Given a global field K and a polynomial φ defined over K of degree at least two, Morton and Silverman conjectured in 1994 that the number of K-rational preperiodic points of φ is bounded in terms of only the degree of K and the degree of φ. In 1997, for quadratic polynomials over K = Q, Call and Goldstine proved a bound which was exponential in s, the number of primes of bad reduction of φ. By ...

In this paper we find a field such that the algebras obtained by the Cayley-Dickson process are division algebras of dimension 2t,∀t ∈ N. Subject Classification: 17D05; 17D99. From Frobenius Theorem and from the remark given by Bott and Milnor in 1958, we know that for n ∈ {1, 2, 4} we find the real division algebras over the real field R. These are: R, C, H(the real quaternion algebra), O(the ...

We consider avoiding squares and overlaps over the natural numbers, using a greedy algorithm that chooses the least possible integer at each step; the word generated is lexicographically least among all such infinite words. In the case of avoiding squares, the word is 01020103 · · · , the familiar ruler function, and is generated by iterating a uniform morphism. The case of overlaps is more cha...

A fast introduction to the the construction of the cohomology of sheaves pioneered by A. Grothendieck and J.L. Verdier. The approach is from the point of view of derived categories, though this concept is never mentioned. 1. Sheaves Let R commutative ring with 1. We denote by RMod the category of R-modules. For any topological space X we denote by OpenpXq the collection of open subsets. It can ...

1 Basic Properties Definition 1. Let X be a scheme. We denote the category of sheaves of OX -modules by OXMod or Mod(X). The full subcategories of qausi-coherent and coherent modules are denoted by Qco(X) and Coh(X) respectively. Mod(X) is a grothendieck abelian category, and it follows from (AC,Lemma 39) and (H, II 5.7) that Qco(X) is an abelian subcategory of Mod(X). If X is noetherian, then ...

The Jacobian Conjecture is the following : If φ ∈ Endk(Ank ) with a field k of characteristic zero is unramified, then φ is an automorphism. In this paper, This conjecture is proved affirmatively in the abstract way instead of treating variables in a polynomial ring. Let k be an algebraically closed field, let Ak = Max(k[X1, . . . , Xn]) be an affine space of dimension n over k and let f : Ak −...

One of the fundamental notions of model theory for modal logics is a bisimulation. Indeed it not only preserves logical value of formulas but it plays the important role for the modal definable classes of models ([2]). The main scope of the paper is to present several types of bisimulations that are proper for the different types of model structures – ordinary Kripke models, their many-valued c...

It was rediscovered several times, can be constructed in many alternative ways and occurs in various fields of mathematics (see the survey [1]). The set of all overlap-free words was studied e. g. by E. D. Fife [8] who described all binary overlapfree infinite words and P. Séébold [13] who proved that the Thue-Morse word is essentially the only binary overlap-free word which is a fixed point of...

where h is the usual absolute logarithmic height. It is reasonably easy to show, from the two properties above, that ĥφ(α) = 0 just in case φ j(α) = φi(α) for some j 6= i. If this is the case, we will say that α is a pre-periodic point for φ. It is natural to ask how small the value of ĥφ can be at points which are not pre-periodic, i.e., wandering points for φ. We will examine this question fo...

Let X be a smooth projective variety over C. Let G be the group O(r,C), or Sp(r,C). We find the natural notion of semistable principal G-bundle and construct the moduli space, which we compactify by considering also principal G-sheaves, i.e., pairs (E,φ), where E is a torsion free sheaf on X and φ is a symmetric (if G is orthogonal) or antisymmetric (if G is symplectic) bilinear form on E. If G...