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Different from the naive algorithm, the fast algorithm is designed based on the monotonic property of the multigranulation rough approximations.
Many region-related attribute reductions are not appropriate for probabilistic rough set models as the monotonic property of regions does not always hold.
In this paper, we first give an monotonic property of -curvature image.
First, we give a monotonic property of -curvature image as follows.
Notice that, from the monotonic property of F α and (3.4), we have.
First, we present the monotonic property of operator function, which is the basis of the following discussion.
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This new formulation leads to symmetric and monotonic properties of the resulting system of nonlinear algebraic equations from an adaptive finite element approximation of the model.
In [52 55], the complete monotonic properties of the function on the right-hand side of (82) were established as follows.
Next, we will provide some important monotonic properties of a positive solution y of (1.1) in cases (2.1) and (2.2), respectively.
In order to prove Theorem 3, the following complete monotonic properties of a function related to a divided difference of the psi function were discovered in [54], the preprint of [55].
It is important to note that, from the Hass diagram on the lattice (mathcal {LCG}), if the concepts of positive and negative borders [23], concerning the anti-monotonic and monotonic properties of the support and item constraints, are added to the algorithm, then the sub-lattices whose closed itemsets satisfy the corresponding constraints will be generated quickly.
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