#Andrey Kolmogorov

[thepastisalreadywritten]

(CNN) — The dappled starlight and swirling clouds of Vincent van Gogh’s “The Starry Night” are thought to reflect the artist’s tumultuous state of mind when he painted the work in 1889.

Now, a new analysis by physicists based in China and France suggests the artist had a deep, intuitive understanding of the mathematical structure of turbulent flow.

As a common natural phenomenon observed in fluids — moving water, ocean currents, blood flow, billowing storm clouds and plumes of smoke — turbulent flow is chaotic, as larger swirls or eddies, form and break down into smaller ones.

It may appear random to the casual observer, but turbulence nonetheless follows a cascading pattern that can be studied and, at least partially, explained using mathematical equations.

“Imagine you are standing on a bridge, and you watch the river flow. You will see swirls on the surface, and these swirls are not random.

They arrange themselves in specific patterns, and these kinds of patterns can be predicted by physical laws,” said Yongxiang Huang, lead author of the study that published Tuesday in the scientific journal Physics of Fluids.

Huang is a researcher at State Key Laboratory of Marine Environmental Science & College of Ocean and Earth Sciences at Xiamen University in southeastern China.

“The Starry Night” is an oil-on-canvas painting that, the study noted, depicts a view just before sunrise from the east-facing window of the artist’s asylum room at Saint-Rémy-de-Provence in southern France.

Van Gogh had admitted himself to an asylum there after mutilating his left ear.

Using a digital image of the painting, Huang and his colleagues examined the scale of its 14 main whirling shapes to understand whether they aligned with physical theories that describe the transfer of energy from large- to small-scale eddies as they collide and interact with one another.

‘The Starry Night’ and turbulence theories

The atmospheric motion of the painted sky cannot be directly measured, so Huang and his colleagues precisely measured the brushstrokes and compared the size of the brushstrokes to the mathematical scales expected from turbulence theories.

To gauge physical movement, they used the relative brightness or luminance of the varying paint colors.

They discovered that the sizes of the 14 whirls or eddies in “The Starry Night,” and their relative distance and intensity, follow a physical law that governs fluid dynamics known as Kolmogorov’s Theory of Turbulence.

In the 1940s, Soviet mathematician Andrey Kolmogorov (1903–1987) described a mathematical relationship between the fluctuations in a flow’s speed and the rate at which its energy dissipates.

Huang and the team also found that the paint, at the smallest scale, mixes around with some background swirls and whirls in a fashion predicted by turbulence theory, following a statistical pattern known as Batchelor’s scaling.

Batchelor’s scaling mathematically represents how small particles, such as drifting algae in the ocean or pieces of dust in the wind, are passively mixed around by turbulent flow.

“This is cool. Indeed this is the type of statistics you would expect from algae blooms being swept around by ocean currents, or dust and particulates in the air,” said James Beattie, a postdoctoral researcher in the Department of Astrophysical Sciences at Princeton University in New Jersey, in an email.

Beattie wasn’t involved in this study but has conducted similar research on the artwork.

“In my paper, I only ever really looked at the large (swirls in the painting), so I didn’t see this second relation,” he said, referring to the Batchelor’s scaling.

‘An amazing coincidence’

Of course, Huang said, van Gogh would not have been aware of such equations but likely he spent a lot of time observing turbulence in nature.

“I think this physical relationship must be embedded in his mind so that’s why when he made this famous ‘Starry Night’ painting, it mimics the real flow,” Huang said.

Beattie agreed: “It’s an amazing coincidence that Van Gogh’s beautiful painting shares many of the same statistics as turbulence,” he said.

“This makes some sense — the models have been constructed to try to capture the statistics of eddies and swirls on multiple scales, each swirl communicating with other swirls through the turbulent cascade.

In some sense, Van Gogh painted something that represents this phenomenon, so why shouldn’t there be some convergence between the theoretical models and the statistics of Van Gogh’s swirls?”

The study team performed the same analysis and detected the same phenomenon in two other images:

— a painting, “Chain Pier, Brighton,” created by British artist John Constable in 1826-7;

— a photograph of Jupiter’s Great Red Spot, taken by NASA’s Voyager 1 spacecraft on 5 March 1979.

