|
Beck's Theorem (geometry)
In discrete geometry, Beck's theorem is any of several different results, two of which are given below. Both appeared, alongside several other important theorems, in a well-known paper by József Beck. The two results described below primarily concern lower bounds on the number of lines ''determined'' by a set of points in the plane. (Any line containing at least two points of point set is said to be ''determined'' by that point set.) Erdős–Beck theorem The Erdős–Beck theorem is a variation of a classical result by L. M. Kelly and W. O. J. Moser involving configurations of ''n'' points of which at most ''n'' − ''k'' are collinear, for some 0 < ''k'' < ''O''(). They showed that if ''n'' is sufficiently large, relative to ''k'', then the configuration spans at least ''kn'' − (1/2)(3''k'' + 2)(''k'' − 1) lines. Elekes and Csaba Toth noted that the Erdős–Beck theorem does not easily extend to high ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Although 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 studie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
József Beck
József Beck (Budapest, Hungary, February 14, 1952) is a Harold H. Martin Professor of Mathematics at Rutgers University. His contributions to combinatorics include the partial colouring lemma and the Beck–Fiala theorem in ''discrepancy theory'', the algorithmic version of the Lovász local lemma, the Beck's theorem (geometry), two extremes theorem in combinatorial geometry and the second moment method in the theory of positional games, among others. Beck was awarded the Fulkerson Prize in 1985 for a paper titled ''"Roth's estimate of the discrepancy of integer sequences is nearly sharp"'', which introduced the notion of discrepancy on hypergraphs and established an upper bound on the discrepancy of the family of arithmetic progressions contained in , matching the classical lower bound up to a polylogarithmic function, polylogarithmic factor. Jiří Matoušek (mathematician), Jiří Matoušek and Joel Spencer later succeeded in getting rid of this factor, showing that the bound ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leroy Milton Kelly
Leroy Milton Kelly (May 8, 1914 – February 21, 2002 for Leroy M Kelly: Holt, Michigan.) was an American whose research primarily concerned . In 1986 he settled a conjecture of by proving that n points in complex 3-space, not all lying on a plane, determine an [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skew Lines
In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the same plane must either cross each other or be parallel, so skew lines can exist only in three or more dimensions. Two lines are skew if and only if they are not coplanar. General position If four points are chosen at random uniformly within a unit cube, they will almost surely define a pair of skew lines. After the first three points have been chosen, the fourth point will define a non-skew line if, and only if, it is coplanar with the first three points. However, the plane through the first three points forms a subset of measure zero of the cube, and the probability that the fourth point lies on this plane is zero. If it does not, the lines defined by the points will be skew. Similarly, in three-dimensional space a very small perturbati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered around discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Beck Theorem
In discrete geometry, Beck's theorem is any of several different results, two of which are given below. Both appeared, alongside several other important theorems, in a well-known paper by József Beck. The two results described below primarily concern lower bounds on the number of lines ''determined'' by a set of points in the plane. (Any line containing at least two points of point set is said to be ''determined'' by that point set.) Erdős–Beck theorem The Erdős–Beck theorem is a variation of a classical result by L. M. Kelly and W. O. J. Moser involving configurations of ''n'' points of which at most ''n'' − ''k'' are collinear, for some 0 < ''k'' < ''O''(). They showed that if ''n'' is sufficiently large, relative to ''k'', then the configuration spans at least ''kn'' − (1/2)(3''k'' + 2)(''k'' − 1) lines. Elekes and Csaba Toth noted that the Erdős–Beck theorem does not easily extend to high ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Szemerédi–Trotter Theorem
The Szemerédi–Trotter theorem is a mathematical result in the field of Discrete geometry. It asserts that given points and lines in the Euclidean plane, the number of incidences (''i.e.'', the number of point-line pairs, such that the point lies on the line) is O \left ( n^ m^ + n + m \right ). This bound cannot be improved, except in terms of the implicit constants. As for the implicit constants, it was shown by János Pach, Radoš Radoičić, Gábor Tardos, and Géza Tóth that the upper bound 2.5n^ m^ + n + m holds. Since then better constants are known due to better crossing lemma constants; the current best is 2.44. On the other hand, Pach and Tóth showed that the statement does not hold true if one replaces the coefficient 2.5 with 0.42. An equivalent formulation of the theorem is the following. Given points and an integer , the number of lines which pass through at least of the points is O \left( \frac + \frac \right ). The original proof of Endre Szemeréd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each successive term can be obtained by multiplying the previous term by 1/2. In general, a geometric series is written as a + ar + ar^2 + ar^3 + ..., where a is the coefficient of each term and r is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the Taylor series, the complex Fourier series, and the matrix exponential. The name geometric series indicates each term is the geometric mean of its two neighboring terms, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighboring terms. The sequence of geometric series term ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Plane Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logical system in which each result is '' proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the ''Elements'' states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geomet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Discrete Geometry
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.PNG "Click for more on -> Theorems In Discrete Geometry")