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Covariance And Contravariance (other)
Covariance and contravariance may refer to: * Covariance and contravariance of vectors, in mathematics and theoretical physics * Covariance and contravariance of functors, in category theory * Covariance and contravariance (computer science), whether a type system preserves the ordering ≤ of types * An informal synonym for invariance (physics) See also * Covariance, in probability theory and statistics, the measure of how much two random variables vary together * Covariance (other) {{Mathematical disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance And Contravariance Of Vectors
In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. Briefly, a contravariant vector is a list of numbers that transforms oppositely to a change of basis, and a covariant vector is a list of numbers that transforms in the same way. Contravariant vectors are often just called ''vectors'' and covariant vectors are called ''covectors'' or ''dual vectors''. The terms ''covariant'' and ''contravariant'' were introduced by James Joseph Sylvester in 1851. Curvilinear coordinate systems, such as cylindrical coordinates, cylindrical or spherical coordinates, are often used in physical and geometric problems. Associated with any coordinate system is a natural choice of coordinate basis for vectors based at each point of the space, and covariance and contravariance are particularly important for understanding how the coordinate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance And Contravariance Of Functors
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a linguistic context; see function word. Definition Let ''C'' and ''D'' be categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D'', * associates each morphism f \colon X \to Y in ''C'' to a morphism F(f) \colon F(X) \to F( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance And Contravariance (computer Science)
Many programming language type systems support subtyping. For instance, if the type (computer science), type is a subtype of , then an expression (computer science), expression of type Liskov_substitution_principle, should be substitutable wherever an expression of type is used. Variance is the category of possible relationships between more composite type, complex types and their components' subtypes. A language's chosen variance determines the relationship between, for example, a list (programming), list of s and a list of s, or a Function (computer science), function return value, returning and a function returning . Depending on the variance of the type constructor, the subtyping relation of the simple types may be either preserved, reversed, or ignored for the respective complex types. In the OCaml programming language, for example, "list of Cat" is a subtype of "list of Animal" because the list type constructor is covariant. This means that the subtyping relation of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariance (physics)
In theoretical physics, an invariant is an observable of a physical system which remains unchanged under some transformation. Invariance, as a broader term, also applies to the no change of form of physical laws under a transformation, and is closer in scope to the mathematical definition. Invariants of a system are deeply tied to the symmetries imposed by its environment. Invariance is an important concept in modern theoretical physics, and many theories are expressed in terms of their symmetries and invariants. Examples In classical and quantum mechanics, invariance of space under translation results in momentum being an invariant and the conservation of momentum, whereas invariance of the origin of time, i.e. translation in time, results in energy being an invariant and the conservation of energy. In general, by Noether's theorem, any invariance of a physical system under a continuous symmetry leads to a fundamental conservation law. In crystals, the electron density is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one variable mainly correspond with greater values of the other variable, and the same holds for lesser values (that is, the variables tend to show similar behavior), the covariance is positive. In the opposite case, when greater values of one variable mainly correspond to lesser values of the other (that is, the variables tend to show opposite behavior), the covariance is negative. The magnitude of the covariance is the geometric mean of the variances that are in common for the two random variables. The Pearson product-moment correlation coefficient, correlation coefficient normalizes the covariance by dividing by the geometric mean of the total variances for the two random variables. A distinction must be made between (1) the covariance of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
