Apollonian Equilateral Triangles

Given an equilateral triangle with $a$ the square of its side length and a point in its plane with $b$, $c$, $d$ the squares of the distances from the point to the vertices of the triangle, it can be computed that $a$, $b$, $c$, $d$ satisfy $3(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$. This paper derives properties of quadruples of nonnegative integers $(a,\, b,\, c,\, d)$, called triangle quadruples, satisfying this equation. It is easy to verify that the operation generating $(a,\, b,\, c,\, a+b+c-d)$ from $(a,\, b,\, c,\, d)$ preserves this feature and that it and analogous ones for the other elements can be represented by four matrices. We examine in detail the triangle group, the group with these operations as generators, and completely classify the orbits of quadruples with respect to the triangle group action. We also compute the number of triangle quadruples generated after a certain number of operations and approximate the number of quadruples bounded by characteristics such as the maximal element. Finally, we prove that the triangle group is a hyperbolic Coxeter group and derive information about the elements of triangle quadruples by invoking Lie groups. We also generalize the problem to higher dimensions

Impurity effects on the grain boundary cohesion in copper

Segregated impurities at grain boundaries can dramatically change the mechanical behavior of metals, while the mechanism is still obscure in some cases. Here, we suggest an unified approach to investigate segregation and its effects on the mechanical properties of polycrystalline alloys using the example of 3$sp$ impurities (Mg, Al, Si, P, or S) at a special type $\Sigma 5(310)[001]$ tilt grain boundary in Cu. We show that for these impurities segregating to the grain boundary the strain contribution to the work of grain boundary decohesion is small and that the chemical contribution correlates with the electronegativity difference between Cu and the impurity. The strain contribution to the work of dislocation emission is calculated to be negative, while the chemical contribution to be always positive. Both the strain and chemical contributions to the work of dislocation emission generally become weaker with the increasing electronegativity from Mg to S. By combining these contributions together we find, in agreement with experimental observations, that a strong segregation of S can reduce the work of grain boundary separation below the work of dislocation emission, thus embrittling Cu, while such an embrittlement cannot be produced by a P segregation because it lowers the energy barrier for dislocation emission relatively more than for work separation

Mock Maass Forms Revisited

In this paper, we use theta integrals to give a different construction of mock Maass forms studied by Sander Zwegers. With this method, we construct new real-analytic modular forms, whose Fourier coefficients are logarithms of algebraic numbers in a real quadratic field.Comment: 17 page

Examining the Effects of Amino and Thiolate Ligands on the Reactivity and Selectivity of Palladium on Carbon in Hydrogenation Reactions

Heterogeneous catalysts are used widely by chemical and energy industries because they show high reactivity but often suffer from lack of selectivity. On the other hand, ligands are commonly used in homogeneous catalysts to control the reactivity and selectivity; however, the effects of the ligands on the steric and electronic properties of heterogeneous catalysts are less understood. We examine the effects of four different ligands: 1-adamantanethiol, 1-adamantylamine, 1-dodecanethiol, and 1-dodecylamine, for the commercial hydrogenation catalyst palladium on carbon. Hydrogenation reactions are used as a screening tool to see the behavior that the different catalysts exhibit in the presence of unsaturated functional groups. Specifically, we study the hydrogenation of alkynes, carbonyls, nitriles, and nitro functional groups as well as the reductive amination between an aldehyde and an amine. Trends across the reactions are observed and are related back to the properties of the different ligands. In the hydrogenation of diphenylacetylene, the catalyst with 4 equivalents of 1-adamantanethiol as the ligand with respect to palladium shows 73% selectivity towards the alkene product whereas the non-ligated palladium on carbon goes 100% to alkane. This shows how ligands can affect the selectivity of heterogeneous catalysts. As ligands effects are understood more thoroughly, more effective catalysts can be designed for industrial reactions

Systems biology for identifying liver toxicity pathways

Drug-induced liver toxicity is one of the leading causes of acute liver failure in the United States, exceeding all other causes combined. The objective of this paper is to describe systems biology methods for identifying pathways involved in liver toxicity induced by free fatty acids (FFA) and tumor necrosis factor (TNF)-α in human hepatoblastoma cells (HepG2/C3A). Systems biology approaches were developed to integrate multi-level data, i.e., gene expression, metabolite profile, toxicity measurements and a priori knowledge to identify gene targets for modulating liver toxicity. Targets that modulate liver toxicity, in vitro, were computationally predicted and some targets were experimentally validated

BEPS and the Emerging Global Approach to Taxing Multinational Enterprises

The G20/OECD Base Erosion and Profit Shifting (BEPS) Project aimed at revising existing tax rules to align the taxation of profit with the location of economic activities and value creation (the “value creation principle”). It has received much commentary in literature and general debates and has been regarded as “the most significant re-write of the international tax rules in a century”, an opportunity to “rebuild a healthy scheme for allocating taxation rights”, having the potential to significantly alter the contours of the international tax regime, transforming the international tax regime, signaling a “new struggle over international taxation”, or representing the “emergence of a new international tax regime”. The OECD claimed that the revised rules represent “the first substantial – and overdue - renovation of the international tax standards in almost a century”