Sentence examples for minimal members from inspiring English sources

Jia [7] studied the minimal members of sublinear expectations and their related properties, established the relationship between linear expectations and the minimal members of sublinear expectations, and obtained the so-called sandwich theorem for sublinear expectations and superlinear expectations.

Analogously to the argument for the minimal members of convex expectations, we have the following result on concave expectations.

In the following, we will discuss the minimal members of three kinds of subsets of (S^{mathrm{cv}}).

Unfortunately, they cannot assert that the minimal members of convex expectations must be sublinear in that lemma (see line 13 of p.48).

Furthermore, Huang and Jia [8] devoted their efforts to the study of the minimal members of convex expectations reaching conclusions similar to those in [7].

The natural questions arise: do the results in [8] hold for convex expectations and what are the minimal members of convex expectations with some constraints?

(mathcal{E}_{0} ) is a minimal member of S; (mathcal{E}_{0} ) is a linear expectation.

(mathcal{E} ) is a minimal member of (S^{mathrm{sl}}); (mathcal{E} ) is a linear expectation.

Then E is a minimal member of (S^{mathrm{cv}}) and a maximal member of (S^{mathrm{conca}}).

Since (S^{mathrm{l}} subseteq S ), Proposition 2.1 shows that S has at least a minimal member and (ii) ⇒ (i) holds.

Then the following statements are equivalent: (i) (mathcal{E} ) is a minimal member of (S^{mathrm{sl}});   (ii) (mathcal{E} ) is a linear expectation.  .