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Clifford Displacement
In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4. In this article ''rotation'' means ''rotational displacement''. For the sake of uniqueness, rotation angles are assumed to be in the segment except where mentioned or clearly implied by the context otherwise. A "fixed plane" is a plane for which every vector in the plane is unchanged after the rotation. An "invariant plane" is a plane for which every vector in the plane, although it may be affected by the rotation, remains in the plane after the rotation. Geometry of 4D rotations Four-dimensional rotations are of two types: simple rotations and double rotations. Simple rotations A simple rotation about a rotation centre leaves an entire plane through (axis-plane) fixed. Every plane that is completely orthogonal to intersects in a certain point . For each such point is the centr ... [...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] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smoothness, smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the circle group. Rotating a circle is an example of a continuous symmetry. For an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Covering Group
In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous group homomorphism. The map ''p'' is called the covering homomorphism. A frequently occurring case is a double covering group, a topological double cover in which ''H'' has index 2 in ''G''; examples include the spin groups, pin groups, and metaplectic groups. Roughly explained, saying that for example the metaplectic group Mp2''n'' is a ''double cover'' of the symplectic group Sp2''n'' means that there are always two elements in the metaplectic group representing one element in the symplectic group. Properties Let ''G'' be a covering group of ''H''. The kernel ''K'' of the covering homomorphism is just the fiber over the identity in ''H'' and is a discrete normal subgroup of ''G''. The kernel ''K'' is closed in ''G'' if and only if ''G'' is Hausdorff (and if and only if ''H'' is Hausdorff). Going i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factor Group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For example, the cyclic group of addition modulo ''n'' can be obtained from the group of integers under addition by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory. For a congruence relation on a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written , where G is the original group and N is the normal subgroup. This is read as '', where \text is short for modulo. (The notation should be interpreted with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G is normal in G if and only if gng^ \in N for all g \in G and n \in N. The usual notation for this relation is N \triangleleft G. Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of G are precisely the kernels of group homomorphisms with domain G, which means that they can be used to internally classify those homomorphisms. Évariste Galois was the first to realize the importance of the existence of normal subgroups. Definitions A subgroup N of a group G is called a normal subgroup of G if it is invariant under conjugation; that is, the conjugation of an element of N by an element of G is always in N. The usual notation fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Product Of Groups
In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted . This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted G \oplus H. Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups. Definition Given groups (with operation ) and (with operation ), the direct product is defined as follows: The resulting algebraic object satisfies the axioms for a group. Specifically: ;Associativity: The binary operation on is associative. ;Identity: The direct product has an identity element, namely , where is the identi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quaternions is often denoted by (for ''Hamilton''), or in blackboard bold by \mathbb H. Quaternions are not a field, because multiplication of quaternions is not, in general, commutative. Quaternions provide a definition of the quotient of two vectors in a three-dimensional space. Quaternions are generally represented in the form a + b\,\mathbf i + c\,\mathbf j +d\,\mathbf k, where the coefficients , , , are real numbers, and , are the ''basis vectors'' or ''basis elements''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, robotics, magnetic resonance i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred to as multiplication. In the case of a field ''F'', the group is , where 0 refers to the zero element of ''F'' and the binary operation • is the field multiplication, *the algebraic torus GL(1). Examples *The multiplicative group of integers modulo ''n'' is the group under multiplication of the invertible elements of \mathbb/n\mathbb. When ''n'' is not prime, there are elements other than zero that are not invertible. * The multiplicative group of positive real numbers \mathbb^+ is an abelian group with 1 its identity element. The logarithm is a group isomorphism of this group to the additive group of real numbers, \mathbb. * The multiplicative group of a field F is the set of all nonzero elements: F^\times = F -\, under the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Torus
In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the Cartesian product of two circles and (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Clifford. The Clifford torus is embedded in , as opposed to in . This is necessary since and each exists in their own independent embedding space and , the resulting product space will be rather than . The historically popular view that the Cartesian product of two circles is an torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis available to it after the first circle consumes and . Stated another way, a torus embedded in is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in . The relationship is similar to that of projecting the edges of a cube onto a sheet of paper. Such a proj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Torus
In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefore isomorphic to the standard torus T''n''). A maximal torus is one which is maximal among such subgroups. That is, ''T'' is a maximal torus if for any torus ''T''′ containing ''T'' we have ''T'' = ''T''′. Every torus is contained in a maximal torus simply by dimensional considerations. A noncompact Lie group need not have any nontrivial tori (e.g. R''n''). The dimension of a maximal torus in ''G'' is called the rank of ''G''. The rank is well-defined since all maximal tori turn out to be conjugate. For semisimple groups the rank is equal to the number of nodes in the associated Dynkin diagram. Examples The unitary group U(''n'') has as a maximal torus the subgroup of all diagonal matrices. That is, : T = \left\. ''T'' is c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugation Of Isometries In Euclidean Space
In a group, the conjugate by ''g'' of ''h'' is ''ghg''−1. Translation If ''h'' is a translation, then its conjugation by an isometry can be described as applying the isometry to the translation: *the conjugation of a translation by a translation is the first translation *the conjugation of a translation by a rotation is a translation by a rotated translation vector *the conjugation of a translation by a reflection is a translation by a reflected translation vector Thus the conjugacy class within the Euclidean group ''E''(''n'') of a translation is the set of all translations by the same distance. The smallest subgroup of the Euclidean group containing all translations by a given distance is the set of ''all'' translations. So, this is the conjugate closure of a singleton containing a translation. Thus ''E''(''n'') is a direct product of the orthogonal group ''O''(''n'') and the subgroup of translations ''T'', and ''O''(''n'') is isomorphic with the quotient group of ''E''(''n' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
