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Invariant Decomposition
The invariant decomposition is a decomposition of the elements of pin groups \text(p,q,r) into orthogonal commuting elements. It is also valid in their subgroups, e.g. orthogonal, pseudo-Euclidean, conformal, and classical groups. Because the elements of Pin groups are the composition of k oriented reflections, the invariant decomposition theorem readsEvery k-reflection can be decomposed into \lceil k/2 \rceil commuting factors. It is named the invariant decomposition because these factors are the invariants of the k-reflection R \in \text(p,q,r). A well known special case is the Chasles' theorem, which states that any rigid body motion in \text(3) can be decomposed into a rotation around, followed or preceded by a translation along, a single line. Both the rotation and the translation leave two lines invariant: the axis of rotation and the orthogonal axis of translation. Since both rotations and translations are bireflections, a more abstract statement of the theorem reads "Eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pin Group
In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group. In general the map from the Pin group to the orthogonal group is ''not'' surjective or a universal covering space, but if the quadratic form is definite (and dimension is greater than 2), it is both. The non-trivial element of the kernel is denoted -1, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted -I. General definition Let V be a vector space with a non-degenerate quadratic form Q. The pin group \operatorname(V, Q) is the subset of the Clifford algebra \operatorname(V, Q) consisting of elements of the form v_1 v_2 \cdots v_k, where the v_i are vectors such that Q(v_i) = \pm 1. The spin group \operatorname(V, Q) is defined similarly, but with k restricted to be even; it is a subgroup of the pin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by Function composition, composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of orthogonal matrix, orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose invertible matrix, inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact group, compact. The orthogonal group in dimension has two connected component (topology), connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted . It consists of all orthogonal matrices of determinant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-Euclidean Space
In mathematics and theoretical physics, a pseudo-Euclidean space of signature is a finite- dimensional real -space together with a non- degenerate quadratic form . Such a quadratic form can, given a suitable choice of basis , be applied to a vector , giving q(x) = \left(x_1^2 + \dots + x_k^2\right) - \left( x_^2 + \dots + x_n^2\right) which is called the ''scalar square'' of the vector . For Euclidean spaces, , implying that the quadratic form is positive-definite. When , then is an isotropic quadratic form. Note that if , then , so that is a null vector. In a pseudo-Euclidean space with , unlike in a Euclidean space, there exist vectors with negative scalar square. As with the term ''Euclidean space'', the term ''pseudo-Euclidean space'' may be used to refer to an affine space or a vector space depending on the author, with the latter alternatively being referred to as a pseudo-Euclidean vector space (see point–vector distinction). Geometry The geometry of a pseudo-E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Group
In mathematics, the conformal group of an inner product space is the group (mathematics), group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space. Several specific conformal groups are particularly important: * The conformal orthogonal group. If ''V'' is a vector space with a quadratic form ''Q'', then the conformal orthogonal group is the group of linear transformations ''T'' of ''V'' for which there exists a scalar ''λ'' such that for all ''x'' in ''V'' *:Q(Tx) = \lambda^2 Q(x) :For a definite quadratic form, the conformal orthogonal group is equal to the orthogonal group times the group of Homothetic transformation, dilations. * The conformal group of the sphere is generated by the inversive geometry, inversions in circles. This group is also known as the Möbius group. * In Euclidean space E''n'', , the conformal group is generated by inversions in hyperspher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Group
In mathematics, the classical groups are defined as the special linear groups over the reals \mathbb, the complex numbers \mathbb and the quaternions \mathbb together with special automorphism groups of Bilinear form#Symmetric, skew-symmetric and alternating forms, symmetric or Bilinear form#Symmetric, skew-symmetric and alternating forms, skew-symmetric bilinear forms and Sesquilinear form#Hermitian form, Hermitian or Sesquilinear form#Skew-Hermitian form, skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the Simple_Lie_group#Exceptional_cases, exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Her ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pin Group
In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group. In general the map from the Pin group to the orthogonal group is ''not'' surjective or a universal covering space, but if the quadratic form is definite (and dimension is greater than 2), it is both. The non-trivial element of the kernel is denoted -1, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted -I. General definition Let V be a vector space with a non-degenerate quadratic form Q. The pin group \operatorname(V, Q) is the subset of the Clifford algebra \operatorname(V, Q) consisting of elements of the form v_1 v_2 \cdots v_k, where the v_i are vectors such that Q(v_i) = \pm 1. The spin group \operatorname(V, Q) is defined similarly, but with k restricted to be even; it is a subgroup of the pin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chasles' Theorem (kinematics)
