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If the domain X is a metric space, then f is said to have a local (or relative) maximum point at the point x ∗, if there exists some ε > 0 such that f(x ∗) ≥ f(x) for all x in X within distance ε of x ∗. Similarly, the function has a local minimum point at x ∗, if f(x ∗) ≤ f(x) for all x in X within distance ε of x ∗.
Further, critical points can be classified using the definiteness of the Hessian matrix: If the Hessian is positive definite at a critical point, then the point is a local minimum; if the Hessian matrix is negative definite, then the point is a local maximum; finally, if indefinite, then the point is some kind of saddle point.
The 2-D plot shows the minima points where we find reactants, the products and the saddle point or transition state. The transition state is a maximum in the reaction coordinate and a minimum in the coordinate perpendicular to the reaction path. The advance of time describes a trajectory in every reaction. Depending on the conditions of the ...
Graph cut optimization is a combinatorial optimization method applicable to a family of functions of discrete variables, named after the concept of cut in the theory of flow networks. Thanks to the max-flow min-cut theorem , determining the minimum cut over a graph representing a flow network is equivalent to computing the maximum flow over the ...
The general graph Steiner tree problem can be approximated by computing the minimum spanning tree of the subgraph of the metric closure of the graph induced by the terminal vertices, as first published in 1981 by Kou et al. [18] The metric closure of a graph G is the complete graph in which each edge is weighted by the shortest path distance ...
graph intersection: G 1 ∩ G 2 = (V 1 ∩ V 2, E 1 ∩ E 2); [1] graph join: . Graph with all the edges that connect the vertices of the first graph with the vertices of the second graph. It is a commutative operation (for unlabelled graphs); [2] graph products based on the cartesian product of the vertex sets:
In mathematics, Fermat's theorem (also known as interior extremum theorem) is a method to find the local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point (the function's derivative is zero at that point).
The residual graph represents the remaining capacity available in the network. Find the Shortest Path: Use a shortest path algorithm (e.g., Dijkstra's algorithm, Bellman-Ford algorithm) to find the shortest path from the source node to the sink node in the residual graph. Augment the Flow: Find the minimum capacity along the shortest path.