Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Soviet
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
who contributed to the mathematics of
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 ...
,
topology
In mathematics, topology (from the Greek words , and ) 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 ...
,
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 ...
,
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 ...
,
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 classi ...
,
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 str ...
and
computational complexity.
Biography
Early life
Andrey Kolmogorov was born in
Tambov
Tambov (, ; rus, Тамбов, p=tɐmˈbof) is a city and the administrative center of Tambov Oblast, central Russia, at the confluence of the Tsna and Studenets Rivers, about south-southeast of Moscow. Population: 280,161 ( 2010 Census); 29 ...
, about 500 kilometers south-southeast of
Moscow
Moscow ( , US chiefly ; rus, links=no, Москва, r=Moskva, p=mɐskˈva, a=Москва.ogg) is the capital and largest city of Russia. The city stands on the Moskva River in Central Russia, with a population estimated at 13.0 millio ...
, in 1903. His unmarried mother, Maria Y. Kolmogorova, died giving birth to him. Andrey was raised by two of his aunts in
Tunoshna (near
Yaroslavl
Yaroslavl ( rus, Ярослáвль, p=jɪrɐˈsɫavlʲ) is a city and the administrative center of Yaroslavl Oblast, Russia, located northeast of Moscow. The historic part of the city is a World Heritage Site, and is located at the confluenc ...
) at the estate of his grandfather, a well-to-do
nobleman
Nobility is a social class found in many societies that have an aristocracy. It is normally ranked immediately below royalty. Nobility has often been an estate of the realm with many exclusive functions and characteristics. The characteris ...
.
Little is known about Andrey's father. He was supposedly named Nikolai Matveevich Kataev and had been an
agronomist
An agriculturist, agriculturalist, agrologist, or agronomist (abbreviated as agr.), is a professional in the science, practice, and management of agriculture and agribusiness. It is a regulated profession in Canada, India, the Philippines, the ...
. Kataev had been exiled from
St. Petersburg
Saint Petersburg ( rus, links=no, Санкт-Петербург, a=Ru-Sankt Peterburg Leningrad Petrograd Piter.ogg, r=Sankt-Peterburg, p=ˈsankt pʲɪtʲɪrˈburk), formerly known as Petrograd (1914–1924) and later Leningrad (1924–1991), i ...
to the Yaroslavl province after his participation in the revolutionary movement against the
tsar
Tsar ( or ), also spelled ''czar'', ''tzar'', or ''csar'', is a title used by East and South Slavic monarchs. The term is derived from the Latin word ''caesar'', which was intended to mean "emperor" in the European medieval sense of the ter ...
s. He disappeared in 1919 and was presumed to have been killed in the
Russian Civil War
{{Infobox military conflict
, conflict = Russian Civil War
, partof = the Russian Revolution and the aftermath of World War I
, image =
, caption = Clockwise from top left:
{{flatlist,
*Soldiers ...
.
Andrey Kolmogorov was educated in his aunt Vera's village school, and his earliest literary efforts and mathematical papers were printed in the school journal "The Swallow of Spring". Andrey (at the age of five) was the "editor" of the mathematical section of this journal. Kolmogorov's first mathematical discovery was published in this journal: at the age of five he noticed the regularity in the sum of the series of odd numbers:
etc.
In 1910, his aunt adopted him, and they moved to Moscow, where he graduated from
high school
A secondary school describes an institution that provides secondary education and also usually includes the building where this takes place. Some secondary schools provide both '' lower secondary education'' (ages 11 to 14) and ''upper seconda ...
in 1920. Later that same year, Kolmogorov began to study at
Moscow State University
M. V. Lomonosov Moscow State University (MSU; russian: Московский государственный университет имени М. В. Ломоносова) is a public research university in Moscow, Russia and the most prestigious ...
and at the same time
. Kolmogorov writes about this time: "I arrived at Moscow University with a fair knowledge of mathematics. I knew in particular the beginning of
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
. I studied many questions in articles in the
Encyclopedia of Brockhaus and Efron, filling out for myself what was presented too concisely in these articles."
Kolmogorov gained a reputation for his wide-ranging erudition. While an undergraduate student in college, he attended the seminars of the Russian historian S. V. Bakhrushin, and he published his first research paper on the fifteenth and sixteenth centuries'
landholding
In real estate, a landed property or landed estate is a property that generates income for the owner (typically a member of the gentry) without the owner having to do the actual work of the estate.
In medieval Western Europe, there were two compet ...
practices in the
Novgorod Republic
The Novgorod Republic was a medieval state that existed from the 12th to 15th centuries, stretching from the Gulf of Finland in the west to the northern Ural Mountains in the east, including the city of Novgorod and the Lake Ladoga regions of mod ...
. During the same period (1921–22), Kolmogorov worked out and proved several results in
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
and in the theory of
Fourier series.
Adulthood
In 1922, Kolmogorov gained international recognition for constructing a
Fourier series that
diverges almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
. Around this time, he decided to devote his life to
mathematics.
In 1925, Kolmogorov graduated from
Moscow State University
M. V. Lomonosov Moscow State University (MSU; russian: Московский государственный университет имени М. В. Ломоносова) is a public research university in Moscow, Russia and the most prestigious ...
and began to study under the supervision of
Nikolai Luzin
Nikolai Nikolaevich Luzin (also spelled Lusin; rus, Никола́й Никола́евич Лу́зин, p=nʲɪkɐˈlaj nʲɪkɐˈlaɪvʲɪtɕ ˈluzʲɪn, a=Ru-Nikilai Nikilayevich Luzin.ogg; 9 December 1883 – 28 January 1950) was a Soviet/Ru ...
.
[ He formed a lifelong close friendship with ]Pavel Alexandrov
Pavel Sergeyevich Alexandrov (russian: Па́вел Серге́евич Алекса́ндров), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote about three hundred papers, ma ...
, a fellow student of Luzin; indeed, several researchers have concluded that the two friends were involved in a homosexual relationship, although neither acknowledged this openly during their lifetimes. Kolmogorov (together with Aleksandr Khinchin
Aleksandr Yakovlevich Khinchin (russian: Алекса́ндр Я́ковлевич Хи́нчин, french: Alexandre Khintchine; July 19, 1894 – November 18, 1959) was a Soviet mathematician and one of the most significant contributors to th ...
) became interested in 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 ...
. Also in 1925, he published his work in 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 ...
, "On the principle of the excluded middle", in which he proved that under a certain interpretation, all statements of classical formal logic can be formulated as those of intuitionistic logic. In 1929, Kolmogorov earned his Doctor of Philosophy
A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...
(Ph.D.) degree, from Moscow State University.
In 1930, Kolmogorov went on his first long trip abroad, traveling to Göttingen
Göttingen (, , ; nds, Chöttingen) is a university city in Lower Saxony, central Germany, the capital of the eponymous district. The River Leine runs through it. At the end of 2019, the population was 118,911.
General information
The ori ...
and Munich
Munich ( ; german: München ; bar, Minga ) is the capital and most populous city of the German state of Bavaria. With a population of 1,558,395 inhabitants as of 31 July 2020, it is the third-largest city in Germany, after Berlin and Ha ...
, and then to Paris
Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), ma ...
. He had various scientific contacts in Göttingen, first with Richard Courant
Richard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician. He is best known by the general public for the book '' What is Mathematics?'', co-written with Herbert Robbins. His research focused on the areas of real ...
and his students working on limit theorems, where diffusion process
In probability theory and statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diff ...
es turned out to be the limits of discrete random processes, then with Hermann Weyl in intuitionistic logic, and lastly with Edmund Landau
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis.
Biography
Edmund Landau was born to a Jewish family in Berlin. His father was Leopol ...
in function theory. His pioneering work, ''About the Analytical Methods of Probability Theory,'' was published (in German) in 1931. Also in 1931, he became a professor at Moscow State University
M. V. Lomonosov Moscow State University (MSU; russian: Московский государственный университет имени М. В. Ломоносова) is a public research university in Moscow, Russia and the most prestigious ...
.
