Let K be a p-adic field of characteristic zero. For polynomials A,B ∈ K[x] we consider decompositions A(x)f(x) +B(x)g(x) = 1 of entire functions f, g on K and try to improve an impossibility result due to A. Boutabaa concerning transcendental f, g. Also, an independent proof of a p-adic diophantic statement due to D. N. Clark is given.

In 1679, Leibniz wrote nine manuscripts on three different arithmetical models of Aristotelian logic. This was a part of his project of a “calculus universalis”. First we show the precise relations of these three models to each other by presenting three criteria which serve the purpose of classifying models of Aristotelian logic. This facilitates the understanding of Leibniz’ constructions. Our...

If G is the category of finite simple graphs G = (V,E), the linear space V of valuations on G has a basis given by the f-numbers vk(G) counting complete subgraphs Kk+1 in G. The barycentric refinement G1 of G ∈ G is the graph with Kl subgraphs as vertex set where new vertices a 6= b are connected if a ⊂ b or b ⊂ a. Under refinement, the clique data transform as ~v → A~v with the upper triangula...

Formal AI systems traditionally represent knowledge using logical formulas. We will show, however, that for certain kinds of information, a modelbased representation is more compact and enables faster reasoning than the corresponding formula-based representation. The central idea behind our work is to represent a large set of models by a subset of characteristic models. More specifically, we ex...

In this paper, we study the ε-lemniscate of the characteristic polynomial in relation to the pseudospectrum of the associated matrix. It is natural to investigate this question because these two sets can be seen as generalizations of eigenvalues. The question of numerical determination of the ε-lemniscate raises the problem of computing the characteristic polynomial p. We can express the coeffi...

We illustrate an efficient new method for handling polynomial systems with degenerate solution sets. In particular, a corollary of our techniques is a new algorithm to find an isolated point in every excess component of the zero set (over an algebraically closed field) of any n by n system of polynomial equations. Since we use the sparse resultant, we thus obtain complexity bounds (for converti...

The Cauchy problem for hyperbolic equations is well known since many years ago. We can take the wave equation in Minkowski space-time as an example. If we choose an infinite space-like surface where it is known the function and its normal derivative (with respect to the surface) the solution is fully obtained in the rest of the space. The situation becomes more delicate when the surface where w...

1. Table of symbols 2 2. Transcendence for Drinfeld modules 2 2.1. Wade’s results 2 2.2. Drinfeld modules 3 2.3. The Weierstraß-Drinfeld correspondence 3 2.4. Carlitz 5 2.5. Yu’s work 6 3. t-Modules 7 3.1. Definitions 7 3.2. Yu’s sub-t-module theorem 8 3.3. Yu’s version of Baker’s theorem 8 3.4. Proof of Baker-Yu 8 3.5. Quasi-periodic functions 9 3.6. Derivatives and linear independence 12 3.7....

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