Object, property, activity: models and ways to build them

This article combines the results we obtained in previous articles and brings the theoretical reasoning made in them to a practical level. I introduced enough terms to look at the concept of a property and explain how to build a property model. This article can be read independently of others, therefore I will repeat a part of the reasoning made earlier, I will part a part, and I will add a part.


Introduction


Those mathematicians or physicists who are starting to study business analysis have a hard time. There is a huge difference between basic science and those practices that are set out in different standards for business analysis. Attempts are being made periodically to familiarize the business analysts community with the point of view of modern philosophers, but such attempts have so far failed. Because of this, a mathematician or physicist, plunging into the study of business analysis standards, is experiencing a light shock. I will try to bridge the gap between what the physicist or mathematician used to work with and the models that analysts are building.


For this, I formulated a body of knowledge, which I called projection modeling, because the method outlined in it is reminiscent of drawing. In drawing lessons, we learn to model spaces. At the same time, the space model is separated from the interpretation of this space. Depending on the problem being solved, the simulated space can be interpreted as a piece of aluminum, and as part of the water, and as an airplane wing: the space model does not depend on its interpretation. In projection modeling, we do the same: first we create a model of space, but already in time, because our world is four-dimensional, if we consider time as a separate dimension, and then we treat this space-time in one way or another. Just as in drawing, the simulated 3-D volume can be interpreted in different ways,


For example, one subject may treat 4-D volume as a car, another subject may treat this same 4-D volume as a piece of iron, the other - as a function of transporting passengers. The only difference from drawing is that the model of space and time is more complicated than the model of space. Therefore, modeling tools must also be more difficult. As a result, the model of our ideas turns into a two-level model:


  1. At the first level, a model of space-time parts and the relations between them is built.
  2. At the second level, a model of subjective atomic representations and the relations between them is built.

Why do you need to simulate space-time?


Case 1


Suppose that two different people were asked to tell about one event. One said: a hammer hit a nail head, another said: a nail hit a hammer. They told about the same event, but from different points of view.


However, what is an event if you can look at it from different sides? Hammer banged on the nail head - this event? No, because this is a story about an event from one point of view, or, more simply, an interpretation of an event. If this is not an event, but an interpretation of an event, then what then is an event?


Case 2


Suppose that two different people were asked to describe the same object. One said: this is a car, another said: this is a boat. They told about the same object, but from different points of view.


However, what is an object if you can look at it from different sides? Is the machine an object? No, because this is a story about an object from one point of view, or, more simply, an interpretation of an object. If it is not an object, but an interpretation of an object, then what then is an object?


Explanation


Both of these cases are united by one thing: the impossibility to express a thought correctly in words. Hammer hit the cap - this event. A machine is an object. And this is hard to argue. But what then do the subjects treat as an event and what exactly do the subjects treat as an object? What exactly are they looking at when they make their interpretations? They perceive the same space-time volume, and in this they agree. In cinema, actors often ask the question: are we seeing the same thing? This question means: are we now looking at the same space and interpreting it in the same way?


It is correct to say that there are two different interpretations of the space-time volume. This explanation is correct and accurate. If we do not understand this, our reasoning will be like a snake biting its tail. That is why, if we want to build a model of interpretations, we must begin it with a model of what we see - with a model of space and time, and only then endow this model with different interpretations.


Examples of two-level models


Example 1


Pick up a ball of aluminum. You see a rough surface, you feel the weight and see the shape of the ball. To create a model of such a view, you need:


  1. Build a model of space, which can then be interpreted as a matte surface
  2. Give the interpretation of this space as a matte surface.
  3. Build a model of space, which can then be interpreted as a piece of aluminum
  4. Give the interpretation of this space as a piece of aluminum
  5. Build a space model, which can then be interpreted as a ball shape
  6. Give the interpretation of this space as a form of a ball
  7. Identify the relationship between the three spaces, treated as a rough surface, as a piece of aluminum and as a form of a ball. I would suggest this:
    1. The space treated as a rough surface is the boundary of the surface treated as a piece of aluminum.
    2. The space treated as a sphere is the idealized boundary of a surface treated as a piece of aluminum.
  8. Specify the relationship between the three interpretations of three different spaces. I would suggest such:
    1. A piece of aluminum has a surface, an idealized representation of which looks like a sphere
    2. A piece of aluminum has a rough surface.
    3. A piece of aluminum has a weight

Example 2


You look at the stage and see a dancer dancing a dance. To create a model of such a view, you need:


  1. Build a model of space-time, which can then be interpreted as a dancer
  2. Give the interpretation of this space-time as a dancer
  3. Build a model of space-time, which can then be interpreted as a dance
  4. Give the interpretation of this space-time as a dance
  5. Identify the relationship between two space-times, treated as a dancer and as a dance. I would suggest this:
    1. The space-time interpreted as a dancer coincides with the space-time interpreted as a dance.
  6. Specify the relationship between two interpretations of two different space-times. I would suggest this:
    1. Dancer dancing dance

The relationship between the space-time model and its interpretation


The space-time model depends on how it will be interpreted later. Such a model is created for its specific interpretation, or, in other words, for a specific type of property. Two different types of properties will produce different models of space-time volumes, even when it may seem that these volumes are the same. For example, the sphere in the case of a piece of aluminum is an idealization of a real form and differs from its real surface. Therefore, building a model of space based on the statement that the piece has the shape of a ball, we obtain a surface that is different from the real surface of the piece.