In 1933, Kolmogorov published his book, ''Foundations of the Theory of Probability'', laying the modern axiomatic foundations of probability theory and establishing his reputation as the world's leading expert in this field. In 1935, Kolmogorov became the first chairman of the department of probability theory at Moscow State University. Around the same years (1936) Kolmogorov contributed to the field of ecology and generalized the Lotka–Volterra model of predator–prey systems.
During the Great Purge
The Great Purge or the Great Terror (russian: Большой террор), also known as the Year of '37 (russian: 37-й год, translit=Tridtsat sedmoi god, label=none) and the Yezhovshchina ('period of Yezhov'), was Soviet General Secret ...
in 1936, Kolmogorov's doctoral advisor Nikolai Luzin
Nikolai Nikolaevich Luzin (also spelled Lusin; rus, Никола́й Никола́евич Лу́зин, p=nʲɪkɐˈlaj nʲɪkɐˈlaɪvʲɪtɕ ˈluzʲɪn, a=Ru-Nikilai Nikilayevich Luzin.ogg; 9 December 1883 – 28 January 1950) was a Soviet/Ru ...
became a high-profile target of Stalin's regime, in what is now called the "Luzin Affair". Kolmogorov and several other students of Luzin testified against Luzin, accusing him of plagiarism, nepotism, and other forms of misconduct; the hearings eventually concluded that he was a servant to "fascistoid science" and thus an enemy of the Soviet people. Luzin lost his academic positions, but curiously, he was not arrested nor expelled from the Academy of Sciences of the Soviet Union
The Academy of Sciences of the Soviet Union was the highest scientific institution of the Soviet Union from 1925 to 1991, uniting the country's leading scientists, subordinated directly to the Council of Ministers of the Soviet Union (until 1946 ...
. The question of whether Kolmogorov and others were coerced into testifying against their teacher remains a topic of considerable speculation among historians; all parties involved refused to publicly discuss the case for the rest of their lives. Soviet-Russian mathematician Semën Samsonovich Kutateladze concluded in 2013, after reviewing archival documents made available during the 1990s and other surviving testimonies, that the students of Luzin had initiated the accusations against Luzin out of personal acrimony; there was no evidence that the students were coerced by the state, nor was there any evidence to support their allegations of academic misconduct. Soviet historian of mathematics A.P. Yushkevich surmised that, unlike many of the other high-profile persecutions of the era, Stalin did not personally initiate the persecution of Luzin, and eventually concluded that he was not a threat to the regime, which would explain the unusually mild punishment relative to other contemporaries.
In a 1938 paper, Kolmogorov "established the basic theorems for smoothing and predicting stationary stochastic processes"—a paper that had major military applications during the Cold War. In 1939, he was elected a full member (academician) of the USSR Academy of Sciences
The Academy of Sciences of the Soviet Union was the highest scientific institution of the Soviet Union from 1925 to 1991, uniting the country's leading scientists, subordinated directly to the Council of Ministers of the Soviet Union (until 1946 ...
.
During World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposing ...
Kolmogorov contributed to the Russian war effort by applying statistical theory to artillery fire, developing a scheme of stochastic distribution of barrage balloons intended to help protect Moscow from German bombers.
In his study of stochastic processes, especially Markov process
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. Informally, this may be thought of as, "What happe ...
es, Kolmogorov and the British mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
Sydney Chapman independently developed the pivotal set of equations in the field, which have been given the name of the Chapman–Kolmogorov equations.
Later, Kolmogorov focused his research on 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 ...
, where his publications (beginning in 1941) influenced the field. In 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 classi ...
, he is best known for the Kolmogorov–Arnold–Moser theorem
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 ...
, first presented in 1954 at the International Congress of Mathematicians. In 1957, working jointly with his student Vladimir Arnold
Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov– ...
, he solved a particular interpretation of Hilbert's thirteenth problem
Hilbert's thirteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails proving whether a solution exists for all 7th-degree equations using algebraic (variant: continuous) f ...
. Around this time he also began to develop, and was considered a founder of, algorithmic complexity theory
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 produ ...
– often referred to as Kolmogorov complexity theory.
