Experimenting with the flip up of an ideal and slightly non-ideal coin in order to fix the non-ideality. 3r33586. 3r33587.

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3r3618. It is seen that only after

3r33357. 3r3-3597. The “cloud” of observed mean values begin to be clearly divided. Well, for household use, we can assume that the coin is a good generator of a random choice of two equally probable options.

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3r3618. 3r33535. Tourist Law 3r3-3030.

3r3618. The equivalence of the geometric and frequency definition of probability reveals the riddle of a single meanness law, known among tourists, geologists, and all those that use topographic maps: 3r3605. 3r3618. 3r33555. The place where the tourist goes is most often either on the bend of the map or on the edge of the sheet. 3r33558. Suppose that we are equally often interested in objects located in all parts of the map. But we are rarely interested in objects of zero measure - the whole point of using the map is to review 3r-3584. surroundings 3r33587. object, that is, some final area. Let it be enough for us to have a small fraction of 3r3-3595. 3r33334. 3r3-3597. from the area of the whole map

3r3-3597. to figure out how to get to the object. This means that if an object approaches a bend or edge at some critical distance of 3r-3595. 3r3-3597. , we find the law of the tourist fulfilled. The share of border areas in the total area of the map will give us the chance to try this law of meanness for ourselves. This is how the unpleasant parts of the map look like at 3r3-3595. 3r3-3597. and one fold.

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3r33333. 3r31414. 3r33584. 3r33585. Gray marked "bad" areas. A section with a half-percentage area for a 40 cm wide map is shown separately; it has a diameter slightly larger than 3 cm. 3r38686. 3r33587.

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3r3618. For the square card

3r3013. 3r3-3597. . Unpleasant stripes will have an area of 3r33595. 3r3404. 3r3-3597. . Four lanes, two vertical and two horizontal, will be located at the edge, any additional bend, horizontal or vertical, will add another strip. In this case, the intersecting stripes add extra squares of 3r33595. 3r3407. 3r3-3597. . Folding the map so that it turns out

3r33410. 3r3-3597. horizontal and 3r39595. 3r3-3597. vertical bends, we get the total area of the unpleasant zone equal to: 3r3-3595. 3r31616. 3r3-3597. . Taking it to the area of the whole map, we get an unpleasant share of the total area:

3r3618. 3r? 3594. 3r3-3598. 3r? 3594. 3r33595. 3r33424. 3r3-3597. 3r3-3598.

3r3618. The figure shows the areas in which this proportion exceeds r3-33595. 3r33430. 3r3-3597. for different values of r3r3595. 3r33334. 3r3-3597. .

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3r33440. 3r31414. 3r33584. 3r33585. Areas where there is an increased likelihood of being on the bend of a card or on its edge. The numbers indicate the proportion of the area of the considered neighborhood from the area of the entire map. 3r33586. 3r33587.

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3r3618. It turns out that the map, folded in half, twice already can formally be considered dishonest in relation to the tourist. Most often, the cards have three vertical and three horizontal folds, which makes it possible to fulfill the law of meanness with a probability of about 3r359595. 3r3-3597. at 3r33595. 3r3-3597.

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3r3618. 3r33535. Where does the accident come from? 3r33535.

3r3618. 3r33434. In the souvenir shops you can find magnetic pendulums for "choice of desires." They, too, are mechanical generators of chance and are sometimes mistakenly called "chaotic pendulums." Starting from some initial position and speed, the pendulum makes a series of “unpredictable” oscillations and finally stops in one of the sectors. However, fluctuations are not unpredictable here, they are just very sensitive to the initial conditions. For each sector in which the pendulum can stay, there is 3r-3584. region of gravity 3r33587. in coordinate-speed space. This is a set of such initial conditions under which the pendulum will necessarily be attracted to a certain point in the specified sector. The stopping point of the pendulum is called

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3r37474. 3r31414. 3r33584. 3r33585. The regions of attraction of attractors for the one-dimensional pendulum of desires — the Duffing oscillator. 3r33586. 3r33587.

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3r3618. As in the case of the coin, slightly shifting the initial conditions, we get from one attractor to another. The dice and the roulette also act in the same way, but they are not in themselves generators of randomness. These are not truly chaotic systems and their behavior can be accurately calculated.

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3r3618. And what is the real accident? A good example of a truly stochastic system is the appearance of cars on the road. People do not agree, do not coordinate their plans, each element of the ensemble outside the road acts independently. And although there are certain patterns in people's behavior - peak hours in the morning and in the evening, empty roads at night, etc., we do not have and will never have enough information about each participant in the movement to predict the appearance of any of them. Also stochastic are the mechanics of elementary particles at the quantum level, the decay of unstable atoms, changes in the genetic code, apparently, earthquakes and quotations of securities on the stock exchange. The only thing that remains for the researcher is to consider them as random variables and describe them in terms of probability theory.

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3r3618. But there is another source of accidents - 3r-3584. dynamic chaos [/i] . Chaotic systems differ from stochastic in that they are described by exact equations and parameters that do not contain randomness. However, their behavior is not just difficult, but chaotic and truly unpredictable. If we begin to oscillate the pendulum of desires, very carefully, with precisely controlled frequency and amplitude, then we will find that its smooth movements cannot be calculated for a long time. No algorithms on arbitrarily exact computing machines will be able to calculate the exact behavior of the pendulum for an arbitrarily distant future. It will not stop at any sector, but will make smooth movements, but will never return to the same point in the coordinate-speed space twice. Another example of an extremely simple chaotic system is an ideal ball bouncing in a gravity field on an ideal table with a spring. The relatively simple Lorentz equations showed that we can never predict the weather for more than a couple of weeks - this is also a chaotic system.

