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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] |
Bernstein's Theorem On Monotone Functions
In real analysis, a branch of mathematics, Bernstein's theorem states that every real number, real-valued function (mathematics), 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 function, 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 (mathematics), 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 ... [...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] |
Zoran Vondraček
Zoran Vondraček (born July 28, 1959) is a Croatian mathematician specializing in Lévy processes, transformed Brownian motions, and probabilistic potential theory. He is a professor of mathematics at the University of Zagreb. Education and career Vondraček graduated from the University of Zagreb in 1982, and obtained a master's degree in mathematics there in 1986. He completed a Ph.D. at the University of Florida in 1990 under the supervision of Murali Rao. In 1992 Vondraček was made assistant professor of mathematics at the University of Zagreb. He became an associate professor in 1997, and a full professor in 2002. Vondraček is a member of the Croatian Academy of Sciences and Arts, the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ..., and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonic Function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus and analysis In calculus, a function f defined on a subset of the real numbers with real values is called ''monotonic'' if it is either entirely non-decreasing, or entirely non-increasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is termed ''monotonically increasing'' (also ''increasing'' or ''non-decreasing'') if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'') if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y\right), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclical Monotonicity
In mathematics, cyclical monotonicity is a generalization of the notion of monotonicity to the case of vector-valued function. Definition Let \langle\cdot,\cdot\rangle denote the inner product on an inner product space X and let U be a nonempty subset of X. A correspondence f: U \rightrightarrows X is called ''cyclically monotone'' if for every set of points x_1,\dots,x_ \in U with x_=x_1 it holds that \sum_^m \langle x_,f(x_)-f(x_k)\rangle\geq 0. Properties For the case of scalar functions of one variable the definition above is equivalent to usual monotonicity. Gradients of convex functions are cyclically monotone. In fact, the converse is true. Suppose U is convex and f: U \rightrightarrows \mathbb^n is a correspondence with nonempty values. Then if f is cyclically monotone, there exists an upper semicontinuous convex function F:U\to \mathbb such that f(x)\subset \partial F(x) for every x\in U, where \partial F(x) denotes the subgradient of F at x. See also * Absolutel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous function, continuous or unitary operator, unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems in general (although one usually is also interested in the actual difference of two numbers, which is ... [...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 (established) real number system. The real number system consists of an uncountable set (\mathbb), together with two binary operations denoted and \cdot, and a total 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, completenes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
