Resolvents of R-Diagonal Operators

We consider the resolvent $(\lambda-a)^{-1}$ of any $R$-diagonal operator $a$ in a $\mathrm{II}_1$-factor. Our main theorem gives a universal asymptotic formula for the norm of such a resolvent. En route to its proof, we calculate the $R$-transform of the operator $|\lambda-c|^2$ where $c$ is Voiculescu's circular operator, and give an asymptotic formula for the negative moments of $|\lambda-a|^2$ for any $R$-diagonal $a$. We use a mixture of complex analytic and combinatorial techniques, each giving finer information where the other can give only coarse detail. In particular, we introduce {\em partition structure diagrams}, a new combinatorial structure arising in free probability.Comment: 29 pages, 12 figures, used gastex.st

Design Philosophy Study on CFRP RC Structures

The present dissertation was developed with the main purpose of comparing the different design philosophies existing for prestressed reinforced concrete members, as well as analyzing a solution that is both safe and economically viable. Since one of the major problems in civil engineering infrastructures is due to corrosion, a study was conducted to examine the feasibility of replacing the material used as reinforcement and prestress. The steel, which for many decades has been used shall be replaced by FRP, particularly CFRP. However, since carbon presents a brittle behavior, a new design methodology suggested by fib and by the Canadian code is presented so that the failure is not due to the reinforcement but due to the concrete, which, although it is not a ductile material, presents more ductility than materials with carbon fibers. In this way, a code has been developed so that, through the balance of the cross section, it calculates the amount of reinforcement required, following the American (ACI) and European code (Eurocode) design methodology. Another code for prestressed reinforced concrete members was elaborated to analyze the prestress system used and the material used as reinforcement and tendons. The European code has proved to be economically viable since, for the same applied moment, it requires a smaller amount of reinforcement. As for the bonded and unbonded system the results show that the systems with unbonded tendons are only feasible when the concrete members are subjected to high moments. It is also observed that depending on the initial force applied on the prestress, the results may be favorable regarding to the amount of reinforcement required, however the amount of reinforcement can be significantly increased when the initial force of the prestress is highly increased