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Prof. Heikki Hyötyniemi AS-74.4192
Lecture 2: Research on Complex Systems
Helsinki University of Technology, 30.1.2009
(v.2009.04.10, cleaned from a rough machine translation to only about line 400 yet! marked with #)
[0:00 / 1]
Welcome back to our Friday's meeting.
This time the theme is complexity theory generally, because this is the framework in which this cybernetics, and in particular neocybernetics is studied.
[0:24 / 2]
If you wonder what complex systems are, here is one depiction.
John Holland, which is a very important researcher in this field, explains that any difficult problem, one could almost say that any problem which can become complex, is a valid object for inquiry.
The idea is that perhaps there could be some general principles beyond all complex systems.
If we talk about trade imbalances, AIDS, genetic defects, mental well-being, or computer viruses, it may start looking a bit questionable whether anything in common can be found.
And the next slide is surely a little sensational here.
[1:45 / 3]
The question is that what do you know about alchemy, and do you laugh at these alchemists' doings? -- then maybe 500 years ago when they practiced it.
You could ponder this in your lecture journal, for example.
But I urge you to keep in mind that Newton was an alchemist as well.
After Newton had invented these easy things, such as theory of gravitation and laws of motion, then Newton concentrated on such a topic what he believed would earn him an eternal reputation and where one could obtain significant results.
At the time, chemistry was not yet known, and the world was in that sense quite a bit unknown yet.
[2:56 / 4]
In a way the "confession" of complexity research is that there is something universal behind all the complex systems.
In a sense, others believe in this and others do not, but before you say that you do not believe so, --
[3:30 / 5]
before you join with, for example, this line of thinking as Kari Enqvist, just to cite an example -- i.e. if you have read this book on complexity, then there he is reasonably negative on the need for any complexity theory at all.
He assumes that the basic theories are already in place, and it is enough if one only fills the gaps, there, between these theories.
His starting point is, that all that there exists in the world, is energy, or matter, as an other manifestation of energy.
And the only problem we have when we try to model the world, is that we should be able to write these Hamilton's functions, that define the energy surface.
And then, if these Hamiltonians have been accomplished, then even all these higher-level phenomena can be directly reduced to mathematics -- i.e. some cognitive phenomena among others.
And, of course, non-linearities result in things being difficult, and equations are extremely complex.
But the basic problem is as if swept under the carpet.
It is not accepted that anything stronger would be required than what we currently have in use, already.
[5:18 / 6]
Here are a few slides, which are intended to show that we are just, in this moment, prisoners of the current understanding.
Even wise people, five hundred or a thousand years ago thought, for example, that heart is the home of the soul.
The justification was that, for example, speech comes from the chest -- and where else does it come from than from the innermost organ that there is, that is from the heart.
And we know that when we get excited, then heart rate increases.
And really the excitement -- the only organ that reacts is the heart.
The brain displays no change of emotional state.
For Aristotle, the brain was only needed to cool the blood.
One might wonder, why even for a thousand years, this theory of Aristotle -- also for the other aspects than the role of the heart -- remained at the surface, or alive, it is due to the fact that it was at the time consistently the most appropriate explanation for this.
If we think of ourselves as though 150 years backward, to the 19th century, when people are beginning to argue about the theory of evolution.
In a specific way, in a sense, this issue has been quite clear -- Darwin was not the first who presented an idea of evolution of some kind.
But at that time there were so many counterexamples, that the simplest explanation was some kind of a divine explanation.
That God had created the species such as they are.
And, perhaps most important, or the main explanation or justification for this was, that if the sun consists of coal, there is no way this coal can be enough to fuel the burning for millions of years, the time evolutionary development would require so that species could evolve.
So, for example, it required the discovery of nuclear physics, before the world of ours was so complete, that a coherent theory of evolution could be developed.
Darwin himself admitted at the time that -- or pulled many of his arguments back -- just because it was written a little bit too early this book of his, as an example, at the time there was no theory on the character of these genetic factors, or in what form they were inherited.
If they were only some kind of continuous substance, so then, these genetic factors, or the genetic material, would only dissolve all the time, and -- in some way would result in ever more uniform population, instead of drifting towards some sort of differentiation of populations.
[9:20 / 7]
Well, we can now think about ourselves, five hundred years in the future.
In exactly the same way as it is now possible to consider these flogiston-theoreticians or alchemists 500 years ago, so in exactly the same way, we can imagine what we contemporaries are thought about, 500 years in the future.
Certainly, many things are very different then.
And, frankly, today there are so many contradictory findings, that surely those living after five hundred years may wonder, that how could a consistent worldview have been possible for a human in this period.
In former times there were much less controversial observations of nature.
Let's take a few examples --
[10:25 / 8]
or rather, on this next slide is an example that in these explanations that currently are offered in the natural sciences, there is still something to ponder on -- everything is not, all the theories are maybe not final.
Here a very simple model of enzyme is depicted.
You know that enzymes are those that take care of chemical processes in the body, they are catalysts to processes.
Catalyzing is based on the fact that -- or the traditional model to think about it is that -- they are like a lock and a key, to which this enzyme fits.
It fits into the lock accurately as a unique key, and it enables, it opens the molecule in some way, so that it is able to take part in reactions.
However, if you think about this molecule, you can well wonder how it could act like a very complicated key system, because the only thing those molecules can sense, of this enzyme, is only its electric field -- or according to the current theory, the electric field which it produces around itself.
And then these molecules blindly poke at each other -- their electric fields interact, and they either attract or repel each other.
It is roughly the same as someone blind would poke with a stick at your face and try to identify you.
An additional problem is, of course, that the molecules do not have any pattern recognition ability, and there is no conscious need to identify somebody.
They just settle in to where is the minimum of their electronic potential, in some way, to be found.
In that sense, one may well wonder how this biochemical process is so precise.
Not to mention these genetic processes.
How can these function so well on the chemical level, and so quickly yet.
