Quartic Forms in Many Variables

We show that a quartic $p$-adic form with at least $3192$ variables possesses a non-trivial zero. We also prove new results on systems of cubic, quadratic and linear forms. As an example, we show that for a system comprising two cubic forms $132$ variables are sufficient.Comment: 8 page

pp-adic Zeros of Quintic Forms

It is shown that a quintic form over a p-adic field with at least 26 variables has a non-trivial zero, providing that the cardinality of the residue class field exceeds 9.Comment: Minor correction

Evidence for Quantum Interference in SAMs of Arylethynylene Thiolates in Tunneling Junctions with Eutectic Ga-In (EGaIn) Top-Contacts

This paper compares the current density (J) versus applied bias (V) of self-assembled monolayers (SAMs) of three different ethynylthiophenol-functionalized anthracene derivatives of approximately the same thickness with linear-conjugation (AC), cross-conjugation (AQ), and broken-conjugation (AH) using liquid eutectic Ga-In (EGaIn) supporting a native skin (~1 nm thick) of Ga2O3 as a nondamaging, conformal top-contact. This skin imparts non-Newtonian rheological properties that distinguish EGaIn from other top-contacts; however, it may also have limited the maximum values of J observed for AC. The measured values of J for AH and AQ are not significantly different (J ≈ 10-1 A/cm2 at V = 0.4 V). For AC, however, J is 1 (using log averages) or 2 (using Gaussian fits) orders of magnitude higher than for AH and AQ. These values are in good qualitative agreement with gDFTB calculations on single AC, AQ, and AH molecules chemisorbed between Au contacts that predict currents, I, that are 2 orders of magnitude higher for AC than for AH at 0 < |V| < 0.4 V. The calculations predict a higher value of I for AQ than for AH; however, the magnitude is highly dependent on the position of the Fermi energy, which cannot be calculated precisely. In this sense, the theoretical predictions and experimental conclusions agree that linearly conjugated AC is significantly more conductive than either cross-conjugated AQ or broken conjugate AH and that AQ and AH cannot necessarily be easily differentiated from each other. These observations are ascribed to quantum interference effects. The agreement between the theoretical predictions on single molecules and the measurements on SAMs suggest that molecule-molecule interactions do not play a significant role in the transport properties of AC, AQ, and AH.