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Hausdorff Moment Problem
In mathematics, the Hausdorff moment problem, named after Felix Hausdorff, asks for necessary and sufficient conditions that a given sequence be the sequence of moments :m_n = \int_0^1 x^n\,d\mu(x) of some Borel measure supported on the closed unit interval . In the case , this is equivalent to the existence of a random variable supported on , such that . The essential difference between this and other well-known moment problems is that this is on a bounded interval, whereas in the Stieltjes moment problem one considers a half-line , and in the Hamburger moment problem one considers the whole line . The Stieltjes moment problems and the Hamburger moment problems, if they are solvable, may have infinitely many solutions (indeterminate moment problem) whereas a Hausdorff moment problem always has a unique solution if it is solvable (determinate moment problem). In the indeterminate moment problem case, there are infinite measures corresponding to the same prescribed moment ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamburger Moment Problem
In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence , does there exist a positive Borel measure (for instance, the measure determined by the cumulative distribution function of a random variable) on the real line such that :m_n = \int_^\infty x^n\,d\mu(x)? In other words, an affirmative answer to the problem means that is the sequence of moments of some positive Borel measure . The Stieltjes moment problem, Vorobyev moment problem, and the Hausdorff moment problem are similar but replace the real line by Hankel kernel on the nonnegative integers : A = \left(\begin m_0 & m_1 & m_2 & \cdots \\ m_1 & m_2 & m_3 & \cdots \\ m_2 & m_3 & m_4 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end\right) is positive definite, i.e., : \sum_m_c_j\overline\ge0 for every arbitrary sequence of complex number">positive definite kernel">positive definite, i.e., : \sum_m_c_j\overline\ge0 for every arbitrary se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Problems
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th ed., (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', vol. 1, 3rd ed., (1968), Wiley, . This number is often expressed as a percentage (%), ranging from 0% to 100%. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These concepts have been given an axiomatic mathematical formaliza ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacob Tamarkin
Jacob David Tamarkin (, ; 11 July 1888 – 18 November 1945) was a Russian-American mathematician, best known for his work in mathematical analysis. Biography Tamarkin was born in Chernigov, Russian Empire (now Chernihiv, Ukraine), to a wealthy Jewish family. His father, David Tamarkin, was a physician and his mother, Sophie Krassilschikov, was from a family of a landowner. He shares a common ancestor with the Van Leer family, sometimes spelled Von Löhr or Valar. He moved to St. Petersburg as a child and grew up there. In high school, he befriended Alexander Friedmann, a future cosmologist, with whom he wrote his first mathematics paper in 1906, and remained friends and colleagues until Friedmann's sudden death in 1925. Vladimir Smirnov was his other friend from the same gymnasium. Many years later, they coauthored a popular textbook titled "A course in higher mathematics". Tamarkin studied in St. Petersburg University where he defended his dissertation in 1917. His advi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Alexander Shohat
James Alexander Shohat (aka Jacques Chokhate (or Chokhatte), 18 November 1886, Brest-Litovsk – 8 October 1944, Philadelphia) was a Russian-American mathematician at the University of Pennsylvania who worked on the moment problem. He studied at the University of Petrograd and married the physicist Nadiascha W. Galli, the couple emigrating from Russia to the United States in 1923. He was an Invited Speaker of the ICM in 1924 at Toronto. Selected works * * * with J. Sherman: * * * * with J. D. Tamarkin: * 18 Aug. 2012 email from R. Askey: "Norman Levinson Norman Levinson (August 11, 1912 in Lynn, Massachusetts – October 10, 1975 in Boston) was an American mathematician. Some of his major contributions were in the study of Fourier transforms, complex analysis, non-linear differential equations, ... give the following paper a very strong review. On van der Pol's and non-linear differential equations, J. Appl. Phys15 (1944), 568-574 long with giving a very strong ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Monotonicity
In real analysis, a branch of mathematics, Bernstein's theorem states that every real-valued function on the half-line that is totally monotone is a mixture of exponential functions. In one important special case the mixture is a weighted average, or expected value. Total monotonicity (sometimes also ''complete monotonicity'') of a function means that is continuous on , infinitely differentiable on , and satisfies (-1)^n \frac f(t) \geq 0 for all nonnegative integers and for all . Another convention puts the opposite inequality in the above definition. The "weighted average" statement can be characterized thus: there is a non-negative finite Borel measure on with cumulative distribution function such that f(t) = \int_0^\infty e^ \, dg(x), the integral being a Riemann–Stieltjes integral. In more abstract language, the theorem characterises Laplace transforms of positive Borel measures on . In this form it is known as the Bernstein–Widder theorem, or Hausdorff–Berns ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolutely And Completely Monotonic Functions And Sequences
In mathematics, the notions of an absolutely monotonic function and a completely monotonic function are two very closely related concepts. Both imply very strong monotonicity properties. Both types of functions have derivatives of all orders. In the case of an absolutely monotonic function, the function as well as its derivatives of all orders must be non-negative in its domain of definition which would imply that the function as well as its derivatives of all orders are monotonically increasing functions in the domain of definition. In the case of a completely monotonic function, the function and its derivatives must be alternately non-negative and non-positive in its domain of definition which would imply that function and its derivatives are alternately monotonically increasing and monotonically decreasing functions. Such functions were first studied by S. Bernshtein in 1914 and the terminology is also due to him. There are several other related notions like the concepts of alm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called the Domain of a function, domain of the function and the set is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. History of the function concept, Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable function, differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly increased the possible applications of the concept. A f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Integral
In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, named after france, French mathematician Henri Lebesgue, is one way to make this concept rigorous and to extend it to more general functions. The Lebesgue integral is more general than the Riemann integral, which it largely replaced in mathematical analysis since the first half of the 20th century. It can accommodate functions with discontinuities arising in many applications that are pathological from the perspective of the Riemann integral. The Lebesgue integral also has generally better analytical properties. For instance, under mild conditions, it is possible to exchange limits and Lebesgue integration, while the conditions for doing this with a Riemann integral are comparatively baroque. Furthermore, the Lebesgue integral can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Difference Operator
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stieltjes Moment Problem
In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions for a sequence (''m''0, ''m''1, ''m''2, ...) to be of the form :m_n = \int_0^\infty x^n\,d\mu(x) for some measure ''μ''. If such a function ''μ'' exists, one asks whether it is unique. The essential difference between this and other well-known moment problems is that this is on a half-line /nowiki>0, ∞), whereas in the bounded interval [0, 1">Hausdorff moment problem one considers a Interval_(mathematics)#Definitions">bounded interval [0, 1 and in the Hamburger moment problem one considers the whole line (−∞, ∞). Existence Let :\Delta_n=\left[\begin m_0 & m_1 & m_2 & \cdots & m_ \\ m_1 & m_2 & m_3 & \cdots & m_ \\ m_2& m_3 & m_4 & \cdots & m_ \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ m_ & m_ & m_ & \cdots & m_ \end\right] be a Hankel matrix In linear algebra, a Hankel matrix (or catalecticant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment Problem
In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure \mu to the sequence of moments :m_n = \int_^\infty x^n \,d\mu(x)\,. More generally, one may consider :m_n = \int_^\infty M_n(x) \,d\mu(x)\,. for an arbitrary sequence of functions M_n. Introduction In the classical setting, \mu is a measure on the real line, and M is the sequence \. In this form the question appears in probability theory, asking whether there is a probability measure having specified mean, variance and so on, and whether it is unique. There are three named classical moment problems: the Hamburger moment problem in which the support of \mu is allowed to be the whole real line; the Stieltjes moment problem, for ,\infty); and the Hausdorff moment problem for a bounded interval, which without loss of generality may be taken as ,1/math>. The moment problem also extends to complex analysis as the trigonometric moment problem in which the Hankel matric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