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3r3618. The theory of dynamic chaos, was able to explain the nature of such unpredictability. The simple one-dimensional pendulum of desires, which we considered, had two stable stationary points — two attractors, and one unstable, from which the system tries to escape, it is shown in white circle. In the chaotic mode, instead of a set of attractors, an infinite set of unstable stationary trajectories appears in the system. This set is infinite, but has

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3r3618. 3r? 3510.

3r3618. 3r33584. 3r33585. Left: Poincare section for the trajectory of a ball bouncing on a spring-loaded table. The set of points belongs to the surface of the sphere corresponding to the law of conservation of energy. Right: a volumetric region that contains a strange attractor, which is born when forced toOlebaniye thick plate. 3r33586. 3r33587.

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3r3618. The smoothness of the chaotic trajectory allows you to still look a bit into the future. This explains one annoying observation: on the one hand, weather forecasters, at times, cannot confidently predict the weather for a week, but on the other hand, if you say that tomorrow will be the same weather as today, you will not be mistaken in about three cases out of four . In general, the jokes about weather forecasters are unfair and we must pay tribute to human thought and perseverance, which allowed us to predict the weather at the present level!

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3r3618. Dynamic chaos is very complex and beautiful as a theory, it generates images of amazing elegance, but it can also be useful. For example, the algorithms by which random numbers are generated in computers are also determined. For the examples in this book, I used a pseudo-random number generator, which did not start the real stochastic process (alpha decay, or counting machines on the road), but calculated the following “random” number based on the previous ones it had previously received.

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3r3618. 3r33535. From coins to butterflies and the very fate of r3r3530.

3r3618. Observing how small deviations grow into global changes in the system lead to the thought of the “butterfly effect”. Let me remind you that this effect implies a chain of far-reaching dramatic consequences from some seemingly insignificant event. The butterfly crushed by researchers of the past in Ray Bradbury's story “And Thunder Has Been Boiled” led to a radical restructuring of the future. And Edward Lorenz, the creator of the theory of dynamic chaos, entitled one of his lectures: “Can the flap of a butterfly wing in Brazil cause a tornado in Texas?” 3r3605. 3r3618.

3r3618. We implicitly refer to this effect, complaining: “If I didn’t turn the corner, everything would be different!”, “If he didn’t get on this train, there would have been no catastrophe with him!” Or “Because of such trifles, they quarreled and gone !! ”But we see that truly stochastic quantum world and ultra-precise atomic clocks, stable Hamiltonian systems in the world of stars and galaxies and chaos of Saturn’s or Kuiper’s belts, thermal movement of molecules and amazing accuracy of biological systems or automobile mechanisms coexist . No, the flap of a butterfly's wing does not generate hurricanes, but disappears without a trace, generating a chain of whirlwinds that transmit energy and information to more and more smaller whirlwinds, as long as both energy and information do not disappear in the chaos of fluctuations. It should be clearly understood that small deviations lead to a radical restructuring of the system only if it is unstable or if the system is on the threshold of 3r3584. bifurcations of r3r3587. or 3r33584. disaster [/i] - so in the language of mathematics are called global restructuring in the behavior of the system with small changes in parameters. And bifurcations always form sets of zero measure in the parameter space — these are points or boundaries. Small perturbations do not lead to catastrophes 3r3584. [/i] almost everywhere. , (this is also an exact term meaning “everywhere, except for a set of zero measure”), and unstable states in nature are rarely observed without passing the “test of time”.

3r3618.

3r3618. If the couple broke up "because of the nonsense," she was destined to break up in any case, she was unstable. Sustainable couples go through wars and famine, and then, sometimes, break up, not because of small things, but as a result of profound changes that can happen to a person during life. In the chain of events that led to the train crash, it is not easy to single out the key event (a specific error or fatal accident) and, most likely, the key will not be an event, but a systematic violation of the rules, leading the system to an unstable state. If there are a lot of parameters in the system, and a number of them are random, and our life is arranged exactly this way, then the information in such a system tends to be lost, and it’s impossible to restore at any particular moment in our life “everything went wrong”. Do not torment yourself with regrets about what happened, but look at what is happening to you now, so as not to miss the real point of bifurcation.

3r3618.

3r3618. In this regard, we can recall one of the laws of merphology, called a certain Dreisen 3r3-3511. Restoration Act 3r33552. :

3r3618. 3r33555. Time to improve the situation is inversely proportional to the time of its deterioration.

3r3618. 3r33558. The following observation is given as an example: 3r33584. 3r33560. It takes more time to glue a vase than to break it. 3r33586. 3r33587. This law surprisingly accurately describes the relationship between the characteristic rates for the 3r33584 process. relaxation 3r33587. a stable system that can be described by the decreasing exponential law 3-33595. 3r3566. 3r3-3597. and the speed of development 3r33584. catastrophic 3r33587. process in an unstable system, in the linear approximation, an exponential growth of a small perturbation 3r33571. 3r3-3597. . These speeds, indeed, are inversely proportional to each other. The example with a vaza, however, is not a relaxation - a transition to the most likely state. He is closer to another process - to 3r33584. self organization 3r33587. This process, as a first approximation, is described by 3r33584. logistics 3r33587. by law and closer in speed to relaxation than to catastrophe.

3r3618.

3r3618.

3r33582. 3r31414. 3r33584. 3r33585. Typical non-stationary processes: catastrophe, relaxation and self-organization, having the same characteristic time. 3r33586. 3r33587.

3r3618.

3r3618. 3r? 3594. 3r3-3598. 3r? 3594. 3r33595. 3r3-3596. 3r3-3597. 3r3-3598.

3r3618.

