The contribution of this paper lies in two aspects. The first one deals with a natural notion of generic Gröbner (or standard) bases on an irreducible affine scheme for an ideal depending on parameters. This takes place in rings of differential operators and concerns the algebraic and the formal case. This gives a generalization of some results of V. Weispfenning obtained in a commutative and a...

— We investigate the representation theory of the polynomial core T S of the quantum Teichmüller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S. Our main result is that irreducible finite-dimensional representations of T S are classified, up to finitely many choices, by group hom...

We give a representation theorem for Hilbert algebras by means of ordered sets and characterize the homomorphisms of Hilbert algebras in terms of applications defined between the sets of all irreducible deductive systems of the associated algebras. For this purpose we introduce the notion of order-ideal in a Hilbert algebra and we prove a separation theorem. We also define the concept of semi-h...

Let A be a Dedekind domain, K the fraction field of A, and f ∈ A[x] a monic irreducible separable polynomial. For a given non-zero prime ideal p of A we present in this paper a new method to compute a p-integral basis of the extension of K determined by f . Our method is based on the use of simple multipliers that can be constructed with the data that occurs along the flow of the Montes Algorit...

For every hyperbolic group, we construct an ideal bicombing: this is a homological analogue of the geodesic flow on negatively curved manifolds. We then construct a cohomological invariant which implies that several Measure Equivalence and Orbit Equivalence rigidity results established in [26] hold for all non-elementary hyperbolic groups and their non-elementary subgroups. For any subgroup Γ o...

Complex systems profoundly change human activities of the day and may be of strategic interest. As a result, it becomes increasingly important to have confidence in the theory of complex systems. Ultimately, this calls for clear explanations why the foundations of the theory are valid in the first place. The ideal situation would be to have an irreducible theory of complex systems not requiring...

— We investigate the representation theory of the polynomial core T S of the quantum Teichmüller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S. Our main result is that irreducible finite-dimensional representations of T S are classified, up to finitely many choices, by group hom...

After the introduction of fuzzy sets by Zadeh [20] in 1965, several researchers have been working on the generalization of the fuzzification. In 1986, Atanassov [2] introduced the notion of intuitionistic fuzzy sets. For more details on the intuitionistic fuzzy sets, we refer to [2-4]. Fuzzy sets are intuitionistic fuzzy sets but the converse is not necessarily true [18]. Kim and Jun [7, 8] int...