“Unlike ‘The Starry Night,’ this painting lacks well-defined swirling patterns, but the clouds are rich of structures with different scales, resembling those frequently seen in the sky,” the study noted of Constable’s artwork.

On display at the Museum of Modern Art in New York, “The Starry Night” is an enormously popular work of art that has been recreated in Lego bricks, drones and dominoes.

Huang said that scientists had long struggled to describe turbulent flow in fluid dynamics in a way that would allow them to predict the phenomenon and that a complete explanation remains a prevailing mystery of physics.

A thorough understanding would help with weather forecasting, flight turbulence and many other processes, he said.

“Even after more than 100 years (of) study, we even don’t know how to define this complex phenomenon,” Huang said.

“It’s extremely important, but it’s extremely difficult.”

"The fact that “The Starry Night” matched statistical models of turbulence even though the artwork doesn’t actually move could suggest that the statistical methods and tools are less precise than scientists may have thought," Beattie said.

"The painting can’t be precisely measured because it’s “actually not turbulence. … (I)t has no kinetic energy,” he said.

However, Beattie said that he was a huge fan of the work of art and that it reflected universality and the beauty of turbulence.

“I deeply love the fact that I can take my understanding of the turbulence in the plasma between galaxies and apply it to the turbulence between stars, between Earth and the Sun or in our own lakes, oceans and atmosphere,” he said.

“What I take away from studies like this is that (van Gogh) captured some of this universality in the beautiful (‘Starry Night’),” Beattie added.

“And I think people know this. They know that something wonderful has been embedded in this painting and we are drawn to it.”

[wordacrosstime]

Limit Distributions for Sums of Independent Random Variables

[Limit Distributions for Sums of Independent Random Variables, by BV Gnedenko (Author), AN Kolmogorov (Author), KL Chung (Translator). Paperback (Reprint of the 1954 Edition) - 5 Aug 2021. Publisher - Martino Fine Books (5 Aug 2021). Language - English. Paperback - 276 pages. ISBN-10 - 1684225795. ISBN-13 - 978-1684225798. Dimensions - 15.6 x 1.57 x 23.39 cm. (details thanks to Amazon)]

[Author full names: Boris Vladimirovich Gnedenko (Russian: Бори́с Влади́мирович Гнеде́нко; 1 January 1912 – 27 December 1995); Andrey Nikolaevich Kolmogorov (Russian: Андре́й Никола́евич Колмого́ров, IPA: [ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf], 25 April 1903 – 20 October 1987) (details thanks to Wikipedia)]

***

A number of years ago I set about a course of self-improvement. For me that meant reading as much as possible from works written by world-class thinkers. This list of august minds was often, though not solely, driven by Nobel Prize winners and other winners of similar or equivalent awards in other areas (mathematics, for example, where the Fields Medal and Abel Prize are the highest honors). However, I am a statistician among other things, and so I have also kept a keen eye out for important works by the giants of that field.

I am not well-acquainted with Boris Gnedenko, though his name has certainly come up from time to time over the years that I’ve both studied and practiced statistics. The name that has come up repeatedly in reverential tones is Andrey Kolmogorov, who is one of a handful of mathematicians who are credited with giving the theoretical underpinnings of statistics a solid and rigorous mathematical foundation.

The volume under review here is one of Kolmogorov’s most important publications. In it, he and Gnedenko elicit the basis and application of results like the Central Limit Theorem (CLT) and the Law of Large Numbers (LLN). These are two of the most important results in all of statistics and probability theory and the slightly more general theory of stochastic (random) processes.

I should mention that this volume was originally published in the Soviet Union in 1949 in Russian. The edition I read here was translated into English in 1954 by KL Chung, and the actual ISBN citation is this particular reprint; there are others from different time periods, and of course if one is a true purist, then one should definitely read the original Russian text. However, according to the preface, all of the formulas and other special figures in the text are actually photostats from the original monograph.

