Model companions of theories with an automorphism

For a theory $T$ in $L, T_\sigma$ is the theory of the models of $T$ with an automorphism $\sigma$. If $T$ is an unstable model complete theory without the independence property, then $T_\sigma$ has no model companion. If $T$ is an unstable model complete theory and $T_\sigma$ has the amalgamation property, then $T_\sigma$ has no model companion. If $T$ is model complete and has the fcp, then $T_\sigma$ has no model completion

On Central Gap Numbers of Symmetric Groups

g(G) denotes the central gap number of a group G. We show that for n &#8805; 8, g(Sn) &#8805; n and g(An) &#8805; n-2. We give exact values of g(Sn) and g(An) for small n's. In particular, g(S9) = 9 and g(A9) = 7. Therefore, for any positive integer n &#8800; 1, 3, 5 there is a group G such that n = g(G). G can be finite or infinite.</p

On the Structure of Hrushovski's Pseudoplanes Associated to Irrational Numbers (Model theoretic aspects of the notion of independence and dimension)

Let α be an irrational number, and a/b a reduced fraction. Suppose 2/3 < α < a/b < 3/4 and b is sufficiently large. Let B be a canonical twig for a/ b and A the set of all leaves in B. Let p ∈ B be a good vertex of B over A. Let M be the generic structure for (K[f], <) where f is the Hrushovski's log-like function associated to a. Assume that B is a closed subset of M. Let D be the orbit of p over A in M. Then M = cl(D). Actually, we can prove this only assuming O < α < a/b < 1

Efficient iPS Cell Production with the MyoD Transactivation Domain in Serum-Free Culture

A major difficulty of producing induced pluripotent stem cells (iPSCs) has been the low efficiency of reprogramming differentiated cells into pluripotent cells. We previously showed that 5% of mouse embryonic fibroblasts (MEFs) were reprogrammed into iPSCs when they were transduced with a fusion gene composed of Oct4 and the transactivation domain of MyoD (called M3O), along with Sox2, Klf4 and c-Myc (SKM). In addition, M3O facilitated chromatin remodeling of pluripotency genes in the majority of transduced MEFs, including cells that did not become iPSCs. These observations suggested the possibility that more than 5% of cells had acquired the ability to become iPSCs given more favorable culture conditions. Here, we raised the efficiency of making mouse iPSCs with M3O-SKM to 26% by culturing transduced cells at low density in serum-free culture medium. In contrast, the efficiency increased from 0.1% to only 2% with the combination of wild-type Oct4 and SKM (OSKM) under the same culture condition. For human iPSCs, M3O-SKM achieved 7% efficiency under a similar serum-free culture condition, in comparison to 1% efficiency with OSKM. This study highlights the power of combining the transactivation domain of MyoD with a favorable culture environment

ON mathrm{K}_{f} IN IRRATIONAL CASES (Model theoretic aspects of the notion of independence and dimension)

Consider an ab initio amalgamation class mathrm{K}_{f} with an unbounded increasing concave function f. We conjecture that if mathrm{K}_{f} has the free amalgamation property then the generic structure for mathrm{K}_{f} has a model complete theory. We consider the case where the predimension function has an irrational coefficient. We show some statements which seem to be useful to show our conjecture