An immensely influential paper in the history of the study of chaos was the work of Ruelle and Takens [1], later extended with Newhouse [2] on the robustness of quasiperiodic dynamics with 3 or more incommensurate frequencies. This addressed the question of whether complexity in dynamics was likely to occur through the accumulation of Hopf-like bifurcations adding additional frequencies (and pr...

The use of radiotherapy and or chemotherapy can result in a wide range of acute and chronic side effects. These are generic concerns for cancer treatment in general. Side effects include bone marrow suppression, and cardiac and lung damage. A full discussion is beyond the scope of this document. These guidelines deal with complications that are of greater significance in younger patients with l...

The representation given in Example 3 has an interesting property. The set of matrices {ρ(e), ρ(g)} have common invariant subspaces (other than the trivial ones, namely (0, 0) and C). Indeed, we can eas­ ily verify that the (orthogonal) one­dimensional subspaces given by (t, t) and (t, −t) are invariant under the action of these matrices. Therefore, the restriction of ρ to those subspaces also ...

1 Overview In this lecture, we will • Wrap up Alon-Edmonds-Luby's graph theoretical codes: construction and analysis of distance • Introduce the interplay between codes and computational complexity: open problems and pseudorandomness 2 Alon-Edmonds-Luby The construction of the code depends on three main ingredients: • A binary code C 0 = [d, R 0 d, δ 0 d] 2 , where R 0 and δ 0 are our target ra...

Humans, like all social species, face various collective action problems (difficulties achieving potential benefits from cooperating when coordination is required or individuals have incentives to defect). Humans solve these problems through various means: communication, monitoring, enforcement, and selective incentives. This chapter summarizes the theory and evidence on human cooperation found...