The development of US naval aviation was much in doubt, especially during its formative years between 1910 and 1913. The lack of interest and allocation of money for aviation changed in 1916, when it became clear that the United States could be drawn into the war in Europe. Yet, the naval and marine air sections were woefully unprepared for warfare when the United States declared war against the Central Powers and had to rely on its European allies to provide bombers and pursuit aircraft throughout the war. There was initially tension between the Navy and Marine Corps as the naval hierarchy saw no need to provide land-based aircraft as the duty of aviation was to patrol and for antisubmarine operations by using seaplanes and flying boats. Negotiations brought forth the concept of the Northern Bombing Group, a land-based unit equipped primarily with British and Italian aircraft. This historical narrative encompasses the formation and development of the US Navy and Marine Corps air services from 1911 to 1918. It includes pre-war and wartime training, aircraft development, operations, and personalities such as the Navy’s Theodore “Spuds” Ellyson and Marine Alfred A. Cunningham. Both pioneers continuously fought to maintain and grow their service’s air arms. Their tenacity would, within seven years, lead to the organization of wartime scouting, pursuit, and bombing units.

This book is about extending index theory to some examples where non-Fredholm operators arise. It focuses on one aspect of the problem of what replaces the notion of spectral flow and the Fredholm index when the operators in question have zero in their essential spectrum. Most work in this topic stems from the so-called Witten index that is discussed at length here. The new direction described in these notes is the introduction of `spectral flow beyond the Fredholm case'. Creating a coherent picture of numerous investigations and scattered notions of the past 50 years, this work carefully introduces spectral flow, the Witten index and the spectral shift function and describes their relationship. After the introduction, Chapter 2 carefully reviews Double Operator Integrals, Chapter 3 describes the class of so-called p-relative trace class perturbations, followed by the construction of Krein's spectral shift function in Chapter 4. Chapter 5 reviews the analytic approach to spectral flow, culminating in Chapter 6 in the main abstract result of the book, namely the so-called principal trace formula. Chapter 7 completes the work with illustrations of the main results using explicit computations on two examples: the Dirac operator in Rd, and a differential operator on an interval. Throughout, attention is paid to the history of the subject and earlier references are provided accordingly. The book is aimed at experts in index theory as well as newcomers to the field.