Recursive definitions on surreal numbers

Let No be Conway's class of surreal numbers. I will make explicit the notion of a function f on No recursively defined over some family of functions. Under some "tameness" and uniformity condition, f must satisfy some interesting properties; in particular, the supremum of the class of element greater or equal to a fixed d in No is actually an element of No. For similar reasons, the concatenation function x:y cannot be defined recursively in a uniform way over polynomial functions

Expansions of the reals which do not define the natural numbers

We study first-order expansions of the reals which do not define the set of natural numbers. We also show that several stronger notions of tameness are equivalent to each others.Comment: 16 pages, version 1.

O-minimal spectrum

Let X be a definable sub-set of some o-minimal structure. We study the spectrum of X, in relation with the definability of types

Tame structures and open cores

We study various notions of "tameness" for definably complete expansions of ordered fields. We mainly study structures with locally o-minimal open core, d-minimal structures, and dense pairs of d-minimal structures.Comment: Version 3.2, 64 page

Groups and rings definable in d-minimal structures

We study groups and rings definable in d-minimal expansions of ordered fields. We generalize to such objects some known results from o-minimality. In particular, we prove that we can endow a definable group with a definable topology making it a topological group, and that a definable ring of dimension at least 1 and without zero divisors is a skew field.Comment: 19 page

Generic derivations on o-minimal structures

Let $T$ be a complete, model complete o-minimal theory extending the theory RCF of real closed ordered fields in some appropriate language $L$. We study derivations $\delta$ on models $\mathcal{M}\models T$. We introduce the notion of a $T$-derivation: a derivation which is compatible with the $L(\emptyset)$-definable $\mathcal{C}^1$-functions on $\mathcal{M}$. We show that the theory of $T$-models with a $T$-derivation has a model completion $T^\delta_G$. The derivation in models $(\mathcal{M},\delta)\models T^\delta_G$ behaves "generically," it is wildly discontinuous and its kernel is a dense elementary $L$-substructure of $\mathcal{M}$. If $T =$ RCF, then $T^\delta_G$ is the theory of closed ordered differential fields (CODF) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that $T^\delta_G$ has $T$ as its open core, that $T^\delta_G$ is distal, and that $T^\delta_G$ eliminates imaginaries. We also show that the theory of $T$-models with finitely many commuting $T$-derivations has a model completion.Comment: 29 page

On Gr\"obner Basis for certain one-point AG codes

In this work we present a way to construct the so-called root diagram for one-point AG codes $C$ arising from certain types of curves $\mathcal{X}$ over $\mathbb{F}_q$ with plane model $f(y)=g(x)$. Using this root diagram we can get an algorithm to obtain a Gr\"obner basis for the submodule $\overline{C}$ associated to $C$Comment: 12 pages, every comment is welcom

On J-Colorability of Certain Derived Graph Classes

A vertex $v$ of a given graph $G$ is said to be in a rainbow neighbourhood of $G$, with respect to a proper coloring $C$ of $G$, if the closed neighbourhood $N[v]$ of the vertex $v$ consists of at least one vertex from every colour class of $G$ with respect to $C$. A maximal proper colouring of a graph $G$ is a $J$-colouring of $G$ if and only if every vertex of G belongs to a rainbow neighbourhood of $G$. In this paper, we study certain parameters related to $J$-colouring of certain Mycielski type graphs.Comment: 12 pages, 8 figure

Logarithmic De Rham, Infinitesimal and Betti Cohomologies

In this article, we analyze the connection between the Log De Rham Cohomology of an fs (not necessary log smooth) log scheme $Y$ over $\mathbb C$ (for $Y$ admitting an exact closed immersion into an fs log smooth log scheme over $\mathbb C$), its Log Infinitesimal Cohomology $H^{^.}(Y^{log}_{inf}, \mathcal O_{Y^{log}_{inf}})$, and its Log Betti Cohomology, which is the Cohomology of its associated Kato-Nakayama topological space $Y^{an}_{log}$, and we prove that they are isomorphic. These results are the log scheme analogues of two classical comparison theorems.Comment: 22 page

A dichotomy for expansions of the real field

A dichotomy for expansions of the real field is established: Either the set of integers is definable or every nonempty bounded nowhere dense definable subset of the real numbers has Minkowski dimension zero