The main thing is not to confuse the property and the type of properties. For example, a white car and a white steamer are different properties, different "white" ones. For one property there will be one model of space-time, for another - another. Combines their property type "white". As a rule, we are not able to distinguish a property from a property type. This is one of the problems of the language: the language does not allow us to do this. But in projection modeling, this differentiation of meanings must be realized very clearly by the analyst. Do not confuse the property and its type. This means that the white steamer and the white car will have in common not the properties, as we used to think, but the type of properties. Properties will be different. This means that one white is not at all the same as another white! These whites are different shades, shapes, position in space and time.


Therefore, the space-time model that we will build will be related to the type of properties that gave rise to it. We call this type of properties generic for both space-time and its model.


The generic type of space-time properties is such a type of properties, on the basis of which a given space-time was selected from the total space-time volume.


The generic type of properties of a space-time model is such a type of properties, on the basis of which a given space-time was selected from the total space-time volume and its model was built.


Property model


We came to the conclusion that any type of properties can become generic for a space-time volume and its model. The property model is the space-time volume model, for which the type of the modeled property acted as a generic one. Therefore, if there is a “white” property, the type of this “white” property acts as a generic for some space-time volume and its model.


Name of space-time volumes


To denote space-time volumes we use the name of the generic property. And, since all properties are grouped into property types, the name of the type of generic properties becomes the name of the volume. For example, the type of generic properties “white” becomes the name for the “white” property, which we associated with the steamer, and for the other “white” property, which we associated with the machine. These are different properties and therefore must have different names, for example, "white # 123", or "white # 124". The analogy with the machines: the machine with the number # 123 and the machine with the number # 234 are different space-time parts, which we treat in the same way as machines. Similarly, "white # 123" and "white # 124" are different whites, which we treat in the same way as white. The same applies to the property "


Representation of space and time


To build a model of space-time, you first need to figure out what space-time is. Let us briefly repeat the theses of previous articles and formulate them in a formal way. At the same time, I apologize for the mistakes I made. This is especially true of the spikes, which led me to the indistinguishability of the property and the type of property.


Usually, a story about space and time begins with a story about space, and then they say that time is a change in that space. Why is space considered out of time? Because we easily imagine a frozen space in time: it is a slice of the space-time volume across time and consideration of this slice. But we do not understand what is frozen time in space. If we want to cut across the space in order to study time, we must select a point, line, or surface and consider its dynamics over time. Logically, if the cut across time is called space, then the cut across space should be called time. Agree unusual?


Both space and its changes are different points of view on the same area of ​​the studied space-time, but according to the rules of the language, the changes should be tied to the space, but the space to changes should not be. We cannot tell about changes without space, but about space without changes, supposedly, we can. In fact, we always look at changes in space, even when we think that nothing changes. Just sometimes we believe that these changes can be neglected. In order not to be confused, I will speak about space, having in view of its changes, which were insignificant within the framework of the task we are solving.


We introduce the term "dance of space", or simply "dance" as a synonym for space-time. If I say simply "space", I will mean the dance of space, the changes in which are minor.


In any dance chosen for modeling, there is a minimum spatial resolution (atomic point), a maximum spatial volume (volume of the studied space), a minimum temporal resolution (atomic instant), and a maximum time interval (the amount of time studied).


What dance can be given meaning?


Suppose that you have lost the ability to see part of the space. This can be imagined because each of us has a blind spot. You can become aware of it through special exercises, but then you adapt again and cease to realize it. This is because our consciousness is able to smooth the visible image. Our consciousness does not work with a picture, but with a spline - functions that smooth it out. The same thing with time. If you show the 25th frame, you will not notice it. Therefore, we make the following statement:


Only a dance that is continuous, or, equivalently, homogeneous, can be given the property.


The question arises: how to build a model of a uniform dance, so that later it can be given meaning, or interpreted?


First we need to formally define the concept of continuity for dance. The first thing that comes to mind is to remember the definition of continuity from mathematical analysis: continuity is when at two close points the attribute values ​​differ slightly. Everything seems logical and beautiful, but the question arises, what is the point?


For example, you are holding a crystal. What is a point on its surface? You can say that a dot is an atom. But, if you argue about the color of the crystal, then the atom does not possess color. Color has a surface of a huge number of atoms. Suppose for this they must be a million. This means that the point on the surface of the crystal, which has color, contains a million atoms. From this it follows that the points may intersect, because neighboring points may have common atoms. It turns out that the definition of the mathematical analysis does not suit us.


Formal definition of homogeneous space


Take a homogeneous space endowed with a property. Divide it into parts. The properties of each of the parts of this space will be similar to the property of the entire homogeneous space (any part of the crystal surface is similar to the entire surface). Whatever part of a homogeneous space we take, the properties of this part are similar to the properties of another part of this space and to the properties of the entire space as a whole. This will be the basis for the formal definition of a homogeneous space.


The homogeneous space for a given type of property is the set of all possible parts of the space, for each of which a property of a given type is defined.


The model of a homogeneous space looks quite impressive: for this you need to consider all possible parts of it, and there may be a lot of them.


If we consider parts of a homogeneous space, directing their size to zero, at some point we will come to the limit beyond which the obtained parts can no longer be endowed with generic properties. This means that there is a limit to the partitioning of space. This limit determines the size of the points of homogeneity of the space endowed with the property. The resolution of the instruments may allow us to see the structure of the points of uniformity. The very same point of homogeneity can not be seen, because its borders intersect with the borders of other points of homogeneity. It can only be imagined.


