### The space of Surface group representations.

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In this paper, we investigate Egoroff’s theorem with respect to monotone set function, and show that a necessary and sufficient condition that Egoroff’s theorem remain valid for monotone set function is that the monotone set function fulfill condition (E). Therefore Egoroff’s theorem for non-additive measure is formulated in full generality.

The integral inequalities known for the Lebesgue integral are discussed in the framework of the Choquet integral. While the Jensen inequality was known to be valid for the Choquet integral without any additional constraints, this is not more true for the Cauchy, Minkowski, Hölder and other inequalities. For a fixed monotone measure, constraints on the involved functions sufficient to guarantee the validity of the discussed inequalities are given. Moreover, the comonotonicity of the considered functions...

In this paper, we analyze and characterize all solutions about $\alpha $-migrativity properties of the five subclasses of 2-uninorms, i. e. ${C}^{k}$, ${C}_{k}^{0}$, ${C}_{k}^{1}$, ${C}_{1}^{0}$, ${C}_{0}^{1}$, over semi-t-operators. We give the sufficient and necessary conditions that make these $\alpha $-migrativity equations hold for all possible combinations of 2-uninorms over semi-t-operators. The results obtained show that for $G\in {C}^{k}$, the $\alpha $-migrativity of $G$ over a semi-t-operator ${F}_{\mu ,\nu}$ is closely related to the $\alpha $-section of ${F}_{\mu ,\nu}$ or the ordinal sum representation of t-norm...

We discuss the rigidity of Einstein manifolds and generalized quasi-Einstein manifolds. We improve a pinching condition used in a theorem on the rigidity of compact Einstein manifolds. Under an additional condition, we confirm a conjecture on the rigidity of compact Einstein manifolds. In addition, we prove that every closed generalized quasi-Einstein manifold is an Einstein manifold provided μ = -1/(n-2), λ ≤ 0 and β ≤ 0.

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