Homology of holomorphs of free groups

The holomorph of a free group $F_n$ is the semidirect product $F_n \rtimes Aut(F_n)$. Using the methods of Hatcher and Vogtmann, we derive stability results and calculate the mod-$p$ homology of these holomorphs for odd primes $p$ in dimensions 1 and 2, and their rational homology in dimensions 1 through 5. Calculations of the twisted (where $Aut(F_n)$ acts by first projecting to $Gl_n(\Z)$ and then including in Gl_n(\Q)) homology H_*(Aut(F_n); \Q^n) follow in corresponding dimensions.Comment: Revised using helpful comments of referee. Accepted by J. Algebra. J. Algebra has agreed to host the Maple program files (currently on my web page) on their sit

Stable rational cohomology of automorphism groups of free groups and the integral cohomology of moduli spaces of graphs

It is not known whether or not the stable rational cohomology groups \tilde H^*(Aut(F_\infty);\Q) always vanish. We show that either the rational cohomology does not vanish in certain dimensions, or the integral cohomology of a moduli space of pointed graphs does not stabilize in certain other dimensions. Similar results are stated for groups of outer automorphisms. This yields that $H^5(\hat Q_m; \mathbb{Z})$, $H^6(\hat Q_m; \mathbb{Z})$, and $H^5(Q_m; \mathbb{Z})$ never stabilize as $m \to \infty$, where the moduli spaces $\hat Q_m$ and $Q_m$ are the quotients of the spines $\hat X_m$ and $X_m$ of ``outer space'' and ``auter space'', respectively, introduced by Culler and Vogtmann and by Hatcher and Vogtmann.Comment: To appear in Publicacions Matematique

The Euler characteristic of the Whitehead automorphism group of a free product

A combinatorial summation identity over the lattice of labelled hypertrees is established that allows one to gain concrete information on the Euler characteristics of various automorphism groups of free products of groups.Comment: 19 pages, 3 figures, to appear in Trans. Amer. Math. So

Design and Implementation of a Fuzzy Logic Speed Controller for an Internal Combustion Engine

Internal combustion engines are challenging to model and control. Uncertainties and nonlinearities pose operating problems for classical controllers. Delays inherent to the engine combustion cycle tend to introduce overshoot and oscillations in most control schemes. Most work that has been done in dealing with delays requires the designer to have extensive knowledge of the system to be controlled. Engines are very difficult to model accurately, thereby ruling out most of these techniques. Fuzzy logic is well suited to this problem, since an accurate model is not needed for design, and it is known to be robust to nonlinearities and parameter variations. The objective of this thesis was to design and implement a fuzzy logic controller to control the speed of a Honda EM3500S portable generator. This new fuzzy controller maintains the robustness of traditional fuzzy logic to nonlinearities and it is also more robust to delays. The control scheme uses dual fuzzy logic control modules in parallel. One of the modules is a traditional fuzzy scheme and the other is a simple two membership fuzzy scheme tuned to reduce oscillations. For optimal performance this second module requires dynamic adjustment of parameters such as input and output gains in response to the system’s current operating condition. The result is a control scheme that offers reduced overshoot and oscillations. The new control scheme was compared to the classical PID and the traditional fuzzy logic controllers. These comparisons were done via computer simulations and laboratory implementation and testing. A windows based C++ program was developed to realize and test the new controller. The better performance of the new control scheme was illustrated