Is it implicit that without randomness, there is no such a thing as scale invariance? He showed fractals and the atmospheric convection, is chaotic behaviour enough for this type of invariance? What other examples exists of such property? He chose constraints or boundaries as practical limits of infinity of real physical systems, and then showed the scale invariance of brownian motion. Is it assumed you only need to choose a scale interval large enough but not theoretical infinity to prove invariance? Does scale invariance means that out of the physical range of scales there is a moment where even the most chaotic systems become smooth and boring? At least the ones modelling real physical phenomena?
"Is it implicit that without randomness, there is no such a thing as scale invariance?" In general, the answer is no. Hairer decided to narrow scope of this talk to system with randomness, because he is interested in this topic and it is big enough to full whole talk.
Normaly scale invariance means that system behave the same at every possible scale. Which is clearly not the case, as shown by Hairer on the example of bug and human being trying walk on water. Point of this talk is rather, that there is a wide range of phenomena that you can rescale in such way that they looks, at this particular scale, as brownian motion. At the smaller scale such systems would usually looks different, so we don't have scale invariance in previously mentioned sense. At least, this is what I understand from his talk.
"Does scale invariance means that out of the physical range of scales there is a moment where even the most chaotic systems become smooth and boring?" If I understand this question correctly, the answer is no. Starting from the fact that brownian motion is about behavior of some random quntity as time progress. Example: what was stock price at 3 o'clock today? In such cases brownian motion looks very universal. But, what happens if your quantity depends on more things that time? Simple example of such quantity is random tetris shown around 40:25. Quantity that you look at is a maximal height that one tetris block reach at given time and given position on the line. So we have two quantities: time and position on line. As Hairer discuss at 45:15, in the case that you have random phenomena that depends on time and one parameter describing position in the space, our best guess is that this phenomena look NOT like brownian motion, but as one of two systems. Either as Edwards-Wilkinson system, which is generalization of brownian motion (kind of) or as Kardar-Parisi-Zhang system. But in our world we have time and THREE space coordinates, which means that we should expect much more such phenomena that in systems with time and ONE space coordinate. This is only tipe of the iceberg and I can spend much more time on this topic, even when I don't have 1/1000 knowledge that Hairer has. chempedia.info/info/edwards_wilkinson/
My own reply to this is: I am still very naïve with the power of randomness and how to spot patterns and symmetries even in the most chaotic systems when the scale changes dramatically. Work to do: To study Ramsey theory, as it seems it can answer (at least partially) my questions.
@@franciscodanieldiazgonzale2096 I have a question, did you study probability theory? Since it is cornerstone to all discussion about randomness in mathematics.
There is a branch of mathematics, functional analysis, that is very similar in name to one subject studied at musics schools. Imagin some musician listing for the course, that starts with remainder from set theory and topology and he wonder where he is?
"Reasonable" "Unreasonable" Unity Eternity Quantum Reciproction-recirculation Singularity relative-timing axial-tangential i-r interference positioning, or the eternal-instantaneous potential possibilities of real-time e-Pi-i sync-duration connectivity function. Flash recognition of Mathematical residual certainty in uncertainty => naturally occurring Disproof Methodology, a Reasonable empirical probability Philosophy of some condensed matter of Precedent in infinite potential possibilities. Limit 1-0-infinity rate-of-timing of/by e-Pi-i sync-duration. "A Rose by any other name will smell as Sweet" or, "There's no new news only new angles". Everywhen everywhere all-ways all-at-once sync-duration Eternity-now sync-print positioning zero-infinity Interval.., the context of imaginary mathematical reasoning or i-reflection containment Origin of elemental pure potential motion-> Reciproction-recirculation Singularity positioning. This lecture is providing the fractal Form prelude concepts to an assembly of observational high quality seeing.
I love Martin Hairer's talks.
mathematics is very important for us all
More public lecture!
Thank you so much 🙏
Thank you!
I love math
Me too.
