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Erdős–Ulam Problem
In mathematics, the Erdős–Ulam problem asks whether the plane contains a dense set of points whose Euclidean distances are all rational numbers. It is named after Paul Erdős and Stanislaw Ulam. Large point sets with rational distances The Erdős–Anning theorem states that a set of points with integer distances must either be finite or lie on a single line. However, there are other infinite sets of points with rational distances. For instance, on the unit circle, let ''S'' be the set of points :(\cos\theta,\sin\theta) where \theta is restricted to values that cause \tan\tfrac to be a rational number. For each such point, both \sin\tfrac and \cos\tfrac\theta 2 are themselves both rational, and if \theta and \varphi define two points in ''S'', then their distance is the rational number : \left, 2\sin\frac \theta 2 \cos\frac \varphi 2 -2\sin\frac \varphi 2 \cos\frac \theta 2 \. More generally, a circle with radius \rho contains a dense set of points at rational distances to e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Set
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, A is dense in X if the smallest closed subset of X containing A is X itself. The of a topological space X is the least cardinality of a dense subset of X. Definition A subset A of a topological space X is said to be a of X if any of the following equivalent conditions are satisfied: The smallest closed subset of X containing A is X itself. The closure of A in X is equal to X. That is, \operatorname_X A = X. The interior of the complement of A is empty. That is, \operatorname_X (X \setminus A) = \varnothing. Every point in X either ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete & Computational Geometry
'' Discrete & Computational Geometry'' is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal publishes articles on discrete geometry and computational geometry. Abstracting and indexing The journal is indexed in: * ''Mathematical Reviews'' * ''Zentralblatt MATH'' * ''Science Citation Index'' * ''Current Contents''/Engineering, Computing and Technology Notable articles The articles by Gil Kalai with a proof of a subexponential upper bound on the diameter of a polyhedron and by Samuel Ferguson on the Kepler conjecture, both published in Discrete & Computational geometry, earned their author the Fulkerson Prize The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at e .... References External link ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Problems Of Plane Geometry
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th century, Italian mathematician Giuseppe Peano formalized arithmetic with his Peano axioms, which are highly important to the field of mathematical logic today. History The prehistory of arithmetic is limited to a small number of artifacts, which may indicate the conception of addition and subtraction, the best-known being the Ishango bone from central Africa, dating from somewhere between 20,000 and 18,000 BC, although its interpretation is disputed. The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations: addition, subtraction, multiplication, and division, as early as 2000 BC. These artifacts do not always reveal the specific process used for solving problems, but the ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called a pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harborth's Conjecture
In mathematics, Harborth's conjecture states that every planar graph has a planar drawing in which every edge is a straight segment of integer length.. This conjecture is named after Heiko Harborth, and (if true) would strengthen Fáry's theorem on the existence of straight-line drawings for every planar graph. For this reason, a drawing with integer edge lengths is also known as an integral Fáry embedding.. Despite much subsequent research, Harborth's conjecture remains unsolved. Special classes of graphs Although Harborth's conjecture is not known to be true for all planar graphs, it has been proven for several special kinds of planar graph. One class of graphs that have integral Fáry embeddings are the graphs that can be reduced to the empty graph by a sequence of operations of two types: *Removing a vertex of degree at most two. *Replacing a vertex of degree three by an edge between two of its neighbors. (If such an edge already exists, the degree three vertex can be remo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monatshefte Für Mathematik
'' Monatshefte für Mathematik'' is a peer-reviewed mathematics journal established in 1890. Among its well-known papers is " Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" by Kurt Gödel, published in 1931. The journal was founded by Gustav von Escherich and Emil Weyr in 1890 as ''Monatshefte für Mathematik und Physik'' and published until 1941. In 1947 it was reestablished by Johann Radon under its current title. It is currently published by Springer in cooperation with the Austrian Mathematical Society. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.58, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... was 0.764. External links *''Monatshefte für Mathematik und ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abc Conjecture
The ''abc'' conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of three positive integers ''a'', ''b'' and ''c'' (hence the name) that are relatively prime and satisfy ''a'' + ''b'' = ''c''. The conjecture essentially states that the product of the distinct prime factors of ''abc'' is usually not much smaller than ''c''. A number of famous conjectures and theorems in number theory would follow immediately from the ''abc'' conjecture or its versions. Mathematician Dorian Goldfeld described the ''abc'' conjecture as "The most important unsolved problem in Diophantine analysis". The ''abc'' conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves, which involves more geometric structures in its statement than the ''abc'' conjecture. The ''abc'' conjecture was sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hector Pasten
In Greek mythology, Hector (; grc, Ἕκτωρ, Hektōr, label=none, ) is a character in Homer's Iliad. He was a Trojan prince and the greatest warrior for Troy during the Trojan War. Hector led the Trojans and their allies in the defense of Troy, killing countless Greek warriors. He was ultimately killed in single combat by Achilles, who later dragged his dead body around the city of Troy behind his chariot. Etymology In Greek, is a derivative of the verb ἔχειν ''ékhein'', archaic form * grc, ἕχειν, hékhein, label=none ('to have' or 'to hold'), from Proto-Indo-European *'' seɡ́ʰ-'' ('to hold'). , or as found in Aeolic poetry, is also an epithet of Zeus in his capacity as 'he who holds verything together. Hector's name could thus be taken to mean 'holding fast'. Description Hector was described by the chronicler Malalas in his account of the ''Chronography'' as "dark-skinned, tall, very stoutly built, strong, good nose, wooly-haired, good beard, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bombieri–Lang Conjecture
In arithmetic geometry, the Bombieri–Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variety of general type. Statement The weak Bombieri–Lang conjecture for surfaces states that if X is a smooth surface of general type defined over a number field k, then the points of X do not form a dense set in the Zariski topology on X. The general form of the Bombieri–Lang conjecture states that if X is a positive-dimensional algebraic variety of general type defined over a number field k, then the points of X do not form a dense set in the Zariski topology. The refined form of the Bombieri–Lang conjecture states that if X is an algebraic variety of general type defined over a number field k, then there is a dense open subset U of X such that for all number field extensions k' over k, the set of points in U is finite. History The Bombieri–Lang conjecture was independent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Terence Tao
Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes topics in harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing and analytic number theory. Tao was born to ethnic Chinese immigrant parents and raised in Adelaide. Tao won the Fields Medal in 2006 and won the Royal Medal and Breakthrough Prize in Mathematics in 2014. He is also a 2006 MacArthur Fellow. Tao has been the author or co-author of over three hundred research papers. He is widely regarded as one of the greatest living mathematicians and has been referred to as the "Mozart of mathematics". Life and career Family Tao's parents are first-generation immigrants from Hong Kong to Australia.''Wen Wei Po'', Page A4, 24 Au ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve. If the curve is contained in an affine space or a projective space, one can take a projection for such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance. These names come from the ancient Greek mathematicians Euclid and Pythagoras, although Euclid did not represent distances as numbers, and the connection from the Pythagorean theorem to distance calculation was not made until the 18th century. The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known for computing distances between different types of objects, such as the distance from a point to a line. In advanced mathematics, the concept of distance has been generalized to abstract metric spaces, and other distances than Euclidean have been studied. In some applications in statistic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
