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The canonical application of topological sorting is in scheduling a sequence of jobs or tasks based on their dependencies.The jobs are represented by vertices, and there is an edge from x to y if job x must be completed before job y can be started (for example, when washing clothes, the washing machine must finish before we put the clothes in the dryer).
In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every non-zero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite .
The principal axis theorem concerns quadratic forms in R n, which are homogeneous polynomials of degree 2. Any quadratic form may be represented as =where A is a symmetric matrix.
When the characteristic is not 2, the quadratic form Q on V can be extended to a quadratic form on all of Cl(V, Q) (which we also denoted by Q). A basis-independent definition of one such extension is Q ( x ) = x t x 0 {\displaystyle Q(x)=\left\langle x^{\mathrm {t} }x\right\rangle _{0}} where a 0 denotes the scalar part of a (the degree- 0 ...
A vertex with a large degree, also called a heavy node, results in a large diagonal entry in the Laplacian matrix dominating the matrix properties. Normalization is aimed to make the influence of such vertices more equal to that of other vertices, by dividing the entries of the Laplacian matrix by the vertex degrees.
In the theory of quadratic forms, the parabola is the graph of the quadratic form x 2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x 2 + y 2 (or scalings), and the hyperbolic paraboloid is the graph of the indefinite quadratic form x 2 − y 2. Generalizations to more variables yield ...
Most instances of geometric algebras of interest have a nondegenerate quadratic form. If the quadratic form is fully degenerate, the inner product of any two vectors is always zero, and the geometric algebra is then simply an exterior algebra. Unless otherwise stated, this article will treat only nondegenerate geometric algebras.
If the angle at the vertex on the real axis is less than π / 3 then the major axis of the ellipse lies on the real axis, as do its foci and hence the roots of the derivative. If that angle is greater than π / 3 , the major axis is vertical and its foci, the roots of the derivative, are complex conjugates.