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The vertex-connectivity statement of Menger's theorem is as follows: . Let G be a finite undirected graph and x and y two nonadjacent vertices. Then the size of the minimum vertex cut for x and y (the minimum number of vertices, distinct from x and y, whose removal disconnects x and y) is equal to the maximum number of pairwise internally disjoint paths from x to y.
The edge connectivity of is the maximum value k such that G is k-edge-connected. The smallest set X whose removal disconnects G is a minimum cut in G . The edge connectivity version of Menger's theorem provides an alternative and equivalent characterization, in terms of edge-disjoint paths in the graph.
In computer science and optimization theory, the max-flow min-cut theorem states that in a flow network, the maximum amount of flow passing from the source to the sink is equal to the total weight of the edges in a minimum cut, i.e., the smallest total weight of the edges which if removed would disconnect the source from the sink.
Let Π ⊆ R n be the Euclidean plane spanned by x, y and z and let C ⊆ Π be the unique Euclidean circle in Π that passes through x, y and z (the circumcircle of x, y and z). Let R be the radius of C. Then the Menger curvature c(x, y, z) of x, y and z is defined by (,,) =.
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Menger conjectured that in ZFC every Menger metric space is σ-compact. A. W. Miller and D. H. Fremlin [3] proved that Menger's conjecture is false, by showing that there is, in ZFC, a set of real numbers that is Menger but not σ-compact. The Fremlin-Miller proof was dichotomic, and the set witnessing the failure of the conjecture heavily ...
Control Center (or Control Centre in British English, Australian English, and Canadian English) is a feature of Apple Inc.'s iOS, iPadOS, macOS, and visionOS operating systems. It was introduced as part of iOS 7 , released on September 18, 2013. [ 1 ]
WiMAX release 2.1, popularly branded as WiMAX 2+, is a backwards-compatible transition from previous WiMAX generations. It is compatible and interoperable with TD-LTE . Newer versions, still backward compatible, include WiMAX release 2.2 (2014) and WiMAX release 3 (2021, adds interoperation with 5G NR ).