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An alternative algorithm for topological sorting is based on depth-first search.The algorithm loops through each node of the graph, in an arbitrary order, initiating a depth-first search that terminates when it hits any node that has already been visited since the beginning of the topological sort or the node has no outgoing edges (i.e., a leaf node):
In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space (,) is the finest locally convex topological vector space (TVS) topology on for which every order interval is bounded, where an order interval in is a set of the form [,]:= {:} where and belong to . [1]
A standard scenario in many computer applications is a collection of points (measurements, dark pixels in a bit map, etc.) in which one wishes to find a topological feature. Homology can serve as a qualitative tool to search for such a feature, since it is readily computable from combinatorial data such as a simplicial complex.
In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exponential of a matrix , which is involved in the closed-form solution of systems of linear differential equations .
The left and right order topologies can be used to give counterexamples in general topology. For example, the left or right order topology on a bounded set provides an example of a compact space that is not Hausdorff. The left order topology is the standard topology used for many set-theoretic purposes on a Boolean algebra. [clarification needed]
The proof is bijective: a matrix A is an adjacency matrix of a DAG if and only if A + I is a (0,1) matrix with all eigenvalues positive, where I denotes the identity matrix. Because a DAG cannot have self-loops , its adjacency matrix must have a zero diagonal, so adding I preserves the property that all matrix coefficients are 0 or 1.
In mathematics, specifically in order theory and functional analysis, a filter in an order complete vector lattice is order convergent if it contains an order bounded subset (that is, a subset contained in an interval of the form [,]:= {:}) and if {: ()} = {: ()}, where is the set of all order bounded subsets of X, in which case this common value is called the order limit of in . [1]
In the matrix notation, the adjacency matrix of the undirected graph could, e.g., be defined as a Boolean sum of the adjacency matrix of the original directed graph and its matrix transpose, where the zero and one entries of are treated as logical, rather than numerical, values, as in the following example: