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Incidence Theorem
In projective geometry, an intersection theorem or incidence theorem is a statement concerning an incidence structure – consisting of points, lines, and possibly higher-dimensional objects and their incidences – together with a pair of objects and (for instance, a point and a line). The "theorem" states that, whenever a set of objects satisfies the incidences (''i.e.'' can be identified with the objects of the incidence structure in such a way that incidence is preserved), then the objects and must also be incident. An intersection theorem is not necessarily true in all projective geometries; it is a property that some geometries satisfy but others don't. For example, Desargues' theorem can be stated using the following incidence structure: *Points: \ *Lines: \ *Incidences (in addition to obvious ones such as (A,AB)): \ The implication is then (R,PQ)—that point is incident with line . Famous examples Desargues' theorem holds in a projective plane if and only if is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted . An element of the form v \otimes w is called the tensor product of v and w. An element of V \otimes W is a tensor, and the tensor product of two vectors is sometimes called an ''elementary tensor'' or a ''decomposable tensor''. The elementary tensors span V \otimes W in the sense that every element of V \otimes W is a sum of elementary tensors. If bases are given for V and W, a basis of V \otimes W is formed by all tensor products of a basis element of V and a basis element of W. The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from V\times W into another vector space Z factors uniquely through a linear map V\otimes W\to Z (see the section below ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''projective space'') and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "Point at infinity, points at infinity") to Euclidean points, and vice versa. Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translation (geometry), translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. Unlike in Euclidean geometry, the concept of an angle does not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incidence Structure
In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the Point (geometry), points and Line (geometry), lines of the Euclidean plane as the two types of objects and ignore all the properties of this geometry except for the heterogeneous relation, relation of which points are incident (geometry), incident on which lines for all points and lines. What is left is the incidence structure of the Euclidean plane. Incidence structures are most often considered in the geometrical context where they are abstracted from, and hence generalize, planes (such as affine plane (incidence geometry), affine, projective plane, projective, and Möbius planes), but the concept is very broad and not limited to geometric settings. Even in a geometric setting, incidence structures are not limited to just points and lines; higher-dimensional objects (Plane (mathematics), planes, Solid geometry, sol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. 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 mainstream 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 (ZFC), or of a less powerful theory, such as Peano arithmetic. 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'' and ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formal system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Desargues' Theorem
In projective geometry, Desargues's theorem, named after Girard Desargues, states: :Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''. Denote the three vertices of one triangle by and , and those of the other by and . ''Axial perspectivity'' means that lines and meet in a point, lines and meet in a second point, and lines and meet in a third point, and that these three points all lie on a common line called the ''axis of perspectivity''. ''Central perspectivity'' means that the three lines and are concurrent, at a point called the ''center of perspectivity''. This intersection theorem is true in the usual Euclidean plane but special care needs to be taken in exceptional cases, as when a pair of sides are parallel, so that their "point of intersection" recedes to infinity. Commonly, to remove these exceptions, mathematicians "complete" the Euclidean plane by adding points at infinity, following Jean-Victor Poncelet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Division Ring
In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element usually denoted , such that . So, (right) ''division'' may be defined as , but this notation is avoided, as one may have . A commutative division ring is a field. Wedderburn's little theorem asserts that all finite division rings are commutative and therefore finite fields. Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields". In some languages, such as French, the word equivalent to "field" ("corps") is used for both commutative and noncommutative cases, and the distinction between the two cases is made by adding qualificatives such as "corps commutatif" (commutative field) or "corps gauche" (skew field). All division rings are simple. That is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Desarguesian Plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus ''any'' two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by , RP2, or P2(R), among other notations. There are many other projective planes, both infinite, such as the complex projective plane, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shimshon Amitsur
Shimshon Avraham Amitsur (born Kaplan; ; August 26, 1921 – September 5, 1994) was an Israeli mathematician. He is best known for his work in ring theory, in particular PI rings, an area of abstract algebra. Biography Amitsur was born in Jerusalem and studied at the Hebrew University under the supervision of Jacob Levitzki. His studies were repeatedly interrupted, first by World War II and then by the 1948 Arab–Israeli War. He received his M.Sc. degree in 1946, and his Ph.D. in 1950. Later, for his joint work with Levitzki, he received the first Israel Prize in Exact Sciences. He worked at the Hebrew University until his retirement in 1989. Amitsur was a visiting scholar at the Institute for Advanced Study from 1952 to 1954. He was an Invited Speaker at the ICM in 1970 in Nice. He was a member of the Israel Academy of Sciences, where he was the Head for Experimental Science Section. He was one of the founding editors of the ''Israel Journal of Mathematics'', and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Identity
Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ability, as in a rational animal, to a psychological process, like reasoning, to mental states, such as beliefs and intentions, or to persons who possess these other forms of rationality. A thing that lacks rationality is either ''arational'', if it is outside the domain of rational evaluation, or ''irrational'', if it belongs to this domain but does not fulfill its standards. There are many discussions about the essential features shared by all forms of rationality. According to reason-responsiveness accounts, to be rational is to be responsive to reasons. For example, dark clouds are a reason for taking an umbrella, which is why it is rational for an agent to do so in response. An important rival to this approach are coherence-based accou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pappus's Hexagon Theorem
In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that *given one set of collinear points A, B, C, and another set of collinear points a,b,c, then the intersection points X,Y,Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear, lying on the ''Pappus line''. These three points are the points of intersection of the "opposite" sides of the hexagon AbCaBc. It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring. Projective planes in which the "theorem" is valid are called pappian planes. If one considers a pappian plane containing a hexagon as just described but with sides Ab and aB parallel and also sides Bc and bC parallel (so that the Pappus line u is the line at infinity), one gets the ''affine version'' of Pappus's theorem shown in the second diagram. If the Pappus line u and the lines g,h have a point in common, one gets the so-called little version of Pappus's theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
