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An example is the compareTo method: a. compareTo (b) checks whether a comes before or after b in some ordering, but the way to compare, say, two rational numbers will be different from the way to compare two strings. Other common examples of binary methods include equality tests, arithmetic operations, and set operations like subset and union.
A third concept related to covariance and contravariance is invariance. A scalar (also called type-0 or rank-0 tensor) is an object that does not vary with the change in basis. An example of a physical observable that is a scalar is the mass of a particle.
The explicit form of a covariant transformation is best introduced with the transformation properties of the derivative of a function. Consider a scalar function f (like the temperature at a location in a space) defined on a set of points p , identifiable in a given coordinate system x i , i = 0 , 1 , … {\displaystyle x^{i},\;i=0,1,\dots ...
Examples: is a model for 3-dimensional space. The metric is equivalent to the standard dot product., =, equivalent to dimensional real space as an inner product space with =. In Euclidean space, raising and lowering is not necessary due to vectors and covector components being the same.
In probability theory and statistics, the covariance function describes how much two random variables change together (their covariance) with varying spatial or temporal separation. For a random field or stochastic process Z ( x ) on a domain D , a covariance function C ( x , y ) gives the covariance of the values of the random field at the two ...
With any number of random variables in excess of 1, the variables can be stacked into a random vector whose i th element is the i th random variable. Then the variances and covariances can be placed in a covariance matrix, in which the (i, j) element is the covariance between the i th random variable and the j th one.
The type coercion for function types may be given by f'(t) = coerce S 2 → T 2 (f(coerce T 1 → S 1 (t))), reflecting the contravariance of parameter values and covariance of return values. The coercion function is uniquely determined given the subtype and supertype. Thus, when multiple subtyping relationships are defined, one must be careful ...
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.