3r3618. Sometimes, walking in the snow, I am surprised that the snowflake falls on my nose. I am surprised that the probability of this event was negligible. If judged, she was born high in the sky over the Pacific Ocean, circling in indiscriminate turbulent flows in a cloud, falling continuously changing direction of movement to get to the tip of my nose! And what a stunning path photons passed from a distant star !? Tens of thousands of years they swept through the universe, they were not absorbed by the dust they did not meet an asteroid! They were born in the quantum world of a distant star, and ended up in the quantum world of the protein opsin on the retina in my eye. Even to consider the probability of this event makes no sense, it is zero, but an event happens, and I see the flickering light of a star. Now it is clear that this is all because the area of my nose and even the molecules have a non-zero measure, but it is still surprising: what almost certainly should not have happened still happens!

3r3618.

3r3618. Philosophers may argue about the predestination or chance of fate, about the truth or illusiveness of our knowledge of nature. I urge the reader to look at the world from the height of mathematical abstractions and admire its beauty and consistency. 3r31414. 3r3618. 3r3618. 3r3618. 3r3611. ! function (e) {function t (t, n) {if (! (n in e)) {for (var r, a = e.document, i = a.scripts, o = i.length; o-- ;) if (-1! == i[o].src.indexOf (t)) {r = i[o]; break} if (! r) {r = a.createElement ("script"), r.type = "text /jаvascript", r.async =! ? r.defer =! ? r.src = t, r.charset = "UTF-8"; var d = function () {var e = a.getElementsByTagName ("script")[0]; e. ): d ()}}} t ("//mediator.mail.ru/script/2820404/"""_mediator") () (); 3r3612. 3r3618. 3r31414. 3r3618. 3r3618. 3r3618. 3r3618.

3r3618.

3r3618.

Published chapters: 3r33552. 3r311. • Introduction to Merrology

3r3618. • The law of watermelon peel and normality of abnormality 3r33333.

3r3618. •

The law of the zebra and foreign line

3r3618. • The curse of the director and damn printers 3r3333311.

3r3618. •

Thermodynamics of class inequality

3r3618. 3r31414. 3r31414.

3r3618.

3r31414.

3r3618. In this chapter, we discuss the predestination of the flight of a coin, topographic maps, mathematical disasters, and the nature of randomness. And along the way we will look at such areas of mathematics as the theory of measures and the theory of dynamic chaos.

3r3618.

3r33333.

3r3618. Talking about the laws of meanness, as the source of everyday turmoil, often begins with the famous 3r3551. sandwich law 3r???. . It is simply formulated, easily verified, and widely known:

3r3618. 3r33555. A sandwich always drops buttered down. 3r33558. It is clear that the word "always" here is an exaggeration. It is easy to imagine the conditions in which the sandwich will fall, leaving the buttered side intact. What do people understand by this law? Most likely, the sandwich drops the oil down often enough to make it noticeable. But does the unfavorable outcome of a fall occur more often than a favorable one? Sandwiches are different, they fall under different circumstances, from different heights There are so many parameters that it makes no sense to talk about patterns in such a task. Anything can happen. It happens that the oil falls down, then it becomes a shame, we remember the law and remember it. And if a sandwich falls uninteresting - butter up, or if it was left without butter at all, then there is nothing to talk about - it’s clear that the law is comic! In the end, a sandwich is like a coin that mathematicians use to produce random variables with two possible values: "eagle" and "tails". If the coin is “honest,” then it absolutely does not matter which side to fall, and we say that the probability of a tail and tails are the same and equal to 3r359595. 3r3-3597. . In theory, with sandwiches, things should be the same. We will return to them in the next chapter, but for now let's take a closer look at the most probable simple probabilistic system - the coin.

3r3618.

3r3618. A coin in theoretical probabilistic experiments is thrown in some special magical way, so that the choice of the initial position, initial speed and speed of twisting when tossing does not affect the probability of a particular outcome. But obviously, this is impossible! A coin is a mechanical system and obeys the laws of mechanics, but they do not contain random variables. The future in the laws of motion of such a simple body as a coin is uniquely determined by the past state of this body. If the coin is tossed by a robot, or a Laplace demon, a mythical creature that has complete information about the coordinates and speeds of any mechanical system, then with constant initial data, identical results will be obtained. We, of course, are not robots and not demons, but do people really throw up coins so carelessly and unpredictably that the laws of mechanics can lead to accidents?

3r3618.

3r3618. And from where, in general, is randomness taken in the world described by the laws of mechanics? Where does the accident come from? What is the difference between truly chaotic or stochastic systems, fundamentally unpredictable, and systems in which it is simply difficult to guess the behavior, but it can be calculated?

3r3618.

3r3618. The coin problem was considered in 1986 by Joseph Keller. We will give a simple explanation for the occurrence of uncertainty in this process, based on the reasoning from Keller's article. Which side a coin falls on depends on its flight time 3r33595. 3r3-3597. and from the angular velocity

3r3-3597. . If you measure the angular velocity in revolutions per unit of time, then the number of revolutions performed by a coin expresses extremely simply 3r3-3595. 3r3-3597. . This dependence sets the lines of equal speed in coordinates 3r33595. 3r3-3597. , and they, in turn, limit the areas corresponding to even and odd number of turns.

3r3618.

3r3618.

3r31414. 3r33584. 3r33585. Chart showing the parity of the number of turns made by a coin in flight. The rectangle shows the area in which the process of fortune-telling on a coin occurs most often. 3r33586. 3r33587.

3r3618.