Is it implicit that without randomness, there is no such a thing as scale invariance? He showed fractals and the atmospheric convection, is chaotic behaviour enough for this type of invariance? What other examples exists of such property? He chose constraints or boundaries as practical limits of infinity of real physical systems, and then showed the scale invariance of brownian motion. Is it assumed you only need to choose a scale interval large enough but not theoretical infinity to prove invariance? Does scale invariance means that out of the physical range of scales there is a moment where even the most chaotic systems become smooth and boring? At least the ones modelling real physical phenomena?
"Is it implicit that without randomness, there is no such a thing as scale invariance?"
In general, the answer is no. Hairer decided to narrow scope of this talk to system with randomness, because he is interested in this topic and it is big enough to full whole talk.
Normaly scale invariance means that system behave the same at every possible scale. Which is clearly not the case, as shown by Hairer on the example of bug and human being trying walk on water. Point of this talk is rather, that there is a wide range of phenomena that you can rescale in such way that they looks, at this particular scale, as brownian motion. At the smaller scale such systems would usually looks different, so we don't have scale invariance in previously mentioned sense.
At least, this is what I understand from his talk.
"Does scale invariance means that out of the physical range of scales there is a moment where even the most chaotic systems become smooth and boring?"
If I understand this question correctly, the answer is no.
Starting from the fact that brownian motion is about behavior of some random quntity as time progress. Example: what was stock price at 3 o'clock today? In such cases brownian motion looks very universal. But, what happens if your quantity depends on more things that time? Simple example of such quantity is random tetris shown around 40:25. Quantity that you look at is a maximal height that one tetris block reach at given time and given position on the line. So we have two quantities: time and position on line. As Hairer discuss at 45:15, in the case that you have random phenomena that depends on time and one parameter describing position in the space, our best guess is that this phenomena look NOT like brownian motion, but as one of two systems. Either as Edwards-Wilkinson system, which is generalization of brownian motion (kind of) or as Kardar-Parisi-Zhang system.
But in our world we have time and THREE space coordinates, which means that we should expect much more such phenomena that in systems with time and ONE space coordinate.
This is only tipe of the iceberg and I can spend much more time on this topic, even when I don't have 1/1000 knowledge that Hairer has.
chempedia.info/info/edwards_wilkinson/
My own reply to this is: I am still very naïve with the power of randomness and how to spot patterns and symmetries even in the most chaotic systems when the scale changes dramatically. Work to do: To study Ramsey theory, as it seems it can answer (at least partially) my questions.
@@franciscodanieldiazgonzale2096 I have a question, did you study probability theory? Since it is cornerstone to all discussion about randomness in mathematics.
The sum of two consecutive numbers is 35.if first number are 3\2 of second number.find the number.
If both the numbers are consecutive then how can one be 3/2 of the second?
@@mohammadareeb1289 tell me answer. Don't teach me English...
14 and 21?
@@maliniaravindan5349 Yes
My dream Oxford...but it is impossible now....
It's not. We are watching interresting things. There's more qualitatity lectures, cources online then you could ever master🙃
Why is this impossible now?
thought this was going to be about music scales
Me too
That music scale use word "scale" like this topic, is not coincidence.
There is a branch of mathematics, functional analysis, that is very similar in name to one subject studied at musics schools. Imagin some musician listing for the course, that starts with remainder from set theory and topology and he wonder where he is?
👍👏👍👏👍
"Reasonable" "Unreasonable"
Unity Eternity Quantum Reciproction-recirculation Singularity relative-timing axial-tangential i-r interference positioning, or the eternal-instantaneous potential possibilities of real-time e-Pi-i sync-duration connectivity function. Flash recognition of Mathematical residual certainty in uncertainty => naturally occurring Disproof Methodology, a Reasonable empirical probability Philosophy of some condensed matter of Precedent in infinite potential possibilities. Limit 1-0-infinity rate-of-timing of/by e-Pi-i sync-duration.
"A Rose by any other name will smell as Sweet" or, "There's no new news only new angles".
Everywhen everywhere all-ways all-at-once sync-duration Eternity-now sync-print positioning zero-infinity Interval.., the context of imaginary mathematical reasoning or i-reflection containment Origin of elemental pure potential motion-> Reciproction-recirculation Singularity positioning.
This lecture is providing the fractal Form prelude concepts to an assembly of observational high quality seeing.