Let R be a regular local ring, K its field of fractions and (V, φ) a quadratic space over R. In the case of R containing a field of characteristic zero we show that if (V, φ)⊗R K is isotropic over K, then (V, φ) is isotropic over R. 1 Characteristic zero case 1.0.1 Theorem (Main). Let R be a regular local ring, K its field of fractions and (V, φ) a quadratic space over R. Suppose R contains a f...

To every morphism of X∗ a congruence fR on X ∗, called kernel congruence, and defined by ufRv iff f(u) = f(v) can be associated. We characterize morphisms having a commutative kernel congruence and, introducing the notion of the index of a morphism, we give a classification of the morphisms of X∗. Moreover, a congruence with special properties, called hp-congruence is considered; every kernel c...

A word is called primitive if it cannot be expressed as the power of another word. Morphisms preserving primitive words are investigated. Similarly to the word case, each square-free morphism is a primitive morphism but the converse does not hold. A precise characterization of primitive morphisms is provided in terms of pure codes. An easily testable characterization is given for uniform morphi...

We consider stacks of filtered φ-modules over rigid analytic spaces and adic spaces. We show that these modules parametrize p-adic Galois representations of the absolute Galois group of a p-adic field with varying coefficients over an open substack containing all classical points. Further we study a period morphism (defined by Pappas and Rapoport) from a stack parametrising integral data and de...

We prove a variant of Läuchli’s completeness theorem for intuitionistic predicate calculus. The formulation of the result relies on the observation (due to Lawvere) that Läuchli’s theorem is related to the logic of the canonical indexing of the atomic topos of Z-sets. We show that the process that transforms Kripke-counter-models into Läuchli-counter-models is (essentially) the inverse image of...

We describe the relationship between different parameters of the initial word and its image obtained by application of a uniform marked morphism. The functions described include the subword complexity, frequency of factors, and the recurrence function. The relations obtained for the image of a word can be used also for the image of a factorial language. Using induction, we give a full descripti...

In this paper we investigate a class of problems permitting a good characterisation from the point of view of morphisms of oriented matroids. We prove several morphism-duality theorems for oriented matroids. These generalize LP-duality (in form of Farkas’ Lemma) and Minty’s Painting Lemma. Moreover, we characterize all morphism duality theorems, thus proving the essential unicity of Farkas’ Lem...

The “noncommutative geometry” of complex algebraic curves is studied. As first step, we clarify a morphism between elliptic curves, or complex tori, and C-algebras Tθ = {u, v | vu = e2πiθuv}, or noncommutative tori. The main result says that under the morphism isomorphic elliptic curves map to the Morita equivalent noncommutative tori. Our approach is based on the rigidity of the length spectra...