Q-system Cluster Algebras, Paths and Total Positivity

In the first part of this paper, we provide a concise review of our method of solution of the Ar Q-systems in terms of the partition function of paths on a weighted graph. In the second part, we show that it is possible to modify the graphs and transfer matrices so as to provide an explicit connection to the theory of planar networks introduced in the context of totally positive matrices by Fomin and Zelevinsky. As an illustration of the further generality of our method, we apply it to give a simple solution for the rank 2 affine cluster algebras studied by Caldero and Zelevinsky

Renormalization Group and Infinite Algebraic Structure in D-Dimensional Conformal Field Theory

We consider scalar field theory in the D-dimensional space with nontrivial metric and local action functional of most general form. It is possible to construct for this model a generalization of renormalization procedure and RG-equations. In the fixed point the diffeomorphism and Weyl transformations generate an infinite algebraic structure of D-Dimensional conformal field theory models. The Wilson expansion and crossing symmetry enable to obtain sum rules for dimensions of composite operators and Wilson coefficients.Comment: 16 page

Strings with Discrete Target Space

We investigate the field theory of strings having as a target space an arbitrary discrete one-dimensional manifold. The existence of the continuum limit is guaranteed if the target space is a Dynkin diagram of a simply laced Lie algebra or its affine extension. In this case the theory can be mapped onto the theory of strings embedded in the infinite discrete line $\Z$ which is the target space of the SOS model. On the regular lattice this mapping is known as Coulomb gas picture. ... Once the classical background is known, the amplitudes involving propagation of strings can be evaluated by perturbative expansion around the saddle point of the functional integral. For example, the partition function of the noninteracting closed string (toroidal world sheet) is the contribution of the gaussian fluctuations of the string field. The vertices in the corresponding Feynman diagram technique are constructed as the loop amplitudes in a random matrix model with suitably chosen potential.Comment: 65 pages (Sept. 91

Engineering biomolecular microenvironments for cell instructive biomaterials

Engineered cell instructive microenvironments with the ability to stimulate specific cellular responses is a topic of high interest in the fabrication and development of biomaterials for application in tissue engineering. Cells are inherently sensitive to the in vivo microenvironment that is often designed as the cell “niche”. The cell “niche” comprising the extracellular matrix and adjacent cells, influences not only cell architecture and mechanics, but also cell polarity and function. Extensive research has been performed to establish new tools to fabricate biomimetic advanced materials for tissue engineering that incorporate structural, mechanical and biochemical signals that interact with cells in a controlled manner and to recapitulate the in vivo dynamic microenvironment. Bioactive tunable microenvironments using micro and nanofabrication have been successfully developed and proven to be extremely powerful to control intracellular signaling and cell function. This review is focused in the assortment of biochemical signals that have been explored to fabricate bioactive cell microenvironments and the main technologies and chemical strategies to encode them in engineered biomaterials with biological information.The authors thank Fundacao para a Ciencia e Tecnologia for C.A.C.'s PhD grant (SFRH/BD/61390/2009). This work was carried out under the scope of the European Union's Seventh Framework Programme (FP7/2007-2013) under grant agreement no REGPOT-CT2012-316331-POLARIS