Even in such a way that this gene expression, or reading of genes, it can happen so consistently in chains.
It is absolutely incomprehensible.
We will return to this issue towards the end of the course.
[13:12 / 9]
Another, a little simpler, example is this:
If you have been reading the [Finnish popular science] magazine Tiede, there was some time ago a topic, a brief writing about how these snowflakes get their form.
It was explained that although these snowflakes are very different, or virtually all different, one snowflake is, however, always born in similar circumstances, in the way that each of its branches has experienced a similar history.
And that is the reason that every branch forms similarly.
So this symmetry is justified by these environmental conditions.
However, just a simple analysis shows that if you take a snowflake, it can be seen that those are in various stages those different branches.
And yet, if it is allowed to develop, that snowflake, they complete later, these incomplete branches, although the circumstances -- or even if the assumption is, that they should have been in exactly the same stage already by then, so that they could be alike.
So how is it explained that they in some way seem to communicate with each other, those branches -- that they are guided by each other's development?
What could be the basis of this?
The current theory cannot answer this, either.
Therefore, one wants to fit -- or explanations are claimed with the present theory in a way that at least an apparent explanation is obtained.
The point, that they have been in the same circumstances, it seems, quite -- to some extent an invented one, or a bad explanation, even.
Let these examples be an introduction to this statement, that indeed, some novel theory is needed too.
And at least those people who think -- or who have drifted into this complexity theory, they see that something new is needed.
This is exactly the same as in artificial intelligence research -- or artificial intelligence is a part of complexity research -- i.e. some scientists see some idea in self-organizing maps, for example, and some scientists do not see.
The field is a bit divided regarding these things.
[16:18 / 10]
Well, then if we assume that a new theory is needed, can start to think that what kind of theory it is and how it could be approached.
If we look at this research field as a complex system, we can start to observe, what kind of research has been done on this field.
And it is a very broad field.
You can easily find by searching the web different research areas which from their perspective shed light on this idea of complexity.
But actually, in this course only one perspective to complexity is chosen, and studied carefully, consistently, and therefore we will not cover these different areas so precisely.
[17:40 / 11]
Well, this is a very old problem in itself, because a very large number of complex systems -- or complex systems have always been studied or at least people have wanted to find some kind of models for them.
Traditionally the approach has been to construct hierarchies -- i.e. one has aimed at this engineering-like approach.
A complex problem is divided into parts and the parts are analyzed separately, and after that these parts are piled up back together, and it is said that this set of partial models is then the model of this whole thing.
And this approach has indeed been practiced since Aristotle.
He defined taxonomies, and then, for example, Linnaean taxonomy -- or this classification approach, which sought to classify all things in nature, in particular the plants and animals -- so it is instructive of this approach.
But also later, for example, the Nobel Prize winner Simon, determined that the architecture of complex systems is based on hierarchy thinking.
He explained, that these are evolutionary resilient structures on the grounds that they are robust -- he explained the reasons behind this in his book.
And also, of course, in our field, control engineering, models are these kind of hierarchical models just because also in our field we have to control complex systems, even if there did not exist any fancy theory.
Therefore, it has been necessary to approach problems explicitly in an engineering-like way.
But in this approach there is precisely the problem that this sort of raw engineering-like approach -- in some way -- "kills", the essential.
In some way -- approaches to artificial intelligence, typically, use such a brisk machinery, in certain -- the thought, the idea of intelligence, or the idea of consciousness or awareness, drowns under the machinery.
And accordingly, if one examines a living machine, a living mechanism, or biological device, as such a device, then this idea of life itself is flooded under the machinery.
And in our field if we start building some optimal robust control system, or an optimal hierarchical system, it is this robustness that first begins to suffer there in the whole.
Something new is needed.
And let's see now what the present day has to offer, new tools, just what Aristotle at the time could not even imagine.
[21:06 / 12]
The manner in which the modern world differs from the times past is in particular the matter that we have this computer that can perform repetitive calculation, in a completely different manner than a researcher in the former times.
So, now that this sort of infinite quadrature is approved, i.e. a non-Greek-based approach is permissible, then the computer can be put crunching -- to build this kind of computational models -- and one can look forward to what appears.
And now begins again divide this field is that niistäkin who believe kompleksisuusteoriaan, so some believe that it komputationaaliset approaches may at some stage, for example, lead to intelligence.
Artificial side has traditionally been the thinking that within twenty years of the computer must be smarter when the programmer, because its calculation capacity is so huge, already twenty years.
But it can be argued that it is not enough, it will simply fall in power.
Because if the methodology is wrong, or the model of cognition, or intelligence in the design is wrong, so there is never any emergoidu.
For example, to imagine that the already existing machinery should be able to fly to mimic the intelligence of, if not a human intelligence can not.
And if we're still at its infancy.
Yes, one can obtain a kind of rational behavior, if it is more or less programmed, but, however, the fly live inability has an entirely different category of complexity even compared to a computer program.
Well, actually, this field is divided in such a way that others do not believe in this, that komputationalistiset approaches would suffice.
However, now this luennossa looking at this approach.
Assume that the computer is able to give us something new.
[23:49 / 13]
But in order to be able to give, so we have to have somebody kind of accuracy it, and someone kind of think it what we are fed into a computer.
Because if we are fed garbage machine, so it is the framework period, and then crap out.
For example, if there is any konvergoimaton algorithm, so rounding errors eventually grow so large that they completely cover the alleen any meaningful information.
So, we must know how to program the machine correctly, and is considered now that the manner in which the machine could be programmed correctly.
[24:29 / 14]
This is achieved in this chaos theory ago.
You have certainly heard of the matter.
Chaos theory in particular, there are many approaches.
But this course now goes through, as if our basis since, in this run.
We mean the field of systemic thinking, and back connections and interactions, that is the source for the idea that we have a model of this sort, which is feedback involved.
