arXiv subjects
Publications and source records attributed to Mohsen Khani.
For any irrational number $ \alpha>\frac{3+\sqrt{5}}{2}\approx2.618$ and given a positive $ n\in\mathbb{N} $, we use elementary number theory to introduce a necessary and sufficient condition for a natural number $ x $ to be in the $n$th iterate of the Beatty sequence of modulus $\alpha$.
We give an example of a valued field $(K,A)$ such that the valuation ring $A$ is definable by an $L_{\text{ring}}$-formula without parameters, but there is no $\exists\forall\exists$ or $\forall\exists\forall$-formula in $L_{\text{ring}}$ to define it.
We introduce a recursive theory that completely axiomatizes the structure $\langle \mathbb{Z},<, +,f,0\rangle$ where $f$ is the function that maps each $x$ to the integer part of $\varphi x $, with $\varphi$ the golden ratio. We prove that our axiomatization is model-complete in a language expanded with a function which we which we refer as the Fibonacci floor function.
We have provided a pure model-theoretic proof for the decidability of the additive structure of the integers together with the function {f} sending {x} to {[ϕx]} where ϕ is the golden ratio.
We have considered a Fraisse class of finitely generated ordered real fields with a colour predicate. A predimension map is defined on finite sets and the Fraisse limit of the class is axiomatized by a theory $T$, which is proved to be dependent. The theory is proved to be non-distal with $dp-rank=\aleph_0$.
This paper concerns the study of Bi-colored expansions of geometric theories in the light of the Fraïssé-Hrushovski construction method. Substructures of models of a geometric theory $T$ are expanded by a color predicate $p$, and the dimension function associated with the pre-geometry of the $T$-algebraic closure operator together with a real number $0<α\leqslant 1$ is used to define a pre-dimension function $δ_α$. The pair $(\mathcal{K}_α^{+},\leqslant_α)$ consisting of all such expansions with a hereditary positive pre-dimension along with the notion of substructure $\leqslant_α$ associated to $δ_α$ is then used as a natural setting for the study of generic bi-colored expansions in the style of Fraïssé-Hrushovski construction. Imposing certain natural conditions on $T$, enables us to introduce a complete axiomatization $\mathbb{T}_α$ for the class of rich structures in this class. We will show that if $T$ is a dependent theory (NIP) then so is $\mathbb{T}_α$. We further prove that whenever $α$ is rational the strong dependence transfers to $\mathbb{T}_α$. We conclude by showing that if $T$ defines a linear order and $α$ is irrational then $\mathbb{T}_α$ is not strongly dependent.
We introduce a model-complete theory which completely axiomatizes the structure $Z_{\alpha}=(Z, +, 0, 1, f)$ where $f : x \to \lfloor{\alpha} x \rfloor $ is a unary function with $\alpha$ a fixed transcendental number. When $\alpha$ is computable, our theory is recursively enumerable, and hence decidable as a result of completeness. Therefore, this result fits into the more general theme of adding traces of multiplication to integers without losing decidability.