# Pafnuty Chebyshev

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Pafnuty Chebyshev
Pafnuty Lvovich Chebyshev
Born16 May 1821
Died8 December 1894 (aged 73)
Nationality Russian
Other namesChebysheff, Chebyshov, Tschebyscheff, Tschebycheff
Alma mater Moscow University
Known forWork on probability, statistics, mechanics, analytical geometry and number theory
Awards Demidov Prize (1849)
Scientific career
Fields Mathematician
Institutions St. Petersburg University
Notable students Dmitry Grave
Aleksandr Korkin
Aleksandr Lyapunov
Andrey Markov
Konstantin Posse

Pafnuty Lvovich Chebyshev (Russian :Пафну́тий Льво́вич Чебышёв,IPA: ) (16 May [ O.S. 4 May] 18218 December [ O.S. 26 November] 1894) [1] was a Russian mathematician. His name can be alternatively transliterated as Chebysheff, Chebychov, Chebyshov; or Tchebychev, Tchebycheff (French transcriptions); or Tschebyschev, Tschebyschef, Tschebyscheff (German transcriptions). Chebychev, mixture between English and French transliterations, is sometimes erroneously used.

Russian is an East Slavic language, which is official in the Russian Federation, Belarus, Kazakhstan and Kyrgyzstan, as well as being widely used throughout Eastern Europe, the Baltic states, the Caucasus and Central Asia. It was the de facto language of the Soviet Union until its dissolution on 25 December 1991. Although nearly three decades have passed since the breakup of the Soviet Union, Russian is used in official capacity or in public life in all the post-Soviet nation-states, as well as in Israel and Mongolia.

Old Style (O.S.) and New Style (N.S.) are terms sometimes used with dates to indicate that the calendar convention used at the time described is different from that in use at the time the document was being written. There were two calendar changes in Great Britain and its colonies, which may sometimes complicate matters: the first was to change the start of the year from Lady Day to 1 January; the second was to discard the Julian calendar in favour of the Gregorian calendar. Closely related is the custom of dual dating, where writers gave two consecutive years to reflect differences in the starting date of the year, or to include both the Julian and Gregorian dates.

Russians are a nation and an East Slavic ethnic group native to European Russia in Eastern Europe. Outside Russia, notable minorities exist in other former Soviet states such as Belarus, Kazakhstan, Moldova, Ukraine and the Baltic states. A large Russian diaspora also exists all over the world, with notable numbers in the United States, Germany, Brazil, and Canada.

## Biography

### Early years

One of nine children, [2] Chebyshev was born in the village of Okatovo in the district of Borovsk, province of Kaluga, into a family which traced its roots back to a 17th-century Tatar military leader named Khan Chabysh. [3] His father, Lev Pavlovich, was a Russian nobleman and wealthy landowner. Pafnuty Lvovich was first educated at home by his mother Agrafena Ivanovna (in reading and writing) and by his cousin Avdotya Kvintillianovna Sukhareva (in French and arithmetic). Chebyshev mentioned that his music teacher also played an important role in his education, for she “raised his mind to exactness and analysis.”

Borovsk is a town and the administrative center of Borovsky District of Kaluga Oblast, Russia, located on the Protva River just south from the oblast's border with Moscow Oblast. Population: 12,283 (2010 Census); 11,917 (2002 Census); 13,405 (1989 Census); 12,000 (1969).

Trendelenburg's gait affected Chebyshev's adolescence and development. From childhood, he limped and walked with a stick and so his parents abandoned the idea of his becoming an officer in the family tradition. His disability prevented his playing many children's games and he devoted himself instead to mathematics.

The Trendelenburg gait is an abnormal gait caused by weakness of the abductor muscles of the lower limb, gluteus medius and gluteus minimus. People with a lesion of superior gluteal nerve have weakness of abducting the thigh at the hip.

In 1832, the family moved to Moscow, mainly to attend to the education of their eldest sons (Pafnuty and Pavel, who would become lawyers). Education continued at home and his parents engaged teachers of excellent reputation, including (for mathematics and physics) P.N. Pogorelski, held to be one of the best teachers in Moscow and who had taught (for example) the writer Ivan Sergeevich Turgenev.

Moscow is the capital and most populous city of Russia, with 13.2 million residents within the city limits, 17 million within the urban area and 20 million within the metropolitan area. Moscow is one of Russia's federal cities.

Ivan Sergeyevich Turgenev was a Russian novelist, short story writer, poet, playwright, translator and popularizer of Russian literature in the West.

### University studies

In summer 1837, Chebyshev passed the registration examinations and, in September of that year, began his mathematical studies at the second philosophical department of Moscow University. His teachers included N.D. Brashman, N.E. Zernov and D.M. Perevoshchikov of whom it seems clear that Brashman had the greatest influence on Chebyshev. Brashman instructed him in practical mechanics and probably showed him the work of French engineer J.V. Poncelet. In 1841 Chebyshev was awarded the silver medal for his work “calculation of the roots of equations” which he had finished in 1838. In this, Chebyshev derived an approximating algorithm for the solution of algebraic equations of nth degree based on Newton's method. In the same year, he finished his studies as "most outstanding candidate".

Nikolai Dmitrievich Brashman was a Russian mathematician of Jewish-Austrian origin. He was a student of Joseph Johann Littrow, and the advisor of Pafnuty Chebyshev and August Davidov.

Jean-Victor Poncelet was a French engineer and mathematician who served most notably as the Commanding General of the École Polytechnique. He is considered a reviver of projective geometry, and his work Traité des propriétés projectives des figures is considered the first definitive text on the subject since Gérard Desargues' work on it in the 17th century. He later wrote an introduction to it: Applications d’analyse et de géométrie.

In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots of a real-valued function.

In 1841, Chebyshev’s financial situation changed drastically. There was famine in Russia, and his parents were forced to leave Moscow. Although they could no longer support their son, he decided to continue his mathematical studies and prepared for the master examinations, which lasted six months. Chebyshev passed the final examination in October 1843 and, in 1846, defended his master thesis “An Essay on the Elementary Analysis of the Theory of Probability.” His biographer Prudnikov suggests that Chebyshev was directed to this subject after learning of recently published books on probability theory or on the revenue of the Russian insurance industry.

In 1847, Chebyshev promoted his thesis pro venia legendi “On integration with the help of logarithms” at St Petersburg University and thus obtained the right to teach there as a lecturer. At that time some of Leonhard Euler’s works were rediscovered by P. N. Fuss and were being edited by V. Ya. Bunyakovsky, who encouraged Chebyshev to study them. This would come to influence Chebyshev's work. In 1848, he submitted his work The Theory of Congruences for a doctorate, which he defended in May 1849. He was elected an extraordinary professor at St Petersburg University in 1850, ordinary professor in 1860 and, after 25 years of lectureship, he became merited professor in 1872. In 1882 he left the university and devoted his life to research.

Leonhard Euler was a Swiss mathematician, physicist, astronomer, logician and engineer who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory.

Viktor Yakovlevich Bunyakovsky was a Russian mathematician, member and later vice president of the Petersburg Academy of Sciences.

Professor is an academic rank at universities and other post-secondary education and research institutions in most countries. Literally, professor derives from Latin as a "person who professes" being usually an expert in arts or sciences, a teacher of the highest rank.

During his lectureship at the university (1852–1858), Chebyshev also taught practical mechanics at the Alexander Lyceum in Tsarskoe Selo (now Pushkin), a southern suburb of St Petersburg.

His scientific achievements were the reason for his election as junior academician (adjunkt) in 1856. Later, he became an extraordinary (1856) and in 1858 an ordinary member of the Imperial Academy of Sciences. In the same year he became an honorary member of Moscow University. He accepted other honorary appointments and was decorated several times. In 1856, Chebyshev became a member of the scientific committee of the ministry of national education. In 1859, he became an ordinary member of the ordnance department of the academy with the adoption of the headship of the commission for mathematical questions according to ordnance and experiments related to ballistics. The Paris academy elected him corresponding member in 1860 and full foreign member in 1874. In 1893, he was elected honorable member of the St. Petersburg Mathematical Society, which had been founded three years earlier.

Chebyshev died in St Petersburg on 26 November 1894.

## Mathematical contributions

Chebyshev is known for his work in the fields of probability, statistics, mechanics, and number theory. The Chebyshev inequality states that if ${\displaystyle X}$ is a random variable with standard deviation σ > 0, then the probability that the outcome of ${\displaystyle X}$ is no less than ${\displaystyle a\sigma }$ away from its mean is no more than ${\displaystyle 1/a^{2}}$:

${\displaystyle \Pr(|X-{\mathbf {E} }(X)|\geq a\,\sigma )\leq {\frac {1}{a^{2}}}.}$

The Chebyshev inequality is used to prove the weak law of large numbers.

The Bertrand–Chebyshev theorem (1845,1852) states that for any ${\displaystyle n>1}$, there exists a prime number ${\displaystyle p}$ such that ${\displaystyle n. This is a consequence of the Chebyshev inequalities for the number ${\displaystyle \pi (n)}$ of prime numbers less than ${\displaystyle n}$, which state that ${\displaystyle \pi (n)}$ is of the order of ${\displaystyle n/\log(n)}$. A more precise form is given by the celebrated prime number theorem: the quotient of the two expressions approaches 1.0 as ${\displaystyle n}$ tends to infinity.

Chebyshev is also known for the Chebyshev polynomials and the Chebyshev bias – the difference between the number of primes that are congruent to 3 (modulo 4) and 1 (modulo 4).

