Doctor of Philosophy

dissertationTropical geometry connects the fields of algebraic and polyhedral geometry. This connection has been used to discover much simpler proofs of fundamental theorems in algebraic geometry, including the Brill-Noether theorem. Tropical geometry has also found applications outside of pure mathematics, in areas as diverse as phylogenetic models and auction theory. This dissertation seeks to answer the question of when the minors of a symmetric matrix form a tropical basis. The first chapter introduces the relevant ideas and concepts from tropical geometry and tropical linear algebra. The second chapter introduces different notions of rank for symmetric tropical matrices. The third chapter is devoted to proving all the cases, outside symmetric tropical rank three, where the minors of a symmetric matrix form a tropical basis. The fourth chapter deals with symmetric tropical rank three. We prove that the 4 × 4 minors of an n×n symmetric matrix form a tropical basis if n ≤ 5, but not if n ≥ 13. The question for 5 < n < 13 remains open. The fifth chapter is devoted to when the minors of a symmetric matrix do not form a tropical basis. We prove the r × r minors of an n × n symmetric matrix do not form a tropical basis when 4 < r < n. We also prove that, when the minors of a matrix (general or symmetric) define a tropical variety and tropical prevariety that are different, then, with one exception, the two sets differ in dimension. The exception is the 4 × 4 minors of a symmetric matrix, where the question is still unresolved. The sixth chapter explores tropical conics. A correspondence between a property of the symmetric matrix of a quadric and the dual complex of that quadric is demonstrated for conics, and proposed for all quadrics. The seventh chapter reviews the results and proposes possible questions for further study. The first appendix is devoted to correcting a proof in a paper cited by this dissertation. The second appendix is a transcript of the Maple worksheets used to perform the computer calculations from the fifth chapter

Porphyrins as building blocks for single-molecule devices

Direct measurement of single-molecule electrical transparency by break junction experiments has become a major field of research over the two last decades. This review specifically and comprehensively highlights the use of porphyrins as molecular components and discusses their potential use for the construction of future devices. Throughout the review, the features provided by porphyrins, such as low level misalignments and very low attenuation factors, are shown with numerous examples, illustrating the potential and limitations of these molecular junctions, as well as differences emerging from applied integration/investigation techniques

Intense Molar Circular Dichroism in Fully Conjugated All‐Carbon Macrocyclic 1,3‐Butadiyne Linked pseudo‐ meta [2.2]Paracyclophanes**

The synthetic access to macrocyclic molecular topologies with interesting photophysical properties has greatly improved thanks to the successful implementation of organic and inorganic corner units. Based on recent reports, we realized that pseudo-meta [2.2]paracyclophanes (PCPs) might serve as optimal corner units for constructing 3D functional materials, owing to their efficient electronic communication, angled substituents and planar chirality. Herein, we report the synthesis, characterization and optical properties of four novel all-carbon enantiopure macrocycles bearing three to six pseudo-meta PCPs linked by 1,3-butadiyne units. The macrocycles were obtained by a single step from enantiopure, literature-known dialkyne pseudo-meta PCP and were unambiguously identified and characterized by state of the art spectroscopic methods and in part even by x-ray crystallography. By comparing the optical properties to relevant reference compounds, it is shown that the pseudo-meta PCP subunit effectively elongates the conjugated system throughout the macrocyclic backbone, such that already the smallest macrocycle consisting of only three subunits reaches a polymer-like conjugation length. Additionally, it is shown that the chiral pseudo-meta PCPs induce a remarkable chiroptical response in the respective macrocycles, reaching unprecedented high molar circular dichroism values for all-carbon macrocycles of up to 1307 L mol$^{−1}$ cm$^{−1}$

Intense Molar Circular Dichroism in Fully Conjugated All-Carbon Macrocyclic 1,3-Butadiyne Linked pseudo-meta [2.2]Paracyclophanes

The synthetic access to macrocyclic molecular topologies with interesting photophysical properties has greatly improved thanks to the successful implementation of organic and inorganic corner units. Based on recent reports, we realized that pseudo-meta [2.2]paracyclophanes (PCPs) might serve as optimal corner units for constructing 3D functional materials, owing to their efficient electronic communication, angled substituents and planar chirality. Herein, we report the synthesis, characterization and optical properties of four novel all-carbon enantiopure macrocycles bearing three to six pseudo-meta PCPs linked by 1,3-butadiyne units. The macrocycles were obtained by a single step from enantiopure, literature-known dialkyne pseudo-meta PCP and were unambiguously identified and characterized by state of the art spectroscopic methods and in part even by x-ray crystallography. By comparing the optical properties to relevant reference compounds, it is shown that the pseudo-meta PCP subunit effectively elongates the conjugated system throughout the macrocyclic backbone, such that already the smallest macrocycle consisting of only three subunits reaches a polymer-like conjugation length. Additionally, it is shown that the chiral pseudo-meta PCPs induce a remarkable chiroptical response in the respective macrocycles, reaching unprecedented high molar circular dichroism values for all-carbon macrocycles of up to 1307 L mol-1 cm-1

Mechanical compression in cofacial porphyrin cyclophane pincers

Intra- and intermolecular interactions are dominating chemical processes, and their concerted interplay enables complex nonequilibrium states like life. While the responsible basic forces are typically investigated spectroscopically, a conductance measurement to probe and control these interactions in a single molecule far out of equilibrium is reported here. Specifically, by separating macroscopic metal electrodes, two π-conjugated, bridge-connected porphyrin decks are peeled off on one side, but compressed on the other side due to the covalent mechanical fixation. We observe that the conductance response shows an exceptional exponential rise by two orders of magnitude in individual breaking events during the stretching. Theoretical studies atomistically explain the measured conductance behavior by a mechanically activated increase in through-bond transport and a simultaneous strengthening of through-space coupling. Our results not only reveal the various interacting intramolecular transport channels in a molecular set of levers, but also the molecules\u27 potential to serve as molecular electro-mechanical sensors and switches

Unravelling the conductance path through single-porphyrin junctions

Porphyrin derivatives are key components in natural machinery enabling us to store sunlight as chemical energy. In spite of their prominent role in cascades separating electrical charges and their potential as sensitizers in molecular devices

Mechanical conductance tunability of a porphyrin–cyclophane single-molecule junction

The possibility to study quantum interference phenomena at ambient conditions is an appealing feature of molecular electronics. By connecting two porphyrins in a cofacial cyclophane, we create an attractive platform for mechanically controlling electric transport through the intramolecular extent of π-orbital overlap of the porphyrins facing each other and through the angle of xanthene bridges with regard to the porphyrin planes. We analyze theoretically the evolution of molecular configurations in the pulling process and the corresponding changes in electric conduction by combining density functional theory (DFT) with Landauer scattering theory of phase-coherent elastic transport. Predicted conductances during the stretching process show order of magnitude variations caused by two robust destructive quantum interference features that span through the whole electronic gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). Mechanically-controlled break junction (MCBJ) experiments at room temperature verify the mechanosensitive response of the molecular junctions. During the continuous stretching of the molecule, they show conductance variations of up to 1.5 orders of magnitude over single breaking events. Uncommon triple- and quadruple-frequency responses are observed in periodic electrode modulation experiments with amplitudes of up to 10 Å. This further confirms the theoretically predicted double transmission dips caused by the spatial and energetic rearrangement of molecular orbitals, with contributions from both through-space and through-bond transport

Mechanical conductance tunability of a porphyrin–cyclophane single-molecule junction

The possibility to study quantum interference phenomena at ambient conditions is an appealing feature of molecular electronics. By connecting two porphyrins in a cofacial cyclophane, we create an attractive platform for mechanically controlling electric transport through the intramolecular extent of π-orbital overlap of the porphyrins facing each other and through the angle of xanthene bridges with regard to the porphyrin planes. We analyze theoretically the evolution of molecular configurations in the pulling process and the corresponding changes in electric conduction by combining density functional theory (DFT) with Landauer scattering theory of phase-coherent elastic transport. Predicted conductances during the stretching process show order of magnitude variations caused by two robust destructive quantum interference features that span through the whole electronic gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). Mechanically-controlled break junction (MCBJ) experiments at room temperature verify the mechanosensitive response of the molecular junctions. During the continuous stretching of the molecule, they show conductance variations of up to 1.5 orders of magnitude over single breaking events. Uncommon triple-and quadruple-frequency responses are observed in periodic electrode modulation experiments with amplitudes of up to 10 Å. This further confirms the theoretically predicted double transmission dips caused by the spatial and energetic rearrangement of molecular orbitals, with contributions from both through-space and through-bond transport. This journal is QN/van der Zant La

Characterization and Management of Food Loss and Waste in North America

Policies and programs on food loss and waste (FLW) are gaining momentum across North America as awareness of the issue continues to grow. The Commission for Environmental Cooperation (CEC) established the North American Initiative on Food Waste Reduction and Recovery as part of its Green Economy and Climate Change project areas. This white paper characterizes FLW in Canada, Mexico and the United States and identifies opportunities for the industrial, commercial and institutional (ICI) sector, governments, and nongovernmental organizations (NGOs) to take action across the three countries. The scope of this research included post-harvest to pre-consumer stages of the food supply chain (i.e., post-harvest food production; processing; distribution; retail; and food service). Pre-harvest food production and the consumer stages of the food supply chain are beyond the scope of this study. This project complements the CEC's North American Initiative on Organic Waste Diversion and Processing, which examines composting, anaerobic digestion, and other industrial processes (e.g. rendering, biofuel) for FLW and other organic waste. The content of this white paper was compiled from primary and secondary sources of information in Canada, Mexico, the United States and countries outside of North America. Primary sources included interviews and email exchanges with 167 stakeholders representing various locations, organization types and sizes, and stages of the food supply chain. Secondary sources included reports, white papers, academic papers, news articles, media recordings and government databases, as well as a review of on-the-ground programs and projects implemented by the ICI sector, governments and NGOs. North American and international experts on the subject matter also vetted key findings during a three-day stakeholder session held in Canada, in February 2017

Preparation of Unsymmetrical Disulfides from Thioacetates and Thiosulfonates

A method for the transformation of organic thioacetates, a widely used functionality for the preparation of self-assembled monolayers on gold surfaces, into unsymmetrical disulfides is reported. Disulfides are readily immobilized on gold in contrast to thioacetates, which usually require a deprotection step prior to bonding to the metal surface. The potential of the method for the controlled preparation of unsymmetrical disulfides has been demonstrated with model compounds comprising several thioacetates, which were readily converted into the corresponding unsymmetrical disulfides