If the size of the point of homogeneity is less than the size of the atomic point of the studied space, we observe an absolutely smooth space. If the size of the point of homogeneity is greater than the size of the atomic point of the studied space, we see it as a rough space.


Let me explain by example. Consider the surface of the carpet. He is fleecy, we see every pile. Any part of the carpet is also fleecy and looks like any other part of it. We will reduce the size of the parts. At some point in one part of the carpet there will be only one villus. Is it possible to call such a part fleecy? No, because one pile does not have the property of fleecy. Therefore, the size of the homogeneity point for a carpet is larger than the size of the atomic point of the studied space and therefore the surface of the carpet looks rough.


An example of a homogeneous space can be found in the drawing. The shaded area in the drawing models a homogeneous space that can be treated as a substance. It is modeled using all the possible parts that can be obtained from this space. There are incredibly many such parts, they intersect, the size of the point of homogeneity is comparable to the size of a group of a billion atoms.


Knowing the generic type of space properties, one can introduce the notion of continuity: for closely spaced points of homogeneity, the attribute values ​​should also be close.


Those who are familiar with functional analysis can see that this definition of homogeneity can be interpreted differently: with the help of Fourier expansion. Then the definition of homogeneity will be:


A space that is homogeneous for a given type of property is a space that has a pronounced burst (or sets of bursts) in the spatial spectral decomposition of properties of a given type.


Depending on convenience, you can use either one or another definition of spatial homogeneity.


For example, we talked about an absolutely smooth space. To determine it, we can know nothing about the points of homogeneity. It is enough to know that the peak in the spectrum of properties of a given type in a given space corresponds to the size of the atomic point of the studied space. Peak blur gives us an idea of ​​the spread of property values. If the peak in the spectrum corresponds to a size larger than the size of the atomic point, we can see several bursts at different frequencies, proportional to the base frequency. This will give us an idea of ​​the second and third harmonics, and if we combine this knowledge with knowledge of the phases of spectral decomposition, we can conclude about the shape of the periodic structure. If there are several peaks at frequencies that are very different from each other, for example, dozens of times, this tells us that there are periodic structures, which, in turn, consist of periodic structures. That is, the stone wall consists of blocks, each of which consists of bricks. Today there are plenty of ways to analyze such spatial structures. As a result, our entire understanding of space can move into the field of spectral analysis, thereby fundamentally changing our understanding of reality. Imagine the AI, which will be flashed the idea of ​​space, based on spectral analysis. What will he see? Only waves, no space in our understanding! Interestingly, in real projects and done? By the way, can such an interpretation help us understand the essence of quantum physics? Today there are plenty of ways to analyze such spatial structures. As a result, our entire understanding of space can move into the field of spectral analysis, thereby fundamentally changing our understanding of reality. Imagine the AI, which will be flashed the idea of ​​space, based on spectral analysis. What will he see? Only waves, no space in our understanding! Interestingly, in real projects and done? By the way, can such an interpretation help us understand the essence of quantum physics? Today there are plenty of ways to analyze such spatial structures. As a result, our entire understanding of space can move into the field of spectral analysis, thereby fundamentally changing our understanding of reality. Imagine the AI, which will be flashed the idea of ​​space, based on spectral analysis. What will he see? Only waves, no space in our understanding! Interestingly, in real projects and done? By the way, can such an interpretation help us understand the essence of quantum physics? What will he see? Only waves, no space in our understanding! Interestingly, in real projects and done? By the way, can such an interpretation help us understand the essence of quantum physics? What will he see? Only waves, no space in our understanding! Interestingly, in real projects and done? By the way, can such an interpretation help us understand the essence of quantum physics?


There is a feature in our reasoning. As soon as we started talking about decomposition into a spectrum, we began to write about the mathematical apparatus of functional analysis. But the trick is that we do not have the right to do this, because the points of homogeneity have completely different properties than the points in the functional analysis. Therefore, I gave a high-quality picture without a claim for a mathematical justification.


The idea of ​​a homogeneous dance


In order to imagine a homogeneous dance, for each atomic instant of the dance studied, we will assign the studied space to correspond. A stack of such spaces for each atomic instant of the dance under study will give us a dance model. This way of presenting is similar to stills from the movie.


Why dance is considered homogeneous in time


So that we can talk about the uniformity of dance in time, the space for each pair of consecutive atomic moments should be similar to each other. If spaces from moment to moment rapidly change their size, shape or position, we cannot speak of such a series of spaces as a uniform dance.


The following statement seems obvious: if for each pair of consecutive atomic moments we can perform a comparison of two spaces, then we will assume that the dance is uniform in time. But in fact, we are now appealing to the atomic moment. We have defined the absolutely smooth dance of space. But the dance may have rough edges. How to formulate the uniformity of a rough dance without appealing to the moment?


The uniformity of dance in time


Consider the rotation of a ballerina. It consists of movements of the same type: one turn, the second turn, and so on. If we observe this rotation for a long time, we see a uniform dance. If in space the homogeneity was determined by the idea of ​​the properties of space, then the idea of ​​dance homogeneity should be based on the idea of ​​the properties of dance.


