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Observe that this process effectively allows us to treat the derivative as a fraction which can be separated. This allows us to solve separable differential equations more conveniently, as demonstrated in the example below. (Note that we do not need to use two constants of integration, in equation as in
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).
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.
Separable differential equation, in which separation of variables is achieved by various means; Separable extension, in field theory, an algebraic field extension; Separable filter, a product of two or more simple filters in image processing; Separable ordinary differential equation, a class of equations that can be separated into a pair of ...
then is called a separable state. Otherwise, is said to be an entangled state. From the Schmidt decomposition, we can see that is entangled if and only if has Schmidt rank strictly greater than 1. Therefore, two subsystems that partition a pure state are entangled if and only if their reduced states are mixed states.
The notion of when two sets are separated or not is important both to the notion of connected spaces (and their connected components) as well as to the separation axioms for topological spaces. Separated sets should not be confused with separated spaces (defined below), which are somewhat related but different.
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.
In mathematics, a logical matrix may be described as d-disjunct and/or d-separable. These concepts play a pivotal role in the mathematical area of non-adaptive group testing . In the mathematical literature, d -disjunct matrices may also be called super-imposed codes [ 1 ] or d -cover-free families .