In kinematics, Chasles' theorem, or Mozzi–Chasles' theorem, says that the most general rigid body displacement can be produced by a screw displacement. A direct Euclidean isometry in three dimensions involves a translation and a rotation. The screw displacement representation of the isometry decomposes the translation into two components, one parallel to the axis of the rotation associated with the isometry and the other component perpendicular to that axis. The Chasles theorem states that the axis of rotation can be selected to provide the second component of the original translation as a result of the rotation. This theorem in three dimensions extends a similar representation of planar isometries as rotation. Once the screw axis is selected, the screw displacement rotates about it and a translation parallel to the axis is included in the screw displacement. Planar isometries with complex numbers Euclidean geometry is expressed in the complex plane by points p = x + y i where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marcel Riesz
Marcel Riesz ( ; 16 November 1886 – 4 September 1969) was a Hungarian mathematician, known for work on summation methods, potential theory, and other parts of analysis, as well as number theory, partial differential equations, and Clifford algebras. He spent most of his career in Lund, Sweden. Marcel is the younger brother of Frigyes Riesz, who was also an important mathematician and at times they worked together (see F. and M. Riesz theorem). Biography Marcel Riesz was born in Győr, Austria-Hungary. He was the younger brother of the mathematician Frigyes Riesz. In 1904, he won the Loránd Eötvös competition. Upon entering the Budapest University, he also studied in Göttingen, and the academic year 1910-11 he spent in Paris. Earlier, in 1908, he attended the 1908 International Congress of Mathematicians in Rome. There he met Gösta Mittag-Leffler, in three years, Mittag-Leffler would offer Riesz to come to Sweden. Riesz obtained his PhD at Eötvös Loránd Universit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bivector
In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. Considering a scalar as a degree-zero quantity and a vector as a degree-one quantity, a bivector is of degree two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and vector quaternions in three dimensions. They can be used to generate rotations in a space of any number of dimensions, and are a useful tool for classifying such rotations. Geometrically, a simple bivector can be interpreted as characterizing a directed plane segment (or oriented plane segment), much as vectors can be thought of as characterizing '' directed line segments''. The bivector has an ''attitude'' (or direction) of the plane spanned by and , has an area that is a scalar multiple of any reference plane segment with the same attitude (and in geometric algebra, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Hestenes
David Orlin Hestenes (born May 21, 1933) is a theoretical physicist and science educator. He is best known as chief architect of geometric algebra as a unified language for mathematics and physics, and as founder of Modelling Instruction, a research-based program to reform K–12 Science, Technology, Engineering, and Mathematics (STEM) education. For more than 30 years, he was employed in the Department of Physics and Astronomy of Arizona State University (ASU), where he retired with the rank of research professor and is now emeritus. Life and career Education and doctorate degree David Orlin Hestenes (eldest son of mathematician Magnus Hestenes) was born 1933 in Chicago, Illinois. Beginning college as a pre-medical major at UCLA from 1950 to 1952, he graduated from Pacific Lutheran University in 1954 with degrees in philosophy and speech. After serving in the U.S. Army from 1954 to 1956, he entered UCLA as an unclassified graduate student, completed a physics M.A. in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Garret Sobczyck
A garret is a habitable attic, a living space at the top of a house or larger residential building, traditionally small with sloping ceilings. In the days before elevators this was the least prestigious position in a building, at the very top of the stairs. Etymology The word entered Middle English through Old French with a military connotation of watchtower, garrison or billet a place for guards or soldiers to be quartered in a house. Like garrison, it comes from an Old French word of ultimately Germanic languages, Germanic origin meaning "to provide" or "defend". History In the later 19th century, garrets became one of the defining features of Second Empire architecture in Paris, France, where large buildings were stratified socially between different floors. As the number of stairs to climb increased, the social status decreased. Garrets were often internal elements of the mansard roof, with skylights or dormer windows. A "bow garret" is a two-story "outhouse" situa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leo Dorst
Leo is the Latin word for lion. It most often refers to: * Leo (constellation), a constellation of stars in the night sky * Leo (astrology), an astrological sign of the zodiac * Leo (given name), a given name in several languages, usually masculine The terms Leo or Léo may also refer to: Acronyms * Lateral epitaxial overgrowth – a semiconductor substrate technology * Law enforcement officer * Law enforcement organisation * '' Louisville Eccentric Observer'', a free weekly newspaper in Louisville, Kentucky * Michigan Department of Labor and Economic Opportunity * Legal Ombudsman, often informally abbreviated to LEO or LeO in the UK. Arts and entertainment Music * L.E.O. (band), a band by musician Bleu and collaborators * ''Leo'' (soundtrack), soundtrack album by Anirudh Ravichander for the 2023 Indian film Film * ''Leo'' (2000 film), a Spanish film * ''Leo'' (2002 film), a British-American film * ''Leo'', a 2007 Swedish film by Josef Fares * ''Leo'' (2012 fil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