We formulate a sign of temporal homogeneity. If we observe some kind of uniform dance of space, we can divide this dance into temporary pieces (make cuts across time), and see that they resemble the entire dance as a whole. This similarity of parts makes the dance uniform in time. The set of temporal parts of a homogeneous dance, similar to each other, is a model of a homogeneous dance.


The homogeneous dance of space in time for a given type of property is the set of all possible temporary parts of the dance, for each of which a property of a given type is defined.


The star for us is a bright point, it is not uniform in space. But her light is uniform in time. Uniformity in time allows us to see this point and assign it some meaning. If we encounter something that does not have homogeneity either in space or in time, we simply will not notice it.


The model of a dance that is uniform in time looks quite impressive: for this you need to consider all of its possible temporary pieces, and there may be a lot of them.


If we reduce the temporal parts of a dance that is uniform in time, at some point we will come to the limit beyond which the parts obtained can no longer be endowed with properties similar to the property of a uniform dance. This means that there is a limit to such a partition. This limit determines the duration of instants of homogeneity of a dance endowed with a property. The resolution of the device may allow us to see the structure of the moments of uniformity. The instant of homogeneity itself cannot be seen, because its borders intersect with the boundaries of other instants of homogeneity.


If the duration of the instant of homogeneity is less than the atomic instant of the dance studied, we observe an absolutely smooth dance. If the duration of the instant of homogeneity is more than the atomic instant of the dance under study, we see it as a rough dance.


Let me explain by example. Imagine a wave of sea with standing waves and try to describe the dance of its surface. If someone does not believe that this is possible, I can assure you that such a dance exists: it is formed due to the interference of the waves reflected from different piers. This dance can be called fermenting. Any temporary part of this dance is also worrying. We will reduce the duration of observations. At some point, only one period of the wave oscillation will fit into one temporary part. Is it possible to call such a part worried? No, because one period of oscillation does not have the property of a wave surface. Therefore, the duration of the instant of homogeneity for the waves of the sea is longer than the duration of the atomic instant of the dance under study, and therefore the dance of the waves looks rough.


An example of a homogeneous dance in time can be found in music. The note on the stave simulates a time-uniform dance of a certain duration, which can be interpreted as the sound of a note. This dance is modeled using all possible lengths that can be obtained from this sound. There are many such parts, they intersect, the size of the instant of homogeneity is comparable to fractions of a second and contains up to twenty sound vibrations.


Knowing the generic type of properties of a dance homogeneous in time, one can introduce the concept of continuity of this property: for closely spaced instants of homogeneity, the values ​​of an attribute must also be close.


Those familiar with mathematics can see that this definition of dance homogeneity can be interpreted differently: with the help of Fourier expansion. Then the definition of homogeneity will be:


A homogeneous dance of space in time for a given type of property is a dance that has a pronounced splash (or multitude of bursts) in the temporal spectral decomposition of a property of a given type in time.


Depending on convenience, you can use either one or another definition of temporal homogeneity.


Uniform idea of ​​a homogeneous dance


We saw dance as a stack of spaces arranged in time. The time has come to abandon this idea and learn to see dance as a single semantic block.


Let us return to the rolling surface of the sea. The dance with which we model this surface in time is homogeneous in time and in space. Let's try to formulate it without breaking the submission into temporal and spatial.


We can divide the surface of the moving sea into parts both along time and across.


The division along gives us the result of observing some part of the sea. Imagine a helicopter hovering above sea level at some altitude. He has a sector of the review, which snatches under him some area for observation. This is an observation of what can be called a part of the dance of space, carved along time. We put a lot of helicopters at different heights. The results of observations - there are parts of the dance, cut along the time.


The division across time looks like the surface of the whole sea, observed by a single helicopter, the observation time of which is limited by the helicopter's fuel supply. We put a lot of helicopters. They will fly away and fly back. The results of the observations are parts of the dance cut across time.


Now combine these two divisions. Let's do the dance division both along time and across. This means that helicopters now begin to observe when they want, and finish when they want, have the opportunity to rise and fall above the surface, changing the field of view. But the rule is observed: the minimum observation time lasts longer than the instant of dance uniformity, the minimum viewing sector is greater than the uniformity point. The results of such observations will give us many pieces that are similar to each other.


We formulate the concept of a homogeneous space-time


Homogeneous spacetime


Homogeneous space-time for a given type of property is the set of all possible temporal and spatial parts of this space-time, for each of which a property of a given type is defined.


We can interpret this homogeneous space-time in different ways. For example, as a wave of the sea.


Separately, it must be remembered that in such structures it is necessary to learn how to correctly measure the properties that are different from the generic ones. Only those whose duration is longer than the instant of uniformity can be performed, and the sensitivity is greater than the point of uniformity. No measurement with a duration of less than a moment of homogeneity in a given space makes sense. This means that it is impossible to measure the height of the wave, because its time of existence is much less than the time of homogeneity. No measurement of an area smaller than the size of spatial uniformity also makes sense. This means that you cannot measure the wavelength. But we can measure the thickness of space and understand that it is non-zero: it is as thick as the height of the waves. This boundary belongs both to the ocean and the atmosphere! Each thesis strongly beats on intuition, we see the waves! But everything should be formal.


Fourier dance analysis


Fourier analysis of dance can be performed not only on spatial or temporal homogeneity separately, but also by combining them together. Such an analysis will show us the structure of the production line.


