We then extend them to Riemannian manifolds, giving a sequence of inequalities which are equivalent to the curvature dimension inequality, and interpolate between this inequality and the Poincaré inequality for the invariant measure.
Hereafter, C is a positive constant which can change value in a sequence of inequalities.
end{aligned} Now, by the operator monotonicity of (F cdot,v)), since (m 1_{mathcal{K}} leq m_{varphi} 1_{mathcal{K}} leqmathcal {M}_{varphi} leq M_{varphi} 1_{mathcal{K}} leq M 1_{mathcal {K}}), we obtain the desired sequence of inequalities (46).
From Theorem 1, there exists a sequence of variational inequalities (mathit{VI} [F, C _{j}]) ((j = 1, ldots)) induced by the original variational inequality (mathit{VI}[F, X]), where the polytope (C _{j}) is given by the linear inequalities (A_{j}^{T}x = b_{j}), (x, b _{j} inmathbb{R}^{{n}}), and (A _{j}) is an (m times n) matrix.
From Theorem 1 there is a sequence of variational inequalities (mathit{VI} [F, C_{{j}}]) ((j = 1, ldots)) induced by the original variational inequality (mathit{VI}[F, X]).
The algorithm requires the solution of a sequence of linear matrix inequalities (LMIs) of increasing size.
Accordingly, a gain-switched state feedback controller can be obtained by solving a sequence of linear matrix inequalities (LMIs) based optimization problems.
The filter parameters can be calculated by solving a sequence of linear matrix inequalities.
Expanding our algorithm to incorporate different levels of sequence inequality we will demonstrate their consistency with the results on the prevalent set.
In Section 2, we will present some exponential inequalities for a sequence of acceptable random variables, such as Bernstein-type inequality, Hoeffding-type inequality.
