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Geometric Probability
Problems of the following type, and their solution techniques, were first studied in the 18th century, and the general topic became known as geometric probability. * (Buffon's needle) What is the chance that a needle dropped randomly onto a floor marked with equally spaced parallel lines will cross one of the lines? * What is the mean length of a random chord of a unit circle? (cf. Bertrand's paradox). * What is the chance that three random points in the plane form an acute (rather than obtuse) triangle? * What is the mean area of the polygonal regions formed when randomly oriented lines are spread over the plane? For mathematical development see the concise monograph by Solomon. Since the late 20th century, the topic has split into two topics with different emphases. Integral geometry sprang from the principle that the mathematically natural probability models are those that are invariant under certain transformation groups. This topic emphasises systematic development of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Buffon's Needle
In probability theory, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon: :Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips? Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry. The solution for the sought probability , in the case where the needle length is not greater than the width of the strips, is :p=\frac \cdot \frac. This can be used to design a Monte Carlo method for approximating the number , although that was not the original motivation for de Buffon's question. The seemingly unusual appearance of in this expression occurs because the underlying probability distribution function for the needle orientation is rotationally symmetric. Solution The problem in more mathematical terms is: G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Bertrand's Paradox (probability)
The Bertrand paradox is a problem within the classical interpretation of probability theory. Joseph Louis François Bertrand, Joseph Bertrand introduced it in his work ''Calcul des probabilités'' (1889) as an example to show that the principle of indifference may not produce definite, well-defined results for probabilities if it is applied uncritically when the Possibility space, domain of possibilities is infinite. Bertrand's formulation of the problem The Bertrand paradox is generally presented as follows: Consider an equilateral triangle that is Inscribed figure, inscribed in a circle. Suppose a chord (geometry), chord of the circle is chosen at random. What is the probability that the Chord (geometry), chord is longer than a side of the triangle? Bertrand gave three arguments (each using the principle of indifference), all apparently valid yet yielding different results: # The "random endpoints" method: Choose two random points on the circumference of the circle and draw th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Society For Industrial And Applied Mathematics
Society for Industrial and Applied Mathematics (SIAM) is a professional society dedicated to applied mathematics, computational science, and data science through research, publications, and community. SIAM is the world's largest scientific society devoted to applied mathematics, and roughly two-thirds of its membership resides within the United States. Founded in 1951, the organization began holding annual national meetings in 1954, and now hosts conferences, publishes books and scholarly journals, and engages in advocacy in issues of interest to its membership. Members include engineers, scientists, and mathematicians, both those employed in academia and those working in industry. The society supports educational institutions promoting applied mathematics. SIAM is one of the four member organizations of the Joint Policy Board for Mathematics. Membership Membership is open to both individuals and organizations. By the end of its first full year of operation, SIAM had 130 me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Integral Geometry
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformations from the space of functions on one geometrical space to the space of functions on another geometrical space. Such transformations often take the form of integral transforms such as the Radon transform and its generalizations. Classical context Integral geometry as such first emerged as an attempt to refine certain statements of geometric probability theory. The early work of Luis Santaló and Wilhelm Blaschke was in this connection. It follows from the classic theorem of Crofton expressing the length of a plane curve as an expectation of the number of intersections with a random line. Here the word 'random' must be interpreted as subject to correct symmetry considerations. There is a sample space of lines, one on which the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Multivariate Calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables ('' multivariate''), rather than just one. Multivariable calculus may be thought of as an elementary part of calculus on Euclidean space. The special case of calculus in three dimensional space is often called ''vector calculus''. Introduction In single-variable calculus, operations like differentiation and integration are made to functions of a single variable. In multivariate calculus, it is required to generalize these to multiple variables, and the domain is therefore multi-dimensional. Care is therefore required in these generalizations, because of two key differences between 1D and higher dimensional spaces: # There are infinite ways to approach a single point in higher dimensions, as opposed to two (from the positive and negative direct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Stochastic Geometry
In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, hence notions of Palm conditioning, which extend to the more abstract setting of random measures. Models There are various models for point processes, typically based on but going beyond the classic homogeneous Poisson point process (the basic model for '' complete spatial randomness'') to find expressive models which allow effective statistical methods. The point pattern theory provides a major building block for generation of random object processes, allowing construction of elaborate random spatial patterns. The simplest version, the Boolean model, places a random compact object at each point of a Poisson point process. More complex versions allow interactions based in various ways on the geometry of objects. Different directions of application include: the production of mode ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Poisson Process
In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of Point (geometry), points randomly located on a Space (mathematics), mathematical space with the essential feature that the points occur independently of one another. The process's name derives from the fact that the number of points in any given finite region follows a Poisson distribution. The process and the distribution are named after French mathematician Siméon Denis Poisson. The process itself was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and actuarial science. This point process is used as a mathematical model for seemingly random processes in numerous disciplines including astronomy,G. J. Babu and E. D. Feigelson. Spatial point processes in astronomy. ''Journal of st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Eugene Seneta
Eugene B. Seneta (born 1941) is Professor Emeritus, School of Mathematics and Statistics, University of Sydney, known for his work in probability and non-negative matrices, applications and history. He is known for the variance gamma model in financial mathematics (the variance gamma process). He was Professor, School of Mathematics and Statistics at the University of Sydney from 1979 until retirement, and an Elected Fellow since 1985 of the Australian Academy of Science. In 2007 Seneta was awarded the Hannan Medal in Statistical Science page 3. by the Australian Academy of Science, for his seminal work in probability and statistics; for his work connected with branching process In probability theory, a branching process is a type of mathematical object known as a stochastic process, which consists of collections of random variables indexed by some set, usually natural or non-negative real numbers. The original purpose of ...es, history of probability and statistics, and many o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Fields Of Geometry
Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by Sponge from '' Rotting Piñata'' (1994) Businesses * Field's, a shopping centre in Denmark * Fields (department store), a chain of discount department stores in Alberta and British Columbia, Canada Places in the United States * Fields, Louisiana, an unincorporated community * Fields, Oregon, an unincorporated community * Fields (Frisco, Texas), an announced planned community * Fields Landing, California, a CDP Other uses * Fields (surname), a list of people with that name * Fields Avenue (other), various roads * Fields Institute, a research centre in mathematical sciences at the University of Toronto * Fields Medal, for outstanding achievement in mathematics * Caulfield Grammarians Football Club, also known as The Fields * FIE ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |