By Bobylev N. A., Bulatov V.

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**Additional info for A Bound on the Real Stability Radius of Continuous-Time Linear Infinite-Dimensional Systems**

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Otherwise put A0 D C0 \ .! n X0 /. Now ﬁx < 2! A˛ /˛< satisfying conditions (i)–(iii) for all ˛ < . Since p D c, there is B Â ! such that jB n A˛ j < ! for all ˛ < . Choose an inﬁnite C Â B such that ŒC 2 is monochrome with respect to and put ´ if jC \ X j D ! C \X A D C \ .! A˛ /˛<2! , let p D ¹A Â ! W jA˛ n Aj < ! for some ˛ < 2! º: Then p is a nonprincipal Ramsey ultraﬁlter. A point x in a topological space X is called a P -point if the intersection of countably many neighborhoods of x is again a neighborhood of x.

13. ˇD is the Stone–Cech compactiﬁcation of D. 13, let us consider the following general situation. Suppose that X is a dense subset of a space Y and f W X ! Z is a continuous mapping from X into a Hausdorff space Z. Fp / in Z, if exists. It is clear that if f has a continuous extension f W Y ! p for every p 2 Y . 14. x/. Deﬁne f W Y ! p Then f is the continuous extension of f . Proof. q/ 2 Z. B/ Â W . B \ A/ Â W . A/. p/ 2 Z. Since Z is regular, one may suppose that U is closed. A/ Â U . Put V D clY A.

D be a mapping with no ﬁxed points. Ai / D ; for each i < 3. Proof. g/, S ordered by Â. Note that G is nonempty, and if C is a chain in G , then C 2 G . Consequently by Zorn’s Lemma, there is a maximal element g 2 G . g/ D D. g/ and assume on the contrary that there is b 2 D n Dg . b// for each n < !. We distinguish between three cases. Case 1: there is n < ! b/ 2 Dg . b// and pick j < 2 different from i . , are different and do not belong to Dg . b// D 1 otherwise: 32 Chapter 2 Ultraﬁlters Case 3: there are m < n < !

### A Bound on the Real Stability Radius of Continuous-Time Linear Infinite-Dimensional Systems by Bobylev N. A., Bulatov V.

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