3r3618. On such a chart, it is possible to show how the result of a coin flip, twisted by a known number of revolutions per second, and caught after a known flip time, will be. If we fall into a white strip, then the same side will fall out that was on top when tossing, if the orange side is the reverse. Lines of equal speed are hyperboles and it can be seen that as the number of turns increases, the alternation of areas becomes more and more frequent, and the areas themselves become thinner. The human hand is imperfect and a very small variation of the initial values overlaps many areas at once, making the outcome unpredictable. In the range of the hand (the rectangle in the diagram), an offset of 3r3-3595 is sufficient. 3r3-3597. to jump from white stripe to orange. The question remains: how does the “integrity” of a true mechanical coin follow from this construction? How to get the probability of falling heads and tails from the resulting diagram?

3r3618.

3r3618. 3r33535. Take urgent action! 3r33535.

3r3618. Let us dive a bit into such mathematics that we don’t go through in school in order to better understand what we are talking about. We talked in 3r3106. introduction of 3r3107. that mathematicians do not study numbers or geometric shapes, as it may seem after studying a school course. They work with mathematical structures (abstract algebras, semirings, fields, monoids, topological spaces, and other abstract things), describe them, as it seems, completely unattached to practice, define them, study their properties, prove theorems. And then they hone their skills in the search for such structures in the most diverse areas of knowledge, making surprisingly useful breakthroughs, including in purely applied industries. We will now deal a little with such mathematics and consider how the basis of the theory of probability is built, based on a very abstract notion of measure.

3r3618.

3r3618. We described the mechanics of the coin and obtained regions describing solution sets with certain properties. Areas are flat figures, how to move from them to probabilities correctly? We need to measure our areas and we naturally come to their area. The area is 3r33584. measure 3r3587. flat figure. This is an exact mathematical term for a function that associates a set of non-negative number values. Examples of measures are 3r3–3584. quantities in enumerable sets 3r33587. (the number of apples in the bag, for example), as well as 3r33584. length [/i] , 3r33584. square 3r33587. , 3r33584. volume figures [/i] .

3r3618.

3r3618. In mathematics, there is a whole section called 3r33584. theories of measures 3r33587. . This theory was born at the turn of the 9th – 20th centuries (Emile Borel and Henri Lebesgue stood at its source) and opened up wide possibilities for mathematicians to analyze very complex objects: cantor and fractal sets. It formed the basis of functional analysis and modern probability theory, the foundation of which was laid by the remarkable Russian mathematician Andrei Kolmogorov. The definition of probability, as a measure, allows you to see all the basic properties of probability for both discrete and continuous sets.

3r3618.

3r3618. Although our book is not a textbook, but it is worthwhile to dwell a bit on this to take a look at the concepts of the theory of probabilities, as it were, from “bird's-eye view” and get a taste of mathematics. To begin with, we list the basic properties of 3r3353584. any 3r33587. measures. In order to better imagine them, you can use instead of the word “measure” the words “quantity” or “length” or “square”.

3r3618. 3r33555. 1. The measure of the empty set is zero.

3r3618. 2. The measure of the entire measurable set is finite for finite measures.

3r3618. 3. The measure of the subset does not exceed the measure of the set

3r3618. 4. The measure of the union of two arbitrary sets is equal to the sum of the measures of these sets minus the measure of their intersection (additivity).

3r3618. 5. The measure of the complement of a subset is equal to the difference of the measures of the whole set and the measure of the subset. 3r33558. Can any non-negative numeric function be a measure? Not at all. For example, age puts a certain number in correspondence with a person. But the age of two people cannot be defined as the sum of their ages. And the speed of running is not a measure - two people do not run twice as fast. But the impulse (the amount of movement) or energy already has the properties of a measure. The weight, the amount of money, the volume of knowledge, the loudness of the cry, though not always easily measurable things, can also serve as a measure on many people.

3r3618.

3r3618. At the intuitive level, the concept of probability is familiar now, practically, everything. It is assessed by political scientists and journalists on a talk show, it is discussed referring to global warming or tomorrow's rain, jokes about it: 3r-5384. 3r33560. What is the probability to meet on the Tver dinosaur? - One second: either a meeting or not. 3r33586. 3r33587. In modern mathematics, the concept of 3r33584. probability of [/i] is defined as a measure on a special set called 3r3-3584. probability space [/i] . It includes both elementary events and their combinations, which are obtained using join, intersection and exclusion operations. An example of an elementary event: “the loss of a troika when throwing a dice”. An example of an event that is not elementary: "the loss of any even number except two." So, we list the properties of the probability:

3r3618. 3r33555. 1. The probability of an impossible event is zero.

3r3618. 2. The probability for the entire probability space is equal to one.

3r3618. 3. If one event entails another, then the probability of the second does not exceed the probability of the first. The relation “entails” for events is equivalent to “is a subset” for sets.

3r3618. 4. The probability of occurrence of at least one of two arbitrary events is equal to the sum of the probabilities of each of these events, minus the probability that events will happen simultaneously.

3r3618. 5. The probability of non-occurrence of an event is one minus the probability of an event occurring. 3r33558. Look closely at the properties of measures and probabilities and it will be seen that we are talking about the same properties.

3r3618.

3r3618. Not all properties of probability follow from its definition, as measures: the concepts of independence of events and the method of calculating the probability of two or more independent events simultaneously as products from probabilities are introduced through 3r3584. conditional probability 3r33587. , but these concepts are consistent with the Kolmogorov definition.

3r3618.

3r3618. Discrete random variables correspond to finite countable sets, in them a natural measure is an ordinary count of the number of elements. Accordingly, the probability in a discrete probabilistic space is the combinatorial counting of variants familiar to each student. For continuous random variables, probability, as a measure, is more like a length or an area, and here we are talking about r3r3584. probability densities of 3r3587. .

3r3618.

