Full Orbit Sequences in Affine Spaces via Fractional Jumps and Pseudorandom Number Generation

Let $n$ be a positive integer. In this paper we provide a general theory to produce full orbit sequences in the affine $n$-dimensional space over a finite field. For $n=1$ our construction covers the case of the Inversive Congruential Generators (ICG). In addition, for $n>1$ we show that the sequences produced using our construction are easier to compute than ICG sequences. Furthermore, we prove that they have the same discrepancy bounds as the ones constructed using the ICG.Comment: To appear in Mathematics of Computatio

A criterion for separating process calculi

We introduce a new criterion, replacement freeness, to discern the relative expressiveness of process calculi. Intuitively, a calculus is strongly replacement free if replacing, within an enclosing context, a process that cannot perform any visible action by an arbitrary process never inhibits the capability of the resulting process to perform a visible action. We prove that there exists no compositional and interaction sensitive encoding of a not strongly replacement free calculus into any strongly replacement free one. We then define a weaker version of replacement freeness, by only considering replacement of closed processes, and prove that, if we additionally require the encoding to preserve name independence, it is not even possible to encode a non replacement free calculus into a weakly replacement free one. As a consequence of our encodability results, we get that many calculi equipped with priority are not replacement free and hence are not encodable into mainstream calculi like CCS and pi-calculus, that instead are strongly replacement free. We also prove that variants of pi-calculus with match among names, pattern matching or polyadic synchronization are only weakly replacement free, hence they are separated both from process calculi with priority and from mainstream calculi.Comment: In Proceedings EXPRESS'10, arXiv:1011.601

Automorphic compatible systems of Galois representations

This thesis investigates properties of compatible systems of Galois representations, mainly focusing on the compatible systems which are attached to certain classes of automorphic representations of GLn. We develop a general method to prove independence results for algebraic monodromy groups in abstract compatible systems of representations, and give applications both in characteristic zero and in positive characteristic settings. In the case of automorphic compatible systems (and actually for a slightly larger class of geometric compatible systems), we apply our method to deduce an independence result, assuming a classical irreducibility conjecture. In addition, we also deduce an independence result in the case of compatible systems of lisse sheaves on normal varieties over finite fields. We then focus on the study of the geometry of (pseudo)deformation spaces of Galois representations and definite unitary groups eigenvarieties at points corresponding to certain classical automorphic representations. In this context, we present smoothness results known in the literature, and suggest possible implications for automorphic compatible systems.</p

Automorphic compatible systems of Galois representations

This thesis investigates properties of compatible systems of Galois representations, mainly focusing on the compatible systems which are attached to certain classes of automorphic representations of GLn. We develop a general method to prove independence results for algebraic monodromy groups in abstract compatible systems of representations, and give applications both in characteristic zero and in positive characteristic settings. In the case of automorphic compatible systems (and actually for a slightly larger class of geometric compatible systems), we apply our method to deduce an independence result, assuming a classical irreducibility conjecture. In addition, we also deduce an independence result in the case of compatible systems of lisse sheaves on normal varieties over finite fields. We then focus on the study of the geometry of (pseudo)deformation spaces of Galois representations and definite unitary groups eigenvarieties at points corresponding to certain classical automorphic representations. In this context, we present smoothness results known in the literature, and suggest possible implications for automorphic compatible systems.</p