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Translation Plane
In mathematics, a translation plane is a projective plane which admits a certain group of symmetries (described below). Along with the Hughes planes and the Figueroa planes, translation planes are among the most well-studied of the known non-Desarguesian planes, and the vast majority of known non-Desarguesian planes are either translation planes, or can be obtained from a translation plane via successive iterations of Duality (projective geometry), dualization and/or Hall plane#Derivation, derivation. In a projective plane, let represent a point, and represent a line. A ''Homography#Central collineations, central collineation'' with ''center'' and ''axis'' is a collineation fixing every point on and every line through . It is called an ''elation'' if is on , otherwise it is called a ''homology''. The central collineations with center and axis form a group. A line in a projective plane is a ''translation line'' if the group of all elations with axis Group action (mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternative Division Ring
In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have *x(xy) = (xx)y *(yx)x = y(xx) for all ''x'' and ''y'' in the algebra. Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions. The associator Alternative algebras are so named because they are the algebras for which the associator is alternating. The associator is a trilinear map given by : ,y,z= (xy)z - x(yz). By definition, a multilinear map is alternating if it vanishes whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent to : ,x,y= 0 : ,x,x= 0 Both of these identities together imply that: : ,y,x ,x,x ,y,x :- ,x+y,x+y= := ,x+y,-y= := ,x,-y- ,y,y= 0 for all x and y. This is equivalent to the '' flexible identity'' :(xy)x = x(yx). The associator of an alternative algebra is therefore alternating. ... [...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] |
Transversal (combinatorics)
In mathematics, particularly in combinatorics, given a family of sets, here called a collection ''C'', a transversal (also called a cross-section) is a set containing exactly one element from each member of the collection. When the sets of the collection are mutually disjoint, each element of the transversal corresponds to exactly one member of ''C'' (the set it is a member of). If the original sets are not disjoint, there are two possibilities for the definition of a transversal: * One variation is that there is a bijection ''f'' from the transversal to ''C'' such that ''x'' is an element of ''f''(''x'') for each ''x'' in the transversal. In this case, the transversal is also called a system of distinct representatives (SDR). * The other, less commonly used, does not require a one-to-one relation between the elements of the transversal and the sets of ''C''. In this situation, the members of the system of representatives are not necessarily distinct. In computer science, comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperboloid Of One Sheet
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface (mathematics), surface generated by rotating a hyperbola around one of its Hyperbola#Equation, principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scaling (geometry), scalings, or more generally, of an affine transformation. A hyperboloid is a quadric surface, that is, a surface (mathematics), surface defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a conical surface, cone or a cylinder, having a central symmetry, center of symmetry, and intersecting many plane (geometry), planes into hyperbolas. A hyperboloid has three pairwise perpendicular rotational symmetry, axes of symmetry, and three pairwise perpendicular reflection symmetry, planes of symmetry. Given a hyperboloid, one can choose a Cartesian coordinate system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regulus
Regulus is the brightest object in the constellation Leo (constellation), Leo and one of the List of brightest stars, brightest stars in the night sky. It has the Bayer designation designated α Leonis, which is Latinisation of names, Latinized to Alpha Leonis, and abbreviated Alpha Leo or α Leo. Regulus appears singular, but is actually a quadruple star system composed of four stars that are organized into two pairs. The Binary star#Spectroscopic binaries, spectroscopic binary Regulus A consists of a blue-white main-sequence star and its companion, a pre-white dwarf. The system lies approximately 79 light years from the Solar System. HD 87884 is separated from Regulus by and is itself a close pair. Regulus, along with five slightly dimmer stars (Zeta Leonis, Mu Leonis, Gamma Leonis, Epsilon Leonis, and Eta Leonis) have collectively been called 'the Sickle', which is an Asterism (astronomy), asterism that marks the head of Leo. Nomenclature ''α Leonis'' (Latinized ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation (geometry)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same Distance geometry, distance in a given direction (geometry), direction. A translation can also be interpreted as the addition of a constant vector space, vector to every point, or as shifting the Origin (mathematics), origin of the coordinate system. In a Euclidean space, any translation is an isometry. As a function If \mathbf is a fixed vector, known as the ''translation vector'', and \mathbf is the initial position of some object, then the translation function T_ will work as T_(\mathbf)=\mathbf+\mathbf. If T is a translation, then the image (mathematics), image of a subset A under the function (mathematics), function T is the translate of A by T . The translate of A by T_ is often written as A+\mathbf . Application in classical physics In classical physics, translational motion is movement that changes the Position (geometry), positio ... [...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] |
Field (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called scalar (mathematics), ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field (mathematics), field. Vector spaces generalize Euclidean vectors, which allow modeling of Physical quantity, physical quantities (such as forces and velocity) that have not only a Magnitude (mathematics), magnitude, but also a Orientation (geometry), direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix (mathematics), matrices, which ... [...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] |
Spread (projective Geometry)
A frequently studied problem in finite geometry is to identify ways in which an object can be covered by other simpler objects such as points, lines, and planes. In projective geometry, a specific instance of this problem that has numerous applications is determining whether, and how, a projective space can be covered by pairwise disjoint subspaces which have the same dimension; such a partition is called a spread. Specifically, a spread of a projective space PG(d,K), where d \geq 1 is an integer and K a division ring, is a set of r-dimensional subspaces, for some 0 < r < d such that every point of the space lies in exactly one of the elements of the spread. Spreads are particularly well-studied in projective geometries over finite fields, though some notable results apply to infinite projective geometries as well. In the finite case, the foundational work on spreads appears in André and independently in Bruck-Bose in connection with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
