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Borwein Integral
In mathematics, a Borwein integral is an integral whose unusual properties were first presented by mathematicians David Borwein and Jonathan Borwein in 2001. Borwein integrals involve products of \operatorname(ax), where the sinc function is given by \operatorname(x)=\sin(x)/x for x not equal to 0, and \operatorname(0)=1. These integrals are remarkable for exhibiting apparent patterns that eventually break down. The following is an example. : \begin & \int_0^\infty \frac \, dx= \frac \pi 2 \\0pt& \int_0^\infty \frac\frac \, dx = \frac \pi 2 \\0pt& \int_0^\infty \frac\frac\frac \, dx = \frac \pi 2 \end This pattern continues up to :\int_0^\infty \frac\frac\cdots\frac \, dx = \frac \pi 2. At the next step the pattern fails, : \begin \int_0^\infty \frac\frac\cdots\frac \, dx &= \frac~\pi \\ pt &= \frac \pi 2 - \frac~\pi \\ pt &\approx \frac \pi 2 - 2.31\times 10^. \end In general, similar integrals have value whenever the numbers are replaced by positive real numbers such ... [...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] |
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] |
David Borwein
David Borwein (March 24, 1924 – September 3, 2021) was a Lithuanian-born Canadian mathematician, known for his research in the summability theory of series and integrals. He also did work in measure theory and probability theory, number theory, and approximate subgradients and coderivatives. He latterly collaborated with his son, Jonathan Borwein, and with B.A. Mares Jr. on the properties of single-variable and many-variable sinc integrals. Biography Borwein was born in March 1924 in Lithuania to an Ashkenazi Jewish family. He formerly resided and worked in St. Andrews, Scotland, before moving to London, Ontario where he eventually became Head of Mathematics at the University of Western Ontario. He was also the president of the Canadian Mathematical Society (CMS). The David Borwein Distinguished Career Award given out by the CMS is named after him. He was an active researcher in summability theory, classical analysis, inequalities, matrix transformations, and was professo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jonathan Borwein
Jonathan Michael Borwein (20 May 1951 – 2 August 2016) was a Scottish mathematician who held an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia. He was a close associate of David H. Bailey, and they have been prominent public advocates of experimental mathematics. Borwein's interests spanned pure mathematics (analysis), applied mathematics (optimization), computational mathematics (numerical and computational analysis), and high performance computing. He authored ten books, including several on experimental mathematics, a monograph on convex functions, and over 400 refereed articles. He was a co-founder in 1995 of software company MathResources, consulting and producing interactive software primarily for school and university mathematics. Borwein was also an expert on the number pi and especially its computation. Early life and education Borwein was born in St. Andrews, Scotland in 1951 into a Jewish family. His father was mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sinc Function
In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatornamex = \frac. Alternatively, the unnormalized sinc function is often called the sampling function, indicated as Sa(''x''). In digital signal processing and information theory, the normalized sinc function is commonly defined for by \operatornamex = \frac. In either case, the value at is defined to be the limiting value \operatorname0 := \lim_\frac = 1 for all real . The normalization causes the definite integral of the function over the real numbers to equal 1 (whereas the same integral of the unnormalized sinc function has a value of ). As a further useful property, the zeros of the normalized sinc function are the nonzero integer values of . The normalized sinc function is the Fourier transform of the rectangular function with no scaling. It is used in the concep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z which starts at 0, and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, or the price of a fluctuating stock and the financial status of a gambler. Random walks have applications to engineering and many scientific fields including ecology, psychology, computer science, physics, chemistry, biology, economics, and sociology. The term ''random walk'' was first introduced by Karl Pearson in 1905. Lattice random walk A popular random walk model is that of a random walk on a regular lattice, where at each step the location jumps to another site according to some probability distribution. In a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dominated Convergence Theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration. In addition to its frequent appearance in mathematical analysis and partial differential equations, it is widely used in probability theory, since it gives a sufficient condition for the convergence of expected values of random variables. Statement Lebesgue's dominated convergence theorem. Let (f_n) be a sequence of complex-valued measurable functions on a measure space . Suppose that the sequence converges pointwise to a function f and is dominated by some integrable function g in the sense that : , f_n(x), \le g(x) for all numbers ''n'' in the index set of the sequence and all points x\in S. Then ''f'' is integrable (in the Lebesgue sense) and : \lim_ \int_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass Factorization Theorem
In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes. The theorem may be viewed as an extension of the fundamental theorem of algebra, which asserts that every polynomial may be factored into linear factors, one for each root. The theorem, which is named for Karl Weierstrass, is closely related to a second result that every sequence tending to infinity has an associated entire function with zeroes at precisely the points of that sequence. A generalization of the theorem extends it to meromorphic functions and allows one to consider a given meromorphic function as a product of three factors: terms depending on the function's zeros and poles, and an associated non-zero holomorphic function. Motivation The consequences of the fundamental theorem of algebra are twofold.. Firstly, any finite sequence \ in the complex plane ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space, there is no difference between ''almost surely'' and ''surely'' (since having a probability of 1 often entails including all the sample points). However, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, and the continuity of the paths of Brownian motion. The terms almost certainly (a.c.) and almost always (a.a.) are also used. Almost never describes the opposite of ''almost surely'': an event that h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be close to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling ''within a particular range of values'', as opposed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