3r3618. The analogy of probability with the measure does not end there. What is 3r33584. the average is 3r33587. ? This is an analogue of r3r3584. the position of the center of mass [/i] figures consisting of point masses or a solid body with a known density. And these values are calculated equally. And how is the variation of random variables around the average characterized: 3r3353584. dispersion 3r33587. ? Also as 3r33584. moment of inertia [/i] characterizes the distribution of mass around the center of mass. And again, the formulas for calculating the variance for a sample or distribution coincide with the formulas for the moment of inertia of a set of bodies or a solid body of a cunning form.

3r3618.

3r3618. By the way, if we replace the sum in “definitions” and properties of probability with “maximum” and product with “minimum”, then we can construct an alternative theory, it is called 3r384. theory of opportunity 3-333587. . This is how math works. We start with abstract reasoning: numbers form an algebra with operations of addition and multiplication, but on a limited number interval one can construct a similar algebra with operations minimum and maximum. We build the concept of measure on a new algebra and find out that it opens up a new view of the world! In contrast to probability theory, in such a theory, it is possible to construct two coordinated measures - 3r-3584. opportunity 3r33587. and 3r?384. the need for 3r33587. and, unlike probability, they are in good agreement with both the operations of combining and intersection of events. This direction, created by the American Lotfi Zade, an Azerbaijani by origin, serves as a basis for 3-33584. Fuzzy Logic [/i] and is used in systems of automatic pattern recognition and decision making.

3r3618.

3r3618. 3r33535. Unbelievable, but it is a fact! 3r33535.

3r3618. The first property of measures seems trivial, but it is interesting for its asymmetry. If the measure of a subset is zero, it does not mean that it is empty! For example, a line is a subset of points of a plane, but its area (measure) is zero. There are also more exotic examples - Cantor and fractal sets, having a complex structure, containing an infinite number of points, visibly "occupying" a certain area or volume, but nonetheless having a zero measure.

3r3618.

3r3618. 3r3618.

3r3618. While preparing this illustration, I found a wonderful image of a disconnected Julia set on a transparent background with high resolution. Having inserted it into the vector editor, I faced an amusing difficulty - it was very difficult to get a mouse to this image in order to select it. It is so “loose” that the probability of hitting the filled pixel was noticeably less than hitting the transparent background. In probability space, subsets of zero measure may also exist, but this does not mean that events from these subsets are impossible. From the fourth to fifth attempts, I still could select the image, since the pixels have a finite size. But what would happen if I got at my disposal a real incoherent Julia set with infinite resolution?

3r3618.

3r3618. Imagine that you are using a software random number generator, which produces an arbitrary real number from 3-335959. 3r3-3597. to 3r?595. 3r3-3597. . What is the probability of dropping the number 3r33595. 3r3-3597. ? and the numbers 3r359595. 3r3-3597. or 3r39595. 3r3-3597. ? In all these cases, the answer will be - zero! Rather, the smallest positive number available to the computer, the so-called machine epsilon, because the computerter operates with a finite number of decimal places. Wait, you say, in what sense - zero? These same numbers are not impossible. Let's conduct an experiment, as a result we will get some specific number and when we get it, then “by construction” the probability of its occurrence cannot be zero. That's right, but how long do you need to wait until exactly 0 drops out? Almost infinite! The fact is that a single number, like a point on an interval, has zero measure and an honest zero probability. Only the measure of a continuous segment, even a very small one, is nonzero. So we are not talking about probability, but about probability density, which, when multiplied by a finite measure of a subset in a probability space, will give a finite value — the probability to fall into this subset. By the way, if we had the perfect random number generator with infinite accuracy, the probability of obtaining any rational number with it will also be zero.

3r3618.

3r3618. If someone patiently conducts a thousand experiments with a coin and happily tells you that he got as many “eagles” as he did “solved”, feel free to doubt or congratulate him with rare luck. Although throwing a coin and a discrete random process, but as statistics accumulate, the power of the probabilistic space will increase and the measure of the event: “the number of eagles coincides with the number of grids” will decrease. It can be shown, using the Stirling formula, that the probability of this “most probable” event tends to zero as the number of trials rises to 3-33595. 3r3-3597. . For a hundred throws, this is a little more than five percent, for ten thousand - just half a percent. In such cases, the mathematicians say: 3r?384. almost certainly the number of "eagles" will not be equal to the number of "tails" 3r-3587. . Strange as it may sound, “almost certainly” is an exact mathematical term meaning that an event is an addition to a subset of a zero-measure probability space. We will return to these arguments, in one of the following chapters, when we ask ourselves: how much each of us can consider ourselves to be normal.

3r3618.

3r3618. 3r33535. Check the honesty of the real coin

3r3618. Let's return to the coin and to her honesty. The Kolmogorov definition of probability reconciled its frequency definition (as the relative frequency of the events) and the geometric (as a fraction of the "volume" of the event in the total "volume" of possibilities). Thus, the fraction of the area of white stripes on the chart calculated for a rotating coin reflects the probability of falling out of the same side with which we planted it.

3r3618.

3r3618. But the trouble! The area of each strip in our diagram is infinite (if we look at the whole quarter of the coordinate plane). However, the additivity of the measure will allow us to accurately show that this does not prevent the areas of shaded and white areas to be the same. In explicit form, the equations for our curves are of the form 3r359595. 3r3-3597. . If the area under the curve is 3r39595. 3r3-3597. equal to r3r3595. 3r3-3597. then due to the additivity property, the area under the curve is 3r3359595. 3r3-3597. will be equal to 3r39595. 3r3-3597. . In turn, for individual strips we get: 3r33595. 3r3-3597. . It turns out that the area difference does not depend on the “number” of the hyperbola. This is not something special relating to hyperbolas; the same conclusion can be made for any curve like 3r33595. 3r3-3597. , if only function 3r33595. 3r3-3597. was measurable. And if this is the case, then for the whole domain of definition, hitting the white part of the diagram or the shaded one is equally likely, as is expected for an "honest" coin.

