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Exceptional Sets in Mathematical Analysis
Rmoutil, Martin ; Kalenda, Ondřej (advisor)
Title: Exceptional Sets in Mathematical Analysis Author: Martin Rmoutil Department: Department of Mathematical Analysis Supervisor: Doc. RNDr. Ondřej Kalenda, Ph.D., DSc., Department of Mathematical Analysis Abstract: The present thesis consists of four research articles. In the first paper we study the notion of σ-lower porous set; our main result is the existence of two closed sets A, B ⊂ R which are not σ-lower porous, but their product in R2 is lower porous. In the second and third article we use a set-theoretical method of el- ementary submodels involving the Lwenheim-Skolem theorem to prove that certain σ-ideals of sets in Banach spaces are separably determined. In the second article we do so for σ-porous sets and σ-lower porous sets. In the next article we refine these methods obtaining separable determination of a wide class of σ-ideals. In both cases we derive interesting corollaries which extend known theorems in separable spaces to the nonseparable setting; for example, we obtain the following theorem. Any continuous convex function on an Asplund space is Frchet differentiable outside a cone small set. In the fourth article we introduce the following notion. A closed set A ⊂ Rd is said to be c-removable if the following is true: Every real function on Rd is convex whenever it is continuous on Rd...
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Exceptional Sets in Mathematical Analysis
Rmoutil, Martin ; Kalenda, Ondřej (advisor)
Title: Exceptional Sets in Mathematical Analysis Author: Martin Rmoutil Department: Department of Mathematical Analysis Supervisor: Doc. RNDr. Ondřej Kalenda, Ph.D., DSc., Department of Mathematical Analysis Abstract: The present thesis consists of four research articles. In the first paper we study the notion of σ-lower porous set; our main result is the existence of two closed sets A, B ⊂ R which are not σ-lower porous, but their product in R2 is lower porous. In the second and third article we use a set-theoretical method of el- ementary submodels involving the Lwenheim-Skolem theorem to prove that certain σ-ideals of sets in Banach spaces are separably determined. In the second article we do so for σ-porous sets and σ-lower porous sets. In the next article we refine these methods obtaining separable determination of a wide class of σ-ideals. In both cases we derive interesting corollaries which extend known theorems in separable spaces to the nonseparable setting; for example, we obtain the following theorem. Any continuous convex function on an Asplund space is Frchet differentiable outside a cone small set. In the fourth article we introduce the following notion. A closed set A ⊂ Rd is said to be c-removable if the following is true: Every real function on Rd is convex whenever it is continuous on Rd...
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Exceptional Sets in Mathematical Analysis
Rmoutil, Martin ; Kalenda, Ondřej (advisor) ; Holický, Petr (referee) ; Zindulka, Ondřej (referee)
Title: Exceptional Sets in Mathematical Analysis Author: Martin Rmoutil Department: Department of Mathematical Analysis Supervisor: Doc. RNDr. Ondřej Kalenda, Ph.D., DSc., Department of Mathematical Analysis Abstract: The present thesis consists of four research articles. In the first paper we study the notion of σ-lower porous set; our main result is the existence of two closed sets A, B ⊂ R which are not σ-lower porous, but their product in R2 is lower porous. In the second and third article we use a set-theoretical method of el- ementary submodels involving the Lwenheim-Skolem theorem to prove that certain σ-ideals of sets in Banach spaces are separably determined. In the second article we do so for σ-porous sets and σ-lower porous sets. In the next article we refine these methods obtaining separable determination of a wide class of σ-ideals. In both cases we derive interesting corollaries which extend known theorems in separable spaces to the nonseparable setting; for example, we obtain the following theorem. Any continuous convex function on an Asplund space is Frchet differentiable outside a cone small set. In the fourth article we introduce the following notion. A closed set A ⊂ Rd is said to be c-removable if the following is true: Every real function on Rd is convex whenever it is continuous on Rd...
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