# Glossari

Aquest glossari és una mostra dels camps més representatius en què Andrei Kolmogórov va treballar.

#### Algorithmic information theory

Algorithmic information theory (AIT) is a branch of theoretical computer science that concerns itself with the relationship between computation and information of computably generated objects (as opposed to stochastically generated), such as strings or any other data structure. In other words, it is shown within algorithmic information theory that computational incompressibility "mimics" (except for a constant that only depends on the chosen universal programming language) the relations or inequalities found in information theory. According to Gregory Chaitin, it is "the result of putting Shannon's information theory and Turing's computability theory into a cocktail shaker and shaking vigorously."

Citació: "Algorithmic information theory" A: *Wikipedia*. Wikimedia Foundation, 2011. [en línia]. [Consulta: 21d'octubre 2021]. Disponible a: <https://en.wikipedia.org/wiki/Algorithmic_information_theory>

#### Classical mechanics

Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mechanics, if the present state is known, it is possible to predict how it will move in the future (determinism), and how it has moved in the past (reversibility).

The earliest development of classical mechanics is often referred to as Newtonian mechanics. It consists of the physical concepts based on foundational works of Sir Isaac Newton, and the mathematical methods invented by Gottfried Wilhelm Leibniz, Joseph-Louis Lagrange, Leonhard Euler, and other contemporaries, in the 17th century to describe the motion of bodies under the influence of a system of forces. Later, more abstract methods were developed, leading to the reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. These advances, made predominantly in the 18th and 19th centuries, extend substantially beyond earlier works, particularly through their use of analytical mechanics. They are, with some modification, also used in all areas of modern physics.

Citació: "Classical mechanics" A: *Wikipedia*. Wikimedia Foundation, 2011. [en línia]. [Consulta: 21d'octubre 2021. Disponible a: <https://en.wikipedia.org/wiki/Classical_mechanics>

#### Dynamical System

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

At any given time, a dynamical system has a state given by a tuple of real numbers (a vector) that can be represented by a point in an appropriate state space (a geometrical manifold). The evolution rule of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a given time interval only one future state follows from the current state. However, some systems are stochastic, in that random events also affect the evolution of the state variables.

Citació: "Dynamical system" A: *Wikipedia*. Wikimedia Foundation, 2011. [en línia]. [Consulta: 21d'octubre 2021. Disponible a: <https://en.wikipedia.org/wiki/Dynamical_system>

#### Intuitionistic logic

Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not include the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic.

Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic.

Citació: "Intuitionistic logic" A: *Wikipedia*. Wikimedia Foundation, 2011. [en línia]. [Consulta: 21d'octubre 2021. Disponible a: <https://en.wikipedia.org/wiki/Intuitionistic_logic>

#### Kolmogorov complexity

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.

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; hence no single program can compute the exact Kolmogorov complexity for infinitely many texts.

Citació: "Kolmogorov complexity" A: *Wikipedia*. Wikimedia Foundation, 2011. [en línia]. [Consulta: 21d'octubre 2021. Disponible a: <https://en.wikipedia.org/wiki/Kolmogorov_complexity>

#### Kolmogorov–Arnold–Moser theorem (KAM Theory)

The Kolmogorov–Arnold–Moser (KAM) theorem is a result in dynamical systems about the persistence of quasiperiodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the perturbation theory of classical mechanics.

The problem is whether or not a small perturbation of a conservative dynamical system results in a lasting quasiperiodic orbit. The original breakthrough to this problem was given by Andrey Kolmogorov in 1954. This was rigorously proved and extended by Jürgen Moser in 1962 (for smooth twist maps) and Vladimir Arnold in 1963 (for analytic Hamiltonian systems), and the general result is known as the KAM theorem.

Arnold originally thought that this theorem could apply to the motions of the solar system or other instances of the n-body problem, but it turned out to work only for the three-body problem because of a degeneracy in his formulation of the problem for larger numbers of bodies. Later, Gabriella Pinzari showed how to eliminate this degeneracy by developing a rotation-invariant version of the theorem.

