Orbit decidability, applications and variations

We present the notion of orbit decidability into a more general framework, exploring interesting generalizations and variations of this algorithmic problem. A recent theorem by Bogopolski-Martino-Ventura gave a renovated protagonism to this notion and motivated several interesting algebraic applications

Bandwidth reduction in rectangular grids

We show that the bandwidth of a square twodimensional grid of arbitrary size can be reduced if two (but not less than two) edges are deleted. The two deleted edges may not be chosen arbitrarily, but they may be chosen to share a common endpoint or to be non-adjacent. We also show that the bandwidth of the rectangular n × m (n ≤ m) grid can be reduced by k, for all k that are sufficiently small, if m − n + 2k edges are deleted

Tamari Lattices and the symmetric Thompson monoid

We investigate the connection between Tamari lattices and the Thompson group F, summarized in the fact that F is a group of fractions for a certain monoid F+sym whose Cayley graph includes all Tamari lattices. Under this correspondence, the Tamari lattice operations are the counterparts of the least common multiple and greatest common divisor operations in F+sym. As an application, we show that, for every n, there exists a length l chain in the nth Tamari lattice whose endpoints are at distance at most 12l/n.Comment: 35page

On classification of groups generated by 3-state automata over a 2-letter alphabet

We show that the class of groups generated by 3-state automata over a 2-letter alphabet has no more than 122 members. For each group in the class we provide some basic information, such as short relators, a few initial values of the growth function, a few initial values of the sizes of the quotients by level stabilizers (congruence quotients), and hystogram of the spectrum of the adjacency operator of the Schreier graph of the action on level 9. In most cases we provide more information, such as whether the group is contracting, self-replicating, or (weakly) branch group, and exhibit elements of infinite order (we show that no group in the class is an infinite torsion group). A GAP package, written by Muntyan and Savchuk, was used to perform some necessary calculations. For some of the examples, we establish that they are (virtually) iterated monodromy groups of post-critically finite rational functions, in which cases we describe the functions and the limit spaces. There are exactly 6 finite groups in the class (of order no greater than 16), two free abelian groups (of rank 1 and 2), and only one free nonabelian group (of rank 3). The other examples in the class range from familiar (some virtually abelian groups, lamplighter group, Baumslag-Solitar groups BS(1±3), and a free product C2 ∗ C2 ∗ C2) to enticing (Basilica group and a few other iterated monodromy groups)

Benefits of Recombinant Adeno-Associated Virus (rAAV)-Mediated Insulinlike Growth Factor I (IGF-I) Overexpression for the Long-Term Reconstruction of Human Osteoarthritic Cartilage by Modulation of the IGF-I Axis

Administration of therapeutic genes to human osteoarthritic (OA) cartilage is a potential approach to generate effective, durable treatments against this slow, progressive disorder. Here, we tested the ability of recombinant adeno-associated virus (rAAV)-mediated overexpression of human insulinlike growth factor (hIGF)-I to reproduce an original surface in human OA cartilage in light of the pleiotropic activities of the factor. We examined the proliferative, survival and anabolic effects of the rAAV-hIGF-I treatment in primary human normal and OA chondrocytes in vitro and in explant cultures in situ compared with control (reporter) vector delivery. Efficient, prolonged IGF-I secretion via rAAV stimulated the biological activities of OA chondrocytes in all the systems evaluated over extended periods of time, especially in situ, where it allowed for the long-term reconstruction of OA cartilage (at least for 90 d). Remarkably, production of high, stable amounts of IGF-I in OA cartilage using rAAV advantageously modulated the expression of central effectors of the IGF-I axis by downregulating IGF-I inhibitors (IGF binding protein [IGFBP]-3 and IGFBP4) while up-regulating key potentiators (IGFBP5, the IGF-I receptor and downstream mitogen-activated protein kinase/extracellular signal–regulated kinase 1/2 [MAPK/ERK-1/2] and phosphatidylinisitol-3/Akt [PI3K/Akt] signal transduction pathways), probably explaining the enhanced responsiveness of OA cartilage to IGF-I treatment. These findings show the benefits of directly providing an IGF-I sequence to articular cartilage via rAAV for the future treatment of human osteoarthritis