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Convex Position
In discrete and computational geometry, a set of points in the Euclidean plane or a higher-dimensional Euclidean space is said to be in convex position or convex independent if none of the points can be represented as a convex combination of the others. A finite set of points is in convex position if all of the points are vertices of their convex hull. More generally, a family of convex sets is said to be in convex position if they are pairwise disjoint and none of them is contained in the convex hull of the others. An assumption of convex position can make certain computational problems easier to solve. For instance, the traveling salesman problem, NP-hard for arbitrary sets of points in the plane, is trivial for points in convex position: the optimal tour is the convex hull. Similarly, the minimum-weight triangulation of planar point sets is NP-hard for arbitrary point sets, but solvable in polynomial time by dynamic programming for points in convex position. The Erdős–Sze ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Discrete Geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object. Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology. History Polyhedra and tessellations had been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Journal Of The ACM
The ''Journal of the ACM'' (''JACM'') is a peer-reviewed scientific journal covering computer science in general, especially theoretical aspects. It is an official journal of the Association for Computing Machinery. Its current editor-in-chief is Venkatesan Guruswami. The journal was established in 1954 and "computer scientists universally hold the ''Journal of the ACM'' in high esteem". See also * ''Communications of the ACM ''Communications of the ACM'' (''CACM'') is the monthly journal of the Association for Computing Machinery (ACM). History It was established in 1958, with Saul Rosen as its first managing editor. It is sent to all ACM members. Articles are i ...'' References External links * {{DEFAULTSORT:Journal Of The Acm Academic journals established in 1954 Computer science journals Association for Computing Machinery academic journals Bimonthly journals English-language journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) () is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stammbach (1997, 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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European Journal Of Combinatorics
The ''European Journal of Combinatorics'' is an international peer-reviewed scientific journal that specializes in combinatorics. The journal primarily publishes papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and the theories of computing. The journal includes full-length research papers, short notes, and research problems on several topics. This journal has been founded in 1980 by Michel Deza, Michel Las Vergnas and Pierre Rosenstiehl. The current editor-in-chief is Patrice Ossona de Mendez and the vice editor-in-chief is Marthe Bonamy. Abstracting and indexing The journal is abstracted and indexed in *MathSciNet, *Science Citation Index Expanded, *Scopus Scopus is a scientific abstract and citation database, launched by the academic publisher Elsevier as a competitor to older Web of Science in 2004. The ensuing competition between the two databases has been characterized as "intense" and is c . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Operations Research Letters
''Operations Research Letters'' is a bimonthly peer-reviewed academic journal covering operations research. It was established in 1981 and is published by Elsevier. The editor-in-chief is Wolfram Wiesemann. Abstracting and indexing The journal is indexed and abstracted An abstracting service is a service that provides abstracts of publications, often on a subject or group of related subjects, usually on a subscription basis. An indexing service is a service that assigns descriptors and other kinds of access po ... in: References External links *{{Official website, https://www.journals.elsevier.com/operations-research-letters Industrial engineering journals Elsevier academic journals English-language journals Bimonthly journals Academic journals established in 1981 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Projective Transformation
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation. Historically, homographies (and projective spaces) have been introduced to study perspective and projections in Euclidean geometry, and the term ''homography'', which, etymologically, roughly means "similar drawing", dates from this time. At the end of the 19th century, formal definitions of projective spaces were introduced, which extended Euclidean and affine spaces by the addition of new points called points at infinity. The term "projective transformation" originated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Projective Space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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McMullen Problem
The McMullen problem is an open problem in discrete geometry named after Peter McMullen. Statement In 1972, David G. Larman wrote about the following problem: Larman credited the problem to a private communication by Peter McMullen. Equivalent formulations Gale transform Using the Gale transform, this problem can be reformulated as: The numbers \nu of the original formulation of the McMullen problem and \mu of the Gale transform formulation are connected by the relationships \begin \mu(k)&=\min\ \\ \nu(d)&=\max\ \end Partition into nearly-disjoint hulls Also, by simple geometric observation, it can be reformulated as: The relation between \mu and \lambda is \mu(d+1)=\lambda(d),\qquad d\geq1 \, Projective duality The equivalent projective dual statement to the McMullen problem is to determine the largest number \nu(d) such that every set of \nu(d) hyperplanes in general position in ''d''-dimensional real projective space form an arrangement of hyperplanes in which one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Unit Square
In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordinate system with coordinates , a unit square is defined as a square consisting of the points where both and lie in a closed unit interval from to . That is, a unit square is the Cartesian product , where denotes the closed unit interval. Complex coordinates The unit square can also be thought of as a subset of the complex plane, the topological space formed by the complex numbers. In this view, the four corners of the unit square are at the four complex numbers , , , and . Rational distance problem It is not known whether any point in the plane is a rational distance from all four vertices of the unit square. See also * Unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequentl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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General Position
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that are possible, which is referred to as special position. Its precise meaning differs in different settings. For example, generically, two lines in the plane intersect in a single point (they are not parallel or coincident). One also says "two generic lines intersect in a point", which is formalized by the notion of a ''generic point''. Similarly, three generic points in the plane are not collinear; if three points are collinear (even stronger, if two coincide), this is a degenerate case. This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general theorems or giving precise statements thereof, and when writing computer programs (see '' generic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Erdős–Szekeres Theorem
In mathematics, the Erdős–Szekeres theorem asserts that, given ''r'', ''s,'' any sequence of distinct real numbers with length at least (''r'' − 1)(''s'' − 1) + 1 contains a monotonically increasing subsequence of length ''r'' ''or'' a monotonically decreasing subsequence of length ''s''. The proof appeared in the same 1935 paper that mentions the Happy Ending problem. It is a finitary result that makes precise one of the corollaries of Ramsey's theorem. While Ramsey's theorem makes it easy to prove that every infinite sequence of distinct real numbers contains a monotonically increasing infinite subsequence ''or'' a monotonically decreasing infinite subsequence, the result proved by Paul Erdős and George Szekeres goes further. Example For ''r'' = 3 and ''s'' = 2, the formula tells us that any permutation of three numbers has an increasing subsequence of length three or a decreasing subsequence of len ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |