Applications ofnon-degenerate and degenerate perturbation theory to problems in one and three dimensions; study of the nuclear finite-size effect on the spectrum of hydrogen-like atoms; hydrogen atom in an external electric field: Stark effect and induced electric dipole moment; derivation of Brillouin--Wigner perturbation theory; variational calculations of the hydrogen and helium atoms; Born-Oppenheimer approximation; variational calculation of the $H_{\text{s}}^{+}$ molecular ion; estimation of bound-state energies of a Hamiltonian with the variational method.

Modular graph functions and modular graph forms map decorated graphs to complex-valued functions on the Poincaré upper half plane with definite transformation properties. Specifically, modular graph functions are SL(2,Z)-invariant functions, while modular graph forms may be identified with SL(2,Z)-invariant differential forms. Modular graph functions and forms generalize, and at the same time unify, holomorphic and non-holomorphic Eisenstein series, almost holomorphic modular forms, multiple zeta-functions, and iterated modular integrals. For example, non-holomorphic Eisenstein series may be associated with one-loop graphs and represent a special class of modular graph functions. The expansion of modular graph forms at the cusp includes Laurent polynomials whose coefficients are combinations of Riemann zeta-values and multiple zeta-values, while each modular graph form may be expanded in a basis of iterated modular integrals. Eisenstein series and modular graph functions and forms beyond Eisenstein series occur naturally and pervasively in the study of the low-energy expansion of superstring amplitudes. Here we shall present a purely mathematical approach with only minimal reference to physics.

The state space of a quantum system is defined; kets and bras are introduced, and their inner and outer products are defined; adjoint, hermitian , and unitary operators are introduced; representations of states and operators on discrete and/or continuous bases are discussed; the properties of commuting hermitian operators are examined; tensor products are defined.

Elliptic functions are introduced via the method of images following a review of periodic functions, Poisson summation, the unfolding trick, and analytic continuation applied to the Riemann zeta-function. The differential equations and addition formulas obeyed by periodic and elliptic functions are deduced from their Kronecker–Eisenstein series representation. The classic constructions of elliptic functions, in terms of their zeros and poles, are presented in terms of the Weierstrass elliptic function, the Jacobi elliptic functions, and the Jacobi theta-functions. The elliptic function theory developed here is placed in the framework of elliptic curves, Abelian differentials, and Abelian integrals.

Applications of the postulates are discussed in a number of problems; the interaction representation is defined.

In this appendix, we shall define and study complex line bundles over an arbitrary compact Riemann surface, provide their topological classification in terms divisors, and give the Riemann–Roch theorem. We shall prove various dimension formulas, including for the dimension of the moduli space of complex or conformal structures on a Riemann surface. We then discuss sections of line bundles from a more physics-oriented point of view in terms of spaces of vector fields, differential forms, and spinor fields.

In this appendix, we present detailed solutions to each one the 75 exercises provided in the body of the text, namely 5 exercises for each one of the Chapters 2–16. When appropriate, for the more advanced exercises, we also provide references to the literature where the corresponding problems were discussed.

Solar flares are commonly accompanied by coronal mass ejections (CME), and thus CMEs display similar size distributions and waiting time distributions as solar flares do. However, some studies report relatively steep power law slopes with values of , which most likely are caused by a bias due to neglecting background subtraction in GOES data. The datasets from LASCO/SOHO are not affected by this background bias, because the white light background from CMEs appears to be sufficiently faint or nonexisting. Waiting time distributions are sampled from a variety of CME and flare catalogs, such as CDAW, LASCO/SOHO, ARTEMIS, CACTus SEEDS, and CORIMP. These waiting time distributions are found to be consistent with the theoretical prediction of the standard FD-SOC model.

In the spring of 1957, the Weinbergs moved to New York for his job at Columbia University, where important experimental work had taken place throught the 1950s. He writes some (largely unimportant) papers on symmetry principles and weak interactions. His first encounter with Murray Gell-Mann gets off to a rocky start. Weinberg starts building a network of colleagues and friends. He misses the chance of tenure at Columbia, so rather than stay for another year as a postdoc, he decides to take up a research position at Berkeley. Before he leaves, he submits his paper on renormalization and infinities.