Kolmogorov married Anna Dmitrievna Egorova in 1942. He pursued a vigorous teaching routine throughout his life, not only at the university level but also with younger children, as he was actively involved in developing a pedagogy
Pedagogy (), most commonly understood as the approach to teaching, is the theory and practice of learning, and how this process influences, and is influenced by, the social, political and psychological development of learners. Pedagogy, taken ...
for gifted children (in literature, music, and mathematics). At Moscow State University, Kolmogorov occupied different positions, including the heads of several departments: probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
, statistics, and random process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
es; mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
. He also served as the Dean of the Moscow State University Department of Mechanics and Mathematics.
In 1971, Kolmogorov joined an oceanographic
Oceanography (), also known as oceanology and ocean science, is the scientific study of the oceans. It is an Earth science, which covers a wide range of topics, including ecosystem dynamics; ocean currents, waves, and geophysical fluid dynamic ...
expedition aboard the research vessel '' Dmitri Mendeleev''. He wrote a number of articles for the ''Great Soviet Encyclopedia
The ''Great Soviet Encyclopedia'' (GSE; ) is one of the largest Russian-language encyclopedias, published in the Soviet Union from 1926 to 1990. After 2002, the encyclopedia's data was partially included into the later ''Bolshaya rossiyskaya e ...
.'' In his later years, he devoted much of his effort to the mathematical and philosophical relationship between 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 ...
in abstract and applied areas.
Kolmogorov died in Moscow in 1987, and his remains were buried in the Novodevichy cemetery.
A quotation attributed to Kolmogorov is ranslated into English "Every mathematician believes that he is ahead of the others. The reason none state this belief in public is because they are intelligent people."
Vladimir Arnold
Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov– ...
once said: "Kolmogorov – Poincaré – Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
– Euler – Newton, are only five lives separating us from the source of our science".
Awards and honours
Kolmogorov received numerous awards and honours both during and after his lifetime:
* Member of the Russian Academy of Sciences
The Russian Academy of Sciences (RAS; russian: Росси́йская акаде́мия нау́к (РАН) ''Rossíyskaya akadémiya naúk'') consists of the national academy of Russia; a network of scientific research institutes from across ...
[
* Awarded the ]Stalin Prize Stalin Prize may refer to:
* The State Stalin Prize in science and engineering and in arts, awarded 1941 to 1954, later known as the USSR State Prize
The USSR State Prize (russian: links=no, Государственная премия СССР, ...
in 1941
* Elected an Honorary Member of the American Academy of Arts and Sciences
The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, a ...
in 1959
* Elected member of the American Philosophical Society
The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...
in 1961
* Award the Balzan Prize in 1962
* Elected a Foreign Member of the Royal Netherlands Academy of Arts and Sciences in 1963
* Elected a Foreign Member of the Royal Society (ForMemRS) in 1964.
* Awarded the Lenin Prize in 1965
* Elected member of the United States National Academy of Sciences in 1967
* Awarded the Wolf Prize
The Wolf Prize is an international award granted in Israel, that has been presented most years since 1978 to living scientists and artists for ''"achievements in the interest of mankind and friendly relations among people ... irrespective of nati ...
in 1980
* Awarded the Lobachevsky Prize
The Lobachevsky Prize, awarded by the Russian Academy of Sciences, and the Lobachevsky Medal, awarded by the Kazan State University, are mathematical awards in honor of Nikolai Ivanovich Lobachevsky.
History
The Lobachevsky Prize was established ...
in 1986
The following are named in Kolmogorov's honour:
* Fisher–Kolmogorov equation
* Johnson–Mehl–Avrami–Kolmogorov equation
* Kolmogorov axioms
The Kolmogorov axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probabili ...
* Kolmogorov equations In probability theory, Kolmogorov equations, including Kolmogorov forward equations and Kolmogorov backward equations, characterize continuous-time Markov processes. In particular, they describe how the probability that a continuous-time Markov p ...
(also known as the Fokker–Planck equations in the context of diffusion and in the forward case)
* Kolmogorov dimension (upper box dimension
Upper may refer to:
* Shoe upper or ''vamp'', the part of a shoe on the top of the foot
* Stimulant, drugs which induce temporary improvements in either mental or physical function or both
* ''Upper'', the original film title for the 2013 found f ...