3r3618.

3r3618. The arguments we have just led seem simple enough, but they give a very general result, applicable to any additive quantities. The abstract concept of measure allowed us to compare among themselves infinite values, remaining within the framework of logic and common sense.

3r3618.

3r3618. Abstractions are good, but it can be argued that in reality we do not throw up coins with all the possible parameters. As shown by experiments with a high-speed camera, the angular velocities fall in the range of 3r-3595. 3r3-3597. to 3r?595. 3r33333. 3r3-3597. revolutions per second, and the duration of the flight - from half to one second. This area is highlighted by a rectangle in the diagram. In it, the total area of the white stripes is slightly larger than the orange ones, and it can be concluded that the probability of hitting the same side as when throwing up will be 3r33595. 3r3304. 3r3-3597. .

3r3618.

3r3618. In 200? a group of Percy Diaconis and co-authors from Stanford published 3-333310. article

In which the detailed analysis of the process of coin toss is given. A detailed description of the mechanics of a flying and rotating disk, which not only rotates, but also precesses - the axis of rotation rotates itself in flight, shows that with manual throwing from the eagle position from above, the probability of an eagle being dropped is one-hundredth more than half.

3r3618.

3r3618. Is it a lot or a little? How many experiments are needed to notice such a difference? With the accumulation of experimental data, the standard error of the mean, reflecting the error with which the average value can be calculated, decreases in proportion to the square root of the number of tests: 3r359595. 3r3333317. 3r3-3597. , here is 3r39595. 3r3-3597. - standard deviation for the studied distribution. In our case, for the Bernoulli distribution with probability 3r359595. 3r3-3597. which is equal to 3r39595. 3r3-3597. . In order to confidently single out one hundredth average deviation, this deviation must exceed 3r-3595. 3r3-3597. standard deviations. Thus, we can estimate the number of trials: 3r3 -9494. 3r3-3598. 3r? 3594. 3r33595. 3r33333. 3r3-3597. 3r3-3598. So many times you need to flip a coin to notice the mechanistic predestination of the result. To make it clearer what is meant, I will give an example of two hundred trials of an ideal and slightly non-ideal “coins” conducted in order to calculate the probability of a loss of, say, an eagle. Each test consists of 3r33595. 3r3-3597. "Toss". The words "coin" and "toss" are quoted, because in fact it was not a physical coin that was used, but a random number generator obeying the Bernoulli distribution.

3r3618.

3r3618.

3r31414. 3r33584. 3r33585. Experimenting with the flip up of an ideal and slightly non-ideal coin in order to fix the non-ideality. 3r33586. 3r33587.

3r3618.

3r3618. It is seen that only after

3r33357. 3r3-3597. The “cloud” of observed mean values begin to be clearly divided. Well, for household use, we can assume that the coin is a good generator of a random choice of two equally probable options.

3r3618.

3r3618. 3r33535. Tourist Law 3r3-3030.

3r3618. The equivalence of the geometric and frequency definition of probability reveals the riddle of a single meanness law, known among tourists, geologists, and all those that use topographic maps: 3r3605. 3r3618. 3r33555. The place where the tourist goes is most often either on the bend of the map or on the edge of the sheet. 3r33558. Suppose that we are equally often interested in objects located in all parts of the map. But we are rarely interested in objects of zero measure - the whole point of using the map is to review 3r-3584. surroundings 3r33587. object, that is, some final area. Let it be enough for us to have a small fraction of 3r3-3595. 3r33334. 3r3-3597. from the area of the whole map

3r3-3597. to figure out how to get to the object. This means that if an object approaches a bend or edge at some critical distance of 3r-3595. 3r3-3597. , we find the law of the tourist fulfilled. The share of border areas in the total area of the map will give us the chance to try this law of meanness for ourselves. This is how the unpleasant parts of the map look like at 3r3-3595. 3r3-3597. and one fold.

3r3618.

3r3618.

3r33333. 3r31414. 3r33584. 3r33585. Gray marked "bad" areas. A section with a half-percentage area for a 40 cm wide map is shown separately; it has a diameter slightly larger than 3 cm. 3r38686. 3r33587.

3r3618.

3r3618. For the square card

3r3013. 3r3-3597. . Unpleasant stripes will have an area of 3r33595. 3r3404. 3r3-3597. . Four lanes, two vertical and two horizontal, will be located at the edge, any additional bend, horizontal or vertical, will add another strip. In this case, the intersecting stripes add extra squares of 3r33595. 3r3407. 3r3-3597. . Folding the map so that it turns out

3r33410. 3r3-3597. horizontal and 3r39595. 3r3-3597. vertical bends, we get the total area of the unpleasant zone equal to: 3r3-3595. 3r31616. 3r3-3597. . Taking it to the area of the whole map, we get an unpleasant share of the total area:

3r3618. 3r? 3594. 3r3-3598. 3r? 3594. 3r33595. 3r33424. 3r3-3597. 3r3-3598.

3r3618. The figure shows the areas in which this proportion exceeds r3-33595. 3r33430. 3r3-3597. for different values of r3r3595. 3r33334. 3r3-3597. .

3r3618.

3r3618.

3r33440. 3r31414. 3r33584. 3r33585. Areas where there is an increased likelihood of being on the bend of a card or on its edge. The numbers indicate the proportion of the area of the considered neighborhood from the area of the entire map. 3r33586. 3r33587.

3r3618.

3r3618. It turns out that the map, folded in half, twice already can formally be considered dishonest in relation to the tourist. Most often, the cards have three vertical and three horizontal folds, which makes it possible to fulfill the law of meanness with a probability of about 3r359595. 3r3-3597. at 3r33595. 3r3-3597.

