The matrix φ(cd) diag (cl)φ(cd) and hence the expectation ∫ φ(cd) diag (cl)φ(cd)dq(cd) is positive semidefinite for any adversary’s strategy φ(cd). Thus, all eigenvalues λi ≥ 0 are non-negative. The corresponding eigenvectors are additionally eigenvectors of Im + ∫ φ(cd) diag (cl)φ(cd)dq(cd) with eigenvalues 1 + λi > 0. Thus, the inverse matrix exists independently of the choice of φ. This prov...

An efficient variant of the double-negation translation explains the relationship between Shoenfield’s and Gödel’s versions of the Dialectica interpretation. Fix a classical first-order language, based on the connectives ∨, ∧, ¬, and ∀. We will define a translation to intuitionistic (even minimal) logic, based on the usual connectives. The translation maps each formula φ to the formula φ∗ = ¬φ∗...

For every action φ ∈ Hom(G, Autk(K)) of a group G on commutative ring K we introduce two abelian monoids. The monoid Cliffk(φ) consists equivalence classes strongly G-graded algebras type up to Clifford system extensions K-central algebras. $${{\cal C}_k}(\phi )$$ equivariance homomorphisms from the Picard groups (generalized collective characters). Furthermore, for such there is an exact seque...