It is considered that the ability of this feedback in itself bring about any kind of emergens.
In this way, the result would be something other than what we intuitively we can imagine.
In other words, whether the calculation so well-defined goal to come to any semmoista which would be a complexity point of view interesting.
And, indeed, this clause is what looks at this little time, so this is the principle of this sort a simple logistic growth of the clause, which has been used on issues like the growth model, or the growth process model.
If you think the growth models, so most simple growth model in this section, here, this exponential growth in the next population is the last population, multiplied by one of the constant lambda.
This lambda is here that the growth factor now.
This logistics model has been developed to move forward in that way -- it has been added to this sort vaimennustekijä this.
In other words, the larger the population, the greater reduction of this growth.
It can be written in that format.
Is seen that the linear term in addition to an exponential growth, we have such a term neliöllinen here.
In other words, non-linear term.
And, indeed, even though we are perfectly able to know qualitatively how this exponential growth is behaving, this seemingly simple addition is just a huge lisäkompleksisuuden this behavior.
[27:23 / 15]
And here is the chaos theory of a sort of landmark, or some kind of a classic photo.
How many have seen this before?
Approximately half of the saw.
So, here is this bifurkaatio also in this sense.
So, this is the so-called bifurkaatiodiagrammi.
I mean it, that this is the lambdan, ie the growth of the term or kasvuparametrin described a function of the fact that what this system ultimately ends up.
If you follow this lambdaa here from zero up to number one, we could say that growth is too small in the model, it is always a balance between the inflow to zero, the system behavior.
Always eventually happens in such a way that x (k) is zero, and the resulting x (k +1) is zero forever.
So, it is a static equilibrium, and stable equilibrium.
On the other hand, when the lambda ykkösestä will rise higher, that is a growth factor to achieve a certain threshold, then it proves that this balance between what goes into it, is no longer zero, but it is something of this sort meaningful equilibrium value.
So, typically in these values that the logistic growth curve used.
It ends up in a meaningful balance.
But if our lambdaa still, that is the model for more than three, we gonna happen something that is totally unpredictable, this formula simply by looking.
It will not be able to see.
It turns out that this equilibrium, which it has, it will become unstable.
If we go for some arbitrary x the value of the population value, it will not ever find this a point, but it enters into this kind of behavior, that it alternately is here, and the second time here, which is bouncing between the two alternative values, and this is a stable cycle.
So, it has a period two.
Lambdan certain values, when lambda is greater than three, but less than three and a half, it goes into issues like the stable, sykiin with two points.
But then again time to place an astonishing phenomenon -- it is a stable 2-period become stable 4-periodic, it has four points which repeat each other.
Regardless of where the x value of the source for it always ends up to this behavior.
This has drawn all of these periodiset Scores here until four.
Eli lambdan value of up to four.
You see -- if kokeilette the model -- that if you have the lambda larger than four, then it is unstable throughout the model, ie it will explode.
But, if the lambda will remain below nelosessa, so it will always be zero and Ykkönen between the x's value, but its behavior will always be only a more complex, closer to what will be fours.
And this --
[Question: Tuplaantuuko it always is the frequency?]
It is here at the beginning of tuplaantuu.
[Is it the end, or is it just -]
Well this is exactly what a surprise it will reply that it did not go so consistently,
[31:30 / 16]
but first of all tuplaantuu, ie first there is this sort 2-period ago, 4-Section, 8-Section, 16-section, but then when you go a certain limit, above, there begins some reason occur on issues like odd episodes, and it can be shown that a certain lambdan to date, this model has been all the possible cycle lengths in a lambdan value.
[So even three?]
We will come back to this three, just over there at the end of the lecture.
So, this -- apparently you have read these things because --
[No I do not terribly, but sometimes -]
- This is a famous paper in chaos theory, just that the period of three means chaos.
So, it is in some ways the most difficult period of what can be achieved.
Well, however, the behavior is very difficult to see that there what the underlying model is, and what is it lambdan value -- that is, the longer the period is, so the more difficult it is to separate the pure chaos, this behavior.
[32:56 / 17]
Well, this is so interesting that bifurkaatiokäyttäytyminen that comes with also try the more complex computer data -- that is to say that just now it x was always a scalar, ie it was the entire population.
Mares developed in a two-dimensional case, ie, between x and y, variables are terms used.
And this is related to that previous in the way that we can write the two-variable function with the complex numbers, in this format.
Eli z next moment is proportional to zetan second squared in the last minute.
This is essentially the same second with the power, as it just made the logistic growth.
And, indeed, this is simply the complex value a function written in open form, in such a way that reaaliosa and kompleksiosa is written separately.
Observe now that if we assume the initial value of x = 0 and y = 0, and this is a standard part varies, so deemed by the system behaves.
[34:30 / 18]
Remembered that skalaaritapauksessa was that bifurkaatiodiagrammi, now two-dimensional case, this so-called the Mandelbrot set.
How many people, this is familiar?
Yeah, this is the second semmoinen, as if the icon for this kaaosteorialle.
So, this now appears in this reaaliakseli, and then this complex with the shaft in there.
When we choose x0, y0 some space here, so all of these in the red marked areas divergenssiin lead.
Means that the model goes into infinity.
Instead, these black-out areas, will mean that it will remain limited -- it does not necessarily converges to anything, but it did not explode.
So, this sort can be found in the so-called strange attraktori what goes into it, all these black areas, where the source, in view of its initial value then, the black area inside.
Interesting is that when you go to this remote area, both here and gonna always be found only in more subtle structure.
So, this border region is interesting.
So, here divergenssi, this is a certain kind of convergence, or at least find it on issues like the cycle reasonably quickly.
These outlying areas are semmoisia in need of very long iteration sometimes to see how it behaves.
[36:00 / 19]
Here is a slightly different manner described, the same thing.
This is the so-called Julia sets.
In other words, they are also theories of chaos symbol, in one way.
