Complex analytic functions have been a rich, important, and insightful chapter in mathematics and their use in physics is matched only by a few other areas. While analyticity is of critical importance, it is often the points where analyticity breaks down that are most interesting. The singularities of otherwise analytic functions often carry much information about the function itself and thereby provide physical insight. Often the singularities are isolated, but there are occasions when the singularities condense into a dense curve. The curve in the complex plane mapped out by this dense set of singularities is called a natural boundary. Functions whose power series is characterized by “gaps” (or “lacunae”) in progression of terms are called lacunary functions and they constitute a particularly important class of functions that exhibit a natural boundary. Because the natural boundary is difficult to deal with, functions with natural boundaries have not seen extensive use in physics over the years. Nonetheless, lacunary functions have found some recent use in tackling physical problems in quantum mechanics, optics, and statistical mechanics. We will be studying many features of these functions.

Noisy light is simply broad bandwidth quasi continuous-wave (cw) laser light having a random time profile. Such light can be used to probe molecular dynamics and it offers an unique alternative to the more conventional frequency domain (narrow bandwidth) spectroscopies and ultra-short (femtosecond scale) pulse time domain spectroscopies. Like the frequency domain spectroscopies, the cw nature of noisy light allows precise measurement of transition frequencies.

However, like time resolved techniques, its ultra-short noise correlation time offers the time resolution to directly measure the time domain material response. There is a noisy light analog for nearly every current conventional optical spectroscopy, but it is particularly useful for coherent Raman scattering (CRS) spectroscopy. Here, the noisy light technique can provide an extremely precise probe of ground state vibrational dynamics. In short, noisy light techniques offer a third means of studying molecular lineshapes which complements the more familiar cw and ultra-short pulse techniques.

The study of noisy light has brought insight into other light matter phenomena such as cascaded events in ultrafast nonlinear optical spectroscopy and to FRET (Förster resonant energy transfer).

Halogen bonding was first recognized in the mid-1800s. Over the course of the 20th century, the science of halogen bonding was still in its beginning stages. Halogen bond research has expanded greatly in the past 20 years, and applications of halogen bonding are becoming more numerous. Halogen bonding has many uses in science. There are important applications in materials science, chemistry, biology, and biochemistry.

An important characteristic of halogen bond donors is the presence of a region of positive charge found on the halogen atom engaged with the lone pair on a halogen bond acceptor. This region of positive charge is called the 𝜎-hole. We use iodo-perfluoroalkanes because the nature of these compounds enhances the 𝜎-hole present on the back side of the iodine. Iodine has a high polarizability and fluorine atoms are very electronegative, so the electronegative fluorine atoms pull the electron density away from the iodine. This optimizes the 𝜎-hole for a strong interaction. The presence of this 𝜎-hole also sets up a Lewis acid-base interaction very nicely. We use Raman spectroscopy is used to investigate how Lewis basicity affects the extent of the halogen bond.

The concept of a fractional derivative has been receiving a lot of attention in the literature in recent years, with entire journals devoted to fractional analysis. Many of the authors of these papers mention the famous correspondence between Leibniz and L’Hopital in 1695. Over the intervening years many definitions of a fractional derivative have appeared; well known examples being the Riemann-Liouville and the Caputo definitions. The current activity clearly suggests the extension of derivatives of non-integer power is not straightforward to say the least.

The conformable derivative does not contribute “new mathematics.” That said, exploration of the conformable derivative and its generalizations can still be interesting and valuable. We hope to provide some physical insight to assist with use in the applied setting. We focus on application in quantum mechanics but some of the discussion about how to interpret physical units and spaces related via Fourier transformation are relevant in general.