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Chebyshev
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebyshev is known for his fundamental contributions to the fields of probability, statistics, mechanics, and number theory. A number of important mathematical concepts are named after him, including the Chebyshev inequality (which can be used to prove the weak law of large numbers), the Bertrand–Chebyshev theorem, Chebyshev polynomials, Chebyshev linkage, and Chebyshev bias. Transcription The surname Chebyshev has been transliterated in several different ways, like Tchebichef, Tchebychev, Tchebycheff, Tschebyschev, Tschebyschef, Tschebyscheff, Čebyčev, Čebyšev, Chebysheff, Chebychov, Chebyshov (according to native Russian speakers, this one provides the closest pronunciation in English to the correct pronunciation in old Russian), and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Polynomial
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshev polynomials of the first kind T_n are defined by : T_n(\cos \theta) = \cos(n\theta). Similarly, the Chebyshev polynomials of the second kind U_n are defined by : U_n(\cos \theta) \sin \theta = \sin\big((n + 1)\theta\big). That these expressions define polynomials in \cos\theta may not be obvious at first sight, but follows by rewriting \cos(n\theta) and \sin\big((n+1)\theta\big) using de Moivre's formula or by using the angle sum formulas for \cos and \sin repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain T_2(\cos\theta)=\cos(2\theta)=2\cos^2\theta-1 and U_1(\cos\theta)\sin\theta=\sin(2\theta)=2\cos\theta\sin\theta, which are respectively a polynomial in \cos\th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Inequality
In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev 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/''k''2 of the distribution's values can be ''k'' or more standard deviations away from the mean (or equivalently, at least 1 − 1/''k''2 of the distribution's values are less 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. Its practical usage is similar to the 68–95–99.7 rule, which applies only to normal distributions. Chebyshev's inequality is more general, stating th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Linkage
In kinematics, Chebyshev's linkage is a four-bar linkage that converts rotational motion to approximate linear motion. It was invented by the 19th-century mathematician Pafnuty Chebyshev, who studied theoretical problems in kinematic mechanisms. One of the problems was the construction of a linkage that converts a rotary motion into an approximate straight-line motion (a straight line mechanism). This was also studied by James Watt in his improvements to the steam engine, which resulted in Watt's linkage. – Cross link straight-line mechanism Equations of motion The motion of the linkage can be constrained to an input angle that may be changed through velocities, forces, etc. The input angles can be either link ''L''2 with the horizontal or link ''L''4 with the hori ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksandr Lyapunov
Aleksandr Mikhailovich Lyapunov (russian: Алекса́ндр Миха́йлович Ляпуно́в, ; – 3 November 1918) was a Russian mathematician, mechanician and physicist. His surname is variously romanized as Ljapunov, Liapunov, Liapounoff or Ljapunow. He was the son of the astronomer Mikhail Lyapunov and the brother of the pianist and composer Sergei Lyapunov. Lyapunov is known for his development of the stability theory of a dynamical system, as well as for his many contributions to mathematical physics and probability theory. Biography Early life Lyapunov was born in Yaroslavl, Russian Empire. His father Mikhail Vasilyevich Lyapunov (1820–1868) was an astronomer employed by the Demidov Lyceum. His brother, Sergei Lyapunov, was a gifted composer and pianist. In 1863, M. V. Lyapunov retired from his scientific career and relocated his family to his wife's estate at Bolobonov, in the Simbirsk province (now Ulyanovsk Oblast). After the death of his father in 18 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertrand's Postulate
In number theory, Bertrand's postulate is a theorem stating that for any integer n > 3, there always exists at least one prime number p with :n < p < 2n - 2. A less restrictive formulation is: for every , there is always at least one prime such that : Another formulation, where is the -th prime, is: for : This statement was first d in 1845 by (1822–1900). Bertrand himself verified his statement for all integers . His conjecture was completely [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Law Of Large Numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value and tends to become closer to the expected value as more trials are performed. The LLN is important because it guarantees stable long-term results for the averages of some random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. Importantly, the law applies (as the name indicates) only when a ''large number'' of observations are considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be "balanced ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Bias
