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In computational complexity theory, the set splitting problem is the following decision problem: given a family F of subsets of a finite set S, decide whether there exists a partition of S into two subsets S 1, S 2 such that all elements of F are split by this partition, i.e., none of the elements of F is completely in S 1 or S 2.
A split graph may have more than one partition into a clique and an independent set; for instance, the path a–b–c is a split graph, the vertices of which can be partitioned in three different ways: the clique {a, b} and the independent set {c} the clique {b, c} and the independent set {a} the clique {b} and the independent set {a, c}
Suppose some data points, each belonging to one of two sets, are given and we wish to create a model that will decide which set a new data point will be in. In the case of support vector machines , a data point is viewed as a p -dimensional vector (a list of p numbers), and we want to know whether we can separate such points with a ( p − 1 ...
The values of these weights depend on the degrees of the two sets of nodes in the original bipartite network. For example, in a co-authorship network, [ 4 ] the number of observed co-authorships depends on (1) the number of papers each author wrote and (2) the number of authors on each paper.
The linear combination of the smallest two eigenvectors leads to [1 1 1 1 1]' having an eigen value = 0. Figure 2: The graph G = (5,5) illustrates that the Fiedler vector in red bisects the graph into two communities, one with vertices {1,2,3} with positive entries in the vector space, and the other community has vertices {4,5} with negative ...
"Can a ball be decomposed into a finite number of point sets and reassembled into two balls identical to the original?" The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different ...
Since every set is contained in its closure, two separated sets automatically must be disjoint. The closures themselves do not have to be disjoint from each other; for example, the intervals [ 0 , 1 ) {\displaystyle [0,1)} and ( 1 , 2 ] {\displaystyle (1,2]} are separated in the real line R , {\displaystyle \mathbb {R} ,} even though the point ...
Narrowing is based on value sets that allow multiple values to be packaged and considered as a single value. This allows the inverses of functions to always be considered as functions. To achieve this value sets must record the context to which a value belongs. A variable may only take on a single value in each possible world. The value sets ...