Here is a selected x0, y0 of here the last film in the picture, and here is the now actually nine pieces of points here black area, and the clubs tukin now it cycles, so that the cycle is like it becomes, for x, y varies when this iteroidaan.
Proves that the x, y-point mass, or the point set of pairs, is a complex issues like the level designs.
So, if you are just like the middle -- this Kolmonen is out there the middle of the Mandelbrot set of -- so this is very simple, this attraktori.
But for a lot of mares fine structure can be found there.
This has been a very popular study destination for all those who have had a powerful graphics computers, 80's.
[37:23 / 20]
Well, this area is very much the key words, key concepts, and a few semmoisista is fraktaalisuus and itsesimilaarisuus.
They are presented in this film -- see how -- or what they mean.
If it comes to study the peripheral area of the Mandelbrot set of selected in this sort, and increased by it.
You will remember that this is a completely deterministic formula, which defines the set of Mandelbrot -- arbitrary, we can zoom in, having only ever more accurate figures for the slide, and can start to draw accurate and detailed maps of this.
This is a zoom on that area, and so forth.
This is always about ten times greater accuracy, are taken into points here, and then when you have already gone terribly much deeper than what this original image was, so there is surprisingly exactly the same shape.
And this same format can be found in very many point to this image.
This is called itsesimilaarisuudeksi.
In other words, this original image is repeating itself in different skaaloissa, infinitely far.
And on the other hand fraktaalisuus is precisely the fact that different skaaloissa happen -- or when you go out for further scale, so there somehow found essentially the same qualitative behavior.
We will come back to this fraktaalisuuteen moment.
[39:24 / 21]
And now when it comes to watch tuommoista files, such as that used in this section is from the Mandelbrot set of the peripheral zone, so easily comes to mind intuition.
If you start to see here, so this is as if, as if something nikamia, and here is this sort of mesh.
This is easy to start to think that now we have found a formula of life.
It can be restored to a single kompleksilausekkeseen, line up.
This is just semmoinen thing that divides people, that if a particular meaning, whether or not this will have any meaning, that are so similaarisia these two images.
And I will spread -- or I belong to that category now, secondly, that believes that this ultimately is not very much in this that they are, the surface pattern is similar.
Because even if we have the following characters would be a very large number, so the surface of characters, they have finite volume, it is much less than the depth of characters.
And if we manage to bring about -- the fact that two different types of complex system, the surface, when projected to any level, to provide the same type of image, so it does not necessarily tell much yet, that the mechanisms would be the same, in these complex systems.
Just like Alan Turing, in the last lecture at the same time, after all, is not kinnostuksen the subject was not the seepran tracks, but the horse over there.
[41:35 / 22]
Well, then, when those models of chaos has been studied, it has been so that they are typically very sensitive to the initial holding.
So, if I start from a little bit different from the initial state as the result may be very different.
In other words, it grows exponentially -- we talk about the so-called Lyapunov-exponents, which tell how quickly these divergent early modes disperse.
It involves the so-called perhosefekti that if a butterfly wink siipiänsä, Amazonas somewhere, so it could move forward, this siivenräpäytyksen-induced disorder, which means that after two weeks in Texas is a Hurricane.
It may accumulate by the end result, in such a chaotic sääjärjestelmässä, in that way.
Here is a bit desolate aspect, in the sense that if we do nothing ennustamiskykyä models exist, where there is always the way it is the weather forecast in the case, that they divergoi, at some point, so whether the models limited their use at all then.
One starting point right here neokybernetiikassa is assumed that the source is to examine the behaviors that are convergent.
[43:20 / 23]
And -- no, this is divergenssin and the convergence problem in the past been considered.
It is possible to define the issues like, for example, the function sets, which converges, ie to achieve a similar result, although the source of a very different initial situations.
Here is an example of here -- really could talk about the IFS theory of iterated Function Systems theory.
So, we have a set formula, and when that formula among any of the initial value of the set is applied, or insert the original point of the set to the system in, or to the model in, so it always produces, after all, on issues like itsesimilaariset recursive structures, such as the here and now, these journals are.
The basic problem is while it is true that, although these surface forms appear very much natural press, so this mechanism, namely the function of this sort of repetition, it is -- hard to imagine how it would happen naturally.
So, this is, indeed -- if this is this whole page, so it is a lesser form of this, magazine fork, and still, this is minimized in this and so on until infinity.
Here is a hand, this fraktaalisuus and itsesimilaarisuus involved.
[53:25 / 24]
This is chaos theory -- could say that it was when the 80-century, a hot topic -- in this graphic came from the computers -- but very quickly discovered that, this is now, really, chaos in itself, that almost any non-linearity, when we start iterating it, so it produces a chaotic issues like the outcome, but interestingly it is the fact that we could somehow identify the formula, which is a natural system is iteroinut in order to have obtained the results achieved.
Eli kompleksisuusteorian goal is the opposite than in the chaos theory: rather than iteroidaan something early, or on the function of an arbitrary number, and to obtain a natural form of time, so take a natural form, and seeks to find the formula that iterating it have been achieved.
Preferred is that there could find any of the following formula, which is able to explain the variety of natural forms.
Certain way, this is once again dealing with issues like the philosopher's stone with.
If we had a theory, which under the natural shape would be able to return to the original clause, so this would solve a lot of problems, what we have complex systems is.
And in the same manner as if these alkemistit once, tried it the philosopher's stone to find, so kompleksisuusteoreetikot tunnel to find the philosopher's stone.
And if it is found, then the world is no longer the same anymore after that.
In exactly the same way if alkemistit would have found the philosopher's stone, the sciences would have gone exactly the other way.
Difficult to imagine what sort of direction.
Well, so that we can bite this problem -- how it is below a function of complexity, then over, to find -- we need some kind of formulation, and so that we would have something concrete to anchor, so based on the fact that we saw in these natural forms, often on issues like the presence of the feature, which means that this fraktaalisuus and itsesimilaarisuus.
[48:13 / 25]
So, I start to define what are Fractals, or go to mathematically analyze Fractals and itsesimilaarisuutta.
And here it is now, under this definition -- that is fraktaalidimensio is this sort logaritmilauseke, or two-logarithm of the ratio, simply.
To have this quite a mathematical task, and you will see that this has any meaning in this clause.
So, this is, indeed, the numerator is a logarithm, itsesimilaaristen parts by number, and the denominator is a logarithm, magnifier or pienennyskertoimesta.
For example, are now even if this square here.
Is this itsesimilaarinen structure?
Yes it is, and we see that if we share here in this mid-cut, and another crossed the other way around a cross, we see that it is four in exactly the same form, square, inside of which combination of the overall picture today is formed.
So, here are four copies of these elements, and these are each laidaltaan half of the original squares.
It means that this is suurennustekijä deuce, and itsesimilaaristen parts number is four.
Put logarithm divided by four second place logarithm, obtained by two.
So, this fraktaalidimension clause, at least in this case, pelkistyy this ordinary dimension clause.
We know that the square is a two-dimensional object.
Also worth noting is that: -- the logarithm of the properties due to -- is exactly the same, which is the figure we have, it always gives the same result.
Then take the line of this snippet, for example.
Are divided into three even parts of this, you will notice that the three third-size piece of the line is similaarisia its entire line of.
So, here now is really three itsesimilaarista part, and suurennustekijä is Three, which means it is a logarithm of three divided by three of logarithm, number one.
Again the fraktaalidimensio give the same dimensiolausekkeen.
Line is the yksidimensioinen creature.
Well this could then share this even though a similar manner, in three parts this way, and three sections horizontally, then it would be nine itsesimilaarista part, and should skaalaustekijä Kolmonen, and placing this formula still yields the same result, the deuce is this fraktaalidimensio.
[51:20 / 26]
Takes a little interesting example, the so-called Sierpinskin triangle.
See you, that it can be constructed -- that is the end result here is that Sierpinskin triangle -- it can be constructed in such a way that issues like a pile of triangles overlapping in the infinite amount.
Or infinite continued mares triangles heaping, ie first the triangle, then this triangle here, piled up at the same overlap, and we can see that this is the original triangle are shrinking, that is skaalaustekijä is two, that is always below its triangle is half sivultansa shrunk.
But, itsesimilaarisia parts always go down, there are three copies.
So, it is a logarithm of three divided by logarithm of second place.
Now do not get no, no overall figures, but from one of 1.58.
Means that now is being done really genuine extension, this concept of dimension, ie we have -- a pattern of this dimension is a fraction now.
If you get to look at any item in here, so if this is the formation of infinite iteration in the way that this has been repeated an infinite number of the triangles heaping, so any of you to take from here, so there certainly will not have any area, really exists, but all points, even though how much suurennatte, so only tämmösiä reiällisiä characters.
So, for any point not found semmoista uniform color of the surface.
If found, then it automatically would be two-dimensional structure, but now it is not two dimensional.
[53:30 / 27]
And it comes pyörittelemään produce fraktaalidimension form.
We can throw the denominator logaritmilausekkeen here the other side, ie we get a clause in which this fraktaalidimensio D is the unifying factor of two between logaritmilausekkeen, ie it is a linear dependence of two logaritmilausekkeen between.
This gives us the tools to analyze an arbitrary pattern, and find it fraktaalidimensio, this clause.
If you are painting someone measuring stick of this function itsesimilaaristen parts number, logarithm-logarithm scale, so this angle is a direct factor is the fraktaalidimensio.
Ie the logarithm-logarithm scale fraktaalidimensio D combine these two logaritmilauseketta.
Eli fraktaalisuus and the so-called potenssilakimalli is inversely the same thing.
If there is a quantity whose connection to one another suureeseen be written in this way, that it is a power here, and it will power can be interpreted fraktaalidimensioksi -- and these two quantities here, can be interpreted that they have in mind potenssilain married to each other.
It is considered a little examples of what this means in practice.
[55:40 / 28]
Of this sort takes the standard example of fraktaalidimensioiden side.
Here is a map of England, and now the measuring stick length of the first 100 kilometers.
We know that when it comes to the English borders, or the island's boundaries, along the run, a hundred kilometers an accuracy in the way that always has the initial focus to one -- then what is considered a yardstick, it will land, or where it hit the beach, so it is the next point and again next and so on, so see that we get the very image of Krouvin England coast.
It is about that.
So, now measuring stick length is a hundred kilometers, and those parts of what this is, as Nita is -- what this number has now been achieved.
It can thus write the log-log scale, where here is, first measuring stick length, and here is how many semmoista mitallista we have, or how long this yardsticks are used, it is coastal.
This has been 50 kilometers yardstick, and we can see that it goes substantially further on, all these gulfs and tämmöisten vuonojen through.
So, it can be assumed that the exercise of this 50 kilometer measuring sticks, we off the coast of England has a length greater than the use of this too long measuring sticks.
And still, the taking of only a more accurate dipstick, a dipstick kilometers, so it goes back further with all the surface modes, or modes of the coast, through the.
This can be extended in principle very far, because always, but when you go out for further examination, so there can be found on issues like tiny promontory, and finally we have these grains of sand that can start to circulate.
And we get on issues like dipstick-coast length pairs, which can be made to this logarithm-logarithm scale draw, and we can see that it is now here, this curve.
There really is this, firstly -- nojoo, does not go into detail, but nevertheless, seen that this käyrästö, or that a number of points, is set approximately to the straight.
That is, must be taken into account, but that this, roughly.
Again -- a set can be split up.
Others will see that yes, they are straight, others will see that there is not direct.
Now, if you believed that these are straight, so -- the English coast, it has fraktaalipituus, fraktaalidimensio, which can be read directly from the angular coefficient.
Similarly, when you do other countries' borders mares views, and expressly semmoisille borders, which are natural shaped -- man drawn boundaries have a direct yes, it does not Fractals dimension, must be taken of any yardstick, it's length remains the same yardstick, regardless, even though the view from the ground pulled limit, which means it remains in the horizontal, directly, the behavior, but on issues like, just the coast, they have this fraktaalisuutta, and we can see that, for example, the South African coast, it is clearly less than fraktaalinen England coast.
It has less to that slope, and so forth.
So, in this way can be models for the chaos theory sense of these, such as land, water or boundaries, countries.
[59:58 / 29]
Well here is the example of an entirely different subject matter.
Eli looked natural language word frequency.
When the words are arranged in the most-used word for lesser-used in the in the way that they are all in the decreasing order of frequency, so you will find that once again be drawn in this rankkaus logaritmiasteikolle, and then this frequency, ie the density of expression of normal tongue, another logarithmic scale, so that is seen it approximately to the set again for this straight, this is the log-log-levels.
So, for some reason, the language the words are also fraktaalisia, in some way, or their density.
This can go again -- someone will see this interesting issue, one not.
But long before those of chaos theory was talk, talk of Zipf's Law, it is precisely this language of words in behavior observed feature.
Into any language, so typically, this same behavior is valid.
It turns out that if there is any way itsejärjestynyt, so there is a mares fraktaalisuutta involved.
[1:01:47 / 30]
But suffice it fraktaalisuus, or fraktaalidimensio, to us -- still -- a good tool.
Well, it discusses the moment.
However, in relation to these, there are different areas of research of the findings and found that there potenssilaki of this sort is true, that is found in logarithm-logarithm scale of a kind of linear dependence.
And low -- depending on the fact that the study area is about, so have invented the same thing again.
And then there are different names, the inventor of, pretty much.
[1:02:26 / 31]
Well, then here is justiin it that -- to show it -- that the linearity is a little bit semmoinen, kyseenalainenkin.
So back to the future.
[1:02:46 / 32]
However, quite a lot of applications are available.
[1:02:52 / 34]
And when it comes to ruvettu to explore what that may fraktaalisuus, or itsesimilaarisuus comes, it is possible to say that, for example, if you start to build a network of mares in the way that it comes more or less randomly to increase the network noodien between the lines, as some such as the Internet, so it follows of this sort itsesimilaarinen structure, typically, in certain conditions.
[1:03:31 / 33]
And, this is, and here is this -- take this film to intervene -- that is to say, this recent book which was very popular, or talk a lot a couple of years ago about this, so this is kind of new into these fraktaalisuusasioille.
This Barabasi said that all the issues like the networks, it is potenssilaki force, and a certain way, this is a single package, or in a different package, that same fraktaalisuusteoriaa, or chaos theory.
[1:04:15 / 35]
Well, here is the example of our industry.
In other words, the manner in which field we can think of, that is mares fraktaalisuutta.
Typically sisimmät säätösilmukat automation systems are stabilizing adjustment, which is -- the time is something second class parts of a few minutes, maybe, depending on the process.
And these individual adjustments is typically a number of wider perspective, which seeks to regulointia something to do.
Suppose that we have the following settings stabilize, and then built a production management-related regulointi.
Regulointeja These are then developed a number of establishments, typically.
These are typically some hours or days of the time scale.
When you go higher up, so there is again säätösilmukka, which aims at optimizing the production of the country according to demand, for example.
In this production at the level of guidance aims to drive them at a lower level regulointeja, as a way to bring it to production, the calculated optimiin.
There will typically days or even months, aikaskaalassa to obtain the right kind of product is achieved.
So, the first time scale, which goes for less when you go down, on the other hand, controls the amount of which increases when going down.
These could also form a graph, where the draw, although the time scale to another scale and another scale then the controls on the amount, and may find that a more or less, they settle in logarithm-logarithm scale stretch.
In this way, may in some way to characterize this industrial plant.
But somehow this is a bit too qualitative, however, a measure of automation system.
We can no proper planning to make on issues like fraktaalidimension help.
It is a little too abstract for use.
It is a question of whether chaos theory in this way applied to actually use -- you can think about it.
[1:07:25 / 36]
However, if the nature of these natural systems to reach their fraktaaliseen structure, so should we not in technical systems, to try the same, because luonnojärjestelmät is typically robust, and they work well -- not necessarily the best -- but in different situations very well.
Even though there any part of collapse, the natural system, so it is typically the total system does not collapse.
It might be advantageous if the industry could build the system the same principle, and -- now that we can fraktaalidimensiota yes some use the analyze, that, if we find that the bottom is too low, for example, control loops, so we can imagine the growth of the volume controls.
But nevertheless, this remains some way descriptive.
[1:08:36 / 37]
Let's go now to look quite different perspective.
In recent times, been very much in the foreground, this Tungsten, on the other hand, Penrose, and so on, who assume that we can not understand any complex system, if we do not just go there, until the bottom.
Penrose talks about the need for something kvanttitason understanding, in order to be aware of, for example, to understand -- and also speaks Kari Enqvist.
Stephen Wolfram is assuming that we need to understand these cellular machines in order to understand complex systems.
Certain way, it is very clear or consistent in the sense that basically physical systems are built from one of the following structures, one of the atoms, or cells.
So, in that sense this is a well-understood starting point.
But what has this little question the law is that whether the sense to forget all of the far-reaching theories that have been built specifically to issues like abstrahoimalla -- atom systems, and cellular systems.
Again research field is divided into two parts, in the sense that others do not accept the descriptions of effective, some sort of Abstractions bottom of these phenomena, and others from the fact that -- or those who do not agree on issues like Abstractions and require that models be built entirely of such basic phenomena in the bottom.
[1:10:48 / 38]
A couple of years ago was the hot topic this Wolfram "A New Science", in which he states,
[1:11:00 / 39]
that the correct model for all the world's complex systems of this sort is a cellular machine.
