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Bernstein's Theorem On Monotone Functions
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–Ber ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability. Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions. Scope Construction of the real numbers The theorems of real analysis rely on the properties of the real number system, which must be established. The real number system consists of an uncountable set (\mathbb), together with two binary operations denoted and , and an order denoted . The operations make the real numbers a field, and, along with the order, an ordered field. The real number system is the unique ''complete ordered field'', in the sense that any other complete ordered field is isomorphic to it. Intuitively, completeness means ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cumulative Distribution Function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an ''upwards continuous'' ''monotonic increasing'' cumulative distribution function F : \mathbb R \rightarrow ,1/math> satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability that the random variable X takes on a value less tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the derivativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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 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] |
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] |
Laplace Transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ... that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a function of a Complex number, complex variable s (in the complex frequency domain, also known as ''s''-domain, or s-plane). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms ordinary differential equations into algebraic equations and convolution into multiplication. For suitable functions ''f'', the Laplace transform is the integral \mathcal\(s) = \int_0^\infty f(t)e^ \, dt. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Stieltjes Integral
In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an instructive and useful precursor of the Lebesgue integral, and an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continuous probability. Formal definition The Riemann–Stieltjes integral of a real-valued function f of a real variable on the interval ,b/math> with respect to another real-to-real function g is denoted by :\int_^b f(x) \, \mathrmg(x). Its definition uses a sequence of partitions P of the interval ,b/math> :P=\. The integral, then, is defined to be the limit, as the mesh (the length of the longest subinterval) of the partitions approaches 0 , of the approximating sum :S(P,f,g) = \sum_^ f(c_i)\left g(x_) - g(x_i) \right/math> where c_i is in the i-th subinterval _i;x_/math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In 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 ..., an integral assigns numbers to functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with Derivative, differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Measure
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. Formal definition Let X be a locally compact Hausdorff space, and let \mathfrak(X) be the smallest σ-algebra that contains the open sets of X; this is known as the σ-algebra of Borel sets. A Borel measure is any measure \mu defined on the σ-algebra of Borel sets. A few authors require in addition that \mu is locally finite, meaning that \mu(C) 0 and μ(''B''(''x'', ''r'')) ≤ ''rs'' holds for some constant ''s'' > 0 and for every ball ''B''(''x'', ''r'') in ''X'', then the Hausdorff dimension dimHaus(''X'') ≥ ''s''. A partial converse is provided by the Frostman lemma: Lemma: Let ''A'' be a Borel subset of R''n'', and let ''s'' > 0. Then the following are equivalent: *''H''''s''(''A'') > 0, where ''H''''s'' den ... [...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] |
