What is Kolmogorov: Definition and 18 Discussions

Andrey Nikolaevich Kolmogorov (Russian: Андре́й Никола́евич Колмого́ров, IPA: [ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf] (listen), 25 April 1903 – 20 October 1987) was a Soviet mathematician who contributed to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity.

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  1. Killtech

    I Bell vs Kolmogorov: Unravelling Probability Theory Limits

    For some reason on this forums here I found some weird opinions that Bell's inequality were somehow disqualifying classical probability theory in general rather then showing merely the limits of locality. I do no understand where that misunderstanding comes from, so I decided to find out. Let's...
  2. Killtech

    I Quantum physics vs Probability theory

    Because I do have a background in the latter it was originally very difficult for me to understand some aspects of QP (quantum physics) when I initially learned it. More specifically whenever probabilities were involved I couldn’t really make full sense of it while I never had any problems...
  3. F

    A Understanding Kolmogorov-Smirnov Test Results for Random Number Generators

    Hi all, I have some doubts regarding the Kolmogorov test: I made a simple c++ program generating two samples of random numbers following a Landau distribution (I used the "hit and miss" method). I made the Kolmogorov test, in order to check the randomness of the generator, but I'm having some...
  4. ATY

    Structure function in turbulence

    Hey guys, I need your help. I have to write a homework and we are actually looking at structure functions (Kolmogorov 41 and so on), but I have no clue what it does. So I have formulas, but I do not understand what a structure function is. Is it something in the "real world" or is it just a way...
  5. zrek

    I What "language" means in Kolmogorov complexity?

    Please help me to understand the answer I found on mathoverflow. The question was: "Do all uncountable sets contain elements with infinite Kolmogorov complexity?" The reasoning is clear for me, but I'd like to understand every word of the answer, which is the following: "...given a language...
  6. P

    Kolmogorov Entropy: Intuitive Understanding & Relation to Shannon Entropy

    Dear all, I'm trying to have an intuition of what Kolmogorov Entropy for a dynamical system means. In particular, 1. what is Kolmogorov Entropy trying to quantify? (what are the units of KS Entropy, is it bits or bits/seconds). 2. What is the relation of KS entropy to Shannon Entropy? 3. In...
  7. S

    Chapman Kolmogorov Th. - generality?

    Does the Chapman - Kolmogorov equation hold for an arbitrary stochastic process? The current wikipedia article on "Stochastic process" ( https://en.wikipedia.org/wiki/Stochastic_process ) seems to say that the Chapman-Kolmogorov equation is not valid for an arbitrary stochastic process. I...
  8. Fooality

    Help with Kolmogorov Complexity proof

    Hey, has anyone here done Kolmogorov complexity? I hadn't, and I was just reading the Wikipedia page to start learning it. The core concept makes sense: The Kolmogorov complexity for a a string s, K(s) is equivalent to the length of a program that generates that string (It can be any math...
  9. C

    Can the Kolmogorov definition of the conditional probability be proven

    It seems weird that such a relatively complex concept is simply given as a definition in most textbooks and then dismissed for further explanation other than using it intact or as a basis for further proofs.
  10. J

    Steady State Solution of Forward Kolmogorov Equation

    Essentially, this looks like a differential equation problem but being rusty on differential equations I am a little stuck. Homework Statement Consider the following SDE d\sigma = a(\sigma,t)dt + b(\sigma,t)dW The Forward Equation (FKE) is given by \frac{\partial p}{\partial t} =...
  11. J

    Kolmogorov smirnov in r - cannot compute correct p-values with ties

    Hi, I'm trying to use the kolmogorov smirnov test in in R to compare one distribution with another. I'm getting the following error: cannot compute correct p-values with ties I think this is because the dataset that I am using has in the first instance around 2000 values, and the second...
  12. N

    K41 Kolmogorov scaling relation and structure functions (turbulence)

    Hi there. I am having trouble interpreting the Kolmogorov K41 scaling relation for homogeneous and isotropic turbulence: S_{p}(l) = <\delta u(l)^{p}> = <|u(r+l) - u(r)|^{p}> \propto (\epsilon l)^{p/3} where l is the length of displacement between two points under consideration in the...
  13. Z

    Kolmogorov & Fomin's Elements of Theory: Real Analysis or Lebesque?

    I'm looking for a Real Analysis book to start with, besides Spivak. On Amazon, one of the reviewers said it was good as a subsequent book for learning Functional Analysis/Lebesque Integration, while another said it was a good introduction to Real Analysis. For those of you that have read it...
  14. D

    Solution of the Kolmogorov forward equation for a linear process

    Consider the 1-D linear system governed by: "dx/dt = a*x(t) + n(t)" where "a" is a scalar and: x(t) = system state n(t) ~ N(0, sigma^2) **************************** We can write Ito's stochastic differential equation of the previous process as: "dx = a*x*dt + 1/2*sigma^2*dW_t"...
  15. S

    Definition of a Kolmogorov Space

    Homework Statement Can a T_{0} topological space be created from the indiscrete topology of a set with only one element?
  16. D

    What are the differences between Blum and Kolmogorov complexities?

    Are they equivalent in some sense?
  17. S

    Kolmogorov complexity of SAT

    I'm not entirely clear on the concept of kolmogorov complexity. Does it mean that the a certain string is complex if there is no combinatorial (not sequential) circuit which outputs that string or does it mean that a certain string is complex if there is no program which can output that string...
  18. T

    Are Strings Kolmogorov-Random Actually Random?

    Are strings that are Kolmogorov-random actually random in the usual sense? An incompressible string probably won't contain this post anywhere in it, but a random one can.