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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. Work 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, Feller reported be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zagreb
Zagreb ( , , , ) is the capital (political), capital and List of cities and towns in Croatia#List of cities and towns, largest city of Croatia. It is in the Northern Croatia, northwest of the country, along the Sava river, at the southern slopes of the Medvednica mountain. Zagreb stands near the international border between Croatia and Slovenia at an elevation of approximately above mean sea level, above sea level. At the 2021 census, the city had a population of 767,131. The population of the Zagreb urban agglomeration is 1,071,150, approximately a quarter of the total population of Croatia. Zagreb is a city with a rich history dating from Roman Empire, Roman times. The oldest settlement in the vicinity of the city was the Roman Andautonia, in today's Ščitarjevo. The historical record of the name "Zagreb" dates from 1134, in reference to the foundation of the settlement at Kaptol, Zagreb, Kaptol in 1094. Zagreb became a free royal city in 1242. In 1851 Janko Kamauf became Z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lawrence Shepp
Lawrence Alan Shepp (September 9, 1936 Brooklyn, NY – April 23, 2013, Tucson, AZ) was an American mathematician, specializing in statistics and computational tomography. Shepp obtained his PhD from Princeton University in 1961 with a dissertation entitled ''Recurrent Sums of Random Variables''. His advisor was William Feller. He joined Bell Laboratories in 1962. He joined Rutgers University in 1997. He joined University of Pennsylvania in 2010. His work in tomography has had biomedical imaging applications, and he has also worked as professor of radiology at Columbia University (1973–1996), as a mathematician in the radiology service of Columbia Presbyterian Hospital. Awards and honors * 2014: IEEE Marie Sklodowska-Curie Award * 2012: Became a fellow of the American Mathematical Society. * 1992: Elected member of the Institute of Medicine * 1989: Elected member of the National Academy of Sciences * 1979: IEEE Distinguished Scientist Award in 1979 * 1979: Lester R. For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
National Medal Of Science
The National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral and social sciences, biology, chemistry, engineering, mathematics and physics. The twelve member presidential Committee on the National Medal of Science is responsible for selecting award recipients and is administered by the National Science Foundation (NSF). History The National Medal of Science was established on August 25, 1959, by an act of the Congress of the United States under . The medal was originally to honor scientists in the fields of the "physical, biological, mathematical, or engineering sciences". The Committee on the National Medal of Science was established on August 23, 1961, by executive order 10961 of President John F. Kennedy. On January 7, 1979, the American Association for the Advancement of Science (AAAS) passed a resolution propo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof By Intimidation
Proof by intimidation (or argumentum verbosum) is a jocular phrase used mainly in mathematics to refer to a specific form of hand-waving, whereby one attempts to advance an argument by marking it as obvious or trivial, or by giving an argument loaded with jargon and obscure results. It attempts to intimidate the audience into simply accepting the result without evidence, by appealing to their ignorance and lack of understanding. The phrase is often used when the author is an authority in their field, presenting their proof to people who respect ''a priori'' the author's insistence of the validity of the proof, while in other cases, the author might simply claim that their statement is true because it is trivial or because they say so. Usage of this phrase is for the most part in good humour, though it can also appear in serious criticism. A proof by intimidation is often associated with phrases such as: * "Clearly..." * "It is self-evident that..." * "It can be easily shown tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feller–Tornier Constant
In mathematics, the Feller–Tornier constant ''C''FT is the density of the set of all positive integers that have an even number of distinct prime factors raised to a power larger than one (ignoring any prime factors which appear only to the first power). It is named after William Feller (1906–1970) and Erhard Tornier (1894–1982) : \begin C_\text & =+\left( \prod_^\infty \left(1- \right) \right) \\ pt& = \left(1+ \prod_^\infty \left(1 - \right) \right) \\ pt& = \left(1+ \prod_^\infty \left( 1- \right) \right) \\ pt& = + \prod_^\infty \left( 1- \right)= 0.66131704946\ldots \end Omega function The Big Omega function is given by : \Omega(x) = \text x \text See also: Prime omega function. The Iverson bracket is : = \begin 1 & \text P \text \\ 0 & \text P \text \end With these notations, we have : C_\text= \lim_ \frac Prime zeta function The prime zeta function ''P'' is give by : P(s) = \sum_ \frac 1 . The Feller–Tornier constant satisfies : C_\text= ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pareto Distribution