I really enjoyed this book. The writing style of Gnedenko and Kolmogorov is delightfully lofty and at times quite flowery, but their explanations of proofs and motivations for results are really first-rate. I wasn’t sure what to expect, given the high intellectual bar these gentlemen set, but I was very pleasantly surprised. Professor Chung, the translator, added quite a few helpful comments and clarifications where appropriate, but overall he seems to have let the book flow as it was originally set down. One of the Appendices was written by another legendary statistician, JL Doob, providing a bit of bridge material related to some measure-theoretic issues that arise in Chapter 1.

My only complaint (a very small one to be sure) is that there were a number of typos that remained intact despite the extensive editorial review this monograph has surely had over the past 70+ years. I wouldn’t exactly call it a super-power, but I personally seem to have a knack for finding typos and grammatical errors in even very technical material. I’ve taken photos of two of the ones I found early in the text as examples here (see below). For some reason my eye was instinctively drawn to these and others. I would thus advise the reader to pause if something seems slightly amiss – it very well might be. I’m happy to say that these errata were few and far between, and in no case do I recall them rendering the explanations or results indecipherable.

Who would enjoy this tome? Anyone who is interested in the theoretical basis for many of the classic results in modern non-Bayesian statistics. The foreword by the translator advises that a solid background in calculus is a must, and mathematical maturity a definite plus. Beyond that, the book itself is remarkably self-contained. This is likely due to the political environment in which it was written. Post-World War II was a difficult time for Soviet intellectuals since they were generally denied access to journals and results outside their own country.

Here are the two examples of typos that I detected early in the book – I think they speak for themselves, but you can be the judge:

[Image credits numbered from top: (1 & 2) two pages from the book photographed by reviewer- with thanks to © publishers and estates of authors. (3 & 4) book cover front and back - with thanks to © publishers (5) Boris Vladimirovich Gnedenko - with thanks, no details of © copyright holder / photographer known (6) Andrey Nikolaevich Kolmogorov - with thanks, no details of © copyright holder / photographer known]

Kevin Gillette

Words Across Time

12 January 2024

wordsacrosstime

[metamatar]

In statistics, originally in geostatistics, kriging or Kriging, (pronounced /ˌˈkɹiːɡɪŋ/) also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging gives the best linear unbiased prediction (BLUP) at unsampled locations.[1] Interpolating methods based on other criteria such as smoothness (e.g., smoothing spline) may not yield the BLUP. The method is widely used in the domain of spatial analysis and computer experiments. The technique is also known as Wiener–Kolmogorov prediction, after Norbert Wiener and Andrey Kolmogorov.

The theoretical basis for the method was developed by the French mathematician Georges Matheron in 1960, based on the master's thesis of Danie G. Krige, the pioneering plotter of distance-weighted average gold grades at the Witwatersrand reef complex in South Africa. Krige sought to estimate the most likely distribution of gold based on samples from a few boreholes.

[jcmarchi]

Kolmogorov-Arnold Networks: The New Frontier in Efficient and Interpretable Neural Networks

New Post has been published on https://thedigitalinsider.com/kolmogorov-arnold-networks-the-new-frontier-in-efficient-and-interpretable-neural-networks/

Kolmogorov-Arnold Networks: The New Frontier in Efficient and Interpretable Neural Networks

Neural networks have been at the forefront of AI advancements, enabling everything from natural language processing and computer vision to strategic gameplay, healthcare, coding, art and even self-driving cars. However, as these models expand in size and complexity, their limitations are becoming significant drawbacks. The demands for vast amounts of data and computational power not only make them costly but also raise sustainability concerns. Moreover, their opaque, black-box nature hinders interpretability, a critical factor for wider adoption in sensitive fields. In response to these growing challenges, Kolmogorov-Arnold Networks are emerging as a promising alternative, offering a more efficient and interpretable solution that could redefine the future of AI.

In this article, we’ll take a closer look at Kolmogorov-Arnold Networks (KANs) and how they’re making neural networks more efficient and interpretable. But before we dive into KANs, it’s essential to first understand the structure of multi-layer perceptrons (MLPs) so that we can clearly see how KANs differentiate themselves from traditional approaches.