)
* Kolmogorov–Arnold theorem
* Kolmogorov–Arnold–Moser theorem
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 ...
* Kolmogorov continuity theorem
* Kolmogorov's criterion
In probability theory, Kolmogorov's criterion, named after Andrey Kolmogorov, is a theorem giving a necessary and sufficient condition for a Markov chain or continuous-time Markov chain to be stochastically identical to its time-reversed version. ...
* Kolmogorov extension theorem
In mathematics, the Kolmogorov extension theorem (also known as Kolmogorov existence theorem, the Kolmogorov consistency theorem or the Daniell-Kolmogorov theorem) is a theorem that guarantees that a suitably "consistent" collection of finite-di ...
* Kolmogorov's three-series theorem
* Convergence of Fourier series In mathematics, the question of whether the Fourier series of a periodic function converges to a given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in th ...
* Gnedenko-Kolmogorov central limit theorem
* Quasi-arithmetic mean
In mathematics and statistics, the quasi-arithmetic mean or generalised ''f''-mean or Kolmogorov-Nagumo-de Finetti mean is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function f. It is a ...
(it is also called Kolmogorov mean)
* Kolmogorov homology In algebraic topology, Steenrod homology is a homology theory for compact metric spaces introduced by , based on regular cycles.
It is similar to the homology theory introduced rather sketchily by Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov ...
* Kolmogorov's inequality In probability theory, Kolmogorov's inequality is a so-called "maximal inequality (mathematics), inequality" that gives a bound on the probability that the partial sums of a Finite set, finite collection of independent random variables exceed some s ...
* Landau–Kolmogorov inequality In mathematics, the Landau–Kolmogorov inequality, named after Edmund Landau and Andrey Kolmogorov, is the following family of interpolation inequalities between different derivatives of a function ''f'' defined on a subset ''T'' of the real ...
* Kolmogorov integral
* Brouwer–Heyting–Kolmogorov interpretation In mathematical logic, the Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, of intuitionistic logic was proposed by L. E. J. Brouwer and Arend Heyting, and independently by Andrey Kolmogorov. It is also sometimes called the rea ...
* Kolmogorov microscales
In fluid dynamics, Kolmogorov microscales are the smallest scales in the turbulent flow of fluids. At the Kolmogorov scale, viscosity dominates and the turbulence kinetic energy is dissipated into thermal energy. They are defined by
where
* is ...
* Kolmogorov's normability criterion
In mathematics, Kolmogorov's normability criterion is a theorem that provides a necessary and sufficient condition for a topological vector space to be ; that is, for the existence of a norm on the space that generates the given topology. The nor ...
* Fréchet–Kolmogorov theorem
In functional analysis, the Fréchet–Kolmogorov theorem (the names of Riesz or Weil are sometimes added as well) gives a necessary and sufficient condition for a set of functions to be relatively compact in an ''L'p'' space. It can be thou ...
* Kolmogorov space
In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a neighborhood not containing the ...
* 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 produ ...
* 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 wit ...
* Wiener filter
In signal processing, the Wiener filter is a filter used to produce an estimate of a desired or target random process by linear time-invariant ( LTI) filtering of an observed noisy process, assuming known stationary signal and noise spectra, and ...
(also known as Wiener–Kolmogorov filtering theory)
* Wiener–Kolmogorov prediction
* Kolmogorov automorphism In mathematics, a Kolmogorov automorphism, ''K''-automorphism, ''K''-shift or ''K''-system is an invertible, measure-preserving automorphism defined on a standard probability space that obeys Kolmogorov's zero–one law.Peter Walters, ''An Introduct ...
* Kolmogorov's characterization of reversible diffusions
* Borel–Kolmogorov paradox In probability theory, the Borel–Kolmogorov paradox (sometimes known as Borel's paradox) is a paradox relating to conditional probability with respect to an event of probability zero (also known as a null set). It is named after Émile Borel and ...
* Chapman–Kolmogorov equation
* Hahn–Kolmogorov theorem
* Johnson–Mehl–Avrami–Kolmogorov equation
* Kolmogorov–Sinai entropy
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special ...
