Empirically we find that many phenomena in planetary science reveal power law-like size distributions with a power law slope of for differential size distributions, or for cumulative size distributions. These observational results fully agree with theoretical predictions of the FD-SOC model. These observations include lunar craters, Saturn ring particles, near-Earth objects, Jovian Trojans, asteroids, Neptune Trojans, the Kuijper belt, and extrasolar planets. Mars fluvial systems and dust storms reveal fractal structures. Terrestrial gamma-ray flashes indicate also scale-free power law slopes. It appears that the scale-free behavior of planetary phenomena could result from both accumulation and fragmentation processes.

In this final appendix, we shall review the modular geometry of the Siegel half-space at higher rank, Riemann theta-functions of higher rank, the embedding of higher-genus Riemann surfaces into the Jacobian variety via the Abel map, and use these ingredients to construct the prime form, the Szego kernel, and other meromorphic functions and differential forms on higher-genus Riemann surfaces.

Solar flare hard X-ray events are produced by the electron thick-target bremsstrahlung process at electron energies of ~20 keV. Large statistical samples of hard X-ray fluxes, fluences, energies, flare durations, and waiting times have been observed with instruments from three different spacecraft (HXRBS/SMM, BATSE/CGRO, and RHESSI) from three different solar cycles and analyzed with different automated event detection methods. Despite of this large variety of data, all datasets reveal self-consistent results, for instance, power law peak fluxes with a slope of , which match the theoretical prediction of the fractal-diffusive SOC model, that is, . Systematic errors and uncertainties of these datasets include insufficient fitting ranges, spacecraft orbital data gaps, finite-size effects, south Atlantic anomaly data gaps, instrumental sensitivity, incomplete samples, thresholded event selection, and background subtraction.

Scattering in one dimension is discussed in terms of wave packets; reflection, transmission, and tunneling probabilities are defined; the WKB method to calculate these probabilities is introduced; the S-matrix for scattering in one dimension is defined; the phase-shift method for one-dimensional parity-invariant potentials is introduced; applications to various combinations of finite and infinite barriers with delta-function potentials are examined.

Weinberg takes the standard first-year courses for physics students, studying mechanics, heat, light, and electromagnetism. While these are not the fancy modern topics, they are the essential foundations for everything else he would learn in physics. He joins Telluride House, a fraternity. In his sophomore year, he lets his studies slide (except for physics and math). He pulls out of this slump with the determination not to waste any more time and forms the habit of being a compulsive worker. He lands a summer job at Bell Telephone Laboratories. In his senior year, he learns quantum mechanics. He decides to apply to graduate school to study for a PhD in physics, and to marry Louise Goldwasser.

A key result of solar flare statistics is the continuity of size distributions over nine orders of magnitude, consisting of nanoflares, microflares, and large flares, covering a range of ~1024–1033 ergs in energy. The FD-SOC model predicts power law distribution functions with a slope of when the energy of flare events are derived from the flare event 2-D area , but a flatter slope of , if the flare energies are derived from the volume-integrated total flux of the 3-D flare volume. These predictions match the observations of EUV nanoflares and microflares. These scaling laws imply more energy is distributed at large flare sizes , and thus, makes nanoflares less important for coronal heating. Such scaling laws are numerically simulated with cellular automaton codes and are applied to the time evolution of coronal loops, magnetic field line breading, and magnetic reconnection processes.

Among black-hole systems, we find a variety with applications of SOC, such as soft gamma-ray repeaters, magnetars, blazars, black holes in accretion disks, and galactic fast radio bursts. Gamma-ray bursts, soft gamma-ray repeaters, as well as black-hole objects, are found to be self-consistent with the theoretical prediction of the FD-SOC mode. Galactic phenomena that possibly have some characteristics in common with SOC models are: fractal galaxy distributions; cosmic ray energy spectrum; extragalactic fast radio bursts; and extragalactic background fluxes.

Having finished her law degree, Louise takes up work at a Boston law firm; they are not returning to Berkeley after all, so Weinberg resigns his professorship there. At MIT, he continues teaching graduate courses on general relativity, with an emphasis on cosmology. He spends the spring of 1971 in Paris, making comparisons between the academic characters of Paris and Boston. Gerard ’t Hooft and Martinus Veltman renormalize Weinberg’s theory of leptons, showing an experimental route to proving the theory. Weinberg starts to consider the extension of the electroweak theory to strongly interacting theories. Electroweak theory starts to receive a lot more attention from theorists. His first book, Gravitation and Cosmology, is published in 1972. Weinberg is offered the Higgins Professorship at Harvard, and accepts.

Familiarity with chemistry from children’s toy kits leads Weinberg to investigate physics, the subject that underlies all of chemistry. He reads George Gamow’s Mr. Tompkins books, among others. He is admitted to the famous Bronx High School of Science, where he becomes friends with Shelly Glashow and Gary Feinberg, who would also become well-known physicists. He wins a New York state scholarship to Cornell.

Spinor wave functions; classical Lagrangian and Hamiltonian of a charged particle in an electromagnetic field, and its quantum Hamiltonian; gauge invariance; spin magnetic moment in a uniform magnetic field; magnetic resonance; the Stern--Gerlach experiment; neutron interferometry and rotations of spinor wave functions; treatment of a particle in a uniform magnetic field with and without the inclusion of spin degrees of freedom; Ahronov--Bohm effect for a charged spinless particle confined in a cylindrical shell.

Some immediate applications of the theory of elliptic functions and modular forms to problems of physical interest are presented, including the construction of the Green functions and functional determinants for the two-dimensional quantum field theories of the bc fields, the scalar field, and the spinor fields on the torus. In particular, it will be shown how the singular terms in the operator product expansion of holomorphic fields for the bc system essentially determine arbitrary correlation functions on the torus in terms of elliptic functions. Along the way, a brief but reasonably systematic introduction will be presented of two-dimensional conformal field theory methods.

Now in Boston, Weinberg describes how his earlier work on current algebra led to effective field theory. With the Vietnam War going on, JASON work focuses on the war effort. In 1967, Weinberg takes up a lectureship at MIT and published his most-cited paper, “A Model of Leptons,” which heralded electroweak theory. He attends the Solvay conference in Brussels in 1967, but misses being in the group photo. Back in Boston, Weinberg discusses making friends through his election to the American Academy of Arts and Sciences. He becomes involved in an independent study of the US’ anti-ballistic missile program, concluding that this would hasten the arms race between the US and the Soviet Union.