Citació: "Kolmogorov–Arnold–Moser Theorem" A: *Wikipedia*. Wikimedia Foundation, 2011. [en línia]. [Consulta: 21d'octubre 2021. Disponible a: <https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%80%93Moser_theorem>

#### Kolmogorov's zero–one law

In probability theory, Kolmogorov's zero–one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, called a tail event, will either almost surely happen or almost surely not happen; that is, the probability of such an event occurring is zero or one.

Citació: "Kolmogorov's zero–one law" A: *Wikipedia*. Wikimedia Foundation, 2011. [en línia]. [Consulta: 21d'octubre 2021. Disponible a: <https://en.wikipedia.org/wiki/Kolmogorov%27s_zero%E2%80%93one_law>

#### Kolmogorov–Smirnov test

In statistics, the Kolmogorov–Smirnov test (K–S test or KS test) is a nonparametric test of the equality of continuous (or discontinuous, see Section 2.2), one-dimensional probability distributions that can be used to compare a sample with a reference probability distribution (one-sample K–S test), or to compare two samples (two-sample K–S test). It is named after Andrey Kolmogorov and Nikolai Smirnov.

Citació: "Kolmogorov–Smirnov test" A: *Wikipedia*. Wikimedia Foundation, 2011. [en línia]. [Consulta: 21d'octubre 2021. Disponible a: <https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test>

#### Márkov processes

A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). It is named after the Russian mathematician Andrey Markov.

Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics.

Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distributions, and have found application in Bayesian statistics, thermodynamics, statistical mechanics, physics, chemistry, economics, finance, signal processing, information theory and speech processing.

The adjectives Markovian and Markov are used to describe something that is related to a Markov process.

Citació: "Markov process" A: *Wikipedia*. Wikimedia Foundation, 2011. [en línia]. [Consulta: 21d'octubre 2021. Disponible a: <https://en.wikipedia.org/wiki/Markov_chain>

#### Mathematical statistics

Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure theory.

Citació: "Mathematical statistics" A: *Wikipedia*. Wikimedia Foundation, 2011. [en línia]. [Consulta: 21d'octubre 2021. Disponible a: <https://en.wikipedia.org/wiki/Mathematical_statistics>

#### Probability theory

Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion. Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.

As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics or sequential estimation. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics.

Citació: "Probability theory" A: *Wikipedia*. Wikimedia Foundation, 2011. [en línia]. [Consulta: 21d'octubre 2021. Disponible a: <https://en.wikipedia.org/wiki/Probability_theory>

#### Topology

In mathematics, topology is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles.

The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed.

Citació: "Topology" A: *Wikipedia*. Wikimedia Foundation, 2011. [en línia]. [Consulta: 21d'octubre 2021. Disponible a: <https://en.wikipedia.org/wiki/Topology>

#### Turbulence

In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers.

Turbulence is commonly observed in everyday phenomena such as surf, fast flowing rivers, billowing storm clouds, or smoke from a chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent. Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason turbulence is commonly realized in low viscosity fluids. In general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other, consequently drag due to friction effects increases. This increases the energy needed to pump fluid through a pipe.

The onset of turbulence can be predicted by the dimensionless Reynolds number, the ratio of kinetic energy to viscous damping in a fluid flow. However, turbulence has long resisted detailed physical analysis, and the interactions within turbulence create a very complex phenomenon. Richard Feynman has described turbulence as the most important unsolved problem in classical physics.

The turbulence intensity effect many fields, for examples fish ecology, air pollution and precipitation.

Citació: "Turbulence" A: *Wikipedia*. Wikimedia Foundation, 2011. [en línia]. [Consulta: 21d'octubre 2021. Disponible a: <https://en.wikipedia.org/wiki/Turbulence>