* Astronomical seeing
In astronomy, seeing is the degradation of the image of an astronomical object due to turbulence in the atmosphere of Earth that may become visible as blurring, twinkling or variable distortion. The origin of this effect are rapidly changing var ...
described by Kolmogorov's turbulence law
* Kolmogorov structure function In 1973, Andrey Kolmogorov proposed a non-probabilistic approach to statistics and model selection. Let each datum be a finite binary string and a model be a finite set of binary strings. Consider model classes consisting of models of given maximal ...
* Kolmogorov–Uspenskii machine model
* 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, namely a ''tail event of independent σ-algebras'', will either almost surely happen or almost sure ...
* Kolmogorov–Zurbenko filter
Within statistics, the Kolmogorov–Zurbenko (KZ) filter was first proposed by A. N. Kolmogorov and formally defined by Zurbenko.I. Zurbenko. The Spectral Analysis of Time Series. North-Holland Series in Statistics and Probability, 1986. It is a ...
* Kolmogorov's two-series theorem
* Rao–Blackwell–Kolmogorov theorem
* Khinchin–Kolmogorov theorem
* Kolmogorov's Strong Law of Large Numbers
Bibliography
A bibliography of his works appeared in
*
** Translation:
* 1991–93. ''Selected works of A.N. Kolmogorov'', 3 vols. Tikhomirov, V. M., ed., Volosov, V. M., trans. Dordrecht
Dordrecht (), historically known in English as Dordt (still colloquially used in Dutch, ) or Dort, is a city and municipality in the Western Netherlands, located in the province of South Holland. It is the province's fifth-largest city after R ...
:Kluwer Academic Publishers
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
.
* 1925. "On the principle of the excluded middle" in Jean van Heijenoort
Jean Louis Maxime van Heijenoort (; July 23, 1912 – March 29, 1986) was a historian of mathematical logic. He was also a personal secretary to Leon Trotsky from 1932 to 1939, and an American Trotskyist until 1947.
Life
Van Heijenoort was born ...
, 1967. ''A Source Book in Mathematical Logic, 1879–1931''. Harvard Univ. Press: 414–37.
*
*
* Kolmogorov, Andrei N. (2005) ''Selected works''. In 6 volumes. Moscow (in Russian)
Textbooks:
* A. N. Kolmogorov and B. V. Gnedenko. ''"Limit distributions for sums of independent random variables"'', 1954.
* A. N. Kolmogorov and S. V. Fomin. ''"Elements of the Theory of Functions and Functional Analysis"''
Publication 1999
Publication 2012
Kolmogorov, Andrey Nikolaevich; Fomin, Sergei Vasilyevich (1975) 970
Year 970 ( CMLXX) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar, the 970th year of the Common Era (CE) and ''Anno Domini'' designations, the 970th year of the 1st millennium, the 70th yea ...
Introductory real analysis. New York: Dover Publications. ..
References
External links
Portal dedicated to AN Kolmogorov
(his scientific and popular publications, articles about him).
The Legacy of Andrei Nikolaevich Kolmogorov
Biography at Scholar''pedia''
* ttp://www.probabilityandfinance.com/articles/04.pdf The origins and legacy of Kolmogorov's Grundbegriffe
Vitanyi, P.M.B., Andrey Nikolaevich Kolmogorov. Scholarpedia, 2(2):2798; 2007
Collection of links to Kolmogorov resources
Interview with Professor A. M. Yaglom about Kolmogorov, Gelfand and other (1988, Ithaca, New York
Kolmogorov School
at Moscow University
at the Computer Learning Research Centre at Royal Holloway, University of London
Lorentz G. G., Mathematics and Politics in the Soviet Union from 1928 to 1953
* ttp://www.math.nsc.ru/LBRT/g2/english/ssk/case_e.html Kutateladze S. S., The Tragedy of Mathematics in Russia
Video recording of the G. Falkovich's lecture: "Andrey Nikolaevich Kolmogorov (1903–1987) and the Russian school"
*
{{DEFAULTSORT:Kolmogorov, Andrey
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