3r3618.

3r3618. 3r33535. Where does the accident come from? 3r33535.

3r3618. 3r33434. In the souvenir shops you can find magnetic pendulums for "choice of desires." They, too, are mechanical generators of chance and are sometimes mistakenly called "chaotic pendulums." Starting from some initial position and speed, the pendulum makes a series of “unpredictable” oscillations and finally stops in one of the sectors. However, fluctuations are not unpredictable here, they are just very sensitive to the initial conditions. For each sector in which the pendulum can stay, there is 3r-3584. region of gravity 3r33587. in coordinate-speed space. This is a set of such initial conditions under which the pendulum will necessarily be attracted to a certain point in the specified sector. The stopping point of the pendulum is called

*Attractor*- attracting point. In the case of a pendulum from a picture, the space of coordinates and velocities is four-dimensional, and so simply the regions of attraction cannot be shown. But if we confine ourselves to only two sectors and reduce the task to a one-dimensional one (such a pendulum is called the Duffing oscillator), then the space of initial values will turn into a plane, so that the regions of attraction can be seen. They look like an intricate Yin-Yang symbol, quickly turning into narrow strips separating areas of attraction.3r3618.

3r3618.

3r37474. 3r31414. 3r33584. 3r33585. The regions of attraction of attractors for the one-dimensional pendulum of desires — the Duffing oscillator. 3r33586. 3r33587.

3r3618.

3r3618. As in the case of the coin, slightly shifting the initial conditions, we get from one attractor to another. The dice and the roulette also act in the same way, but they are not in themselves generators of randomness. These are not truly chaotic systems and their behavior can be accurately calculated.

3r3618.

3r3618. And what is the real accident? A good example of a truly stochastic system is the appearance of cars on the road. People do not agree, do not coordinate their plans, each element of the ensemble outside the road acts independently. And although there are certain patterns in people's behavior - peak hours in the morning and in the evening, empty roads at night, etc., we do not have and will never have enough information about each participant in the movement to predict the appearance of any of them. Also stochastic are the mechanics of elementary particles at the quantum level, the decay of unstable atoms, changes in the genetic code, apparently, earthquakes and quotations of securities on the stock exchange. The only thing that remains for the researcher is to consider them as random variables and describe them in terms of probability theory.

3r3618.

3r3618. But there is another source of accidents - 3r-3584. dynamic chaos [/i] . Chaotic systems differ from stochastic in that they are described by exact equations and parameters that do not contain randomness. However, their behavior is not just difficult, but chaotic and truly unpredictable. If we begin to oscillate the pendulum of desires, very carefully, with precisely controlled frequency and amplitude, then we will find that its smooth movements cannot be calculated for a long time. No algorithms on arbitrarily exact computing machines will be able to calculate the exact behavior of the pendulum for an arbitrarily distant future. It will not stop at any sector, but will make smooth movements, but will never return to the same point in the coordinate-speed space twice. Another example of an extremely simple chaotic system is an ideal ball bouncing in a gravity field on an ideal table with a spring. The relatively simple Lorentz equations showed that we can never predict the weather for more than a couple of weeks - this is also a chaotic system.

3r3618.

3r3618. The theory of dynamic chaos, was able to explain the nature of such unpredictability. The simple one-dimensional pendulum of desires, which we considered, had two stable stationary points — two attractors, and one unstable, from which the system tries to escape, it is shown in white circle. In the chaotic mode, instead of a set of attractors, an infinite set of unstable stationary trajectories appears in the system. This set is infinite, but has

*zero measure*, and is a very complex disjointed structure. Once on one of these trajectories, in principle, it is impossible to follow it using any finite algorithms. But the most surprising thing was that this infinite number of unstable trajectories is itself attracting! 3r?500. The chaotic system continuously jumps from the neighborhood of one unstable trajectory to another, all the while remaining within this strange attractor. So these sets are called: 3r3r-3584. strange attractors [/i] . This is how fascinating the cross section looks like with the plane of a strange attractor for a pendulum of desires subject to harmonic oscillations. This object for a one-dimensional pendulum can be described in three-dimensional space (coordinate, velocity, phase of the forced oscillation). If you cut the attractor in this space with a plane, then you can see its structure, it is called 3r-3584. Poincare section [/i] . Each point here is a trace of the trajectory, and the color of the points reflects the relative speed at which the trajectories diverge from each other. Here are a couple of beautiful strange attractors:3r3618.

3r3618. 3r? 3510.

3r3618. 3r33584. 3r33585. Left: Poincare section for the trajectory of a ball bouncing on a spring-loaded table. The set of points belongs to the surface of the sphere corresponding to the law of conservation of energy. Right: a volumetric region that contains a strange attractor, which is born when forced toOlebaniye thick plate. 3r33586. 3r33587.

3r3618.

3r3618. The smoothness of the chaotic trajectory allows you to still look a bit into the future. This explains one annoying observation: on the one hand, weather forecasters, at times, cannot confidently predict the weather for a week, but on the other hand, if you say that tomorrow will be the same weather as today, you will not be mistaken in about three cases out of four . In general, the jokes about weather forecasters are unfair and we must pay tribute to human thought and perseverance, which allowed us to predict the weather at the present level!

3r3618.

3r3618. Dynamic chaos is very complex and beautiful as a theory, it generates images of amazing elegance, but it can also be useful. For example, the algorithms by which random numbers are generated in computers are also determined. For the examples in this book, I used a pseudo-random number generator, which did not start the real stochastic process (alpha decay, or counting machines on the road), but calculated the following “random” number based on the previous ones it had previously received.

3r3618.

3r3618. 3r33535. From coins to butterflies and the very fate of r3r3530.