Anisotropy of homogeneous dance


We talked about the points of uniformity of the dance. But these points of uniformity may not be just cubes. They may have elongated shapes. Take velveteen as an example. Its point of uniformity has a pronounced elongated shape. Or the light of a star, for which homogeneity in space is not defined at all. But the space-time anisotropy is much more interesting. It allows us to represent the business function in the form of a set of scenarios of the same type!


Other bases for the decomposition of properties


We took the Fourier series for analysis, assuming that the studied space is divided by a rectangular grid. However, this is not the only way to divide space. Therefore, decomposition is not required to be made in a Fourier series. You can make a decomposition on any of the convenient bases, resulting in the homogeneity of different types.


Simulation of space-time dance homogeneity


There are two ways to determine the spatial and temporal uniformity of the dance:


  1. Through many parts of it
  2. Through the spectrum of generic properties

The second method is good, but the width of the range of spatial and temporal frequencies that we perceive is rather narrow. We are not able to catch frequencies beyond these limits. Therefore, when we need to analyze homogeneity beyond perception, we use a different method of analysis: using sets of elements of the same type. It was with him that I began the discussion of homogeneity. Of course, computer-aided data analysis methods can find the necessary patterns using frequency analysis, but this analysis will not be based on human empirical experience.


Homogeneity analysis using multiple spaces endowed with properties of the same type


For this analysis, we need to be able to:


  1. Create and use the concept of the type of properties
  2. Realize similar items
  3. Aware of many similar elements
  4. Handle sets of similar elements.

If you cannot create concepts, or types, you cannot compare the properties of different parts of a dance. In order to compare them, it is necessary to find common properties in them, and this, by definition, is the concept or type.


Having a type, you should be able to compare the properties of different parts of space-time. If you didn’t know how to do this, then you couldn’t create a type and you couldn’t use it.


To represent a homogeneous space-time, it is necessary to see the set of similar parts as one piece.


But set operations are a more subtle thing. For awareness of homogeneous space-time is not necessary. But it is necessary, if we want to build a complex representation, or to get a new one from the existing ones. This ability, most likely, is peculiar only to highly developed consciousness.


Nevertheless, since the idea of ​​sets arose much later than the idea of ​​objects, I can make the assumption that either we did not notice earlier how our consciousness works, or the mind works by analyzing a spectrum of similar properties.


Homogeneous space modeling


In the future, to simplify the presentation, I will say "space", implying a dance, but this dance looks like a frozen pas.


Suppose you are observing a space endowed with a property. There are two ways to describe it.


Method 1


If the size of the space homogeneity point for a given type of property is less than the atomic point of the studied space, it seems to us that we observe an absolutely smooth homogeneous space endowed with a property of this type, which we imagine consisting of an infinite set of zero-sized points endowed with similar properties. However, as we found out earlier, such a model is approximate, because the representation of these properties in the form of a spectrum will give us a maximum, which corresponds to the size of the atomic point of the studied space.


It can be said that such a space endowed with a property is modeled with the help of atomic points of the studied space, but this will violate the simulation logic. Homogeneous space consists of sub-spaces of homogeneity, and already sub-spaces of homogeneity are modeled with the help of atomic points of the studied space. However, the atomic points of the studied space are larger than the points of homogeneity, which means that in order to build a model of the point of homogeneity we need to take several atomic points, which seems ridiculous. However, we do so, getting a model of homogeneous space is rougher than it could have been if we had more accurate instruments. To build a model of a homogeneous space, we first need to assemble models of homogeneity points from atomic points, from which then we will already assemble a model of a homogeneous space.


If we increase the resolution of the device, then we will receive more and more new details about the structure of a homogeneous space, and at some point we will face the fact that the size of the atomic point of the investigated space will become less than the point of homogeneity. Then we will say that the resolution limit for the given space has been reached. If we continue to increase the resolution further, we will not learn anything new about this homogeneous space.


If we have five atomic points that model a homogeneous space. Can this space be called homogeneous? No, because in order to get a homogeneous space, we need at least five points of uniformity, and for this we need at least 10 atomic points! Therefore, the model of a homogeneous space should include 10 atomic points! What we see can be called a homogeneous point, but not a homogeneous space. What is a homogeneous point in this case: a homogeneous space, or a series of objects of the same type, we do not know, because the resolution of the instruments does not allow us to see this.


Method 2


If the size of the point of homogeneity of space is greater than the atomic point of the investigated space, it seems to us that we see a homogeneous space filled with the same type of elements, or spatial structure.


Imagine that you are looking at a fabric projector screen from a distance of 10 meters. You do not see the structure of the fabric, only the surface. If you come closer, then at a distance of one meter you will see the structure of the fabric. The surface will still remain uniform, but now the point of homogeneity will be much larger than the limit of resolution of the device. At the moment when homogeneous space points appear on our screens, a second way of perception arises, requiring a different way of description. Homogeneous space can now be divided into parts in two fundamentally different ways. The first method, as before, divides space into parts like it: fabric into fabric. But now a different method of division is available: now it is possible to continue splitting, and divide each of the resulting parts into parts, which are also similar to each other, but at the same time differ from parts of a homogeneous space. These parts are called elements of a homogeneous space. Each part, including the point of uniformity, is divided into many similar elements. This means that the volume of an element is much less than the volume of any part of a homogeneous space, including points of homogeneity.


