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James Propp
James Gary Propp is a professor of mathematics at the University of Massachusetts Lowell. Education and career In high school, Propp was one of the national winners of the United States of America Mathematical Olympiad (USAMO), and an alumnus of the Hampshire College Summer Studies in Mathematics. Propp obtained his AB in mathematics in 1982 at Harvard. After advanced study at Cambridge, he obtained his PhD from the University of California at Berkeley. He has held professorships at seven universities, including Harvard, MIT, the University of Wisconsin, and the University of Massachusetts Lowell. Mathematical research Propp is the co-editor of the book ''Microsurveys in Discrete Probability'' (1998) and has written more than fifty journal articles on game theory, combinatorics and probability, and recreational mathematics. He lectures extensively and has served on the Mathematical Olympiad Committee of the Mathematical Association of America, which sponsors the USAMO. In the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jim Propp
James Gary Propp is a professor of mathematics at the University of Massachusetts Lowell. Education and career In high school, Propp was one of the national winners of the United States of America Mathematical Olympiad (USAMO), and an alumnus of the Hampshire College Summer Studies in Mathematics. Propp obtained his AB in mathematics in 1982 at Harvard. After advanced study at Cambridge, he obtained his PhD from the University of California at Berkeley. He has held professorships at seven universities, including Harvard, MIT, the University of Wisconsin, and the University of Massachusetts Lowell. Mathematical research Propp is the co-editor of the book ''Microsurveys in Discrete Probability'' (1998) and has written more than fifty journal articles on game theory, combinatorics and probability, and recreational mathematics. He lectures extensively and has served on the Mathematical Olympiad Committee of the Mathematical Association of America, which sponsors the USAMO. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arlington, Massachusetts
Arlington is a town in Middlesex County, Massachusetts. The town is six miles (10 km) northwest of Boston, and its population was 46,308 at the 2020 census. History European colonists settled the Town of Arlington in 1635 as a village within the boundaries of Cambridge, Massachusetts, under the name Menotomy, an Algonquian word considered by some to mean "swift running water", though linguistic anthropologists dispute that translation. A larger area, including land that was later to become the town of Belmont, and outwards to the shore of the Mystic River, which had previously been part of Charlestown, was incorporated on February 27, 1807, as West Cambridge, replacing Menotomy. In 1867, the town was renamed Arlington, in honor of those buried in Arlington National Cemetery; the name change took effect that April 30. The Massachusett tribe, part of the Algonquian group of Native Americans, lived around the Mystic Lakes, the Mystic River and Alewife Brook. When ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Projective Space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space. Basic properties Construction As with all projective spaces, RP''n'' is formed by taking the quotient of under the equivalence relation for all real numbers . For all ''x'' in one can always find a ''λ'' such that ''λx'' has norm 1. There are precisely two such ''λ'' differing by sign. Thus RP''n'' can also be formed by identifying antipodal points of the unit ''n''-sphere, ''S''''n'', in R''n''+1. One can further restrict to the upper hemisphere of ''S''''n'' and merely identify antipodal points on the bounding equator. This shows that RP''n'' is also equivalent to the closed ''n''-dimensional disk, ''D''''n'', with antipodal points on the boundary, , identified. Low-dimensional examples * RP1 is called the real projective line, which is topologically ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say V is if the dimension of V is finite, and if its dimension is infinite. The dimension of the vector space V over the field F can be written as \dim_F(V) or as : F read "dimension of V over F". When F can be inferred from context, \dim(V) is typically written. Examples The vector space \R^3 has \left\ as a standard basis, and therefore \dim_(\R^3) = 3. More generally, \dim_(\R^n) = n, and even more generally, \dim_(F^n) = n for any field F. The complex numbers \Complex are both a real and complex vector space; we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by \chi ( Greek lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra. Polyhedra The Euler characteristic \chi was classically defined for the surfaces of polyhedra, acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Occurrences Of Grandi's Series
This article lists occurrences of the paradoxical infinite "sum" +1 -1 +1 -1 ... , sometimes called Grandi's series. Parables Guido Grandi illustrated the series with a parable involving two brothers who share a gem. Thomson's lamp is a supertask in which a hypothetical lamp is turned on and off infinitely many times in a finite time span. One can think of turning the lamp on as adding 1 to its state, and turning it off as subtracting 1. Instead of asking the sum of the series, one asks the final state of the lamp. One of the best-known classic parables to which infinite series have been applied, Achilles and the tortoise, can also be adapted to the case of Grandi's series. Numerical series The Cauchy product of Grandi's series with itself is 1 − 2 + 3 − 4 + · · ·. Several series resulting from the introduction of zeros into Grandi's series have interesting properties; for these see Summation of Grandi's series#Dilution. Grandi's series is just one example of a diver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Sign Matrix
In mathematics, an alternating sign matrix is a square matrix of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. These matrices generalize permutation matrices and arise naturally when using Dodgson condensation to compute a determinant. They are also closely related to the six-vertex model with domain wall boundary conditions from statistical mechanics. They were first defined by William Mills, David Robbins, and Howard Rumsey in the former context. Examples A permutation matrix is an alternating sign matrix, and an alternating sign matrix is a permutation matrix if and only if no entry equals . An example of an alternating sign matrix that is not a permutation matrix is : \begin 0&0&1&0\\ 1&0&0&0\\ 0&1&-1&1\\ 0&0&1&0 \end. Alternating sign matrix theorem The ''alternating sign matrix theorem'' states that the number of n\times n alternating sign matrices is : \prod_^\frac = \frac. The f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surcomplex Number
In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers (including the hyperreal numbers) can be realized as subfields of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the '' Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithms
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm can be expressed within a finite amount of space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chain Monte Carlo
In statistics, Markov chain Monte Carlo (MCMC) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain a sample of the desired distribution by recording states from the chain. The more steps that are included, the more closely the distribution of the sample matches the actual desired distribution. Various algorithms exist for constructing chains, including the Metropolis–Hastings algorithm. Application domains MCMC methods are primarily used for calculating numerical approximations of multi-dimensional integrals, for example in Bayesian statistics, computational physics, computational biology and computational linguistics. In Bayesian statistics, the recent development of MCMC methods has made it possible to compute large hierarchical models that require integrations over hundreds to thousands of unknown parameters. In rare even ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs ''now''." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). It is named after the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics. Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