In number theory, Chebyshev's bias is the phenomenon that most of the time, there are more primes of the form 4''k'' + 3 than of the form 4''k'' + 1, up to the same limit. This phenomenon was first observed by Russian mathematician Pafnuty Chebyshev in 1853. Description Let π(''x''; ''n'', ''m'') denote the number of primes of the form ''nk'' + ''m'' up to ''x''. By the prime number theorem (extended to arithmetic progression), :\pi(x;4,1)\sim\pi(x;4,3)\sim \frac\frac. That is, half of the primes are of the form 4''k'' + 1, and half of the form 4''k'' + 3. A reasonable guess would be that π(''x''; 4, 1) > π(''x''; 4, 3) and π(''x''; 4, 1) < π(''x''; 4, 3) each also occur 50% of the time. This, however, is not supported by numerical evidence — in fact, π(''x''; 4, 3) > π(''x''; 4, 1) occurs much more frequently. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrey Markov
Andrey Andreyevich Markov, first name also spelled "Andrei", in older works also spelled Markoff) (14 June 1856 – 20 July 1922) was a Russian mathematician best known for his work on stochastic processes. A primary subject of his research later became known as the Markov chain. Markov and his younger brother Vladimir Andreevich Markov (1871–1897) proved the Markov brothers' inequality. His son, another Andrey Markov (Soviet mathematician), Andrey Andreyevich Markov (1903–1979), was also a notable mathematician, making contributions to constructive mathematics and Recursion#Functional recursion, recursive function theory. Biography Andrey Markov was born on 14 June 1856 in Russia. He attended the St. Petersburg Grammar School, where some teachers saw him as a rebellious student. In his academics he performed poorly in most subjects other than mathematics. Later in life he attended Saint Petersburg Imperial University (now Saint Petersburg State University). Among his t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yegor Ivanovich Zolotarev
Yegor (Egor) Ivanovich Zolotarev (russian: Его́р Ива́нович Золотарёв) (31 March 1847, Saint Petersburg – 19 July 1878, Saint Petersburg) was a Russian mathematician. Biography Yegor was born as a son of Agafya Izotovna Zolotareva and the merchant Ivan Vasilevich Zolotarev in Saint Petersburg, Imperial Russia. In 1857 he began to study at the fifth St Petersburg gymnasium, a school which centred on mathematics and natural science. He finished it with the silver medal in 1863. In the same year he was allowed to be an auditor at the physico-mathematical faculty of St Petersburg university. He had not been able to become a student before 1864 because he was too young. Among his academic teachers were Somov, Chebyshev and Aleksandr Korkin, with whom he would have a tight scientific friendship. In November 1867 he defended his Kandidat thesis ''“About the Integration of Gyroscope Equations”'', after 10 months there followed his thesis pro venia lege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dmitry Grave
Dmitry Aleksandrovich Grave (russian: Дми́трий Алекса́ндрович Гра́ве; September 6, 1863 – December 19, 1939) was an Imperial Russian and Soviet mathematician. Naum Akhiezer, Nikolai Chebotaryov, Mikhail Kravchuk, and Boris Delaunay were among his students. Brief history Dmitry Grave was educated at the University of St Petersburg where he studied under Chebyshev and his pupils Korkin, Zolotarev and Markov. Grave began research while a student, graduating with his doctorate in 1896. He had obtained his master's degree in 1889 and, in that year, began teaching at the University of St Petersburg. For his master's degree Grave studied Jacobi's methods for the three-body problem, a topic suggested by Korkin. His doctorate was on map projections, again a topic proposed by Korkin, the degree being awarded in 1896. The work, on equal area plane projections of the sphere, built on ideas of Euler, Joseph Louis Lagrange and Chebyshev. Grave became professor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksandr Korkin
Aleksandr Nikolayevich Korkin (russian: Александр Николаевич Коркин; – ) was a Russian mathematician. He made contribution to the development of partial differential equations, and was second only to Chebyshev among the founders of the Saint Petersburg Mathematical School. Among others, his students included Yegor Ivanovich Zolotarev Yegor (Egor) Ivanovich Zolotarev (russian: Его́р Ива́нович Золотарёв) (31 March 1847, Saint Petersburg – 19 July 1878, Saint Petersburg) was a Russian mathematician. Biography Yegor was born as a son of Agafya Izoto .... Some publications * * * References External links * *Korkin's Biography the St. Petersburg University Pages (in Russian, but with an image) 1837 births 1908 deaths People from Vologda Oblast People from Vologda Governorate 19th-century mathematicians from the Russian Empire Mathematical analysts {{Russia-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Demidov Prize
The Demidov Prize (russian: Демидовская премия) is a national scientific prize in Russia awarded annually to the members of the Russian Academy of Sciences. Originally awarded from 1832 to 1866 in the Russian Empire, it was revived by the government of Russia's Sverdlovsk Oblast in 1993. In its original incarnation it was one of the first annual scientific awards, and its traditions influenced other awards of this kind including the Nobel Prize. History In 1831 Count Pavel Nikolaievich Demidov, representative of the famous Demidov family, established a scientific prize in his name. The Saint Petersburg Academy of Sciences (now the Russian Academy of Sciences) was chosen as the awarding institution. In 1832 the president of the Petersburg Academy of Sciences, Sergei Uvarov, awarded the first prizes. From 1832 to 1866 the Academy awarded 55 full prizes (5,000 rubles) and 220 part prizes. Among the winners were many prominent Russian scientists: the founder of fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
