Frobenius distributions of elliptic curves in short intervals on average

The Ramanujan Journal - For an elliptic curve $$E/{\mathbb {Q}}$$ , let $$a_p$$ denote the trace of its Frobenius endomorphism over $${\mathbb {F}}_p$$ , where p is a prime of good reduction for E....     

The nearest neighbor random walk on subspaces of a vector space and rate of convergence

LetX be the collection ofk-dimensional subspaces of ann-dimensional vector spaceV _n overGF(q). A metric may be defined onX by letting

Necessary conditions for characterization of laws via mixed sums

SupposeX andY are independent and identically distributed, and independent ofU which satisfies 0U1. Recent work has centered on finding the lawsL(X) for whichX U(X+Y) where denotes equality in law. We show that this equation corresponds to a certain projective invariance property under random rotations. Implicitly or explicitly, it has been assumed that the characteristic function ofX has an expansion property near the origin. We show that solutions may be admitted in the absence of this condition when –logU has a lattice law. A continuous version of the basic problem replaces sums with a Lévy process. Instead we consider self-similar processes, showing that a solution exists only whenU is constant, and then all processes of a given order are admitted.This research was in part supported by NSERC grant A-8466.     

Reaching the Fundamental Limitation in CO2 Reduction to CO with Single Atom Catalysts

The electrochemical CO₂ reduction reaction (CO₂RR) to value-added chemicals with renewable electricity is a promising method to decarbonize parts of the chemical industry. Recently, single metal atoms in nitrogen-doped carbon (MNC) have emerged as potential electrocatalysts for CO₂RR to CO with high activity and faradaic efficiency, although the reaction limitation for CO₂RR to CO is unclear. To understand the comparison of intrinsic activity of different MNCs, two catalysts are synthesized through a decoupled two-step synthesis approach of high temperature pyrolysis and low temperature metalation (Fe or Ni). The highly meso-porous structure results in the highest reported electrochemical active site utilization based on in situ nitrite stripping; up to 59±6% for NiNC. Ex situ X-ray absorption spectroscopy (XAS) confirms the penta-coordinated nature of the active sites. The catalysts are amongst the most active in the literature for CO₂ reduction to CO. The density functional theory calculations (DFT) show that their binding to the reaction intermediates approximates to that of Au surfaces. However, it is found that the turnover frequencies (TOFs) of the most active catalysts for CO evolution converge, suggesting a fundamental ceiling to the catalytic rates.     

The impact of polystyrene resins in solid-phase organic synthesis

Summary A major objective of the DIVERSOMER^® technology is to provide pure and characterized compounds for biological testing in order to prevent false negatives in our libraries. On several occasions, analysis of the final products by¹H-NMR and MS, has revealed by-products from the polystyrene solid support. Subsequently, three alternative methods were studied to remove polystyrene by-products; (i) prewashing of the resin prior to execution of the synthesis; (ii) pretreatment of the resin with the cleavage conditions consistent with the solid-phase synthesis reaction scheme; and (iii) parallel purification.