|
Racks And Quandles
In mathematics, racks and quandles are sets with binary operations satisfying axioms analogous to the Reidemeister moves used to manipulate knot diagrams. While mainly used to obtain invariants of knots, they can be viewed as algebraic constructions in their own right. In particular, the definition of a quandle axiomatizes the properties of conjugation in a group. History In 1943, Mituhisa Takasaki (高崎光久) introduced an algebraic structure which he called a ''Kei'' (圭), which would later come to be known as an involutive quandle. His motivation was to find a nonassociative algebraic structure to capture the notion of a reflection in the context of finite geometry. The idea was rediscovered and generalized in (unpublished) 1959 correspondence between John Conway and Gavin Wraith, who at the time were undergraduate students at the University of Cambridge. It is here that the modern definitions of quandles and of racks first appear. Wraith had become interested in these s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space X is denoted by \pi_1(X). Intuition Start with a space (for example, a surface), and some point in it, and all the loops both starting and ending at this point— paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breakin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension (mathematics), dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient History of geometry#Greek geometry, Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the Greek mathematics, ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of ''mathematical proof, proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as eviden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Three-dimensional Space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (mathematics), point). This is the informal meaning of the term dimension. In mathematics, a tuple of Real number, numbers can be understood as the Cartesian coordinates of a location in a -dimensional Euclidean space. The set of these -tuples is commonly denoted \R^n, and can be identified to the -dimensional Euclidean space. When , this space is called three-dimensional Euclidean space (or simply Euclidean space when the context is clear). It serves as a model of the physical universe (when relativity theory is not considered), in which all known matter exists. While this space remains the most compelling and useful way to model the world as it is experienced, it is only one example of a large variety of spaces in three dimensions called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tame Knot
Tame may refer to: *Taming, the act of training wild animals *River Tame, Greater Manchester *River Tame, West Midlands and the Tame Valley *Tame, Arauca, a Colombian town and municipality * "Tame" (song), a song by the Pixies from their 1989 album ''Doolittle'' *TAME (IATA code: EQ), flag carrier of Ecuador *tert-Amyl methyl ether, an oxygenated chemical compound often added to gasoline. * Tame.it, a context search engine for Twitter *Tame, a variety of the Idi language of Papua New Guinea *Tame (surname), people with the surname *Tame Impala Tame Impala is the psychedelic music project of Australian multi-instrumentalist Kevin Parker. In the recording studio, Parker writes, records, performs, and produces all of the project's music. As a touring act, Tame Impala consists of Parke ..., the psychedelic music project of Australian multi-instrumentalist Kevin Parker. {{disambig, geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Knot Theory And Its Ramifications
The ''Journal of Knot Theory and Its Ramifications'' was established in 1992 by Louis Kauffman and was the first journal purely devoted to knot theory. It is an interdisciplinary journal covering developments in knot theory, with emphasis on creating connections between with other branches of mathematics and the natural sciences. The journal is published by World Scientific. , retrieved 2015-03-02. According to the '''', the journal has a 2020 |
Roger Fenn
Roger is a given name, usually masculine, and a surname. The given name is derived from the Old French personal names ' and '. These names are of Germanic origin, derived from the elements ', ''χrōþi'' ("fame", "renown", "honour") and ', ' ("spear", "lance") (Hrōþigēraz). The name was introduced into England by the Normans. In Normandy, the Frankish name had been reinforced by the Old Norse cognate '. The name introduced into England replaced the Old English cognate '. ''Roger'' became a very common given name during the Middle Ages. A variant form of the given name ''Roger'' that is closer to the name's origin is ''Rodger''. Slang and other uses Roger is also a short version of the term "Jolly Roger", which refers to a black flag with a white skull and crossbones, formerly used by sea pirates since as early as 1723. From up to , Roger was slang for the word "penis". In ''Under Milk Wood'', Dylan Thomas writes "jolly, rodgered" suggesting both the sexual double entend ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Colin Rourke
Colin Rourke (born 1 January 1943) is a British mathematician who worked in PL topology, low-dimensional topology, differential topology, group theory, relativity and cosmology. He is an emeritus professor at the Mathematics Institute of the University of Warwick and a founding editor of the journals ''Geometry & Topology'' and ''Algebraic & Geometric Topology'', published by Mathematical Sciences Publishers, where he is the vice chair of its board of directors. Early career Rourke obtained his Ph.D. at the University of Cambridge in 1965 under the direction of Christopher Zeeman. Most of Rourke's early work was carried out in collaboration with Brian Sanderson. They solved a number of outstanding problems: the provision of normal bundles for the PL category (which they called "Block bundles"), the non-existence of normal microbundles (top and PL), and a geometric interpretation for all (generalized) homology theories (joint work with Sandro Buoncristiano, see bibliography) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set(mathematics)
Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electronics and computing *Set (abstract data type), a data type in computer science that is a collection of unique values ** Set (C++), a set implementation in the C++ Standard Library * Set (command), a command for setting values of environment variables in Unix and Microsoft operating-systems * Secure Electronic Transaction, a standard protocol for securing credit card transactions over insecure networks * Single-electron transistor, a device to amplify currents in nanoelectronics * Single-ended triode, a type of electronic amplifier * Set!, a programming syntax in the scheme programming language Biology and psychology * Set (psychology), a set of expectations which shapes perception or thought *Set or sett, a badger's den *Set, a small tuber ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object. Definition In the context of abstract algebra, a mathematical object is an algebraic structure such as a group, ring, or vector space. An automorphism is simply a bijective homomorphism of an object with itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator.) The identity morphism (identity mapping) is called the trivial automorphism in some contexts. Respectively, other (non-identity) automorphisms are called nontrivial automorphisms. The exact definition of an automorphism depends on the type of "mathematical ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Egbert Brieskorn
Egbert Valentin Brieskorn (7 July 1936, in Rostock – 11 July 2013, in Bonn) was a German mathematician who introduced Brieskorn spheres and the Brieskorn–Grothendieck resolution. Education Brieskorn was born in 1936 as the son of a mill construction engineer in East Prussia. He grew up in Freudenberg (Siegerland) and studied mathematics and physics at the Ludwig-Maximilians-Universität München and the Rheinische Friedrich-Wilhelms-Universität Bonn. In 1963 he received his doctorate at Bonn under Friedrich Hirzebruch with thesis ''Zur differentialtopologischen und analytischen Klassifizierung gewisser algebraischer Mannigfaltigkeiten'', followed by his habilitation in 1968. Career From 1969 until 1973 he was professor ordinarius at Georg-August-Universität Göttingen and from 1973 to 1975 at the Sonderforschungsbereich Theoretische Mathematik in Bonn (since 1980 called the Max-Planck-Institut für Mathematik). From 1975 until his retirement as professor emeritus in 2001 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
