SLₖ-Tilings and Paths in ℤᵏ

An SLₖ-frieze is a bi-infinite array of integers where adjacent entries satisfy a certain diamond rule. SL₂-friezes were introduced and studied by Conway and Coxeter. Later, these were generalized to infinite matrix-like structures called tilings as well as higher values of k. A recent paper by Short showed a bijection between bi-infinite paths of reduced rationals in the Farey graph and SL₂-tilings. We extend this result to higher kby constructing a bijection between SLₖ-tilings and certain pairs of bi-infinite strips of vectors in ℤᵏ called paths. The key ingredient in the proof is the relation to Plucker friezes and Grassmannian cluster algebras. As an application, we obtain results about periodicity, duality, and positivity for tilings

Determining Material Structures and Surface Chemistry by Genetic Algorithms and Quantum Chemical Simulations

With the advent of modern computing, the use of simulation in chemistry has become just as important as experiment. Simulations were originally only applicable to small molecules, but modern techniques, such as density functional theory (DFT) allow extension to materials science. While there are many valuable techniques for synthesis and characterization in chemistry laboratories, there are far more materials possible than can be synthesized, each with an entire host of surfaces. This wealth of chemical space to explore begs the use of computational chemistry to mimic synthesis and experimental characterization. In this work, genetic algorithms (GA), for the former, and DFT calculations, for the latter, are developed and used for the in silico exploration of materials chemistry. Genetic algorithms were first theorized in 1975 by John Holland and over the years subsequently expanded and developed for a variety of purposes. The first application to chemistry came in the early 1990’s and surface chemistry, specifically, appeared soon after. To complement the ability of a GA to explore chemical space is a second algorithmic technique: machine learning (ML) wherein a program is able to categorize or predict properties of an input after reviewing many, many examples of similar inputs. ML has more nebulous origins than GA, but applications to chemistry also appeared in the 1990’s. A history perspective and assessment of these techniques towards surface chemistry follows in this work. A GA designed to find the crystal structure of layered chemical materials given the material’s X-ray diffraction pattern is then developed. The approach reduces crystals into layers of atoms that are transformed and stacked until they repeat. In this manner, an entire crystal need only be represented by its base layer (or two, in some cases) and a set of instructions on how the layers are to be arranged and stacked. Molecules that may be present may not quite behave in this fashion, and so a second set of descriptors exist to determine the molecule’s position and orientation. Finally, the lattice of the unit cell is specified, and the structure is built to match. The GA determines the structure’s X-ray diffraction pattern, compares it against a provided experimental pattern, and assigns it a fitness value, where a higher value indicates a better match and a more fit individual. The most fit individuals mate, exchanging genetic material (which may mutate) to produce offspring which are further subjected to the same procedure. This GA can find the structure of bulk, layered, organic, and inorganic materials. Once a material’s bulk structure has been determined, surfaces of the material can be derived and analyzed by DFT. In this thesis, DFT is used to validate results from the GA regarding lithium-aluminum layered double hydroxide. Surface chemistry is more directly explored in the prediction of adsorbates on surfaces of lithiated nickel-manganese-cobalt oxide, a common cathode material in lithium-ion batteries. Surfaces are evaluated at the DFT+U level of theory, which reduces electron over-delocalization, and the energies of the surfaces both bare and with adsorbates are compared. By applying first-principles thermodynamics to predict system energies under varying temperatures and pressures, the behavior of these surfaces in experimental conditions is predicted to be mostly pristine and bare of adsorbates. For breadth, this thesis also presents an investigation of the electronic and optical properties of organic semiconductors via DFT and time-dependent DFT calculations

Mathematical surfaces models between art and reality

In this paper, I want to document the history of the mathematical surfaces models used for the didactics of pure and applied “High Mathematics” and as art pieces. These models were built between the second half of nineteenth century and the 1930s. I want here also to underline several important links that put in correspondence conception and construction of models with scholars, cultural institutes, specific views of research and didactical studies in mathematical sciences and with the world of the figurative arts furthermore. At the same time the singular beauty of form and colour which the models possessed, aroused the admiration of those entirely ignorant of their mathematical attraction

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more