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setting and preliminary applications
Published in Mircea Sofonea, Stanisław Migórski, Variational-Hemivariational Inequalities with Applications, 2017
Mircea Sofonea, Stanisław Migórski
Our aim in what follows is to provide sufficient conditions such that P∩ $ {\mathcal{P}}\mathop \cap \nolimits $ Gr(S) ≠ ∅ and, alternatively, such that P∩Gr(S) $ {\mathcal{P}}\mathop \cap \nolimits Gr(S) $ has a single element, i.e., is a singleton. Clearly, this problem is a particular case of the problem previously studied, in which Q=Gr(S) $ {\mathcal{Q}} = Gr(S) $ . Since S is a univalued map defined on X $ {\mathcal{X}} $ , it is clear that for each u∈X $ u \in {\mathcal{X}} $ there exists a unique η∈Y $ \eta \in {\mathcal{Y}} $ such that η = Su or, equivalently, (u,η)∈Q $ (u, \eta ) \in {\mathcal{Q}} $ . We deduce from here that condition (1.14) is satisfied and, moreover Bu=Su. $$ Bu = Su . $$
Combining belief functions taking into consideration error in judgement
Published in International Journal of General Systems, 2020
Sunay P. Pai, Rajesh S. Prabhu Gaonkar
Let be the set containing N elements corresponding to N finite set of mutually exclusive and exhaustive hypothesis. This set is called as frame of discernment. The power set of is defined as the set which contains all the possible subsets A of and is represented by P(). The cardinality of this set is 2N, each representing the event “the object is possible in A”. where Ø denotes the empty set. Subsets containing only one element are called singletons. A Basic Probability Assignment (BPA) is a function from P() to [0, 1] defined by A → m(A) and which satisfies the following conditions:
Optimality conditions for nonsmooth vector problems in normed spaces
Published in Optimization, 2020
In this section we assume that int . Consider the problem (P). For convenience, if a subset is a singleton, then it is identified with its unique element. When f admits a first-order approximation at a point , we denote by the cone of critical directions at . Particularly, when f is Fréchet differentiable at we set