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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 (probability), 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 emph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Buffon's Needle
In mathematics, 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 (geometry), parallel strips of wood, each the same width, and we drop a Sewing needle, 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 ''p'', in the case where the needle length ''ℓ'' is not greater than the width ''t'' of the strips, is :p=\frac \cdot \frac\ell t. This can be used to design a Monte Carlo method for approximating the number pi, , although that was not the original motivation for de Buffon's question. Solution The problem in more mathematical terms is: Given a needle of length \ell dropped on a plane ruled with parallel lines ''t'' units apart, what is the probability that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertrand's Paradox (probability)
The Bertrand paradox is a problem within the classical interpretation of probability theory. 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 domain of possibilities is infinite. Bertrand's formulation of the problem The Bertrand paradox is generally presented as follows: Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the 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 the chord joining them. To calculate the probability in question imagine the triangle rotated so its vertex coi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 memb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 affin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 several variables, rather than just one. Multivariable calculus may be thought of as an elementary part of advanced calculus. For advanced calculus, see calculus on Euclidean space. The special case of calculus in three dimensional space is often called vector calculus. Typical operations Limits and continuity A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions. For example, there are scalar functions of two variables with points in their domain which give different limits when approached along different paths. E.g., the function. :f(x,y) = \frac approaches zero whenever the point (0,0) is approached along lines through the origin (y=kx). However, when the origin is appr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 models ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. The Poisson point process is often called simply the Poisson process, but it is also called a Poisson random measure, Poisson random point field or Poisson point field. This point process has convenient mathematical properties, which has led to its being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy,G. J. Babu and E. D. Feigelson. Spatial point processes in astronomy. ''Journal of statistical planning and inference'', 50(3):311–326, 1996. biology,H. G. Othmer, S. R. Dunbar, and W. Alt. Models of dispersal in biological systems. ''Journal of mathematical biology'', 26(3):263–298, 1988. ecology,H. Thompson. Spatial point processes, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wendel's Theorem
In geometric probability theory, Wendel's theorem, named after James G. Wendel, gives the probability that ''N'' points distributed uniformly at random on an (n-1)-dimensional hypersphere all lie on the same "half" of the hypersphere. In other words, one seeks the probability that there is some half-space with the origin on its boundary that contains all ''N'' points. Wendel's theorem says that the probability is : p_=2^\sum_^\binom. The statement is equivalent to p_ being the probability that the origin is not contained in the convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ... of the ''N'' points and holds for any probability distribution on that is symmetric around the origin. In particular this includes all distribution which are rotationally invariant around ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, Oregon, an unincorporated community * Fields (Frisco, Texas), an announced planned community 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 * FIELDS, a spacecraft instrument on the Parker Solar Probe See also * Mrs. Fields, an A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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