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Complete Monotonicity
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 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 ... 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 ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...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 (''m''0, ''m''1, ''m''2, ...), does there exist a positive Borel measure ''μ'' (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) \text In other words, an affirmative answer to the problem means that (''m''0, ''m''1, ''m''2, ...) is the sequence of moments of some positive Borel measure ''μ''. The Stieltjes moment problem, Vorobyev moment problem, and the Hausdorff moment problem are similar but replace the real line by ,+\infty) (Stieltjes and Vorobyev; but Vorobyev formulates the problem in the terms of matrix theory), or a bounded interval (Hausdorff). Characterization The Hamburger moment problem is solvable (that is, (''m''''n'') is a sequence of moments) if and only if the corresponding Hankel kernel on the nonnegative integers : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Problems
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."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, . The higher the probability of an event, the more likely it is that the event will occur. 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 conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacob Tamarkin
Jacob David Tamarkin (russian: Я́ков Дави́дович Тама́ркин, ''Yakov Davidovich 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 Chernihiv, Imperial Russia 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 Unive ... [...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 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 negative comment on Shohat's earlier paper on von der Pol's equation" See also * Shohat expansion *Shohat–Favard theorem In mathematics, Favard's theorem, also called the Shohat–Favard theorem, states that a sequence of polynomials satisfying a suit ... [...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–Bernstei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. 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. 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 (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Integral
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined. Long before the 20th century, mathematicians already understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the ''area under the curve'' could be defined as the integral, and computed using approximation techniques on the region by polygons. However, as the need to consider more irregular functions arose—e.g., as a result of the limiting processes of mathematical analysis and the mathematical theory of probability—it became clear that more careful approximation techniques were needed to define a suitable integral. Also, one might ... [...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 0, ∞),_whereas_in_the_0, ∞),_whereas_in_the_Hausdorff_moment_problem">/nowiki>0, ∞),_whereas_in_the_Hausdorff_moment_problem_one_considers_a_Interval_(mathematics)#Terminology.html" "title="Hausdorff_moment_problem.html" ;"title="/nowiki>0, ∞), whereas in the Hausdorff moment problem">/nowiki>0, ∞), whereas in the Hausdorff moment problem one considers a Interval_(mathematics)#Terminology">bounded interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying betwe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Felix Hausdorff
Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, and functional analysis. Life became difficult for Hausdorff and his family after Kristallnacht in 1938. The next year he initiated efforts to emigrate to the United States, but was unable to make arrangements to receive a research fellowship. On 26 January 1942, Felix Hausdorff, along with his wife and his sister-in-law, died by suicide by taking an overdose of veronal, rather than comply with German orders to move to the Endenich camp, and there suffer the likely implications, about which he held no illusions. Life Childhood and youth Hausdorff's father, the Jewish merchant Louis Hausdorff (1843–1896), moved with his young family to Leipzig in the autumn of 1870, and over time worked at various companies, including a linen-and cotton goo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other examples of intervals are the set of numbers such that , the set of all real numbers \R, the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element). Real intervals play an important role in the theory of integration, because they are the simplest sets whose "length" (or "measure" or "size") is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure. Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical app ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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