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1

A Term Assignment for Polarized Bi-intuitionistic Logic and its Strong Normalization

Biasi C., Aschieri F.

Fundamenta Informaticae

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2008

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Vol. 84, nr 2

185-205

EN

We propose a term assignment (let calculus) for Intuitionistic Logic for Pragmatics ILP_AC, a polarized sequent calculus which includes ordinary positive intuitionistic logic LJ its dual LJ and dual negations ( )^ which allow a formula to "communicate" with its dual fragment. We prove the strong normalization property for the term assignment which follows by soundly translating the let calculus into simply typed l calculus with pairings and projections. A new and simple proof of strong normalization for the latter is also provided.

2

On the extension of certain maps with values in spheres

Biasi C., Libardi A. K., Pergher P. L., Spież S.

Bulletin of the Polish Academy of Sciences. Mathematics

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2008

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Vol. 56, no 2

177-182

EN

Let E be an oriented, smooth and closed m-dimensional manifold with m ≥ 2 and V ⊂ E an oriented, connected, smooth and closed (m - 2)-dimensional submanifold which is homologous to zero in E. Let Sn[sup]n-2 ⊂ S[sup]n be the standard inclusion, where S[sup]n is the n-sphere and n ≥ 3. We prove the following extension result: if h : V → S[sup]n-2 is a smooth map, then h extends to a smooth map g : E → S[sup]n transverse to S[sup]n-2 and with g[sup]-1(S[sup]n-2) = V. Using this result, we give a new and simpler proof of a theorem of Carlos Biasi related to the ambiental bordism question, which asks whether, given a smooth closed n-dimensional manifold E and a smooth closed m-dimensional submanifold V ⊂ E, one can find a compact smooth (m + 1)-dimensional submanifold W ⊂ E such that the boundary of W is V.

3

A non-standard version of the Borsuk-Ulam theorem

Biasi C., Mattos D. de

Bulletin of the Polish Academy of Sciences. Mathematics

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2005

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Vol. 53, no 1

111-119

EN

E. Pannwitz showed in 1952 that for any n≥2, there exist continuous maps φ : Sn → Sn and f : Sn → R2 such that f(x) ≠ f(φ(x)) for any x ∈ Sn. We prove that, under certain conditions, given continuous maps ψ,φ : X → X and f : X → R2, although the existence of a point x ∈ X such that f(ψ(x)) = f(φ(x)) cannot always be assured, it is possible to establish an interesting relation between the points f(φψ(x)), f(φ2(x)) and f(ψ2(x)) when f(φ(x)) ≠ f(ψ(x)) for any x ∈ X, and a non-standard version of the Borsuk–Ulam theorem is obtained.