Homogeneous space element


Consider an element of homogeneous space. How many element types can I get by looking at a homogeneous space? Assume that the test space consists of alternating segments of white and black. You can say that the element will be a space that can be interpreted as a pair of segments: black plus white. It seems that you have solved the problem, but no, because from the sequence white plus black you can also get an element of this space. Well, well, you say, we found two different standard elements, and that's enough! No, because it is possible to build such a typical element: black segment + skip + skip + white segment, or come up with two different typical elements: black + skip + black and white + skip + white. A typical element can be started from the middle of a segment: half white + black + half white. The same with atoms: they can also be broken apart. We have not one typical element, as our intuition tells us, but an unimaginable number of them! Therefore, it is correct to say that a homogeneous space can be divided into elements in countless ways. Often this fact is neglected, telling us the obvious way of breaking a homogeneous space, for example, they say that a crystal consists of atoms. But this is one way of countless ways to divide a crystal into elements. There is no indication as to which of the typical elements to be considered better, or worse. For example, you can say. that the crystal consists of two halves of atoms, "leaning" to each other with their backs. Therefore, it is correct to say that a homogeneous space can be divided into elements in countless ways. Often this fact is neglected, telling us the obvious way of breaking a homogeneous space, for example, they say that a crystal consists of atoms. But this is one way of countless ways to divide a crystal into elements. There is no indication as to which of the typical elements to be considered better, or worse. For example, you can say. that the crystal consists of two halves of atoms, "leaning" to each other with their backs. Therefore, it is correct to say that a homogeneous space can be divided into elements in countless ways. Often this fact is neglected, telling us the obvious way of breaking a homogeneous space, for example, they say that a crystal consists of atoms. But this is one way of countless ways to divide a crystal into elements. There is no indication as to which of the typical elements to be considered better, or worse. For example, you can say. that the crystal consists of two halves of atoms, "leaning" to each other with their backs. But this is one way of countless ways to divide a crystal into elements. There is no indication as to which of the typical elements to be considered better, or worse. For example, you can say. that the crystal consists of two halves of atoms, "leaning" to each other with their backs. But this is one way of countless ways to divide a crystal into elements. There is no indication as to which of the typical elements to be considered better, or worse. For example, you can say. that the crystal consists of two halves of atoms, "leaning" to each other with their backs.


Elements of a homogeneous space are required to form a periodic structure. This period can be observed very precisely, and can be floating. However, the spectrum of such a structure should have a pronounced maximum, otherwise such a structure cannot be called homogeneous! Therefore, a homogeneous space considered by us as consisting of elements is called a periodic homogeneous space. The space that we see as absolutely smooth is called a smooth homogeneous space. Let me remind you that a smooth can become periodic if you increase the resolution of the device. What unites different typical elements of a periodic homogeneous space? Period size All typical elements are divided into groups with the same period length. The group with the smallest size of typical elements corresponds to the most compact elements. With them, as a rule, we work. However, there are still very, very many typical elements of the same size. Therefore, it is easier to combine them into one type and say that all of these are elements of the same type up to a transformation (parallel translation and rotation). Then the number of typical elements will be reduced to a finite number, and in the particular case to one. A model of such a typical element will be a fragment of a periodic structure up to a transformation. When we analyze the Fourier spectrum obtained on a uniform periodic structure, we see peaks. Each peak corresponds to the type of atomic element up to a transformation. The deviation of the peak from the central axis gives us an idea of ​​the size of this typical atomic element. Homogeneous space can be divided but the elements of the same type, and maybe - into elements of different types. Periods for elements of different types may be in integer relations, but they may not. Now we are slowly sinking into crystallography. Now it is clear why spectral analysis is much easier to apply in practice than the analysis of structural elements.


There is one more important aspect to remember! The manifestation of the structure can be the result of an increase in resolution, or it can be the result of a change in the flows that are used to explore the space. For example, instead of a stream of light, you can use acoustic waves. Together with the change of flows, the homogeneous space and the principle of registration of flows will change. It is possible that this will manifest those structures that were invisible in streams of another type. Therefore, there is no such rule that a crystal consists only of atoms. It may consist of domains that manifest themselves in a magnetic field. It can consist of anything, including the fact that we still do not know. Therefore, when we are talking about the structure, we need to indicate the type of device that was used to register the properties and detect the structure.


Boundary situations


Sometimes we are confronted with borderline situations where we can see one thing or another. For example, if you look at a building, you can see each window separately, if there are only two of them in your field of view (the third way of perception), or to observe a homogeneous space, if there are 100 of them in your field of view (the second way of perception). There is a borderline between them, when you can switch the way of your perception from individual objects to the perception of many objects. The same switching can occur between the first method and the second method of perception. You can look at the monitor screen and see a picture from pixels (the second way of perception), or you can - a continuous picture (the first way of perception).


Simulation of homogeneity over time


Suppose you observe a dance and see its properties in time. For modeling a dance that is uniform in time, the reasoning is similar to modeling a dance that is uniform in space. There are two ways:


Method 1


If the instant of uniformity of dance for the properties of this type is less than the atomic instant of the studied dance, we see smooth movements, or even a frozen picture. It is believed that many instantaneous states of the power of the continuum are used to simulate smooth movements. However, as we found out earlier, this is only an approximate model, far from the real idea. Fourier series decomposition of such a set of atomic states will give us a peak over a duration equal to the duration of an atomic instant.


Homogeneous dance consists of a set of intervals of homogeneity. Each such interval is similar to the other and is similar to the entire uniform dance as a whole. The model of the point of homogeneity is a few atomic moments. At the same time, it cannot be said that the model of a uniform dance consists of atomic moments, only from intervals of homogeneity.