Understanding Multi-Layered Perceptron (MLP)

Multi-layer perceptrons (MLPs), also known as fully connected feedforward neural networks, are fundamental to the architecture of modern AI models. They consist of layers of nodes, or “neurons,” where each node in one layer is connected to every node in the next layer. The structure typically includes an input layer, one or more hidden layers, and an output layer. Each connection between nodes has an associated weight, determining the strength of the connection. Each node (except those in the input layer) applies a fixed activation function to the sum of its weighted inputs to produce an output. This process allows MLPs to learn complex patterns in data by adjusting the weights during training, making them powerful tools for a wide range of tasks in machine learning.

Introducing Kolmogorov-Arnold Networks (KANs)

Kolmogorov-Arnold Networks are a new type of neural networks making a significant shift in how we design neural networks. They are inspired by Kolmogorov-Arnold representation theorem, a mid-20th century mathematical theory developed by renowned mathematicians Andrey Kolmogorov and Vladimir Arnold. Like MLPs, KANs have a fully connected structure. However, unlike MLPs, which use fixed activation functions at each node, KANs utilize adjustable functions on the connections between nodes. This means that rather than merely learning the strength of the connection between two nodes, KANs learn the entire function that maps input to output. The function in KANs is not fixed; it can be more complex—potentially a spline or a combination of functions—and varies for each connection. A key distinction between MLPs and KANs lies in how they process signals: MLPs first sum the incoming signals and then apply non-linearity, whereas KANs first apply non-linearity to the incoming signals before summing them. This approach makes KANs more flexible and efficient, often requiring fewer parameters to perform similar tasks.

Why KANs are more Efficient than MLPs

MLPs follow a fixed approach to transform input signals into outputs. While this method is straightforward, it often requires a larger network—more nodes and connections—to handle the complexities and variations in data. To visualize this, imagine solving a puzzle with pieces of a fixed shape. If the pieces don’t fit perfectly, you need more of them to complete the picture, leading to a larger, more complex puzzle.

On the other hand, Kolmogorov-Arnold Networks (KANs) offer a more adaptable processing structure. Instead of using fixed activation functions, KANs employ adjustable functions that can change themselves to the specific nature of the data. To put it in the context of the puzzle example, think of KANs as a puzzle where the pieces can adapt their shape to fit perfectly into any gap. This flexibility means KANs can work with smaller computation graphs and fewer parameters, making them more efficient. For example, a 2-layer width-10 KAN can achieve better accuracy and parameter efficiency compared to a 4-layer width-100 MLP. By learning functions on the connections between nodes rather than relying on fixed functions, KANs demonstrate superior performance while keeping the model simpler and more cost-effective.

Why KANs are More Interpretable than MLPs

Traditional MLPs create intricate layers of relationships between incoming signals, which can obscure how decisions are made, particularly when handling large volumes of data. This complexity makes it difficult to trace and understand the decision-making process. In contrast, Kolmogorov-Arnold Networks (KANs) offer a more transparent approach by simplifying the integration of signals, making it easier to visualize how they are combined and contribute to the final output.

KANs make it easier to visualize how signals are combined and contribute to the output. Researchers can simplify the model by removing weak connections and using simpler activation functions. This approach can sometimes result in a concise, intuitive function that captures the KAN’s overall behavior and, in some cases, even reconstructs the underlying function that generated the data. This inherent simplicity and clarity make KANs more interpretable compared to traditional MLPs.

Potential of KANs for Scientific Discoveries

While MLPs have made significant advances in scientific discovery, such as predicting protein structures, forecasting weather and disasters, and aiding in drug and material discovery, their black-box nature leaves the underlying laws of these processes shrouded in mystery. In contrast, the interpretable architecture of KANs has the potential to reveal the hidden mechanisms that govern these complex systems, providing deeper insights into the natural world. Some of the potential use cases of KANs for scientific discoveries are:

Physics: Researchers have tested KANs on basic physics tasks by generating datasets from simple physical laws and using KANs to predict these underlying principles. The results demonstrate KANs’ potential to uncover and model fundamental physical laws, revealing new theories or validating existing ones through their ability to learn complex data relationships.