3r3618. Observing how small deviations grow into global changes in the system lead to the thought of the “butterfly effect”. Let me remind you that this effect implies a chain of far-reaching dramatic consequences from some seemingly insignificant event. The butterfly crushed by researchers of the past in Ray Bradbury's story “And Thunder Has Been Boiled” led to a radical restructuring of the future. And Edward Lorenz, the creator of the theory of dynamic chaos, entitled one of his lectures: “Can the flap of a butterfly wing in Brazil cause a tornado in Texas?” 3r3605. 3r3618.

3r3618. We implicitly refer to this effect, complaining: “If I didn’t turn the corner, everything would be different!”, “If he didn’t get on this train, there would have been no catastrophe with him!” Or “Because of such trifles, they quarreled and gone !! ”But we see that truly stochastic quantum world and ultra-precise atomic clocks, stable Hamiltonian systems in the world of stars and galaxies and chaos of Saturn’s or Kuiper’s belts, thermal movement of molecules and amazing accuracy of biological systems or automobile mechanisms coexist . No, the flap of a butterfly's wing does not generate hurricanes, but disappears without a trace, generating a chain of whirlwinds that transmit energy and information to more and more smaller whirlwinds, as long as both energy and information do not disappear in the chaos of fluctuations. It should be clearly understood that small deviations lead to a radical restructuring of the system only if it is unstable or if the system is on the threshold of 3r3584. bifurcations of r3r3587. or 3r33584. disaster [/i] - so in the language of mathematics are called global restructuring in the behavior of the system with small changes in parameters. And bifurcations always form sets of zero measure in the parameter space — these are points or boundaries. Small perturbations do not lead to catastrophes 3r3584. [/i] almost everywhere. , (this is also an exact term meaning “everywhere, except for a set of zero measure”), and unstable states in nature are rarely observed without passing the “test of time”.

3r3618.

3r3618. If the couple broke up "because of the nonsense," she was destined to break up in any case, she was unstable. Sustainable couples go through wars and famine, and then, sometimes, break up, not because of small things, but as a result of profound changes that can happen to a person during life. In the chain of events that led to the train crash, it is not easy to single out the key event (a specific error or fatal accident) and, most likely, the key will not be an event, but a systematic violation of the rules, leading the system to an unstable state. If there are a lot of parameters in the system, and a number of them are random, and our life is arranged exactly this way, then the information in such a system tends to be lost, and it’s impossible to restore at any particular moment in our life “everything went wrong”. Do not torment yourself with regrets about what happened, but look at what is happening to you now, so as not to miss the real point of bifurcation.

3r3618.

3r3618. In this regard, we can recall one of the laws of merphology, called a certain Dreisen 3r3-3511. Restoration Act 3r33552. :

3r3618. 3r33555. Time to improve the situation is inversely proportional to the time of its deterioration.

3r3618. 3r33558. The following observation is given as an example: 3r33584. 3r33560. It takes more time to glue a vase than to break it. 3r33586. 3r33587. This law surprisingly accurately describes the relationship between the characteristic rates for the 3r33584 process. relaxation 3r33587. a stable system that can be described by the decreasing exponential law 3-33595. 3r3566. 3r3-3597. and the speed of development 3r33584. catastrophic 3r33587. process in an unstable system, in the linear approximation, an exponential growth of a small perturbation 3r33571. 3r3-3597. . These speeds, indeed, are inversely proportional to each other. The example with a vaza, however, is not a relaxation - a transition to the most likely state. He is closer to another process - to 3r33584. self organization 3r33587. This process, as a first approximation, is described by 3r33584. logistics 3r33587. by law and closer in speed to relaxation than to catastrophe.

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3r3618.

3r33582. 3r31414. 3r33584. 3r33585. Typical non-stationary processes: catastrophe, relaxation and self-organization, having the same characteristic time. 3r33586. 3r33587.

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3r3618. 3r? 3594. 3r3-3598. 3r? 3594. 3r33595. 3r3-3596. 3r3-3597. 3r3-3598.

3r3618.

3r3618. Sometimes, walking in the snow, I am surprised that the snowflake falls on my nose. I am surprised that the probability of this event was negligible. If judged, she was born high in the sky over the Pacific Ocean, circling in indiscriminate turbulent flows in a cloud, falling continuously changing direction of movement to get to the tip of my nose! And what a stunning path photons passed from a distant star !? Tens of thousands of years they swept through the universe, they were not absorbed by the dust they did not meet an asteroid! They were born in the quantum world of a distant star, and ended up in the quantum world of the protein opsin on the retina in my eye. Even to consider the probability of this event makes no sense, it is zero, but an event happens, and I see the flickering light of a star. Now it is clear that this is all because the area of my nose and even the molecules have a non-zero measure, but it is still surprising: what almost certainly should not have happened still happens!

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3r3618. Philosophers may argue about the predestination or chance of fate, about the truth or illusiveness of our knowledge of nature. I urge the reader to look at the world from the height of mathematical abstractions and admire its beauty and consistency. 3r31414. 3r3618. 3r3618. 3r3618. 3r3611. ! function (e) {function t (t, n) {if (! (n in e)) {for (var r, a = e.document, i = a.scripts, o = i.length; o-- ;) if (-1! == i[o].src.indexOf (t)) {r = i[o]; break} if (! r) {r = a.createElement ("script"), r.type = "text /jаvascript", r.async =! ? r.defer =! ? r.src = t, r.charset = "UTF-8"; var d = function () {var e = a.getElementsByTagName ("script")[0]; e. ): d ()}}} t ("//mediator.mail.ru/script/2820404/"""_mediator") () (); 3r3612. 3r3618. 3r31414. 3r3618. 3r3618. 3r3618. 3r3618.