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In mathematics, a Lie group is a group that is also a differentiable manifold.
In the mathematical field of group theory, Lagrange's theorem is a theorem that states that for any finite group G, the order of every subgroup of G divides the order of G. The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup of a finite group , not only is an integer, but its value is the index , defined as the number of left cosets of in .
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the group. The group is so named because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.
In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields, discovered by the French mathematician Évariste Galois.
In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group.
Smoking cessation, usually called quitting smoking or stopping smoking, is the process of discontinuing tobacco smoking. Tobacco smoke contains nicotine, which is addictive and can cause dependence. As a result, nicotine withdrawal often makes the process of quitting difficult.
In mathematics, a linear algebraic group is a subgroup of the group of invertible matrices that is defined by polynomial equations. An example is the orthogonal group, defined by the relation where is the transpose of .
A cohort study is a particular form of longitudinal study that samples a cohort, performing a cross-section at intervals through time. It is a type of panel study where the individuals in the panel share a common characteristic.
In mathematics the spin group, denoted Spin(n), is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n, R), such that there exists a short exact sequence of Lie groups (when n ≠ 2)
In mathematics, especially in the area of algebra known as group theory, the Fitting subgroupF of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it represents the smallest subgroup which "controls" the structure of G when G is solvable. When G is not solvable, a similar role is played by the generalized Fitting subgroupF*, which is generated by the Fitting subgroup and the components of G.
In mathematics, a maximal compact subgroupK of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups.
In mathematics, a Gelfand pair is a pair (G,K ) consisting of a group G and a subgroup K (called an Euler subgroup of G) that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to the theory of Riemannian symmetric spaces in differential geometry. Broadly speaking, the theory exists to abstract from these theories their content in terms of harmonic analysis and representation theory.
Tobacco products, especially when smoked or used orally, have negative effects on human health, and concerns about these effects have existed for a long time. Research has focused primarily on cigarette smoking.
In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. 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 historical linguistics, a linkage is a network of related dialects or languages that formed from a gradual diffusion and differentiation of a proto-language.
Nicotine dependence is a state of dependence upon nicotine. Nicotine dependence is a chronic, relapsing disease defined as a compulsive craving to use the drug, despite social consequences, loss of control over drug intake, and emergence of withdrawal symptoms. Tolerance is another component of drug dependence. Nicotine dependence develops over time as a person continues to use nicotine. The most commonly used tobacco product is cigarettes, but all forms of tobacco use and e-cigarette use can cause dependence. Nicotine dependence is a serious public health problem because it leads to continued tobacco use, which is one of the leading preventable causes of death worldwide, causing more than 8 million deaths per year.
Plateosauria is a clade of sauropodomorph dinosaurs which lived during the Late Triassic to the Late Cretaceous. The name Plateosauria was first coined by Gustav Tornier in 1913. The name afterwards fell out of use until the 1980s.
In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of Rn, this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively well understood.
NGC 428 is a barred spiral galaxy in the constellation of Cetus, with its spiral structure distorted and warped, possibly the result of the collision of two galaxies. There appears to be a substantial amount of star formation occurring within NGC 428 and it lacks well defined arms — a telltale sign of a galaxy merger. In 2015 the Hubble Space Telescope made a close-up shot of the galaxy with its Advanced Camera for Surveys and its Wide Field and Planetary Camera 2. The structure of NGC 428 has been compared to NGC 5645.
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