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Isotropic
In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describe situations where properties vary systematically, dependent on direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. Mathematics Within mathematics, ''isotropy'' has a few different meanings: ; Isotropic manifolds: A manifold is isotropic if the geometry on the manifold is the same regardless of direction. A similar concept is homogeneity. ; Isotropic quadratic form: A quadratic form ''q'' is said to be isotropic if there is a non-zero vector ''v'' such that ; such a ''v'' is an isotropic vector or null vector. In complex geometry, a line through the origin in the direction of an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anisotropy
Anisotropy () is the structural property of non-uniformity in different directions, as opposed to isotropy. An anisotropic object or pattern has properties that differ according to direction of measurement. For example, many materials exhibit very different physical or mechanical properties when measured along different axes, e.g. absorbance, refractive index, conductivity, and tensile strength. An example of anisotropy is light coming through a polarizer. Another is wood, which is easier to split along its grain than across it because of the directional non-uniformity of the grain (the grain is the same in one direction, not all directions). Fields of interest Computer graphics In the field of computer graphics, an anisotropic surface changes in appearance as it rotates about its geometric normal, as is the case with velvet. Anisotropic filtering (AF) is a method of enhancing the image quality of textures on surfaces that are far away and viewed at a shallow angle. Older ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropic Line
In the geometry of quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any pair of its points is zero. An isotropic line occurs only with an isotropic quadratic form, and never with a definite quadratic form. Using complex geometry, Edmond Laguerre first suggested the existence of two isotropic lines through the point that depend on the imaginary unit : Edmond Laguerre (1870) "Sur l’emploi des imaginaires en la géométrie" Oeuvres de Laguerre2: 89 : First system: (y - \beta) = (x - \alpha) i, : Second system: (y - \beta) = -i (x - \alpha) . Laguerre then interpreted these lines as geodesics: :An essential property of isotropic lines, and which can be used to define them, is the following: the distance between any two points of an isotropic line ''situated at a finite distance in the plane'' is zero. In other terms, these lines satisfy the differential equation . On an arbitrary surface one can stu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropic Quadratic Form
In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise it is a definite quadratic form. More explicitly, if ''q'' is a quadratic form on a vector space ''V'' over ''F'', then a non-zero vector ''v'' in ''V'' is said to be isotropic if . A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form. Suppose that is quadratic space and ''W'' is a subspace of ''V''. Then ''W'' is called an isotropic subspace of ''V'' if ''some'' vector in it is isotropic, a totally isotropic subspace if ''all'' vectors in it are isotropic, and a definite subspace if it does not contain ''any'' (non-zero) isotropic vectors. The of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces. Over the real numbers, more generally in the case where ''F'' is a real closed field (so that the signature is defined), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropic Vector Field
In differential geometry, an isotropic vector field is a concept that refers to a vector field that maintains the same properties in all directions at each point in space. Definition A vector field V on a manifold M is said to be isotropic if, for every point p \in M , the vector V(p) has the same magnitude and directionality properties in all directions around p . This implies that the vector field does not prefer any particular direction, and its characteristics are invariant under rotations about any point. Properties * Uniformity: An isotropic vector field exhibits uniform behavior across the manifold. This means that its magnitude and orientation are consistent in all directions at any given point. * Symmetry: The isotropy of the vector field implies a high degree of symmetry. In physical contexts, this often corresponds to systems that are invariant under rotations, such as isotropic materials in elasticity or cosmological models in general relativity. * In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropic Coordinates
In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of ''nested round spheres''. There are several different types of coordinate chart which are ''adapted'' to this family of nested spheres; the best known is the Schwarzschild chart, but the isotropic chart is also often useful. The defining characteristic of an isotropic chart is that its radial coordinate (which is different from the radial coordinate of a Schwarzschild chart) is defined so that light cones appear ''round''. This means that (except in the trivial case of a locally flat manifold), the angular isotropic coordinates do not faithfully represent distances within the nested spheres, nor does the radial coordinate faithfully represent radial distances. On the other hand, angles in the constant time hyperslices are represented without distortion, hence the name of the chart. Isotropic charts are most often applied to static spherically symmetric spacetimes in metric theories of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropic Vector
In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which . In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector. A quadratic space which has a null vector is called a pseudo-Euclidean space. The term ''isotropic vector v'' when ''q''(''v'') = 0 has been used in quadratic spaces, and anisotropic space for a quadratic space without null vectors. A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces ''A'' and ''B'', , where ''q'' is positive-definite on ''A'' and negative-definite on ''B''. The null cone, or isotropic cone, of ''X'' consists of the union of balanced spheres: \bigcup_ \. The null cone is also the union of the isotropic lines through the origin. Split algebras A comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a fixed field , such as the real or complex numbers, and one speaks of a quadratic form ''over'' . Over the reals, a quadratic form is said to be '' definite'' if it takes the value zero only when all its variables are simultaneously zero; otherwise it is '' isotropic''. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory ( orthogonal groups), differential geometry (the Riemannian metric, the second fundamental form), differential topology ( intersection forms of manifolds, especially four-manifolds), Lie theory (the Killing form), and statistics (where the exponent of a zero-mean multivariate normal distribution has the quadratic form -\mathbf^\math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropic Position
In the fields of machine learning, the theory of computation, and random matrix theory, a probability distribution over vectors is said to be in isotropic position if its covariance matrix is proportional to the identity matrix. Formal definitions Let D be a distribution over vectors in the vector space \mathbb^n. Then D is in isotropic position if, for vector v sampled from the distribution, \mathbb\, vv^\mathsf = \mathrm. A ''set'' of vectors is said to be in isotropic position if the uniform distribution over that set is in isotropic position. In particular, every orthonormal set of vectors is isotropic. As a related definition, a convex body K in \mathbb^n is called isotropic if it has volume , K, = 1, center of mass at the origin, and there is a constant \alpha > 0 such that \int_K \langle x, y \rangle^2 dx = \alpha^2 , y, ^2, for all vectors y in \mathbb^n; here , \cdot, stands for the standard Euclidean norm. See also * Whitening transformation A whitening transf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropic Manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Motivation Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Space
In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and topological groups. More precisely, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ''G'' are called the symmetries of ''X''. A special case of this is when the group ''G'' in question is the automorphism group of the space ''X'' – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, ''X'' is homogeneous if intuitively ''X'' looks locally the same at each point, either in the sense of isometry (rigid geometry), diffeomorphism (differential geometry), or homeomorphism (topology). Some authors insist that the action of ''G'' be faithful (non-identity elements act non-trivially), althou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotropic Radiation
Isotropic radiation is radiation that has the same intensity regardless of the direction of measurement, such as what would be found in a thermal cavity.Bohren, Craig F.., Clothiaux, Eugene E.. Fundamentals of Atmospheric Radiation: An Introduction with 400 Problems. Germany: Wiley, 2006. Page 207 This can be electromagnetic radiation, sound, or elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. The Standard Model presently recognizes seventeen distinct particles—twelve fermions and five bosons. As a c ...s. References Radiation {{Physics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial function replacing the binary operation; * '' Category'' in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation on the morphisms, called ''inverse'' by analogy with group theory. A groupoid where there is only one object is a usual group. In the presence of dependent typing, a category in general can be viewed as a typed monoid, and similarly, a groupoid can be viewed as simply a typed group. The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed , , say. Composition is then a total function: , so that . Special cases include: * '' Setoids'': sets that come with an equivalence relation, * '' G-sets'': sets equippe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