If I say that the car is parked, I mean atomic instant, interval of homogeneity, or homogeneous dance in time? Depending on the context, you can think of one or the second, or the third, but in the language you will not find a way to differentiate these concepts. To separate them, I will say: an atomic state, a uniform point state and a homogeneous state. Why in space did we not encounter the need to differentiate such terms? Because in space, as I told in one of the articles, we cannot imagine the analogue of state in time.


The instantaneous state is what you can see instantly: size, position, speed, color. A homogeneous state is what we otherwise call a state: a state of rest, movement, transformation, and so on.


Method 2


If the instant of homogeneity of the dance we observe lasts longer than the atomic instant, we see a rhythmic dance.


Let's say that you observe an engine piston making fast movements. It moves so fast that you do not see it, but you see a solid cylinder. It seems fixed and slightly transparent. For you, it is at rest, and its description will be done in the first way. Change the sensitivity of the device. At some point you will notice that there is a piston and it is moving. This will be the moment when you notice the rhythmic dance of space. I call such a dance a regular activity, because the dance we observe has a regular period.


The description of a regular activity is similar to the description of a spatial structure, but now the period is not a spatial element, but a time element: a typical scenario consisting of atomic instants up to a shift in time.


Often forget about the shift and believe that the beginning and end of the period are obvious. But in fact, it must be remembered that the beginning and end of a typical scenario can be shifted arbitrarily. This means that it does not matter from what moment to start a typical scenario: from one or another piston position. The same goes for describing any regular activity.


Let there is a turner who sharpens the details. The pattern of his movements is rhythmic, and we can assume that we see regular activity. But, if so, can we find a typical period? Here it is: hold the part, grind the part, throw the part into the basket, rest. We remember that the period is determined to within shear. This means that another sequence will also be a period: throw the item into the basket, rest, clamp the workpiece and carve the part! It doesn’t matter where we start the period or where we end it. However, there are many standards for business analysis, in which many words are written about the correct and incorrect modeling of typical scenarios, which in these standards are called processes. And nowhere is it said that the typical scenario can be shifted along the cycle. There are three reasons for this:


  1. Often confused space-time modeling and modeling activities, mixing these models in one bottle. When you see a drawing of a detail, you understand that you see a model of space-time, which you interpret as a detail. But when you see a model of an operation, you somehow forget that you see in front of you a model of space-time, interpreted by you as an operation!


  2. Often, we don’t think that with the help of common standards of regular activity modeling we model typical elements of regular activity, but not its elements. It seems that this could be a revelation to many.


  3. It seems obvious that the start of actions in a cycle is determined by the appearance of some new accounting object, for example, details. At the same time, they forget that the accounting object can be changed. Suppose there is a warehouse in which the containers are stacked. Let there be regular activity, the typical cycle of which consists of two operations: to fold the container, to give the container. It seems obvious to start the cycle by accepting the container, not by returning it, because how can one give away what is not. At the same time, they forget that the initial conditions may be such that at the very beginning of the study of regular activity the warehouse was filled to capacity with containers. And the first action was to give the container. Or you can argue otherwise: if we take as an object of accounting not a container, but an empty space. Then it seems logical that the appearance of this place begins with the operation to give the container and ends with the operation to accept the container. There is nothing that would give us a reason to say where to start a typical cycle. It is possible that a change of perspective will give a simpler solution to the problem, for example, it is more convenient to take into account your empty places in the warehouse than to keep records of foreign containers.

The group of typical periods obtained by a shift in time will be combined into one type and called the typical period with an accuracy to shift. A typical period is called an element of regular activity and we say that regular activity can be divided into parts in two fundamentally different ways: homogeneity intervals that are similar to regular activity itself (the movement is divided into a series of movements), and typical elements whose properties are different from regular activity but resemble each other.


Regular activity can also be studied using spectral analysis. To do this, it is necessary to determine the typical state, build a model of their position in time and subject this model to Fourier analysis. In the same way, as in spatial structures, we obtain the spectral frequency distribution of states of certain types. Then everything is decided by the analysis of the distribution of these frequencies. Ideally, they say that the company should work like a clock. This means that for each state there are pronounced peaks in the spectrum. If the frequencies for different states are amenable to integer comparison, typical scenarios can be created based on these states. A well-established enterprise has pronounced frequency peaks, a poorly debugged enterprise — weakly expressed. The task of the enterprise designer is to do so so that typical states are defined and their frequency distribution would be pronounced. This analysis shows the rhythms of typical conditions. And how can you not remembersongs of sailors , invented to synchronize the work. Thus, the main analyst tool, which is engaged in designing a regular activity of an enterprise, after the modeling tool of model scenarios should be spectral analysis, and perhaps the main one, since it solves the problem in ways that are much simpler than analysis of sets.


Simultaneous homogeneity in both space and time


There are 3 types of homogeneity in space and 3 types of homogeneity in time:


  1. No uniformity
  2. Smooth structure
  3. Periodic structure

If you multiply the options, you get the following combinations:


  1. 1-1 There is no homogeneity in space, there is no homogeneity in time. A point that flashed for a moment. Unable to track.


  2. 1-2 There is no homogeneity in space, smoothness in time. Star light


  3. 1-3 There is no homogeneity in space, periodicity in time. Quasar.


