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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 the Soviet school of probability theory. Life and career He was born in the village of Kondrovo, Kaluga Governorate, Russian Empire. While studying at Moscow State University, he became one of the first followers of the famous Luzin school. Khinchin graduated from the university in 1916 and six years later he became a full professor there, retaining that position until his death. Khinchin's early works focused on real analysis. Later he applied methods from the metric theory of functions to problems in probability theory and number theory. He became one of the founders of modern probability theory, discovering the law of the iterated logarithm in 1924, achieving important results in the field of limit theorems, giving a definition of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kondrovo, Kaluga Oblast
Kondrovo (russian: Ко́ндрово) is a town and the administrative center of Dzerzhinsky District in Kaluga Oblast, Russia, located on the Shanya River (a tributary of the Ugra in the Oka's basin), northwest of Kaluga, the administrative center of the oblast. Population: History Originally known as Kondyrevo (), it was a ''votchina'' of Dmitry Kondyrin, a Russian voyevoda who was granted these lands for his service during the second war with the Grand Duchy of Lithuania in 1500–1501. The settlement's name gradually changed to Kondrovo by the 1840s. In 1790, a paper mill was built in the village, and by the end of the 19th century it became famous for its high quality paper. Town status was granted to Kondrovo in 1938. During World War II, Kondrovo was occupied by the German Army from October 9, 1941 to January 19, 1942. Administrative and municipal status Within the framework of administrative divisions, Kondrovo serves as the administrative center of Dzerzh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Limit Theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern general form, this fundamental result in probability theory was precisely stated as late as 1920, thereby serving as a bridge between classical and modern probability theory. If X_1, X_2, \dots, X_n, \dots are random samples drawn from a population with overall mean \mu and finite variance and if \bar_n is the sample mean of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener–Khinchin Theorem
In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary random process has a spectral decomposition given by the power spectrum of that process. History Norbert Wiener proved this theorem for the case of a deterministic function in 1930; Aleksandr Khinchin later formulated an analogous result for stationary stochastic processes and published that probabilistic analogue in 1934. Albert Einstein explained, without proofs, the idea in a brief two-page memo in 1914. The case of a continuous-time process For continuous time, the Wiener–Khinchin theorem says that if x is a wide-sense stochastic process whose autocorrelation function (sometimes called autocovariance) defined in terms of statistical expected value, r_(\tau) = \mathbb\big (t)^*x(t - \tau)\big/math> (the asterisk denotes comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pollaczek–Khinchine Formula
In queueing theory, a discipline within the mathematical theory of probability, the Pollaczek–Khinchine formula states a relationship between the queue length and service time distribution Laplace transforms for an M/G/1 queue (where jobs arrive according to a Poisson process and have general service time distribution). The term is also used to refer to the relationships between the mean queue length and mean waiting/service time in such a model. The formula was first published by Felix Pollaczek in 1930 and recast in probabilistic terms by Aleksandr Khinchin two years later. In ruin theory the formula can be used to compute the probability of ultimate ruin (probability of an insurance company going bankrupt). Mean queue length The formula states that the mean number of customers in system ''L'' is given by : L = \rho + \frac where *\lambda is the arrival rate of the Poisson process *1/\mu is the mean of the service time distribution ''S'' *\rho=\lambda/\mu is the utilization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Of Lenin
The Order of Lenin (russian: Орден Ленина, Orden Lenina, ), named after the leader of the Russian October Revolution, was established by the Central Executive Committee on April 6, 1930. The order was the highest civilian decoration bestowed by the Soviet Union. The order was awarded to: * Civilians for outstanding services rendered to the State * Members of the armed forces for exemplary service * Those who promoted friendship and cooperation between people and in strengthening peace * Those with meritorious services to the Soviet state and society From 1944 to 1957, before the institution of a specific length of service medals, the Order of Lenin was also used to reward 25 years of conspicuous military service. Those who were awarded the titles " Hero of the Soviet Union" and " Hero of Socialist Labour" were also given the order as part of the award. It was also bestowed on cities, companies, factories, regions, military units, and ships. Various educational institu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
USSR State Prize
The USSR State Prize (russian: links=no, Государственная премия СССР, Gosudarstvennaya premiya SSSR) was the Soviet Union's state honor. It was established on 9 September 1966. After the dissolution of the Soviet Union, the prize was followed up by the State Prize of the Russian Federation. The State Stalin Prize ( Государственная Сталинская премия, ''Gosudarstvennaya Stalinskaya premiya''), usually called the Stalin Prize, existed from 1941 to 1954, although some sources give a termination date of 1952. It essentially played the same role; therefore upon the establishment of the USSR State Prize, the diplomas and badges of the recipients of Stalin Prize were changed to that of USSR State Prize. In 1944 and 1945, the last two years of the Second World War 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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 – to the Council of People's Commissars of the Soviet Union). In 1991, by the decree of the President of the Russian Soviet Federative Socialist Republic, the Russian Academy of Sciences was established on the basis of the Academy of Sciences of the Soviet Union. History Creation of the Academy of Sciences of the Soviet Union The Academy of Sciences of the Soviet Union was formed by a resolution of the Central Executive Committee and the Council of People's Commissars of the Soviet Union dated July 27, 1925 on the basis of the Russian Academy of Sciences (before the February Revolution – the Imperial Saint Petersburg Academy of Sciences). In the first years of Soviet Russia, the Institute of the Academy of Sciences was perceived rath ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Queuing Theory
Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service. Queueing theory has its origins in research by Agner Krarup Erlang when he created models to describe the system of Copenhagen Telephone Exchange company, a Danish company. The ideas have since seen applications including telecommunication, Traffic engineering (transportation), traffic engineering, computing and, particularly in industrial engineering, in the design of factories, shops, offices and hospitals, as well as in project management. Spelling The spelling "queueing" over "queuing" is typically encountered in the academic research field. In fact, one of the flagship journals of the field is ''Queueing Systems''. Single queue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Theory
Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. The field is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering. A key measure in information theory is entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip (with two equally likely outcomes) provides less information (lower entropy) than specifying the outcome from a roll of a die (with six equally likely outcomes). Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy. Important sub-fields of information theory include sourc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Physics
Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic nature. Its applications include many problems in the fields of physics, biology, chemistry, and neuroscience. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics develop the phenomenological results of thermodynamics from a probabilistic examination of the underlying microscopic systems. Historically, one of the first topics in physics where statistical methods were applied was the field of classical mechanics, which is concerned with the motion of particles or objects when subjected to a force. Scope Statistical physics explains and quanti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Khinchin's Constant
In number theory, Aleksandr Yakovlevich Khinchin proved that for almost all real numbers ''x'', coefficients ''a''''i'' of the continued fraction expansion of ''x'' have a finite geometric mean that is independent of the value of ''x'' and is known as Khinchin's constant. That is, for :x = a_0+\cfrac\; it is almost always true that :\lim_ \left( a_1 a_2 ... a_n \right) ^ = K_0 where K_0 is Khinchin's constant :K_0 = \prod_^\infty ^ \approx 2.6854520010\dots (with \prod denoting the product over all sequence terms). Although almost all numbers satisfy this property, it has not been proven for ''any'' real number ''not'' specifically constructed for the purpose. Among the numbers whose continued fraction expansions apparently do have this property (based on numerical evidence) are π, the Euler-Mascheroni constant γ, Apéry's constant ζ(3), and Khinchin's constant itself. However, this is unproven. Among the numbers ''x'' whose continued fraction expansions are known ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
