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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.
If this infinite continued fraction converges at all, it must converge to one of the roots of the monic polynomial x 2 + bx + c = 0. Unfortunately, this particular continued fraction does not converge to a finite number in every case. We can easily see that this is so by considering the quadratic formula and a monic polynomial with real ...
When a partial fraction term has a single (i.e. unrepeated) binomial in the denominator, the numerator is a residue of the function defined by the input fraction. We calculate each respective numerator by (1) taking the root of the denominator (i.e. the value of x that makes the denominator zero) and (2) then substituting this root into the ...
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: (′) (′) =.
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 ...
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. [1]
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]
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).