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Stochastic Nature Of Quantum Measurements
Stochastic (; ) is the property of being well-described by a random probability distribution. ''Stochasticity'' and ''randomness'' are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; in everyday conversation, however, these terms are often used interchangeably. In probability theory, the formal concept of a ''stochastic process'' is also referred to as a ''random process''. Stochasticity is used in many different fields, including image processing, signal processing, computer science, information theory, telecommunications, chemistry, ecology, neuroscience, physics, and cryptography. It is also used in finance (e.g., stochastic oscillator), due to seemingly random changes in the different markets within the financial sector and in medicine, linguistics, music, media, colour theory, botany, manufacturing and geomorphology. Etymology The word ''stochastic'' in English was originally used as an adjective with the definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random
In common usage, randomness is the apparent or actual lack of definite pattern or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if there is a known probability distribution, the frequency of different outcomes over repeated events (or "trials") is predictable.Strictly speaking, the frequency of an outcome will converge almost surely to a predictable value as the number of trials becomes arbitrarily large. Non-convergence or convergence to a different value is possible, but has probability zero. Consistent non-convergence is thus evidence of the lack of a fixed probability distribution, as in many evolutionary processes. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4. In this view, randomness is not haphaza ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cryptography
Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of Adversary (cryptography), adversarial behavior. More generally, cryptography is about constructing and analyzing Communication protocol, protocols that prevent third parties or the public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, information security, electrical engineering, digital signal processing, physics, and others. Core concepts related to information security (confidentiality, data confidentiality, data integrity, authentication, and non-repudiation) are also central to cryptography. Practical applications of cryptography include electronic commerce, Smart card#EMV, chip-based payment cards, digital currencies, password, computer passwords, and military communications. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Matrix
In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, ''substitution matrix'', or Markov matrix. The stochastic matrix was first developed by Andrey Markov at the beginning of the 20th century, and has found use throughout a wide variety of scientific fields, including probability theory, statistics, mathematical finance and linear algebra, as well as computer science and population genetics. There are several different definitions and types of stochastic matrices: *A right stochastic matrix is a square matrix of nonnegative real numbers, with each row summing to 1 (so it is also called a row stochastic matrix). *A left stochastic matrix is a square matrix of nonnegative real numbers, with each column summing to 1 (so it is also called a column stochastic matrix). *A ''doubly stochastic matrix'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harald Cramér
Harald Cramér (; 25 September 1893 – 5 October 1985) was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. John Kingman described him as "one of the giants of statistical theory".Kingman 1986, p. 186. Biography Early life Harald Cramér was born in Stockholm, Sweden on 25 September 1893. Cramér remained close to Stockholm for most of his life. He entered the Stockholm University as an undergraduate in 1912, where he studied mathematics and chemistry. During this period, he was a research assistant under the famous chemist, Hans von Euler-Chelpin, with whom he published his first five articles from 1913 to 1914. Following his lab experience, he began to focus solely on mathematics. He eventually began his work on his doctoral studies in mathematics which were supervised by Marcel Riesz at the Stockholm University. Also influenced by G. H. Hardy, Cramér's research led to a PhD in 1917 for his thesi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfgang Doeblin
Wolfgang Doeblin, known in France as Vincent Doblin (17 March 1915 – 21 June 1940), was a French-German mathematician. Life A native of Berlin, Wolfgang was the son of the Jewish-German novelist and physician, Alfred Döblin, and Erna Reiss. His family escaped from Nazi Germany to France where he became a French citizen. Studying probability theory at the Université de Paris under Fréchet, he quickly made a name for himself as a gifted theorist. He received his doctorate at age 23. Drafted in November 1938, after refusing to be exempted from military service, he was a soldier in the French army when World War II broke out in 1939, and was stationed at Givet, in the Ardennes, as a telephone operator. There, he wrote down his latest work on the Chapman–Kolmogorov equation, and sent this as a "pli cacheté" (sealed envelope) to the French Academy of Sciences. His company, sent to the sector of the Saare on the ligne Maginot in April 1940, was caught in the German attack ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Lévy (mathematician)