And, cellular machines by examining we will be all systems, after all, understand.
The basic problem here is that the cell-machine theory is too strong a theory -- or it is very much a strong theory in the sense that it gives enough of all the cool-time simulation, it can be shown up, that these cellular automata are universaalilaskimia, ie they can take everything what the computer can do .
[1:11:52 / 40]
But, just the problem is that they are too powerful a tool, that is, you know -- or if you have laskettavuuden familiar with the theory -- so you know that this Gödelin epätäydellisyyslause biting tämmöisiin too strong formalism.
In other words, if there is just this sort of universal computing ability formalism, so you can never be any proper math, or analysis, the foundation to build on.
The only thing you can do, is a simulation.
[1:12:31 / 41]
Well, Stephen Wolfram says in fact that it does not matter, we just have to invent a new science that is not based on mathematics.
This is a little semmoinen bold assertion in the sense that, so far, all that upheaval has occurred in science, they have agreed to such, the scientific paradigm in.
Although there are new findings that are inconsistent with the old findings, then the scientific regeneration in the way that, the new and the old are compatible, after all.
Build a more comprehensive theory.
Now, this cell machines based on the theory assumes that all of the old science is old-fashioned.
In other words, the automatic blood cell theory is incompatible with the terms of the old with the science.
And everything must be based on simulation.
[1:13:53 / 42]
And this is -- now will not be so long, then in this, that can already speak of the end of science.
And this, for example, John Horganin book is a fun time for a description of how the chaos theory -- that he speaks kaopleksisuudesta with chaos theory and kompleksisuusteoria combined -- how science is beginning to be more key words and promises that can not be considered.
Each time only the new investigators will bring new pledges, but that age has not been any way to prove.
Well, those researchers who use the new science, so assume that is not the old certificate of mechanism required, but it is a fact that simulates -- the computer is the reality of what pyöritämme these cellular machines -- and the end result which is obtained, so it just has a natural shape, and are no analysis is not available.
What is interesting in itself, and kyberneettisesti worrying is the fact that when they have been so popular these kompleksisuuspuheet and others, but also in other fields of science, it was felt that in order for the fields of science remain attractive also -- so that they would receive the money -- they have to some extent, go According to this same philosophy.
Even those of the strongest hard sciences -- will be even physics -- so there is going a little meta physics direction -- even if the view of cosmology as an example of hard physics, it actually talks about issues, issues such as black holes, or madonrei'istä, or even further when you go, talk rinnakkaisuniversumeista, and so on.
These are the meta physics, these rinnakkaisuniversumit, for example, in the sense that by definition they are in our universe beyond.
We never can get them any information, no measurement data.
Traditional physics was based on empiricism, or the fact that the environment is measured, and attempts to build explanations for them what behavior is observed -- then all the theories which use rinnakkaisuniversumeita, they can not be a science in the sense that they can not be built on the measurement data, and the mere hypothesis, to a mathematical construction.
[1:17:19 / 43]
No, but something good in these kompleksisuusteorioissa is, and I think it is good it is this, that to some extent, given the freedom to intuition.
Richard Feynman, who invented kvanttielektrodynamiikkaa, namely the huge fine, or at least effective, the model alkeishiukkasten behavior, has said that it is simply can not understand it alkeishiukkasten world.
Must only rely on mathematics and models.
So, in this course, at least, and in general these kompleksisuustutkimuksissa, rely on the fact that simple formulas -- they have no value in itself, if they do not reflect the nature, or if they do not give any kind of further understanding of the natural behavior.
It is not enough to describe the nature, or is able -- or even that is already something if they are able to produce the measurements, or if they are able to make predictions of behavior, but the objective should be that they, indeed, in some way could it the philosopher's stone to shed light.
So, this is the approach.
This course is accepted now that in some sense, this is an acceptable approach, and an intuitive approach.
But nevertheless, it is considered that the extent to which, this math, the old, traditional mathematics is able to give us the tools for this intuition to flesh.
Well of course the problem is that all of us are just your own intuition.
[1:19:42 / 44]
This is precisely the problem, the verge, that how can we get anything other than the mares discourse about.
This is an old example, that if blind people have to sign on the elephant, so depending on the fact that what they are exploratory, so they get quite a different image from the elephant nature.
Someone who will test the tail, understand that this is now the reality must be described somehow köytenä or köysinä.
Someone who will try its feet, to understand that it is impossible for the forest.
And who will try the body, sees it as if kivimuurina, and the snout appears to käärmeeltä and brings fang or tusk appears keihäältä.
Everyone has their own approach and all of it because of seeing the same thing differently, from another perspective.
And that is why we need some kind of a concrete framework and that's now developing the next time.
In order to have a common language, regardless of the fact that the approaches are very different.
[1:21:08 / 45]
Well here is of this sort, we report on the laboratory, which is a chaos of these stories easy to find.
[1:21:18 / 46]
Indeed, returns to the now back to three, which here was the speech already, that is the way it is this, which has already been mentioned here, is the chaos theory of sustainable results.
Without taking a position on whether the theory of chaos in some way a dead end or not, so there are some lasting results that will be permanent, interest, achievements.
One is perhaps the Feigenbaumin bifurkaation universality, that is very different systems -- they have this bifurkaatio, even the repetition of the respect given a numerical law.
But perhaps the more attractive on this Sharkovskiin theorem that goes this way:
First, write the natural numbers in such a policy, where the first, we have all the odd numbers, up to infinity, and infinity then we will start even on issues like what the figures are deuces only once -- when it divided the figure into chapters beginning -- and so on will continue their two growing potency, up to infinity again here.
In this way, that all of their two power up to infinity has been reviewed and have been made to these incredibly chains.
Since then, the end is put to these Kakkonen power in descending order, as the final number one.
So, it really starts to three of the chain.
