This thesis is concerned with the area of algebraic set theory. Algebraic set theory was invented by Joyal and Moerdijk [18] with the aim to study set theory from the perspective of category theory. The central notion is a category of classes, given by a triple (E ,S,Ps), consisting of a Heyting pretopos E , a particular class S of arrows of E that are called small maps and an endofunctor Ps : ...

Given a Hilbert space (H, 〈·, ·〉), and interval Λ ⊂ (0, +∞) and a map K ∈ C(H,R) whose gradient is a compact mapping, we consider the family of functionals of the type: I(λ, u) = 1 2 〈u, u〉 − λK(u), (λ, u) ∈ Λ×H. As already observed by many authors, for the functionals we are dealing with the (PS) condition may fail under just this assumptions. Nevertheless, by using a recent deformation Lemma ...

We show that propositional logic and its extensions can support answer-set programming in the same way stable logic programming and disjunctive logic programming do. To this end, we introduce a logic based on the logic of propositional schemata and on a version of the Closed World Assumption. We call it the extended logic of propositional schemata with CWA (PS + , in symbols). An important feat...

Let P be an elementary closed semi-algebraic set in R, i.e., there exist real polynomials p1, . . . , ps (s ∈ N) such that P = ̆ x ∈ R : p1(x) ≥ 0, . . . , ps(x) ≥ 0 ̄ ; in this case p1, . . . , ps are said to represent P . Denote by n the maximal number of the polynomials from {p1, . . . , ps} that vanish in a point of P. If P is non-empty and bounded, we show that it is possible to construct n ...