Gonenc Onay

• Doctor of Philosophy
• D4C

Stochastic gradient descent method and its variants constitute the core optimization algorithms that achieve good convergence rates for solving machine learning problems. These rates are obtained especially when these algorithms are fine-tuned for the application at hand. Although this tuning process can require large computational costs, recent wo...

Abraham Robinson's method for finding model completions is refined and and generalized for model companions and is applied to the theory of fields equipped with both a valuation and an automorphism.

In this article we study modules endowed with a ultrametric, from the point of view of the geometric notion $C$-minimality. We give a complete characterization of $C$-minimal valued modules over non-commutative rings of skew polynomials of the form $R:=K[t;\varphi]$, where $K$ is a field, $\varphi$ an endomorphism of $K$ and $R$ is the $K$-algebra...

Following our first article, we continue to investigate ultrametic modules over a ring of twisted polynomials of the form $[K;\vfi]$, where $\vfi$ is a ring endomorphism of $K$. The main motivation comes from the the theory of valued difference fields (including characteristic $p>0$ valued fields equipped with the Frobenius endomorphism). We introd...

Let $R$ be the (non commutative-) ring of additive polynomials over the field $K:=\mathbb{F}_p(X)^h$, the henselization of the field $\mathbb{F}_p(X)$. We show that the (right-) $R$-module theory of the field $\mathbb{F}_p((X))$ is decidable. Moreover, we provide a recursively enumerable axiom system $T_1$ in the language $L_{\mathcal{O}}$, the lan...

Abraham Robinson's method for finding model completions is refined and and generalized for model companions and is applied to the theory of fields equipped with both a valuation and an automorphism.

We introduce a notion of valued module which is suitable to study valued fields of positive characteristic. Then we built-up a robust theory of henselianity in the language of valued modules and prove Ax-Kochen Ershov type results.

We introduce a notion of valued module which is suitable to study valued fields of positive characteristic. Then we built-up a robust theory of henselianity in the language of valued modules and prove Ax-Kochen Ershov type results.

We provide axiomatization and relative quantifier elimination for valued fields equipped with an automorphism, in residue characteristic zero. Similar results are known under strong assumptions on the interaction between the automorphism and the valuation. We remove such assumptions and provide general treatment. As a consequence we obtain an axiom...

Cette thèse étudie les modules values sur des anneaux de polynômes tordus de la forme R := K[t:φ] où φ est un endomorphisme du corps K . Les exemples motivants sont les corps values (M/,v) de caractéristique p>(). où R = K [t ;x ↦ XP ], l'anneau des polynômes additifs à coefficient dans un sous-corps K de M. , Le coeur de la thèse se trouve dans le...