Paul Pierre Lévy (15 September 1886 – 15 December 1971) was a French mathematician who was active especially in probability theory, introducing fundamental concepts such as local time, stable distributions and characteristic functions. Lévy processes, Lévy flights, Lévy measures, Lévy's constant, the Lévy distribution, the Lévy area, the Lévy arcsine law, and the fractal Lévy C curve are named after him. Biography Lévy was born in Paris to a Jewish family which already included several mathematicians. His father Lucien Lévy was an examiner at the École Polytechnique. Lévy attended the École Polytechnique and published his first paper in 1905, at the age of nineteen, while still an undergraduate, in which he introduced the Lévy–Steinitz theorem. His teacher and advisor was Jacques Hadamard. After graduation, he spent a year in military service and then studied for three years at the École des Mines, where he became a professor in 1913. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maurice Fréchet
Maurice may refer to: *Maurice (name), a given name and surname, including a list of people with the name Places * or Mauritius, an island country in the Indian Ocean * Maurice, Iowa, a city * Maurice, Louisiana, a village * Maurice River, a tributary of the Delaware River in New Jersey Other uses * ''Maurice'' (2015 film), a Canadian short drama film * Maurice (horse), a Thoroughbred racehorse * ''Maurice'' (novel), a 1913 novel by E. M. Forster, published in 1971 ** ''Maurice'' (1987 film), a British film based on the novel * ''Maurice'' (Shelley), a children's story by Mary Shelley *Maurice, a character from the Madagascar ''franchise'' *Maurices, an American retail clothing chain *Maurice or Maryse, a type of cooking spatula See also *Church of Saint Maurice (other) * *Maurice Debate, a 1918 debate in the British House of Commons *Maurice Lacroix, Swiss manufacturer of mechanical timepieces, clocks, and watches *Mauricie, Quebec, Canada *Moritz (other) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Feller
William "Vilim" Feller (July 7, 1906 – January 14, 1970), born Vilibald Srećko Feller, was a Croatian–American mathematician specializing in probability theory. Early life and education Feller was born in Zagreb to Ida Oemichen-Perc, a Croatian–Austrian Catholic, and Eugen Viktor Feller, son of a Polish–Jewish father (David Feller) and an Austrian mother (Elsa Holzer). Eugen Feller was a famous chemist and created ''Elsa fluid'' named after his mother. According to Gian-Carlo Rota, Eugen Feller's surname was a "Slavic tongue twister", which William changed at the age of twenty. This claim appears to be false. His forename, Vilibald, was chosen by his Catholic mother for the saint day of his birthday. Career and later life Feller held a docent position at the University of Kiel beginning in 1928. Because he refused to sign a Nazi oath, he fled the Nazis and went to Copenhagen, Denmark in 1933. He also lectured in Sweden (Stockholm and Lund). As a refugee in Sweden, Fe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Doob
Joseph Leo Doob (February 27, 1910 – June 7, 2004) was an American mathematician, specializing in analysis and probability theory. The theory of martingales was developed by Doob. Early life and education Doob was born in Cincinnati, Ohio, February 27, 1910, the son of a Jewish couple, Leo Doob and Mollie Doerfler Doob. The family moved to New York City before he was three years old. The parents felt that he was underachieving in grade school and placed him in the Ethical Culture School, from which he graduated in 1926. He then went on to Harvard where he received a BA in 1930, an MA in 1931, and a PhD (''Boundary Values of Analytic Functions'', advisor Joseph L. Walsh) in 1932. After postdoctoral research at Columbia and Princeton, he joined the department of mathematics of the University of Illinois in 1935 and served until his retirement in 1978. He was a member of the Urbana campus's Center for Advanced Study from its beginning in 1959. During the Second World War, he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrey Kolmogorov
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 who played a central role in the creation of modern probability theory. He also contributed to the mathematics of topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and Analysis of algorithms, computational complexity. Biography Early life Andrey Kolmogorov was born in Tambov, about 500 kilometers southeast of Moscow, in 1903. His unmarried mother, Maria Yakovlevna Kolmogorova, died giving birth to him. Andrey was raised by two of his aunts in Tunoshna (near Yaroslavl) at the estate of his grandfather, a well-to-do Russian nobility, nobleman. Little is known about Andrey's father. He was supposedly named Nikolai Matveyevich Katayev and had been an Agronomy, agronomist. Katayev ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksandr Khinchin
Aleksandr Yakovlevich Khinchin (, ), July 19, 1894 – November 18, 1959, was a Soviet mathematician and one of the most significant contributors to the Soviet school of probability theory. Due to romanization conventions, his name is sometimes written as "Khinchin" and other times as "Khintchine". 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph L
Joseph is a common male name, derived from the Hebrew (). "Joseph" is used, along with "Josef (given name), Josef", mostly in English, French and partially German languages. This spelling is also found as a variant in the languages of the modern-day Nordic countries. In Portuguese language, Portuguese and Spanish language, Spanish, the name is "José". In Arabic, including in the Quran, the name is spelled , . In Kurdish language, Kurdish (''Kurdî''), the name is , Persian language, Persian, the name is , and in Turkish language, Turkish it is . In Pashto the name is spelled ''Esaf'' (ايسپ) and in Malayalam it is spelled ''Ousep'' (ഔസേപ്പ്). In Tamil language, Tamil, it is spelled as ''Yosepu'' (யோசேப்பு). The name has enjoyed significant popularity in its many forms in numerous countries, and ''Joseph'' was one of the two names, along with ''Robert'', to have remained in the top 10 boys' names list in the US from 1925 to 1972. It is especiall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
