»ã±¨±êÌâ (Title)£ºThe relations among the notions of various kinds of stability and their applications

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»ã±¨ÈË (Speaker)£º¹ùÌúÐÅ£¨ÖÐÄÏ´óѧ£©

»ã±¨¹¦·ò (Time)£º2024Äê4ÔÂ3ÈÕ(ÖÜÈý) 10:00

»ã±¨µØÖ· (Place)£ºÐ£±¾²¿GJ303

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»ã±¨ÌáÒª£ºFirst, we prove that a random metric space can be isometrically embedded into a complete random normed module, as an application of which, it is easy to see that the notion of d_¦Ä-stability introduced for a nonempty subset of a random metric space can be regarded as a special case of the notion of ¦Ä-stability introduced for a nonempty subset of a random normed module, as another application we give the final version of the characterization for a d_¦Ä-stable random metric space to be stably compact. Second, we prove that an L^¡Þ-module is an L^p-normed L^¡Þ-module iff it is generated by a complete random normed module, from which it is easily seen that the gluing property of an L^p-normed L^¡Þ-module can be derived from the ¦Ä-stability of the generating random normed module, as applications the known and new basic facts of module duals for L^p-normed L^¡Þ-modules can be obtained, in a simple and direct way, from the theory of random conjugate spaces of random normed modules. Third, we prove that a random normed space is order complete iff it is complete with respect to the (¦Å,¦Ë)-topology, as an application it is proved that the d-decomposability of an order complete random normed space is exactly its d-¦Ä-stability. Finally, we prove that an equivalence relation on the product space X¡ÁB of a nonempty set X and a complete Boolean algebra B is regular iff it can be induced by a B-valued Boolean metric d on X, as an application it is proved that a nonempty subset of a Boolean set (X,d) is universally complete iff it is a B-stable set defined by a regular equivalence relation.