Biology and Genomics: KANs can be used to uncover the complex relationships between genes, proteins, and biological functions. Their interpretability also offers researchers the ability to trace gene-trait connections, opening new avenues for understanding gene regulation and expression.

Climate Science: Climate modeling involves the simulation of highly complex systems that are influenced by many interacting variables, such as temperature, atmospheric pressure, and ocean currents. KANs could enhance the accuracy of climate models by efficiently capturing these interactions without the need for excessively large models.

Chemistry and Drug Discovery: In chemistry, particularly in the field of drug discovery, KANs could be utilized to model chemical reactions and predict the properties of new compounds. KANs could streamline the drug discovery process by learning the intricate relationships between chemical structures and their biological effects, potentially identifying new drug candidates more quickly and with fewer resources.

Astrophysics: Astrophysics deals with data that is not only vast but also complex, often requiring sophisticated models to simulate phenomena like galaxy formation, black holes, or cosmic radiation. KANs could help astrophysicists model these phenomena more efficiently by capturing the essential relationships with fewer parameters. This could lead to more accurate simulations and help uncover new astrophysical principles.

Economics and Social Sciences: In economics and social sciences, KANs could be useful for modeling complex systems like financial markets or social networks. Traditional models often simplify these interactions, which can lead to less accurate predictions. KANs, with their ability to capture more detailed relationships, might help researchers better understand market trends, policy impacts, or social behaviors.

The Challenges of KANs

While KANs present a promising advancement in neural network design, they come with their own set of challenges. The flexibility of KANs, which allows for adjustable functions on connections rather than fixed activation functions, can make the design and training processes more complex. This added complexity can lead to longer training times and may require more advanced computational resources, which could diminish some of the efficiency benefits. This is primarily because, presently the KANs are not designed to take advantage of GPUs. The field is still relatively new, and there are not yet standardized tools or frameworks for KANs, which can make them harder for researchers and practitioners to adopt compared to more established methods. These issues highlight the need for ongoing research and development to address the practical hurdles and fully leverage the advantages of KANs.

The Bottom Line

Kolmogorov-Arnold Networks (KANs) offer a significant advancement in neural network design, addressing the inefficiencies and interpretability issues of traditional models like multi-layer perceptrons (MLPs). With their adaptable functions and clearer data processing, KANs promise greater efficiency and transparency, which could be transformative for scientific research and practical applications. While still in the early stages and facing challenges such as complex design and limited computational support, KANs hold the potential to reshape how we approach AI and its use in various fields. As the technology matures, it may provide valuable insights and improvements across many domains.

[jhavelikes]

In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that produces the object as output. It is a measure of the computational resources needed to specify the object, and is also known as algorithmic complexity, Solomonoff–Kolmogorov–Chaitin complexity, program-size complexity, descriptive complexity, or algorithmic entropy. It is named after Andrey Kolmogorov, who first published on the subject in 1963[1][2] and is a generalization of classical information theory. The notion of Kolmogorov complexity can be used to state and prove impossibility results akin to Cantor's diagonal argument, Gödel's incompleteness theorem, and Turing's halting problem. In particular, no program P computing a lower bound for each text's Kolmogorov complexity can return a value essentially larger than P's own length (see section § Chaitin's incompleteness theorem); hence no single program can compute the exact Kolmogorov complexity for infinitely many texts.

Kolmogorov complexity - Wikipedia

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Was it a chance encounter when you met that special someone or was there some deeper reason for it? What about that strange dream…. Powered by AutoBlogger.co

[siumerghe]

Hi! 16 and 26. Thanks

16. which stereotype about your country you hate the most and which one you somewhat agree with?

Most hated stereotype: vodka! Yeah, there are alcoholics in Russia, same as in other countries, and there are problems with alcohol in the Russian province, among poor people and especially among indigenous people of Siberia and Arctica who biologically lack alcohol-processing ferments.

However, personally, I don't know any hard-drinking people in Russia, neither among ethnic Russians nor among other nationalities. No one in my family drank vodka, even during various celebrations. People in the cities, especially younger generations, being more health-conscious, usually don't drink much. "But you are Russian! Why are you not drinking/drinking so little? Drink more!" Ugh. Me being from Russia doesn't mean I enjoy drinking or have a high alcohol tolerance!