  4. 2-3 Smooth in space, no uniformity over time. 25 frame. Elusive


  5. 2-2 In space, smoothness, smoothness in time. The surface of the table.


  6. 2-3 Smooth in space, periodicity in time. Dancer dance


  7. 3-3 In space, periodicity, in time periodicity. The work of the conveyor line for bottling beer.

Features of the detection of homogeneity in the dance, both homogeneous in time and in space.


If the space is simultaneously homogeneous both in space and in time, it is possible to take measurements in which the lack of information in space compensates for the excess of information in time and vice versa.


The spatial observation area is less than the studied homogeneity.


Suppose that the area of ​​our observation is so limited that the point of homogeneity does not fit in it. To find the homogeneity we need in space, we need to be able to generalize our ideas. We move in space, limited by our area of ​​perception, gradually sweeping out more and more spaces. Next we use the method of generalizing our data. Methods can be very different. With the help of one method of generalization, homogeneity can be detected, with the help of another - it is impossible. The limitation of this method is the assumption that the properties of space during our crawling on it will change slightly, that is, it must be uniform in time. This method compensates for the small size of the observation area by increasing the observation time.


The accuracy of the device is too rough for the studied area of ​​space


The inverse problem, when using a too coarse detector to try to find uniformity at a very small scale for a given detector, is solved with the help of a fast survey. Then the analysis of the phases in the spectral decomposition will allow us to detect small details of space. The same technique is used to detect the crystal structure of a substance using radiation, the wavelength of which is greater than the size of uniformity. This method compensates for the spatial coarseness of the detector by increasing the temporal sensitivity.


Features of detection of uniformity in time


Observation time is less than uniformity interval


If the time interval during which the observation is conducted is less than a moment of homogeneity, we can compensate for this with the size of the observed area. We can observe a large amount of space in order to highlight typical states of typical elements in it. This will allow us to assume that in the observed time interval, each element is frozen in one or another state. By approximating time back and forth, we can assume the past and future states of the elements we observe. Thus, we can use homogeneity in space to study homogeneity in time.


The registration speed of the device is not fast enough to register the interval of uniformity.


Following the obtained logic, in order to solve this problem, we must increase the spatial sensitivity of the instrument. I have not yet found the right example.


The impact of additional information on the interpretation of homogeneity


If we look at the dance, the limits of homogeneity of which are beyond the limits of our observation window, we may have discrepancies in the interpretation of what we have seen. For example, waves in the ocean - is it homogeneity in time, or homogeneity in space? I will explain the difference. Suppose we have a string. Suppose that it has the form of a sinusoid. The question is: does it have such a form, or are we in the frame of reference associated with the traveling wave? If we do not have additional data, we cannot say which of the answers is correct. But, if we know the boundary conditions - the boundaries where the string is fixed, or we know the structure of the string and see that we are moving relative to this structure (relative to the string substance), then we can say for sure: whether it is homogeneity in space, or in time. If the string is sinusoidal, and this sinusoid is fixed relative to the edges of the string or its substance, then we have spatial homogeneity. If the string runs, then we have temporary homogeneity. In some cases, we cannot say for sure what it is: spatial, or temporal homogeneity. And only experience can give us the right answer. And then the waves in the ocean will be temporary homogeneity, and the waves in the picture - spatial.


Two properties of any regular dance


Since a regular dance in space or time can be divided into parts in two ways, we can interpret them in two ways. If we endow parts of a space with properties of the same type as the entire space as a whole, we are talking about one property. If the elements of this dance have properties different from those of the dance, we have a different property. In other words, if I say that the flow of oil consists of flows of oil, it will be one property, if I say that every part of the flow of oil consists of molecules, it will be a different property of the same flow. Often it is the second type of properties - the composition becomes generic to build a uniform dance!


Property classification


Any dance uniform in space and (or) time can simulate a property. And vice versa: any property requires a model in the form of a uniform in space and (or) dance time.


At first I wanted to classify all the properties. This was supposed to be a rather interesting story, but so far I have no time for a full account of this issue. I will only give one example that allows you to understand how the representations constructed above help to conduct business analysis.


Business function


The definition of a function is based on the definition of a stream. But when the spatial observation area is small, we cannot say what became the basis for the flow formation: spatial homogeneity, or temporal? For example, when a stream of parts flows past, we cannot say for sure: it flows through a space filled with details, or parts are created before entering the perception window and are destroyed upon leaving it. Maybe so and so. Therefore, speaking of flows in the definition of a business function, we can proceed from both the temporal uniformity of the dance: the uniformity of the events that occur (customers regularly come in) and the spatial homogeneity of the dance: the regularity of the flowing flow (the flow of oil). Both methods fit the definition of a function, however, temporary homogeneity is often forgotten,


When a business function is endowed with many regular flows over time, they forget to say that each flow has a different structure and different moments of homogeneity. By definition, this leads to different models of homogeneous dances. Homogeneous dances, corresponding to different streams, occupy the same volume of explored space, but have completely different structure. These are different dances! Combining them together should be accompanied by a model of relations between them and their interpretations. I have not seen anything like this, and this greatly impoverishes the model of enterprise activity. The problem occurs due to the violation of the condition of uniform homogeneity for all flows. For example, the instant of homogeneity for multiple streams should certainly be greater than the longest instant of homogeneity among all streams. I see the same part how this condition is grossly violated in those or other models built by business analysts who do not feel this restriction. Now you don't have to feel it, you just need to know it.


Thanks


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