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In which case, a property is said to be local if it can be detected from the local rings. For instance, being a flat module over a commutative ring is a local property, but being a free module is not. For more, see Localization of a module.
Note that in the one-variable case, the Hessian condition simply gives the usual second derivative test. In the two variable case, (,) and (,) are the principal minors of the Hessian. The first two conditions listed above on the signs of these minors are the conditions for the positive or negative definiteness of the Hessian.
A common special case is bivariate interpolation or two-dimensional interpolation, based on two variables or two dimensions. When the variates are spatial coordinates, it is also known as spatial interpolation. The function to be interpolated is known at given points (,,, …
The least squares regression line is a method in simple linear regression for modeling the linear relationship between two variables, and it serves as a tool for making predictions based on new values of the independent variable. The calculation is based on the method of the least squares criterion. The goal is to minimize the sum of the ...
The image of a function f(x 1, x 2, …, x n) is the set of all values of f when the n-tuple (x 1, x 2, …, x n) runs in the whole domain of f.For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value.
Typically, property testing algorithms are used to determine whether some combinatorial structure S (such as a graph or a boolean function) satisfies some property P, or is "far" from having this property (meaning that an ε-fraction of the representation of S must be modified to make S satisfy P), using only a small number of "local" queries ...
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information " (in units such as shannons ( bits ), nats or hartleys ) obtained about one random variable by observing the other random ...
Values of other variables can be found subsequently in the same way. In total, n+1 runs of the algorithm are required, where n is the number of distinct variables in Φ. This property is used in several theorems in complexity theory: NP ⊆ P/poly ⇒ PH = Σ 2 (Karp–Lipton theorem) NP ⊆ BPP ⇒ NP = RP P = NP ⇒ FP = FNP