And Sharkovskiin theorem says that if we are able to find any constant iteration -- or if we have one of the function f is continuous, and iteroimme this function f, if we can find then in the cycle, whose length can be found here now, Sharkovskiin chain, so it means that everyone in the chain , in the chapter the right of the period lengths, are also on that iteration.
Selecting a suitable starting point, so keep track of it and its dimensions period, or it and its scale cycles.
This involves just this, so I mentioned that if the period is near the three, going from one iteration, all, all-scale cycles can be found for it.
[1:24:25 / 47]
An example of this is of this sort himself invented a continuous description, in other words, a simple model, in this next film is a description of its shape, ie, it goes this way, it is the second part, and then when it comes to this zero point, then goes, will continue downward.
Now we can see that if iteroimme -- the source for a point number one, moving, and iteroidaan it three times, then reached number one.
This is a continuous description, and this iteroidaan, and that really are three step-scale cycle.
According to the argument here -- if iteroidaan a suitable initial state -- can be found in any cycle length.
And it can be shown, or we can construct the chain, now this function, and it is here.
[1:25:42 / 48]
See you, that if we look at only those points for themselves, and not be interested in this convergence, so that we can move forward to do iteration, but will also go backwards, because if we are in a cycle, but also backwards to come to uniquely -- or should we find this same cycle length.
Eli ruvetaankin to examine the käänteiskuvausta.
Instead of examining this description of the non-linear representation, in its käänteiskuvausta, or rather, the descriptions, because this is, of course, not reversible, because this is not monotonic.
But we can write two monotone, and even a linear function, for this -- when this kind of reversal, it is seen that this is divided into two function which are both linear.
And now if iteroimme this -- the selection of a point here, and iteroidaan this -- so if it is a cycle, so it enters into a certain number of step, after this same point, or the same value.
So, now it is marked f1 has the ylähaaraa, and f2 has alahaaraa.
A function of number one is the ylähaara and function of two alahaara.
Since going back here, so we can choose an arbitrary, of either branch of it -- back into the right way iteroitaessa them both, however, returns to the same point.
We can rewind to look at the latest chain branches, in which either the elected leader in a function or a function of two, and requires that a certain x given a step beyond, returns to the same point.
[1:28:15 / 49]
This has now claimed four-step scale chain.
Suppose that at the first X applies the function ykköstä, then a function of deuces, a function of deuces and the function of deuces, and requires that, after the x is the same value.
Since this is a linear function, or both, f1 and f2, it is linear, we can write the open, in this case, and obtained in this manner, x is directly solved.
And this works, regardless of how long we have this chain, or how long the cycle is requested.
Now, the solution was that the four-step scale cycle from the point of 4 / 9.
You can try it.
And indeed, this four-step to reach the same point.
How is it possible that this can be found in all the cycles?
It now, but it is so real that has wonderful qualities that one can fit between the sufficient range of scores.
[1:29:34 / 50]
Well, could be yet another example of what kind of lasting results in chaos theory, namely the Benford law is the law of this sort that -- time semmoinen interesting, and quite useful in certain applications.
Technology & the economy was a question that was the accountant, who was given the task to examine whether any of the Group's financial statements credible.
It was a couple of dozen very occasional figures, which illustrate how this group of companies turnover -- just one-off women looking figures.
And it could be inferred, the time with great confidence that they were in the fictitious numbers.
Because the numbers were actually taken from a normal random number.
Natural systems, on issues like the numbers, are scattered so-called Benford law.
Frankly, on issues like itsesimilaariset -- may be to think that someone from the economy is itsesimilaarinen in the sense that it is small businesses, large businesses, etc., and is a fine structure, business exchanges, in all sizes.
Small businesses have a large number, this is like a fraktaalinen structural check.
And it is the manner in which the companies' turnover set up -- if they assumed a kind of fraktaalirakenteeksi, so they are in logarithmic scale occasionally.
So, if you look downward, or sticks, so you can see that the logarithmic scale is a scale of zero to infinity is quite different-looking than the standard scale.
Logaritmiasteikolla Ykkönen, the second interval is much longer than their two and three of the interim, not to mention some eight and nine joint.
If you randomly throw the company's turnover logaritmiasteikolle, there is a greater probability that it ends up excluding a number one first.
And this indeed -- for example, natural constants in this way as if the natural random numbers to cast any of the thread, and it can be argued that they form fraktaalirakenteen, and indeed, if you look at these figures, both here and has been number one to a much in this first number.
If, instead, look at prices or any other so this is no longer the case, but the nine that are very prominent in all prices.
[Question: How is this true, if the turnovers can be changed to another currency, so there was always the structure? ]
That same structure is found at all times.
It does not matter, because it is independent of scale -- you can change the scale in this exchange factor, and yet the same structure is maintained.
This scale independence is precisely this feature fraktaalisuuden that you take any of the zoom level, so there will always be qualitatively similar structure.
[Question: What if the number is changed the system? Disclose to another number system? ]
You mean binäärijärjestelmää or something ...?
[Yeah, or something to the system of four or five systems. ]
Binäärijärjestelmä is extremely simple in the sense that it always starts ykkösellä all.
[If we add, even though the view of one of the 50 systems or 60 system? ]
Yes, the same shall remain in force, although this number one, this frequency should be lower, but instead here -- that it would begin the 49 plays in the system, so it is highly unlikely.
Yes, the same laws are still valid, but you can -- it can really shape this like in figure leads to the system, just by looking directly to the tiuhassa logaritmiasteikolla these selected numbers.
Well, this was what all this time I thought to make.
Next time, based on the fact that instead of this now as if the mass, crowd, or the audience, perhaps bifurkoituivat, in the sense that can be understood, to accept any hypothesis, the goal would be just the reverse of this the other way around, trying to find semmoiset laws which also converges to in the way that everybody could accept them, and after then the next time I start to lead models ever since.
[1:35:42 / -]
(v.2009.04.10, cleaned from a rough machine translation to only about line 400 yet! marked with #)