The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial, and many other types of observable phenomena; the principle originally applied to describing the distribution of wealth in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population. The Pareto principle or "80-20 rule" stating that 80% of outcomes are due to 20% of causes was named in honour of Pareto, but the concepts are distinct, and only Pareto distributions with shape value () of log45 ≈ 1.16 precisely reflect it. Empirical observation has shown that this 80-20 distribution fits a wide range of cases, including natural phenomena and human activities. Definitions If ''X'' is a random variable with a Pareto (Type I) distribution, then the probability that ''X'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lindeberg's Condition
In probability theory, Lindeberg's condition is a sufficient condition (and under certain conditions also a necessary condition) for the central limit theorem (CLT) to hold for a sequence of independent random variables. Unlike the classical CLT, which requires that the random variables in question have finite variance and be both independent and identically distributed, Lindeberg's CLT only requires that they have finite variance, satisfy Lindeberg's condition, and be independent. It is named after the Finnish mathematician Jarl Waldemar Lindeberg. Statement Let (\Omega, \mathcal, \mathbb) be a probability space, and X_k : \Omega \to \mathbb,\,\, k \in \mathbb, be ''independent'' random variables defined on that space. Assume the expected values \mathbb\, _k= \mu_k and variances \mathrm\, _k= \sigma_k^2 exist and are finite. Also let s_n^2 := \sum_^n \sigma_k^2 . If this sequence of independent random variables X_k satisfies Lindeberg's condition: : \lim_ \frac\sum_^n \mathbb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Palm Calculus
In the study of stochastic processes, Palm calculus, named after Swedish teletrafficist Conny Palm, is the study of the relationship between probabilities conditioned on a specified event and time-average probabilities. A Palm probability or Palm expectation, often denoted P^0(\cdot) or E^0cdot/math>, is a probability or expectation conditioned on a specified event occurring at time 0. Little's formula A simple example of a formula from Palm calculus is Little's law L=\lambda W, which states that the time-average number of users (''L'') in a system is equal to the product of the rate (\lambda) at which users arrive and the Palm-average waiting time (''W'') that a user spends in the system. That is, the average ''W'' gives equal weight to the waiting time of all customers, rather than being the time-average of "the waiting times of the customers currently in the system". Feller's paradox An important example of the use of Palm probabilities is Feller's paradox, often associ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feller-continuous Process
In mathematics, a Feller-continuous process is a continuous-time stochastic process for which the expected value of suitable statistics of the process at a given time in the future depend continuously on the initial condition of the process. The concept is named after Croatian-American mathematician William Feller. Definition Let ''X'' : bounded, continuous and Σ-measurable function ''g'' : R''n'' → R, E''x'' 'g''(''X''''t'')depends continuously upon ''x''. Examples * Every process ''X'' whose paths are almost surely constant for all time is a Feller-continuous process, since then E''x'' 'g''(''X''''t'')is simply ''g''(''x''), which, by hypothesis, depends continuously upon ''x''. * Every Itô diffusion with Lipschitz-continuous drift and diffusion coefficients is a Feller-continuous process. See also * Continuous stochastic process In probability theory, a continuous stochastic process is a type of stochastic process that may be said to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feller's Coin-tossing Constants
Feller's coin-tossing constants are a set of numerical constants which describe asymptotic probabilities that in ''n'' independent tosses of a fair coin, no run of ''k'' consecutive heads (or, equally, tails) appears. William Feller showed that if this probability is written as ''p''(''n'',''k'') then : \lim_ p(n,k) \alpha_k^=\beta_k where α''k'' is the smallest positive real root of :x^=2^(x-1) and :\beta_k=. Values of the constants For k=2 the constants are related to the golden ratio, \varphi, and Fibonacci numbers; the constants are \sqrt-1=2\varphi-2=2/\varphi and 1+1/\sqrt. The exact probability ''p''(n,2) can be calculated either by using Fibonacci numbers, ''p''(n,2) = \tfrac or by solving a direct recurrence relation leading to the same result. For higher values of k, the constants are related to generalizations of Fibonacci numbers In mathematics, the Fibonacci numbers form a sequence defined recursively by: :F_n = \begin 0 & n = 0 \\ 1 & n = 1 \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feller Process
In probability theory relating to stochastic processes, a Feller process is a particular kind of Markov process. Definitions Let ''X'' be a locally compact Hausdorff space with a countable base. Let ''C''0(''X'') denote the space of all real-valued continuous functions on ''X'' that vanish at infinity, equipped with the sup-norm , , ''f'' , , . From analysis, we know that ''C''0(''X'') with the sup norm is a Banach space. A Feller semigroup on ''C''0(''X'') is a collection ''t'' ≥ 0 of positive linear maps from ''C''0(''X'') to itself such that * , , ''T''''t''''f'' , , ≤ , , ''f'' , , for all ''t'' ≥ 0 and ''f'' in ''C''0(''X''), i.e., it is a contraction (in the weak sense); * the semigroup property: ''T''''t'' + ''s'' = ''T''''t'' o''T''''s'' for all ''s'', ''t'' ≥ 0; * lim''t'' → 0, , ''T''''t''''f'' − ''f'' , , = 0 for every ''f'' in ''C''0('' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