Oh, and also Russians never say "Na zdorovie!" when toasting. It means "My pleasure" or "You are welcome" and is a response to "Thank you". A mistake I'm tired of correcting my non-Russian-speaking friends. As a toast, it sounds incredibly stupid. Thanks, Hollywood, for inventing this nonsense.

Russia is not a country of vodka, it's a country of black tea, with an average Russian citizen (including ethnic Russians and other nationalities) drinking 3 cups of tea per day. Every meal ends with tea (brewed in a small kettle from loose tea leaves and then mixed with hot water), and it’s not just tea - it’s tea with something: sweets, sandwiches, pies, blini, dried fruits... Tea without food is called пустой чай - “empty tea”. Tea is also drunk in-between the meals, with something or on its own. So, you wanna picture a stereotypical Russian? Forget the vodka, go for the black tea instead!

The stereotype I somewhat agree with: Russians are good at math. This one is true. The mathematical education in the USSR was largely shaped by Andrey Kolmogorov, one of the world's most prominent mathematicians. The Soviet education system was stellar in exact sciences, and its heritage is still alive in present-day Russia. Russian schools put extra effort into teaching math while focusing on problem-solving (not “see and repeat”), Russian math program is in general more difficult than in the West (what is a school-level material in Russia, in the US is taught in college), there are prestigious (and free) math schools for gifted children, and Russian universities have the best math courses (which are available on youtube - but in Russian). 

26. does your nationality get portrayed in Hollywood/American media? what do you think about the portrayal?

It does - unfortunately. I wish Hollywood forget that Russia exists. There are all kinds of offensive and harmful stereotypes (all Russian men are either alcoholics or KGB agents, all Russian women are prostitutes), incorrectly understood folklore and culture, crazy errors in geography, history, fashion, and whatnot. All Hollywood Russians are evil. And if not, then a typical Hollywood Russian man dies painfully as a way to atone for the crime of being Russian. A typical Hollywood Russian woman betrays her country and/or becomes the lover of the hero (who is American, of course). And all of them drink vodka like water and say "Na zdorovie!"

It’s also a bit strange when ethnic Russians are portrayed by actors with distinct White American or Western-European facial features. Native people easily notice such details. It’s like seeing a Japanese being portrayed by a Chinese, or an Afro-French being portrayed by an Afro-American. 

Western movies about Russia and Russians that I like: Eastern Promises 2007, Chernobyl 2019. The latter even got ethnic Russian and other Soviet nationalities types right (except for 1 - only 1! - actor with a rather Anglo-Saxon face).

“hi, I’m not from the US” ask set

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[thedalatribune]

© Paolo Dala

Illustration by Paolo Dala (2019)

Turbulence X Vincent Van Gogh

One of the most remarkable aspects of the human brain is its ability to recognize patters and describe them. Among the hardest patterns we’ve tried to understand is the concept of turbulent flow in fluid dynamics. The German Physicist Werner Heisenberg said:

“When I meet God I’m going to ask him two questions: Why relativity and why turbulence? I really believe he will have an answer for the first.”

As difficult as turbulence is to understand mathematically, we can use art to depict the way it looks.

In June 1889, Vincent van Gogh painted the view just before sunrise from the window of his room at the St. Paul de Mausole Asylum in Saint Rémy de Provence (where he admitted himself after mutilating his own ear in a psychotic episode). In the ‘Starry Night’, his circular brushstrokes crate a night sky filled whit swirling clouds and eddies of stars. 

Van Gogh and other impressionists represented light in a different way than their predecessors, seeming to capture its motion... The effect is caused by luminance, the intensity of the light in the colors of the canvas. The more primitive part of our visual cortex, which sees light contrast, and motion, but not color, will ben two differently colored areas together if they have the same luminance; but out brains’ primate subdivision will the contrasting colors without blending. With these two interpretations happening at once, the light in many Impressionist works seems to pulse, flicker and radiate oddly...

