Classifying the Jacobian Groups of Adinkras

Supersymmetry is a theoretical model of particle physics that posits a symmetry between bosons and fermions. Supersymmetry proposes the existence of particles that we have not yet observed and through them, offers a more unified view of the universe. In the same way Feynman Diagrams represent Feynman Integrals describing subatomic particle behaviour, supersymmetry algebras can be represented by graphs called adinkras. In addition to being motivated by physics, these graphs are highly structured and mathematically interesting. No one has looked at the Jacobians of these graphs before, so we attempt to characterize them in this thesis. We compute Jacobians through the 11-cube, but do not discover any significant discernible patterns. We then dedicate the rest of our work to generalizing the notion of the Jacobian, specifically to be sensitive to edge directions. We conclude with a conjecture describing the form of the directed Jacobian of the directed $n$-topology. We hope for this work to be useful for theoretical particle physics and for graph theory in general

Kinetics and Mechanism of the Ligand Exchange Reaction Between Tetradentate Schiff Base N,N’-ethylen-bis (salicylaldimine) and Ni(N,N’-propylen-bis(salicylaldimine))

Visible spectrophotometry is used to study the kinetics of ligand exchange in the system Ni(salpn)/H2salen with or without triethylamine (NEt3) and H2O in dimethylformamide (DMF) solvent at 25 ± 0.1 °C and 0.01M NaNO3 [H2salen and H2salpn are N,N’-ethylen-bis(salicylaldimine) and N,N’-propylen-bis(salicylaldimine), respectively]. It was found that the reaction rate is of the first-order with respect to Ni(salpn). In addition, the effect of NEt3 andH2O on the rate of the reaction was examined. The rate of the ligand exchange reaction was accelerated by adding NEt3 to the reaction mixture. However, the ligand exchange rate was not changed by adding H2O to the mixture reaction. The effects of NEt3 and H2O on the ligand exchange rate show that deprotonation/protonation of the H2salen ligand and anionic form of H2salen is important. On the basis of these results, the reaction mechanism is discussed.KEYWORDS Kinetic, mechanism, Schiff base, Ni complexes, ligand exchange

Aharonov-Bohm differential conductance modulation in defective metallic single-wall carbon nanotubes

Using a perturbative approach, the effects of the energy gap induced by the Aharonov-Bohm (AB) flux on the transport properties of defective metallic single-walled carbon nanotubes (MSWCNTs) are investigated. The electronic waves scattered back and forth by a pair of impurities give rise to Fabry-Perot oscillations which constitutes a coherent backscattering interference pattern (CBSIP). It is shown that, the CBSIP is aperiodically modulated by applying a magnetic field parallel to the nanotube axis. In fact, the AB-flux brings this CBSIP under control by an additional phase shift. As a consequence, the extrema as well as zeros of the CBSIP are located at the irrational fractions of the quantity $\Phi_\rho={\Phi}/{\Phi_0}$, where $\Phi$ is the flux piercing the nanotube cross section and $\Phi_{0}=h/e$ is the magnetic quantum flux. Indeed, the spacing between two adjacent extrema in the magneto-differential conductance (MDC) profile is decreased with increasing the magnetic field. The faster and higher and slower and shorter variations is then obtained by metallic zigzag and armchair nanotubes, respectively. Such results propose that defective metallic nanotubes could be used as magneto-conductance switching devices based on the AB effect.Comment: 11 pages, 4 figure

Environmental, developmental, and genetic factors controlling root system architecture

A better understanding of the development and architecture of roots is essential to develop strategies to increase crop yield and optimize agricultural land use. Roots control nutrient and water uptake, provide anchoring and mechanical support and can serve as important storage organs. Root growth and development is under tight genetic control and modulated by developmental cues including plant hormones and the environment. This review focuses on root architecture and its diversity and the role of environment, nutrient, and water as well as plant hormones and their interactions in shaping root architecture