## Legacy

Chebyshev is considered to be a founding father of Russian mathematics. Among his well-known students were the mathematicians Dmitry Grave, Aleksandr Korkin, Aleksandr Lyapunov, and Andrei Markov. According to the Mathematics Genealogy Project, Chebyshev has 12,422 mathematical "descendants" as of July 2018. [4]

The lunar crater Chebyshev and the asteroid 2010 Chebyshev were named to honor his major achievements in the mathematical realm. [5]

## Publications

• Tchebychef, P. L. (1899), Markov, Andreĭ Andreevich; Sonin, N. (eds.), Oeuvres, I, New York: Commissionaires de l'Académie impériale des sciences, MR   0147353, Reprinted by Chelsea 1962
• Tchebychef, P. L. (1907), Markov, Andreĭ Andreevich; Sonin, N. (eds.), Oeuvres, II, New York: Commissionaires de l'Académie impériale des sciences, MR   0147353, Reprinted by Chelsea 1962
• Butzer (1999), "P. L. Chebyshev (1821–1894): A Guide to his Life and Work", Journal of Approximation Theory, 96: 111–138, doi:10.1006/jath.1998.3289

## Related Research Articles

Andrey Andreyevich Markov was a Russian mathematician best known for his work on stochastic processes. A primary subject of his research later became known as Markov chains and Markov processes.

In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a collection of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. They have applications in many disciplines including sciences such as biology, chemistry, ecology, neuroscience, and physics as well as technology and engineering fields such as image processing, signal processing, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance.

Andrey Nikolaevich Kolmogorov was a Soviet mathematician who made significant contributions to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity.

In probability theory, Chebyshev's inequality guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean. Specifically, no more than 1/k2 of the distribution's values can be more than k standard deviations away from the mean. The rule is often called Chebyshev's theorem, about the range of standard deviations around the mean, in statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers.

Mikhail Vasilyevich Ostrogradsky was a Russian mathematician, mechanician and physicist of Ukrainian descent. Ostrogradsky was a student of Timofei Osipovsky and is considered to be a disciple of Leonhard Euler and one of the leading mathematicians of Imperial Russia.

In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant. It is named after the Russian mathematician Andrey Markov, although it appeared earlier in the work of Pafnuty Chebyshev, and many sources, especially in analysis, refer to it as Chebyshev's inequality or Bienaymé's inequality.

In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Markov.

In probability theory, the Vysochanskij–Petunin inequality gives a lower bound for the probability that a random variable with finite variance lies within a certain number of standard deviations of the variable's mean, or equivalently an upper bound for the probability that it lies further away. The sole restrictions on the distribution are that it be unimodal and have finite variance. The theorem applies even to heavily skewed distributions and puts bounds on how much of the data is, or is not, "in the middle."

Nikolay Yakovlevich Sonin was a Russian mathematician.

In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure μ to the sequences of moments

In probability theory, the method of moments is a way of proving convergence in distribution by proving convergence of a sequence of moment sequences. Suppose X is a random variable and that all of the moments

In probability theory, an ergodic dynamical system is one that, broadly speaking, has the same behavior averaged over time as averaged over the space of all the system's states in its phase space. In physics the term implies that a system satisfies the ergodic hypothesis of thermodynamics.

In mathematics, the Markov brothers' inequality is an inequality proved in the 1890s by brothers Andrey Markov and Vladimir Markov, two Russian mathematicians. This inequality bounds the maximum of the derivatives of a polynomial on an interval in terms of the maximum of the polynomial. For k = 1 it was proved by Andrey Markov, and for k = 2,3,... by his brother Vladimir Markov.

In mathematics, the Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev in 1875 and rediscovered by Wolfgang Hahn. The Hahn class is a name for special cases of Hahn polynomials, including Hahn polynomials, Meixner polynomials, Krawtchouk polynomials, and Charlier polynomials. Sometimes the Hahn class is taken to include limiting cases of these polynomials, in which case it also includes the classical orthogonal polynomials.

In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced by Pafnuty Chebyshev (1864) and rediscovered by Gram (1883).

August Yulevich Davidov was a Russian mathematician and engineer, professor at Moscow University, and author of works on differential equations with partial derivatives, definite integrals, and the application of probability theory to statistics, and textbooks on elementary mathematics which were repeatedly reprinted from the 1860s to the 1920s. He was president of the Moscow Mathematical Society from 1866 to 1885.

Pavel Alekseevich Nekrasov (1853–1924) was a Russian mathematician and a Rector of the Imperial University of Moscow.

In the mathematical theory of random processes, the Markov chain central limit theorem has a conclusion somewhat similar in form to that of the classic central limit theorem (CLT) of probability theory, but the quantity in the role taken by the variance in the classic CLT has a more complicated definition.

## References

1. Paul Butzer & François Jongmans, P. L. Chebyshev (1821-1894) : A Guide to his Life and Work, Journal of Approximation Theory, vol. 96, p. 112 (1999)
2. Schmadel, Lutz D. (2007). "(2010) Chebyshev". Dictionary of Minor Planet Names – (2010) Chebyshev. Springer Berlin Heidelberg. p. 163. doi:10.1007/978-3-540-29925-7_2011. ISBN   978-3-540-00238-3.