Sixty (60) years later, Russian Mathematician Andrey Kolmogrov furthered our mathematical understanding of turbulence when he proposed that energy in a turbulent fluid at length R varies in proportion to the five-thirds (5/3) power of R. Experimental measurements show Kolmogorov was remarkably close to the way turbulent flow works, although a complete description of turbulence remains one of the unsolved problems  in physics. A turbulent flow is self-similar if there is an energy cascade. In other words, big eddies transfer their energy to smaller eddies, which do likewise at other scales. Examples of this includes Jupiter’s ‘Great Red Spot’, cloud formations, and interstellar dust particles. 

In 2004 using the Hubble Space Telescope, scientists saw the eddies of dust and gas around a star, and it reminded them of van Gogh’s ‘Starry Night’. This motivated scientists from Mexico, Spain, and England to study the luminance in van Gogh’s paintings in detail. They discovered that there is a distinct patter of turbulent fluid structures close to Kolmogorov’s Equation hidden in may of van Gogh’s paintings. The researchers digitized the paintings, and measured how brightness varies between any two pixels. From the curves measured for pixel separations, they concluded that paintings from van Gogh’s period of psychotic agitation behave remarkably similar to fluid turbulence. His ‘Self-portrait with a Pipe’, from a calmer period in van Gogh’s life, showed no sign of this correspondence. And neither did other artists’ work that seemed equally turbulent at first glance, like Munch’s ‘The Scream’. 

While it’s too easy to say van Gogh’s turbulent genius enabled him to depict turbulence, it’s also far too difficult to accurately express the rousing beauty of the fact that in a period of intense suffering, van Gogh was somehow able to perceive and represent one of the most supremely difficult concepts nature has ever brought before mankind, and to unite his unique mind’s eye with the deepest mysteries of movement, fluid and, light.

Natalya St. Clair The Unexpected Math behind van Gogh's ‘Starry Night’

[semechkablack]

the same old story about andrey kolmogorov, my friends. the amount of shit he came up with? incredible. i will not get into it but no matter what class i had... he was everywhere. when i had trouble with functional analysis i picked up the book he wrote and let me tell you. he is the most comprehensible soviet mathematician out there. and what do you think. he lived his whole life with his friend pavel alexandrov. you know how they met?

"Their friendship began in the summer of 1929, when Andrey Nikolaevich [Kolmogorov], who by that time had some experience with boat trips, became the organizer of a trip along the Volga from Yaroslavl to Samara. The second participant in the 1300 km trip was one of Kolmogorov's school friends. "It is still not entirely clear to me how I managed to offer Pavel Sergeevich [Alexandrov] to be the third companion. However, he agreed immediately", writes Andrey Nikolaevich. They returned from the trip with a ready-made decision to live together." (x) lmao

kolmgorov was also a founder of a large scientific school! "...they [he and Alexandrov] wanted to recreate at their dacha in Komarovka near Moscow. They invited their students there for hiking and skiing, listening to music and making mathematical conversations" :')

some more excerpts from an article about him:

"The scientific world perceived Kolmogorov and Alexandrov as a couple. They worked together, rested together in the sanatoriums of the Academy of Sciences and sent food parcels to besieged Leningrad." (x)

"The man with whom Andrey Kolmogorov shared a home from 1929 until the end of his life was the topologist Pavel Alexandrov. Five years after they began to live together, male homosexuality in the USSR was outlawed. Kolmogorov and Alexandrov, who called themselves friends, practically did not make a secret of their relationship and nevertheless had no problems with the law." (x)

in 1981, Alexandrov wrote: “My friendship with A. N. Kolmogorov occupies an absolutely exclusive, unique place in my life; in 1979 this friendship had its fiftieth anniversary, and over the entire half-century period, not only did it not give a crack, but was not even accompanied by any quarrel..." (x)

Kolmogorov said, “For me, these fifty-three years of our close and unbreakable friendship were the basis for the fact that my entire life as a whole turned out to be filled with happiness, and the basis of my well-being was the unremitting solicitude on the part of Pavel Sergeevich.” :'))

AND kolmogorov started his letters to alexandrov with “Милый Пусик!” which is... a lot. basically, a really rough translation here, it means “my cute babe/sweetie” :’)

like, yeah, these iconic gay math kings! good for them

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