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However, not all sets of four points, no three collinear, are linearly separable in two dimensions. The following example would need two straight lines and thus is not linearly separable: Notice that three points which are collinear and of the form "+ ⋅⋅⋅ — ⋅⋅⋅ +" are also not linearly separable.
This equation is an equation only of y'' and y', meaning it is reducible to the general form described above and is, therefore, separable. Since it is a second-order separable equation, collect all x variables on one side and all y' variables on the other to get: (′) (′) =.
A separable partial differential equation can be broken into a set of equations of lower dimensionality (fewer independent variables) by a method of separation of variables. It generally relies upon the problem having some special form or symmetry .
In field theory, a branch of algebra, an algebraic field extension / is called a separable extension if for every , the minimal polynomial of over F is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated roots in any extension field). [1]
The splitting field of x 2 + 1 over F 7 is F 49; the polynomial has no roots in F 7, i.e., −1 is not a square there, because 7 is not congruent to 1 modulo 4. [3] The splitting field of x 2 − 1 over F 7 is F 7 since x 2 − 1 = (x + 1)(x − 1) already splits into linear factors. We calculate the splitting field of f(x) = x 3 + x + 1 over F 2.
Separable polynomials are used to define separable extensions: A field extension K ⊂ L is a separable extension if and only if for every α in L which is algebraic over K, the minimal polynomial of α over K is a separable polynomial. Inseparable extensions (that is, extensions which are not separable) may occur only in positive characteristic.
Conversely, a metrizable space is separable if and only if it is second countable, which is the case if and only if it is Lindelöf. To further compare these two properties: An arbitrary subspace of a second-countable space is second countable; subspaces of separable spaces need not be separable (see below).
If ρ is separable, it can be written as = In this case, the effect of the partial transposition is trivial: = () = As the transposition map preserves eigenvalues, the spectrum of () is the same as the spectrum of , and in particular () must still be positive semidefinite.