The "Plug and Play" template can be individually or successively grafted by dual-responsive molecules on the α-CD modified channels by host-guest interactions and can be peeled off by UV irradiation. The artificial channels present six kinds of responses cycling among four states responding to three environment stimuli, as light, pH, and temperature.

In this work we study the reverse order laws for {1, 2, 3}-and {1, 2, 4}-inverses of a product of two matrices by using the expressions for maximal and minimal ranks of the generalized Schur complement. The necessary and sufficient conditions for B{1, 2, 3}A{1, 2, 3} ⊆ (AB){1, 2, 3} and B{1, 2, 4}A{1, 2, 4} ⊆ (AB){1, 2, 4} are presented.

In this article, we study the reverse order laws for {1, 2, 3}-and {1, 2, 4}-inverses of multiple matrix products by using the maximal and minimal ranks of the generalized Schur complements. The necessary and sufficient conditions for the inclusions A n f1, 2, 3gA nÀ1 f1, 2, 3g Á Á Á A 1 f1, 2, 3g ðA 1 A 2 Á Á Á A n Þf1, 2, 3g and A n f1, 2, 4gA nÀ1 f1, 2, 4g Á Á Á A 1 f1, 2, 4g ðA 1 A 2 Á Á Á A n Þf1, 2, 4g, are presented.

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In this paper, we study mixed-type reverse-order laws for the Moore-Penrose inverse of an operator product AB, and obtain necessary and sufficient conditions for these mixed-type reverse-order laws. Results related to other generalized inverses are also proved.Keywords Mixed-type reverse-order law · Moore-Penrose inverse · Generalized inverse · Linear bounded operators · Operator product · Hilbert space Mathematics Subject Classification (2010) 47A05 · 15A09 · 15A24